Paper Title: Relation between the geometric shape and rotation of Galactic globular clusters.

 







Geometric shape of Galactic globular clusters


   
  

   Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, Chemin Pegasi 51b, 1290 Versoix, Switzerland  

    mauricio.cruzreyes@epfl.ch, richard.anderson@epfl.ch   


    
    



  We homogeneously measured the elliptical shapes of 163 globular clusters (GCs) using the on-sky distribution of their cluster members and the third data release of the ESA mission  (DR3).  The astrometry enables the differentiation of stars within clusters from those in the field.  This feature is particularly valuable for clusters located in densely populated areas of the sky, where conventional methods for measuring the geometry of the GCs are not applicable.  The median axial ratio of our full sample is ⟨ b/a ⟩ =  0.935^+0.033_-0.090 and 0.986^+0.009_-0.004 for the subset of 11 GCs previously studied based on Hubble Space Telescope imaging.  We investigated whether the minor axis of the ellipses can be interpreted as a pseudo-rotation axis by comparing it to measurements of cluster rotation.   Using the radial velocities from , we detected rotation for three clusters, NGC 5139, NGC 104, and NGC 6341, and observed an alignment between the pseudo-rotation axis and the 2D projection of the real rotation axis. To expand the set of clusters for which rotation has been detected, we analyzed multiple literature references. Depending on the reference used for comparison, we observed an alignment in between 76% to 100% of the clusters. The lack of an alignment observed in some clusters may be linked to different scales analyzed in various studies. Several studies have demonstrated that the orientation of rotation varies with the distance from the center.  We estimate that the next  release will increase the number of stars with radial velocities in GCs from  ∼ 10,000 in  DR3 to ∼ 55,000 in  DR4. This will enable the measurement of rotation and ellipticities at identical angular scales for additional clusters, which will help us to clarify whether the previously mentioned alignment occurs in all clusters.
 


   

   Relation between
the geometric shape and 
 rotation of Galactic globular clustersTables <ref> and <ref> are available in electronic form at the CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5)
or via https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A/
    Mauricio Cruz Reyes 10000-0003-2443-173X
          Richard I. Anderson10000-0001-8089-4419
    Received August 5, 2024
======================================================================================================================================================================================================================================================================






§ INTRODUCTION

Globular clusters (GCs) are often thought of as spherical distributions of stars by nonspecialists. Nonetheless, the first reports of nonsphericity date back to more than a century ago, when <cit.> examined the distribution of stars in five GCs that were observed using photographic plates. The authors concluded that ellipsoids provided a more suitable approximation of their real geometry. This insight gained further support from <cit.>, who compiled parameters for 157 GCs and showed that deviations from sphericity are rather common: 94 of the 157 GCs exhibited them. However, the quality of these measurements was somewhat limited by the spatial resolution of the observations, which primarily came from the catalog of <cit.>, which used the Palomar <cit.> and the Southern Sky survey <cit.> photographic plates.   Nevertheless, <cit.> provided further evidence of nonsphericity by analyzing the spatial distribution of sources with Two Micron All-Sky Survey (2MASS) photometry <cit.>, revealing similar deviations in 116 clusters, although this time with CCD measurements. Using the astrometry from the Early Data Release 3 (EDR3) of , <cit.> presented an interesting analysis of the ellipticity in NGC 5139. The study illustrated that the cluster is more spherical at its core, while its ellipticity increases with the distance from the center.


Cluster rotation is also common. It was first detected in NGC 5139 (ω Centauri) and NGC 104 (47 Tucanae) by <cit.> using radial velocity (RV) measurements. The respective rotation velocities are v_max = 8  km  s^-1 and 6.5  km  s^-1. Further investigations by <cit.> and <cit.> revealed that the rotation axes of both clusters are inclined with respect to the line of sight at angles of 39.2 ± 4.4^∘ and 33.6 ± 1.8^∘, respectively.


A more recent analysis for 25 GCs carried out by <cit.> using 500,000 spectra of 200,000 stars obtained with the Multi Unit Spectroscopic Explorer (MUSE) <cit.>, revealed that at least 21 of them rotate. It is important to note that the different studies analyzing rotation did not necessarily analyze the same angular scales. As an example, the angular scales analyzed in this study are larger by 4 to 18 times than those investigated by M23. This distinction is crucial because the orientation of the rotation can vary depending on the specific scale that is studied. For NGC 5139, this is so extreme that its center rotates in the direction opposite to the global rotation <cit.>. 

Considering that the vast majority of GCs are not spherical and that large-scale rotation has been detected in most of them, it is natural to ask whether these two phenomena are correlated. The first observational evidence suggesting a correlation between these two variables was found by <cit.>. By comparing RV maps obtained with the VIRUS-W[Wide field visual integral-replicable unit spectrograph ] Spectrograph <cit.> with the ellipticities of 11 GCs observed using the ACS/WFC camera of the Hubble Space Telescope (), F14 concluded that the geometry of clusters is primarily determined by their rotation. These findings were confirmed by K18, who concluded that this correlation indeed exists, based on a larger sample of GCs and RV measurements. However, <cit.> concluded that there is no significant evidence of a relation between the geometry and the rotation of GCs. This conclusion was derived based on a compilation of 45,561 RVs of individual stars from a sample of 109 GCs, including data from  DR2 and multiple ground-based spectrographs, along with the ellipticities measured by C10. The main difference between these studies is that both F14 and K18 measured the ellipticities of the GCs using  images, while S19 used ellipticity measurements previously obtained by C10 based on the 2MASS catalog <cit.>.  As we show below, it is likely that this difference contributes to the observed difference among their results.


As the distance increases, the precision in the measurement of the cluster ellipticities (and position angles) decreases because it becomes increasingly difficult to detect the cluster members and observe stars near the cluster center. C10 stated that the resolution of 2MASS (1 arcsec per pixel) only allowed them to explore the outer region of each cluster. This limitation similarly affects the measurements of <cit.>, as the data were obtained using photographic plates in a 1.2 m Oschin Schmidt telescope.  Among the telescopes used for measuring the ellipticities of GCs, the  stands out with the highest resolution, with ∼ 0.05 arcseconds per pixel in the F606W and F814W filters. 

In this context, it might be useful to employ  <cit.> to measure the ellipticities for GCs to reanalyze the correlation between the orientation of the ellipses and rotation.   requires a separation of 0.23 arsec to resolve two equal-brightness sources in the along-scan direction <cit.>. Even though its resolution is lower than that of the ,  presents distinct advantages. It provides parallaxes and proper motions for a larger number of sources, which enables us to distinguish cluster members from field stars. This is particularly useful for clusters located in regions with a high star density and for distinguishing the shape of the clusters at their borders, where field star contamination is high. The cluster FSR 1758 serves as a prime example because it is located within the Galactic bulge <cit.>. Its position makes it difficult to visually distinguish the stars that belong to the cluster, and therefore, the traditional methods used to determine its geometry are not accurate.

This paper provides a detailed analysis of the ellipticities that were derived and used in <cit.> to determine the membership of RR Lyrae stars in Galactic GCs. These measurements made use of the  astrometry to assess the ellipticities of most of the GCs detected within the Milky Way. Our paper is structured as follows. Section <ref> describes the data and the method we used to measure the ellipticities.  Section <ref> provides an analysis of the ellipticities and a comparison with previous results from the literature. The relation between RVs and the geometry of GCs is discussed in Sect. <ref>. Section <ref> discusses the potential of  for measuring the rotation of GCs, and Sect. <ref> provides a summary of the paper. Appendix <ref> compiles the ellipticities and position angles from the literature.



§ DESCRIPTION OF THE DATA AND METHOD

The sample of clusters and their members used in this study for our analysis is exactly the same as was used in  to measure the ellipticity of GCs and to identify their population of RR Lyrae stars. This dataset was compiled by 
<cit.>, who analyzed the astrometric parameters (parallax, positions, and proper motions) of all sources located in the vicinity of 170 galactic clusters[The sample of cluster members is available on 
 https://zenodo.org/records/4891252Zenodo.  ] previously identified in the literature using  EDR3.  As explained in , we updated the VB21 dataset by cross-matching it with  DR3 <cit.>. This update had no impact on the astrometry because  DR3 and EDR3 share the same astrometric parameters. However, DR3 includes more sources with measured RVs.

Since the publication of the VB21 dataset, the  collaboration has provided additional observations for NGC 5139 <cit.>, acquired using the Sky Mapper instrument. This instrument allows us to capture images of the cluster, which enables a detailed exploration of its center and the observation of approximately half a million sources that were previously undetectable using the standard observation mode. The majority of sources observed with the Sky Mapper are associated with NGC 5139, but some of them are field stars.  The  area examined with the Sky Mapper spans 0.81 degrees in radius, while the stars in the VB21 sample, with probabilities higher than 50%, only extends up to 0.67 degrees. Distinguishing between cluster members and field stars in the sample provided by the Sky Mapper would require a membership analysis, which is beyond the scope of this paper. As a result, we did not include these stars in the measurement of the ellipticity of NGC 5139.


The procedure we used to measure the ellipticities of the GCs was described in Section 3.1 of . We therefore explain the method only in general terms here and focus on comparing our results with prior research in the literature.  The approach used to measure the ellipticities involves modeling the 2D geometry in right ascension and the declination of each GC using a principal component analysis (PCA). The PCA determines the length and orientation of two orthogonal vectors on the sky plane, and we used these two vectors to define an ellipse.  We denote the major axis of the ellipse as a and the minor axis as b. The eigenvectors (ν_a,ν_b) correspond to the axes of the ellipse, and the eigenvalues (λ_a,λ_b) provide a measure of their length.  The position angle was measured counterclockwise from the x-axis and is equal to θ = arctan(ν_a,y/ν_a,x), and the ellipticity is ϵ = 1 - √(λ_b/λ_a) = 1 - b/a.  To determine the mean value and uncertainties of the parameters, we bootstrapped the sample of cluster members 1000 times. The ellipticity and position angle were calculated as the mean value of the distribution resulting from the bootstrap, and the uncertainties correspond to the standard deviation. Due to projection effects, our measurements set a lower limit to the true ellipticity. For example, if the major axis of an ellipse is aligned with the line of sight, the measured ellipticity corresponds to that of a circle.




§ RESULTS

Using the method presented in the previous section, we were able to measure the ellipticity of 163 GCs.   Figure <ref> displays the distribution of cluster ellipticities in the sky.  Based on  astrometry, we measured the ellipticities of FSR 1758 and BH 140 for the first time with a significance (defined as ϵ/σ_ϵ) greater than three and with more than 1000 detected members for both of them. Previously, the ellipticity of these clusters could not be measured because of the high density of stars surrounding them. BH 140 is located near the Galactic plane <cit.>, and FSR 1758 resides in the Galactic bulge <cit.>.




Figure <ref> illustrates the relation between the number of stars and the significance of the ellipticity for all clusters. As anticipated, the significance ratio increases with the number of cluster members.  The median number of stars used to determine the ellipticities per cluster is 1393, but it can reach 147,516 for clusters such as NGC 5139 and be as low as 8 for Ryu 879. The specific number of stars used for each cluster can be found in Table <ref>.  Out of the 163 clusters for which we measured the ellipticity, only 46 clusters have ϵ/σ_ϵ > 3. We caution that clusters with a small number of members and a high eccentricity may require a further detailed inspection. Two of the 22 clusters with fewer than 100 stars and ϵ/σ_ϵ > 2 have previously published ellipticity measurements: UKS 1 and Terzan 12. These measurements were taken by C10. Our results for these clusters are consistent within 1σ, although this might be attributed to the large uncertainties in the C10 data.  Additionally, Fig. <ref> suggests that clusters with particularly few stars have a particularly high ϵ. 




Figure <ref> shows the ellipticities of 157 out of 163 clusters in our dataset, plotted against their galactocentric distance (R_G). The distances were obtained from the Galactic GC Database Version 4[<https://people.smp.uq.edu.au/HolgerBaumgardt/globular/>]. The figure shows a clear correlation between the number of stars that are detected in a cluster and the ellipticity that is determined. At the same time, the number of detected stars is highest near R_G≈ 8 kpc, that is, around the Sun's galactocentric distance, likely because member stars are more readily resolved in GCs close to the Sun while also featuring smaller astrometric uncertainties. This also explains the trend toward fewer member stars in GCs located at galactocentric distances below ∼ 4 and above ∼ 12 kpc, which leads to a cone-shaped distribution. We therefore consider the trend toward higher ellipticities at short galactocentric radii spurious and caused by observational selection effects, and we cannot confirm the claim by <cit.> that clusters are more elliptical near the bulge. Our results suggest that a minimum number of stars of about 10^3 is required to measure ellipticities faithfully. 

A particularly interesting cluster is NGC 6544, for which we find an ellipticity of 0.142 ± 0.010. It is located at a distance of R_G = 5.62 ± 0.06 kpc. Clusters with a similar number of stars (10^3 - 10^4) and with R_G < 10 kpc typically have a much lower ellipticity, with an average of ϵ = 0.09.







In high-density regions where field stars are not easily distinguishable from cluster members, it is preferable to use  data to assess the geometry of GCs. However, for clusters in which  detects only a limited number of stars, traditional methods might offer more suitable alternatives for measuring their ellipticities.




Figure <ref> provides a comparison with previously reported results in the literature. However, we note that this comparison is not straightforward because the ellipticity may change as a function of the angular distance from the cluster center and because the ellipticity values reported in different studies were measured at different angular scales. For example, according to <cit.>, the ellipticity of NGC 5139 extends from ϵ∼ 0.02 at 40 arcmin from the center, and increases to ϵ∼ 0.6 at the tidal tails of the cluster, 240 arcmin apart from the center. Previously, <cit.> measured ϵ = 0.05 ±   0.010 at 28 arcmin and <cit.>  ϵ = 0.045 at 30 arcmin. Our measurement indicates ϵ = 0.064 ± 0.002 at 40.2 arcmin.  


Figure <ref> shows the good agreement between the ellipticities of the 11 GCs studied by F14 using  imaging and our results based on , aside from a small mean difference of Δϵ = 0.026 ± 0.013 (our results tend to be slightly more elliptical).  Fortunately, all  datasets contain the 11 clusters examined by F14 with the , making it possible to clearly see the differences in the ellipticity values reported by each study.  Figure <ref> shows that the comparison of our results with ground-based studies exhibits large scatter, likely due to the lower spatial resolution of the telescopes. 










Figure <ref> shows a comparison between the sample of stars used to measure the geometry of NGC 5024 and NGC 6218 in this study and the sample provided by F14. For the clusters in common, the radius covered by  is about seven times smaller on average than that covered by , but the median number of stars in the  sample is nine times larger than that obtained with . This difference can be attributed to two main factors: completeness and resolution. The  observations are subject to a complex selection function that is the subject of ongoing research (see <cit.> for the example of NGC 1261). A dominant factor of the selection function is that when the  observational capacity of 1,050,000 objects per square degree <cit.> is exhausted, the brightest stars are prioritized. As a result, the  completeness is lowest at the center of clusters and for sources with G>18. Additionally, the higher spatial resolution of the  better distinguishes sources in the very busy GC centers, resulting in much larger numbers of stars despite the significantly smaller footprint.



Unfortunately, it is not possible to make a comparison similar to that shown in Fig. <ref> for the sample of clusters in C10 and H10 because the data used in these papers are not available. 




Based on , we measured the ellipticities of 163 GCs with a similar precision as .  The median axial ratio (b/a) for H10, C10, F14, and our sample are shown in Table <ref>. Individual ellipticities and position angles are presented in Tables <ref> and <ref>. 







§ RADIAL VELOCITIES AND THEIR CONNECTION WITH THE ELLIPTICITIES

As mentioned in the introduction, F14 identified a connection between the rotation of GCs and their ellipticities, revealing that the position angles of the ellipses, derived from both RV measurements and  images, are generally aligned. To compare the position angles measured by F14 with ours, we first transformed them to the same reference system. This can be achieved by using the following equation:

    θ^T = (90^∘ - θ) 180^∘,

where θ^T is the F14 angle in our coordinate system, and θ is the angle originally measured by F14. When comparing our angles (θ_0) with the F14 angles, it is important to note that both are only defined from 0^∘ to 180^∘. This implies that the maximum difference that can exist between them is  90^∘. Consequently, the difference between them is given by


    Δθ  = min (  |θ_0 - θ^T |, 180 - |θ_0 - θ^T|   ).


On the other hand, both S19 and M23 measured the position angle of the rotation axis for the GCs in their sample. Thus, if there is a relation between the geometry of GCs and rotation, our angles should be correlated with the angle that is perpendicular to those measured by S19 and M23. These angles are defined from 0^∘ to 360^∘, and when we compare them with ours, we therefore need to remove the orientation. In this way, we avoid overestimating the differences between them.  Additionally, we need to transform them to our reference system. For these two transformations, we used the following equation:

    θ^T =  θ 180 ^∘,

where θ^T is the angle that we need for the comparison, and θ represents the angles originally measured by S19 or M23. To estimate the difference between our angles and those measured by S19 and M23, we used Eq. <ref>.

Figures <ref> and <ref> summarize the comparison between our position angles and the corresponding ones from the literature obtained using RV measurements and   photometry. The data necessary to reproduce these plots can be found in Table <ref>. We defined the angle differences as statistically significant when they exceeded three standard deviations, with the standard deviation being the sum in quadrature of the uncertainties from both measurements. These clusters are highlighted in all panels, even when the difference is above our threshold with respect to only one dataset. Figure <ref> shows that the position angles obtained with  are consistent within ∼ 2 σ with those measured by F14 using  photometry. By comparing our angles with the angles of F14 obtained using RVs,  we observed that for NGC 6254, this difference exceeds our threshold. However, this cluster does not stand out as an outlier when compared with the results of M23. None of the clusters in the S19 sample exhibit differences beyond our threshold.

Discrepancies in position angles may arise due to the use of different angular scales in the measurements. As shown by <cit.> for NGC 7078, the position angle derived from RVs varies depending on the angular separation from the cluster center. The angular scales employed for measuring ellipticities in this paper are nine times larger on average than those used for M23 to measure rotation. In the M23 sample, six clusters show statistically significant differences: NGC 6266, NGC 6656, NGC 5286, NGC 6541, NGC 6293, and NGC 6752.  For the first two clusters, this difference disappears when using the angles from F14 or S19.  The remaining four GCs are not included in either the F14 or S19 datasets.  In particular, there is a difference of 42.64 ± 9.69 ^∘ between our angle and the one reported by M23 for NGC 5286. For the clusters NGC 6541, NGC 6293, and NGC 6752, our angles are aligned perpendicularly to those reported by M23 within the uncertainties. It should be noted that the statistical confidence in the ellipticity of NGC 6293 is particularly low, being only 2 σ. We therefore considered that more evidence is necessary before we can conclude that the angles are perpendicular for this cluster. 

Figure <ref> shows a comparison of the angular scales examined in this study to determine the position angles with those analyzed by M23 for two GCs. The data collected by M23 primarily target the central regions of clusters, while  allows the analysis of entire clusters.  However, as shown in the figure, for NGC 104, the completeness of  decreases near the region analyzed by M23 due to its limited angular resolution. For NGC 6441, the M23 observations are restricted to a radius that is smaller by about five times than that of , but the number of stars in the M23 sample is four times higher. 



The difference in the projection of the rotation axis and the position angles of our ellipses remains below three standard deviations for 100%, 90%, and 76% of the clusters in the S19, F14, and M23 samples, respectively.  This suggest that there is a correlation between the geometric shape of GCs and their rotation. However, as shown previously, the  ellipticities and rotation were measured at different angular scales. As we show in the next section, there are three clusters for which we can measure rotation with  at the same angular scale as the ellipticities. For these clusters, the projection of the rotation axis and the position angle derived from the PCA are aligned.











§ GLOBULAR CLUSTER ROTATION AND 

The RV spectrometer on board the  satellite measured average RVs of 33,812,183 stars brighter than G_RVS = 14 mag across the full sky, including 10084 in GCs. The performance verification paper for RVs <cit.> demonstrated that the RVs were abundant and precise enough to detect the rotation of NGC 104. This section analyzes the potential of  to detect rotation in a larger sample of clusters. 

We analyzed all clusters and cluster members with membership probabilities above 90% in the VB21 sample. For 133 clusters, there is at least one RV measurement in  DR3.  The distribution of the number of RV measurements in clusters can be found in Fig. <ref>. 



We divided the members of each cluster into two subsets using the minor axis of the ellipses as a reference and estimated the difference in the median RV among the two subsets.  We considered rotation to be detected when this difference exceeded three times the square-summed standard error on the median of the subsets. Following this procedure for all clusters in our sample, we detected the rotation of NGC 5139  and NGC 104 with a statistical confidence of seven and eight standard deviations.  By further dividing the sample of cluster members for NGC 5904 into different sections, we detected its rotation with a 3  σ confidence.  The absolute values of the rotation amplitudes for NGC 104, NGC 5139, and NGC 5904 are 5.01 ± 0.61, 6.65 ± 0.97, and 6.41± 1.95  km s^-1, respectively. These amplitudes are consistent within two standard deviations with those found by S19.

The RV maps for these clusters are shown in Fig. <ref>. The stars were rotated to align the orientation of the ellipse with the plot axes. If the measured ellipticity were caused by rotation, the minor axis would be expected to align with the projected rotation axis. The figure shows that they are  indeed aligned. 
<cit.> concluded that for NGC 5139, the axes are perpendicular, although their study analyses the central region of the cluster. As mentioned in their paper, this difference may arise from the fact that the position angles derived from RVs change relative to the cluster center distance.







We inspected the remaining 130 clusters individually to account for cases in which the axis of rotation is not aligned with the minor axis of the ellipses. We found no cluster for which this applied. However,  NGC 6341, for which F14 previously reported a detection of rotation, is an interesting case. Fig. <ref> shows that the RV map exhibits a pattern similar to the patterns found in the clusters for which we detected rotation. However, the statistical uncertainties on the RVs do not allow us to conclude that this cluster rotates. This could be related to the fact that there are only 103 stars with measured RVs for this cluster in  DR3. 





The main reason why rotation has not been detected in more GCs is the low number of stars with RV measurements available in . The median number of RV measurements per cluster is 24. In comparison, for NGC 5139, NGC 104, and NGC 5904, the number of cluster members with measured RVs in  are 1696, 1501, and 194, respectively. Conversely, all clusters in S19 have a minimum of 50 stars with RVs, some containing up to 2500. This contrasts strongly with what can be obtained using integral field spectrographs (IFS). For example, using MUSE, <cit.> was able to measure up to 20,000 RVs per GC, although at much smaller angular scales than those accessible with .  

An instrument such as the WEAVE[William Herschel Telescope Enhanced Area Velocity Explorer] telescope <cit.> could incorporate both functionalities. WEAVE has a multi-object spectrograph (MOS) with a field of view of 2 square degrees and an IFS of 90x78 arcsec^2  (almost twice larger than the MUSE IFS). Together, these components would enable IFS-based studies of the innermost cores of northern GCs complemented by the MOS at larger separations. The proposed wide-field spectroscopic telescope <cit.> would simultaneously probe much larger core areas (3x3 arcmin^2) using its IFS and more than 20 000 stars in the outer regions of GCs in the southern hemisphere.


The third  data release  is the result of the analysis of 34 months of data, and it was limited to measuring RVs for stars with G_RVs<14 mag. The next data release will be based on 66 months, and this will allow the  collaboration to push this limit to G_RVs = 16 mag <cit.>.  By using the photometric transformations from G to G_RVs provided by <cit.> and examining the cluster members with membership probabilities above 90% in the sample used in this paper, we can establish that ∼ 70,000 new stars will fall within the  observable magnitude range due to the increased depth of observed magnitudes. In  DR3, ∼ 63% of all stars in GCs within the  observable magnitude range have measured RVs. If this trend persists in  DR4, it would imply the presence of ∼ 45,000 new stars with measured RVs.

The improvements in the astrometry and the increase in the number of stars with measured RVs in  DR4 may facilitate the detection of rotation in a larger number of GCs. Based on the results of this study, we conclude that ∼ 200 individual RVs uniformly distributed throughout a cluster are necessary to measure its rotation, assuming their rotation amplitude is ∼ 6  km s^-1. 



§ SUMMARY

Our study showed the potential of  of measuring the geometry (ellipticity) and rotation of GCs.  astrometry allows us to measure the geometry of GCs in regions of the sky where discerning cluster members from background or foreground stars was once challenging, for example, for clusters located in the Galactic plane. It is worth mentioning that astrometry is also useful for studying the outskirts of clusters in greater detail because in these areas, the field contamination has a more significant impact on the measurement of ellipticities than at the core of GCs. The clusters BH 140 and FSR 1758 are notable examples of the  capabilities, as we have measured their ellipticities for the first time. This was previously impossible due to their location within high-density star regions. 

Our results are consistent with those previously found using the   but they were determined for a sample of GCs that was 15 times larger. Our dataset represents the most extensive collection of ellipticities for GCs in the literature. However, it is essential to note that for clusters located at large distances from us, the detection of member stars becomes increasingly challenging because the uncertainties associated with the astrometry increase. Consequently, the number of stars used in the analysis decreases with distance, which in turn increases the statistical uncertainties on the ellipticities. 

By comparing the orientation of the minor axes here with measurements of the projected on-sky rotation axis from the literature, we demonstrated that depending on the reference used for comparison, they are aligned for 76% to 100% of the GCs. The lack of an alignment might arise because the ellipticities and rotation were measured at different angular scales. Using the RVs from , we have confirmed the rotation of NGC 104, NGC 5139, and NGC 5904. For these clusters, we found that their geometry is aligned with the orientation of the rotation.


The fourth data release of  will provide RVs for stars up to two magnitudes fainter than in the third realese. Together with an additional 32 months of observations, this will enhance the ability of  to measure the rotation of GCs.






This research received support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement No. 947660). RIA is funded by the SNSF through a Swiss National Science Foundation Eccellenza Professorial Fellowship (award PCEFP2_194638). 

We thank Henry Leung for designing the https://milkyway-plot.readthedocs.io/en/stable/milkyway-plot python package used to generate Fig. <ref>.

This work has made use of data from the European Space Agency (ESA) mission
Gaia (<https://www.cosmos.esa.int/gaia>), processed by the Gaia
Data Processing and Analysis Consortium (DPAC,
<https://www.cosmos.esa.int/web/gaia/dpac/consortium>). Funding for the DPAC
has been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. 

This research has made use of NASA's Astrophysics Data System; the SIMBAD database and the VizieR catalog access tool[<http://cdsweb.u-strasbg.fr/>] provided by CDS, Strasbourg; Astropy[<http://www.astropy.org>], a community-developed core Python package for Astronomy <cit.>; TOPCAT[<http://www.star.bristol.ac.uk/ mbt/topcat/>] <cit.>.





aa







§ CLUSTER PARAMETERS

In this appendix, we compile the cluster parameters discussed in this paper, covering ellipticities in Table <ref>, the median axial ratio for the cluster in our sample and literature in Table <ref> and position angles in Table <ref>.

















 Paper Title: MeerKLASS L-band deep-field intensity maps: entering the HI dominated regime.




firstpage–lastpage
Weak ℤ_2 Supertopology
    Andreas P. Schnyder
    August 5, 2024
=======================




§ ABSTRACT 
We present results from MeerKAT single-dish  intensity maps, the final observations to be performed in L-band in the MeerKAT Large Area Synoptic Survey (MeerKLASS) campaign. The observations represent the deepest single-dish  intensity maps to date, produced from 41 repeated scans over 236 ^2, providing 62 hours of observational data for each of the 64 dishes before flagging. By introducing an iterative self-calibration process, the estimated thermal noise of the reconstructed maps is limited to ∼ 1.21 mK (1.2 × the theoretical noise level). This thermal noise will be sub-dominant relative to the  fluctuations on large scales (k ≲ 0.15 h Mpc^-1), which demands upgrades to power spectrum analysis techniques, particularly for covariance estimation. In this work, we present the improved MeerKLASS analysis pipeline, validating it on both a suite of mock simulations and a small sample of overlapping spectroscopic galaxies from the Galaxy And Mass Assembly (GAMA) survey. Despite only overlapping with ∼ 25% of the MeerKLASS deep field, and a conservative approach to covariance estimation, we still obtain a > 4 σ detection of the cross-power spectrum between the intensity maps and the 2269 galaxies at the narrow redshift range 0.39 < z < 0.46. We briefly discuss the  auto-power spectrum from this data, the detection of which will be the focus of follow-up work. For the first time with MeerKAT single-dish intensity maps, we also present evidence of  emission from stacking the maps onto the positions of the GAMA galaxies. 






cosmology: large scale structure of Universe – cosmology: observations – radio lines: general – methods: data analysis – methods: statistical








§ INTRODUCTION


Probing density fluctuations in large-scale cosmic structure is a cornerstone of precision cosmology. Constructing maps that span wide areas and reach deep redshifts provides a rich resource for testing and constraining the standard
ΛCDM cosmological model. Surveys at optical and near-infrared wavelengths have made impressive progress in this pursuit <cit.>. However, extending experiments to a broader range of wavelengths boosts the quantity of available data. Furthermore, multi-wavelength observations provide robust checks of systematic influence due to the alternative instrumental approach required to survey other wavelengths. There can also be vastly different complex astrophysics for tracers at different wavelengths. Thus alternative tracers can validate the consistency of cosmological conclusions.

Of particular interest for surveying large-scale structure is mapping redshifted neutral hydrogen () gas at 21cm radio wavelengths. In the post-reionisation Universe, at redshifts z ≲ 6, the remaining  is overwhelmingly contained within dark matter halos <cit.>, meaning the 21cm radiation spontaneously emitted from  acts as a tracer of the large-scale structure. However, at the redshifts of interest for large-scale structure cosmology (z ≳ 0.1), the 21cm emission becomes faint and hard to detect with the high signal-to-noise required for a comprehensive mapping survey. This problem can be circumvented by abandoning the requirement to resolve individual galaxies and instead integrate the combined unresolved emission with low-aperture radio telescopes. This process is now commonly known as  intensity mapping <cit.>.

The novel approach of intensity mapping presents some challenges. Since it records all radiation entering the telescope receiver within its frequency bandpass, the contribution to the signal will not come from cosmological  alone; in fact, only a small percentage is. There are multiple sources of foregrounds in the frequency ranges corresponding to redshifted 21cm emission. These mostly originate from our Galaxy and come from diffuse synchrotron and free-free emission, but there can also be strong extragalactic point sources <cit.>. These sources dominate the cosmological  brightness temperature by three to five orders of magnitude. Furthermore, instrumental effects can distort the foreground spectra, making their removal additionally challenging <cit.>. In addition to foregrounds, the telescopes will detect terrestrial, human-made radio frequency interference (RFI) <cit.>. This can dominate even more than the foregrounds, but vary with time and sky position so can therefore be flagged and deleted from the final maps. Residual RFI will also average down if repeated scans are performed over the same patch of sky, as is the case with MeerKLASS observations. 

Despite foreground contamination and RFI challenges, the potential benefits from  intensity mapping are too large to ignore, thus justifying the ongoing endeavour to develop the technique into a tool for precision cosmology. Experiments have already delivered promising results, demonstrating how detections are possible even without purpose-built instruments <cit.>. New experiments are starting to arrive <cit.>, many designed specifically with  intensity mapping in mind, creating a bright future ahead.

The MeerKAT telescope,[https://www.sarao.ac.za/science/meerkat/] an array of 64 dishes in the Upper Karoo region of South Africa, has already showcased its capability to perform  intensity mapping both as an interferometer, claiming the first detection of the  power spectrum <cit.>, and also in single-dish mode, where the field-of-view can reach cosmological scales. Using the single-dish pilot survey <cit.>, a detection of the cross-correlation power spectrum was achieved with overlapping WiggleZ galaxies <cit.>.

The MeerKAT Large Area Synoptic Survey (MeerKLASS) <cit.> will continue adding to these observations until it is merged with the SKA Observatory (SKAO) <cit.> on the same site.[https://www.skao.int/en] MeerKAT can observe in two frequency bands relevant for 21cm-cosmology: L-band (900 MHz < ν < 1420 MHz, reaching redshifts z < 0.58) and UHF-band (580 MHz < ν < 1000 MHz, 1.45 > z > 0.4).

In this work, we present results from MeerKAT single-dish observations in the MeerKLASS L-band deep field, a set of observations performed over a single 236 ^2 patch with 41 repeated scans for each of the 64 dishes. Integrating over the sky in this way will average down time-varying contributions from thermal noise and RFI, thus boosting the signal-to-noise of time-constant sky signals, which includes the foregrounds and the cosmological . Previous analysis <cit.> exploited the fact that the foreground cleaned intensity maps were thermal noise dominated when estimating error bars on the power spectrum measurements. For this MeerKLASS deep-field and future MeerKLASS observations, it can no longer be assumed that the thermal noise will be dominant, therefore demanding more emphasis on covariance estimation. We therefore explore using the scatter in the foreground transfer function <cit.>, primarily used to correct for signal loss from foreground cleaning, as a measure of the covariance in measurements. The transfer function is constructed using mock signal injection into the real data, hence its scatter should encapsulate cosmic variance, thermal noise fluctuations, residual RFI contributions, foreground residuals and signal loss correction uncertainties.

We validate the precision and accuracy of the analysis pipeline to ensure estimator accuracy is within the statistical error associated with these MeerKLASS observations. We formalise the main steps in our pipeline, such as beam reconvolution, foreground cleaning and sampling to a Cartesian grid, presenting validation steps on a suite of simulations which emulate the MeerKLASS L-band deep-field observations. We also use a small sample of overlapping spectroscopic galaxies from the Galaxy And Mass Assembly (GAMA) <cit.> survey to study the cross-power spectrum and its covariance under different analysis choices.

The paper is outlined as follows: sec:ObsData outlines the observations for the MeerKLASS deep field, our new calibration strategy (with an iterative self-calibration process) and overlapping GAMA galaxies. sec:Mocks presents the methods for generating our suite of mock data which emulate the observations. sec:FGcleaning provides details of our foreground cleaning technique. sec:PkEst focuses on the power spectra, formalising our estimation and modelling process, and presents the results. sec:stack shows the results from stacking the MeerKLASS deep-field maps onto the locations of GAMA galaxies and discusses the process and complications that come with this. We then finally conclude in sec:conclusion.



§ OBSERVATIONAL DATA




 §.§ MeerKLASS L-band deep-field


Using the same scan strategy as the MeerKLASS L-band pilot survey in <cit.>, a series of observations were conducted using the MeerKAT telescope between September 1st and December 29th 2021. 
Observations with single dishes typically require a scanning strategy where the dishes are rapidly moved across the sky. The MeerKAT antennas were set to scan in azimuth at constant elevation to minimise fluctuations of ground spill and airmass. To make sure the projected speed on the sky is 5 arcmin s^-1, the telescope scan speed was set to 5/cos(el) arcmin s^-1 (where el is the elevation angle) along the azimuth. This speed also ensures that the telescope pointing does not move significantly compared to the width of the primary beam (∼ 1deg) during a single time dump (2 s).

With a time resolution of 2 s, this gives a scan speed of no more than 10 arcmin per time sample, which is well within the beam size. Noise diodes attached to each receiver were fired for 0.585 s once every 19.5 s during the observation to provide a relative time-ordered data (TOD) calibration reference. The dishes are moved back and forth with a slew of about 10 deg sky projection in each direction, corresponding to an observing time of about 120 s per stripe. The duration of each set of scans with 48 strips is about 100 minutes, and before and after each scan we spent  ∼ 8 minutes tracking a nearby celestial point source to use as a bandpass calibrator and absolute flux calibrator.

At fixed elevation, two scans can be performed per night, corresponding to when the field is rising and setting respectively. The two scans will cross each other to achieve good sky coverage in the region of overlap.
There were 41 observations in total (each one referred as a block), the details of each are listed in the Appendix in <ref>. The observations repeated scans covering a patch of sky approximately 236 ^2 in the southern sky, spanning in R.A. (330^∘,360^∘) and Dec. (-36^∘, -25^∘). The 62 hours obtained for each of the 64 dishes before flagging, provide a cumulative 3968 hours of observations,[This number is reduced for the final useable data since some observation blocks were entirely flagged and some dishes failed in observation. We provide details of this later in this section.] likely to be the deepest single-dish intensity maps, in terms of observational hours per square degree, in the MeerKLASS campaign.



 §.§ Standard MeerKLASS calibration


 

We employ the MeerKAT single-dish calibration pipeline KATcali


to calibrate the deep-field observations and create the combined 3D data cubes. KATcali includes several cycles of RFI flagging, calibration, and map-making. The details of the calibration are thoroughly described in <cit.>. Here, we provide a brief overview of the main points. We take a two-step approach to the calibration process. The first step is to use the flux model of a strong point source during tracking observations to calibrate the bandpass and absolute flux, as well as the injection power of the noise diode. The second step then uses the noise diode injections to calibrate the TOD during the constant-elevation scanning phase.

To calibrate the scanning data, we build the model as 


    T_ model(t, ν)= T_ diffuse (t, ν)+ T_ el (t, ν)
        + T_ diode(t, ν)  + T_ rec(t, ν) ,


where T_ el is the antenna temperature model of elevation-dependent terrestrial emission (atmosphere and ground spill),  T_ diode is the noise diode contribution, T_ rec is the receiver temperature, while 
the diffuse celestial components T_ diffuse includes Galactic emission and the cosmic microwave background (CMB), 


    T_ diffuse(t,ν)= T_ Gal(t, ν) +T_ CMB ,


where the CMB emission is set to be a uniform T_ CMB=2.725K <cit.>, and the Galactic emission T_ Gal (t, ν) is derived from PySM
() that is based on the reprocessed Haslam 408 MHz total intensity map (). 

To compare with the raw data from the correlator, we multiply the signal model by the gain, g(t, ν), so that


    T̂_ model(t, ν)= g(t, ν)  T_ model(t, ν) ,


where the hat indicates a temperature in the (arbitrary) correlator units. 

The signal and gain models are then fitted (with a Bayesian framework; see more details in Section 3.3 of ) to the TOD for each polarisation, frequency channel, dish, and observation scan, all of which are treated independently. Using the gain solution, we can then obtain the calibrated temperature,


    T_ cal(t,ν)≡T̂_ raw(t, ν)/g(t, ν) ,


where T̂_ raw is the raw data collected by MeerKAT.

The residual temperature difference between the signal model and calibrated data is then


    T_ res(t,ν)≡ T_ cal(t,ν) - T_ model(t, ν) .


In standard calibration, we assumed that T_ res is noise-dominated during the fitting, ignoring the faint point sources in the target field.  This was a fair assumption when the noise level was relatively high <cit.>. In this work, we seek a better calibration approach by implementing a self-calibration method. 






 §.§ Self-calibration based on the MeerKLASS sky map




The higher signal-to-noise from the MeerKLASS deep-field data may allow us to deal more carefully with the point sources, which were ignored in the standard calibration, to achieve a better calibration result. Additionally, we aim to use a more accurate sky model to replace the T_ Gal model that is based on the Haslam map. 
The 408 MHz radio continuum all-sky map, which has been widely used to study diffuse synchrotron radiation from our Galaxy and works as a Galactic foreground model for cosmological research, combines data from four different surveys that happened more than 40 years ago (; ). The instrument performance in the 1980s limited the data quality, especially for the Galactic substructures and the zero level of the maps <cit.>. The selection of the spectral index map, used to generate Galactic maps at MeerKAT frequencies, is also a matter of debate <cit.>. 

We therefore extend the calibration process described in sec:StandardCalibration and test the behaviour of the results by introducing an iterative self-calibration process with several loops.
This includes 

(1) replacing T_ diffuse (t, ν) with  T_ sky (t, ν), obtained from the previous Level-6 T_ sky result, when the corresponding Level-6 pixel has a hit-count>40 (which ensures that T_ sky values are obtained from high signal-to-noise data); 

(2) setting the time-order weight w(t_i)=1 for all the scan time samples; instead of setting a higher weight for the diode (main calibrator during the standard calibration) injection samples (see more details in Section 3.3 of ).
Thus <ref> becomes 


    T^ upd_ model(t, ν)= T_ sky (t, ν)+ T_ el (t, ν)
        + T_ diode(t, ν)  + T_ rec(t, ν) .


In this way, we have all components (especially point sources and Galactic substructures) included in the input “model”. The difference between the data and our calibrated sky data is


    T^ upd_ res(t,ν)≡ T_ cal(t,ν) - T^ upd_ model(t, ν) ,


which should be noise-dominated compared to that in <ref>. The self-calibration process is iterative for five loops (see <ref>), in which the T_ sky (t, ν) always uses the Level-6 result of the previous calibration loop. Each iteration refines the calibration, reducing the discrepancies between the observed data and the model. We stopped the self-calibration iteration after five loops because the standard deviation of T_ res converged to a stable value, as shown in <ref>.












 §.§ Characteristics of final MeerKLASS data cubes




  §.§.§ Final data cubes


The combined calibrated temperature cubes (with a pixel size of 0.3 deg and frequency resolution of 0.2 MHz) are derived from all available scans and dishes. However, predominantly due to RFI, an identified repeated source of which we discuss in the following sub-section, only 27 of the 41 observation blocks yielded well-calibrated results. We mark them in the last column of <ref>. In <ref> we present the number count of good (flagged time <50%) channels within the 971–1075 MHz band (channels 550–1050) for each block and each dish. In particular, we found that antenna m059 has a higher chance of surviving calibration even when all other dishes fail.[For consistency with other MeerKAT work, we indicate MeerKAT's 64 antennae with this notation spanning m000-m063.] This may be because m059 is the easternmost dish of MeerKAT, located far from the array core, providing it with a more isolated environment.

We show the final sky temperature map from the combination of the remaining 27 scans after 5 loops of self-calibration at 1023 MHz in the middle panel of <ref>. For comparison, the top panel shows the PySM diffuse Galactic emission model with the same sky coverage. We also show in the bottom panel the total number of time samples (across all dishes and observations) that were combined in each pixel.

To demonstrate the improvement in data quality from the deep-field intensity maps, <ref> plots the amplitude of each pixel along the line-of-sight as a function of frequency. We compare this to the previous MeerKLASS L-band pilot survey (top-panel) relative to the deep field used for this work. The maps should be dominated by the continuum synchrotron spectra, thus their smoothness is a rough proxy for how dominant any systematics are. There are numerous perturbations in the pilot survey spectra that have consequences for the effectiveness of foreground cleaning, required to access the cosmological . However, for the deep field, the spectra appear smoother and well-calibrated. In addition to the improvements shown in <ref>, we also find a marginal performance improvement on the line-of-sight rms when using the self-calibration technique relative to the previous standard calibration in the deep-field. The average variance along the line-of-sight of each pixel decreases to 977.2 mK^2 for the self-calibration, down from 984.3 mK^2 for the standard calibration. 













  §.§.§  RFI contaminations

From the 41 observation blocks, we had to remove 14 blocks due to excessive, low-level RFI contaminations. In affected blocks, these contaminations were sometimes recognisable on the map level of individual dish data as horizontal stripes. After careful investigations, we have now identified the source of the RFI contamination as the downlink of the mobile communication tower (∼ 54 km away) that enters through the sidelobes and causes a non-linear gain effect in the entire band. Our aggressive flagging approach in this study, with the entire removal of 14 blocks, was to ensure high data quality for our analysis. We are currently developing a method to correct for this effect, which should allow us to use the currently flagged data for future projects.




  §.§.§ Noise level estimation


Here we estimate the noise levels from the final maps to check if they are consistent with theoretical expectations. We use the final data cube, T_ sky, to estimate the noise level using the difference between four neighbouring channels (the so-called ABBA technique). By adopting the ABBA pattern, this difference statistic removes both constant and linearly varying signals to cancel signals detected by the receiver, leaving only the noise.  The noise level is estimated by


    Δ T_i(ν^⋆)= 1/2( T^i_ sky(ν) + T^i_ sky(ν+δν)  ) 
                     -1/2 ( T^i_ sky(ν-δν)+T^i_ sky(ν+2δν)),


where the index i goes over all pixels in the map and ^⋆ means the combined result from the four channels [ν-δν, ν, ν+δν, ν+2δν]. 
From the observation data, we take a weighted rms,


    Δ T_ RMS^2(ν^⋆) = 
         N_pix/N_pix - 1 (∑_i w_iΔ T^2_i/∑_i w_i -
         [∑_i w_iΔ T_i/∑_iw_i]^2 ),


where the sum is over the number of pixels in the map, N_pix, and we have suppressed the dependence of w_i and Δ T_i on ν^⋆. For the weight, w_i, we used the theoretical expected variance itself, w_i(ν^⋆)=1/σ^2_ th,i(ν^⋆).
The corresponding variance in each pixel would be,


    σ^2_ th,i(ν^⋆)=1/4(σ^2_ th,i(ν)+σ^2_ th,i(ν+δν)+σ^2_ th,i(ν-δν)
                        + σ^2_ th,i(ν+2δν) ).


Here the noise in the initial polarised TOD is given by the radiometer equation 


    σ^2_ th,pol(t,ν) = T^2_ sys(t,ν)/δν δ t,


where δν=0.209 MHz is the frequency width of the MeerKAT L-band channels and δ t=2 s is the length of each time stamp recording. 
For the system temperature we use the calibrated data itself, T_ sys = T_ cal,HH(t,ν) or T_ cal,VV(t,ν)  for horizontal and vertical polarisation data, respectively. 
Thus the noise in the initial intensity TOD is


    σ^2_ th(t,ν) = T^2_ cal,HH(t,ν)+T^2_ cal,VV(t,ν)/4 δν δ t.


In order to get the final data cube, T^i_ sky(ν), the data goes through several stages of averaging. We propagate the variance taking into account this averaging in order to get to the final σ^2_ th,i(ν).

The theoretical rms which can be compared to Δ T_ RMS (ν^⋆) in <ref>, is then simply


    σ^2_ th (ν^⋆) = N_pix ( ∑_i σ^-2_ th,i(ν^⋆)   )^-1.


More details can be found in Section 5.6 of <cit.>. 


 

In the upper panel of <ref>, we show the noise levels (from data and theory) in the 971–1075 MHz frequency band (channels 550–1050). Note here both the real and theory values are related to the real T_ sys and hit counts in corresponding map pixels. The count reduction due to RFI flagging, and unflagged low-level RFIs that are hiding in the data would cause wiggles. Four groups of spikes are present at higher frequencies, while below 1023 MHz, there is much less scatter compared to the previous MeerKLASS pilot survey result. We therefore chose to restrain our cosmological analysis to the 971–1023 MHz region, for which we show the noise level in the middle panel, finding a median of Δ T_ RMS=1.21mK. In the lower panel of <ref>, we also show the histogram of the ratio 
R_ RMS=Δ T_ RMS (ν^⋆) / σ_ th (ν^⋆) in the 971–1023 MHz band. We find a median value of R_ RMS=1.2, while some channels are noisy and R_ RMS can reach 1.5. 





 §.§ GAMA spectroscopic galaxies


Overlapping with the MeerKLASS deep field, is a catalogue of spectroscopic galaxies recorded in the Galaxy and Mass Assembly (GAMA) survey. GAMA began observing in February 2008 <cit.>. The survey aimed to construct catalogues of galaxies to test the cold dark matter paradigm and probe galaxy evolution <cit.>. For our purposes, we exploit the fact that these galaxies will trace dark matter and should therefore correlate with the overlapping MeerKLASS intensity maps after foreground cleaning.  

We use the fourth and final data release from the Galaxy and Mass Assembly survey, GAMA DR4 <cit.>. This includes galaxies from the GAMA II survey phase which enlarged the GAMA I equatorial survey, by including extra survey regions. One of these extra regions was the 23 hr (G23) field, which completely overlaps in area with the MeerKLASS L-band deep field, shown by the blue scatter points in the bottom panel of <ref>, which represent the GAMA G23 galaxies relative to the observational time footprint map for the L-band deep field. The G23 field covered 339  < R.A. < 351 and -35  < Dec. < -30. The magnitude limit of the field was reduced from the planned 19.8 in r-band to 19.2 in i-band due to limited observational time, but this allowed the GAMA II survey to obtain a high 94.2% redshift completion in the G23 region  <cit.>.

The GAMA DR2 only provides random catalogues that emulate the survey selection function of the real galaxies for the G09, G12, and G15 regions, (not the G23 used in this work). Ideally, these would be used to construct a survey completion window function, which would be used in the cross-power spectrum estimation. In the absence of this, we assume a uniform survey completion across the G23 area and a constant distribution with redshift, given the very flat distribution for the redshift range used in this analysis (green region of <ref>). Given our signal-to-noise levels, survey masks will be a sub-dominant effect in the cross-power, thus this is an adequate approximation to make.

Due to our aggressive flagging strategy on the MeerKLASS L-band data, the overlap region in redshift is restricted to 0.39 < z < 0.46. As shown by <ref>, this catches the GAMA survey galaxy redshift distribution, N_g(z), near the end of its optimal range, but still provides 2269 galaxies to cross-correlate with. At this redshift range, the comoving (volumetric) number density of the GAMA galaxies is n̅_g = 4.8 × 10^-4 h^3 Mpc^-3.









§ SIMULATION OF MOCK DATA


To accompany the calibrated observations outlined in sec:ObsData, we produce a suite of 500 mock simulations, used for validating the analysis pipeline, calculating signal loss from foreground cleaning and estimating covariance in the power spectrum measurements. To create this large number of mocks we generate lognormal density fields <cit.> from which a map of  brightness temperatures can be cut, and galaxies Poisson sampled onto for the spectroscopic galaxy counterpart.

An important stage of the analysis pipeline is regridding the intensity mapping voxels with celestial sky coordinates (R.A., Dec., ν) onto a regular 3D Cartesian grid with lengths l_x, l_y, l_z in h^-1Mpc, on which a Fast Fourier Transform (FFT) can be applied for the power spectrum estimation <cit.>. We therefore require the mocks to begin in celestial coordinates so we can validate the regridding process. To achieve this, we begin by fitting a Cartesian cube around the full MeerKLASS deep-field footprint, creating a grid with dimensions l_x,l_y,l_z = 774.9,335.7,252.2 h^-1Mpc over n_cell = 512^3 cells. We use a large number of cells for this initial grid to approximately emulate a continuous sky, aiming to avoid the simulated maps resolving the finite input grid structure. Onto this grid, we simulate a lognormal density field. This is done by interpolating k-modes from a model power spectrum onto the FFT grid. The  model power spectrum is given by


    P_(k)≡ P_(k,μ) = T_^2 b_^2 (1 + f μ^2/b_)^2 P_m(k)/1 + (k μ σ_v/H_0)^2 .


The anisotropy in this model makes it convenient to parameterise k using μ, defined as the cosine of the angle between the line-of-sight and k. The  parameters are the mean  gas temperature (T_) and the  bias (b_). f is the growth rate of structure and is included to model linear redshift space distortions (RSD) which enhance clustering across all scales. The denominator is included to approximately model the non-linear RSD; the so-called Fingers-of-God (FoG), which suppresses clustering at small scales. The FoG are modulated by σ_v, which is the velocity dispersion of the tracer, in this case, the  gas. All of these quantities have a redshift dependence, which we omit for brevity; they will also have little variation given our narrow redshift range. Lastly, H_0 is the Hubble parameter at z = 0, known as the Hubble constant. An alternative process for generating brightness temperatures is pursued for our stacking analysis discussed later in the paper.
More realistic lognormal line intensity mapping mocks are available. For example, in future work, we intend to adopt the approach in <cit.> where luminosities from the spectral lines are painted given a luminosity function. This has been shown to reproduce better RSD for galaxy distributions and models the whole temperature distribution expected in the maps.

Inverse Fourier transforming the model in <ref> provides the  correlation function ξ_ which is then log-transformed via ξ^'_ = ln(1 + ξ_). Fourier transforming ξ^'_ back gives the modified lognormal power spectrum P^'_. Gaussian random numbers are then sampled into each cell of the Fourier grid δ_G(k) with variance σ^2(k) ∝ P^'_(k). This Gaussian random field is then inverse Fourier transformed to real space giving δ_G(x) from which the lognormal  brightness temperature fluctuation field is given by


    δ T_(x) = exp[δ_G(x) - σ^2_G/2] - 1 ,


where σ_G^2 is the measured variance of the Gaussian random number field δ_G. Further details on this process can be found in the Appendix A of <cit.>. For the galaxy over-density field δ_g(x), we follow the same procedure except the input model is for the galaxy power spectrum, given by


    P_g(k)≡ P_g(k,μ) = b_g^2 (1 + f μ^2/b_g)^2 P_m(k)/1 + (k μ σ_v/H_0)^2 ,


where b_g is the bias for the galaxies. We assume that the velocity dispersion σ_v for the  gas and galaxies is the same for simplicity. This will be sufficient for this work where signal-to-noise is low at the small scales where σ_v has the main effect.

The galaxy over-density δ_g is unit-less whereas the intensity maps δ T_ are brightness temperature fluctuations measured in mK. We use the same seed in the random number generation to ensure that both the δ T_ and δ_g correlate. The lognormal code we adopt utilises some functions from the publically available package  <cit.>.[https://github.com/steven-murray/powerbox]



 §.§ Transforming between Cartesian grids and sky maps


As already mentioned, the galaxies and the intensity mapping voxels from observations are in celestial coordinates at a certain frequency (or redshift) i.e. (θ,ν) ≡ (θ_R.A.,θ_Dec.,ν). However, for the final power spectrum estimation, we require the fields be in a Cartesian space with dimensions (l_x,l_y,l_z) in h^-1Mpc. For this we follow the recent work in <cit.> which we discuss further in sec:PkEst.

Creating lognormal mocks in celestial coordinates that emulate observations requires the inverse of this step i.e. δ T_(x) → δ T_(θ,ν), since the lognormal fields are generated onto Cartesian grids measured in h^-1Mpc. Transforming mock galaxy coordinates (the Poisson sampling of which is discussed in sec:Galmocks) from (l_x,l_y,l_z) [h^-1Mpc] → (θ_R.A.,θ_Dec.,ν) is trivial. We can use [https://docs.astropy.org/en/stable/index.html] <cit.> transformation routines on each galaxy's coordinate. However, transforming the tessellating field of Cartesian cells, each with an associated brightness temperature, into a gap-less map of voxels with identical celestial coordinates and frequency binning as the real MeerKAT maps, requires a resampling of the field. For this we also utilise the techniques and public code[https://github.com/stevecunnington/gridimp] outlined <cit.>. 

This process involves a Monte-Carlo particle sampling technique where each Cartesian cell is filled with N_p particles, randomly distributed within the cell. The particles are assigned the brightness temperature of the cell they fall within. The particles are then transformed to celestial coordinates (θ_R.A.,θ_Dec.,z) and added into the map voxels associated with their celestial coordinate and the frequency channel corresponding to their redshift. Each voxel is normalised by the number of particles falling within it. Providing enough sampling particles per cell are used (in this work we use N_p = 10) and the Cartesian grid is of sufficient resolution (hence our choice of n_cell = 512^3), it will provide an intensity map in celestial sky coordinates, that traces the fluctuations generated by the lognormal field. 

Throughout this work, we do not require the necessary extensions of higher-order particle assignment or field interlacing to suppress the effects of aliasing <cit.>, since these effects are below the current statistical noise from this data. We are however, still required to implement a correction to the power spectrum estimation from the Nearest-Grid-Point assignment which smooths the field, which we formalise in sec:PkEst.



 §.§  intensity map mocks


Using the regridding process described in the previous section we are provided with  intensity mapping mocks δ T_(θ,ν), with identical coordinates and dimensions as the observed MeerKAT intensity maps. From here we can add in observational effects and simulate various systematics. The most dominant effect comes from foreground contamination. Systematics will have a large influence on foreground contamination, and this can be extremely complex. Therefore, this is not something we attempt to directly simulate in this work since we risk under-estimating the problem. Instead, we use the observational data itself which has the relevant systematics contained within, albeit only one realisation of them. However, adding these directly onto different realisations of  mocks, still recreates stochasticity in the  signal's response to systematics. We discuss this further in sec:FGcleaning.

Another important addition to the mocks will be a smoothing to the intensity maps corresponding to the angular pixelisation. This is necessary to emulate the radio telescope's beam, which for single-dish intensity mapping is quite broad giving a low angular resolution to the maps and damping small-scale fluctuations perpendicular to the line-of-sight. In this work, we assume the MeerKAT beam profile to be Gaussian, which is a simplification but still valid given the signal-to-noise at small scales where the beam suppresses signal below the noise. We will investigate implementing a more realistic MeerKAT beam profile in future work <cit.>. The assumed Gaussian beam profile has a frequency (ν) dependent standard deviation in radians given by


    σ_beam(ν) = 1/2√(2ln 2)c/ν D_dish ,


where we use D_dish = 13.5 m for the diameter of the MeerKAT dishes and c is the speed of light. To emulate the beam effects we smooth the mock maps by creating a 2D Gaussian array kernel centred on the median angular pixel θ_0 given by


    B_beam(θ,ν_i) = exp[-1/2(θ - θ_0/σ_beam(ν_i))^2] .


This is normalised such that ∑_θB_beam=1, so that convolution with this beam kernel does not change the total brightness temperatures within the maps. The  intensity map mocks are convolved with <ref> at each frequency ν_i to give the beam-smoothed  map;


    δ T_⊛(θ,ν_i) = δ T_(θ,ν_i)⊛B_beam(θ,ν_i) ,


where the ⊛ symbol represents a convolution.


To realistically emulate additional components such as foregrounds, noise and their response to systematics, we add the real observations to each mock iteration. This is to avoid the requirement of simulating the complex and poorly understood systematics. 

In some cases, mostly for studying the  auto-correlation, it will be useful to simulate just the thermal noise originating from the thermal signature of the receivers.

We do this by generating Gaussian random noise the power of which can then be measured with the same pipeline as the rest of the analysis, providing a robust estimate of the thermal noise floor. We begin with an analytical description of the thermal noise fluctuations per pixel using the radiometer equation (as in sec:ObsData but now at map-level)


    σ_N(θ,ν) = T_sys(ν)/√(2 δν t_p(θ,ν)) ,


where t_p is the summed observation time contained in each voxel, which can be taken directly from the time stamp counts as shown in <ref>, with each count representing δ t =2sec. This emulates the real data by creating a non-uniform noise map, generating higher noise around the edges where scan coverage is lower. For the system temperature, we follow <cit.> and use a model tuned to coincide with the estimated system temperature of the MeerKAT telescope <cit.>[See https://skaafrica.atlassian.net/wiki/x/AYCHE], assuming 


    T_sys(ν)=T_rx(ν)+T_spl+T_CMB+T_gal(ν) ,


where we assume the contribution from our own galaxy is T_gal=25K (408 MHz / ν)^2.75, the CMB temperature is T_CMB=2.725 K, as in sec:ObsData. The spill-over temperature is assumed as T_spl=3 K and the receiver temperature we model as


    T_rx(ν) = 7.5K+10K(ν/GHz-0.75)^2 .


Due to the narrow frequency band we use, this gives a near-constant system temperature with the range 15.97<T_sys<16.02 [K]. Gaussian random numbers can then be generated across the MeerKLASS deep-field map with the voxel dependent rms of σ_N(θ,ν).



 §.§ Galaxy mock catalogues


To accompany the mock intensity maps, we also require corresponding galaxy mocks with the same correlated clustering.  Rather than transforming the Cartesian galaxy over-density field δ_g(x) into sky coordinates, it is more realistic to instead Poisson sample the field to generate a catalogue of galaxies with the same number density as the GAMA G23 survey. This also allows additional validity checks such as testing the stacking pipeline which cannot be done without a catalogue of galaxy coordinates. 

We generate the galaxy catalogue coordinates initially in the Cartesian domain to exploit the high n_cell = 512^3 resolution. We first calculate the average galaxy counts in cells ⟨n_g⟩ for the real GAMA G23 region in the redshift range where there is overlap with the intensity map frequency range. We then take the galaxy survey selection window function, which for the GAMA G23 region, we have to assume is constant between 339 < R.A. < 351 and -35 < Dec. < -30 (see discussion in sec:GAMAobs). We then transform this window function into the high resolution Cartesian space. A galaxy number counts field n_g(x) is then given by


    n_g(x) =  ⟨ n_g⟩(δ_g(x) + 1) W_g(x) ,


where we use the transformed window function W_g(x) and the real GAMA average counts ⟨ n_g⟩ to ensure the mock galaxy counts will share the same survey selection as the real galaxy survey, with the same number density. In this process we normalise the window function W_g such that its maximum value is 1, so for the GAMA G23 survey, this becomes a binary mask of 0 and 1. We then Poisson sample the n_g field to introduce some stochasticity which we expect to be present in the real galaxy catalogues. This provides an integer value in each Cartesian cell for the galaxy counts. The last step is to assign random Cartesian coordinates within the cell boundaries for the n_g(x) galaxies within each cell. At this stage we rely on the Cartesian field having sufficiently high resolution such that the random assignment of a coordinate within the cell does not shift the galaxies away from the peaks of the density fluctuations. The catalogue of galaxy coordinates (x, y, z [h^-1Mpc]) can then be transformed into celestial coordinates (θ_R.A.,θ_Dec.,z). 


In this work, we generate 500 mock intensity maps with corresponding mock GAMA galaxy catalogues. As some early validation of these mocks, we show in <ref> how the power spectra for both the  intensity map and galaxy auto-correlations, along with their cross-correlation, agree with the fiducial model (black dashed line). This demonstrates that our estimator is unbiased and processes such as the regridding are performing to within an acceptable tolerance. The power spectrum estimation process used here on the mocks is identical to that used on the real data and is detailed later in sec:PkEst, along with the modelling.



The mocks used in the test of <ref> do not include any additional systematics beyond the telescope beam in the intensity maps. We will investigate the inclusion of foregrounds and systematics in sec:FGcleaning. The large scatter shown by the shaded blue region, which represents the 1 σ range across the 500 mocks, is therefore driven by cosmic variance due to the small volume covered by the data. We see that the errors for the galaxy power spectrum appear larger than the  intensity maps. This is because we have not introduced any thermal noise (or other observational or systematic effects) into the intensity maps. However, the galaxy catalogues will have shot noise contributing to the uncertainties. This nicely showcases the potential for  intensity mapping if noise and systematics can be controlled, however, this is not the purpose of this exercise so is not a point we will labour. There are also additional caveats when considering high-density tracers such as . As shown by <cit.>, despite the low shot noise, stochastic noise and non-linear bias in the  field can limit the potential of constraints inferred from the power spectrum.

The green dashed lines in the residual panels mark the regions where there is > 90% accuracy relative to the model which we can see is achieved at all scales for all three cases. This is also well within the statistical uncertainty (blue-shaded regions). The largest residual comes in the cross-correlation where discrepancies seem to increase with k nearly reaching the -10% boundary. We speculate that this may be caused by the Poisson sampling of mock galaxies followed by random assignment of coordinates within the cell. This extra stochasticity is not repeated in the  intensity map mocks and could therefore be generating a pseudo-cross-correlation coefficient, which would depart from r = 1 at smaller scales (higher-k). However, this appears to be a minor effect. Furthermore, this could be seen as a feature, since we expect r < 1 in the real observations, which will also have a scale dependence. For these reasons, we deem this to be a suitable result for the purposes of this work, and defer a higher precision investigation and validation to future work.  



§ FOREGROUND CLEANING




Foreground cleaning  intensity maps is a well-studied problem <cit.>, and generally blind cleaning techniques are the currently preferred method for single-dish experiments. As their name suggests, blind methods do not rely on prior knowledge of the foreground structure. This is advantageous for current experiments because the foregrounds at these frequencies lack observational constraints. Furthermore, blind techniques avoid placing overwhelmingly stringent calibration requirements on the data, since the dominant foregrounds require beyond sub-percent precision modelling for a successful subtraction. Blind techniques instead work by exploiting the foreground dominance and their frequency-correlated structure to assume that they can be reduced to a distinct set of correlated modes separated from the oscillating, sub-dominant fluctuations which will contain most of the  information.

We outline the formalism for our blind foreground cleaning approach in sec:PCAcleaning, but first we discuss two pre-processing steps before foreground cleaning. These are designed to improve the foreground clean by reducing the impact from residual RFI or systematics, plus any known frequency-dependent observational effects. 



 §.§ Reconvolution
 

All radio intensity mapping experiments will have a chromatic beam whose size evolves with frequency. This can add to the number of degrees of freedom in the radial foreground structure, inhibiting foreground removal since more modes need to be removed to perform a suitable clean. Smoothing the maps differently in each frequency channel can homogenise the angular resolution, giving them a consistent effective beam size at all frequencies. This reconvolution should also suppress residual contamination manifesting as localised amplitude spikes in the temperature maps, smoothing them out so there are fewer discontinuous fluctuations which will impact the PCA and Fourier transforms for power spectrum estimation.

The reconvolution step involves convolving the intensity maps, one frequency channel ν_i at a time with the kernel



    K(θ,ν_i)=
        
        exp[-1/2(θ - θ_0)^2/γσ_max^2-σ_beam^2(ν_i)] ,


where σ_beam is the beam size as previously defined in <ref>, σ_max is the maximum σ_beam(ν_min) value at the lowest frequency,
and γ is a scaling factor which governs how much the final effective resolution decreases. The reconvolution kernel is normalised such that the sum over all pixels is unity i.e. Σ_θK(θ,ν_i) = 1. 

As in previous work <cit.>, we use a weighted reconvolution to mitigate the impact of edge pixels with low scan coverage. From hereon, the weights for the intensity maps derive directly from the time stamp count in each pixel, as shown in <ref>. The weighted reconvolved intensity maps are given by


    δ T_⊛(θ,ν_i) = [δ T(θ,ν_i)w(θ,ν_i) ] ⊛K(θ,ν_i)/w(θ,ν_i) ⊛K(θ,ν_i)


The weights themselves must also be reconvolved to keep them consistent and optimal for the new δ T_⊛ maps. The reconvolved weights are given by 


    w_⊛(θ,ν_i) = [w(θ,ν_i) ⊛K(θ,ν_i)]^2/w(θ,ν_i) ⊛K^2(θ,ν_i) .


In this work we use a value of γ = 1.4 for the reconvolution kernel in <ref>. We find that neglecting this resmoothing step entirely increases the influence of systematics in the results which is likely due to residual RFI or contaminants not being smoothed and depressing their dominance. We discuss this further in sec:ResmoothingAppendix but also plan a dedicated study into this aspect of the analysis where a more robust model of the MeerKAT beam will also be implemented.

The resmoothed, uncleaned map is shown in <ref> (top panel). Careful comparison with <ref> reveals how some of the foreground structure has been smoothed, albeit subtly, which is the aim. Too much reconvolution will suppress power in high k_⊥ modes in the final power spectrum estimate. The reconvolution is more noticeable in the time stamp count field, which we use as the weights w_⊛, shown by the bottom panel of <ref>.






 §.§ Map trimming


Shown by the time stamp count field of <ref> (and the resmoothed version in <ref>), the MeerKLASS scanning strategy of combining numerous rising and setting scans does not generate a uniform coverage. The edges of the patch result in relatively low recorded time stamps due to a lack of overlapping scans in these regions. 

We found that implementing an edge trim to the field improved the quality of the foreground clean and final power spectrum estimates. This is motivated and explored in greater detail in Carucci et al. (in prep.). The improvement from map trimming is because pixels are removed where residual time-varying contamination has been averaged down less, due to fewer scans. 

In <ref>, the black contour shows the cut to the patch we make, removing everything outside the trimmed lines. This amounts to all pixels at 334^∘ < R.A. < 357^∘ being set to zero in both the weights and intensity map field.



 §.§ Blind foreground cleaning with PCA


We represent the intensity maps with matrices  which are equivalent to the T(θ,ν) temperature fields, discretised into arrays on which matrix algebra can be performed.

Blind foreground cleaning assumes the matrix of the observational data cube _obs, which contains the intensity maps with n_θ pixels at each of the n_ν frequency channels, can be generalised by the linear equation


    _obs = + .


 is referred to as the mixing matrix and is the set of N_fg spectral functions estimated to project out the foregrounds.  are the N_fg source maps obtained by projecting  along the data;


    = ^T_obs .


The dot product  thus has shape [n_ν,n_θ], along with the other terms in <ref>, and represents a subset of the observations estimated to contain the bulk of the foreground contamination. The residual term  will contain the remaining  signal and thermal noise. In reality, the situation is more complex and there is cross-term leakage. Both  signal will leak into the subtracted  term causing signal loss, and foreground will leak into the remaining  causing foreground residuals. The control of both these is crucial and we discuss them further throughout the remaining sections.

Different blind foreground techniques only vary in their approach to estimating the mixing matrix .
In this work, we adopt the straightforward principal component analysis (PCA) approach, since it is well-tested and understood. Implementing more sophisticated blind techniques is planned in future work. PCA foreground cleaning begins by computing the frequency covariance matrix of the weighted observations


    =(∘_obs)^T(∘_obs)/n_θ-1 ,


where we have used the intensity mapping weights to optimise the estimation and ∘_obs represents the element-wise product between the weights and the observational data matrix. Since the weights are the stacked average of the counts along the line-of-sight, they are constant in frequency by design to avoid increasing the rank of _obs i.e. not inject additional structure into the frequency spectra. Weighting in this way means  is not a formally normalised covariance estimation, however, this is not a concern for us since we are only extracting unitless eigenvectors from the covariance and using these to project out modes in the real observed data so sensible amplitudes are recovered in the component separated pieces. We discuss this further, with some examples in sec:FGweights.  is then eigen-decomposed by


    = Λ^T ,


where  is the matrix of eigenvectors with the corresponding eigenvalues contained in Λ. The mixing matrix is then formed by taking the first N_fg column vectors u_i from  i.e. = [u_1,u_2,...,u_N_fg]. The first 9 eigenmodes are shown in the Appendix plot <ref>. It is at this stage a choice is required for N_fg, which determines how many modes are removed and deemed as foregrounds. If the N_fg is too high, the  signal loss increases; on the contrary, if N_fg is too low, residual contamination will dominate more. The estimate of the foregrounds is provided by _fg ≡ which is subtracted to provide the residuals, revealing the final cleaned data used in the analysis i.e. _clean ≡ where


    _clean = _obs -   _fg .


<ref> shows the resulting cleaned maps by removing N_fg=10 PCA modes. We show the maps split into six frequency sub-bands (given in the bottom-left of each panel) to illustrate the changing intensity structures. Each panel is the frequency average of the 42 channels in each sub-band showing a reasonably consistent amplitude of temperature fluctuations across the sub-bands. The amplitudes in the maps are still likely driven by non-cosmological contributions, mostly thermal noise, but they are not far from the 0.1 mK fluctuations we expect from . As we will show statistically in sec:PkEst, over large-scale fluctuations, the predominant contribution comes from the cosmological .







 §.§ Foreground cleaning tests using mocks


In order to validate our foreground removal and estimate the signal loss it induces, we run the same PCA cleaning technique on mock intensity maps. As discussed in sec:HIIMmocks, simulating foreground contamination that emulates the systematics and complex observational effects is beyond current capabilities. We therefore use the real observations themselves and inject mock  intensity maps into these. We then apply the foreground removal pipeline on the mock-injected data and estimate the signal loss from the injected mock whose original signal is known.

We follow the formalism outlined in <cit.>, beginning with a  mock, whose matrices we notify by  to distinguish from the real observation . The mock is injected into the real observations i.e. _obs + _. This is then cleaned by projecting out a subset of modes defined by the mixing matrix, as in the previous sections, such that


    _clean = (_obs + _) - ^T(_obs + _) ,


The data in _clean will now contain signal loss emulating that of the real data clean, along with residual systematics and noise from the real data.

We show the results of mocks contaminated by the real data in <ref>. We show measured power spectra averaged over all 500 mocks under different scenarios (details of power spectrum estimation are provided in the following sec:PkEst). Firstly, the grey dotted line shows the -only mocks, _, with no foreground cleaning effects, representing the true power. The red solid line shows the result of adding the real observations to the mock, then PCA cleaning as in <ref>. The power is higher than the grey line, despite the inevitable signal loss the clean will have caused. This is evidence that the _clean maps contain residual systematics and thermal noise, which are additively biasing the power[There will also be a small contribution from the true  signal in the observed data.]. We can demonstrate this by considering the orange solid line which is the cross-correlation of the cleaned mocks _clean with the -only mocks _, a test only possible with simulations. Since the residual systematics and noise will be uncorrelated with the -only mocks, the distortion in power comes purely from signal loss, hence why it is lower than the -only power at all scales.

To isolate the contribution from unknown systematics, we need to account for thermal noise and signal loss from foreground cleaning. The thermal noise contribution can be approximated using <ref> to generate 500 noise maps and add these to the -only mocks. The black-dashed line shows this and demonstrates the boost in power relative to the grey-dotted line, more so at small scales where, due to the beam, the  signal is low relative to the scale-independent noise[The noise is not completely scale-independent because it has gone through the same regridding and reconvolution processes as the data which causes some subtle damping.]. The last result presented in <ref> is the reconstructed power (i.e. correcting for foreground cleaning signal loss) shown by the blue solid line. This utilises the signal-loss evaluated from the cross-power (orange line) to reconstruct the _clean auto-power. This is discussed more in the following sub-section. The discrepancy between the blue-solid and black-dashed line must be caused by systematics, since signal-loss and thermal noise have now been accounted for. We find a similar excess power with the real observations, which we discuss in the following sec:PkEst. The results in <ref> are to validate the realism of the mock pipeline, and also to motivate the signal loss reconstruction process, which we formalise in the next sub-section.





 §.§ Signal loss correction with transfer functions


We can utilise the method of mock signal injection from the previous sec:FGcleanMocks for estimating the signal loss in the power spectrum of the real data. This is now a well-understood process <cit.> and used in previous single-dish intensity mapping analyses <cit.>.

We begin with the _clean term in <ref>, which as demonstrated, will contain signal loss that should emulate the real data. In some cases, the cleaned real data _clean can be subtracted from _clean. This accelerates the convergence of the transfer function calculation by removing the additional noise caused by the real data but has no bearing on the signal loss calculation. Leaving this piece in the calculation of _clean is however crucial when using the transfer function for estimating errors in the power spectrum. We also find that due to the large number of mocks used in this work, an equivalent accuracy of the transfer function is obtained whether subtracting _clean or not. For simplicity, we therefore only use the case as defined by <ref>, without the _clean subtraction.

The transfer function is defined by the power spectrum, signified by the operator P(), of the cleaned mock _clean with the foreground-free -only mock _, and the ratio of this with the power spectrum of the -only mock


    T_(k)=⟨P(_clean,_)/P(_,_)⟩_N_mock .


The angled brackets indicate an averaging over a large number of mocks such that a converged result is obtained. In this work, we use all mocks in our suite, thus N_mock=500. The operator P() can represent any power spectrum estimator i.e. we use this same recipe for estimating signal loss in the spherically averaged power and also in the cylindrical (k_⊥, k_∥) power spectrum, which we later utilise in some analysis.

In this work, we will mostly focus on the MeerKLASS deep field's cross-correlation with the overlapping GAMA galaxies. The signal loss from PCA cleaning in the cross-power spectrum should be identical to that in the auto-power. This was established empirically in <cit.> which showed how removing a subset of PCA modes entirely but leaving their contribution in the other half of the cross-power product (e.g. in the galaxy maps which have no cleaning, thus no signal loss) generates no more power than the auto-. We present further reasoning for why this is the case in sec:TF_signalloss_autoVcross. To summarise, we believe it is explained by the PCA modes being orthogonal, thus maximally uncorrelated. Therefore, the additional remaining modes in the uncleaned field are linearly uncorrelated with what survives in the cleaned , producing no extra power. 

Despite the approximately equal signal loss between auto and cross-correlations, we still define a separate transfer function for the cross-power spectrum purely for covariance estimation purposes whereby the only difference is we switch in the GAMA galaxy mocks _g instead which will emulate the signal strength and shot-noise of the GAMA galaxies, thus providing a more robust error estimate in the cross-power.


    T_,g(k)=⟨P(_clean,_g)/P(_,_g)⟩_N_mock .


Any measured power spectra should be divided by T(k), thus the transfer function represents a reconstruction of signal at the power spectrum level. 



§ POWER SPECTRUM ESTIMATION


In this section we present power spectra estimations from the foreground cleaned MeerKLASS L-band deep field. We first outline the pipeline for performing these measurements then present the results. By default, we use maps that have been foreground cleaned by removing =10 PCA modes, as outlined in the previous section. However, we also examine the results from varying . 



 §.§ Transforming to a Cartesian grid


The first task to perform in order to measure a power spectrum is to transform the foreground cleaned intensity maps onto a Cartesian grid with comoving h^-1Mpc coordinates. This allows for a Fast Fourier transform to be calculated on the Cartesian array, from which the power at different Fourier bandpasses k, can be averaged and presented. 

We discussed the transformation between sky maps and Cartesian grids already in sec:RegriddingMocks when we performed the inverse routine of sampling Cartesian cells from lognormal mocks into celestial sky map voxels that emulate the MeerKLASS intensity maps. We again utilise the recent work and code in <cit.> and first define a grid of size l_x,l_y,l_z = 632.2,341.5,231.5 h^-1Mpc with n_x,n_y,n_z = 88,48,168 cells along each dimension. The choice of cells is a factor of 1.5 smaller than the number of map voxels in R.A., Dec., and ν. Making the Cartesian grid slightly lower resolution than the map and using 10 sampling particles per map voxel ensures each Cartesian cell is well-sampled, avoiding empty cells.

The foreground cleaned maps δ T_(θ,ν) are transformed to this Cartesian grid, giving δ T_(x). The intensity mapping weights also follow an identical transformation process w_(θ,ν) → w_(x). The catalogue of GAMA galaxies can be directly gridded by transforming their celestial coordinates into Cartesian, then assigning them onto the Cartesian grid n_g(x). For all sampling particles and galaxies we use a straightforward nearest grid point (NGP) assignment. <cit.> presented extensions to this method for avoiding aliased power and obtaining sub-percent accuracy. However, given the current signal-to-noise of our data at small scales, aliasing effects will be negligible, thus higher-order interpolations and interlacing are overkill and serve only to increase compute demand. However, we correct for the dominant effect which is the field's smoothing by the (NGP) particle assignment. This requires deconvolving the field, achieved by dividing through by the NGP mass assignment window function in Fourier space <cit.>


    δ̃ ̃T̃_(k) →δ̃ ̃T̃_(k)/W_ngp(k) ,


where the window function W_ngp accounts for the sampling particles discretisation into cells of size H. These are separable functions along each Cartesian dimension, thus modelled by a Fourier transform of a top-hat function, which becomes a sinc function; sinc(x)=sin(x)/x and therefore


    W̃_ngp(k)=sinc(k_x H_x/2) sinc(k_y H_y/2) sinc(k_z H_z/2) .


Previous work <cit.> employed a tapering along the radial direction of the regridded field with the primary purpose of reducing edge effects in the Fourier transform. Our modelling approach should however account for these effects since the model is also convolved with the window function <cit.>. We still found results were worse when omitting a radial tapering though, which we suspect is instead caused by edge-effects from the foreground clean <cit.> which are suppressed by the tapering. We experimented with a less aggressive taper by using a Tukey window, also known as a cosine window (this is also explored in Carucci et al. in prep). This has a shape parameter α which is simply a top-hat with no apodisation when α=0 and equivalent to a Hann window when α=1. We show these window functions in <ref>. By opting for less aggressive Tukey windows we still effectively suppress the extreme edges where foreground cleaning effects may be present but avoid unnecessary tapering on the central data. We will explore this further with future data in MeerKAT's UHF band where the larger useable frequency ranges may render this less of an issue. In this work we found optimal results when using α=0.8 for the Tukey window which is applied to the intensity map weights w_(x). We do not apply this to the galaxy data as we found this increased the correlations between k-bins in the power spectrum. 





 §.§ Power spectrum formalism


With the fields assigned to a Cartesian space and appropriately tapered, the power spectra can then be measured. The Fourier transformed field of the  temperature fluctuation maps δ T_ is given by


    F̃_(k)=∑_xδ T_(x) w_(x) exp (i k·x) ,


where the weights, now denoted by w_ to distinguish them from the galaxy weights, still derive from the field of time-stamp counts, which are regridded following the same process as outlined in the previous sub-section. The Fourier transformed field of the the galaxy count field n_g is given by


    F̃_g(k)=∑_x n_g(x) w_g(x) exp (i k·x) - N_gW̃_g(k) ,


where N_g = ∑ n_g is the total number of galaxies in the optical maps and W̃_g is the weighted Fourier transform of the selection function


    W̃_g(k)=∑_x W_g(x) w_g(x) exp (i k·x) .


The galaxy window function W_g is normally constructed by stacking mock galaxies that carefully follow the survey selection characteristics of the experiment to account for its incompleteness. As discussed previously, there are no GAMA G23 mocks available for this purpose so we therefore construct a simple window function which is 1 within the survey boundaries and 0 everywhere else. Some inclusion of the redshift evolution of N(z) could have been included, but as <ref> showed, the variation over our narrow redshift range is minimal. The window function is normalised so that ∑_xW_g(x) = 1. Since we lack this proper galaxy window function we do not implement the optimal weighting as per <cit.>, but instead use uniform weighting, so equivalent of assuming w_g(x)≡W_g(x). 

The cross-power spectrum estimator is then given by


    P̂_,g(k) = V_cell/∑_x w_(x)w_g(x)W_g(x)Re{F̃_(k)F̃^*_g(k)}1/N_g ,


where V_cell=l_xl_yl_z/(n_xn_yn_z). The hat notation (P̂) indicates that this is the estimated power from the data, to distinguish from the modelled power, which we discuss shortly. The  auto power spectrum estimator is more straightforwardly given by


    P̂_(k) = V_cell/∑_x w_^2(x)|F̃_(k)|^2 ,



















These power spectra are either spherically averaged into bandpowers |k| ≡ k to provide 1D power spectrum results or alternatively cylindrically averaged into 2D bandpowers defined by modes perpendicular (k_⊥) and parallel (k_∥) to the radial direction z, which we assume is the line-of-sight in a plane-parallel approximation, valid for the current MeerKLASS sky sizes. They are finally divided by their relevant foreground transfer functions (<ref> and <ref>) to reconstruct signal loss for each k-bin.



  §.§.§ Modelling the power spectrum


Here we outline the model of the power spectrum estimations for which we mostly follow the work in <cit.>, with some extensions. We model the  power spectrum using the equation already introduced in <ref> when discussing mock simulations. We repeat the model here for convenience;


    P_(k)≡ P_(k,μ) = T_^2 (b_ + f μ^2)^2P_m(k)/1 + (kμσ_v/H_0)^2 .


Additional modelling of observational and mapping effects is also necessary since these will distort the power.  To model the MeerKAT beam and the additional smoothing from the reconvolution process we assume the field has undergone a constant smoothing by a 2D Gaussian kernel whose standard deviation is given by R_beam=√(γ) R^max_beam in comoving units. γ=1.4 accounts for the reconvolution to a common effective resolution and R^max_beam is the Gaussian approximation of the MeerKAT beam size at the deepest redshift (and lowest frequency ν_min)


    R^max_beam = χ(z_max) σ_beam(ν_min) ,


where σ_beam is given as before from <ref> and χ(z) is the comoving distance to redshift z, which in this case, is the maximum redshift for our frequency range. The perpendicular modes will then be damped according to


    B̃_beam(k) = exp[-k^2(1-μ^2) R_beam ^2/2] .


For this work, we find R_beam=14.1 h^-1Mpc. 

Similar to the beam, there will also be distortions to the power caused by the finite resolution of the intensity maps.  We approximate the pixel size in comoving units by s_pix=χ(z_eff)Δθ where Δθ=0.3 is the pixel size used for the MeerKLASS L-band mapping. The frequency channel depth is approximated by taking the mean across all spacings between redshifts corresponding to channel boundaries i.e. s_chan=Σ_i [χ(z_i) - χ(z_i+1)]/N_chan. These quantities are used in damping functions to model the suppressed resolution from mapping into pixels and frequency channels, the former being the far more dominant effect for the wide-pixel MeerKLASS intensity maps. These functions are given as


    B̃_pix(k) = sinc(k√(1-μ^2)  s_pix/2) ,


and


    B̃_chan (k)=sinc(k μ  s_chan /2) ,


The model and damping functions are discretised onto the same 3D grid of k-modes as the FFT for the real data, then convolved with the survey window function. The model for the observed  auto-power spectrum therefore becomes


    P_^obs(k) = W^2_(k) ⊛[P_(k)  B_beam^2(k) B_pix^2(k) B_chan^2(k)]/∑_x w^2_(x) W^2_(x) ,


where we have normalised by the sum of the weights and window function, W_, to obtain an amplitude that matches the weighted and windowed estimator. For the intensity mapping window function, we assume a simple binary function; 1 where we have scan coverage and 0 everywhere else. There is scope for improving this approximation and further work is needed to examine how intensity mapping surveys are impacted by any selection effects. 

A similar approach is followed for modelling the cross-power with GAMA. The cross-correlation power spectrum is given by


    P_,g(k)= T_[r b_ b_g+b_ f μ^2+b_g f μ^2+f^2 μ^4] P_m(k)/1 + (kμσ_v/H_0)^2 .


where we introduce r, the correlation coefficient between the  fluctuations and the GAMA galaxy over-densities. This accounts for the different astrophysics in each tracer and non-linear clustering, which can mean the two fields are not exact multiples of each other and there is instead some stochasticity which will suppress their cross-power <cit.>. We assume r to be constant, which is sufficient for the current signal-to-noise, but in reality it will exhibit some scale dependence. The correlation coefficient only enters on the single term in <ref> because it only influences the biased tracer fields, not their unbiased velocity fields which are modulated by f. The observational cross power is then similarly convolved with the survey window functions but only requires single factors of the damping functions B; 


    P_,g^obs   (k) = 
       W_(k)W^*_g(k) ⊛[P_,g(k)  B_beam(k) B_pix(k) B_chan(k)]/∑_x w_(x) w_g(x) W_(x) W_g(x)  .


We do not include modelling of uncertainties in the GAMA galaxy positioning, such as redshift estimation. The GAMA radial kernel equivalent of <ref> will likely have a larger impact due to spectroscopic redshift estimation errors. However, this will still be a sub-dominant contribution to the overall modelling relative to the beam and intensity map pixelisation, thus it is reasonable to neglect observational effects from the GAMA galaxies.

We perform a simple least-squares fit for an overall amplitude to our cross-power, equivalent to assuming fixed values for all parameters except for a fitted T_. This is deliberately simplistic to keep the model and fitting in a low-dimensional form. The small overlap sample in the cross-correlation means the statistical noise from such measurements is not yet suited to high-dimensional modelling. We are not therefore extending much from previous work <cit.> in this regard. Furthermore, the auto- still has evidence of additive bias, as we will discuss, and therefore can only be trusted for upper limits in parameter inference. 

For the power amplitude fitting, we assume the parameter values outlined by <ref>. The  bias comes from interpolated results from hydrodynamical simulations <cit.>, which can be approximated by the polynomial


    b_(z)=0.842+0.693 z-0.046 z^2 .


The galaxy bias is obtained by fitting the amplitude of the GAMA auto-power spectrum. The growth rate is obtained from f=Ω_m(z_eff)^0.545 for our Planck 18 cosmology. The correlation coefficient r and velocity dispersion σ_v are poorly constrained nuisance parameters tuned by eye to provide an adequate fit. We discuss model and parameter fitting further in sec:ParamConstraints where we present results from a simultaneous MCMC fit on several parameters, demonstrating the degeneracies between them and discuss plans for breaking such degeneracies, but this is not a central focus of this work. 










 §.§ Error estimation with the transfer function


In previous analysis <cit.>, an analytical approach was primarily used for estimating errors on the power spectrum measurements. However, this approach makes various assumptions, mainly that thermal noise dominates the error budget and that Gaussian statistics are valid. Since we are now reaching regions in k-space where thermal noise no longer dominates with MeerKLASS, we are required to relax these assumptions. Furthermore, the assumption of Gaussianity is arguably not solid, given the additive systematics we know to be still be present within the data.

A more robust approach of estimating the errors is to use the transfer function scatter across the different mock realisations T_i(k). This approach has been used in previous analysis <cit.> and has also been validated in simulations <cit.>. For it to perform effectively, the mocks in the transfer function calculation must emulate the real data effectively which we have strived to do so, as outlined in sec:Mocks.

The transfer function for one single realisation is defined by 


    T_i^,g(k)=P(_clean,_g)/P(_,_g) ,


i.e. equivalent to <ref> but without the averaging over N_mock. To reconstruct signal in a measured power spectrum, we take the ensemble average of <ref> i.e. T(k)=⟨T_i(k)⟩_N_mock, and divide the estimated power by this; P̂_rec(k)=P̂(k)/T(k). Note we are dropping the subscript ,g for brevity and also because the formalism we present here is identically applicable to the -auto. The assumption made when using the transfer function for error estimation is that the scatter in the reconstructed power comes from the transfer function distribution T(k)+δT_i(k) contributed by each ith mock. We can isolate the scatter component from the array of calculated transfer functions by simply subtracting the ensemble mean for each ith mock


    δT_i(k) = T_i(k) - T(k) .


Assuming any errors add in quadrature, error propagation suggests


    δP̂^rec_i(k)/P̂_rec(k) = √((δP̂_i(k)/P̂(k))^2 + (δT_i(k)/T(k))^2) ,


however, we assume all quantifiable error lies in the transfer function distribution, hence δP̂_i=0 and so


    δP̂^rec_i(k) = P̂_rec(k)δT_i(k)/T(k) .



Our setup thus provides N_mock realisations of δP̂_i^rec and the covariance can be computed from this for any k-bin.
We reiterate that this formalism is valid whichever way the k-modes are binned e.g. for a 2D cylindrical averaged or 1D spherically averaged power. In this work, our error bars in any 1D power spectrum estimate represent the limits of the 68th percentile regions of the distribution of δP̂^rec_i(k). If the uncertainties are Gaussian symmetric, this is equivalent to 1 σ error bars, however, as we will later show, this is not perfectly the case for all k-modes. 



 §.§ Spherical averaging k-cuts


Before presenting spherically averaged power spectra from the MeerKLASS data, we first present results from the cross-correlation with GAMA galaxies in cylindrical-(k_⊥,k_∥) space. This is shown in <ref> and motivates a discussion on where in k-space we have the most signal-to-noise so we can make optimal spherically averaged P̂(k) measurements. The dashed gray lines on all panels represent the |k|-bin boundaries we use for the spherically averaged power spectra, restricting this to k<0.3 to avoid highly non-linear cosmology and increasingly poor signal-to-noise. The orange solid lines represent boundaries for potential k-cuts which we will discuss.



The first panel of <ref> shows the cross-power between GAMA G23 galaxies and the MeerKLASS deep field, cleaned by removing =10 PCA modes. The k_⊥ k_∥ scaling is for demonstrative purposes so the high power at low-k does not saturate the plot. At k≳0.15 the large MeerKAT beam begins to have an impact and will damp  fluctuations, which explains why the cross-power signal tends to zero at high-k_⊥. Except for the extreme low-k regions, the rest of the k-space reveals a reasonably consistent positive cross-correlation signal, providing strong evidence of consistent clustering between the MeerKLASS maps and GAMA galaxies. As discussed in the previous sec:TFsignalloss, the signal loss in this power has been corrected with the foreground transfer function. The second panel shows the corresponding transfer function for the =10 clean. Low values of T_,g signify high signal loss. This is evidence that nearly all signal is lost for the lowest-k_∥ modes which is expected since these modes should be highly degenerate with the removed long radial modes of the continuum foregrounds. The high signal loss at low-k_⊥ is interesting and not seen in previous MeerKLASS work <cit.> with a similar method, albeit with less robust mocks. This is likely caused by the small GAMA footprint (see <ref>) which means the large-ranging modes at these low-k are not adequately resolved and deliver little signal which is easily coupled to foreground modes and stripped away in the clean. This can be explored further in follow-up work with wider surveys where we expect this not to be a problem.

Due to the noisy power at low-k and damped power at high-k_⊥ in <ref>, we can choose to cut these regions of k-space from our spherically averaged power to obtain a more robust result. The orange solid lines represent chosen boundaries, where modes beyond these boundaries are excluded in the averaging of each k-bin (shown by the dashed gray lines). Providing the modelling of any power spectra also incorporates the k-cuts, then it will not bias any results. The k-cuts approach is further motivated by considering the signal-to-noise of the cross-power which we show in the third panel of <ref>. Here the errors σ(P̂_,g) are computed using the transfer function by taking the standard deviation from the δP̂_i^rec(k_⊥,k_∥) distribution (see previous sec:TFforErrors). This result provides a strikingly clear window where signal-to-noise is optimal; the orange to red regions in the third panel. The orange k-cut boundary lines have been chosen to target this region. One limitation of this approach is that it excludes the regions of k-space with the most modes, as shown by the fourth and final panel. We will present comparisons between the spherically averaged P(k) with and without these k-cuts in the upcoming sec:GAMAcross. 



 §.§  auto-correlation power spectrum


We next examine results for the auto-correlation  power spectrum from the MeerKLASS L-band deep field. Previous single-dish intensity mapping results have been thermal noise-dominated across all scales. We are now crossing an important threshold with MeerKLASS where the  fluctuations should be dominating the noise at large scales. We demonstrate this in <ref>, where we show the estimated  power spectrum from the MeerKLASS deep field (blue data points) and overlay an estimate of the thermal noise (red solid line). As outlined in sec:HIIMmocks, this noise is modelled by generating Gaussian random fields with the expected rms given by <ref> which utilises the observational hit map. Despite being the  auto-power, we still restrict the measurement to the GAMA-field footprint, for consistency with the cross-correlation results. We tested using the full field but found no improvement in the auto-power. We leave exploration of the full-field to follow-up work which focuses on the -auto power spectrum. The results in <ref> include the scale k-cuts discussed in the previous section.

<ref> also shows the expected  power (grey dotted line) from <ref>. For this auto- model, we do not fit the amplitude to avoid additive bias from residual systematics over inflating the model estimate. Instead we use the scale-independent T_ fit from the upcoming cross-power measurement. This model shows how the cosmological  power surpasses the thermal noise on large scales (k≲0.15). The thermal noise will be present even in the absence of clustering, thus the total expected power can be reasonably approximated as a simple addition of the  power, plus the thermal noise (shown by the black dashed line).



The small discrepancy between the deep-field power estimate and the model represents the lowest upper limit from a  auto-correlation on these large scales. We find the mean residual factor between model and data to be just 2.36 across all k. Whilst the model and data are close in agreement, inferred parameter values for the bias and mean  temperature from fitting the amplitude of the auto power yields inconsistent values with the GAMA cross-correlation (discussed next). This is therefore evidence of low-level residual systematics that are additively biasing the power. In this evaluation, we inherently assume that the transfer function reconstructs any signal loss in the Gaussian thermal noise. If signal loss in the Gaussian noise is underestimated, the residual between the model and data may be higher than we report here. We therefore intend to robustly test this assumption when pursuing a more dedicated study of the  auto-correlation.

In follow-up work, we will focus on improving this upper limit from the  auto power which can be done in two ways: Firstly, by implementing some corrections to remove low-level systematic contributions such as the residual RFI structures discussed in sec:zebra. Secondly, we can split the observations into sub-sets i.e. combinations of different maps from different scan times and/or dishes. This is useful for isolating dish- and time-dependent systematics and removing their biasing contributions to the power. Since each scan and dish is observing the same patch of sky, we would ideally expect all maps to agree, thus any differences will represent thermal noise, or systematics such as residual RFI, varying polarisation leakage etc. We do not pursue these ideas in this paper since they warrant their own detailed investigation.



 §.§ Cross-correlation power with GAMA


We now focus on the cross-correlation with the GAMA galaxies for the remainder of the paper. In <ref> we present the cross-correlation power spectrum between the GAMA galaxies and the MeerKLASS deep field. All points are positive so none are lost to the log scaling, indicating a strong, positive correlation. Generally, a good agreement with the model is obtained. We performed a null test on the result by substituting the GAMA mock galaxies (from sec:Galmocks) in place of the real galaxies. The mocks should not correlate with the intensity maps, and indeed, the resulting cross-power converges to zero once averaged over a sufficient number of mocks. The fitted amplitude of the model is equivalent to an inferred parameter of T_=0.166 mK, with all other values in <ref> assumed. We discuss this further in sec:ParamConstraints, but do not focus on parameter inference in detail due to the large statistical noise owing to the small overlap footprint between MeerKLASS and GAMA.



The error bars for the cross-power in <ref> are obtained using the scatter from the foreground transfer function as discussed in sec:TFforErrors. The error size appears to be over-predicted based on the scatter of the data points, which have a reasonably tight agreement with the model relative to what the error bars reflect. This is supported by examining the reduced χ^2 which we find to be χ^2_dof≡χ^2/N_dof=0.42. A χ^2_dof<1 is conventionally indicative of overinflated errors. Using the reduced χ^2 however is not a reliable test where there are non-linear contributions, as is likely the case for these observations that are still prone to residual systematics. In these cases, the number of degrees of freedom cannot be robustly determined <cit.>. For our case we simply assume N_dof=13, for our 14 k-bins minus a single fitted degree of freedom, representing the amplitude fit. Furthermore, noise within the data causes high uncertainty in χ^2 estimates, which despite the relatively low noise levels we are now reaching, may still be a concern. Alternatives to χ^2 have been proposed, and specifically for 21cm applications 
<cit.>, but these have not been pursued in this work. We only use the χ^2 as an approximate guide, mainly for relative comparisons between different cases.

For the estimation of χ^2, we include the full contribution from off-diagonals in the k-bin covariance , which can be trivially estimated for our error estimation method, simply the covariance over P̂_i^rec(k)=P̂_rec(k)+δP̂^rec_i(k) (see sec:TFforErrors). The χ^2 is then estimated from






    χ^2=(p̂-p_ mod)^T ^-1  (p̂-p_ mod) ,


where p̂ is the data vector and p_mod the corresponding model. We present the normalised covariance matrix (or correlation matrix) in <ref>, showing evidence that the k-bins are slightly correlated, especially at low-k. This could be contributing to the large error bars, hence why we include the full covariance in the χ^2 estimation.



To investigate the errors from the cross-power further, we plot the full distributions for the reconstructed power spectra P̂_i^rec from each ith mock in the foreground transfer function computation. <ref> shows this for each k-bin, including an additional low-k bin which we otherwise exclude due to the large error on these scales. We centre the distributions on the model and normalise by it, to aid comparison with the case of no k-cuts (red results), which includes the areas of k-space with low signal-to-noise (discussed in sec:kcuts). The plot shows how the distributions have some non-Gaussian features, which could also impact error estimation, along with the correlated k-bins shown by the covariance. We find the average kurtosis across all k-bins for the k-cut case to be -0.476, indicating the distribution is flatter than a normal distribution with more contribution in the tails. This could also widen the central percentiles and thus increase the error bar size. 



<ref> shows a mild improvement in performance for the case with k-cuts. The distributions around the model are less broad at low-k, which is expected since this was one of the regions where signal-to-noise was poorest (see third panel of <ref>). In terms of the Gaussianity of the distributions, we find a better kurtosis score for the case with k-cuts, suggesting the profiles are more Gaussian. In contrast, we find the results without k-cuts to be less skewed, although this is marginal and in general, the skewness of the distributions is not concerning. Any improvement coming from the k-cuts is small and we also find little difference in detection significance, as we will explore shortly. As the volume of data grows beyond this small overlap sample, the k-cut results may prove more optimal since we will no longer be cutting modes from an already sample variance-limited data set. This is something that future data sets with MeerKLASS will reveal.



  §.§.§ Varying the foreground clean


So far we have only considered the chosen case of removing =10 PCA modes for the foreground clean. <ref> shows how the χ^2 (which we continue to use as a rough goodness-of-fit metric) varies with . As seen in previous work <cit.>, too low an  does not sufficiently clean the foregrounds, causing a poor fit. Going too high with  also diminishes the fit. Ideally, we would expect high  to cause more signal loss which should be reconstructed by the transfer function, giving unbiased results and thus not diminishing the fit. A higher uncertainty could be expected with greater signal loss, but this should be accounted for by the increasing error bars i.e. the χ^2 should stabilise and not start to rise again at high . On further scrutiny, we found that it is a single data point at k∼0.2 with very little power distorting the fits at higher . This is evidence of low-level residual systematics that become more influential as signal is stripped away in the more aggressive foreground clean. Therefore, with future results where systematics are more controlled, this is expected to be less of a problem. For the case with no k-cuts (thin dashed lines), there is a plateau in χ^2 at higher , as we argued should be the expected situation. This is because the no k-cut power spectra were less impacted by the k∼0.2 anomaly at high  which is potentially suggesting that the k-cuts are overly aggressive for this scale. With larger overlapping fields, the more accessible k-bins will naturally mitigate this discrepency. The no k-cut results in <ref> are therefore encouraging that a model and data fit should reach a plateau for high  as expected with the current analysis pipeline. 



The right hand side (red) y-axis in <ref>, also shows the detection significance defined by √(Δχ^2)≡√(χ^2 - χ_null^2), where χ^2_null is the result's χ^2 with a null model with zero power. The cross-correlation with GAMA is returning a modest > 4σ detection for the most optimal choices of foreground clean, lower than previous MeerKLASS detections <cit.>. This is however still impressive, given that galaxies only overlap with approximately a quarter of the full MeerKLASS deep field. The detection significance is also restricted by the more robust approach to error estimation, which as we have already discussed, provides conservative estimates potentially inflated by residual systematics and correlated k-bins. To demonstrate this point, we artificially construct a scenario where a χ^2_dof∼1 is obtained for the =10 case. This is achieved by assuming a perfectly diagonal covariance and
decreasing the diagonal errors by 40%. This boosts the detection significance to 9.8 σ, verifying that it is the conservative approach to errors lowering the detection significance, rather than the data-model residual. This is of course only a toy demonstration but it validates the argument that if large errors, which are suspected to be driven by systematics and k-bin correlations, can be contained, then detection robustness should improve. This is expected in future work where we will pursue dedicated studies into systematic mitigation. Some mode coupling is always likely, driven at least by non-linear clustering. However, we are confident that improvements can be made in covariance. MeerKLASS observations ongoing now are covering wider areas with more overlap with Stage-IV optical galaxy surveys. These observations are also using MeerKAT's UHF-band, which will allow a deeper usable redshift range, extending the 0.39<z<0.46 used in this work to approximately 0.4<z<1.4. This will naturally improve the large statistical noise due to the small volume overlap in this work.





§ STACKING
 

In this section, we explore the possibility of a stacking detection of the  signal in the MeerKLASS L-band deep-field intensity map. The technique of spectral line stacking <cit.> can be used to detect the  content of galaxies below noise limit. In combination with spectroscopic galaxy surveys, spectral line stacking probes the average emission line profile of selected galaxy samples. Complimentary to the intensity mapping technique, the stacking measurement of the emission line profile probes the  mass function of galaxy samples (e.g. ), the relation of  content with its host environment (e.g. ), and the baryonic Tully-Fisher relation (e.g. ). Interferometric observations of MeerKAT using the MeerKAT International GigaHertz Tiered Extragalactic Exploration survey (MIGHTEE; ) have been used to produce  measurements with regard to the scaling relation <cit.> with galaxy properties and the dependency on the large-scale structure environment <cit.>. See also <cit.> for direct detections of  galaxies in the MIGHTEE survey.

For intensity mapping experiments, stacking analysis has been performed on the  intensity maps using the Parkes telescope <cit.> and the CHIME telescope <cit.>, as well as on CO intensity maps from the CO Mapping Array Project <cit.> and Lyα emission of [O III]-selected galaxies from the Hobby-Eberly Telescope Dark Energy Experiment <cit.>.

MeerKLASS, on the other hand, faces unique challenges. The ∼1deg instrument power beam introduces heavy contamination from nearby sources. For the GAMA galaxy sample used, the number density of the sources is ∼ 35  deg^-2 when integrated along the frequency sub-band. Therefore, when stacking each source, the stacking procedure will inevitably include emissions from more than one GAMA galaxy. The velocity width of the  sources further scatter the 21 cm emission into nearby channels. The double-counting of sources from the beam creates a plateau of excess signal as shown in <ref>, and can be resolved by PCA cleaning, which we discuss in [apdx:stack]Appendix <ref>. However, the residual after PCA cleaning also incorporates systematic effects. The chromatic structure of the systematic effects produces oscillations in the stacked profile, with their amplitudes comparable to the signal. For this work, we are only interested in using stacking as part of the validation test. As discussed later, we obtain a detection of a stacked signal in both angular and spectral space, which can be validated by performing a null test with random galaxy positions. The existence of a stacked signal further demonstrates that we have reached the depth needed for the possible detection of the  power spectrum using the MeerKLASS L-band intensity maps. We leave a more detailed study into refining the stacking technique for probing  science to follow-up work.



 §.§ The stacking procedure of the intensity maps

We use the PCA-cleaned map described in [sec:FGcleaning]Section <ref> and the positions of the GAMA galaxies to perform a 3D stacking in (δα,δϕ,δν) space so that

    I(δα,δϕ,δν) = ∑_i I(α_i+δα, ϕ_i+δϕ, ν_i+δν) w_i/N_ g∑_i w_i,

where (α_i,ϕ_i) are the right ascension and declination of the map pixel in which the i^ th galaxy resides, I is the flux density, N_ g is the number of galaxies, and w_i=w_(α_i+δα,ϕ_i+δϕ,ν_i+δν) is the reconvolved weight of each pixel defined in [subsec:reconvolution]Section <ref>. We choose the angular and the spectral bins to be consistent with the map resolution with Δα=Δϕ=0.3deg. The frequency resolution corresponds to an effective velocity resolution of Δ v=62.83km s^-1 at the centre frequency of the sub-band 997.37MHz. Note that no Doppler correction has been performed on the data. For our analysis, the intensity maps are coherently averaged in frequencies for PCA cleaning and therefore can not be averaged in velocity channels.

In order to calculate the statistical noise for the stacked cube, we perform the same stacking routine on random galaxy positions, similar to the shuffling null test in sec:GAMAcross. The positions are randomly sampled uniformly within the region of the GAMA galaxy sample. The redshifts are randomly sampled following the redshift distribution of the survey. The random process is repeated for 100 realisations to calculate the mean and the standard deviation of the reference cube. The reference cube is used for performing the null test, i.e. checking that the stacked emission signal in the reference cube is consistent with zero.

After obtaining the stacked cubes, we create the angular and spectral stacking by summing over the cubes along the radial and transverse directions. To control the systematics generated by the foreground cleaning, we limit the voxels that go into the summation. For angular stacking, only voxels that are within 150 km s^-1 of the central channel are considered, corresponding to the typical width of the HI emission line profile. For spectral stacking, only the voxels whose centre positions are within the 1deg area of the centre voxel in each channel are included, corresponding to the size of the primary beam. The same summation is performed on the random realisation to calculate the mean and the standard deviation of the reference image and spectrum. We note that there is signal loss due to an insufficient number of voxels included. However, as the random realisations follow the same procedure, it does not affect the validation of the detection which is our focus in this work. 







 §.§ The detection of stacked  signal


In <ref>, we show the results for the angular stacking of the GAMA galaxies on the  intensity map. The stacked map shows a clear detection at the centre of the image, with the centre pixel achieving a detection with a statistical significance of 8.35 σ. The stacked  emission follows the structure of the beam. 

The extended structure of emission outside the beam FWHM is a result of the relatively large thermal noise fluctuation and contamination from nearby sources in the stacking procedure as discussed above. We note that, as shown in <ref>, the lower part of the GAMA galaxy sample region sits near the edge of the observed intensity maps. As a result, the lower half of the stacked image has a higher level of statistical fluctuations as shown in the right panel of <ref>.
Furthermore, the reference image is consistent with a null detection, demonstrating that the stacked signal does not arise from systematic effects in the map.



In <ref>, we show the stacked spectrum along the line-of-sight. 
The peak of the stacked spectrum shows a detection with a significance of 6.30 σ. The detection is validated by the reference spectrum which is consistent with a null detection. However, as seen in the reference spectrum, there is a systematic noise component with fluctuations on large frequency intervals. The systematics can be most visibly seen at the peaks and troughs of the stacked signal at v ∼ 2500, 4500, and 7000km/s. We have identified the source of this fluctuation as the residual systematics in the map after PCA cleaning, which is demonstrated in [apdx:stack]Appendix <ref>.
As stated previously, we are only interested in validating the detection against the null test and leave the suppressing of the systematic effects for future work.

The stacking results show a clear detection of  signal in the positions of the GAMA galaxies. The detection of an emission line signal is statistically significant against the null test.

At the same time, the residual systematics in the data, while being sub-dominant comparing to the  signal, induce structures in the stacked spectrum. We intend to follow up on the findings in this work and develop analysis techniques to extract the underlying emission line signal in the future, as well as using the stacked signal to understand the residual systematics.




§ CONCLUSIONS


In this work we have presented L-band observations from the MeerKLASS intensity mapping campaign. The survey covered a single patch of approximately 200 ^2, repeatedly scanned 41 times (27 scans remained after aggressive RFI flagging), which we have referred to as the MeerKLASS L-band deep field. The observations used in our analysis came from only a small section of the full L-band (971.2<ν<1023.6 at redshift 0.39<z<0.46) to avoid RFI which we find is more present in this band relative to the lower frequency UHF-band (580<ν<1000) which all future MeerKLASS surveys will be conducted in. We detailed the calibration strategy used on these observations which included a new approach of self-calibration (sec:SelfCal), exploiting the data's higher signal-to-noise to devise a less model-dependent calibration. The improvement in data quality relative to previous MeerKLASS pilot surveys is evident (see e.g. <ref>) and encouraging for future MeerKLASS surveys gathering ever-larger data sets.

We demonstrated the enhanced signal-to-noise from the calibrated deep-field intensity maps by presenting the auto-correlation power spectrum (<ref>) which revealed the lowest upper-limit from a  power spectrum at these cosmological scales k<0.3. This remains a factor of ∼ 2 above where we would expect the  power to lie. We will pursue this limit further in follow-up work where we will explore a systematic removal strategy to reduce the quantity of flagged data plus the cross-correlation of divided data sets to mitigate the additive bias from time-dependent residual systematics. 

We also presented results from the cross-correlation power spectrum between the MeerKLASS deep-field and 2269 overlapping GAMA galaxies with spectroscopic redshifts. Despite the small field of overlap (∼25% of the deep field patch), we still obtained a clear > 4 σ detection of the cross-power spectrum (<ref>). Furthermore, we presented a detection of  emission from stacking the MeerKLASS maps onto the positions of the GAMA galaxies. This yielded a 8.35 σ and 6.30 σ detection from angular and spectral stacking respectively, the first stacking detections with MeerKAT in single-dish mode (see sec:stack). 

We implemented several novel strategies into the power spectrum analysis pipeline, including a selective procedure to target the region of k-space with the highest signal-to-noise. The third panel of <ref> demonstrated the clear window of Fourier modes where power dominates over the error. By implementing some minima and maxima cuts on k_⊥ and k_∥ we were able to exclude regions of k-space dominated by error. This provides a more robust result since error-dominated modes will not be included in final spherical averaging. Comparison of the error profiles for each k-bin (<ref>) demonstrated a mild improvement in terms of Gaussianity when using the k-cuts, however, the sacrifice of many modes contained at high-k could be restricting this improvement. We plan to investigate this strategy in future work where larger data sets will reveal the optimal approach.

We formalised our approach to covariance estimation for the power spectra measurements which utlises the foreground transfer function distribution from each mock realisation. This distribution will be determined by all foreseeable sources of error; residual systematics, thermal noise fluctuations, galaxy shot-noise, cosmic variance and any uncertainty in signal-loss correction. This should therefore be a robust method for error estimation. However, we found the errors in the spherically averaged cross-power spectrum appear to be large relative to their scatter about the model. This was supported by a χ^2 analysis. Further inspection revealed that the k-bins are mildly correlated and the error profiles for each k-bin are not perfectly Gaussian, both of which could be inflating the errors. We hypothesise that this should be naturally mitigated with future MeerKLASS analysis where systematics can be more reliably controlled. 

This work also presented validation tests of our improved analysis pipeline. This was achieved by generating a suite of simulations that emulate the real data products. The mock data was then put through the same pipeline as the real data where we could validate its performance. Results in <ref> showed good agreement with the expected model based on the known inputs to generate the mocks, thus validating processes such as Cartesian field regridding and power spectrum estimation. Measured power from the mocks in <ref> which included the foreground cleaning also showed good consistency with real data.  



The MeerKLASS campaign is ongoing and will continue to be so in the pre-SKAO era. Further observations have already begun in the telescope's UHF-band. These lower frequencies should be less prone to RFI, meaning far fewer channels will be flagged. Furthermore, the lower frequency and wider wave-band mean a much larger portion of redshift space will be covered, providing intensity maps of unprecedented volume.



§ ACKNOWLEDGEMENTS


The authors would like to acknowledge the ongoing contributions to the MeerKLASS collaboration from members Suman Chatterjee, Karin Fornazier, Tamirat Gogo, Wenkai Hu, Piyanat Kittiwisit and Sifiso Mahlalela which provided valuable insights for this paper. We would also like to thank Jacob Burba, whose UK SKA Regional Centre project involved a full run-through of our post-calibration analysis pipeline, providing useful validation. 

JW acknowledges support from the National SKA Program of China (No. 2020SKA0110100).

SCu is supported by a UK Research and Innovation Future Leaders Fellowship grant [MR/V026437/1]. 

This result is part of a project that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 948764; PB).

JLB acknowledges funding from the Ramón y Cajal Grant RYC2021-033191-I, financed by MCIN/AEI/10.13039/501100011033 and by
the European Union “NextGenerationEU”/PRTR, as well as the project UC-LIME (PID2022-140670NA-I00), financed by MCIN/AEI/ 10.13039/501100011033/FEDER, UE.
SCa acknowledges support from the Italian Ministry of University and Research, PRIN 2022 `EXSKALIBUR – Euclid-Cross-SKA: Likelihood Inference Building for Universe Research', from the Italian Ministry of Foreign Affairs and International
Cooperation (grant no. ZA23GR03), and from the European Union – Next Generation EU. 

IPC is supported by the European Union within the Next Generation EU programme [PNRR-4-2-1.2 project No. SOE0000136, RadioGaGa].

JF acknowledges support of Fundação para a Ciência e a Tecnologia through the Investigador FCT Contract No. 2020.02633.CEECIND/CP1631/CT0002, the FCT project PTDC/FIS-AST/0054/2021, and the research grants UIDB/04434/2020 and UIDP/04434/2020. 

AP is a UK Research and Innovation Future Leaders Fellow [grant MR/X005399/1]. ZC's research is supported by a UK Research and Innovation Future Leaders Fellowship [grant MR/X005399/1].

MGS acknowledges support from the South African Radio Astronomy Observatory and National Research Foundation (Grant No. 84156).

LW is a UK Research and Innovation Future Leaders Fellow [grant MR/V026437/1]. 

The MeerKAT telescope is operated by the South African Radio Astronomy Observatory, which is a facility of the National Research Foundation, an agency of the Department of Science and Innovation. We acknowledge the use of the Ilifu cloud computing facility, through the Inter-University Institute for Data Intensive Astronomy (IDIA).

GAMA is a joint European-Australasian project based around a spectroscopic campaign using the Anglo-Australian Telescope. The GAMA input catalogue is based on data taken from the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey. Complementary imaging of the GAMA regions is being obtained by a number of independent survey programmes including GALEX MIS, VST KiDS, VISTA VIKING, WISE, Herschel-ATLAS, GMRT and ASKAP providing UV to radio coverage. GAMA is funded by the STFC (UK), the ARC (Australia), the AAO, and the participating institutions. The GAMA website is https://www.gama-survey.org/https://www.gama-survey.org/.

For the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript version arising from this submission. 




§ DATA AVAILABILITY


The data underlying this article will be shared upon reasonable request to the corresponding author. Access to the raw data used in the analysis is public (for access information please contact archive@ska.ac.za).





mnras
 



















§ OBSERVATION DETAILS


<ref> provides the details for the MeerKLASS L-band deep field observations, for all 41 attempted observation blocks.





§ PCA FOREGROUND CLEANING




 §.§ Impact of reconvolution


As discussed in the main text, the reconvolution of the maps before foreground cleaning (outlined in subsec:reconvolution), slightly improves the foreground clean performance. This is not necessarily just because it homogenises the effective resolution of the beam, as is the primary aim of this technique. The smoothing will also suppress systematics in the data. Residual contributions, for example from RFI, will distort the true amplitude of the sky map. If these contributions vary with frequency, which will be the case for most systematics, then this will create small perturbations to the spectra along each pixel. Despite smoothing the maps in the angular direction only, this will still suppress perturbations along the line of sight, hence providing slightly smoother spectra, which may be linked to the improved foreground clean. 

We demonstrate this in <ref> which plots the amplitudes of each pixel along frequency, as previously shown in <ref> (here we also trim the edges of the map, as outlined in sec:maptrim). The top panel shows the spectra without reconvolution. A clear improvement in the smoothness of the spectra is evident when reconvoltion is used (bottom panel). 

Future work is planned where a more realistic MeerKAT beam will be implemented into our modelling. In this work, we will also investigate this technique of reconvolution and aim to conclude if the smoothing to a common resolution is the source of any improvement or if it is just entirely from the suppression of systematics, reducing the perturbations they cause in the spectra which aids the PCA process.





 §.§ Weighted frequency covariance estimation


Here we explore the small impact that opting to use weights in the covariance estimation has on the PCA cleaning process. The weights used are constant along the frequency direction, so not to add structure to the modes and diminish the PCA performance. Furthermore, after trimming the map edges, the time stamps have a more uniform coverage (see <ref>). This means the weights do not vary considerably across our whole field, so we would expect only minor changes by including them. 

<ref> shows the first 9 eigenmodes from , extracted from the diagonalisation of the covariance matrix computed using <ref>. We show the difference between the weighted case (blue solid) and the unweighted (orange dashed), equivalent to omitting the  factor in the covariance calculation. As expected, the differences are marginal, especially when considering that in some modes the discrepancy is mainly an arbitrary change of sign. However, some small differences are still noticeable, particularly for the higher eigenmodes where noise will become more of a component which is effectively what we are weighting for. 

We found little evidence of improved performance in the final results whether weights were used or not in the foreground clean. The difference between the cleaned maps was at the sub-1μK level, and testing the process on the mocks showed no clear performance gain. However, we included them in our final results for completeness. Furthermore, the weighted formalism may be useful in future work focussing on  auto-correlations, where cross-correlating sub-sets of maps from different dishes or time blocks will be pursued. Here, different map weights can be used in a combined cross-covariance computation; then asymmetric singular vectors can be extracted from this, as in the Singular Value Decomposition (SVD) approach of <cit.>.



To accompany the eigenmode plot of <ref>, we also show the projected map from the first 7 eigenmodes in <ref> for the weighted PCA case. We omit the first eigenmode since it is indistinguishable from the synchrotron sky maps we have shown previously. These show at map level, each component that is being removed in the PCA clean. For example, a N_fg=7 clean would remove all these components. No obvious systematic structure is apparent which could otherwise be targetted for removal by other means, rather than relying on an aggressive PCA clean. However, as we discussed in sec:zebra, when cleans are performed individually on smaller sub-sets of time blocks, constant declination stripes become apparent in some of the components which we will aim to correct for in calibration in future work.












 §.§ Power index of the transfer function for signal resconstruction in auto- and cross-correlations


The transfer function defined in <ref> is applied to the measured power spectrum to reconstruct the loss of information due to foreground cleaning. A priori, intuition would lead to applying the transfer function as defined for the cross-power spectrum of  and galaxies, and the square of the transfer function for the  auto-power spectrum, as signal loss occurs twice in that calculation <cit.>. However, <cit.> found empirically that squaring the transfer function over-corrected the signal loss and that the transfer function should be applied equally to cross- and auto-power spectra for correct modelling. In this appendix, we provide simplified analytic arguments to support such a claim. 

Let us assume the idealised scenario where we can decompose the cosmological fluctuations δ T_(k)≡ r(k)+l(k), where r and l correspond to the residual and loss modes of the cosmological signal, respectively. Under this separation, the transfer function can be understood as


    T_(k) = ⟨ r_ m(k)[r_ m(k')+l_ m(k')]^*⟩/⟨[r_ m(k)+l_ m(k)][r_ m(k')+l_ m(k')]^*⟩ ,


where the subscript `m' refers to quantities obtained from mock injection. 
Due to the component separation process in PCA,
r and l are a combination of independent components, and are therefore uncorrelated. The power spectrum of their sum is


    ⟨[r_ m(k)+l_ m(k)].    . [r_ m(k')+l_ m(k')]^*⟩ = 
     =     1/(2π)^3δ_D(k-k')[P_ m^(r)(k)+P_ m^(l)(k)] 
    =     1/(2π)^3δ_D(k-k')P_ m^(k) .


Note that the equation above applies equally to the actual observed cosmological signal. Therefore, the transfer function can be now understood as


    T_(k) ∼P_ m^(r)(k)/P_ m^(r)(k)+P_ m^(l)(k) .


Let us now consider the application of this conceptual transfer function to the power spectra measured from the clean maps. In this case, we have


    ⟨ r(k) r^*(k')⟩/T_(k)∼ P^(r)(k)[ P_ m^(r)(k) + P_ m^(l)(k)/P_ m^(r)(k)]


for the auto-power spectrum, and


    ⟨ r(k)g^*(k')⟩/T_(k)∼ P^(rg)(k)[P_ m^(r)(k) + P_ m^(l)(k)/P_ m^(r)(k)]


for the cross-power spectrum. Assuming P^(r)∼P_m^(r), it is clear that a single power of the transfer function (i.e. T^1) correctly reconstructs the auto-power to the desired P^(r)+P^(l) level. For the cross-power spectrum, let us simplify the situation and assume linear theory, for which δ_g∝δ T_∼ r+l. Then, P^(rg)∝ P^(r); if we denote the proportionality factor generically as A, we have 


    ⟨ r(k)g^*(k')⟩/T_(k)∼AP^(r)(k)[1+P_ m^(l)(k)/P_ m^(r)(k)] ,


which similarly shows that a single power of the transfer function must also be applied to the cross-power spectrum.




§ PARAMETER INFERENCE FROM THE CROSS-POWER


We have not dedicated a lot of attention to the modelling of the power spectra in this work. This is mostly due to the large errors in the cross-power due to the small sample overlap and the small additive bias still present in the auto- power spectrum. This makes high dimensional models unnecessary and we opted instead for a simple amplitude fit of a fiducial model with some assumptions on parameters (see <ref>). Some of these parameters are poorly constrained however, so here we explore what happens if we extend the model and include an MCMC fit of four parameters; φ={Ω_,b_,b_g,σ_v}. Ω_ is the dimensionless  density parameter and can be directly linked to the mean  temperature through


    T_(z)=180 Ω_(z)   h (1+z)^2/√(Ω_m(1+z)^3+Ω_Λ) mK ,


where Ω_m and Ω_Λ are the density parameters for matter and the cosmological constant respectively.

We assume wide and flat priors for the φ parameters given by 


	
  * 0<10^3Ω_<2
	
  * 0.5<b_<2.0
	
  * 0<σ_v<600 [km s^-1]
	
  * 1.8<b_g<2.0


A tighter prior is assumed for the GAMA galaxy bias since we fit the galaxy auto-correlation power spectrum, finding a best-fit bias value of b_g∼1.9. The remaining priors are sufficiently wide, even for these poorly constrained parameters. We stress that this is still an overly simplistic approach, but our aim here is a demonstration of current modelling capabilities, not rigorous parameter inference.

<ref> shows the inferred parameter constraints from the MCMC fit. We use  <cit.> with 500 iterations of 200 walkers. We checked the chains had converged and discarded a burn-in of 40 iterations. Since it is entirely prior dominated in the cross power, we marginalise over b_g and do not show its parameter contour. The nuisance parameter σ_v is also completely prior dominated and suggests that a more refined model is needed on smaller scales, something explored in <cit.>. There is a constraint on Ω_ but this is only due to the restricted prior on b_. Relaxing the  bias prior would inflate the Ω_ constraint indefinitely. 










<ref> thus demonstrates the degeneracy between Ω_ and b_ which has led previous efforts to merge this parameter fit in different ways <cit.>
. This degeneracy can be broken by utilising RSD which introduces an anisotropic dependence on b_ but not on Ω_. We tested a measurement of the power spectrum quadrupole (a smoking-gun of RSD) <cit.> and found a clear non-zero measurement. However, we found that this was overwhelmingly driven by observational effects and higher signal-to-noise is required to make a distinction between this and a true RSD-induced anisotropy. Further discussion on the degeneracies between astrophysics and cosmology and the capabilities of the LIM power spectrum multipoles to constrain them can be found in <cit.>, where optimal parametrisations are proposed



§ EFFECTS OF PCA IN THE STACKED SIGNAL

In this section, we use mock  galaxy signals to generate signal-only maps for stacking, and illustrate the effects of PCA in creating additional systematic noise structure.



As we are only interested in simulating the effects for stacking in this section, we use generate Poisson realisations of  sources with no clustering information as shown in <ref>. For galaxies inside the survey coverage of GAMA, we use the exact same positions as the GAMA galaxies to replicate the double-counting effects. The rest of the map is then filled with galaxies at random positions assuming the same number density as the GAMA field. Random redshifts are assigned to the galaxies following the redshift kernel shown in <ref>. Each galaxy is then assigned a  mass following the  mass function reported in <cit.>. Furthermore, we assign the velocity full width half maximum of the emission line profile w_50 to each galaxy using the Tully-Fisher relation reported in <cit.>. The emission line profile of each galaxy is then calculated and added to the mock signal cube following the simplified busy function <cit.>

    B_2(v) = a/2×(  erf[ b(w_50^2-v^2)  ] +1 ) × (cv^2+1),

where erf is the error function and a,b,c are parameters to be set. For each galaxy, we randomly choose b logarithmically uniformly in [10^-3,10^-2]  km^-2s^2 and c in [10^-2,10^0]  km^-2s^2. The amplitude parameter a is then set by the  mass of the galaxy. We find that varying the choice of the busy function parameters does not significantly impact the results due to the relatively low velocity resolution. The signal is then convolved with the resmoothed beam described in [subsec:reconvolution]Section <ref>.




As discussed in [sec:stack]Section <ref>, the low angular resolution of the map induces leakage from nearby sources in the stacked profile. To illustrate this, we perform the stacking routine on the signal-only mock cube and compare against the input mock signal in <ref>. The input signal is calculated by averaging the emission line profile of each source according to <ref>. The double-counting introduces a large plateau of contamination across the frequencies. We note that the increasing slope of the plateau is due to the redshift distribution of sources. We then apply the PCA cleaning to the mock signal cube and repeat the stacking procedure. Instead of calculating the mixing matrix from the mock cube, we use the exact same mixing matrix obtained from the map for PCA and show the results in <ref> (orange dashed line). The PCA removes the plateau to recover an emission line profile. However, it introduces several effects. There is signal loss due to the cleaning as well as the fact that we did not sum over an area much larger than the beam. Moreover, the PCA extends the width of the profile as seen in the lower panel of <ref>. 

Apart from the effects on the recovered signal, PCA also creates a systematic noise component as seen in the stacked spectrum. To further demonstrate this, we repeat the reference spectrum calculation described in [sec:stack]Section <ref> on the mock signal and present the results in <ref>. 

Compared to the reference spectrum, the stacked mock signal outside the centre area of the peak shows deviations. The deviation from the reference spectrum appears to be coherent over large velocity intervals Δ v∼5000km/s. The scale of the interval is consistent with the systematic effects seen in the stacked spectrum in <ref>. We emphasize that this is purely the combination of the  signal and the PCA cleaning since the mock map does not contain any other component. We further find that, when increasing the angular area for summation from (1  deg)^2 to (3  deg)^2, the amplitude of the coherent structure increases. 

The structure of the PCA cleaning is caused by the residual systematics in the data. Following the structure of the stacked spectrum in <ref>, we generate random Gaussian realisations of systematic noise components and convolve the systematics to the mock signal. We then repeat the stacking procedure as well as reference spectrum calculation and show the results in <ref>. As one can see, the mock signal affected by the systematics shows similar oscillating structure as seen in the data. The reference spectrum, on the other hand, shows consistentcy with a null detection, and demonstrates the fact that the detection reported in [sec:stack]Section <ref> is a result of the detection  signal with residual systematics, and does not arise purely from contamination in the data.



We intend to simulate the mock data cube with foregrounds, noise and systematics to fully understand the impact of residual systematics on the stacked spectrum in the follow-up work.










 Paper Title: Separating deterministic and stochastic gravitational wave signals in realistic pulsar timing array datasets.

 

    
   




    Dipartimento di Fisica “G. Occhialini", Universitá degli Studi di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy

    i.ferranti@campus.unimib.it
    INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy
    INAF - Osservatorio Astronomico di Cagliari, via della Scienza 5, 09047 Selargius (CA), Italy
    INAF - Osservatorio Astronomico di Brera, via Brera 20, I-20121 Milano, Italy
    ASTRON, Netherlands Institute for Radio Astronomy, Oude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands
    
Not anymore
Ferranti et al





    











  Recent observations by pulsar timing arrays (PTAs) suggest the presence of gravitational wave (GW) signals, potentially originating from supermassive black hole binaries (SMBHBs). These binaries can generate two kinds of signals: a stochastic gravitational wave background (GWB) or deterministic continuous gravitational waves (CGW). Being able to correctly recognize and separate them is crucial for accurate signal recovery and astrophysical interpretation. This paper aims at investigating the interaction between stochastic GWB and deterministic CGW signals within current analysis pipelines. We focus on understanding potential misinterpretations and biases in the parameter estimation when these signals are analysed separately or together. To this end, we performed several realistic simulations based on the European PTA 24.8yr dataset. We first injected either a GWB or a CGW into five datasets (three GWB realisations and two CGW realisations) with identical noise and analysed each signal type independently, and then analysed data sets containing both a stochastic GWB and a single resolvable CGW. We compared parameter estimation using different search models, including Earth term (ET) only or combined Earth and pulsar term (ET+PT) CGW templates, and correlated or uncorrelated power law GWB templates. We show that, when searched independently, the GWB and CGW signals can be misinterpreted as each other, and only a combined search is able to recover the true signal present. For datasets containing both a GWB and a CGW, failure to account for the latter biases the recovery of the GWB, but when performing a combined search, both GWB and CGW parameters can be recovered without any strong bias. Care must be taken with the method used to perform combined searches on these multi-component datasets, as the CGW PT can be misinterpreted as a common uncorrelated red noise. However, this can be avoided by direct searches for a correlated GWB plus a CGW (ET+PT). Our study underscores the importance of combined searches to ensure unbiased recovery of GWB parameters in the presence of strong CGWs. This is crucial to interpret the signal recently found in PTA data, and is a first step towards a robust framework for disentangling stochastic and deterministic GW components in future, more sensitive datasets.
    
    
 


  

  

  

  

  





   Separating deterministic and stochastic gravitational wave signals in realistic pulsar timing array datasets
    Irene Ferranti1,2, Golam Shaifullah1,2,3, Aurelien Chalumeau1,2,5 & Alberto Sesana1,2,4
    August 5, 2024
============================================================================================================







§ INTRODUCTION


Pulsar Timing Array (PTA) collaborations search for gravitational waves (GWs) with frequencies in the nanoHertz (nHz) band (10^-9-10^-7Hz). The primary target of PTA experiments is a stochastic gravitational wave background (GWB) generated by the incoherent superposition of the GWs emitted by all the super-massive black hole binaries (SMBHBs) that populate our universe <cit.>. The signal manifests itself in the data as a red noise process common to all the pulsars in the array, with a specific angular correlation, first derived by <cit.>. Recently, the PTA community found evidence for a signal with both these characteristics: the collaborations reporting this finding were the European PTA collaboration along with the Indian PTA collaboration <cit.>, the North American Nanohertz Observatory for Gravitational Waves collaboration <cit.>, the Parkes PTA collaboration <cit.> and the Chinese PTA collaboration <cit.>. 

While the Hellings and Downs (HD hereinafter) correlation is distinctive of a GW origin of the signal, separating it from possible correlated noise of different nature <cit.>, its source cannot yet be confidently established. In fact, besides SMBHBs, a nHz cosmic GWB can also be produced by a number of physical processes occurring in the early universe, such as cosmic strings networks <cit.>, primordial curvature perturbations <cit.>, or QCD phase transitions in the early universe <cit.>, all of which can provide a viable explanation of the observed signal <cit.>.
Moreover, the HD correlation is not unique of stochastic GWBs; as shown in <cit.> and in <cit.>, the signal generated by a single SMBHB and observed by an array of pulsars results in an angular correlation with the HD expectation value, albeit with a somewhat different variance. Note that the quality of the current data does not allow a description of the signal variance and  therefore, the possibility that a particularly loud source – a massive and nearby SMBHB – can be mistaken by a GWB is real. By loud source, we mean a source that can be singled out from the noise and, if present, from the GWB by means of a deterministic, template based search. Since the SMBHBs producing such signals are far from mergers and persistent in the data, their signal is generally referred to as continuous gravitational wave <cit.>.

CGWs searches have been performed on the latest data release of EPTA <cit.> and on the NANOGrav 15yr dataset <cit.>. They both found evidence in favour of the presence of a CGW with a 2.5-3σ significance, compared to a model where only pulsar noise is present in the data. This value is comparable to the evidence in favour of the presence of a GWB, which has a 3-to-4σ significance). Some simulations carried out in <cit.> already showed how a CGW can be misinterpreted as a GWB and, vice versa, a GWB can be misinterpreted as a CGW. In both the EPTA and NANOGrav analyses, when a GWB is also searched for, evidence for a CGW significantly decreases, indicating that its presence is at the very least not necessary to explain the observations. However, the degeneracy between the pulsar response to the two signals does not allow to conclusively rule out the presence of a CGW. 

The presence of these resolvable sources in the PTA data on the one hand poses a number of interesting challenges, on the other hand opens the door to invaluable opportunities. In fact, while the analysis pipelines currently used to search for a GWB rely on the assumption that the background is a Gaussian, isotropic, and stationary process characterised by a power-law Fourier spectrum <cit.>, the presence of a CGW can result in an excess of power at specific frequencies, which produces some level of anisotropy and non Gaussianity <cit.>. Thus, the standard analysis may depend on inapplicable assumptions and yield biases on the estimated GWB parameters <cit.>. This is a recognised issue within the PTA community and analysis techniques are being developed to go beyond the power law, isotropic Gaussian assumption <cit.> and to pull out of that data GWBs and CGWs simultaneously <cit.>.

Within this context, it is of paramount importance to study the interplay of CGWs and GWB when both are present in the data and what challenges arise in the interpretation of the analysis results. From an exquisitely scientific perspective, conversely, the prospects of detecting CGWs is particularly appealing. In fact due to their deterministic nature, PTAs are able to estimate the source parameters, in particular the sky location via triangulation <cit.> and possibly the distance to the source <cit.>. The detection of one or more resolvable sources <cit.> would be very precious as it would allow to track the orbital evolution of the binary and thus obtain direct information on the Massive Black Holes (MBHs) coupling to the environment <cit.>. This information cannot be well constrained by a measure of the GWB power spectral density, because it is strongly degenerate with the MBH mass and eccentricity distribution. In addition, CGWs offer the exciting opportunity to search for electromagnetic counterparts of the GW signal, opening the prospects of low frequency GW multimessenger astronomy <cit.>. Although PTAs can realistically localise the CGW source only within several deg^2 <cit.> they must be originated by particularly massive and/or nearby SMBHBs, making the search of their galaxy host viable <cit.>. Identifying the host and detecting accretion activity onto the binary will shed light about the physics governing the formation and (thermo)dynamical evolution of SMBHBs and the accretion discs surrounding them. 

In this paper, we carry out the first detailed investigation of the interplay between stochastic GWBs and deterministic CGWs in realistic PTA datasets, assessing the performance of currently used analysis pipelines when both signals are present in the data. We do so by building state of the art mock datasets capturing the complexity of real data, and by injecting them with as realistic as possible GW signals, mimicking the one found in the recent EPTA analyses <cit.>. Realistic GWBs are injected as the incoherent sum of individual waves consistent with astrophysical observations <cit.>, and CGW are modelled according to the candidate identified in <cit.>. Datasets are then analysed with the standard models implemented in the  <cit.> analysis suite, which models the GWB as a common red process characterized by a single power law spectrum, while we also test CGW models including only the Earth term or the full Earth plus pulsar components. 

The paper is organized as follows. In Section <ref> we describe in detail the simulations: the properties of the adopted datasets, the injected signals and the model used for the recovery. In Section <ref> we present the main results of our analysis. Using the methods described in the previous section, we carried out several investigations. We first assess the importance of including the pulsar term in the CGW model (Section <ref>). We then study the degeneracy of signal recovery on datasets containing only a GWB or a CGW (Section <ref>), showing that datasets injected only with a CGW can produce evidence of a GWB (Section <ref>) and vice versa (Section <ref>), and that a joint search is able to pin down the correct nature of the signal (Section <ref>). The performance of a combined analysis of a GWB and a CGW on datasets containing both signals is then evaluated in Section <ref> and we discuss our main findings in Section <ref>.
 Throughout the paper we assume a Λ cold dark matter cosmology with parameters (H_0, Ω_m, Ω_Λ)=(70 km s^-1Mpc^-1, 0.3, 0.7). 
















  


§ METHODS

In this section, we describe both the simulations structure and the models and methods used in the analysis.



 §.§ Simulations




Our simulations are based on the second data release of EPTA <cit.>. In particular, we have reproduced the DR2full dataset: 25 pulsars with a maximum observation time of 24.8 yr. The main properties of the real time of arrivals (ToAs) of each pulsar are reproduced: the observation time, the number of ToAs, the timing model (TM), the multiband measures – the frequency coverage is identical to that of the EPTA DR2full pulsars – and the noise levels. The uneven sampling of the ToAs is also reproduced, but large gaps, which are common in the real data streams, are not present in the simulations (see Fig. <ref> for a comparison between real and simulated residuals).
All the properties of the arrival times were injected in the dataset using the  package <cit.>. The same package is used to inject the three components of the noise budget: white noise (WN), achromatic – independent of the observation radio frequency –  red noise (RN) and chromatic red noise (due to dispersion measure variations, and customarily referred to as DM). 

To model the white noise, the ToAs are sampled from a Gaussian distribution centred at the value predicted by the TM and with a width given by:

    σ_i = √(EFAC_i^2σ_ ToA^2 + EQUAD_i^2),

where σ_ ToA is the errorbar associated to each ToA according to the template-fitting errors obtained in EPTA DR2, the EFAC takes into account for the ToA measurement errors and the EQUAD accounts for any putative extra source of white noise. The two parameters EFAC and EQUAD are specific for each observing backend. In the simulations, each pulsar's backend, as well as the ToAs' uncertainties, are identical those of DR2full; therefore, each pulsar has ToAs taken from ∼12 different backends, which means 12 different couples of values for EFAC and EQUAD. The values used for EFAC and EQUAD are the maximum likelihood values obtained by the single pulsar noise analysis of EPTA DR2full <cit.>.

In addition to the white noise, achromatic and chromatic red noise contribute to the single-pulsar noise budget. These are time-correlated noise components modelled as a stationary Gaussian process with a power-law power spectral density (PSD) of the form

    P(f; A, γ) = A^2/12π^2(f/ yr^-1)^-γ yr^3.

For achromatic red noise, in each ToA, the delay induced in the residuals is proportional to the red noise PSD, 

    Δ_ RN∝√(P(f; A_ RN, γ_ RN)).


This noise component accounts for the long-term variability of the pulsar spin along with other sources, such as unmodelled objects orbiting the pulsar. For chromatic red noise, on the other hand, the induced delay also depends on the observing radio frequency because it is due to the interaction of the radio signal with the ionised interstellar medium (IISM), the interplanetary medium in the solar system, and the Earth's ionosphere. This effect is taken into account during the observations and inside the timing model which considers its value at a reference epoch together
with its first and second derivatives. However, the inhomogeneous and turbulent nature of the IISM also induces stochastic variations in the DM value, which are modelled as chromatic red noise. The residual DM delay is modelled as 

    Δ_ DM∝√(P(f; A_ DM, γ_ DM))×ν^-2

where ν is the observing radio frequency. The values of A_ RN, γ_ RN, A_ DM and γ_ DM used in the simulations are  reported in Table <ref>, together with the number of Fourier components used in the modelling of these processes. These values – amplitude, slope and number of Fourier components – correspond to maximum likelihood values obtained from the single pulsar noise analysis of EPTA DR2full. An example of simulated data compared to the real EPTADR2 dataset is shown in Fig. <ref>.






  §.§.§ Gravitational wave signal injection


Having specified the properties of the adopted PTA, we now turn to the description of the injected GW signals. The signal produced by a SMBHB is composed by two terms, customarily referred to as Earth term (ET) and pulsar term (PT), which are respectively associated to the GW strain hitting the Earth at the ToA of the considered pulse and the strain hitting the pulsar at the time of emission of the same pulse.

Throughout the paper we will always assume circular SMBHBs, evolving under the effect of GW backreaction only (i.e. we neglect any possible coupling with the local stellar or gaseous environment). Under these assumptions, the delays induced in the ToAs at an epoch t are fully defined by 8 parameters: the chirp mass ℳ[Note that in the quadrupolar approximation employed here for the injected waveform, the signal does not depend on the individual binary masses.], the observed frequency f_gw, the redshift z, the inclination angle , the sky location (θ, ϕ) – respectively polar and azimuthal angle – , the initial phase ϕ_0 and the polarization angle ψ. Following <cit.>, we model them as:


    s(t, Ω̂) = F^+(Ω̂)Δ s_+(t) +F^×(Ω̂)Δ s_×(t),

where 

    Δ s_A(t) = s_A(t_p) - s_A(t_e),

the functions F^A(Ω̂) – with A indexing the polarization and Ω̂ being the unit vector pointing from the GW source to the Solar System Barycenter ˆ
(SSB) – are the antenna pattern functions that depend on the relative position of the pulsar and the source <cit.>. s_A(t_p) and s_A(t_e) are the PT and ET respectively. They both take the form


    s_+(t) = ℳ^5/3/d_Lω(t)^1/3(-sin(2(Φ(t)-Φ_0))(1+cos^2)cos(2ψ) 
       - 2cos(2(Φ(t)-Φ_0))cos sin(2ψ))
       s_×(t) = ℳ^5/3/d_Lω(t)^1/3(-sin(2(Φ(t)-Φ_0))(1+cos^2)sin(2ψ) 
       +2cos(2(Φ(t)-Φ_0))cos cos(2ψ)),

where 

    Φ(t) = Φ_0 + 1/32ℳ^5/3(ω_0^-5/3-ω(t)^-5/3),

and 

    ω(t) = (ω_0^-8/3-256/5ℳ^5/3t)^-3/8.

The only difference between PT and ET is that while the latter is the delay induced at the time of observation on Earth t_e, the former has to be evaluated at the time of emission of the radio pulse at the pulsar t_p = t_e - L(1 + Ω̂·p̂), where L is the pulsar distance and p̂ is the pulsar position in the sky. We inject in our data two different GW components: a stochastic GWB and a resolvable CGW. Since the evolution of the signal over the observation time is small, the phase approximation is used both in the injection and in the recovery: ω_ Earth = ω_0 at each epoch t.

Stochastic GWB. The GWB is injected as a realistic superposition of GWs coming from a cosmic population of SMBHBs. The injection pipeline starts from a catalog of SMBHBs, computes the delays induced by each binary in each pulsar ToA according to Eq. (<ref>) and then sums all the delays in the time domain. The distribution of the sources' ℳ, f_gw, z and  determines the observed power spectrum generated by the SMBHBs population: the binned total characteristic strain, where the frequency bin width is Δ f = 1/T with T the time span of the pulsar array, can be evaluated as <cit.>

    h_c^2(f_i) = ∑_j∈Δ f_ih_j^2(f_r)f_r/Δ f_i,

where f_i is the central frequency of the bin Δ f_i, and the sum runs over all the binaries for which the observed frequency f=f_r/(1+z)∈Δ f_i, where f_r is the GW frequency in the binary rest frame. The strain of each GW signal is given by 

    h(f_r) = 2√(1/2(a()^2 + b()^2))(Gℳ)^5/3(π f_r)^2/3/c^4d(z),
 
where a() = 1+cos^2, b() = -2cos and d(z) is the comoving distance of the binary, which is computed as 

    d(z) = c/H_0∫_0^z1/√(Ω_m (1+z)^3 + Ω_Λ)dz.

For the population of circular GW driven SMBHBs, assumed here, the characteristic strain given by Eq. (<ref>) can be approximated as a power law. It is therefore customary to describe the signal as

    h_c(f) = A_ GW(f/ yr^-1)^α,

where α=(3-γ)/2=-2/3 being γ the PSD induced in the timing residual (cf. Eq. <ref>).

The power spectrum h_c^2(f_i) thus depends on the number of emitting sources per unit redshift, mass and frequency d^3N/(dzdℳdf_r). A list of ∼ 120k binaries is constructed by extracting ℳ, f_gw, z as Monte Carlo samples of a numerical distribution d^3N/(dzdℳdf_r), obtained from the observation-based models described in <cit.> and <cit.>. In those papers, the authors construct hundreds of thousands of SMBHB populations in agreement with current constraints on the galaxy merger rate, the relation between SMBHs and their hosts, the efficiency of SMBH coalescence and the accretion processes following galaxy mergers. From this set of models, we selected 100 populations producing a GWB amplitude A ∼ 2.4×10^-15, consistent with the signal observed by EPTA <cit.>. The other parameters of the binaries are then randomly sampled such that sources are isotropically distributed in the sky – cos θ∈Uniform(-1, 1) and ϕ∈Uniform(0, 2π) –, their orbital angular momenta are uniformly distributed on the unit sphere – cos ι∈Uniform(-1, 1) –, the phase and the polarization angle are random – ϕ_0 ∈Uniform(0, 2π), ψ∈Uniform(0, π). 


Resolvable CGW. In some of the dataset, an individually resolvable CGW is injected. The waveform is again injected in each pulsar in the time domain residual according to Eq. (<ref>). The only difference, compared to the SMBHBs contributing to the unresolved GWB is that the amplitude of the signal is large enough that the source can be identified above the pulsar noise and the unresolved GWB by a signal search employing a deterministic filter to match the signal (see next Section). We stress that we always inject both the ET and the PT for every SMBHB, regardless of whether it provides an unresolvable contribution to the GWB or it is a resolvable CGW.

Datasets. For the purpose of this investigation, seven different datasets were produced, with the exact same noise realization, but different gravitational signals:

    
  * dataset injGWB01, injGWB02, injGWB03. These three datasets contain a GWB-only. They are produced injecting three background selected from the 100 populations mentioned above. The three populations are chosen among those displaying a smooth behaviour in the lowest frequency bins, without particularly loud sources.

    
  * dataset injCGW5, injCGW20. These two datasets only contain a single loud CGW-only. The injection procedure is similar to the GWB datasets, but the population file used contains only one binary. The two injected sources have Earth term GW frequencies of 5nHz (injCGW5) and 20nHz (injCGW20). The parameters of the 5nHz source are chosen to approximately reproduce frequency and amplitude of the candidate CGW found in the EPTA DR2new, <cit.>, which in fact has a frequency of ∼5nHz and an amplitude h∼1.1× 10^-14. The source is then located in the Virgo cluster; this choice determines sky location and distance, and consequently the chirp mass. Inclination, initial phase and polarization angles are randomly extracted as in the GWB case. The source injected in injCGW20 has identical parameters except for the higher frequency (20nHz instead of 5nHz), thus resulting in a higher strain (h∼ 2.7× 10^-14). The delays induced in pulsar PSR J1022+1001 by these two CGWs are shown in Fig. <ref>.
    
    
  * dataset injGWB03+CGW5, injGWB03+CGW20. These two datasets are injected with a GWB and a CGW. The GWB is in both cases the one resulting from population 03, while the CGWs are either at 5nHz or 20nHz.










 §.§ Signal modelling and analysis methods

All the analyses are carried using the  <cit.> package. The starting point of the analysis is the likelihood function of the timing residuals δt

    P(δ𝐭|ξ, ζ) = exp[-1/2(δ𝐭 - 𝐌ϵ - 𝐬)^T𝐂^-1(δ𝐭 - 𝐌ϵ - 𝐬)]/√((2π)^ldet𝐂).

The term Mϵ is the timing model error expressed as a linear function of the offset from the nominal timing model parameters, ϵ. ξ are the timing model parameters and ζ are the stochastic components parameters. The likelihood function sampled to search for the gravitational signal is marginalized over the timing model parameters as described in <cit.>. 𝐂=𝐂(ζ) is the covariance matrix that contains all the contributions from stochastic processes, which are:

    
  * WN, modelled according to Eq. (<ref>). All the white noise parameters are fixed at the injected values in the GW searches;
    
  * RN, modelled as a Gaussian process with amplitude of the Fourier components given by Eq. (<ref>)  (<cit.>;
    
  * DM, modelled as a Gaussian process with amplitude of the Fourier components given by Eq. (<ref>) and amplitude depending on the observed radio frequency (see Eq. <ref>). As for RN, the DM is not present in all the pulsars and, if present, the number of Fourier bins used in the Gaussian process depends on the pulsar and it is equal to the number of components used in the injection. For a summary of the noise model characteristics, see Table <ref>.
    
  * common red noise (CRN), if searched for. A red noise process which is common to all the pulsars can be accounted for in the covariance matrix. If included in the analysis, it is also modelled as a Gaussian process. The amplitude of the components are modelled in two ways: as a power law function of the frequency, and therefore the parameters are the slope γ_ CRN and the amplitude A_ CRN; or as a binned spectrum, and thus the parameters are the amplitudes of the common process in each frequency bin f_i. In both cases, the number of components used in the Gaussian process is 30.
    

Although we inject the GWB as  a superposition of unresolved SMBHBs, we follow the standard PTA analysis practice and model it in our likelihood function as a CRN characterized by an amplitude and a spectral index, as in Eq. (<ref>), which can be directly translated in a characteristic strain of the form given by Eq. (<ref>). Note that, depending on the dataset, we use two different flavours of CRN. In most searches, the GWB is modelled as a common uncorrelated red noise (CURN). This means that inter-pulsar correlations induced by the GWB are neglected and the GWB posterior distribution is then reconstructed a posterirori with the reweighting technique <cit.>, when possible. The advantage of this model is that it is much faster since it keeps the correlation matrix in the likelihood function diagonal and it naturally provides the Bayes factor of the reconstructed model versus the  sampled model, besides the posterior distribution of both models. However, the reweighting technique is not always efficient and, as we will see later, using a CURN to model a GWB is not always appropriate; therefore, when needed, we use the HD-correlated CRN model which we referred to as GWB model. In this case, GWB induced inter-pulsar spatial correlation are also modelled, according to the HD curve <cit.>

    Γ_ab=3/21-cosθ_ab/2ln( 1-cosθ _ab/2)-1/41-cosθ_ab/2 +1/2

where a and b are indexes identifying two pulsars in the array and θ_ ab is their angular separation in the sky. Note that searching for an HD-correlated CRN is much more computationally expensive, since the correlation matrix is not diagonal anymore.

The last term in the likelihood function is s, which is the delay induced by an individually resolvable CGW, and is analytically described by Eqs.(<ref>)–(<ref>) which are implemented in the CGW model of . An approximation that can be adopted is to consider in the CGW model the ET only, neglecting the PT. This is computationally very convenient, since the former only depends on the 8 source parameters. Conversely, the latter also depends on the distance of each pulsar in the array. Since those  distances are generally known with large uncertainties (of the order of 20%), they must be considered as additional free parameters in the search. Therefore, the complete model has at least one more parameter per each pulsar with respect to the ET only model. Moreover, as prescribed in <cit.>, in order to ease the search of the best parameters, one extra parameter is added per each pulsar, which is an initial phase at the pulsar, Φ_p. We will assess the performance of an ET only search on realistic signals including ET and PT in Section <ref>.

We mainly adopt Bayesian techniques in our analysis, although we also exploit the frequentist optimal statistics, when useful. The priors used in the search are listed in table <ref> and are all uninformative, except for the pulsar distances' priors that exploit information obtained from independent electromagnetic observations. The prior distributions have a width which is taken as the measurement error listed in <cit.>; for the pulsars that are not included in this paper, the relative error is fixed at 20%. The mean of the prior is chosen such that the pulsar distance used for the injection is within 1σ of the prior distribution for each pulsar. Parameter estimation is then performed by sampling the posterior distribution with the Markov
Chain Monte Carlo (MCMC) sampler , <cit.>. For the 25 pulsar PTA we employ, the dimensionality of the posterior distribution is 62 (2 parameters – slope and amplitude – for each red noise and dispersion measure process included in the model) plus the GW model dimension (2 for the GWB power-law model, 8 or 58 for an ET only or a full ET+PT CGW model search respectively). Model selection is done by computing the Bayes factor, either with the reweighting technique, when it is efficient, or with the product space method implemented in  <cit.>.


































  


§ RESULTS


Using the methods described in the previous section, we carried out several investigations. In Section <ref>, we assess the importance of modelling a CGW including both the Earth and pulsar terms. In Section <ref>, we investigate the degeneracy between the analysis results obtained with a GWB-only dataset and with a CGW-only dataset. In Section <ref>, it is shown how CGW-only datasets can produce strong evidence in favour of an HD-correlated GWB. Viceversa, in Section <ref>, it is shown how a GWB-only dataset can be misinterpreted as a continuous wave. This degeneracy is solved in Section <ref>, by means of a combined analysis that searches simultaneously for a GWB and a CGW on both GWB-only and CGW-only datasets . Finally, in Section 
<ref>, we evaluate the performance of a combined analysis of a GWB and a CGW on datasets containing both signals. 






 §.§ On the importance of the pulsar term in CGW modelling





















As described in section <ref>, the signal imprinted by a CGW in the ToAs is the sum of two terms: the Earth term and the pulsar term. The ET is common to all pulsars and only depends on the 8 parameters describing the source. Conversely, the PT depends on the distance of each pulsar and the phase of the GW signal at the pulsar's location. 
This means that modelling of the full CGW ET+PT in a search performed on an array of 25 pulsars (like the one we are considering here) requires 50 extra free parameters, that must be added to the 8 source parameters and to the 62 parameters of the individual pulsars noise components. It is therefore interesting to compare searches of CGW modelled as ET only and ET+PT on realistic datasets, to see whether the computational efficiency of the ET only search bears any significant drawback.


The two templates have been tested on both CGW-only datasets (injCGW5 and injCGW20), and the main results are shown in Fig. <ref>. From the figure it is clear that the performance of the ET only template depends on the CGW frequency. In fact, when applied to the injCGW20 dataset (right panel of Fig. <ref>), the ET only model performs well on all parameters that it is sensitive to, i.e. all source parameters except the chirp mass and the luminosity distance. Conversely, the same model applied to the injCGW5 dataset results in some non-negligible biases in the recovery. As it can be seen from the left panel of Fig. <ref>, the injected values for the frequency and sky location are not compatible with the 90% C.I. of the recovered posterior distributions. 

The different performance of the ET only template in the two cases is due to the different degree of separability of the ET and PT terms for the two injected sources. For the injected system chirp mass, ℳ=1.16· 10^9M_⊙, the average frequency difference between ET and PT is Δ f∼ 
3.8×10^-6nHzyr^-1Δ t∼ 1.0× 10^-2 nHz∼ 0.02Δ f_bin for the 5nHz CGW and Δ f∼ 1.4× 10^-3nHzyr^-1Δ t∼ 4.0 nHz∼ 3.2Δ f_bin for the 20nHz CGW. Therefore, while in the 20nHz CGW, ET and PT falls at very distinct frequencies, they blend together in the same frequency bin for the 5nHz CGW. In the former case, the ET only template only picks the common power produced by the ET at its frequency, disregarding the PT term; while in the latter, the ET only template is trying to fit a superposition of ET and PT essentially overlapping at the same frequency. This concept is demonstrated in Fig. <ref>, where the reconstructed waveforms are compared to the injected ones. In the 20nHz case (injCGW20 dataset), the ET only template almost perfectly fits the ET of the full injected waveform, while in the 5nHz case (injCGW5), the ET only template absorbs power from both ET and PT and the reconstructed waveforms sits, for many pulsars, somewhat in between the ET and the full injected signal and is also way more uncertain as shown by the much larger credible region.





Conversely, the full ET+PT template is able to correctly reconstruct the signal and, because it is sensitive to the frequency evolution, it also allows to constrain both the source chirp mass and the luminosity distance. The performance is very different for the 5 nHz CGW and the 20 nHz CGW: for the former the chirp mass recovery is ℳ=0.52^+9.98_-0.48× 10^8M_⊙, while for the latter it is ℳ=1.01^+0.07_-0.06×10^9M_⊙, where we give the median and the 68% credible interval. The superior recovery of the chirp mass in the 20 nHz case is due to the much larger frequency evolution of the source between ET and PT, allowing to place a much tighter constrain on ḟ and thus on M.
Another consequence of the large frequency evolution of the 20 nHz CGW is that the model is sensitive to the pulsar distances. This is shown in Fig.<ref> where the reconstructed waveform (left) and the posterior distribution of the pulsar distance (right) are shown for PSR J1713+0747 for the injCGW20 (top row) and injCGW5 (bottom row) datasets. In the first case, the posterior distribution on the distance is narrower than the prior and centred at the injected value: this is allowed by a correct modelling of the frequency evolution, which is reflected in the excellent reconstruction of the waveform. Conversely, for the 5 nHz case, the posterior distribution of the pulsar distance is almost indistinguishable from the prior, but signal reconstruction still matches the injection, as the frequency evolution is too small for the search to be sensitive to it. We note that the GW signal reconstruction and distance determination is not always as good for the 20 nHz CGW. This is shown in the middle row of Fig.<ref> for pulsar PSR J1918-0642. In this case the posterior distribution resembles the prior, and while the reconstructed signal does not strongly disagree with the injection, the median fails to capture the signal shape accurately. In fact, only about 13 pulsars, whose evolution is accurately reconstructed, significantly contribute to the chirp mass recovery.



In addition to better and unbiased recovery of the CGW parameters, there is another rationale for preferring the complete CGW template over the ET-only template: in a joint CGW + GWB search, the contribution to the CGW signal due to the pulsar term, if not taken into account by the CGW model, is absorbed by the GWB model, since it is a common red noise process. It will therefore add extra spurious power to the GWB, biasing the recovery of its parameters. An example of this behaviour is shown in Fig. <ref>, where the GWB+CGW search is applied to the dataset injCGW5: when the CGW is searched with the full template, the posterior distribution of the GWB parameters is unconstrained, when the search is done with the ET-only template, the posterior distribution of the GWB parameters remains well constrained because it absorbs all the power coming from the unmodelled pulsar term.

We therefore conclude that while the ET-only template can be used to search for possible CGW candidates and provides a reasonable estimate of the source parameters <cit.>, the full ET+PT model must be preferred, especially in a joint search with a stochastic GWB. Despite the much lower computational efficiency, we will therefore always model our CGWs as ET+PT in the following.











 §.§ Degeneracy between GWB and CGW

In this section, searches for GWB, CGW and combined searches of the two signals together are applied to the 3 GWB realizations (injGWB01, injGWB02, injGWB03) and to the 2 CGW realizations (injCGW5, injCGW20). The aim is to study the degeneracy between the two signals and how it can be solved.





  §.§.§ CGW misinterpreted as a GWB


We start by performing a GWB analysis on the three injGWB and the two injCGW datasets. To this end we make use of both Bayesian and frequentist techniques following this procedure:

    
  * a power law-shaped and HD-correlated stochastic process is assumed for the GWB, then the slope γ_ GWB and the amplitude log_10A_ GWB of the powerlaw process are sampled, together with the noise parameters to obtain the posterior distribution;
    
  * the Bayes factor ℬ^ GWB_ CURN is evaluated to compare the evidence for an HD-correlated and an uncorrelated red noise process. The bayes factor is computed either with the reweigthing method or with the product space when reweighting is not efficient;
    
  * the averaged angular correlation between the pulsars is estimated with the optimal statistics method averaging over 10 angular bins, each one containing 30 pulsar pairs. A powerlaw-shaped background with variable γ is assumed;
    
  * the SNR of the HD cross-correlation marginalized over the noise parameters is obtained with the optimal statistics method, assuming a powerlaw-shaped background with variable γ.


1.4 





The results of the powerlaw spectrum parameter estimation are listed in the last two columns of Table <ref> for all five datasets and two examples of the recovered GWB parameters' posterior distributions are shown in Fig. <ref> for datasets injGWB02 and injCGW5. In general, the GWB search returns well constrained posteriors regardless of the actual content of the dataset, which in practice means that a CGW can easily be detected as (and mistaken for) a GWB. However, the signal recovered from the two CGW-only datasets (injCGW5, injCGW20) is higher and flatter than the one obtained from the background datasets. This can be clearly seen both from the A and γ values of Table <ref> and from the 2D contours in Fig. <ref>. This is expected since the extra power due to the two single sources is at frequencies which are higher than the loudest GWB frequency bins that dominate the powerlaw spectrum recovery in the GWB-only datasets.






The Bayes factors reported in Table <ref> show that an HD-correlated common process (i.e. a GWB) is generally strongly favoured regardless of the content of the data. In fact, dataset injCGW5 produces a Bayes factor ℬ^ GWB_ CURN of ∼ 10 and HD S/N of 3.0^+0.5_-0.5 (fixing γ to 3), while injCGW20 
gives ℬ^ GWB_ CURN of ∼ 60 and S/N = 3.2^+0.6_-0.6 (fixing γ to 2). Interestingly, injGWB03 does not favour an HD correlated process, displaying ℬ^ GWB_ CURN of ∼ 0.15. This is consistent with <cit.>, who found that, due to the stochastic nature of the signal, individual realizations of the same SMBHB population can result in Bayes factor that differ by several orders of magnitude.

Although the Bayes factors obtained on the  injCGW5 and injCGW20 are compatible with a GWB detection, the resulting HD S/N is significant only in a small range of values of γ that are consistent with the posterior distributions obtained by the Bayesian search, as shown in Fig. <ref>. By fixing γ to the nominal value of 13/3, the HD S/N drops to 1.3^0.4_-0.4 and to 1.1^1.0_-0.9 for the injCGW5 and injCGW20 datasets respectively. Conversely, the 
injGWB01 and injGWB02 datasets (also shown in Fig. <ref>) result in distributions of the HD S/N that are consistently high for all values of γ≥ 4.


Moreover, at higher values of γ, the CGW-only dataset has an increasing evidence for a dipolar-correlated signal, whose S/N surpasses that of HD. This behaviour is not observed in the GWB-only datasets, for which the dipole S/N always stays lower than the HD S/N. Finally, two examples of HD reconstruction are shown in Fig. <ref> for datasets injGWB02 and injCGW5. While the quadrupolar pattern can be seen in both datasets, the scatter around the expected relation appears to be much larger when a CGW only is present in the data, consistent with <cit.> and <cit.>. Moreover, the HD correlation for the injCGW5 dataset is reconstructed assuming γ=3, while it tends to disappear for higher values of γ.














  §.§.§ GWB misinterpreted as a CGW


We now search for a CGW in the same 5 datasets analysed in the previous section. In the search, we estimate the posterior distributions of all the noise parameters – except for the WN, which is fixed – and the CGW parameters – 8 source parameters plus 50 additional parameters for the pulsars' distances and phases. We then use hypermodeling to evaluate the Bayes factor between a model containing a CGW versus a model featuring intrinsic pulsar noise only. 

We can see from Table <ref> that also in this case, regardless on whether the dataset contains a CGW or a GWB, a CGW is recovered with high confidence, with a Bayes factor ℬ^ CGW_ PSRN>1000 in all cases. Table <ref> also shows that the CGW frequency and amplitude are fairly well constrained. The performance of the CGW search on the injCGW datasets has already been assessed in Section <ref>. As shown in Fig. <ref> when a full ET+PT template is used for the CGW, the model can correctly recover the injected signals and estimate their parameters, although with differences depending on the frequency of the source. Here it is particulary interesting to check the CGW recovery on the three injGWB datasets, which is shown in Fig. <ref>: all source parameters are fairly well constrained, including the sky location, which is different in each dataset. These parameters, however, do not correspond to any specific source present in the datasets, as can be seen in the three panels of Fig. <ref>. The recovered CGW frequency always corresponds to the frequency bin that is the most constrained by the HD-correlated free spectrum search; moreover, the amplitude recovered is always compatible with the GWB amplitude in that frequency bin. Thus, it can be concluded that the CGW parameters are constrained because the CGW model absorbs part of the HD-correlated common red noise produced by the incoherent superposition of all the SMBHBs composing the GWB. 

Finally, we note that the CGW amplitudes recovered from the three injGWB datasets are respectively 1.2^+1.5_-0.7× 10^-14, 1.4^+0.6_-0.5× 10^-14  and 1.8^+0.5_-0.4× 10^-14 (median and 68% credible interval). These values are all well compatible with the amplitude of the DR2new CGW candidate, which is 1.0^+0.5_-2.6× 10^-14 (median and 90% credible interval) <cit.>.














  §.§.§ Breaking the degeneracy: combined search for a GWB and a CGW




We now perform a joint CGW + GWB search on the five datasets considered thus far. We perform two flavours of combined searches: (i) common uncorrelated red noise plus CGW (CURN+CGW), and (ii) HD-correlated common red noise plus CGW (GWB+CGW). The CGW is always modelled as the sum of ET and PT, and together with the signal parameters, all the RN and DM parameters are also searched over.

The results of these two searches are reported in Table <ref> for all datasets, while posterior distributions of selected parameters of the model components are shown in Fig. <ref> for datasets injGWB02 and injCGW5. In figure, orange and light purple distributions are obtained respectively from the GWB+CGW and CURN+CGW searches, and results of GWB-only and CGW-only searches are also shown for comparison. To see how searching for both type of signals simultaneously allows to properly characterize the actual content of the data, let us discuss the injGWB and injCGW separately. 

In the injGWB cases, when a common red noise component is added to the model alongside the CGW, the evidence for the latter strongly decreases. This can be seen both from the drop of the Bayes factors in the first three lines of Table <ref>, and from the posterior distributions of the CGW parameters in the left panel of  Fig. <ref>. In Section <ref>, when searching for a CGW on a dataset containing only a GWB, we found  ℬ^ CGW_ PSRN>1000. This is because the GWB produces a common noise that the model prefers to assign to the CGW signal (common to all pulsars) rather than to the individual pulsars' RN components. 
When a CURN is included in the model, ℬ^ CURN+CGW_ CURN drops significantly but is still slightly larger than unity, suggesting that the presence of CGW component is still preferred. This can also be seen in the posterior distribution of the CGW amplitude shown in purple in the left panel of  Fig. <ref> that, while featuring a long tail that extends to the lower end of the prior range, still shows a significant peak around logA≈-14.3. This behaviour is shared by all three injGWB datasets and is due to the fact that the GWB manifests itself primarily as a common red noise process, with a weaker, emerging 'off diagonal' HD-correlated component. The CURN models it as a common red noise process, but is unable to account for the emerging HD correlation, which the model then assigns to the CGW.
Conversely, when a GWB is added to the model, we get ℬ^ GWB+CGW_ GWB<1 (first three lines of Table <ref>), strongly suggesting that the GWB component is now able to describe the full content of the data. This is also clear by looking at the CGW posterior parameters in   the left panel of  Fig. <ref>. The orange distributions show that both the frequency and amplitude posteriors are almost perfectly flat, with the latter displaying a cut-off, which we find to be at a similar value for all three realizations. From this we can extract  a 95% upper limit of the CGW amplitude which is ≈4.0×10^-15, ≈3.8×10^-15 and ≈4.6×10^-15 for datasets injGWB01, injGWB02 and injGWB03 respectively. Note that these upper limits are lower than the GWB level, since the total GW content of the dataset is correctly accounted for by the GWB component. Finally, the recovered source is not localized in the sky, with the posterior distributions of the sky location almost uniform across the sphere, consistent with the prior. 







Differently from the CGW parameters, the GWB parameters' posterior distributions are not significantly affected by the addition of the CGW component to the model: the GWB posterior distributions obtained from the GWB+CGW search and with the GWB-only search are well consistent with each other (see left panel of Fig. <ref> for the injGWB02 dataset and table <ref> for all the datasets) confirming again that the GWB component of the model is correctly accounting for all the HD-correlated power. Only a small variation can be noticed in the CURN+CGW recovery of the background parameters: the posterior distributions are slightly wider with a longer low amplitude tail. This is because the CURN cannot absorb any HD correlated power, which is (spuriously) assigned to the CGW component.  


 
Conversely, when the same analysis is applied to the injCGW datasets, the posterior distributions of the CGW parameters are always well constrained, both with the CURN+CGW and with the GWB+CGW search, as shown in the right panel of Fig. <ref>. Searching for CGW-only, CURN+CGW or GWB+CGW, in this case, gives consistent results for the CGW parameter, with no significant changes in the posterior distributions' shape and width (see right panel of Fig. <ref>). Moreover, GWB+CGW is able to recover the sky location with a precision which is comparable to the one obtained with the CGW-only search and the model comparison strongly favours the GWB+CGW model over the GWB-only model (see table <ref>). As expected, the GWB posterior distribution is instead completely unconstrained when the model applied in the analysis contains both a common red noise and a CGW: the power that generates the GWB signal recovered with the GWB-only search is now correctly completely absorbed by the CGW component (see Fig. <ref>).





  §.§.§ Noise-only datasets

As a further check, we tested the analysis pipelines on a dataset containing only the noise and no GW signal, neither deterministic nor stochastic. The noise realization is the one contained in all the datasets analysed. In this case, neither a GWB nor a CGW is detected. As for the GWB, we find a value of the Bayes factor ℬ^ HD_ CURN∼1.5 and the posterior distributions of A and γ are poorly constrained, with a 95% upper limit on the GWB amplitude of ∼ 4.47× 10^-15. As for the CGW, the Bayes factor ℬ^ CGW_ PSRN is ∼ 0.5 and the CGW parameters are totally unconstrained, with the amplitude showing a 95% upper limit of ∼4.5×10^-15. Although many more noise realizations should be tested to draw robust statistical conclusions, these tests show on fully realistic datasets that current PTA detection pipelines are unlikely to return a spurious detection in absence of an actual GW signal in the data.





 §.§ Joint GWB+CGW analysis on combined CGW+CGW datasets


In this section we present the results of simultaneous searches for a GWB and a CGW, applied to datasets injGWB03+CGW5 and injGWB03+CGW20. We remind the reader that both datasets contain the same noise and GWB realisation from dataset injGWB03. 

In each dataset a loud CGW is superimposed to the GWB: in one case the CGW is at 5nHz (injGWB03CGW5), in the second one it is at 20nHz (injGWB03CGW20). This is done in order to see how the joint analysis performs in presence of a slowly evolving source (the 5nHz CGW) and on a fast evolving one (the 20nHz CGW), and how the two of them interact with the GWB.









  §.§.§ Bias induced in the GWB recovery by the presence of a resolvable source in the SMBHBs population

As already discussed by <cit.> and  <cit.>, the presence of a particularly loud source among the systems contributing to the GWB, if not accounted for in the analysis, can significantly bias the recovery of the overall GWB signal. As a preliminary step in our investigation, we therefore perform a GWB only search on the injGWB03CGW5 and injGWB03CGW20 datasets. As expected, the analysis produces a biased recovery of the GWB parameters: the results are shown in the right panels of Fig. <ref> and <ref>: in both cases, the recovered background is flatter and higher than the injected signal due to the extra power introduced by the loud source. In the two following sections, we demonstrate how this bias can be solved by performing a joint analysis of the GWB and the CGW components.





  §.§.§ Results for the injGWB03+CGW5 dataset


















The results presented here are obtained by sampling the posterior distributions of the CURN+CGW model and then reweighting them to obtain the posteriors of the GWB+CGW model.
The results of the combined search applied to the dataset injGWB03+CGW5 provides good overall results, with the majority of the CGW parameters recovered within 2σ of the injected values. 

The posterior distributions of the CGW parameters are shown in the left panel of Fig. <ref>, while the recovery of the background parameters is shown in the right panel. The recovery of the CGW parameters is successful: the frequency and the sky localization of the source are well constrained – the width of the posterior distributions is ≲10% of the prior width for all three parameters – and compatible within 68% with the injected values: injected frequency is f = 5.02nHz, recovered frequency is f = 5.04^+0.04_-0.04nHz; injected sky location is (12^ h19^ min50^ s, 12.38^∘), recovered sky location is (12^ h 33^+20  min_-19, 10^+5_-5^∘). However, the 3D source location cannot be properly constrained, since the luminosity distance posterior distribution is  extremely wide due to its strong covariance with the chirp mass. Both posterior distributions are roughly bimodal, with a narrow peak close to the injected value – in particular for the chirp mass – and a long, bumpy tail extending to low values of luminosity distance and chirp mass. Note that the posterior of the luminosity distance spans the interval (0.02, 24)Mpc within the 90% credible region; for such a nearby putative source, ancillary information coming from galaxy catalogues can effectively enable the identification of the source host <cit.>. The inclination angle, initial phase and polarization angle are well constrained and compatible with the injected values.

Focusing now on the GWB parameters, the posterior distribution in the right panel of Fig. <ref> shows how accounting for the resolvable source in the model solves the bias in the GWB parameters' recovery. In fact, the posterior distribution obtained with the combined search is well consistent with the one obtained with the GWB-only search applied to the GWB-only dataset injGWB03: they peak at similar values and have comparable widths, showing that the parameters of the stochastic and of the deterministic components are not significantly correlated and the interplay between the two signals does not affect the search.





  §.§.§ Results for the injGWB03+CGW20 dataset













 





The joint search applied to the injGWB03CGW20 dataset unveils some new interesting behaviour of the recovery pipeline. In fact, in this case, when a joint CURN+CGW (ET+PT) search is performed and the GWB is obtained with the reweighting technique, the estimate of the GWB parameters is completely off, as demonstrated by the filled purple distributions shown in the right panel of 
Fig. <ref>. The result is indeed quite similar to a GWB search only (shown in red), meaning that a significant fraction of the CGW power is absorbed in the common background component and the CURN+CGW search followed by reweighting fails in this case, producing strong biases in the results.



The reason of this failure is the following: as shown in Section <ref>, due to the significant difference in the 20nHz CGW between the Earth and the pulsar term GW frequency, in order to correctly account for both terms with the CGW model, the majority of the 50 PT parameters (i.e. 25 pulsar distances and 25 individual PT GW phases) have to be constrained. This is a difficult task for the sampler, which in the CURN+CGW case converges to an easier, alternative solution: the pulsar terms are all absorbed by the common uncorrelated red noise component, which is possible because they are indeed a common sinusoidal process that is uncorrelated, since the pulsars are all at different sky locations and distances from Earth. This justifies the much higher and flatter recovered background shown in the right panel of Fig. <ref>: the recovered amplitude is 1.83^+0.28_-0.26×10^-14, while the injected value is 2.4×10^-15. 

This interpretation is confirmed by the recovered CGW parameters (not shown). Most of them are biased and in particular the chirp mass, which determines the frequency evolution of the signal, is much lower than the injected value: the obtained value is 6. 9^+18.0_-5.0×10^6 M_⊙, which corresponds to a 95% upper limit on the frequency derivative of 4.2×10^-6nHz·yr^-1, ∼300 times smaller than the ḟ of the injected SMBHB. This is consistent with the fact that the values of the CGW parameters are such that the recovered time domain signal matches the injected Earth term and completely neglects the pulsar term contribution. This is confirmed by the shape of the reconstructed CGW shown in  Fig.<ref>. In the top panel we can see that in the CURN+CGW search the recovered CGW waveform for pulsar PSR J1713+0747 reproduces exactly the injected ET contribution. In fact, being HD-correlated, in presence of a CGW component in the model, the Earth term is not absorbed by the CURN component. Indeed, the recovered background is slightly lower than the one obtained by a GWB-only search on the same dataset (shown by the red distributions in the right panel of Fig. <ref>), because in that case both the Earth and pulsar terms are absorbed by the background component. 

We stress that this bias indeed corresponds to the maximum of the posterior distribution. This can be explained by the fact that the CURN model does not exactly match the injected common red noise, which is HD correlated, and therefore it prefers to absorb the pulsar term, since it is much cheaper to describe it as a two-parameter CURN signal, rather than to determine the 2×25 parameters describing each individual sinusoidal pulsar term.

In this dataset, the injected signal components are  properly recovered only by a direct GWB+CGW (ET+PT) search, which is more computationally expensive due to the off-diagonal HD terms that must be computed when evaluating the correlation matrix in Eq. (<ref>). In fact, in a GWB+CGW search the pulsar term is not absorbed by the HD-correlated common red noise component and both signals are properly reconstructed. In the right panel of Fig.<ref> the GWB recovery obtained by this search is shown in filled purple: it agrees very well with that obtained with the GWB-only template applied to the GWB-only dataset (empty purple), although it is slightly larger, which can be explained by the fact that the GWB+CGW model contains 58 more parameters describing the single source. A tiny correlation with this high number of parameters can easily explain the less precise recovery of the background parameters.

The correct recovery of the GWB parameters is a direct consequence of the correct recovery of the CGW parameters. In the left panel of Fig. <ref>, the posterior distributions of the 8 source parameters are shown and in Fig. <ref> the median and 90% credible region of the recovered CGW signal is shown for pulsar PSR J1713+4707. From the recovery, it can be seen how the model is able to account for the frequency evolution of the signal: the frequency recovered is 19.99^+0.21_0.18nHz (injected is 20nHz) and the chirp mass 1.14^+0.11_0.10×10^9M_⊙ (injected is 1.16× 10^9M_⊙). This allows the luminosity distance to be much better constrained than in the low frequency case, with a median and 68% credible interval of 32.7^+15.1_-9.3Mpc. Since the injected value is 21.6Mpc, the value obtained is a mild overestimate, which is probably due to the anti correlation with the inclination angle and the initial phase, which are both slightly underestimated. The 2D sky location recovery is also improved with respect to the 5nHz case, resulting in a declination angle of 12^ h38^+17 min_-16 and a right ascension of 11.5^+3.0_-3.4^∘.

  


§ DISCUSSION AND CONCLUSIONS


In this paper, we have carried out the first comprehensive study of the interplay between stochastic GWBs and deterministic CGWs in realistic PTA datasets, assessing the performance of currently used analysis pipelines. To this end we constructed mock datasets featuring all the complexity of EPTA DR2full <cit.>; we simulated up to 24.8 years of unevenly sampled, multi frequency ToAs for the 25 pulsars of DR2full and injected WN, RN and DM consistent with the maximum likelihood values found in the DR2full single pulsar analysis <cit.>. The injected GW signals were also as realistic as possible, tailored to mimic the findings of the recent EPTA analyses <cit.>. A GWB with nominal amplitude of A_ GW=2.4× 10^-15 was injected as the incoherent sum of individual waves coming from observationally based cosmic populations of SMBHBs <cit.>, while the deterministic CGW was modelled according to the candidate signal identified in the analysis of EPTA DR2new <cit.>. In all cases, SMBHBs were assumed to be circular and both the ET and PT of the GW induced time delays were injected. These GW signals were searched with the standard models implemented in the  analysis suite. The GWB was therefore modelled as a common red process characterized by a single power law spectrum, while the model CGW matched the one used for injecting the signal (i.e., circular, GW driven SMBHB), either including only ET or ET+PT. With this set-up we carried a number of investigations, whose results are summarized in the following.

We first compared the performance of ET-only and ET+PT templates on two datasets containing only a CGW either at 5nHz (injCGW5) or at 20nHz (injCGW20). In both cases, the ET+PT template was able to fully recover the signal, providing an unbiased estimate of all its parameters, including chirp mass, distance and sky location. Things were not so straightforward when the ET only template was employed. In this case, the 5nHz CGW dataset resulted in a biased recovery of the frequency and the sky location, whereas in the 20nHz case, the recovery was unbiased. In fact, in the latter case Earth and pulsar term can be easily disentangled and the ET-only template correctly matches the ET component of the signal. Conversely, in the 5 nHz case, ET and PT fall in the same frequency bin and the ET template tries to fit a mixture of the two, leading to a bias in the signal reconstruction. In both cases, the ET-only model is not sensitive to the chirp mass and therefore to the luminosity distance, preventing 3D source localization.

We also stress that in both cases, modelling ET only leaves behind some unmodelled residual (e.g. the full PT in the 20 nHz case), which will likely be absorbed by the GWB template in a joint search, leading to a biased recovery of the background parameters. These are the reasons why the complete CGW template (ET+PT) must be preferred to the ET-only, even if this means having 2 times the number of pulsars extra parameters (50 in our case) and therefore much larger computational costs for the sampling.

We therefore adopted the full ET+PT template for the CGW and turned to the investigation of datasets in which only one of the two types of signal was injected (either a GWB or a CGW) and the analyses were performed separately with one model at the time (eiter a GWB or a CGW). We analysed 5 datasets: 3 realizations of a realistic GWB (dataset injGWB01,injGWB02, injGWB03) and 2 realizations of single resolvable sources (datasets injCGW5, injCGW20). All datasets contained the same noise realization. These searches showed that a GWB can easily be misinterpreted as a CGW and vice versa. In fact, all sources returned evidence in favour of either GW signal and well constrained recovery of the model parameters. However, when a combined search is performed – meaning that the model used for the recovery contains both an HD-correlated GWB and a CGW – this degeneracy is broken: when only a GWB is present in the data, the posterior distributions of the CGW parameters obtained with the combined search are unconstrained and the CGW amplitude shows an upper limit; on the contrary, when a CGW is the only signal injected, the combined search gives unconstrained posterior distributions for the GWB parameters. No CGW evidence is absorbed by the HD-correlated background if the single source is present in the data. Therefore, the combined search is able to correctly recover the signal that is actually present in the data. Focusing on GWB-only datasets, we highlight that the CGW evidence is totally absorbed when an HD-correlated GWB is present in the model, while it does not completely disappear if only a CURN component is modelled. In this case, even if poorly constrained, the CGW parameters posterior distributions still peak at specific values. Therefore, modelling the common red noise as HD-correlated is necessary to totally absorb the spurious CGW. These results are similar to the ones obtained with the real EPTA DR2new data <cit.>: CURN + CGW searches peaking at a CGW amplitude of ∼10^-14, and peak disappearing once the background is modelled as HD-correlated and the Bayes factor ℬ^ GWB+CGW_ GWB being of order unity. Our simulations show that this behaviour arises when a CGW is indeed not present in the data, strengthening 
  the conclusion reached in 
the EPTA DR2 analysis.

We then analysed datasets containing both a stochastic GWB and a CGW, either at 5 nHz (injGWB03+CGW5) or at 20 nHz (injGWB03+CGW20). The low frequency CGW is an almost monochromatic source, while the high frequency one is drifting and the signal frequency significantly changes between the  PT and the ET. Performing a combined search on these types of datasets is particularly important because a single resolvable source, if not properly taken into account in the search, will induce a bias in the GWB parameters recovery <cit.>.

The results of the combined searches indicate that the bias in the GWB recovery induced by the presence of the single source is resolved when the CGW is correctly accounted for by the deterministic CGW model, so long as it contains both ET and PT. This result is obtained with both datasets, injGWB03CGW5 and injGWB03CGW20. However, in the injGWB03CGW5 case, the same unbiased result can be obtained with two methods: by performing a CURN+CGW search and then reweighing to obtain the HD+CGW posterior distribution, and by directly sampling the HD+CGW posterior distribution. The first method is computationally advantageous, as the CURN model is considerably faster to compute than the HD one. In the injGWB03CGW20 case, however, the first method (reweighting) is unable to recover the unbiased posterior distribution, as the CGW template solely accounts for the ET contribution, while the PT is more easily absorbed by the CURN. The posterior distribution on the CURN parameters is therefore strongly biased with respect to the GWB injection. Performing a direct sampling of the HD+CGW model, on the contrary, gives an unbiased recovery of the GWB parameters: the posterior distributions of the CURN and HD parameters are consequently disjoint, which makes the reweighing technique extremely inefficient. Therefore, while computationally much more expensive, the only method that is able to converge to unbiased results is the direct HD+CGW search. As for the GWB, the CGW recovery is successful, with most of the parameters constrained and compatible with the injection.

Given the recent PTA results indicating the presence of a gravitational signal in the data, it is crucial to be able to distinguish between the two types of signal that can be produced by SMBHBs. This work represents a first thorough study of this problem. Despite the limited parameter exploration, which is constrained by the high computational cost of the analyses, the investigation decisively points towards the importance of performing combined searches. Indeed, in the cases analysed, combined searches were able to both identify which signal was present in the data, without misinterpretation, and to guarantee an unbiased recovery of the GWB parameters in the presence of a loud individual source in the SMBHB population. Looking ahead, it will be important to extend this analysis to richer and more sensitive datasets, investigating the possibility of disentangling multiple searches, also in the presence of eccentricity or environmental coupling. In fact, the high amplitude of the currently observed signal implies that the future SKA might be able to resolve several tens of individual SMBHBs at S/N>5 <cit.>. Being able to successfully identify and disentangle those signals will have invaluable payouts for the understanding of SMBHB evolution and dynamics in the context of multimessenger astrophysics. Given the complexity and high dimensionality of these sources, it is of paramount importance to design more efficient search techniques <cit.>.


       I.F. thanks Stanislav Babak for his hospitality and mentoring at the APC, where this investigation was initiated.
       We thank the B-Massive group at Milano-Bicocca University for useful discussions and comments. A.C., A.S., G.M.S. acknowledge the financial support provided under the European Union’s H2020 ERC Consolidator Grant “Binary Massive Black Hole Astrophysics” (B Massive, Grant Agreement: 818691). A.C. acknowledges financial support provided under the European Union's Horizon Europe ERC Starting Grant "A Gamma-ray Infrastructure to Advance Gravitational Wave Astrophysics" (GIGA).

  

aa







 Paper Title: DESI Massive Post-Starburst Galaxies at $\mathbf{z\sim1.2}$ have compact structures and dense cores.





0000-0001-6454-1699]Yunchong Zhang 
Department of Physics and Astronomy and PITT PACC, University of Pittsburgh, Pittsburgh, PA 15260, USA

0000-0003-4075-7393]David J. SettonBrinson Prize Fellow
Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA

0000-0002-0108-4176]Sedona H. Price
Department of Physics and Astronomy and PITT PACC, University of Pittsburgh, Pittsburgh, PA 15260, USA

0000-0001-5063-8254]Rachel Bezanson
Department of Physics and Astronomy and PITT PACC, University of Pittsburgh, Pittsburgh, PA 15260, USA

0000-0002-3475-7648]Gourav Khullar
Department of Physics and Astronomy and PITT PACC, University of Pittsburgh, Pittsburgh, PA 15260, USA

0000-0001-8684-2222]Jeffrey A. Newman
Department of Physics and Astronomy and PITT PACC, University of Pittsburgh, Pittsburgh, PA 15260, USA


Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA

0000-0001-6098-7247]Steven Ahlen
Physics Dept., Boston University, 590 Commonwealth Avenue, Boston, MA 02215, USA

0000-0001-8085-5890]Brett H. Andrews
Department of Physics and Astronomy and PITT PACC, University of Pittsburgh, Pittsburgh, PA 15260, USA


Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK


Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA

0000-0002-1769-1640]Axel de la Macorra
Instituto de Física, Universidad Nacional Autónoma de México,  Cd. de México  C.P. 04510,  México

0000-0002-5665-7912]Biprateep Dey
Department of Physics and Astronomy and PITT PACC, University of Pittsburgh, Pittsburgh, PA 15260, USA


Department of Physics & Astronomy, University College London, Gower Street, London, WC1E 6BT, UK


Institut d'Estudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain
Institute of Cosmology and Gravitation, University of Portsmouth, Dennis Sciama Building, Portsmouth, PO1 3FX, UK
Institute of Space Sciences, ICE-CSIC, Campus UAB, Carrer de Can Magrans s/n, 08913 Bellaterra, Barcelona, Spain

0000-0003-3142-233X]Satya Gontcho A Gontcho
Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA

0000-0002-5612-3427]Jenny E. Greene
Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA


NSF NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA


Department of Physics, Southern Methodist University, 3215 Daniel Avenue, Dallas, TX 75275, USA

0000-0003-3510-7134]Theodore Kisner
Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, CA 94720, USA

0000-0002-7613-9872]Mariska Kriek
Leiden Observatory, Leiden University, P.O. Box 9513, NL-2300 AA Leiden, The Netherlands

0000-0001-6755-1315]Joel Leja
Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA
Institute for Computational & Data Sciences, The Pennsylvania State University, University Park, PA 16802, USA
Institute for Gravitation and the Cosmos, The Pennsylvania State University, University Park, PA 16802, USA

0000-0003-4962-8934]Marc Manera
Departament de Física, Serra Húnter, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra Barcelona, Spain

0000-0002-1125-7384]Aaron Meisner
NSF NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA


Institució Catalana de Recerca i Estudis Avançats, Passeig de Lluís Companys, 23, 08010 Barcelona, Spain
Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra Barcelona, Spain

0000-0002-2733-4559]John Moustakas
Department of Physics and Astronomy, Siena College, 515 Loudon Road, Loudonville, NY 12211, USA

0000-0001-7145-8674]Francisco Prada
Instituto de Astrofísica de Andalucía (CSIC), Glorieta de la Astronomía, s/n, E-18008 Granada, Spain


Department of Physics and Astronomy, Sejong University, Seoul, 143-747, Korea

0000-0002-9646-8198]Eusebio Sanchez
CIEMAT, Avenida Complutense 40, E-28040 Madrid, Spain


Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA
University of Michigan, Ann Arbor, MI 48109, USA

0000-0002-2949-2155]Małgorzata Siudek
Institute of Space Sciences, ICE-CSIC, Campus UAB, Carrer de Can Magrans s/n, 08913 Bellaterra, Barcelona, Spain

0000-0003-3256-5615]Justin Spilker
Department of Physics and Astronomy and George P. and Cynthia Woods Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University,
4242 TAMU, College Station, TX 77843-4242, USA


NSF NOIRLab, 950 N. Cherry Ave., Tucson, AZ 85719, USA

0000-0002-1714-1905]Katherine A. SuessHubble Fellow
Kavli Institute for Particle Astrophysics and Cosmology and Department of Physics, Stanford University, Stanford, CA 94305, USA

0000-0003-1704-0781]Gregory Tarlé
University of Michigan, Ann Arbor, MI 48109, USA

0000-0002-6684-3997]Hu Zou
National Astronomical Observatories, Chinese Academy of Sciences, A20 Datun Rd., Chaoyang District, Beijing, 100012, P.R. China


§ ABSTRACT 

Post-starburst galaxies (PSBs) are young quiescent galaxies that have recently experienced a rapid decrease in star formation, allowing us to probe the fast-quenching period of galaxy evolution. In this work, we obtained HST WFC3/F110W imaging to measure the sizes of 171 massive (log(M_*/M_⊙)∼ 11) spectroscopically identified PSBs at 1<z<1.3 selected from the DESI Survey Validation Luminous Red Galaxy sample. This statistical sample constitutes an order of magnitude increase from the ∼20 PSBs with space-based imaging and deep spectroscopy. We perform structural fitting of the target galaxies with  and compare them to quiescent and star-forming galaxies in the 3D-HST survey. We find that these PSBs are more compact than the general population of quiescent galaxies, lying systematically ∼ 0.1 dex below the established size-mass relation. However, their central surface mass densities are similar to those of their quiescent counterparts (log(Σ_1 kpc/(M_⊙/kpc^2))∼ 10.1). These findings are easily reconciled by later ex-situ growth via minor mergers or a slight progenitor bias. These PSBs are round in projection (b/a_median∼0.8), suggesting that they are primarily spheroids, not disks, in 3D. We find no correlation between time since quenching and light-weighted PSB sizes or central densities. This disfavors apparent structural growth due to the fading of centralized starbursts in this galaxy population. Instead, we posit that the fast quenching of massive galaxies at this epoch occurs preferentially in galaxies with pre-existing compact structures. 








§ INTRODUCTION
 
It is well established that galaxies can be divided into two categories in terms of their star-formation activities: star-forming galaxies, whose star-formation rates (SFRs) scale with their stellar masses; and quiescent galaxies, which form very few stars relative to their existing stellar masses <cit.>. Furthermore, as revealed by extra-galactic surveys in recent years <cit.>, quiescent galaxies are more compact than their star-forming counterparts at fixed stellar masses across all epochs up to z∼ 4. Such a bimodal distribution in star-formation activities and structures among galaxy populations implies a connection between the structural transformation and the shutting off of star formation in those galaxies. 

It has become increasingly evident that the timescales over which galaxies quench can be subdivided into two modes, which dominate the transformations at different cosmic epochs. Galaxies can shut down slowly by gradually assembling their stellar masses and exhausting their gas reserve, which is common at low redshifts or they can decrease in star formation rates within just millions of years. The latter “fast mode" typically occurs after an extreme starburst and is more prevalent at higher redshifts <cit.>. Simulations have shown that gas-rich dissipative mergers or interactions can induce both central starburst and compaction of structure in galaxies <cit.>, which can be later followed by rapid quenching if paired with active galactic nuclei (AGN) feedback or starburst-driven winds <cit.>. Alternatively, a number of compact star-forming galaxies exist at high redshifts <cit.> and their number densities decrease over cosmic time, along with the increase in quiescent galaxies' number densities <cit.>. This suggests that morphological transformation could have already occurred during the star-forming phase for some galaxies. These plausible scenarios point out ways to link morphological compaction to the cessation of star formation among the fast-quenching galaxies. However, it is still under debate what exact physical mechanisms affect the rapid suppression of star formation and whether those physical mechanisms are universally dominant. Hence, placing observational constraints on the properties of galaxies on the fast quenching track is vital to our understanding of how exactly their quenching process relates to their morphological transformation. 

Post-starburst galaxies (PSBs), which are also known as “K+A" galaxies, provide us with a unique view of this fast-quenching process. Typically, these galaxies experienced a dramatic drop in star formation in the most recent 1   Gyr and are not currently forming stars. As a result, their spectra are dominated by the flux from late-type B and type A stars <cit.>. In common practice, PSBs are selected by methods that aim to identify spectral energy distribution (SED) features that represent such unique stellar population composition, which includes searching for the existence of strong Balmer absorption along with a lack of nebular emission lines <cit.>, “K+A" template fitting <cit.>, photometric selection methods (UVJ color space <cit.>; super colors <cit.>, or unsupervised machine learning techniques <cit.>). A detailed review of PSBs selection methods can be found in <cit.>.

To date, structural studies of PSBs have yielded interesting insights into their evolutionary histories. At intermediate redshifts (z∼0.7), studies have shown that PSBs have systematically smaller half-light radii than the general population of quiescent galaxies at the same mass and the same epoch <cit.> while having similar central densities as quiescent galaxies <cit.>. In addition, spatially-resolved slits or integral field unit spectra have revealed younger stellar population in the center than outskirts in PSBs at both intermediate redshifts <cit.> (z∼0.7) and local universe <cit.>. In consonance with these findings in PSBs, starburst components of small sizes (∼ 0.1  kpc) have been observed in massive starburst galaxies <cit.>. These results jointly favor central starbursts as the mechanism of the morphological transformation in the fast quenching path. On one hand, the studies of the most massive PSBs (log(M_*/M_⊙) > 10) at intermediate redshifts found no significant color or age gradients <cit.>. On the other hand, negative color gradients are found in both typical star-forming and quiescent galaxies <cit.>, suggesting that the half-mass radii are smaller than the half-light radii in these galaxies. If PSBs evolve from those typical star-forming galaxies, the central starburst can drastically lower the mass-to-light ratio in the center of the galaxy and potentially explain the flattening of color gradients. By incorporating quenching time inferred from spectroscopy, <cit.> examine the PSB size as a function of t_q. They found that PSBs at z ∼ 0.7 were compact and their sizes hardly evolved relative to the coeval quiescent galaxies in the first 1  Gyr since quenching. This result suggests that the starburst occurred at a spatial scale comparable to the half-light radius of the PSB progenitors.  Taken together, the physical conditions necessary to produce fast quenching may be different for the progenitors of PSBs in different mass regimes at earlier epochs. For those massive PSB galaxies, their star-forming progenitors would already have been compact. 

Since the PSB number densities only start to increase at z>1 <cit.> and fast-quenching likely dominates the earlier universe, we need to further study PSBs at earlier cosmic times to better understand the role of fast-quenching in galaxy evolution. Firstly, high signal-to-noise rest-frame optical spectra are needed to pinpoint the time since quenching (t_q) while high-resolution imaging is required for precise characterization of morphologies. Given the rarity of massive (log(M_*/M_⊙) > 10.8) PSBs at z>1 <cit.>, only a handful of objects have been studied with deep spectroscopy at these redshifts in the full CANDELS/3D-HST fields <cit.>. However, the recent Dark Energy Spectroscopic Instrument <cit.> Survey Validation dataset, which was released publicly as a part of the Early Data Release <cit.> and contains 1-5 hour integrations of ∼20000 LRGs from z=0.4-1.3, has enabled us to select PSBs by their Balmer absorption strength (H_δ, a > 4 Å) and resulted in an order of magnitude increase in the number of spectroscopically confirmed post-starburst galaxies at z>1 <cit.>. While the DESI Legacy Survey imaging <cit.> that this sample was selected on is insufficiently high-resolution to perform structural analysis, HST data (HST SNAP 17710, PI: D. Setton) for 171 out of the (H_δ, a > 4 Å) 409 galaxies at z>1 are recently obtained and allow us to perform joint analysis of the structures and star formation histories for a previously unprecedented sample.

In this work, we study the sizes, structures, and merger signatures of 171 PSBs selected from 1<z<1.3 DESI Luminous Red Galaxies, using WFC3/F110W imaging from HST in conjunction with star-formation history derived from DESI spectroscopy. This represents an order of magnitude increase in the sample size of massive (log(M_*/M_⊙) > 10.8) recently quenched galaxies at z>1. The statistical power of such a sample size enables us to robustly characterize these galaxies' morphology and test whether there's any structural transformation within the population of fast-quenching galaxies in the first 1  Gyr since they became quiescent. In Section <ref>, we describe the target selection and reference sample from the 3D-HST survey. We describe our methodology for size measurements and structural analysis in Section <ref> and the results of this work in Section <ref>. In Section <ref>, we discuss the results and their implication to the fast quenching pathway of galaxy evolution. Throughout this paper, we assume 
 a flat Λ CDM cosmology with Ω_Λ = 0.69, Ω_m = 0.31 and H_0 =  67.66   km s^-1 Mpc^-1 as reported in <cit.> and quote AB magnitudes. All reported effective radii (R_e) are measurements of the semi-major axis and are not circularized. 



§ DATA
 


 §.§ The DESI Survey Validation Luminous Red Galaxy Spectroscopic Sample
 



The galaxies in this work were observed as a part of the Luminous Red Galaxy component of the DESI Survey Validation Sample <cit.>, which became public as a part of the DESI Early Data Relase <cit.>. We note that the sample selection of DESI Validation Sample LRG is limited by a z-band magnitude cut of z_fiber<21.6 and biased towards brighter objects <cit.> at z>0.8. The Dark Energy Spectroscopic Instrument (DESI) is a robotic, fiber-fed, highly multiplexed spectroscopic surveyor that operates on the Mayall 4-meter telescope at Kitt Peak National Observatory <cit.>. DESI, which can obtain simultaneous spectra of almost 5000 objects over a ∼ 3^∘ field <cit.>, is currently conducting a five-year survey of about a third of the sky. This campaign will obtain spectra for approximately 40 million galaxies and quasars <cit.>.

The goal of the Dark Energy Spectroscopic Instrument (DESI) is to determine the nature of dark energy through the most precise measurement of the expansion history of the universe ever obtained <cit.>. DESI was designed to meet the definition of a Stage IV dark energy survey with only a 5-year observing campaign. Forecasts for DESI <cit.> predict a factor of approximately five to ten improvement on the size of the error ellipse of the dark energy equation of state parameters w0 and wa relative to previous Stage-III experiments. 

The sheer scale of the DESI experiment necessitates multiple supporting software pipelines and products, which include significant imaging from the DESI Legacy Imaging Surveys (; Schlegel et al. 2024 in preparation), an extensive spectroscopic reduction pipeline <cit.>, a template-fitting pipeline to derive classifications and redshifts for each targeted source (Redrock; Bailey et al. 2024 in preparation, and for the special case of QSOs, ), a pipeline to assign fibers to targets (Raichoor et al. 2024 in preparation), a pipeline to tile the survey and to plan and optimize observations as the campaign progresses <cit.>, an online exposure time calculator (Kirkby et al. 2024 in preparation), a pipeline to select targets for spectroscopic follow-up (desitarget; ), and Maskbit to generate “clean" photometry <cit.>. 

We make use of the joint spectrophotometric spectral energy distribution (SED) fits from <cit.>, which modeled the spectro-photometric data for the entire Survey Validation LRG sample at 0.4<z<1.3. The data included the Milky Way extinction-corrected g/r/z/W1/W2 photometry and the spectra covering 5800 Å < λ_obs < 9824 Å, in which the resolution R   (λ/Δλ) varies from ∼ 3200 to 5100. Briefly, this spectro-photometric fitting used the Bayesian stellar population synthesis code  <cit.>, assuming a non-parametric star-formation history (SFH) optimized for fitting post-starburst galaxies <cit.>. The fits assumed a <cit.> initial mass function and used the mass–metallicity prior described in <cit.>.  We refer the reader to <cit.> for further details.

In Figure <ref>, we show an example of the spectro-photometric fitting. Using the derived SFH, we calculated the fraction of mass formed in the most recent 1  Gyr (f_1Gyr) and the time since quenching (t_q). We define t_q as the time elapsed since the last time the specific star formation rate (sSFR; SFR divided by stellar mass) dropped below 10^-10 yr^-1. This sSFR threshold was chosen such that new stellar mass formed is insignificant relative to the existing stellar mass, which is commonly used to define passive galaxies in literatures <cit.>. We report the uncertainties of t_q for each object as the time when 16 or 84 percent of the posterior drop below the aforementioned sSFR threshold. We note that 32 of the total 171 H_δ, a > 4 Å galaxies are determined to be currently star-forming based on their sSFR. These galaxies are excluded from the subsequent analysis but will be included in the catalog that accompanies this paper.




 §.§ HST WFC3/F110W Imaging


Because the resolution (pixel scale ∼ 0.262") and seeing of the ground-based DESI Legacy Survey Imaging <cit.> is insufficient to measure robust sizes in compact quiescent galaxies at z>1 <cit.>, we instead utilize HST/WFC3 F110W imaging that was measured as a part of HST/SNAP 17110 (PI: D. Setton). The statistical sample program obtained imaging for a random subset of the 409 DESI Survey Validation LRGs with equivalent width H_δ,a>4  Å, empirically selecting for all galaxies which may show signs of A-star or early B-star dominated spectra <cit.>. F110W was selected as the imaging band to maximize signal-to-noise in the 23.3 minute exposures while also observing at red enough wavelengths (λ_rest∼5000 Å) to measure light-weighted sizes that are reasonable proxies for the stellar mass-weighted sizes. In total, 182 galaxies in our sample were observed, however, the telescope lost locking on the acquired guide star and drifted during the exposure for 17 galaxies, resulting in artifacts in their final images. We remove 9 objects in the sample that have these artifacts in all four pre-drizzled single-shot images. For the remaining affected objects, we recombine the pre-drizzled single-shot images that are unaffected and perform analysis on the reprocessed image[Two additional objects are removed from this sample due to their spectra being low signal-to-noise and we cannot infer their star formation histories properly.].

We obtained images using a 4-point dither pattern using the WFC3-IR-DITHER-BOX-MIN pattern to achieve optimal PSF-sampling, and observed using the MULTIACCUM mode with NSAMP=12 and SAMP-SEQ=STEP50, with 349.2 seconds/integration for a total integration time of 1396.9 seconds. The spatial resolution of these images is around 0.13”, which translates to ∼ 1  kpc at z∼ 1. In Appendix <ref>, we present a selected gallery of image cutouts, demonstrating the combination of sensitivity and resolution across the full range of stellar masses in our sample.

The PSFs on the HST/WFC3 IR detector are well characterized and stable, but vary spatially across the detector.  This spatial variation is described by a 3x3 array of empirically constructed PSF models in <cit.>. Since the target galaxies are always centered in the imaging, we choose the center PSF in the 3x3 array to best represent the PSF of the target galaxy in each single-shot image. As these single-shot images are processed through  <cit.> to make the final image products on which we perform the galaxy structure fitting, the empirically constructed PSFs are further distorted by the drizzle pattern in the image pipeline. To account for such distortion, we inject the center empirical PSF stamp into each pre-drizzled single-shot image such that the target galaxy is replaced by the empirical PSF. We then repeat the drizzling processes on these PSF-injected images, using the same  inputs as documented in the drizzle log files. We finally extract the PSFs, which have captured the drizzle patterns, from the re-drizzled final outputs and use them for the aforementioned structure fitting analysis.



 §.§ 3D-HST Comparison Sample

We choose the galaxies in the 3D-HST survey <cit.> to contextualize the size and structures of PSBs in this sample relative to the coeval star-forming and quiescent galaxies, given the availability of its auxiliary morphology catalog based on HST imaging data. We select only the 3D-HST galaxies that are within the same stellar mass (log(M*/M_⊙) > 10.65) and redshift (1<z<1.3) range as the PSBs in this sample, using the stellar mass derived with  <cit.> and photometric redshifts in <cit.>. The photometric redshifts of 3D-HST galaxies are derived with multiple photometric bands and are sufficiently accurate, with a normalized median absolute deviation <2.7% when compared to the available spectroscopic redshifts in the fields. Since the angular diameter distance does not vary drastically at these redshifts (1<z<1.3), the systematic errors in derived physical sizes are small and the comparison of physical sizes is not affected significantly by the difference between photometric and spectroscopic redshifts. We include their best-fitting Sérsic parameters measured in the F125W filter from the catalog of <cit.>, which are the closest to the wavelength range of the F110W filter among the available catalog data. We further omit the galaxies flagged as bad fits. We note that although these best-fitting parameters were measured using a different code (), we find no systematic biases between fits done with Pysersic and Galfit, see details in Section <ref> and Appendix <ref>.
We separate star-forming and quiescent galaxies in the 3D-HST sample with the UVJ rest-frame colors derived in <cit.> and the color cut in <cit.>.

The post-starburst galaxies in this sample have high stellar masses (10.65 < log(M_*/M_⊙) < 11.8), a regime where the number densities for PSBs, quiescent and star-forming galaxies are low <cit.>. Due to the size limitation of the 3D-HST survey footprint and the relative scarcity of objects in this mass range, we lack the number of high stellar mass quiescent galaxies or star-forming galaxies to assemble a mass-matched coeval comparison group for the PSBs in this sample. We instead choose to extrapolate the properties of the quiescent and star-forming population as a smooth function of stellar mass in the subsequent analysis. We divide the quiescent galaxies and star-forming galaxies into six mass bins over the range log(M_*/M_⊙): 10.65-11.8, with each bin to be roughly ∼ 0.2   dex wide. We calculate the median sizes of star-forming galaxies and quiescent galaxies in each bin and the corresponding median errors via the bootstrapping method implemented by standard routine in  <cit.>. We fit a power-law scaling relation similar to the one in <cit.> to the median stellar mass and median sizes:

    R_e/(kpc) = A(M_*/10^11.1M_⊙)^α,

where the best-fitting slope α and normalization A are determined via Markov chain Monte Carlo (MCMC) sampling implemented by  <cit.>, assuming a likelihood function:

    ln p(R_e|M_*,σ,A,α) = 
    
         -1/2∑( (R_e/(kpc)- R(M_*,A,α))^2/σ^2),

where R(M_*,A,α) is Equation <ref> and σ are the errors on median R_e. In the following figures, we visualize the errors in our fitted relation by drawing the 16th and 84th percentile of the best-fit A and α parameter posterior distributions and plotting the corresponding scaling relationship. We extrapolate the mass-central density relation similarly for quiescent galaxies and star-forming galaxies.



§ STRUCTURE FITTING
 


 §.§ Size Measurements
 

 

We use  <cit.>, a Bayesian structural fitting tool, to fit Sérsic profiles to the galaxies in this sample. We use  <cit.> to identify and deblend sources in each image and generate corresponding segmentation maps. Based on the segmentation maps, we create masks for sources that are either 2 mags dimmer than the target galaxy or visually identified as a point source. We enlarge the masks by 3 pixels through kernel convolution in standard  routines, in order to fully mask light from all interloping sources. We assume a single Sérsic profile for each unmasked source and a flat sky background, which we fit simultaneously. 

We perform structural fitting with the same single Sérsic profile setup using  <cit.> as a cross-check to the measurement we obtained with . We find no significant systematic biases in the measurements of these two fitting routines. (median (log(R_e, pysersic)-log(R_e, GALFIT)) ∼ 0.03   dex) A detailed comparison of results from the two fitting routine is included in Appendix <ref>. In the subsequent analysis of this paper, we choose to report the PSB morphological properties measured by . 

We measure the fraction of flux enclosed in the center region (an ellipse with the best-fitting axis-ratio and a semi-major axis of 1kpc) for all galaxies using their best-fitting Sérsic profiles. Assuming a uniform mass-to-light ratio, we compute the fraction of stellar mass contained in the same area. Then we derive the stellar mass surface density within one kiloparsec (Σ_ 1kpc) for these galaxies: 

    Σ_ 1kpc = M_*f_ 1kpc/f_ tot/π( 1 kpc)^2(b/a).




 §.§ Accounting for Deviations from Sérsic Profiles

We find symmetric low surface brightness features (eg., blobs, strips) close to the target galaxies in several cases, through visual inspection of image cutout and modeling residuals, see Appendix <ref> for examples. The presence of these features in some galaxies suggests that they are tidally disrupted by recent merger events or have nearby companions. Given that these faint features are close to or completely blended in the target galaxies, we could not properly extract them in the segmentation map or mask them during the modeling process. As a result, the fitting routine can be driven to compensate for the additional flux brought by these features in the galaxy outskirts, biasing the half-light radius and/or the Sérsic index. In addition, the morphology of post-starburst galaxies may not necessarily follow a perfect single Sérsic profile. The systematic choice of using a single profile in modeling can result in bias in the subsequent analysis.

We follow the procedure in <cit.> to account for the impact of any non-single-Sérsic flux components in our size measurements. After running the structural fitting routines described in Section <ref>, we extract the model residual and apply the same mask used in the fitting to the residual for each instance. We then perform annular photometry on the masked residual, using the best-fitting x-y position, angular position, and axis ratio from each 2d Sérsic model, to extract the radial profile of residual flux. We add this residual profile (PSF-convolved) back into the analytical 1D model profile (non-PSF-convolved). Finally, we measure the half-light radius of the combined profile in 1D space through numerical integration.

In general, single Sérsic profiles accurately capture the 2D light distribution of the target galaxies in this sample, even for those that are tidally disrupted. In Figure <ref>, we show the example fit for a galaxy with obvious tidal disruption features. For this object, the radial surface brightness residuals are minimal (max {μ_obs - μ_model} < 0.2   mags/arcseconds^2) and the corrected R_e do not deviate significantly from the R_e measured directly (|R_e-corrected - R_e-direct | ∼ 0.04   arcseconds). A complete evaluation of the impact of these residual corrections is available in Appendix <ref>.




 §.§ Close Pairs

Five objects in this sample resolve into close pairs of galaxies in the HST imaging. In all cases, the DESI fibers enclose light from both components. We find no evidence of multiple sets of spectral features occurring at different wavelengths in the spectra of these pairs, suggesting that the enclosed components in each pair have similar redshifts or drastically different redshifts. We assume these pairs are interacting systems in the following analysis, though we cannot fully rule out the possibility that these pairs are superpositions of isolated objects. We note that the stellar population properties and star formation histories derived from SED fitting with  in these cases are composite results of both components. We are unable to specify the stellar population properties of each component or to determine whether both components are “Post-Starburst". 

In the aforementioned structure fitting analysis, we model each component separately as a single Sérsic profile. We find that in all cases, the components in each pair have similar magnitudes (differences are within 1 mag). We choose to report the fitting results of the component that is closest to the fiber-pointing for each pair. We expect the reported sizes of these systems to be underestimated and not reflect their true location on the mass-size plane, since we report the stellar mass that is the sum of each pair. Throughout this paper, we denote close pairs with different symbols and derive scaling relations without that sub-sample.




§ RESULTS



 §.§ PSBs Exhibit Compact Sizes
 



On the mass-size (M_*-R_e) plane, galaxies' sizes scale with their masses and different galaxy populations display distinct scaling relations across different epochs. The quiescent mass-size relation is generally steeper and has a lower normalization than that of star-forming galaxies <cit.>, reflecting the more compact morphology of quiescent galaxies. By comparing the sizes (R_e) of PSBs with their star-forming and quiescent counterparts from the same epoch, we aim to empirically constrain the evolutionary pathways of PSBs and shed light on their quenching modes.

In the upper panel of Figure <ref>, we show the effective radius versus stellar mass for the galaxies in this sample in green. The 3D-HST star-forming and quiescent galaxies at (1<z<1.3), for which sizes are measured in the HST/F125W imaging, are plotted in blue and red respectively in the same panel <cit.>. We fit power-law mass-size relations for star-forming galaxies, quiescent galaxies, and PSBs following the procedure described in Section <ref> and show these relations as blue, red, and green dashed lines respectively. In addition, we mark the galaxies identified as having clear tidal or merger features during the visual inspection as green star symbols. The masses used here for star-forming and quiescent galaxy populations are derived through  while assuming a non-parametric SFH <cit.> and have median values about 0.2 dex higher than those in <cit.>. 

We confirm that these massive PSBs have much smaller half-light radii than star-forming galaxies and have sizes similar to, but even smaller than, the quiescent population, as found in several previous studies <cit.>. However, we do note that this comparison is limited by the small number of star-forming galaxies and quiescent galaxies available in the most massive regime (log(M_*/M_⊙) > 11.3) and the extrapolation becomes less reliable. To probe the subtle difference in size between quiescent galaxies and PSBs, we measure the vertical offset of PSBs from the quiescent mass–size relation, same as equation 2 in <cit.>:

    Δ log(R_e) = log(R_e) - log(r̂_e,Q(logM_*/M_⊙)).


In the lower panel of Figure <ref>, we show log(M_*) versus Δ log(R_e) for these PSBs. Overall, the PSBs in this sample are slightly more compact than their contemporary quiescent population, lying systematically ∼ 0.075   dex below the quiescent mass-size relation while the scatter in sizes is ∼ 0.1   dex. If the same analysis is performed with PSB sizes measured with GALFIT, the PSBs then lie systematically ∼ 0.08   dex below the quiescent mass-size relation. Such exceptional compactness is consistent with PSBs at slightly lower stellar masses (10 < log(M_*/M_⊙) < 11.2) at similar redshifts <cit.> and PSBs at lower redshifts (z ∼ 0.7) with similar stellar masses <cit.>.



 §.§ Dense Cores within Post-Starburst Galaxies




Similar to half-light radii, the central densities (Σ_1kpc) of galaxies also scale with masses <cit.>. Typically, the star-forming relation has a lower normalization but a steeper slope than the quiescent relation, reflecting the growth of the bulge during the star-forming phase. In the upper panel of Figure <ref>, we present the central surface mass densities (Σ_1kpc) versus stellar mass for PSBs (greens), star-forming galaxies (blue), and quiescent galaxies (red). The extrapolated star-forming, quiescent, and post-starburst relations are plotted as blue, red, and green dashed lines respectively. The color and marker signs of PSBs are similar to those in Figure <ref>. In the lower panel, we show the distribution of relative central surface mass densities (Δ log(Σ_1kpc)) versus stellar mass. This quantity is defined as the vertical offset of PSBs to the quiescent relation in the logΣ_1kpc-M_* plane, similar to Δ log(R_e).

The PSBs follow a relatively shallow M_* - Σ_1kpc scaling relationship and have a median log(Σ_1kpc) ∼ 10.1. Overall, the median value and scaling relation of PSB central densities are similar to those of quiescent galaxies in this comparison sample(ΔΣ_1kpc, PSB-Quiescent∼ - 0.07   dex), and to those of quiescent galaxies found in the literature <cit.>. Furthermore, these PSBs have much denser inner structures than the star-forming galaxies around the same mass (ΔΣ_1kpc, PSB-SFG∼  0.5   dex). Such similarity between PSBs and quiescent galaxies at z∼ 1.1 is consistent with the PSBs above z∼ 0.7 studied in <cit.> and indicates that the structural differences between the two populations are due to the more extended outskirts of the older quiescent population.



The mass profiles derived in observed space capture and demonstrate any potential deviations from the Sérsic profiles. In the upper panels of Figure <ref>, we show the light-weighted median mass surface density profiles of quiescent galaxies, star-forming galaxies, and PSBs in red, blue, and black solid lines and mark the scatter of their profiles with red, blue dashed lines, and a gray-shaded region. In the lower panels, we show the residual profiles in black and their scatters in gray by subtracting the median quiescent profile or the median star-forming profile from the median PSB profile. To obtain these light-weighted median mass density profiles, we first measure the empirical surface brightness profile with annular photometry. Assuming the mass-to-light ratio is the same along the profile, we then derive the mass-to-light ratio individually for each galaxy by taking the SED-inferred stellar mass over the integrated flux using the brightness profile. Finally, we renormalize the surface brightness profile of each galaxy by the corresponding mass-to-light ratio. For each stacked PSB, we match and stack a quiescent as well as a star-forming 3D-HST galaxy that has similar stellar mass (|log(M_*, PSB) - log(M_*, Quiescent/Star-forming)| < 0.1 dex). Since we do not have enough 3D-HST galaxies on the high mass end for matching, we only choose galaxies at log(M_*/M_⊙) < 11.2 and match 60 sets of galaxies in total. We note that the profiles in this analysis are derived from raw observational data and therefore include the PSF. The overall flux distribution is altered by the PSF and the subsequently derived mass density profiles have lower numerical values at a = 1  kpc than the Σ_1kpc values reported in previous sections, which are derived in non-PSF-convolved space. All the star-forming galaxies and quiescent galaxies matched are from the same redshift interval 1<z<1.3 as this sample. The angular diameter distance does not vary significantly within this redshift interval and the galaxies stacked in this analysis have similar PSF HWHM effective sizes. We overplot the half width at half maximum (HWHM) of the WFC3/F110W imaging PSF, which we derived using the angular diameter distance at the median redshit of this sample (z=1.15), as a gray-shaded region in each panel. The PSF HWHM of F125W imaging is similar to that of F110W. We deem the comparison of the overall mass profile shape to be sufficiently robust for a > 1  kpc for these different galaxy populations.

The comparison of these profiles validates the previous conclusion that both the typical quiescent galaxies and PSBs have much smaller half-light radii and higher central densities than the typical star-forming galaxies. The systematic difference in the light-weighted mass profiles further reveals what is driving the fits of PSBs to have overall smaller sizes than those of quiescent galaxies, despite the fits of these two populations field similar central densities.



 §.§ Other parametric measures of structure





Parameters used in our structure fitting other than half-light radius such as axis-ratio and Sérsic index give us additional insights into the 3D structures of galaxy populations. In Table <ref>, we tabulate the results and uncertainties of measured axis-ratios, Sérsic indices, and half-light radii as well as derived Σ_1 kpc of all PSBs. In the left panel of Figure <ref>, we show the measured projected axis-ratios as a function of stellar mass for Star-forming galaxies (blue), quiescent galaxies (red), and PSBs (green). In the right panel, we over-plot the mass-binned (log(M_*/M_⊙): [10.6-11],[11-11.3], and [11.3-11.8]) medians (filled circles) and population scatter(filled band for the PSB sample, dashed lines for 3D-HST comparison). 

Although incredibly elongated examples exist, the PSBs predominantly have large projected axis ratios (median b/a ∼ 0.8; nearly circular in projection) in all mass bins, which is consistent with the finding in <cit.>. We perform axis ratio modeling to infer the intrinsic 3D shapes of these galaxies, following a similar methodology to <cit.> modified to use a Bayesian framework (using the code the ;  for more details see J. Gibson et al., in press and S. Price et al., in prep.). The modeling suggests that nearly all (≳90%) of these objects are round spheroids rather than disks, in contrast with the wide spread exhibited by randomly-oriented disks amongst the SFG population. Furthermore, given that PSB galaxies differ structurally from both comparison samples, their shapes must either change in transition or the channel only impacts a biased subset of the population. The prevalent large axis-ratios in our fits are not driven by the presence of tidal features; there is no significant distinction in the axis-ratio distribution between isolated and merging PSBs. Overall, our finding of these PSBs being compact spheroids is consistent with previous studies such as <cit.> and <cit.>. 

A number of quiescent galaxies in the comparison sample are elongated in projection (b/a <0.5), suggesting that these galaxies could have experienced color transformation without transforming into spheroids and bias the comparison between quiescent galaxies and PSBs. We note that these galaxies lie systematically ∼ 0.07   dex below the quiescent mass-size relation with a scatter similar to the overall quiescent population. The Σ_1kpc of these galaxies are ∼ 0.2   dex above the median of the entire quiescent galaxy population at fixed masses. However, when removing this population of elongated quiescent galaxies in fitting, the quiescent relation in M_* - Σ_1kpc is only lowered by ∼ 0.1   dex in the low mass regime (log(M_*/M_⊙) < 11.2) and the relation hardly changes at higher masses. Hence, our qualitative conclusions in Section <ref> and Section <ref> would remain similar when considering the bias caused by these galaxies. 

The PSBs in this sample are fitted with relatively high Sérsic indices (n_median∼ 3), similar to quiescent galaxies. However, we do note that the numerical values of Sérsic index can be sensitive to the exact implementation of fitting routine, in the high sersic index regime (n > 2). We choose not to make a quantitative comparison between the Sérsic indices measured for quiescent galaxies () and PSBs ().





§ DISCUSSION



 §.§ Looking backward: How was the compact structure formed?





If the progenitors of these PSBs are the typical star-forming galaxies with extended structures (represented by the star-forming relation in M_*-R_e in Figure <ref>), one likely pathway for them to achieve the compact structures seen in this sample is going through a central starburst phase. If a large fraction of stellar mass (the median f_1Gyr is around 0.4 in this sample) formed within a spatial scale much smaller than the original half-light radius, the centers of the galaxy light profiles would be skewed by the burst. Since the young stellar population is orders of magnitude brighter than the old, the half-light radius of the galaxy would shrink towards the half-light radius of the burst component, assuming a non-negligible burst fraction. The derived star formation histories suggest that the recent bursts also drastically increased stellar masses (∼  0.2  dex, given a typical f_1Gyr∼ 0.4  and a typical log(M_*/M_⊙) ∼ 11.2 ). Thus, such a process could effectively move star-forming galaxies onto the PSB relation.

The formation of compact structures in galaxies due to a central starburst has been seen in several simulation works <cit.>, which can be caused by disruption events due to gas instability in star-forming disks <cit.>. In this case, the dissipative processes, which may be induced by disruptive gas inflow or gas-rich mergers, cause the gas in galactic disks to lose angular momentum, fall towards the inner region of the galaxies, and then trigger a compact burst of star formation in the center. As the stellar surface density reaches a certain physical upper limit <cit.> and the cool gas is depleted at the end of the starburst episode, these galaxies become quiescent if there is no further inflow of cool gas. This scenario is supported by several observational works focusing on a small sample of PSBs at z ∼ 0.8 such as <cit.> and <cit.>, which exhibit inverse age gradients or color gradients that require the superposition of a compact younger stellar population on top of a more extended old stellar population. The compact sizes of starburst components (∼ 0.1  kpc) observed in massive starburst galaxies further bolster this point of view <cit.>.

Alternatively, the PSBs in this sample can be descendants of star-forming massive compact galaxies at higher redshifts, which are morphologically similar to the quiescent massive compact galaxies found at lower redshifts <cit.>. In this scenario, the compact structure is already in place for these galaxies while they are still forming stars. They retain roughly constant sizes and move along a horizontal track until reaching the preferential location in the mass-size plane where they shut off rapidly.

Although, the two physical scenarios would leave clear imprints on the age and therefore color gradients of post-starburst galaxies, their compact sizes and single-band HST imaging hinders such a test. However, the dominance of a central burst on the light profiles would fade on ∼ Gyr timescales, as the M/L ratio of the galaxies would rapidly evolve as the youngest stars die off. Thus, the post-starburst evolution of size and central density relative to their quiescent descendants can also discriminate between the two scenarios. In the case of the central starburst scenario, we expect to see a growth in relative size and a decrease in relative central density as the burst ages. In some extreme cases, the half-light radii growth could be significant (change in log(R_e/kpc) ∼ 0.2 dex) in t_q < 1  Gyr, if the physical scale of the central starburst is smaller than 1 kpc and the burst mass is less than 20% of the total formed mass <cit.>. In contrast, if the star-forming progenitors were already compact, we expect no evolution in relative size or central density in t_q < 1  Gyr. Enabled by the deep spectroscopic data of this sample, we can infer the time since the quenching of every object and statistically examine whether there is a trend in relative size or central density as a function of t_q.


In Figure <ref>, we show the scatter in relative size and central density versus time since quenching (t_q) for PSBs after minor (left, f_1Gyr: [0-0.2]) and major (right, f_1Gyr: [0.2-1.0]) starbursts based on derived burst mass fractions. We note that f_1Gyr is a proxy for the mass formed during the burst period in the SFH, assuming the onset of the starburst episode was within 1  Gyr in the lookback time of these galaxies. This value is likely much lower than the true fraction of mass formed during the burst for those with large t_q. We overplot the median values in each age bin (t_q: [0-0.3],[0.3-0.7], and [0.7-2]) in red. Overall, the median relative size and median central density of these galaxies do not evolve significantly as their youngest, most massive stars fade in the first 1  Gyr, following both major and minor starbursts. The lack of significant growth in sizes or decrease in central densities among these galaxies suggests that the recent burst of star formation likely occurred with a physical radius greater than 1 kpc. This further implies that the bursts occurred in already compact galaxies. Individual PSB can still be growing after quenching, given the scatter in these parameter spaces. However, the scatter is likely dominated by the inherited heterogeneity of the PSB population that they were quenched at different sizes and central densities. 

Although central starbursts within 1 kpc are unlikely the dominant formation pathway for these PSBs, the physical mechanism that drives the rapid quenching of star formation remains elusive. Merger-induced starbursts and the subsequent depletion of cold gas may be able to quench these galaxies abruptly. Previous observational works have unveiled the existence of extreme gas outflows in massive compact starburst galaxies <cit.>, which can even persist in the post-starburst phase <cit.> given that some gas contents can still be retained in the galaxy after rapidly quenching <cit.>. These outflows are driven by stellar radiation pressure and ram pressure contributed by stellar winds from a compact starburst component (∼ 0.1  kpc) and can far exceed the central escape velocities in these galaxies <cit.>. Such extreme stellar feedback is able to expel a critical fraction of the cold gas and quench these galaxies without necessarily invoking AGN feedback <cit.>. But it is unclear whether the extended starburst preferred in this sample could achieve comparable SFR surface densities to power those outflows with sufficient radiation pressure and ram pressure. Additionally, merger-induced starburst can still further funnel gas into the inner region of the galaxies, trigger AGN activities, and quench the star formation with short dynamical times (t_dyn∼ 100  Myrs; ). But it is not sufficient to simultaneously shut off star formation at all radii, especially in the largest (R_e > 4  kpc) galaxies. Given the redshift range of this sample (1<z<1.3), we lack the spectral coverage (5800 Å < λ_obs < 9824  Å) of diagnostic emission line signatures for AGN activities, such as [O III] λ5007/Hβ, and the WISE photometry in our data is not sufficiently deep to identify the existence of AGNs. 

Only a fraction (∼ 40%) of the galaxies are identified as having merger signatures and we find no change in the fraction of mergers with t_q, which suggests mergers are weakly correlated to quenching in these PSBs. Objects with potential faint tidal features are likely misidentified as non-disruptive due to a limited surface brightness sensitivity in these images (μ_max < 22   mags/arcseconds^2), leading to an underestimated merger fraction. Other mechanisms have been proposed to link quenching to the stabilization of gas disk, such as morphological quenching <cit.>, in which the gas disk stabilizes once the galaxy acquires a spheroid-dominated morphology. These mechanisms may explain the globally simultaneous shut-off in these galaxies, but it is not clear whether they could introduce a sufficiently abrupt shutdown to produce the spectra seen in PSBs. Spatially resolved spectra are needed to probe the distribution of stellar ages at different radii in these galaxies. This kind of information will allow us to discover any spatial pattern associated with the rapid decrease of star formation or the lack thereof in future studies, shedding light on the exact mechanism that quenches these galaxies in future studies.

The massive z ∼ 1.1 PSBs in this sample are similar to those at z ∼ 0.8 in <cit.>, who plotted PSB sizes against t_q and did not find any significant changes in PSB sizes since they quenched. The most massive PSBs at z ∼ 1.2 are likely quenched in a similar way as those at z ∼ 0.8. But at the same time, the evolutionary constraints placed on these PSBs seem to raise tension against other works on PSBs at intermediate redshifts, such as <cit.> and <cit.> at z ∼ 0.8, which favor the central starburst scenario. We emphasize that the PSBs in this sample represent a rare population in terms of their stellar mass (log(M_*/M_⊙)_median∼ 11.2), which is about 0.5 dex higher than those in <cit.> and <cit.>. It is reasonable to speculate that these PSBs with contradictory age gradients are descendants of different progenitors, representing variants of the fast-quenching evolutionary pathways that dominate different mass regimes. 

One possible progenitor for the PSBs at higher mass are the compact star-forming galaxies known as “Blue Nuggets" <cit.>. The theory is these galaxies form massive but compact structures through the dissipative compaction of the gaseous disk, similar to the mechanism in a central starburst scenario <cit.>. At z ∼ 1.6 (∼ 1  Gyr before the median redshift of this sample), the number density of compact star-forming galaxies is ∼ 10^-4  Mpc^-3 <cit.>, which is about an order of magnitude higher than the number density of PSBs with the same selection criterion as this sample at z∼ 1.2 (at least 10^-5  Mpc^-3; ). If a subset of these high redshift “Blue Nuggets" experiences an episode of starburst at all radii on top of the existing dense core instead of quenching immediately, they would eventually form galaxies that are consistent with PSBs in this sample. It is unclear whether the drastic star-forming episode can persist, since the gas contents in these “nuggets" are generally low <cit.>, especially in the center <cit.>. Given that these objects are already massive and likely have high halo mass, it's also difficult for cold inflow gas to reach the center of these galaxies. The high gas density at earlier cosmic time may enable sufficient gas inflows that continue to fuel the intense stellar assembly at all spatial scales of those nuggets, after their initial period of starburst. 



 §.§ Looking forward: Minor mergers as the dominant mass growth mode.



Compared to contemporary quiescent galaxies at the same masses, the PSBs in this sample are smaller while having similar central densities, as shown in Section <ref>. At lower redshifts (z ∼ 0.8) where the descendants of these PSBs would reside, the quiescent M_*-R_e relation has a higher normalization than that at z∼ 1.1 <cit.> while the quiescent M_*-Σ_1kpc relation hardly evolves from z∼ 1.1 to z∼ 0.8 <cit.>. If these PSBs stay quenched and indeed evolve into typical quiescent galaxies at lower redshifts, they must increase their sizes while maintaining a constant core structure. Furthermore, we have already shown these PSBs lack any significant growth in relative sizes in the first 1  Gyr since quenching as their recently formed stars fade. This suggests that the mechanism that gives these PSBs the typical quiescent structure is likely some form of mass growth in the outskirts, which operates at a longer timescale than 1  Gyr, rather than passively fading. 

In Figure <ref>, the mass-matched quiescent galaxies appear to have higher median stellar mass densities than the contemporary PSBs starting from a = 6  kpc, and the differences between their light-weighted mass density profiles increase at larger radii. In comparison, the quiescent galaxy and PSB mass profiles become relatively similar within a = 3   kpc before reaching the region dominated by the PSF. One potential mechanism that can explain the evolution from the PSB profiles towards the quiescent galaxy profiles in Figure <ref> is ex-situ growth through dry minor mergers <cit.>, in which the radius of the merged system can grow as the square of the change in mass when the mass of the in-falling dwarf is insignificant in contrast to the main galaxy.

There are several caveats to the interpretation presented here. Firstly, we derive mass density profiles through the light distribution in only one filter for both PSBs (F110W) and quiescent galaxies (F125W) [Given the median redshift of this sample (z_median = 1.15), the difference in rest-frame effective wavelengths between F110W and F125W is small (less than 500Å) and the intrinsic color gradients between these two filters are negligible for our comparison.], assuming there is no color gradient or change in mass-to-light ratio in the radial direction. For PSBs in this sample, this assumption is supported by several studies that find flat color gradients in some massive PSBs at intermediate redshifts <cit.>. In addition, such an assumption is self-consistent with the findings that there is no change in their relative sizes as they fade. The lack of growth suggests the stars that dominate the light from those galaxies are evenly distributed along the radial direction, consequently resulting in relatively flat color gradients. Meanwhile, both star-forming and quiescent galaxies tend to have color gradients that are bluer at large radii <cit.>. These quiescent galaxies could have smaller mass-to-light ratios in the outskirts therefore overestimated mass density profile at larger radii. But the existence of negative color gradients in quiescent galaxies can be also explained by the minor mergers, which would bring in younger ex-situ formed stellar populations in the outskirts on top of the flat color gradient of PSBs. The difference in color gradients between PSBs and the typical star-forming galaxies implies that the progenitors of these PSBs are not among the typical star-forming galaxies if the central starburst was not the dominant evolutionary channel.

Secondly, the mass cut we implement in this analysis excludes the most massive PSBs and the majority of the subset with tidal features. The increased fraction of tidally disrupted features in those PSBs can scale up their mass density profile in the outskirts and compensate for the difference we see in Figure <ref>. However, we may have missed a potential tidally disrupted population in the low mass regime due to the detection limit in these images, if the surface brightness of disruption features scales with the overall brightness of these galaxies. 

Furthermore, the evolution pathways of galaxies have been shown to have complex branching structures <cit.>, reflecting the stochasticity in physical processes that drive their evolutionary behaviors. These PSBs may only represent a biased subset of the progenitors of quiescent galaxies. It is still possible that the descendants of these PSBs only make up the fraction of quiescent galaxies that are relatively compact. If the galaxies through other quenching channels join the quiescent population with larger sizes, they can increase the median size of the overall quiescent galaxy population. Alternatively, if some of these PSBs stay isolated from any merger events or rejuvenation, they can make up the population of red nugget relics discovered at lower redshift (z<1; ) and in the local universe <cit.>, given their similar locations on the mass-size plane. To fully assess the complicated evolutionary pathways of fast-quenching galaxies and disentangle them from progenitor biases, a robust study of the number densities of these galaxies as a function of cosmic time is required. As the DESI SV LRG sample selection is incomplete at z>0.8, we are currently unable to robustly measure the number density of these PSBs <cit.> and draw connections between them and any lower redshift descendent candidates. 



§ CONCLUSION


To shed light on the fast-quenching pathway of galaxy evolution, we spectroscopically select and study a sample of 171 massive (log(M_*/M_⊙) ∼ 11) PSBs at 1<z<1.3 from DESI LRG. Using HST WFC3/F110W imaging, we fit these galaxies as single Sérsic profiles with a Bayesian fitting framework . We obtain robust estimates for their sizes as well as other parametric structural measures, such as Sérsic index and axis ratio. This represents an order of magnitude increase in the number of spectroscopically confirmed PSBs with robust structural measurements above z>1. Combining these results with stellar population properties inferred from SED fitting via  and comparing them with those of quiescent and star-forming galaxies in 3D-HST survey, we reach the following conclusions:


  * We robustly demonstrate that massive post-starburst galaxies are extremely compact at 1<z<1.3. At similar stellar mass, their half-light radii are systematically ∼ 0.1   dex smaller than those of quiescent galaxies and ∼ 0.4   dex smaller than those of star-forming galaxies. 


  * PSBs in this sample have central surface mass densities similar to quiescent galaxies (log(Σ_1kpc) ∼ 10.1), suggesting that the inner structures of PSBs are not significantly affected by any potential evolutionary mechanism after quenching. 


  * These PSBs are round in projection (b/a_median∼ 0.8), suggesting that they are primarily spheroids, not disks, in 3D. 


  * We find no trends in median PSB sizes and central densities relative to the quiescent counterparts (Δ log(R_e) and Δ log(Σ_1kpc)), within time since both major and minor bursts of star formation. This suggests that the previous episode of star formation was not centrally concentrated, but occurred throughout the galaxy.



The PSB number density increases with redshift and the majority of the PSB population is located at earlier times <cit.>, suggesting that the rapid quenching path played a more dominant role in galaxy evolution in the early universe. Hence, it is necessary to find and characterize PSBs at the highest redshift possible. It is difficult to acquire large spectroscopic samples at higher redshifts, where objects are more significantly dimmed and require longer spectral integration time. The sample selection of DESI Validation Sample LRG, which enabled this work, is limited by a z-band magnitude cut of z_fiber<21.6 and biased towards brighter objects <cit.> at z>0.8. In the future, the Prime Focus Spectrograph <cit.> surveys would enable us to spectroscopically select PSB samples that are magnitude complete and to estimate PSB number densities robustly at z>1.  In addition to this work, various samples at 0.5<z<2 <cit.> have reported PSB median sizes that are consistently smaller than median quiescent galaxy sizes at fixed mass in <cit.>. But it is unclear whether this trend will continue at higher redshifts (z>2). To study the sizes and structures of these rare objects at high redshifts, we need space-based images with a wavelength coverage redder than HST and a survey area wider than JWST. The Roman space telescope, which is designed to survey large areas at long wavelengths, would be ideal to provide IR images with high resolution and stable PSF for studying the morphology of high-z PSBs. Combining these data in future studies, we will be able to characterize PSBs around the cosmic noon and constrain the fast quenching path of galaxy evolution at earlier cosmic times.



§ ACKNOWLEDGEMENTS

RB acknowledges support from the Research Corporation for Scientific Advancement (RCSA) Cottrell Scholar Award ID No: 27587 and from the National Science Foundation NSF-AAG grant #1907697 and NSF-CAREER grant # 2144314.

D.S. gratefully acknowledges the support provided by The Brinson Foundation through a Brinson Prize Fellowship grant for this work. The published results were also funded by the Polish National Agency for Academic Exchange (Bekker grant BPN/BEK/ 2021/1/00298/DEC/1) and the European Union's Horizon 2020 research and innovation program under the Maria SkłodowskaCurie (grant agreement No. 754510).

The data points used to create the figures in this work are available at 10.5281/zenodo.12693678.

This material is based upon work supported by the U.S. Department of Energy (DOE), Office of Science, Office of High-Energy Physics, under Contract No. DE–AC02–05CH11231, and by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract. Additional support for DESI was provided by the U.S. National Science Foundation (NSF), Division of Astronomical Sciences under Contract No. AST-0950945 to the NSF’s National Optical-Infrared Astronomy Research Laboratory; the Science and Technology Facilities Council of the United Kingdom; the Gordon and Betty Moore Foundation; the Heising-Simons Foundation; the French Alternative Energies and Atomic Energy Commission (CEA); the National Council of Humanities, Science and Technology of Mexico (CONAHCYT); the Ministry of Science, Innovation and Universities of Spain (MICIU/AEI/10.13039/501100011033), and by the DESI Member Institutions: <https://www.desi.lbl.gov/collaborating-institutions>. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the U. S. National Science Foundation, the U. S. Department of Energy, or any of the listed funding agencies.

The authors are honored to be permitted to conduct scientific research on Iolkam Du’ag (Kitt Peak), a mountain with particular significance to the Tohono O’odham Nation.

This research is based on observations made with the NASA/ESA Hubble Space Telescope obtained from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract  NAS 5–26555. These observations are associated with the program HST/SNAP 17110.

The specific observations analyzed can be accessed via [https://doi.org/10.17909/scn6-hs39]https://doi.org/10.17909/scn6-hs39.

Support for program HST/SNAP 17110 was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555.


Mayall (DESI), 
HST (WFC3)


Astropy <cit.>, AstroDrizzle <cit.>, GALFIT <cit.>, Scipy <cit.>, SEP <cit.>, Photutils <cit.>, Prospector<cit.>, pysersic <cit.>
 

aasjournal





§ CUTOUTS OF SELECTED GALAXIES IN THIS SAMPLE

In Figure <ref>, we present a gallery of cutouts for selected galaxies in this sample. Each cutout is randomly selected from the ensemble of objects occupying the same parameter space in the size-mass plane. Cutouts are arranged according to their location in the PSB mass-size relation. We re-scale the pixel values in each image through an algorithm that mimics the implementation of log color scale in DS9, to better visualize the structures with low surface brightness. Galaxies identified as “interacting" are framed in red, “close pairs" are framed in blue, and “isolated" are in black.




§ SYSTEMATIC DIFFERENCES BETWEEN STRUCTURAL PARAMETERS FITTED WITH  AND 

We show the systematic impact on half-light radii when applying the residual correction in <cit.> in the upper left panel of Figure <ref>, using  results. Correcting the half-light radii with residuals overall increases them by 0.03 dex in log(R_e). We compare the systematic differences between the residual-corrected half-light radii (upper right), axis ratios (lower left), and the derived central surface mass densities (lower right) inferred via  and . The median value of half-light radii (residual-corrected) via  is 0.01 dex in log(R_e) higher than that of . The median value of axis ratios via  is almost identical to that of . The median value of central densities derived with parameters via  is 0.01 dex higher than that of . Overall, the systematic difference between the results produced by  and  is small and has no significant impact on the qualitative conclusions in this work. The half-light radii measurements also do not deviate significantly after applying the residual correction and the deviations from Sérsic profiles in these galaxies do not affect our conclusions qualitatively.





 Paper Title: An ALMA survey of submillimetre galaxies in the Extended Chandra Deep Field South: an unbiased study of SMG environments measured with narrowband imaging.



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Chemical abundance gradients of organic molecules within a protostellar disk
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§ ABSTRACT 
Submillimetre galaxies (SMGs) are some of the most extreme star-forming systems in the Universe, whose place in the framework of galaxy evolution is as yet uncertain. It has been hypothesised that SMGs are progenitors of local early-type galaxies, requiring that SMGs generally reside in galaxy cluster progenitors at high redshift. 
We test this hypothesis and explore SMG environments using a narrowband VLT/HAWK-I+GRAAL study of  and  emitters around an unbiased sample of three ALMA-identified and spectroscopically-confirmed SMGs at z ∼ 2.3 and z ∼ 3.3, where these SMGs were selected solely on spectroscopic redshift.
Comparing with blank-field observations at similar epochs, we find that one of the three SMGs lies in an overdensity of emission-line sources on the ∼4 Mpc scale of the HAWK-I field of view, with overdensity parameter δ_g = 2.6^+1.4_-1.2. A second SMG is significantly overdense only on ≲ 1.6 Mpc scales and the final SMG is consistent with residing in a blank field environment. 
The total masses of the two overdensities are estimated to be 
log(M_h/ M_⊙) = 12.1–14.4, 
leading to present-day masses of 
log(M_h,z=0/ M_⊙) = 12.9–15.9. 
These results imply that SMGs occupy a range of environments, from overdense protoclusters or protogroups to the blank field, suggesting that while some SMGs are strong candidates for the progenitors of massive elliptical galaxies in clusters, this may not be their only possible evolutionary pathway.





galaxies: star formation – galaxies: photometry – galaxies: evolution – submillimetre: galaxies









§ INTRODUCTION



Since their discovery more than two decades ago, submillimetre galaxies <cit.> have proven to be important laboratories for exploring galaxy formation and evolution. 
These galaxies are identified in (sub)millimetre surveys and have typical infrared (IR) luminosities of L_ IR∼ 10^12-13 L_⊙ corresponding to star formation rates of ∼10^2-3  yr^-1 <cit.>. 
They are massive, with stellar masses of M_∗∼ 10^10-11   <cit.>, 
dust masses of ≳10^8   <cit.>, 
and cold gas masses of ∼10^11  <cit.>, 
but with gas depletion times of just a few hundred Myr <cit.>. 
The SMG redshift distribution peaks at z ∼  2.5 <cit.>, making these massive dusty galaxies the most intense star-forming systems in the Universe during its peak epoch of star formation <cit.>. 
SMGs contribute up to ∼20 percent of the cosmic star-formation rate density at z ∼ 2 <cit.>.

The extreme properties of SMGs has long made them a good test of galaxy evolution models <cit.>, yet questions about their evolution and role in the evolution of other galaxies remain. 
SMGs have similar properties to those expected of the progenitors of local massive elliptical galaxies, which formed most of their stars in short bursts at z ≳ 2 <cit.>. Indeed, the dust emission from SMGs is typically compact <cit.>, which is consistent with a scenario in which a gas-rich z≳2 galaxy undergoes a compact starburst, leading to a compact quiescent galaxy, which eventually evolves into a local elliptical galaxy <cit.>. 
Since local ellipticals are predominantly found in galaxy clusters <cit.> then if SMGs are indeed a progenitor phase in their formation, then it is expected that SMGs should reside in early galaxy clusters, or `protoclusters', at  z ≳ 2. 

Galaxy protoclusters <cit.> are typically defined as structures that will collapse and virialise to form a galaxy cluster by z = 0. Simulations have shown that in a Λ CDM universe, protoclusters form hierarchically at the highest density regions of the matter distribution in the universe <cit.> at z ∼ 4–6 <cit.>. As such, protoclusters are characterised by overdensities of galaxies relative to the average galaxy density in the coeval blank field. Conversely to their present-day descendants, galaxies in a protocluster are generally not bound to a single halo; they instead occupy large structures extended over megaparsec (Mpc) scales, with the main halo containing as little as 20 percent of the member galaxies <cit.>.

Unfortunately, observationally identifying protoclusters is challenging. Methods of detecting galaxy clusters from their X-ray emission <cit.> or by searching for their imprint on the cosmic microwave background at millimetre wavelengths <cit.> via the Sunyaev-Zel'dovich effect <cit.> are rendered impractical due to the lack of a hot intracluster medium (ICM). Similarly, searches for high concentrations of passive early-type galaxies occupying a tight `red sequence' in colour-magnitude space <cit.> become ineffective since the stellar populations of galaxies in protoclusters typically have not evolved sufficiently for a significant 4000 Å break to be detected. 
Consequently, the majority of protocluster surveys resort to searching for overdensities of galaxies at high redshift. Such searches depend on the existence of accurate redshift information across large cosmological volumes, and several protoclusters have been discovered serendipitously through large spectroscopic surveys <cit.>. In lieu of expensive large-scale spectroscopic observations, an alternative method is to use wide-field narrowband photometric surveys to search for overdensities of galaxies with strong emission lines at a particular observed-frame wavelength <cit.>. 

Whether SMGs commonly inhabit protoclusters or protocluster-like environments is as yet uncertain. Several examples of SMGs residing in protoclusters have been documented <cit.>, but these systems were selected for detailed follow-up because of prior evidence of high galaxy densities. I.e. they comprise a highly biased subset and therefore cannot be used to make inferences about the general SMG population. 

Clustering studies have been used to obtain statistical measurements indicative of the whole SMG population. Results from single-dish clustering measurements suggest that on average SMGs at z ∼ 2.5 reside in dark matter halos of mass ∼10^13  <cit.>. 
This is marginally lower than expected for the progenitors of massive ellipticals, and implies that SMGs are instead more likely to evolve into 2–3L^∗ galaxies in groups and small clusters. 
However, these halo mass measurements have typical uncertainties of ∼0.5 dex due to the difficulties associated with obtaining accurate photometric redshifts for SMGs. Furthermore, these clustering measurements relied on the statistical identification of optical/near-IR counterparts to submillimetre sources detected in low-resolution single-dish surveys, which are incorrect in ∼30 percent of cases and incomplete in a further ∼30 percent <cit.>. 
More recently, <cit.> and <cit.> measured the clustering of SMGs which had been followed up interferometrically with the Atacama Large Millimetre/submillimetre Array (ALMA). 
Using a small sample of 50 ALMA-identified SMGs with spectroscopic redshifts and employing forward modelling to correct for incompleteness, <cit.> estimated halo masses that are a factor of ∼ 3.8 lower than other studies of SMGs.  
From a significantly larger parent sample, <cit.> selected a complete sample of ∼350 ALMA-identified SMGs with photometric redshifts to measure halo masses consistent with the results from the single-dish studies, particularly at z>2 <cit.>. 
Overall, the picture from clustering measurements is complex, and differing results from different studies may be methodological, due to sample selection or cosmic variance. Other ways of measuring the environments of SMGs are required.

Statistical photometric redshifts have identified galaxy overdensities around ∼5–60 percent of SMGs <cit.>, but these are subject to significant selection biases <cit.>, and few overdensities have been spectroscopically confirmed.
Instead, in order to determine the nature of a `typical' SMG environment, and thus confirm whether SMGs really are the progenitors of massive elliptical galaxies in local clusters, we need targeted observations of individual SMGs, but with no prior knowledge of their environments in order to avoid biases. To this end, we have conducted a wide-field narrowband survey of the environments of three SMGs identified as part of the ALESS project <cit.>,
in which follow-up observations of submillimetre sources detected in the LABOCA ECDFS Submillimetre Survey <cit.> were conducted using ALMA. We search for overdensities of  or  emitters around these three SMGs to assess whether they reside in protocluster-like environments. The target SMGs were selected on the basis of redshift only, and with no prior information about their environments. 
Our method is similar to that employed by <cit.>, which combined narrowband photometry with submillimetre observations to identify an overdensity of  emitters around two SMGs in SSA 13. However, the SMGs targeted by <cit.> were already known to be closely grouped with each other and three optically-faint radio-galaxies. Our study is the first to perform such an analysis around a sample of  SMGs that are selected without prior knowledge of their environment.

The structure of this paper is as follows: in 2 we describe the SMG sample selection, our observations and data reduction, and the identification of candidate companion galaxies for each target SMG; 3 includes our main results, analysis and discussion; our conclusions are presented in 4. Throughout this paper we use a ΛCDM cosmology with Ω_M = 0.315, Ω_Λ = 0.685 and H_0 = 67.4 km s^-1 Mpc^-1 <cit.>. 
All magnitudes are presented in the AB system, where a 1 μJy source has a magnitude of 23.9 <cit.>.





§ OBSERVATIONS AND GALAXY IDENTIFICATION



In this study we use the High Acuity Wide-field K-band Imager <cit.> on the Very Large Telescope (VLT) to investigate the environments of three ALMA-identified SMGs from ALESS. 
As part of ALESS, extensive follow-up was undertaken to obtain spectroscopic redshifts of the SMGs <cit.>, 
which enables a search for galaxies that share environments with these submillimetre sources. 
The wide-field imaging capabilities and narrowband photometric filters of HAWK-I allow for an efficient emission-line survey of their environments, which are expected to span physical scales on the order of several Mpc if consistent with being protoclusters <cit.>.




 §.§ Sample selection


The blank-field LABOCA ECDFS Submillimetre Survey (LESS) observed  0.5×0.5 degrees in ECDFS with  APEX/LABOCA  and detected 126  sources at >3.7σ at 870- <cit.>. Each of these sources was followed-up with ALMA to yield the 131 ALESS sources described in <cit.>, divided into a main catalogue of 99 SMGs and a supplementary catalogue of 32 SMGs. The SMGs in the main catalogue all lay within the ALMA primary beam full width at half-maximum (FWHM) of the highest-quality maps, while those in the supplementary catalogue were either extracted from outside the primary beam or from lower-quality maps <cit.>. 

Spectroscopic redshifts were obtained for 52 of the 131 ALESS SMGs by <cit.>,  
and targets for our study are selected from these 52 ALMA-identified and spectroscopically-confirmed SMGs. 
We require SMGs with spectroscopic redshifts that shift the  or λ5007 emission lines into the wavelength coverage of the HAWK-I  filter; this requires that the SMGs are located at z = 2.299±0.023 or z=3.324±0.060. Of the 52 ALMA-identified SMGs with spectroscopic redshifts from <cit.>, five (ALESS 6.1, 75.2, 87.1, 102.1, and 112.1) have spectroscopic redshifts within the desired range for Hα. A sixth SMG (ALESS 5.1) has a CO-derived spectroscopic redshift of  z = 3.303 <cit.>, which places  in the  coverage.

These six SMGs were the proposed targets for observations in four HAWK-I pointings (PID: 0103.A-0668). The six SMGs were selected purely based on their spectroscopic redshifts, with no consideration of their environments. Of the four pointings, only two were  observed during the service-mode observations and the choice of pointings was random. The two observed pointings contain three of the six proposed targets: ALESS 5.1, ALESS 75.2, and ALESS 102.1, whose spectroscopic redshifts are z = 3.303, 2.294, and 2.296, respectively. 
Details of these three targeted SMGs are provided in Table <ref>. A total of 16 other ALESS SMGs lie within the two HAWK-I pointings, but these are not considered in this study as their redshifts are such that no bright emission lines are expected in the narrowband filter. 
Indeed, those that are detected in our HAWK-I observations fail to meet our criteria for being emission-line galaxies (see <ref> and Figure <ref>).








 §.§ HAWK-I data


Each pointing was imaged using the HAWK-I instrument <cit.> on the VLT in the  (central wavelength: λ_c = 2.146 ; FWHM: Δλ = 0.324 ) and  (λ_c = 2.165 ; Δλ = 0.030 ) filters <cit.>. The FWHM of the  filter is equivalent to Δ z = 0.046 (0.060) at z ∼ 2.3 (3.3), corresponding to a velocity width of Δ v = 4200 km s^-1. All three observing blocks (OBs) for the field containing ALESS 5.1 and 75.2 (hereafter Pointing 5+75) were executed on 2019 August 21, while the OBs for the field containing ALESS 102.1 (hereafter Pointing 102)  were split among three separate nights from 2019 August 21 to 2020 January 01. The total exposure times for Pointing 5+75 (Pointing 102) were 7.2 ks (6.6 ks) and 900 s (660 s) in the  and  filters, respectively. 
Individual exposures of 120 s () and 30 s () were 
taken using the “HAWKI img obs AutoJitter” template, with five random 
dither positions within a 20 box for each filter in each OB. 
Each pointing covers a 75 × 75 area, except for a cross-shaped gap of width 15 between the detector's four 2k × 2k Hawaii 2RG arrays. Using HAWK-I's GRAAL system <cit.>, we achieved point spread functions (PSFs) of ∼04 in  and ∼03 in  (see Table <ref>). 

The data were reduced using a custom Python-based pipeline, with each of the four detector chips treated separately. Briefly, the pipeline begins by dark-subtracting the data and subsequently using twilight flats to perform flat-fielding. We then use SExtractor <cit.> to detect sources in each of the flattened frames and produce individual masks. 
A final flat field is produced for each frame by median combining all masked frames from the same OB except the frame being flattened; the frames are then flattened using their unique final flat fields. 
The astrometry of each flattened frame is then calibrated by using scamp <cit.> to match our detected sources with sources detected in a reference K-band image, correcting for any distortions across the field of view by fitting a third-order polynomial. The reference images used for Pointing 5+75 and Pointing 102 were taken from the Taiwan ECDFS NIR Survey <cit.> and the MUltiwavelength Survey by Yale-Chile <cit.>, respectively. Different reference images were required for each pointing because while TENIS is deeper and has higher resolution, roughly a quarter of the Pointing 102 field of view lies outside of the TENIS coverage. Finally, the astrometrically-corrected frames were median combined using SWarp <cit.>. The resultant stacks in both bands were photometrically calibrated using MUSYC  data <cit.> such that they all had a zeropoint magnitude of 30.0 mag and ensure a median ( – ) colour of 0.



Source detection and photometry were conducted using SExtractor <cit.> operating in dual-image mode; the stacked  images were used to identify the positions of sources, and then photometry was extracted at these positions in both the  and  images to ensure that any difference between the measured  and  photometry is a purely intrinsic property of the sources and not caused by positional offsets. After masking noisy regions near the edges of the stacked images, we detected a total of 2175 sources in Pointing 5+75 and 1754 in Pointing 102. Apertures with a diameter of 125 were used, as this is large enough to contain the majority of the flux for all detected sources while minimising the amount of additional background noise captured. The 3σ limiting magnitudes measured in these 125 diameter apertures are provided for each filter and each HAWK-I detector chip in Table <ref>. To account for the variation in size of the detected sources, we then estimated total magnitudes in each filter by selecting all bright (m_K_s < 19.5) sources and (for each pointing separately) calculating the median difference between their fixed-aperture magnitudes and their magnitudes measured by SExtractor in adaptively-scaled <cit.> apertures (mag_auto; ); this difference was then added to the fixed-aperture magnitudes of all sources in the pointing to obtain their total magnitudes. 
The choice to use the same aperture size for all sources and apply a correction (as opposed to simply using the mag_auto values) ensures that estimates of the total magnitudes are self-consistent whilst also closely matching existing photometry in the same band. 







 §.§ Ancillary data


There exists a wealth of photometric data in the ECDFS, which  supplements our HAWK-I photometry. Archival TENIS <cit.>, MUSYC <cit.> and HAWK-I (Zibetti, priv. comm.)  data were collated by <cit.> (hereafter S14) and then used to calibrate our astrometry and photometry (see <ref>). 

In <ref> and <ref> we fit spectral energy distributions to galaxies in our sample in order to first derive properties such as photometric redshift, stellar mass and star formation rate. For the photometric redshifts we use <https://github.com/gbrammer/eazy-py> – an updated version of the photometric redshift code eazy <cit.> written in Python (see <ref>) – while for the other galaxy properties we use  <cit.>.
To this end, we also make use of existing ECDFS images spanning the ultraviolet (UV) to mid-infrared (MIR; see <ref>). 
These images were sourced either from the MUSYC 2010 Public Data Release <cit.> or from TENIS <cit.>. 
The MUSYC dataset consists of UU_38BVRI broadband images from the Wide Field Imager (WFI) on the MPG/ESO 2.2m telescope <cit.>; 5000 Å narrowband and z^' broadband imaging from the Mosaic-II camera on the CTIO Blanco 4m telescope <cit.>, with JK_s broadband imaging from the Infrared Sideport Imager on the same telescope <cit.>; 18 medium-band (IA427, IA445, IA464, IA484, IA505, IA527, IA550, IA574, IA598, IA624, IA651, IA679, IA709, IA738, IA767, IA797, IA827, IA856) images taken with the Subaru telescope's Suprime-Cam <cit.>; Spitzer/IRAC images at 3.6, 4.5, 5.8 and 8.0  <cit.>H-band data are also available but our pointings are not covered.. 


We also make use of spectroscopic redshifts in the ECDFS from studies whose areas overlap with our pointings, obtained from publicly-available composite catalogues<https://www.eso.org/sci/activities/garching/projects/goods/MasterSpectroscopy.html>^,<http://member.ipmu.jp/john.silverman/CDFS_vlt.html> <cit.>. The spectroscopic redshifts used are from the VIMOS VLT Deep Survey <cit.>; the GOODS/VIMOS survey <cit.>; the Extended Chandra Deep Field-South Survey <cit.>; and <cit.>. 
Additionally we utilise the results of the spectroscopic study conducted as part of ALESS by <cit.>.

Finally, we make use of the <cit.> Chandra point source catalogue for the identification of AGN in our final sample (see <ref>).





 §.§ Emission line galaxy selection


Star-forming galaxies at the same redshifts as our target SMGs (z ∼ 2.295 ± 0.023 for ALESS 75.2 and ALESS 102.1; z ∼ 3.324 ± 0.030 for ALESS 5.1) have emission lines that are redshifted into the narrow wavelength coverage of the  filter. Since the  filter is near the centre of the  transmission a galaxy without line emission at these wavelengths will have a (–) colour of zero. However, due to the narrow width of the  filter relative to the  filter, galaxies with redshifts that place an emission line in the narrow  filter will  have a (–) colour that is significantly greater than zero. 
We employ the same methodology as previous narrowband surveys <cit.> to identify line-emitting galaxies. This methodology uses two parameters to select sources with a significant, physically-driven narrowband excess, as opposed to an excess due to random noise. 

The first of these parameters, Σ, quantifies the significance of the narrowband excess compared to the expected random scatter for a source with zero (–) colour <cit.>. Σ is given by:


    Σ = 1 - 10^-0.4(BB - NB)/10^-0.4(ZP - NB)√(rms_NB^2 + rms_BB^2) ,


where NB and BB are the apparent magnitudes in the narrowband () and broadband () filters, respectively; ZP is the zeropoint magnitude of the narrowband images; rms_NB and rms_BB are the rms counts in 125 apertures for the individual narrowband and broadband quadrants, respectively. 
We require candidate line-emitters  have Σ > 3, which is consistent with previous narrowband studies <cit.>; see Figure <ref>. Note that this Σ does not correspond directly to signal-to-noise (SNR) in the  filter, but is a separate quantity based on counts; Σ > 3 implicitly excludes sources with SNR ≲ 8 in  <cit.>. 

In addition to having Σ > 3 line emitters are required to have an observed equivalent width (EW) > 50Å. The EW is calculated for each source using:


    EW = Δλ_Brγf_Brγ - f_K_s/f_K_s - f_Brγ (Δλ_Brγ/Δλ_K_s) ,


where Δλ_Brγ and Δλ_K_s are the widths of the two filters and f_Brγ and f_K_s are the flux densities of the source in each filter. The 50 Å lower limit on EW for a source to be selected as a line emitter was chosen to lie above the 3σ scatter in (–) colours for bright ( > 19.5 mag) sources in both pointings (Figure <ref>).




Before applying the selection criteria, we first account for sources that are detected with ≥3σ significance in the  filter but <3σ in . We classify these sources as non-detections in , and replace their aperture magnitudes with the relevant 3σ limiting magnitude (see Table <ref>). However, several of these non-detections have counterparts in the S14 catalogue (within a 1 matching radius) and thus have  magnitudes from either TENIS <cit.>, archival HAWK-I observations (Zibetti, priv. comm.), or MUSYC <cit.>. For these sources, we replace our HAWK-I  photometry with values from one of these surveys, preferentially using TENIS photometry as it is the deepest of the three (with a limiting 3σ magnitude of m_ lim^3σ = 24.45 mag); if no TENIS photometry is available then we opt for the archival HAWK-I values (m_ lim^3σ = 24.36 mag), using MUSYC (m_ lim^3σ = 22.55 mag) only when no photometry exists for either of the other two. Note that while MUSYC  observations are the shallowest of all the data considered here (including our own), there are 11 sources for which only MUSYC photometry is available. However, all of these sources reside in regions of Pointing 102 that are (a) outside of the coverage of the TENIS and archival HAWK-I observations, and (b) close to the quadrant edges in our HAWK-I observations where the noise is at its greatest.


Using the Σ > 3 and EW > 50 Å selection criteria, 81 and 80 candidate line emitters are identified in Pointing 5+75 and Pointing 102, respectively (Figure <ref>). Of these candidates, 30 are  non-detections with no  photometry in the S14 catalogue, and thus Σ and EW are calculated by assuming that their  magnitudes are equal to the 3σ limiting magnitudes of our data. Since this can only provide a lower limit for the (–) colour and thereby underestimate Σ for these sources, we do include these sources in our sample of candidate line emitters. 

Finally, we visually inspect all 161 candidate line emitters, removing stars/quasi-stellar objects and image artefacts. 
The final sample consists of 79 and 68 candidate line emitters in Pointing 5+75 and Pointing 102, respectively (147 sources in total).






 §.§ Identifying line emitters associated with the SMGs


Narrowband excess alone is not sufficient to identify star-forming galaxies in the same environments as our target SMGs; such an excess could be caused by a number of possible emission lines at different redshifts (see Figure <ref>). We therefore use the available multi-band photometric data covering our pointings to estimate photometric redshifts for the narrowband emitters in our sample. The S14 catalogue contains photometric redshift estimates for sources across the ECDFS, however after cross-matching with our data (using a matching radius of 1), a significant fraction (>30 percent) of the line emitters identified in <ref> do not have broadband counterparts in this catalogue and thus lack any redshift information. 
We therefore perform our own spectral energy distribution (SED) fitting using .

To maximise the number of sources for which we can derive photometric redshifts, we extract fixed-aperture photometry at their HAWK-I  positions in the UV-to-MIR images described in <ref>. 
Each image is astrometrically calibrated using scamp <cit.> and SWarp <cit.> to match the astrometry of our HAWK-I images, and then photometrically recalibrated so that all images have a zeropoint magnitude of 30.0 mag. Photometry is extracted in fixed apertures using the photutils Python package <cit.>; apertures of diameter 20 are used for all images except those from Spitzer/IRAC, for which we use apertures of diameter 38 due to the larger point spread function (PSF). 
Aperture corrections are determined for each filter by measuring the median difference between the magnitudes measured in these apertures and those measured in adaptively scaled apertures with SExtractor for bright point sources. 
Final corrections are applied to each filter to account for Galactic attenuation, using values from <cit.> and <cit.>.




 operates using a χ^2-minimisation procedure in which linear superpositions of template SEDs are tested at different redshifts to find an optimal fit to the observed fluxes <cit.>. In keeping with other recent studies which implement  <cit.>, we use the “tweak_fsps_QSF_12_v3” set of 12 template SEDs, which cover a wide range of galaxy types and utilise a <cit.> initial mass function (IMF) and a <cit.> dust attenuation law while assuming solar metallicity. An advantage of these templates is that they include emission lines, such that a narrowband excess can provide a relatively tight constraint on the redshift. 

As discussed in <ref>, there have been several spectroscopic studies in the ECDFS, from which spectroscopic redshifts have been obtained for a number of galaxies across the field. Using a matching radius of 15, we cross-match our data with catalogues from VVDS <cit.>, the GOODS/VIMOS survey <cit.>, the ECDFS spectroscopic survey <cit.>, and the spectroscopic studies conducted by <cit.> and <cit.>. This gives spectroscopic redshifts for 163 (∼4.1 percent) of the 3929 sources detected in our HAWK-I imaging, including nine for which photometric redshifts could not be estimated due to insufficient photometry. Seven of the 163 sources with spectroscopic redshifts are emission line galaxies selected in <ref>; the spectroscopic redshifts for these sources are used for our analyses. We compare the photometric and spectroscopic redshifts for our HAWK-I sources in Figure <ref>. There is strong agreement between the photometric and spectroscopic redshifts, which is quantified using the normalised median absolute deviation (NMAD) of Δ z:


    σ_NMAD = 1.48 ×median(|Δ z - median(Δ z)/1 + z_spec|) ,


where z_spec is the spectroscopic redshift and Δ z = z_spec - z_phot. We obtain σ_NMAD = 0.062 when considering all 152 HAWK-I detections with photometric and spectroscopic redshifts. 





Only five (∼3.4 percent) of our 147 emission line galaxies have neither spectroscopic nor photometric redshifts, the latter being due to a lack of photometry with sufficient depth. Figure <ref> shows the redshifts of the remaining 142 emission line galaxies, compared with the significance of their narrowband excess (Σ; Equation <ref>). 
Peaks in the redshift distribution are visible at z ∼ 2.3 (both pointings) and z ∼ 3.3 (Pointing 5+75 only), as expected of  and  in the environments of the target SMGs. We select as  () emitters any galaxies for which 2.23 < z < 2.37 (3.23 < z < 3.41), where these redshift ranges correspond to 3× the FWHM of the  filter when  () has redshifted to the centre. We represent these selection criteria with shaded regions in Figure <ref>; the highlighted galaxies are henceforth assumed to be   and  emitters at similar redshifts as the target SMGs. 
We identify 44  emitters and 4  emitters in Pointing 5+75, and in Pointing 102 there are  11  emitters ( emitters in Pointing 102 are not further considered because there is no SMG at z ∼ 3.3 in this pointing). 
Table <ref> summarises the results of each step in the sample selection. We note that all of these  and  candidates have an SNR > 8.5 in the  filter as a natural consequence of our selection process (see also <ref>). We therefore do not expect the sizes of these samples to be significantly affected by Eddington bias <cit.>.




To identify any AGN in the sample we use a 1 matching radius to locate counterparts in the <cit.>  Chandra point source catalogue. None of the  emitters and only one  of the  emitters (2.3%) is an X-ray luminous AGN, which is consistent with the rate of X-ray AGN in blank-field surveys of  emitters at the same redshift <cit.>. Since the AGN fraction is the same as in field surveys then this galaxy is kept in our sample to enable a fair like-for-like comparison between the SMG fields and blank field  emitters. 




Figure <ref> shows the distributions of emission line galaxies across the HAWK-I pointings, with  and  emitters highlighted. For all three target SMGs, the companion galaxies are spread across the entire field of view and therefore span several physical Mpc. This is consistent with expectations from simulations <cit.>, in which protoclusters are seen to extend over several Mpc, such that the entire structure is unlikely to be captured by a single HAWK-I pointing. We also note the presence of a dense clump of seven  emitters (three of which are spectroscopically confirmed at z ∼ 2.3) in the northeast of Pointing 5+75, which coincides with a photometrically-identified  blob at z ∼ 2.3 <cit.>. This system will be discussed further in <ref>. 

Of the three SMGs targeted, only ALESS 102.1  is identified as an  (or ) emitter in our data. <cit.> did not identify  ALESS 102.1 as an  emitter in their spectroscopy, because the wavelength coverage with VLT/FORS2 and VLT/VIMOS does not cover  at z∼2.3. 
The original spectroscopic redshift for ALESS 5.1 is from CO(4-3) <cit.> and no emission lines were observed in Keck/DEIMOS, Keck/MOSFIRE or VLT/XSHOOTER observations <cit.>; this is likely because the redshifted  line clashes with a bright OH^- sky line for this source <cit.>. For ALESS 75.2 the original spectroscopic redshift was measured, in part, thanks to a faint  line detected in Keck/MOSFIRE observations <cit.>, which is below the detection limit of our data.









§ RESULTS, ANALYSIS & DISCUSSION





 §.§ Measuring luminosity functions


In order to quantify whether the SMGs reside in significant overdensities of  or  emitters, a comparison to the blank field needs to be drawn. The High Redshift (z) Emission Line Survey <cit.> is a large narrowband survey of emission line galaxies, including  emitters at z = 2.23 <cit.> and  emitters at z = 3.24 <cit.> in COSMOS. The HiZELS results are therefore representative of  and  emitters in regions of average density at redshifts similar to those of our target SMGs, and we use their luminosity functions as a blank-field sample for comparison with our results.

To construct luminosity functions for our  and  emitters we bin them according to line luminosities, making corrections to the observed number counts in each bin to account for completeness, contamination from other emission lines, dust attenuation, and the shape of the narrowband filter profile. Each of these steps is described in more detail below.



  §.§.§ Survey volumes


Approximating the  filter profile as a top-hat function with width equal to the FWHM of the filter (Δλ = 0.030 ), the  emission line should be detectable from z_min = 2.276 to z_max = 2.322, corresponding to a comoving volume per square degree of 5.86×10^5 cMpc^3 deg^-2. The redshift range within which  can be detected extends from z_min = 3.294 to z_max = 3.354, which gives 1.04×10^6 cMpc^3 deg^-2. Accounting for the high-noise regions that were masked prior to source detection (see <ref>) and the gaps between the HAWK-I detector chips, the surveyed areas in Pointing 5+75 and Pointing 102 are 0.0117 and 0.0118 deg^2, respectively. The volumes probed are therefore: 6859 cMpc^3 for  in Pointing 5+75; 6891 cMpc^3 for  in Pointing 102; 12180 cMpc^3 for  in Pointing 5+75; 12230 cMpc^3 for  in Pointing 102. 
In <ref> we correct the derived luminosity functions to account for the fact that the  filter is not a perfect top-hat function, which leads to the volume probed being slightly different for sources with different luminosities. 




  §.§.§ Completeness correction


It is possible that real galaxies with weak emission lines were missed in our selection process (<ref>) despite actually meeting the selection criteria: the sample is incomplete at low emission line fluxes. We correct for this using the method employed by <cit.>, applying it separately for each quadrant of each pointing due to the variation in depth between detector chips (see Table <ref>). For each emission line ( and ), we select sources that failed to meet the emission line galaxy selection criteria (i.e. sources for which EW < 50 Å and/or Σ < 3) with redshifts within the range used to identify the targeted emission line (see <ref>). Due to the size of these samples, we generate ∼1000 mock galaxies by randomly varying the  and  magnitudes of the selected galaxies according to their uncertainties (assuming a Gaussian probability distribution for each magnitude) and placing them at random positions within their quadrant, removing any sources for which these changes result in them being classed as a line emitter. We then artificially inject line flux to each galaxy in this bolstered sample of non-line emitters, beginning with 10^-22 erg s^-1 cm^-2 and incrementally increasing it by 0.05 dex. Line fluxes are calculated as


    F_line = Δλ_Brγf_Brγ - f_K_s/1 - (Δλ_Brγ / Δλ_K_s) ,


where f_Brγ and f_K_s are the  and  flux densities, respectively, in units of erg s^-1 cm^-2 Å^-1. With each increment of injected line flux, we recalculate the EW and Σ and reapply the line emitter selection criteria to determine the catalogue completeness as a function of line flux. This is used to estimate the completeness corrections for our luminosity functions. The uncertainty in the completeness at a given line flux is estimated by regenerating the mock galaxies 1000 times and measuring the standard deviation in the completeness across all iterations. 





  §.§.§ Removing  contamination


The  emission line lies in between a doublet of  lines at rest-frame wavelengths of 6548 Å and 6583 Å, which will contribute to the measured  flux density and therefore affect the observed EW and emission line flux. Using spectroscopic data taken with Subaru/FMOS and Keck/MOSFIRE, <cit.> observed an anti-correlation between the λ6583-to- line flux ratio and the rest-frame EW (EW_rest = EW / (1 + z)) for the  emitters in HiZELS, deriving the following empirical relation:


    F_[Nii]/F_Hα = -0.296 ×log_10(EW_rest,Hα+Nii) + 0.8 .


We adopt this relation to apply corrections to the line fluxes of all  emitters in our sample, resulting in a median decrease of 11^+9_-5 percent in  emission line flux. 




  §.§.§ Relative contributions from λ5007, λ4959 and 


Thus far only the λ5007 emisson line has been considered in the discussion of  emitters at z ∼ 3.3. However, this line is part of a doublet, with its counterpart residing at a rest-frame wavelength of 4959 Å, and there is a narrow range of redshifts (z = 3.336–3.344) in which both lines can contribute to a galaxy's  flux. Furthermore, while the  emission line is sufficiently separated from the  doublet to avoid contaminating the measured  line flux, it is still close enough such that there is the potential for  emitters to be misidentified as  emitters (see Figure <ref>). Therefore, rather than try and separate our sample into λ5007, λ4959 and  emitters, we present a combined + luminosity function; this also allows for a consistent comparison with the blank-field +  luminosity function from <cit.>. 

We do however take into account the results of <cit.> when estimating the total volumes probed by the  filter in the search for  emitters: using spectroscopy, <cit.> find that for HiZELS ∼50 percent of photometrically-selected + emitters at z ∼ 1.4 are λ5007, ∼27 percent are λ4959, ∼16 percent are , with the remaining ∼7 percent being simultaneous detections of λ5007 and λ4959. Based on these results, <cit.> then add to the total volume probed (i.e. the volume probed if searching for λ5007 emitters) 16 percent of the volume that would be probed had their search been for , and 25 percent of the volume had they been searching for λ4959. We thus apply similar corrections to our total volume probed for λ5007 emitters.





  §.§.§ Corrections for dust attenuation


Dust in star-forming galaxies reprocesses light emitted in the rest-frame UV and optical, and can therefore reduce the amount of  and  flux observed. In order to estimate the intrinsic brightness of the emission lines (i.e. their integrated luminosities), one has to correct for the effect of dust attenuation. For an attenuation of A_line (mag) at the emission line wavelength, the conversion from line flux to intrinsic line luminosity is


    L_line = 4 π D_L^2F_line× 10^0.4A_line ,


where D_L is the luminosity distance. We follow <cit.> and assume an attenuation at the  wavelength of A_Hα = 1 mag, which is based on previous HiZELS studies <cit.>. <cit.> do not correct for dust attenuation when plotting their luminosity functions, so we also leave our + luminosities uncorrected to ensure a consistent comparison. <cit.> later go on to calculate the dust-corrected star formation rates (SFRs) of their galaxies, where they then assume an attenuation of A_+ = 1.35 mag, derived by assuming A_Hα = 1 mag and using a <cit.> dust attenuation curve. We thus adopt the same correction when calculating our own SFRs (see <ref>).




  §.§.§ Filter profile volume corrections


The comoving volumes used for our luminosity functions (<ref>) are calculated by approximating the  filter as a top-hat filter with width equal to the  FWHM. Since the filter profile is not a top-hat in reality, this introduces two main effects which need to be accounted for when estimating the galaxy number densities. Firstly, bright emitters whose line falls near the edges of the  filter will suffer a significant loss of line flux and thus appear to be fainter than they really are. This produces an overall bias towards faint sources in our sample. Secondly, any faint emitters close to the filter edges might be missed from our sample, and are therefore only detectable over a narrower redshift range (and thus a smaller volume) than their bright counterparts.

To correct for these effects, we follow the method used by <cit.> and <cit.>, as first proposed in <cit.>. An initial Schechter fit is performed to the uncorrected`Uncorrected' only in terms of the filter profile correction; the results have already been corrected for line completeness and dust attenuation by this stage. data. We then generate a mock sample of 10^5 fake sources with a luminosity distribution that is weighted by the uncorrected Schechter function. These sources are randomly assigned redshifts with a uniform distribution across the whole possible  coverage. They are then convolved through the  filter profile such that their luminosities decrease according to their assigned redshift (i.e. according to the position of the redshifted emission line in the filter profile) and rebinned using the same luminosity bins as for the uncorrected data. Comparing the resultant distribution to the input distribution reveals that bright sources are underestimated relative to the fainter sources, as expected. The real data are corrected using the ratio of these distributions.




  §.§.§ Fitting Schechter functions


Finally, we perform fits to the corrected data using a Schechter function <cit.>:


    Φ(L)dL = ln(10) Φ^∗(L/L^∗)^α + 1 e^-(L/L^∗) dlog L ,


where Φ(L) is the number density at luminosity L, Φ^∗ is the normalisation of the luminosity function, L^∗ is the characteristic luminosity, and α is the slope at the faint end of the luminosity function, where the power-law component dominates. 

The faintest bins (open symbols in Figure <ref>) are excluded from each fit due to their low completeness. For the  emitters, we take `low completeness' to mean that the low-luminosity edge of the bin lies below the 30 percent completeness limit. 
The line luminosities of all + emitters lie above the 90 percent completeness threshold; we therefore do not exclude any from the fit.


Due to the small number of bins left available for fitting, it is impossible to reliably constrain all three free parameters of the Schechter function simultaneously. For the  luminosity functions, we therefore fix the faint-end slope, α, to the value of -1.59 obtained by <cit.> at z = 2.23. For the + emitters, we only have one available bin and thus fix both α and log(L^∗/erg s^-1) to the values for the z = 3.24 sample of + from <cit.>, which are -1.60 and 42.83, respectively. Thus our + luminosity function is effectively a renormalised version of the <cit.> result, with Φ^∗ being the only free parameter. In addition to the  luminosity functions of the individual pointings, we also construct fits to the combined sample of  emitters from both SMG fields, as this provides a more general view of SMGs at z ∼ 2.3 with improved statistics. 

The best-fit parameters for each luminosity function are summarised in Table <ref>. Uncertainties are estimated for each free parameter by randomly perturbing the bin heights according to their uncertainties and then recalculating the fit, and repeating this process until 10^5 fits have been made. The 1σ confidence interval for each parameter is then estimated using the 16th and 84th percentiles of the best-fit values.






 §.§ Analysing luminosity functions





We next use the luminosity functions to assess whether the targeted SMGs reside in overdensities of emission-line galaxies. Figure <ref> compares the observed luminosity functions from the SMG fields to those from the blank field surveys of <cit.> and <cit.>. 
The environment of ALESS 75.2 shows signs of being overdense relative to the field at z ∼ 2.3, with most bins lying significantly above the blank field luminosity function. Conversely the environments of ALESS 5.1 and 102.1 are broadly consistent with the blank field luminosity functions at their respective epochs. 
An overdensity remains when the  emitters from both fields at z ∼ 2.3 are combined, implying that on average SMGs at this epoch reside in overdense, protocluster-like environments, which qualitatively consistent with clustering results <cit.>. The contrast between the individual  luminosity functions suggests that there is significant variation across SMG environments, although observations of additional SMGs are required to confirm and quantify the field-to-field variation. Furthermore, as explored later in this section and shown in Figure <ref>, the field around ALESS 102.1 is itself overdense on smaller scales. 


To quantitatively compare the SMG field and blank field luminosity functions we consider the parameters of the Schechter function fits ( <ref>). The parameters of Schechter function fitting are often correlated, so in Figure <ref> we show the uncertainties of the luminosity function parameters in the L^∗–Φ^∗ plane (as described in <ref> the faint-end slope, α, is fixed), comparing our SMG fields with the blank fields at similar redshifts. 
For the individual SMG environments at z ∼ 2.3 the fit parameters are offset from those of the blank field, although for the ALESS 102.1 region the offset is only at the ∼1σ level. 
These separations are driven by a higher L^∗, and,  in the case of ALESS 75.2, by a larger Φ^∗, which implies that this environment is preferentially overdense in bright line emitters compared to the blank field. 
Meanwhile, the environment of ALESS 5.1 exhibits an offset of <1σ relative to the blank-field value of Φ^∗ at z ∼ 3.3, implying this SMG does not reside in an overdnensity.

 
We quantify the galaxy overdensity in each sample of  and + emitters in two ways. Firstly, we calculate the ratio of the Φ^∗ from the best fit Schechter function to those from the relevant blank-field luminosity functions, Φ^∗/Φ^∗_field. This ratio tells us how much higher the `knee' of each SMG-field luminosity function is relative to the blank-field. These ratios are presented in Table <ref>. The value for the + luminosity function suggests that the environment of ALESS 5.1 at z∼3.3 is 1.2^+0.6_-0.4 times as dense as the field, i.e. it is consistent with the blank field. For the  emitters at z∼2.3, L^∗ is also a free parameter in the Schechter fits, and the L^∗–Φ^∗ correlation means that we must first refit the data with L^∗ fixed to the z ∼ 2.3 blank-field value from <cit.>, i.e. log(L^∗ / erg s^-1) = 42.83. 
These fits give Φ^∗/Φ^∗_field values of 3.6^+0.6_-0.6 and 1.7^+0.3_-0.3 for the ALESS 75.2 and ALESS 102.1 fields, respectively. The combined sample of  emitters from both SMG environments suggests that the average SMG environment at z ∼ 2.3 is 2.6^+0.4_-0.4 times more dense than the blank field at this epoch. 




To derive a more representative estimate of the galaxy overdensity in each environment, we also estimate the number of  or + emitters that one would expect to find in a blank field with the volumes probed by our observations. To do this, we integrate the field luminosity functions across the luminosity range covered by our data, excluding low completeness bins; i.e. we integrate across the ranges 42.5 ≤log(L^∗_ / erg s^-1) < 43.8 and 42.4≤log(L^∗_+ / erg s^-1) < 42.7 and multiply by the volumes probed in each HAWK-I pointing to estimate the expected number counts in an equivalent blank field, N_field. Since these field galaxies would have contributed to the observed number counts, we quantify the galaxy overdensity in each environment using


    δ_g = N_total - N_field/N_field


where N_total is the sum of the counts in our complete bins. Uncertainties in N_total are determined by adding in quadrature the uncertainties in the bin counts. For N_field the uncertainties are estimated by randomly permuting the blank-field Schechter parameters within their uncertainties prior to integrating, then repeating the process 10^5 times and using the 16th and 84th percentiles of the resultant number counts to define the 1σ confidence interval.

The values of δ_g for each sample of  and + emitters are summarised in Table <ref> along with the significance of this overdensity, σ_δ. Based on these values, the environments of ALESS 5.1, 75.2, and 102.1 are overdense by factors of 0.2^+2.5_-0.7 (0.3σ_δ), 2.6^+1.4_-1.2 (2.3σ_δ), and 0.2^+0.6_-0.5 (0.5σ_δ), respectively. If the samples of  emitters from both pointings are considered as one, then the SMG environments at z ∼ 2.3 are overdense by a factor of 1.5^+1.0_-0.8 (1.9σ_δ) on average. 
The above uncertainties do not account for cosmic variance, which could cause a factor ∼2 difference in number counts, as based on the  emitters in two equal depth HAWK-I pointings in COSMOS and UDS <cit.>. Including cosmic variance in our calculations does not change our overall conclusions that the ALESS 75.2 and the combined z∼2.3 SMG fields are overdense, nor does it affect the conclusions that ALESS 5.1 and ALESS 102.1 reside in environments consistent with the blank field. 

While Pointing 102 as a whole is not overdense, the majority of the  emitters in this pointing are contained within the same quadrant as the SMG, as can be seen in Figure <ref> (see also <ref>). We therefore recalculate δ_g for this SMG environment, this time considering only the volume probed within that quadrant (1722 cMpc^3), finding δ_g = 3.8^+2.4_-1.8 (2.1σ) in this area, which suggests that ALESS 102.1 actually does reside in an overdense environment with a physical scale of ∼1.6 Mpc. This high concentration of galaxies surrounding the SMG could be indicative of structure formation on smaller scales than those of protoclusters and it is possible that ALESS 102.1 resides in a protogroup <cit.>. We discuss the spatial distribution of companion galaxies in each SMG environment in more detail in <ref>.



The question remains as to whether the target SMGs reside in protoclusters, which will evolve into bound clusters by the present day. 
To learn more, we compare the overdensities in the SMG fields with previous studies of protoclusters. However, protoclusters exhibit a wide range of galaxy overdensities; a `typical' value of δ_g is not well-defined, though we highlight here structures  at similar redshifts to our target SMGs. For example, δ_g = 2.5 for the z = 1.99 protocluster in the GOODS-N field <cit.>. The protoclusters 4C 10.48 (z = 2.35) and 4C 23.56 (z = 2.48), which were both identified using narrowband selection of  emitters around luminous radio galaxies, were found to have overdensities of δ_g = 11^+2_-2 and δ_g = 4^+5_-3, respectively <cit.>. 
Similarly, <cit.> used a narrowband search for  emitters at z = 2.23 to identify overdensities of δ_g∼ 3, 2, and 2 around a quasi-stellar object overdensity, a high-redshift radio galaxy, and an overdensity of SMGs and optically faint radio galaxies, respectively. 
The protocluster Cl J0227-0421 at z = 3.29 is overdense by a factor of 10.5 ± 2.8 <cit.>. Two protoclusters in the COSMOS field at z = 2.10 and z = 2.47 were found to have overdensities of δ_g∼ 8 and δ_g∼ 3.3, respectively <cit.>. <cit.> confirm overdensities of  emitters in two protocluster candidates, BOSS1244 and BOSS1542, with overdensity factors of δ_g = 5.6 ± 0.3 and δ_g = 4.9 ± 0.3, respectively. 
Comparing δ_g for these protoclusters with our values, we posit that the environments of ALESS 75.2 and ALESS 102.1 (and thus of SMGs on average at z ∼ 2.3) are consistent with being protoclusters, albeit at the lower-density end. For the remainder of the analyses we assume that members of these overdensities will form larger structures by z = 0, although we caution that the significance of these overdensities is relatively low (1.3–2.3σ_δ) and thus it is uncertain whether they will coalesce by z∼0 <cit.>.





 §.§ Spatial distribution of line emitters



In order to investigate the role of environment in shaping the evolution of SMGs, and to assess whether the target SMGs reside in special regions within any surrounding structures, we next explore the spatial distributions of coeval line emitters across our HAWK-I pointings. Due to the small size of the  emitter sample, we limit this part of the analysis to the  emitters around ALESS 75.2 and 102.1. 
Figure <ref> compares the surface density of  emitters as measured in annuli centred on each target SMG (where the density calculations account for masked and unobserved regions by assuming the density is the same as in the observed regions)  with the surface densities one would expect based on the blank-field luminosity function from <cit.>. The annuli have inner and outer radii increasing in increments of 20, and are represented by dashed circles in Figure <ref>. Note that these large annuli are necessary due to the sample sizes, but make it difficult to probe the protocluster structures in detail. 
We therefore also show in Figure <ref> the result of smoothing the distributions of  emitters in Pointing 5+75 (left) and Pointing 102 (right) using a 2D Gaussian kernel with width corresponding to 0.5 Mpc at z ∼ 2.3. This method of visualisation clarifies where the SMGs lie in relation to any density peaks and can highlight any substructures. For simplicity we assume that there are no  emitters in the unobserved region between the detector chips, or in regions of the image that have been masked, and thus the densities shown in these regions are potentially underestimated. 

For ALESS 102.1, there is a noticeable decrease in the surface density of  emitters as a function of radial distance from the SMG, with the innermost bin in Figure <ref> being significantly overdense relative to the field despite the environment not being overdense as a whole (see also <ref>). This is also clear from Figure <ref>,  which shows that the SMG lies ∼20 from a ∼325 (∼1.6 Mpc) density peak. Furthermore, Figure <ref> demonstrates that the innermost 20 annulus contains more than half of the  emitters detected across the Pointing 102 pointing.  
Figure <ref> includes a panel showing the location of the SMGs and our  emitter density maps in the context of the overdensity of  emitters (LAEs) at z ∼ 2.3 mapped by <cit.>. This shows that the small-scale overdensity around ALESS 102.1 is in a broader underdense region, and it is therefore unlikely to be a condensed infalling knot within a larger structure. 


In the case of ALESS 75.2 we show two results in Figure <ref>: one where we include all  emitters in the pointing (open squares), and one where we exclude the dense clump of  emitters in the north-east (filled squares; see <ref>). In both cases, there is no significant trend in the  surface density as a function of separation from the SMG, although it does show signs of decreasing at the outermost radii if the dense north-easterly clump is excluded. This lack of trend implies that ALESS 75.2 does not reside in a particularly special region of the structure, and/or the structure extends beyond the HAWK-I field of view. 
The latter hypothesis is supported by the comparison of the  emitter overdensity with that of the LAEs from <cit.> (Figure <ref>, right), which shows that the whole structure around ALESS 75.2 is within a larger region of LAE overdensity. This suggests that the  emitter structure likely spans a physical distance ≳ 3.5 Mpc at z ∼ 2.3, which is consistent with the simulations of e.g. <cit.>, in which protoclusters are expected to extend over ≳ 10 Mpc at z ∼ 2. The <cit.> structure in this region includes the  blob CDFS-LAB03, which coincides with seven  emitters (see also <ref>). The overall picture is consistent with previous findings, in which  blobs are found to be associated with massive dark matter halos and filamentary large-scale structures <cit.>. 











 §.§ SMG companions: SFRs and stellar masses


In this section we investigate the dust-corrected star formation rates (SFR) and stellar masses (M_⋆) of the individual galaxies around each target SMG, to determine whether they lie on the main sequence of star formation at their epochs. This correlation between SFR and M_⋆ has been observed out to z ∼ 6 <cit.> and galaxies significantly above the main sequence are usually considered to be short-lived starbursts, whereas those significantly below the main sequence are typically quenched. The position of galaxies relative to the main sequence provides insights into their evolutionary state and can be used to infer the role of any environmentally-driven mechanisms enhancing or inhibiting star formation activity. 
Note that while ALESS 5.1 does not appear to reside in an overdense structure, the properties of the coeval + emitters in its vicinity are still of interest and we thus include them in this part of the analysis.

We obtain stellar masses and SFRs for our galaxies by using the SED fitting code,   <cit.>, to fit SEDs to the same fixed-aperture photometry used to derive photometric redshifts in <ref>. Figure <ref> compares the relationship between SFR and stellar mass for the  and + emitters that are SMG companions with the main-sequence at similar epochs using the prescription from <cit.>. These galaxies generally scatter about the main sequence at their respective epochs, following a similar trend of increasing SFR with increasing stellar mass. 
We thus find no significant evidence of enhanced star formation in any of these SMG environments, despite the range of overdensities that they span; this is contrary to some previous studies in which enhanced SFRs have been found in overdense environments at z ≳ 1 <cit.>, however it is consistent with several other studies in which no environmentally-driven SFR enhancement is observed at high redshift <cit.>.




In addition to the -derived SFRs, we also calculate the dust-corrected star formation rates for the  emitters using the scaling relation from <cit.>, modified for a <cit.> IMF: 


    SFR (M_⊙ yr^-1) = 4.65 × 10^-42 L_ (erg s^-1) ,


where the  line flux has been corrected for contamination by the nearby  doublet (see <ref>) and we assume a dust attenuation of A_ = 1 mag (<ref>). For the + emitters we assume an attenuation of A_+ = 1.35 mag following <cit.> and use the relation between SFR and L_+ from <cit.>, similarly modified for a <cit.> IMF:


    SFR (M_⊙ yr^-1) = 4.32 × 10^-42 L_+ (erg s^-1) .


The bottom panel of Figure <ref> shows how the ratio of the magphys-derived and line-derived SFR estimates varies with stellar mass. Also shown is the  dust attenuation required for the SFR derived from Equation <ref>, SFR_ Hα, to agree with the -derived value, SFR_magphys, as given by:


    A_ Hα = 2.5log_10( SFR_magphys/ SFR_ Hα) + 0.4 .


An analagous equation for A_+ can be obtained by adding 0.14 mag. It is evident that as one moves to higher stellar mass, the assumption that A_ Hα (A_+) = 1.0 (1.35) mag results in underestimated SFRs compared with the results from SED fitting. We therefore caution that while such an assumption may be suitable for galaxies with low-to-average stellar mass, it becomes less reliable for high-mass galaxies.


We also include the SMGs themselves in Figure <ref>, with the SFRs and stellar masses for these calculated by <cit.> and <cit.> using . As expected for sources selected due to their infrared-brightness, the SMGs are among the most active galaxies in the observed fields. ALESS 102.1 is also one of the most massive galaxies in its environment, which suggests that if it is in a protocluster then it may be brightest cluster galaxy (BCG) progenitor, i.e. a proto-BCG. 
Similarly, ALESS 5.1 is massive relative to other galaxies in the surrounding region, but given the low density of this environment we deem it unlikely that this SMG is a proto-BCG. 
Conversely, ALESS 75.2 has a lower mass, which is not exceptional for its environment, and which points towards it being more likely to evolve into a normal cluster member. This is consistent with the spatial analysis of  emitters and LAEs (<ref>), which showed that ALESS 75.2 is offset from the densest regions of this field. 

Since ALESS 102.1 has a counterpart  emitter in our sample, we also compare our -derived SFR and stellar mass with those derived by <cit.>. Our stellar mass of log(M_⋆/ M_⊙) = 11.49^+0.18_-0.05 is in good agreement with their value of log(M_⋆/ M_⊙) = 11.42^+0.06_-0.06. Conversely, our SFR of log( SFR/ M_⊙ yr^-1) = 2.75^+0.22_-0.25 is significantly higher than their estimate of log( SFR/ M_⊙ yr^-1) = 2.12^+0.25_-0.23. This is likely due to the inclusion of FIR and radio photometry in their SED fitting which are absent from our own fit; the dust component (and thus the SFR) is better constrained in the <cit.> SED fit. We therefore opt to use their values of SFR and stellar mass for this galaxy instead of our own.




 §.§ Stellar mass functions


We next construct the stellar mass functions of the galaxies around each SMG and compare these with the blank field. The stellar mass functions are derived following a similar procedure as for the luminosity functions (see <ref>), minus the corrections that are only relevant to luminosity functions (dust attenuation, line flux contamination, filter profile corrections). Completeness corrections were applied to each mass bin according to the completeness values estimated in <ref> based on the emission line fluxes. We then fit Schechter functions to the data:


    Φ(M_⋆)dM_⋆ = ln(10) Φ^∗(M_⋆/M^∗)^α + 1 e^-(M_⋆/M^∗) dlog M_⋆ ,


where Φ(M_⋆) is the number density at stellar mass M_⋆, Φ^∗ is the normalisation of the stellar mass function, M^∗ is the characteristic stellar mass, and α is the slope at the faint end of the stellar mass function. Mass bins that are less than 50 percent complete are excluded from the fitting procedure. 
As with the luminosity functions, we also fix the faint-end slope α to the  values derived for blank-field stellar mass functions by <cit.> and <cit.> (i.e. α = -1.37 and α = -1.3 for the for the  and + emitters, respectively). For + emitters we also fix the characteristic stellar mass, M^∗, to the blank-field value of log(M^∗ / M_⊙) = 10.96 <cit.>. 

The stellar mass functions are presented in Figure <ref>, with the parameters in  Table <ref>. Uncertainties in each parameter are estimated following the same procedure as for those of the luminosity functions (see <ref>) and the correlation between the parameters and their uncertainties are shown in  Figure <ref>, which demonstrates that at the upper limit the characteristic stellar mass, M^∗, is poorly constrained for all of our samples except the  emitters around ALESS 75.2. 
However, the lower limit is sufficient to show that in the z∼2.3 SMG regions the characteristic stellar mass is significantly higher than the z∼2.3 field, which suggests that the stellar mass build-up in SMG companion galaxies is further advanced than the coeval field <cit.>. Due to our selection of  emitters the galaxies have non-negligible star-formation rates (though many are below the main sequence; <ref>). Observations using a local galaxy density estimator suggest that local environment has minimal effect on the stellar mass function of star-forming or quiescent galaxies at z=1.5–2 <cit.>. However, there is evidence of protocluster environments being skewed towards containing galaxies with higher masses than the field <cit.>, consistent with our results. 














 §.§ Dark matter halo masses and evolution




We next estimate the total halo masses of the SMG environments in order to place them within the context of existing protoclusters and trace their likely evolution, focusing primarily on the overdense environments of the two SMGs at z ∼ 2.3. 
Since these overdensities are unvirialised and lack a detectable intracluster medium, the classic methods for weighing galaxy clusters cannot be used. 
Instead, we use two methods that have been used in protocluster studies, though the underlying assumptions required can lead to significant uncertainties, as discussed in the following subsections.
The first method is detailed in <ref> and uses the stellar-to-halo mass relation (SHMR) to estimate the high-redshift mass of the clusters (hereafter SHMR method) and evolve it to the local Universe using the Millennium and Millennium-II simulations <cit.>. 
The second method follows <cit.> and assumes the region of interest is a homogeneous sphere undergoing spherical collapse and uses the overdensity parameter to estimate the z=0 descendant mass, which we trace back to high-redshift using the Millennium and Millenium-II simulations. This is referred to as the spherical collapse model (SCM) method and detailed in <ref>.
In <ref> we discuss the evolution of the SMG environments compared with other systems and previous measurements. 




  §.§.§ The SHMR method for deriving halo masses



The SHMR method for estimating protocluster masses involves identifying the most massive galaxy in the structure and converting its stellar mass to a halo mass, and taking this to be the halo mass of the whole structure. This method has been employed in recent protocluster studies <cit.> and implicitly assumes that all member galaxies occupy a single halo at the observed redshift of the structure, which may not be the case if some of the galaxies are still infalling. Nevertheless, we deem this assumption preferable to the commonly-used alternative of estimating the halo masses of each individual galaxy and summing them together <cit.>, which risks `double-counting' overlapping dark matter halos to produce an overestimate of the structure halo mass. 
Note that we perform this calculation even for ALESS 5.1 and its surrounding  emitters despite our analyses revealing no signs of their environment being significantly overdense. This is because this method does not explicitly depend on the density of the surrounding environment, and the high stellar mass of ALESS 5.1 (see <ref> and <ref>) suggests it may yet reside in a massive halo. We do however caution that the result obtained here likely represents an extreme upper limit on the mass of any possible structure around ALESS 5.1.

We estimate halo masses for each target SMG and their candidate companion galaxies using the SHMR from <cit.>. We use the relation as defined at z = 2 for galaxies in the environments of ALESS 75.2 and 102.1, and at z = 3 for galaxies in the environment of ALESS 5.1. 
For the SMGs themselves we use the stellar masses from the literature <cit.>. 
Some of our galaxies have stellar masses which lie above the range at which the SHMR is defined and for these we use the stellar-to-halo mass ratio for the largest halo mass at which the relation is defined <cit.> to convert the stellar mass to a halo mass. This affects only two  emitters from P5_75, along with one from P102 which we have identified as a counterpart to ALESS 102.1 (see <ref> and Figure <ref>). None of the  emitters have stellar masses above the range for which the SHMR is defined at z ∼ 3, but ALESS 5.1 does lie above this range.  Uncertainties on individual galaxy halo masses are estimated based on the stellar mass uncertainties and the uncertainties in the SHMR derived by <cit.>.


We derive halo masses of log(M_h/ M_⊙) = 11.45–14.46 for individual  and  emitters, with medians of log(M_h/ M_⊙) = 12.16^+0.21_-0.16, 11.94^+0.46_-0.35, 11.75^+0.26_-0.12 for galaxies in the environments of ALESS 5.1, 75.2 and 102.1, respectively. The halo masses of the corresponding SMGs are log(M_h/ M_⊙) = 13.94^+0.38_-0.21, 12.02^+0.08_-0.25, 14.39^+0.03_-0.37, derived using their stellar masses reported in the literature <cit.>. 
ALESS 5.1 and 102.1 are both the most massive galaxies in their respective environments; we therefore adopt their halo masses as the total masses of the potential structures at the observed redshifts. ALESS 75.2 is not the dominant galaxy in its environment, being surpassed in stellar (and hence inferred halo) mass by ∼40 percent of its companion  emitters. The most massive of these is a spectroscopically confirmed member <cit.> located in the  emitter density peak associated with the  blob CDFS-LAB03 (see Figure <ref>), with a halo mass of log(M_h/ M_⊙) = 14.38^+0.01_-0.40. We thus assume this is the total mass of the surrounding structure. Since high-redshift radio galaxies are commonly found in protocluster cores <cit.>, we search for radio counterparts for this galaxy in the second data release from the Very Large Array 1.4 GHz survey of the ECDFS <cit.>, for which the typical sensitivity is 7.4  per 28×16 beam. We find no counterparts within 30 of this  emitter and thus rule it out as being a high-redshift radio galaxy.



The total halo masses at the observed redshift obtained using the SHMR method are thus log(M_h/ M_⊙) = 13.94^+0.38_-0.21, 14.38^+0.01_-0.40, 14.39^+0.03_-0.37 for the environments of ALESS 5.1, 75.2, and 102.1, respectively. 
We note that these masses may be affected by systematic uncertainties on the stellar masses (due to uncertainties on star-formation histories and resulting mass-to-light ratios, which is particularly relevant for young starbursts; e.g. ) and on the SHMR for very high mass galaxies, which are present in the simulations from which the SHMR is derived <cit.>. Indeed, predictions from halo mass functions suggest that the halo masses inferred from this method should be sufficiently rare that finding three such structures in the ∼0.25 deg^2 ECDFS is unlikely <cit.>. 
Therefore, we consider these SHMR-derived halo masses to be upper limits and as such they are represented by the upper bounds on Figure <ref> (the lower bounds are derived in <ref>), which compares the halo masses of the SMG environments with previously-studied galaxy clusters and protoclusters. 

To assess whether these SMG environments are true protoclusters, we evolve the masses derived from the SHMR method to the present day masses and compare with known galaxy clusters in the local Universe. This is done using the redshift-dependent formula for the mean mass accretion rate derived from the results of the Millennium and Millennium-II simulations <cit.>:


    ⟨Ṁ⟩_mean =     46.1  M_⊙yr^-1(M_z/10^12M_⊙)(1 + 1.11z) 
       ×√(Ω_m,0(1+z)^3 + Ω_Λ,0) ,


where M_z is the halo mass of the structure at its observed redshift, and Ω_m,0 and Ω_Λ,0 are the present-day density parameters for matter and the cosmological constant according to our assumed cosmology <cit.>. 

For each overdensity we begin with the total halo masses estimated using the SHMR method and apply Equation <ref> to incrementally add mass in small time steps until the present day is reached. The present-day masses obtained with this method are log(M_h,z=0/ M_⊙) = 15.93^+0.62_-0.33, 15.81^+0.01_-0.55, 15.82^+0.04_-0.50 for the overdensities containing ALESS 5.1, ALESS 75.2, and ALESS 102.1, respectively.
These masses suggest that these structures would all evolve into some of the most massive clusters in the Universe, rivalling that of the Coma cluster <cit.> and other massive clusters at z ≲ 1 such as those in the CLASH <cit.>. However, given the rarity of such massive structures seen in the local Universe, we posit that the identification of progenitor structures around all three of our target SMGs is due to the systematics in the calculations, rather than a real occurrence. 
In addition to the possible systematics in the stellar masses and SHMR, as previously described, we also note that the mean mass accretion rate given by Equation <ref> is poorly constrained for halos with masses of log(M_h/ M_⊙) ≳ 14 beyond z ∼ 0.5. 
Furthermore, Equation <ref> alone does not account for the diversity of evolutionary paths that real dark matter halos undergo, being the mean result for many halos in the Millennium simulation. 
Therefore, as for the high-redshift SHMR-derived halo masses, we also take these SHMR-derived z=0 and intermediate masses to be upper limits. 
This upper limit on the halo mass at the SMG redshift and the evolution to the present day is shown in Figure <ref> as the upper edges of the shaded regions for ALESS 75.2 and 102.1, and as a single solid line for ALESS 5.1. 





  §.§.§ The SCM method for deriving halo masses



An alternative method for estimating the present-day mass of each SMG environment is obtained following <cit.>, which approximates each SMG environment as a homogeneous sphere undergoing spherical collapse. In this case the total present-day mass is given by:


    M_h,z=0 = ρ̅ V (1 + δ_m) ,


where ρ̅ is the mean comoving matter density of the Universe, δ_m is the dark matter mass overdensity, and V is the comoving volume of the structure. We refer to this method as the spherical collapse model (SCM) method. Since the assumption of spherical collapse is unphysical for environments that are not overdense, we only perform this calculation for the two SMG environments at z ∼ 2.3.

To estimate the volume of each overdensity, we assume that the structures are spherical and use the spatial extent of the structure on the sky to infer the angular diameter of the sphere containing the member galaxies. 
As discussed in <ref>, the overdensity around ALESS 75.2 extends beyond the confines of the HAWK-I pointing and therefore the size of the HAWK-I field-of-view can be used as a lower limit on the angular diameter of the structure. 
Therefore, for this environment we calculate the comoving volume for a spatial extent of ∼ 7.5 <cit.>, which corresponds to a comoving volume of V ∼ 1000 cMpc^3. This volume should be considered a lower limit, and thus the derived halo mass is also a lower limit. 
For the environment of ALESS 102.1, we assume that the structure is confined to the quadrant containing the SMG (<ref>), such that the angular diameter of the sphere is then ∼ 220 <cit.>, which gives a comoving volume of V ∼ 110 cMpc^3.

The dark matter mass overdensity is linked to the observed galaxy overdensity, δ_g, via 


    δ_m = δ_g/b ,


where b is the bias parameter. To estimate the bias parameters for  emitters at z ∼ 2.3, we make use of the relationship between b and L_Hα derived by <cit.> at z = 2.23. Following a similar method to that of <cit.>, we derive a linear fit to this relation and estimate b at the mean value of L_Hα for each SMG environment. This gives b_Hα = 2.9^+0.2_-0.1  and 2.8^+0.2_-0.3 for the candidate companions of ALESS 75.2 and 102.1, respectively.

Using the above values of b and V along with the δ_g values calculated in <ref>, we obtain present-day halo masses of 
log(M_h,z=0/ M_⊙) = 13.90^+0.09_-0.10 and 13.05^+0.12_-0.14 
for the environments of ALESS 75.2 and ALESS 102.1, respectively. Contrary to the previous present-day mass estimates, these masses suggest that descendant of the environment of ALESS 102.1 is more akin to a galaxy group than a galaxy cluster <cit.>, while the environment of ALESS 75.2 may evolve into a `Virgo-like' or `Fornax-like' cluster by z = 0 <cit.>.

The significant disparity between the present-day halo mass estimates from the SCM method compared with the SHMR method likely arises from the assumptions and uncertainties in the calculations, including the systematics discussed in <ref> and the estimates of the structure volumes. 
Note that for ALESS 75.2 the derived z=0 SCM halo mass is a lower limit, due to the volume used being a lower limit. 
Since the masses derived from the SCM method are all lower than the those from the SHMR method and its uncertainties, we proceed with the SCM estimates and adopt the lower bound of the 1–σ confidence intervals as lower limits for the present-day halo masses.
Thus, the present-day halo mass estimates from the two methods gives a range of plausible evolutionary pathways for the two z ∼ 2.3 SMG environments as shown on Figure <ref>.

We also use the present-day halo mass estimates from the SCM method in combination with  Equation <ref> to trace the evolution of these SMG environments back in time to their observed redshifts, thereby obtaining a second estimate of the total mass at these redshifts. These masses are 
log(M_h/ M_⊙) = 12.93^+0.07_-0.08 and 12.24^+0.10_-0.11 
for the potential structures around ALESS 75.2 and ALESS 102.1, respectively. 
This calculation provides an evolutionary track that connects the lower halo mass limit at the z ∼ 2.3 to the corresponding value at z = 0, and this is what defines the bottom edge of each shaded region in Figure <ref>. 












  §.§.§ Halo masses and evolution



The halo mass estimates of each SMG environment are summarised in Table <ref>, and Figure <ref> compares the SMG halo masses and their evolution with previously-studied galaxy clusters and protoclusters.  
As described in <ref> and <ref>, the upper and lower bounds shown for the SMG halos are derived from the masses calculated using the SHMR method and the lower limits from the SCM method, respectively (with the exception of ALESS 5.1, for which only the SHMR method is used). Thus, this region encompasses the full range of possible halo masses and evolution for the SMGs. 
In Figure <ref> these are compared with clusters from the Cluster Lensing And Supernova Survey with Hubble <cit.>; clusters from the Gemini Cluster Astrophysics Spectroscopic Survey <cit.>; clusters and protoclusters from the Clusters Around Radio-Loud AGN program <cit.>. We also show the halo masses calculated by <cit.> for overdense structures known to be rich in dusty star-forming galaxies (DSFGs), including: the GOODS-N protocluster at z = 1.99 <cit.>; the COSMOS protoclusters at z = 2.10 <cit.> and z = 2.47 <cit.>; the `Spiderweb' protocluster, MRC 1138-256, at z = 2.16 <cit.>; the SSA 22 protocluster at z = 3.09 <cit.>; the GN20 overdensity at z = 4.05 <cit.>. 
In this sample of DSFG-rich protoclusters we additionally include the halo mass of the DRC at z = 4 <cit.>. 
Figure <ref> shows that the potential structures surrounding ALESS 75.2 and 102.1 are consistent with being protoclusters at z ∼ 2.3. The upper limit for the halo mass derived for ALESS 5.1 is also consistent with this environment being a protocluster, but we emphasise that our previous analyses suggest it is unlikely to reside in such a structure. 

Both the present-day  and  high-redshift masses obtained using the SHMR method (<ref>) are significantly higher than those obtained using the SCM method (<ref>), and higher than expected for three structures all found in a survey of ∼0.25 deg^2 based on predictions from halo mass functions <cit.>. This is consistent with the SHMR masses being affected by systematic effects that make them upper limits, as discussed in <ref>. The SCM-derived masses are typically lower than the masses of protoclusters at similar redshifts (see Figure <ref>), but are consistent with the results of SMG clustering studies <cit.>, which generally agree that SMGs reside in halos with log(M_h/ M_⊙) ≲ 13 at z = 1–3. 
However, we note that the studies by <cit.> and <cit.> were both conducted using single-dish observations and are thus limited by false counterpart identification and source blending. Although the high-resolution interferometric studies by <cit.> and <cit.> are not afflicted by these limitations, they present conflicting results for the halo masses of the SMG population, likely stemming from the methodological differences described in <ref>.

Figure <ref> also includes the approximate boundaries separating different gas regimes, as proposed by <cit.>: in halos for which log(M_h/ M_⊙) ≲ 12, inflowing gas is predominantly cold and enables the growth of galaxies; in halos above this mass threshold, these gas inflows are shock-heated resulting in strangulation of the galaxy within. However, if these massive halos still fall below some other, redshift-dependent mass threshold (as marked by the `cold in hot' boundary in Figure <ref>), then penetrating cold gas may still be sustaining galaxy growth. At z ∼ 2.3 this mass threshold is log(M_h/ M_⊙) ∼ 12.9, while at z = 3.3 it is log(M_h/ M_⊙) ∼ 14.0.

Based on our mass estimates, the halo of ALESS 5.1 is likely in the `cold in hot' category at its observed redshift, particularly when noting that the mass of this halo is possibly overestimated. It is therefore probable that ALESS 5.1, along with any other galaxy that may share its halo, is undergoing growth sustained by penetrating cold gas inflows. Conversely galaxies in the environment of ALESS 75.2 are more likely to be undergoing strangulation due to shock-heating in the halo at the observed redshift; the SHMR-derived halo mass lies significantly above the limit for `cold in hot' gas inflows, and the 1σ confidence interval for the SCM-derived mass only just crosses below the limit. 
We cannot conclude anything about the gas regime in the environment of ALESS 102.1 as the `cold in hot' boundary is straddled by the mass estimates for this structure.







§ CONCLUSIONS

We have conducted a wide-field narrowband survey of star-forming galaxies in the environments of three SMGs at z ∼ 2.3 and z ∼ 3.3 to determine whether these SMGs reside in protocluster-like environments. By studying individual SMGs selecting based only on their spectroscopic redshifts we have measured `typical' SMG environments. 
Our main conclusions are as follows:

  
  * Using HAWK-I  and  photometry, we identified a total of 147 candidate emission line galaxies in the two HAWK-I pointings containing the three target SMGs. After extracting photometry from archival UV-to-NIR broadband images, we performed SED fitting with  to obtain photometric redshifts for these galaxies and identified 44, 11, and 4 companion galaxies to the SMGs ALESS 75.2 (z_spec = 2.294), ALESS 102.1 (z_spec = 2.296), and ALESS 5.1 (z_spec = 3.303), respectively.
 
  
  * By constructing luminosity functions for each SMG environment and comparing with blank-field luminosity functions from the literature at similar redshifts, we measure overdensity parameters of δ_g = 0.2^+2.5_-0.7, 2.6^+1.4_-1.2, and 0.2^+0.6_-0.5 across the whole ∼4 Mpc HAWK-I field of view for ALESS 5.1, 75.2, and 102.1, respectively. 
    Whilst ALESS 102.1 is not overdense on these large scales, it does sit in a ∼1 Mpc region with δ_g=3.8^+2.4_-1.8. Therefore 2/3 of the target SMGs reside in overdense environments.
 
  
  * We considered the spatial distribution of companion  emitters in the environments of the two SMGs at z ∼ 2.3 (ALESS 75.2 and 102.1) by measuring their density in annuli around the SMGs and by constructing overdensity maps. For ALESS 75.2 the companion galaxies are spread out across the entire HAWK-I field of view, spanning a few Mpc. This is consistent with simulations, in which protoclusters are seen to extend over several Mpc at z ∼ 2–3. The SMG resides near a possible density peak of  emitters, although a greater peak is seen a few arcminutes eastward which appears to be associated with a previously discovered  blob <cit.>. 
    The overdensity around ALESS 102.1 is smaller (∼1 Mpc) and could instead evolve into a galaxy group locally. 
    
  
  * Stellar masses and SFRs were obtained for the companion galaxies in each SMG environment by performing SED fitting with . The galaxies are generally scattered about the star-forming main sequence at their respective epochs, with no evidence of enhanced star formation activity in either environment.
 
  
  * Two methods were used to estimate the total halo mass of each of the two overdense SMG environments, which provided upper and lower bounds on the halo masses at the observed redshifts and evolving to the present day. 
  These reveal that ALESS 75.2 likely resides in a protocluster, while ALESS 102.1 resides in either a protocluster or a protogroup. 
  
	
  * We therefore surmise that 2/3 of these SMGs are strong candidates for the progenitors of massive elliptical galaxies in clusters, although the possibility remains for them to end up in galaxy groups. If these targets are indeed representative of `typical' SMGs then this suggests that SMGs in general are likely to evolve into massive elliptical galaxies by the present day, as suggested by certain evolutionary models <cit.>, but with significant variation in the surrounding environments.


With this study we have demonstrated the efficacy of narrowband surveys as a means of searching for galaxy overdensities around SMGs selected without bias towards particular environments. 
Future followup with larger samples of SMGs and/or spectroscopic confirmation of companion galaxies would confirm the nature of the overdensities that we have detected, and resolved analyses (e.g. with ALMA and/or JWST) will further reveal how the member galaxies evolve. 








§ ACKNOWLEDGEMENTS


We are extremely grateful to Ian Smail for numerous insightful discussions, which significantly improved this work. The research is based on observations collected at the European Southern Observatory under ESO programme 0103.A-0668(A).
TC received support from the Science and Technology Facilities Council (STFC; 2287406) and the Faculty of Science and Technology at Lancaster University. 
JW acknowledges an STFC Ernest Rutherford Fellowship (ST/P004784/2).
HD acknowledges support from the Agencia Estatal de Investigación del Ministerio de Ciencia, Innovación y Universidades (MCIU/AEI) under grant (Construcción de cúmulos de galaxias en formación a través de la formación estelar oscurecida por el polvo) and the European Regional Development Fund (ERDF) with reference (PID2022-143243NB-I00/10.13039/501100011033). 
The authors would also like to acknowledge the publicly available software Topcat <cit.>, as well as the Python packages NumPy <cit.>, SciPy <cit.>, Astropy <cit.>, Matplotlib <cit.>, and Photutils <cit.>.





§ DATA AVAILABILITY


The data used in this publication can be accessed via the http://archive.eso.org/eso/eso_archive_main.htmlESO Science Archive under Program ID: 0103-0668(A), PI: Wardlow.






mnras
 
















 Paper Title: Detection of antifreeze molecule ethylene glycol in the hot molecular core G358.93$-$0.03 MM1.



firstpage–lastpage
Multi-Site Class-Incremental Learning with Weighted Experts in Echocardiography
    Kit M. Bransby1,2, Woo-Jin Cho Kim1, Jorge Oliveira1, Alex Thorley1, Arian Beqiri1, Alberto Gomez1, Agisilaos Chartsias1
    August 5, 2024
============================================================================================================================



§ ABSTRACT 
The identification of complex prebiotic molecules using millimeter and submillimeter telescopes allows us to understand how the basic building blocks of life are formed in the universe. In the interstellar medium (ISM), ethylene glycol ((CH2OH)_2) is the simplest sugar alcohol molecule, and it is the reduced alcohol of the simplest sugar-like molecule, glycolaldehyde (CH2OHCHO). We present the first detection of the rotational emission lines of aGg^' conformer of ethylene glycol ((CH2OH)_2) towards the hot molecular core G358.93–0.03 MM1 using the Atacama Large Millimeter/Submillimeter Array (ALMA). The estimated column density of aGg^'-(CH2OH)_2 towards the G358.93–0.03 MM1 is (4.5±0.1)×10^16 cm^-2 with an excitation temperature of 155±35 K. The abundance of aGg^'-(CH2OH)_2 with respect to H2 is (1.4±0.5)×10^-8. Similarly, the abundances of aGg^'-(CH2OH)_2 with respect to CH2OHCHO and CH3OH are 3.1±0.5 and (6.1±0.3)×10^-3. We compare the estimated abundance of aGg^'-(CH2OH)_2 with the existing three-phase warm-up chemical model abundance of (CH2OH)_2, and we notice the observed abundance and modelled abundance are nearly similar. We discuss the possible formation pathways of aGg^'-(CH2OH)_2 towards the hot molecular cores, and we find that aGg^'-(CH2OH)_2 is probably created via the recombination of two CH2OH radicals on the grain surface of G358.93–0.03 MM1.



ISM: individual objects (G358.93–0.03 MM1) – ISM: abundances – ISM: kinematics and dynamics – stars:
formation – astrochemistry








§ INTRODUCTION


In the past few years, the higher sensitivity of radio and (sub)millimeter telescopes have made it possible to identify simple and complex molecules with an increasing number of atoms in the interstellar medium (ISM). Interstellar molecules, which contain more than six atoms, including carbon, are known as complex organic molecules (hereafter COMs) <cit.>. COMs are mainly found in dense collapsing cloud cores <cit.>, protoplanetary disks <cit.>, hot molecular cores (hereafter HMCs) <cit.>, high-mass protostars <cit.>, and hot corinos <cit.>. The presence of complex prebiotic molecules in the ISM suggests that those molecules are created during the initial stage of star formation and that they are preserved until the development of small bodies. The formation mechanisms of COMs in the star-forming regions are being intensively debated in astrochemistry. Two possible routes have been proposed to create COMs: gas phase and grain surface chemical reactions <cit.>. Only the identification of COMs and their relative abundances and two- or three-phase warm-up chemical models in a large number of star-forming regions and HMCs will help us understand the chemical formation pathways of the COMs <cit.>.

Ethylene glycol (hereafter EG) is known as a di-alcohol molecule that is commonly used in prebiotic sugar synthesis. The EG molecule is the sugar alcohol of the aldehyde sugar molecule glycolaldehyde (hereafter GA). EG is an asymmetric top molecule with coupled rotation around its two C–O bonds and one C–C bond, which generates different conformers <cit.>. There are six conformers with respect to the C–C bond with gauche (G) arrangement of the two H2C = CH2 groups and four low-stable conformers with anti-arrangement of the CH2 groups. In the G conformer, the two OH groups pick the gauche orientation with respect to each other. The `a' and `g' notations are used for the anti (a) and gauche (g) conformers of the OH groups with respect to the two C–O bonds. When the OH group provides the hydrogen for the intramolecular bonding, the gauche orientation is noted as aGg^'. The EG molecule has only two conformers, aGg^' and gGg^'. The 3D molecular structure of aGg^'-EG and gGg^'-EG is shown in Fig. <ref>. The gGg^' conformer of EG is 200 cm^-1 or 290 K higher in energy than aGg^' <cit.>. For aGg^' and gGg^' conformers of EG, the tunneling is observed between two equivalent equilibrium configurations, and it splits each rotational level into two different states, v = 0 and v = 1 <cit.>. For the aGg^' and gGg^' conformers, the v = 1 state is around 7 and 1.4 GHz higher than the v = 0 <cit.>. The emission lines of EG were first detected towards the galactic center along the line of sight to Sgr B2, but the several detected lines of EG were blended <cit.>. Evidence of EG was also found towards the HMCs Orion KL, W51 e2, G34.3+0.2, G31.41+0.31, G10.47+0.03 <cit.>, and hot corinos IRAS 16293-2422 B, NGC 1333 IRAS 2A <cit.>. Additionally, the emission lines of EG were also found towards the comets Hale-Bopp, C/2012 F6 (Lemmon), C/2013 R1 (Lovejoy), 67P/Churyumov-Gerasimenko, and meteorites Murchinson and Murray <cit.>.

In the ISM, HMCs are one of the earliest stages of the high-mass star-formation regions, and most of the complex and prebiotic molecules, including possible glycine (NH2CH2COOH) precursors, are found in these objects <cit.>. COMs in HMCs play a crucial role in increasing the chemical complexity in the ISM <cit.>. HMCs are small, compact objects (≤0.1 pc) with a warm temperature (≥100 K) and a high gas density (≥10^6 cm^-3) <cit.>. In the ISM, these objects are short-lived (10^5 yr to 10^6 yr) <cit.>. Most of the COMs are observed in the warm-inner regions of the envelope of HMCs, where the temperature exceeds the water ice desorption temperature of ∼100 K <cit.>. The fractional abundances of the COMs in the HMCs are found to be between 10^-7 and 10^-11 with respect to molecular H2 <cit.>. In the HMCs, the gas temperature starts to increase due to the collapse of the pre-stellar core. This is called the "warm-up" phase <cit.>. In this phase, several COMs are created in the ice, then sublimated and destroyed in the gas phase <cit.>. So the resultant abundance of the complex molecules in the gas depends on the initial abundances of the precursors <cit.>. Studies at the beginning of the 2000s showed that known (at the time) gas phase reactions were not efficient enough to explain the high abundances of COMs observed in HMCs. However, more recent quantum chemical studies have brought attention back to the gas phase chemistry in forming COMs <cit.>.




The HMC candidate G358.93–0.03 MM1 is situated in the high-mass star-formation region G358.93–0.03 at a distance of 6.75^ +0.37_ -0.68 kpc <cit.>. The luminosity and total gas mass of G358.93–0.03 are ∼7.7×10^3 _⊙ and 167±12 _⊙ <cit.>. In this region, there exist a total of eight sub-millimeter continuum sources, which are defined as G358.93–0.03 MM1 to G358.93–0.03 MM8 in order of decreasing right ascension <cit.>. Recently, <cit.> found two more continuum sources associated with G358.93–0.03 MM1 and G358.93–0.03 MM2. Among the eight continuum sources, G358.93–0.03 MM1 and G358.93–0.03 MM3 are believed to be HMCs because the emission lines of CH3CN and maser lines of CH3OH are evident in those sources <cit.>. The G358.93–0.03 MM1 hot core is more chemically rich than the G358.93–0.03 MM3 <cit.>. Earlier, the maser lines of HDO, HNCO, and CH3OH were detected towards the G358.93–0.03 MM1 using ALMA, TMRT, and VLA <cit.>. Recently, the rotational emission lines of the possible urea precursor molecule cyanamide (NH2CN), simplest sugar-like molecule glycolaldehyde (CH2OHCHO), possible NH2CH2COOH precursor molecule methylamine (CH3NH2), peptide bond molecules formamide (NH2CHO) and isocyanic acid (HNCO) were also detected from G358.93–0.03 MM1 using ALMA bands 6 and 7 <cit.>.

In this article, we present the first detection of the simplest sugar alcohol molecule, EG, towards the HMC object G358.93–0.03 MM1 using the ALMA. In Section <ref>, we discuss the ALMA data and their reduction. The result of the identification and spatial distribution of EG is presented in Section <ref>. The discussion and conclusions of the identification of EG are shown in Section <ref> and <ref>, respectively.







§ OBSERVATIONS AND DATA REDUCTION


The high-mass star-formation region G358.93–0.03 was observed using ALMA band 7 receivers to study the massive protostellar accretion outburst (ID: 2019.1.00768.S., PI: Crystal Brogan). The observations were performed on October 11th, 2019 with on-source integration times of 756 s. The observed phase center of G358.93–0.03 was (α,δ)_ J2000 = 17:43:10.000, –29:51:46.000. A total of 47 antennas were used to observe the G358.93–0.03 with a minimum baseline of 14 m and a maximum baseline of 2517 m. During the observation, J1550+0527 was used as a flux calibrator and bandpass calibrator, and J1744–3116 was used as a phase calibrator. The observed frequency ranges of G358.93–0.03 were 290.51–292.39 GHz, 292.49–294.37 GHz, 302.62–304.49 GHz, and 304.14–306.01 GHz, with a spectral resolution of 977 kHz.

For data reduction and imaging, we used the Common Astronomy Software Application (CASA 5.4.1) with the ALMA data analysis pipeline <cit.>. In the pipeline, we used the task SETJY with the Perley-Butler 2017 flux calibrator model for flux calibration <cit.>. We also applied the pipeline tasks hifa_bandpassflag and hifa_flagdata for bandpass calibration and flagging the bad antenna data. After the initial data reduction, we used the task MSTRANSFORM to separate the target object G358.93–0.03. We created the four continuum images of G358.93–0.03 by using line-free channels at frequency ranges of 290.51–292.39 GHz, 292.49–294.37 GHz, 302.62–304.49 GHz, and 304.14–306.01 GHz using the CASA task TCLEAN with the HOGBOM deconvolver. Recently, <cit.> analyzed this data, and they showed the continuum emission image of G358.93–0.03 at frequency 303.39 GHz (988 μm) (Fig. 1. of <cit.>). <cit.> clearly detected the eight sub-millimeter wavelength continuum sources, G358.93–0.03 MM1 to G358.93–0.03 MM8. <cit.> also detected other two continuum sources associated with G358.93–0.03 MM1 and G358.93–0.03 MM2, which are defined as G358.93–0.03 MM1A and G358.93–0.03 MM2A. Before making the spectral images, we use the task UVCONTSUB to subtract the background continuum emission from the UV plane of the calibrated data. We also made self-calibration multiple times using the tasks GAINCAL and APPLYCAL for better RMS of the final images. We create the spectral images of G358.93–0.03 at frequency ranges of 290.51–292.39 GHz, 292.49–294.37 GHz, 302.62–304.49 GHz, and 304.14–306.01 GHz using the task TCLEAN with the SPECMODE = CUBE parameter. Finally, we apply the task IMPBCOR for the correction of the primary beam pattern in the spectral images of G358.93–0.03.



§ RESULTS





 §.§ Line emission from the G358.93–0.03

In the spectral images of G358.93–0.03, we observe that only the spectra of G358.93–0.03 MM1 and G358.93–0.03 MM3 show any line emission. The synthesized beam sizes of the spectral images of star-forming region G358.93–0.03 at frequency ranges of 290.51–292.39 GHz, 292.49–294.37 GHz, 302.62–304.49 GHz, and 304.14–306.01 GHz are 0.42 arcsec × 0.36 arcsec, 0.42 arcsec × 0.37 arcsec, 0.41 arcsec × 0.36 arcsec, and 0.41 arcsec × 0.35 arcsec, respectively. We generate the molecular spectra from the HMCs G358.93–0.03 MM1 and G358.93–0.03 MM3 by creating a 0.91 arcsec diameter circular region, which is larger than the line-emitting regions of these two sources. The extracted molecular spectra of G358.93–0.03 MM1 and G358.93–0.03 MM3 in the observable frequency ranges are shown in Fig. 2 of <cit.>. We notice that the molecular spectra of G358.93–0.03 MM1 are more chemically rich than the spectra of G358.93–0.03 MM3. In the molecular spectra of G358.93–0.03 MM1, we notice the evidence of an inverse P Cygni profile associated with the CH3OH emission lines. The presence of the inverse P Cygni profile indicates that the G358.93–0.03 MM1 is undergoing infall. The position of G358.93–0.03 MM1 is RA (J2000) = 17^h43^m10^s.101, Dec (J2000) = –29^∘51^'45^''.693. The position of G358.93–0.03 MM3 is RA (J2000) = 17^h43^m10^s.0144, Dec (J2000) = –29^∘51^'46^''.193. The systemic velocities (V_LSR) of the spectra of G358.93–0.03 MM1 and G358.93–0.03 MM3 are –16.5 km s^-1 and –18.2 km s^-1, respectively <cit.>.










  §.§.§ Detection of EG towards the G358.93–0.03 MM1


We assume local thermodynamic equilibrium (LTE) and used the Cologne Database for Molecular Spectroscopy (CDMS) <cit.> to identify the rotational emission lines of EG from the molecular spectra of G358.93–0.03 MM1. We use CASSIS for the LTE modelling <cit.>. The LTE assumption is reasonable in the inner region of the G358.93–0.03 MM1 because the maximum gas density of the warm inner region of the hot core is 1×10^8 cm^-3 <cit.>. We used the Markov Chain Monte Carlo (MCMC) algorithm in CASSIS for fitting the LTE model spectra of EG over the observed molecular spectra of G358.93–0.03 MM1. The details of the MCMC fitting method using the CASSIS are described in <cit.>. After the LTE analysis, we have detected a total of 106 transitions of the aGg^' conformer of EG with ≥2.5σ statistical significance in the spectra of G358.93–0.03 MM1 in the observable frequency ranges. After the spectral analysis using the LTE model, we observe that 35 transitions of aGg^'-EG are non-blended, and all of those non-blended transitions are detected above 5σ. To understand the non-blended transitions of aGg^', we have used more than 200 different molecular transitions in the LTE modelling, which are taken from the CDMS database, including those molecules detected by <cit.>, <cit.>, <cit.>, and <cit.>. We find that the 71 transition lines of aGg^'-EG are blended with the other nearby molecular transitions. The upper-level energies of the detected 106 transitions of aGg^'-EG vary from 76.49 K to 597.41 K. The upper-level energies of the non-blended transitions of aGg^'-EG vary between 206.05 K and 416.16 K. There are no missing high-intensity transitions of aGg^'-EG in the observable frequency ranges. After the LTE modelling, the best-fit column density of aGg^'-EG is (4.5±0.1)×10^16 cm^-2 with an excitation temperature of 155±35 K and a source size of 0.50 arcsec. During the LTE fitting, we used the FWHM of the LTE spectra of aGg^'-EG is 3.21 km s^-1. We have taken this FWHM value for LTE modelling of aGg^'-EG because recently <cit.> estimated the FWHM of CH2OHCHO in the spectra of G358.93–0.03 MM1 using this data was 3.2 km s^-1. The LTE-fitted rotational emission spectra of aGg^'-EG towards the G358.93–0.03 MM1 are shown in Fig. <ref>. The LTE-fitted spectral line parameters of aGg^'-EG are shown in Tab. <ref>. After the detection of the emission lines aGg^'-EG from the spectra of G358.93–0.03 MM1, we also search the emission lines of gGg^'-EG using the LTE modelling spectra. After the spectral analysis, we observe that all detected lines of gGg^'-EG are blended with other nearby molecular transitions. Thus, all detected lines of gGg^'-EG are blended with other molecules, so we estimate the upper limit column density of gGg^'-EG towards the G358.93–0.03 MM1 is ≤(2.9±0.6)×10^16 cm^-2. <cit.> estimated that the temperature of GA is 300 K, and we estimate that the temperature of EG towards the G358.93–0.03 MM1 is 155 K. Previously, <cit.> showed that the temperatures of complex molecules vary between 100 K to 300 K in the inner shell of hot molecular core (see Fig 8 in <cit.>). So, both molecules EG and GA exist in slightly different regions of the inner shell of G358.93–0.03 MM1.



  §.§.§ Abundance of EG towards the G358.93–0.03 MM1

To determine the fractional abundance of aGg^'-EG, we use the column density of aGg^'-EG inside the 0.50 arcsec beam, which we divide by the molecular H2 column density. The fractional abundance of aGg^'-EG with respect to molecular H2 towards the G358.93–0.03 MM1 is (1.4±0.5)×10^-8, where the column density of H2 towards the G358.93–0.03 MM1 is (3.1±0.2)×10^24 cm^-2 which is derived from the dust continuum emission at wavelength 988 μm <cit.>. The column density ratio of aGg^'-EG and GA (hereafter EG/GA) is 3.1±0.5, where the column density of GA towards the G358.93–0.03 MM1 is (1.5±0.9)×10^16 cm^-2 <cit.>. After spectral analysis using the LTE model, we also find the column density and excitation temperature of CH3OH towards G358.93–0.03 MM1, which are (7.5±0.5)×10^18 cm^-2 and 150±28 K, respectively. So, the column density ratio of aGg^'-EG and CH3OH is (6.1±0.3)×10^-3. <cit.> and <cit.> derived the abundance of aGg^'-EG with respect to H2 towards two other hot molecular cores, G10.47+0.03 and G31.41+0.31, to be (3.7±1.5)×10^-8 and (1.5±0.4)×10^-8 respectively, which are close to our derived abundance of aGg^'-EG towards the G358.93–0.03 MM1. This may indicate that the chemical formation pathway(s) of aGg^'-EG towards the G358.93–0.03 MM1 may be similar to those in G10.47+0.03 and G31.41+0.31.



  §.§.§ Searching for EG towards the G358.93–0.03 MM3

We also attempt to search for emission lines of aGg^'-EG and gGg^'-EG towards the G358.93–0.03 MM3 using the LTE-modelled spectra, but we do not detect them. The derived upper limit column densities of aGg^'-EG and gGg^'-EG towards the G358.93–0.03 MM3 are ≤(1.6±0.9)×10^15 cm^-2 and ≤(1.2±0.5)×10^15 cm^-2 respectively. We assumed the excitation temperature of 155 K to estimate the upper limit column density of EG towards the G358.93–0.03 MM3. The upper limits of their fractional abundance are ≤(4.5±1.2)×10^-9 and ≤(3.4±0.7)×10^-9 respectively, where the column density of H2 towards the G358.93–0.03 MM3 is (3.5±0.7)×10^23 cm^-2 <cit.>.






 §.§ Spatial distribution of aGg^'-EG towards the G358.93–0.03 MM1

We produce the integrated emission maps (moment zero maps) of some selected highly intense non-blended emission lines of aGg^'-EG towards the G358.93–0.03 MM1 using the task IMMOMENTS. During the run of the task IMMOMENTS, we use the channel ranges of the spectral images where the emission lines of aGg^'-EG are detected. We create the integrated emission maps of aGg^'-EG for non-blended transitions towards the G358.9–0.03 MM1 at Fig. <ref>. We overlaid the 988 μm continuum emission map of G358.93–0.03 over the integrated emission maps of aGg^'-EG. The continuum emission map of G358.93–0.03 is taken from <cit.>. We found that the integrated emission maps of aGg^'-EG exhibit a peak at the position of the continuum. The integrated emission maps indicate that the emission lines of aGg^'-EG originate from the high density, the warm inner region of G358.93–0.03 MM1. After the extraction, we apply the CASA task IMFIT to fit the 2D Gaussian over the integrated emission maps of aGg^'-EG to estimate the size of the emitting regions. The following equation is used
	

    θ_S=√(θ^2_50-θ^2_beam)

where θ_beam indicates the half-power width of the synthesised beam and θ_50 = 2√(A/π) denotes the diameter of the circle whose area surrounded the 50% line peak of aGg^'-EG <cit.>. The derived emitting regions of aGg^' at several frequencies are shown in Tab. <ref>. The synthesized beam sizes of the integrated emission maps vary between 0.41 arcsec × 0.35 arcsec and 0.42 arcsec × 0.36 arcsec. The derived emitting regions of aGg^'-EG at different frequencies vary between 0.40 arcsec and 0.42 arcsec. After fitting a 2D Gaussian, we observe that the size of the emitting regions of aGg^'-EG is comparable to or slightly greater than the synthesized beam sizes of the integrated emission maps. This result indicates that the detected transition lines of aGg^'-EG are not spatially resolved or are only marginally resolved towards the G358.93–0.03 MM1. As a result, drawing any conclusions about the morphology of the spatial distributions of aGg^'-EG towards the G358.93–0.03 MM1 is impossible. Higher spatial and angular resolution observations are required to understand the spatial distribution of aGg^'-EG towards the G358.93–0.03 MM1.





§ DISCUSSION




 §.§ Comparison of the EG/GA ratio in different star-forming regions

We compared the EG/GA ratio in G358.93–0.03 MM1 with other regions such as the massive star-forming regions Orion KL <cit.>, W51e2 <cit.>, G34.3+0.2 <cit.>, the intermediate-mass hot core NGC 7129 FIRS2 <cit.>, the HMC candidate G31.41+0.31 <cit.>, the low-mass protostars IRAS 16293 B <cit.>, and NGC 1333 IRAS 2A <cit.>. The EG/GA ratios of the above-mentioned sources are shown in Tab. <ref>. The uncertainty of the EG/GA ratio in G358.93–0.03 MM1 can be deduced by considering the propagation of the uncertainty of the column densities of EG and GA. The behaviour of the EG/GA with respect to the luminosity of the different star-forming regions is shown in Fig. <ref>. As per Fig. <ref>, the EG/GA ratios increased with luminosities, with a lower value of 1 for the low-mass protostar IRAS 16293 B and a higher value of 10 for the HMC object G31.41+0.31. The lower limit of the hot cores in Orion KL, G34.3+0.2, and W51e2 is as high as 15. Earlier <cit.> also see the same nature from the variation of the EG/GA ratios with luminosities in the star-forming regions, but they do not describe why these ratios increase with luminosities. We observe different EG/GA ratios because different initial compositions of the ices could produce very different values of the EG/GA ratios <cit.>. Pure CH3OH ices create a ratio of >10, but ices with a composition of CH3OH:CO 1:10 produce a ratio of <0.25. This implies that the initial composition of the grains is significantly different for the high- and low-mass star-forming regions. Earlier, <cit.> computed the two-phase warm-up chemical modelling of EG and GA, and they also observed that the modelled EG/GA ratios decreased with luminosities for different low- and high-mass star-formation regions (see Fig. 6 and 7 in <cit.>). Subsequently, <cit.> revised the physical assumptions in chemical modelling and added the grain mantle desorption. After this correction, they observed that the modelled EG/GA ratios increased with the luminosities (see Fig.  12 in <cit.>). We see that the modelled EG/GA ratios in <cit.> cannot match any sources because the modelled EG/GA ratios with respect to luminosities are larger than the observed EG/GA ratios for different high- and low-mass sources. 






 §.§ Possible formation pathways of EG in HMCs

In HMCs, grain surface chemistry is crucial for the formation of COMs, including EG and GA. Previously, several grain surface chemical routes have been proposed, in which some of the reactions involved thermal and energetic processes <cit.>. In HMCs and hot corinos, the warm-up phase is thought to be important for the production of COMs. The warm-up phase allows the more strongly bound radicals to become mobile at the grain surface <cit.>. Earlier, several experimental studies showed that the photochemistry in CH3OH–CO ice mixtures is sufficient to explain the observed abundance of EG in the star-forming regions <cit.>. Initially, two main formation pathways were proposed for the formation of EG


HCO + CH2OH → CH2OHCHO             (1)


CH2OHCHO + H → CH2OHCHOH        (2)


CH2OHCHOH + H → (CH2OH)_2            (3)


and


HCO + 2H → CH2OH                              (4)


CH2OH + CH2OH → (CH2OH)_2            (5)


Reaction 1 indicates that the reaction between radical HCO and radical CH2OH on the gain surfaces creates GA. Similarly, reactions 2 and 3 indicate that the subsequential hydrogenation of GA on the grain surface produces the EG <cit.>. Earlier, <cit.> raises many questions about the formation of EG using reactions 2 and 3. If EG is created via GA, a relatively constant EG/GA ratio would be expected <cit.>. In Fig. <ref>, we observe that the EG/GA ratio varies by more than an order of magnitude (1 to >15) with increasing luminosity, which suggests that there is no direct link between EG and GA. The large variation in the EG/GA ratio indicates that GA is not a direct precursor of EG, as well as that they do not share a common precursor <cit.>. We observe that EG is more abundant than GA in G358.93–0.03 MM1, which means the unsaturated molecule GA is broken towards the G358.93–0.03 MM1. For that reason, <cit.> claims that reactions 2 and 3 are not sufficient to produce EG from GA. Earlier, <cit.> claimed that the double bond C = O in the unsaturated molecule GA can be easily broken in the gas phase of star-forming regions which indicates there is a possibility that GA can survive on the grain surface and EG can be produced by reactions 2 and 3. Similarly, reactions 4 and 5 indicate that the EG is formed by the recombination of two CH2OH radicals in the grain surface, where radical CH2OH is created via the subsequential hydrogenation of HCO <cit.>. <cit.> also used reaction 5 in the three-phase warm-up chemical models and showed that reaction 5 is the most efficient for the formation of EG in the HMC candidate G10.47+0.03. Previously, <cit.> claimed that reaction 5 is the most efficient pathway to the formation of EG but <cit.> showed that reaction 3 is the most efficient for the formation of EG towards the G31.41+0.31 using the two-phase warm-up chemical modelling. Recently, <cit.> supports the three-phase warm-up chemical model of <cit.> and they show that reaction 5 is the most efficient route to the formation of EG towards the G31.41+0.31.



 §.§ Comparison between the modelled and observed abundance of EG

Recently, <cit.> developed a three-phase (gas + grain + ice mantle) warm-up chemical model using the three-phase astrochemical model code MAGICKAL <cit.> to understand the formation mechanisms and abundance of different COMs, including EG, towards the HMCs. To model a HMC, they first assumed a free-fall collapse of a cloud (phase I), followed by a warm-up phase (phase II). In the first phase, the gas density increases from n_H = 3×10^3 cm^-3 to 2×10^8 cm^-3, and <cit.> assume a constant temperature of 10 K. In the second phase (warm-up phase), the gas density remains constant at 2×10^8 cm^-3 and the temperature increases with time from 8 to 400 K. <cit.> used three different warm-up stages during the chemical model corresponding to a fast, medium, or slow warm-up from 8 to 400 K. The warm-up time scales of this chemical model were 5×10^4 yr (fast), 2×10^5 yr (medium), and 1×10^6 yr (slow). During chemical modelling, Garrod et al. (2022) used different types of chemical reactions for the formation of EG on the grain surface of HMCs (see Tab. 3 of <cit.>). The estimated modelled abundances of EG with respect to H2 in <cit.> was 6.0×10^-8, 1.1×10^-8, and 2.2×10^-12 corresponding to the fast, medium, and slow warm-up models (see Tab. 17 of <cit.>). Similarly, the modelled abundances of EG with respect to CH3OH was 5.5×10^-3, 1.0×10^-3, and 2.7×10^-7 corresponding to the fast, medium, and slow warm-up models (see Tab. 18 of <cit.>). After chemical modelling, <cit.> showed that the EG is formed via the recombination of two CH2OH radicals (reaction 5) on the grain surface of HMCs.
 
To understand the formation pathways of EG towards the G358.93–0.03 MM1, we compare our estimated abundance of EG with the modelled abundance by <cit.>. This comparison is physically reasonable because the maximum gas density and dust temperature of G358.93–0.03 MM1 are 1×10^8 cm^-3 <cit.> and ∼150 K <cit.>, respectively. Hence, the three-phase warm-up chemical model of <cit.> is appropriate for explaining the chemical evolution of EG towards the G358.93–0.03 MM1. We estimate the abundance of EG with respect to H2 towards the G358.93–0.03 MM1 to be (1.4±0.5)×10^-8, which is very close to the medium warm-up model abundance of EG. Similarly, the abundance of EG with respect to CH3OH towards the G358.93–0.03 MM1 is (6.1±0.3)×10^-3, which is very close to the fast warm-up model abundance of EG with respect to CH3OH. This comparison indicates that the simplest sugar alcohol molecule, EG, may form due to the recombination of two CH2OH radicals (reaction 5) on the grain surface of G358.93–0.03 MM1. For G31.41+0.31, <cit.> estimated that the abundance of EG with respect to H2 is (1.5±0.4)×10^-8, which is very close to the medium warm-up model abundance of EG. This means that reaction 5 may be responsible for the production of EG towards the G31.41+0.31. Earlier, <cit.> showed that reaction 5 has an activation barrier, and <cit.> did not include an activation barrier during the chemical modelling. Therefore, it is necessary to verify if the main formation pathway of EG is via the recombination of two CH2OH radicals by using a three-phase warm-up model with the activation barrier.



§ CONCLUSION

 
We present the first confirmed detection of the rotational emission lines of the simplest sugar alcohol molecule, aGg^'-EG, towards the HMC candidate G358.93–0.03 MM1 using ALMA. The derived abundance of aGg^'-EG with respect to H2 towards the G358.93–0.03 MM1 is (1.4±0.5)×10^-8. The EG/GA ratio towards the G358.93–0.03 MM1 is 3.1±0.5. Similarly, the abundance of aGg^'-EG with respect to CH3OH towards the G358.93–0.03 MM1 is (6.1±0.3)×10^-3. We observe that the fractional abundance of EG with respect to H2 towards the G358.93–0.03 MM1 is nearly similar to the abundance of EG towards the other two HMCs, G10.47+0.03 and G31.41+0.31 <cit.>. We discuss the possible formation mechanisms of EG in HMCs. We compare our estimated abundance of EG with respect to H2 towards the G358.93–0.03 MM1 with the three-phase warm-up chemical model abundances of EG from <cit.>, and we find that the derived and medium warm-up modelled abundances of EG are close. Similarly, we also compare the abundance of EG with respect to CH3OH with the modelled abundance values of EG from <cit.>, and we find that the derived and fast warm-up modelled abundances of EG with respect to CH3OH are close. We also notice that the modelled abundance of EG by <cit.> is similar to the observed abundance of EG towards other HMC candidate G31.41+0.31. So, EG may formed via the recombination of two CH2OH radicals on the grain surface of G358.93–0.03 MM1 and G31.41+0.31. Earlier, <cit.> showed that the grain surface reaction between two CH2OH radicals has an activation barrier, and <cit.> did not include the activation barrier during the chemical modelling. As a result, it is essential to confirm whether the main formation mechanism of EG is via the recombination of two CH2OH radicals, using a three-phase warm-up model that incorporates the activation barrier. The identification of high abundances of EG in G358.93–0.03 MM1 suggests that grain surface chemistry is efficient for the formation of other COMs, which should therefore be detectable. A combined spectral line study with an LTE model and three-phase warm-up chemical modelling is required to understand the hot core chemistry towards the G358.93–0.03 MM1, which will be carried out in our follow-up study.
	


§ ACKNOWLEDGEMENTS

We thank the anonymous reviewer and scientific editor Prof. Cecilia Ceccarelli for their helpful comments, which improved the manuscript. A.M. acknowledges the Swami Vivekananda Merit-cum-Means Scholarship (SVMCM) for financial support for this research. S.V.  acknowledges support from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program MOPPEX 833460. This paper makes use of the following ALMA data: ADS /JAO.ALMA#2019.1.00768.S. ALMA is a partnership of ESO (representing its member states), NSF (USA), and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in co-operation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ.
	


§ DATA AVAILABILITY

The plots within this paper and other findings of this study are available from the corresponding author on reasonable request. The data used in this paper are available in the ALMA Science Archive (<https://almascience.nrao.edu/asax/>), under project code 2019.1.00768.S.

aasjournal

	natexlab#1#1
	
	
	
	
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 Paper Title: Photometric properties of classical bulge and pseudo-bulge galaxies at $0.5\le z<1.0$.

 

   
   The classical and pseudo-bulge galaxies
   Hu et al.

   

   
        
   National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China        School of Astronomy and Space Science, University of Chinese Academy of Sciences, Beijing 100049, China        Key Laboratory of Space Astronomy and Technology, National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100101, China        Institute for Frontiers in Astronomy and Astrophysics, Beijing Normal University, Beijing 102206, China    


   



 
  
  
   We compare the photometric properties and specific star formation rate (sSFR) of classical and pseudo-bulge galaxies with M_* ≥ 10^9.5 M_⊙ at 0.5≤ z<1.0, selected from all five CANDELS fields. 
   We also compare these properties of bulge galaxies at lower redshift selected from MaNGA survey <cit.>.
  
   This paper aims to study the properties of galaxies with classical and pseudo-bulges at intermediate redshift, to compare the differences between different bulge types, and to understand the evolution of bulges with redshift.  
   
  
   Galaxies are classified into classical bulge and pseudo-bulge samples according to the Sérsic index n of the bulge component based on results of two-component decomposition of galaxies, as well as the position of bulges on the Kormendy diagram. 
   For the 105 classical bulge and 86 pseudo-bulge galaxies selected, we compare their size, luminosity, and sSFR of various components.
  
   At given stellar mass, most classical bulge galaxies have smaller effective radii, larger B/T, brighter and relatively larger bulges, and less active star formation than pseudo-bulge galaxies. 
   Besides, the two types of galaxies have larger differences in sSFR at large radii than at the central region at both low and mid-redshifts. 
   
  
   The differences between properties of the two types of bulge galaxies are in general smaller at mid-redshift than at low redshift, indicating that they are evolving to more distinct populations towards the local universe. Bulge type is correlated with the properties of their outer disks, and the correlation is already present at redshift as high as 0.5<z<1.
   

   

   Photometric properties of classical bulge and pseudo-bulge galaxies at 0.5≤ z<1.0
    Jia Hu1,2E-mail: hujia@nao.cas.cn,
          Qifan Cui1,3qfcui@nao.cas.cn,
          Lan Wang1,2,
          Wenxiang Pei1,2,4
          Junqiang Ge1,2
        
    Received 00 00, 2024; accepted 00 00, 2024
===================================================================================================================================================================





§ INTRODUCTION



In the central region of a spiral galaxy, there is often a bulge component that contains a higher concentration of stars and light compared to the disk component that extends to a larger radius <cit.>.
Bulges are classified into two main categories based on their properties observed at low redshift.
Like elliptical galaxies, typical classical bulges are dominated by random motion and follow the fundamental plane <cit.>.
In contrast, pseudo-bulges exhibit characteristics similar to disk galaxies, with rotational motion and notable substructures such as nuclear bars, spiral arms, and inner rings <cit.>. 
In general, classical bulge galaxies are more massive, redder <cit.>, and have larger central velocity dispersion <cit.> than pseudo-bulge galaxies.




Theoretically, classical bulges are commonly attributed to violent processes like clumpy collapse <cit.> and galaxy mergers <cit.>, while pseudo-bulges are thought to form slowly through secular evolution <cit.>, involving mild gas inflow and stellar migration triggered by bars or spiral arms <cit.>.
However, simulations, semi-empirical models <cit.> and observations <cit.> indicate that bulge formation and evolution is more complicated than the general picture.


In the previous work of <cit.>, with the help of the MaNGA survey <cit.> and two-component decomposition catalogue <cit.>, we compared the resolved properties of classical and pseudo-bulges and their host galaxies at 0.01<z<0.15. 
We found that at given stellar mass, disks of pseudo-bulge galaxies are younger, have more active star formation, rotate more, and may contain more HI content compared with the disks of classical bulge galaxies.
More interestingly, the differences between the properties of disks for the two types of bulge galaxies are larger than the differences between bulges themselves, indicating that the formation of bulges may be closely related to the evolution of outer disks in galaxies.


While most statistical studies of bulge properties focus on the low redshift range, studying the bulge component of faraway galaxies is more difficult due to the limited resolution of observation.
The central region of higher-redshift galaxies can be resolved only by space telescopes, such as Hubble Space Telescope <cit.>, James Webb Space Telescope <cit.> and China Space Station Telescope <cit.>.
For example, <cit.> explore the growth of different types of bulge galaxies since z ∼ 1 using HST and find that classical bulges are brighter than pseudo ones at all redshift ranges. 
However, there are few studies about the other properties of bulges at mid-redshift.


In this work, to investigate more the properties of bulges and bulge galaxies at intermediate redshift, we study galaxies of different types of bulges at 0.5≤ z<1.0 from five CANDELS <cit.> fields observed by HST. 
To select galaxies with bulge components, we perform two-component decomposition for the galaxies with M_* ≥ 10^9.5 M_⊙. 
Following the previous work of <cit.>, we divide selected samples into classical and pseudo-bulge galaxies using the criteria of bulge Sérsic index combined with the position of bulges on the Kormendy diagram <cit.>.
We compare the photometric properties of bulges and disks including size and luminosity, as well as sSFR calculated by rest-frame UVI colour, and sSFR profiles for the two types of bulge galaxies.



This paper is organised as follows. 
Section <ref> introduces how we select galaxies with different types of bulges from CANDELS. 
Section <ref> studies and compares the photometric properties of bulges and disks.
Section <ref> compares sSFR for the two types of bulges, their galaxies, and the sSFR profile of the galaxies. 
Conclusions and discussions are presented in Section <ref>.
Throughout the paper we adopt flat Λ-CDM concordance model (H_0=70.0 km s^-1 Mpc^-1 and Ω_M=0.30).




§ TWO-COMPONENT DECOMPOSITION AND SAMPLE SELECTION



We perform two-component decomposition fitting for 2178 galaxies with good image quality selected from five CANDLES fields, and select 344 most reliable two-component galaxies from them. 
Then we classify galaxies with classical and pseudo-bulges by criteria of both the bulge Sérsic index and the position of bulges on the Kormendy diagram. 
In the end, 105 classical bulge galaxies and 86 pseudo-bulge galaxies are selected. 



 §.§ Selection of two-component galaxies




For all sources in the five catalogues of CANDELS fields, including COSMOS <cit.>, EGS <cit.>, GOODS-N <cit.>, GOODS-S <cit.> and UDS <cit.>, we first select 2178 galaxies that have image quality good enough for further decomposition fitting, following the criteria presented in the previous works of <cit.> and <cit.>. The details of selection criteria are listed below, and Tab.  <ref> presents the numbers of galaxies selected after applying each of the criteria.


∘ CLASS_STAR < 0.9 are sources identified as galaxies rather than stars, where CLASS_STAR is an output index from Source Extractor <cit.>. 

∘ Only galaxies with F160W(H) band magnitude (M_H) brighter than 24.5 are selected, which have signal-to-noise ratios high enough to get reliable images.

∘ GALFIT FLAG (H) = 0 selects galaxies that can be fitted reliably by GALFIT for F160W band images <cit.>.


∘ Galaxies with redshifts in the range of 0.5 ≤ z < 1.0 are selected (see more detailed redshift measurements in <cit.>).


∘ Small galaxies with M_* < 10^9.5 M_⊙ are removed.
The stellar mass is calculated by fitting SPS models to broadband photometry using the code FAST <cit.>, based on models of <cit.> with initial mass function of <cit.>, exponentially declining τ-models of star formation history, solar metallicity, and a dust law of <cit.>.

∘ The criterion about semi-major axis of whole galaxies <cit.> removes sources that are smaller than the full-width at half-maximum (FWHM) of point spread function (PSF) of HST images (average value in F160W band of 0.18). 
 

∘ In the CANDELS catalogue, we select galaxies that are marked by SExtractor with PhotFlag = 0, indicating that they are not contaminated by foreground sources or neighboring objects, have no bad pixels, and are not located on the borders of the mosaic in the F160W band.


∘ We have also visually inspected the images to ensure that the galaxies are uncontaminated in all bands of F160W, F606W, F814W, and F125W.





Combining the above criteria, we finally select 2178 galaxies to perform the two-component decomposition fitting using GALFIT.
Firstly, we conduct preprocessing for images of sources before the fitting.
We generate multi-band thumbnail images with a size of 401 × 401 pixels from the original CANDELS mosaics for individual targets. 
The sky background using a pixel-by-pixel background estimation method is subtracted. 
In addition, we take photometric parameters provided by SExtractor as initial structural parameter priors for GALFIT <cit.>.

Using GALFIT, we fit F160W images of our selected galaxies by two photometric function models: a single Sérsic profile <cit.>, and a two-component model consisting of a Sérsic profile for the bulge and an exponential profile for the disk (SerExp).
During the fitting process, we set some limitations.
For both models, the range of index n on Sérsic profile is from 0.5 to 8 and the effective radius range is from 0.5 to 50 pixels. 
For SerExp models, we set the effective radii of disks to be larger than those of bulges (R_b < R_d), to ensure domination of disk components in the outer area.
As a result, we get the parameters of each galactic component for each photometric function model, such as effective radius, magnitude, axis ratio, and position angle.


Based on the fitting results of GALFIT, we select the most robust two-component galaxy samples for which the SerExp model is preferred. In Fig. <ref>, we show examples of three galaxies that are identified as classical bulge + disk, pseudo-bulge + disk, and one-component elliptical by our method. The robust two-component galaxies are required to follow all the selection criteria below strictly: 

∘ According to both F160W image and colour image (combined by F606W, F814W, and F160W images), 778 galaxies are selected in which a brighter and more concentrated central component can be visually identified, together with an extended bluer component in the outer region with some structure, like spiral arms or star formation clumps. 
For example, as shown in the leftmost three panels of Fig. <ref>, the top two galaxies exhibit brighter central components and bluer disks with spiral arms, while the bottom one shows a single smooth component.


∘ For a two-component galaxy, its bulge light is required to dominate in the inner region and its disk to dominate in the outer region by analyzing the light profiles of different model fitting results. The one-dimensional residuals, representing the difference between real data and model data, must be smaller than ± 0.1 at all radii. For example, in the third panel of the third column of Fig. <ref>, the light profile shows that its disk does not dominate in the outer region; thus, it is classified as a single-component galaxy. Based on these criteria, we select 657 galaxies for which the SerExp model is preferred.


∘ The two-dimensional SerExp model images of the bulge/disk components reproduce well bulges/disks in the F160W images, showing similar effective radius, axis ratio, and position angle visually between them. 
This also means that the relative residuals from the two-dimensional SerExp modal data must be closer to 0 at all pixels.
For example, in the second and fourth columns of the first and second row in Fig. <ref>, the SerExp model results give good fitting to the images, with much smaller two-dimensional residuals.
Therefore 187 galaxies with inadequate results of SerExp model are discarded.


∘ Robust bulge components are required to have effective radii greater than the half-width at half-maximum (HWHM) of PSF of HST images <cit.>, which has an average value in F160W band of 0.09. 
126 galaxies with small bulges are removed.


Finally, 344 bulge+disk galaxies with reliable fitting results are selected to be analysed in this work.
In addition, we also select elliptical galaxies to set the Kormendy relation criterion for identifying bulge types in the following subsection. 
Similar to the example shown in the third row of Fig. <ref>, 392 elliptical candidates best-fitted by only one smooth component on F160W are selected.
Combined with large Sérsic index (3<n<7.95), 253 typical elliptical galaxies are selected.

We choose the F160W band to be studied and presented in this paper, because the images have higher signal-to-noise ratio than the other bands. 
Besides, F160W band is relatively close to the r-band in SDSS at rest-frame, which is generally considered the good band where bulge features are most prominent observationally<cit.>.
We have also performed the same fitting procedures on images in other bands and have checked that the obtained classification results of galaxy types remain similar.







 §.§ Selection of classical and pseudo-bulge galaxies





For classifying bulge types in low redshift galaxies, there are two common photometric criteria to divide bulge types <cit.>. The first is the Sérsic index of bulge component n_b, because most classical bulges are found to have n_b>2 and most pseudo-bulges have n_b<2 <cit.>. The other criterion is the positions of bulges on the Kormendy diagram <cit.>, where classical bulges are found to follow the Kormendy relation between effective radius R_e and average surface brightness within the effective radius ⟨μ_e ⟩ of elliptical galaxies, while pseudo-bulges normally deviate <cit.>.
For mid-redshift galaxies, only a few works have been done <cit.>.
We visually inspect the central parts of galaxies and find that bulges with structures such as inner ring, nuclear bar, and inner spiral mostly also have small n_b and are located deviating from the Kormendy relation. 
Therefore we select our mid-redshift classical and pseudo-bulge samples also using criteria of n_b=2 combined with the Kormendy relation.

For the 344 two-component galaxies selected as described in the previous subsection, following the classification method of <cit.>, we first use n_b=2 as the divider to classify bulge types, and get 181 classical and 155 pseudo-bulge galaxies for our sample.
In our previous works <cit.>, we select classical bulge galaxies with 3<n_b<7.95 at low redshift, to minimise the effect of uncertainty in fitting bulge Sérsic index.
In this work, to have a larger sample, we use 2<n_b<7.95 to select classical bulge samples. 



For the criterion of positions on the Kormendy diagram, we first fit the Kormendy relation of elliptical galaxies based on the 253 ellipticals selected as described in Section <ref>. Panel (a) of Fig. <ref> shows the result. The black solid line is the best linear fitting of these elliptical galaxies: 

    ⟨μ_e ⟩  = 2.91 ×log(R_e/kpc)+17.76.

The two black dotted lines show the 1σ scatter (σ=0.74). 
We require the pseudo-bulges to lie below the lower dotted line in panel (a) of Fig. <ref>, and the classical bulges to lie above the lower dotted line. 
Combined with the criterion of Sérsic index, we get 86 pseudo-bulges and 105 classical bulges, and their positions on the Kormendy diagram are indicated by blue and red dots in panel (b) of Fig. <ref>. 






In Fig. <ref>, we compare stellar mass and redshift of the selected galaxies with different types of bulges.  
In panel (a), the filled bars show the stellar mass distribution for classical and pseudo-bulge galaxies, and their peak values are 10^10.7 M_⊙ and 10^9.7 M_⊙ respectively.
For comparison, the unfilled bars are over-plotted, which show results of galaxies at low redshift from our previous work <cit.>. The peak stellar masses are 10^11.3 M_⊙ and 10^9.9 M_⊙ for classical and pseudo-bulge galaxies.
The comparisons indicate that both types of bulge galaxies at low redshift are more massive than galaxies at mid-redshift, and classical bulge galaxies are more massive than pseudo-bulge galaxies at all redshifts investigated.
Panels (b) and (c) of Fig. <ref> show that redshifts of two types of bulges both have wide distributions, independent of stellar mass. 






§ PHOTOMETRIC PROPERTIES OF BULGES AND BULGE GALAXIES



In this section, we compare the size and absolute magnitude of the selected classical and pseudo-bulges, and those of their host galaxies, as a function of galaxy stellar mass. 
The results are also compared with the properties of bulges and bulge galaxies studied in our previous work at low redshift <cit.>, to get a clue of the evolution of bulges with cosmic time. 


Note that bulge samples selected in this work from CANDELS and those selected at low redshift by <cit.> are obtained by two-component fitting in bands with different wavelengths. 
F160W band is used for mid-redshift samples from the CANDELS survey, and the corresponding rest-frame wavelength is 8000 Å for galaxies at z=1. For the low redshift MaNGA sample, we use the SDSS r-band (6165 Å) for analyses.
Caution needs to be exercised when interpreting the comparison between the two.
 



 §.§ Size of bulges and bulge galaxies




In Fig.  <ref>, we compare the sizes of bulges of different types and the sizes of their host galaxies and disk components.
The effective radius of each component is adopted to represent the size, which is from the two-component decomposition fitting as described in section <ref>. 


The top left panel of Fig.  <ref> compares the sizes of bulges. 
For CANDELS galaxies, sizes of pseudo-bulges decrease at M_*<10^10 M_⊙, and become almost flat or with a slight increase at larger masses. 
For classical bulges, their sizes remain similar at M_*<10^10.6 M_⊙, and then increase with stellar mass.
Both types of bulges have similar sizes in the overlapped region of stellar mass. 
Compared with the low redshift bulge samples selected by <cit.>, as indicated by light symbols and lines, the sizes of both types of bulges are similar at different redshifts, only with exception for pseudo-bulges in galaxies less massive than ∼ 10^9.8 M_⊙.



In the top right of Fig.  <ref>, for both classical and pseudo-bulge galaxies, the disk effective radius (R_d) increases with stellar mass for M_*>10^10 M_⊙. 
Disks of pseudo-bulge galaxies are a bit larger than disks of classical ones with similar stellar mass. 
This general trend is consistent with low redshift samples, with a larger difference between bulge types at low redshift than at mid-redshift.
Compared with the top left panel of Fig.  <ref>, larger differences exist in disk size than in bulge size between classical and pseudo-bulge galaxies, at both low and mid-redshifts. 
For the effective radius of the whole galaxies as shown in the bottom left of Fig.  <ref>, a similar trend is seen in the radius of disks. 
The difference between galaxies with different bulge types is even larger, mainly due to a larger bulge-to-total ratio of classical bulge galaxies, as will be seen later in Fig. <ref>.


We check the relative size of bulges to the whole galaxies in the bottom right panel of Fig.  <ref>. 
For pseudo-bulge galaxies, the relative bulge size decreases significantly with stellar mass. 
For classical ones, the dependence on stellar mass is much smaller. 
In the overlapped mass range of 10^10.2-10.4 M_⊙, pseudo-bulge galaxies have relatively smaller bulges than classical ones. 
The difference between two bulge type galaxies is much larger for low redshift samples than for mid-redshift samples, mainly due to smaller sizes of classical bulge galaxies at low redshift. 






 §.§ Absolute magnitude and bulge-to-total ratio



We compare in Fig.  <ref> the absolute magnitude of our selected classical and pseudo-bulges, the absolute magnitude of disks of their host galaxies, and the luminosity bulge-to-total ratio (B/T) of their host galaxies.
The absolute magnitude of bulges and disks are from the results of two-component decomposition as described in section <ref>. 
B/T is the ratio between bulge luminosity and total luminosity of both bulge and disk components.

The left panel of Fig.  <ref> shows that for galaxies both at mid- and low redshifts, bulges are brighter in more massive galaxies. 
In addition, classical bulges are brighter than pseudo-bulges in galaxies at given stellar mass at both mid- and low redshifts.
The result is consistent with <cit.>, which shows that classical bulges are brighter than pseudo-bulges in B-band and I-band at 0.4 ≤ z<1.0 and 0.02 ≤ z<0.05.


In the middle panel of Fig.  <ref>, with stellar mass increasing, the absolute magnitude of disks is brighter, at both mid- and low redshifts.
The absolute magnitude of the disks in the two types of galaxies is more similar at mid-redshift than at low redshift, with obviously brighter disks in pseudo-bulge galaxies than in classical bulge galaxies at low redshift. 
The results are also consistent with ones studied in <cit.> in B- and I-band. 



In the right panel of Fig.  <ref>, we compare the B/T for classical and pseudo-bulge galaxies as a function of galaxy stellar mass. 
At mid-redshift, pseudo-bulge galaxies have obviously smaller bulge fractions than classical ones at given stellar mass. 
Similar results have been derived by <cit.> when comparing B/T of classical and pseudo-bulge galaxies in B-band at 0.4<z<1.0.
At low redshift, the difference of B/T between classical and pseudo-bulge galaxies is even larger. 
In particular, for classical bulge galaxies at stellar mass less than 10^10.8 M_⊙, B/T increases with stellar mass at mid-redshift while decreases with stellar mass at low redshift. 
The large B/T of classical bulge galaxies leads to smaller effective radii of the galaxies, and makes the difference in R_e more prominent than the difference in R_d between pseudo-bulge and classical bulge galaxies, as seen in Fig. <ref>. 







§ THE SSFR OF BULGES AND BULGE GALAXIES



In this section, we study and compare spatially resolved sSFR for the selected galaxies with classical and pseudo-bulges. 
For galaxies at mid-redshift from CANDELS, sSFR of each pixel in a given galaxy is estimated based on the position of galaxies on the rest-frame UVI colour diagram, as shown in the top right panel of Fig. 10 in <cit.> and <cit.>.
This new method is used to determine sSFR values, which have been tested to be consistent with results from broadband stellar population fitting from UV to Infrared <cit.>.
Details regarding estimating rest-frame UVI colour and sSFR can be found in Appendix A and B of <cit.>.

For comparison, we also show sSFR for bulge galaxies at low redshift in MaNGA selected by <cit.>. 
SFR on each pixel of these galaxies is calculated according to H α flux by empirical law of <cit.>, where flux of H α and stellar mass on each pixel of these MaNGA galaxies are from full spectrum fitting <cit.>.
As we will see in Fig. <ref> and Fig. <ref>, there exists systematic difference, up to two orders of magnitude, in the sSFR derived from the different methods applied to low and mid-redshift galaxies.
About one order of magnitude difference in sSFR at given stellar mass can be due to the intrinsic difference between galaxies at higher redshift of z=1 and galaxies at z=0, as studied by <cit.> and <cit.>.
<cit.> compare sSFR of the same galaxies estimated from different methods, and find that the difference in sSFR values from H α and rest-frame UVI diagram is small.
Therefore, the other one-order of magnitude differences in sSFR between galaxies at low and mid-redshifts may be caused by different IMFs, flux calibrations, and other possible effects of spectral processing techniques.


In the left panel of Fig. <ref>, we compare the average sSFR within R_e of classical and pseudo-bulge galaxies indicated by red and blue symbols/lines. 
This value can be considered to represent the strength of star formation activities of the whole galaxies, since the average values of sSFR within an ellipse with a given radius vary with increasing radii R and gradually become stable at R ≥ R_e.
For galaxies at both redshifts, we find that galaxies with pseudo-bulges have more active star formation than galaxies with classical bulges at given stellar mass.
In addition, the difference in sSFR between classical and pseudo-bulge galaxies at mid-redshift is much smaller than that of galaxies at low redshift.

For the bulge components, we compare the sSFR within R_b to reflect the bulge star formation activities, and the results are shown in the right panel of Fig. <ref>.
sSFR of classical bulges is a bit smaller than pseudo-bulges at given stellar mass at both mid- and low redshifts.
However, the difference between bulge types at mid-redshift is obviously smaller than that at low redshift, and is also smaller than the difference between galaxies that host different types of bulges as shown in the left panel. 
This may indicate an evolution trend of different types of bulges to somehow deviate more in star formation activities from z =1 to z=0.






In Fig. <ref>, we check further the profiles of the sSFR of galaxies with classical and pseudo-bulge at mid-redshift, to investigate in more detail the star formation activities in different components of galaxies.
The median sSFR for each galaxy sample is shown as a function of radius. 
The influence of PSF on the spectrum is considered, and the innermost points of galaxies are chosen to be greater than their HWHM.
Looking at the blue symbols in Fig. <ref>, for galaxies with stellar mass in the range 10^9.5-10 M_⊙, sSFR profiles are in general flat, with slightly increasing for both types of bulge galaxies.
For galaxies more massive than 10^10 M_⊙, the sSFR profiles increase with increasing radius. 
Comparing the two types of bulge galaxies within the same stellar mass range, the sSFR of pseudo-bulge galaxies is similar and becomes larger than the classical ones from the center to the outer regions.


Comparing the sSFR profiles of galaxies at mid-redshift as shown in Fig. <ref> with the results of low redshift galaxies as shown in the middle panel of Fig. 10 from <cit.>, the difference between the two bulge type galaxies is smaller at mid-redshift than at low redshift, consistent with the trend seen in Fig. <ref>. 
Besides, the slopes of sSFR for galaxies at mid-reshift are basically monotonic, while at low redshift pseudo-bulge galaxies mostly have peak sSFR at radius of around 0.8-1.4R_e. 
The evolution of sSFR slopes of pseudo-bulge galaxies may give hint on the co-evolution of the bulge and the disk components. 









§ SUMMARY



In this paper, we compare photometric properties and sSFR of classical and pseudo-bulges and their host galaxies at 0.5≤ z<1.0 selected from the CANDELS survey. We also compare the results with those derived at low redshift, based on the samples selected from MaNGA survey in an earlier work of <cit.>.
The galaxies with classical bulge and pseudo-bulge are selected based on the Sérsic index n of bulge component by two-component decomposition fitting, combined with the criterion of the position of bulges on the Kormendy diagram.


Both at mid- and low redshifts, classical bulge galaxies are more massive than pseudo-bulge galaxies. 
At given stellar mass, most classical bulge galaxies have smaller effective radii of all components, larger B/T, brighter and relatively larger bulges, and lower sSFR at all radii than pseudo-bulge galaxies. 
However, the differences in most properties between two types of bulge galaxies are in general smaller at mid-redshift than at low redshift, indicating that these two types of bulges and their galaxies are evolving to more distinct populations towards the local universe. 


Besides, the difference in central sSFR between classical and pseudo-bulge galaxies is smaller than the difference at larger radii for mid-redshift galaxies, which is consistent with what was previously discovered in <cit.>. 
Therefore, the co-evolution of bulges with their host galaxies and outer disks is already present at redshift as high as 0.5 ≤ z<1.


While it is difficult to simulate the formation and evolution of galactic bulges precisely due to the numerical effect suffered <cit.>,
in future works, we will study further the observational properties of bulge galaxies at higher redshift, based on analysis of two-component decomposition fitting for galaxies at 1.0 ≤ z<3.0 selected from JWST within five CANDELS fields. 







This work is supported by the National Natural Science Foundation of China (grant No. 11988101), the National SKA Program of China (Nos. 2022SKA0110200, 2022SKA0110201), the National Key Research and Development Program of China (No. 2023YFB3002500), the Strategic Priority Research Program of Chinese Academy of Sciences, (grant No. XDB0500203), and K.C. Wong Education Foundation. JG acknowledges support from the Beijing Municipal Natural Science Foundation (No. 1242032), the Youth Innovation Promotion Association of the Chinese Academy of Sciences (No. 2022056), and the science research grants from the China Manned Space Project.




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