Extragalactic Stellar Tidal Streams: Observations meet Simulation
Juan Miro-Carretero Maria A. Gomez-Flechoso David Martinez-Delgado Andrew P. Cooper Santi Roca-Fabrega Mohammad Akhlaghi Annalisa Pillepich Konrad Kuijken Denis Erkal Tobias Buck Wojciech A. Hellwing Sownak Bose
 



   

   

    Departamento de Física de la Tierra y Astrofísica, Universidad Complutense de Madrid, Plaza de las Ciencias 2, E-28040 Madrid, Spain 
    Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands
    Instituto de Física de Partículas y del Cosmos (IPARCOS), Fac. CC. Físicas, Universidad Complutense de Madrid, Plaza de las Ciencias, 1, E-28040 Madrid, Spain
    Centro de Estudios de Física del Cosmos de Aragón (CEFCA), Unidad Asociada al CSIC, Plaza San Juan 1, 44001 Teruel, Spain
    ARAID Foundation, Avda. de Ranillas, 1-D, E-50018 Zaragoza, Spain
     Instituto de Astrofísica de Andalucía, CSIC, Glorieta de la Astronomía, E-18080, Granada, Spain    
    Institute of Astronomy and Department of Physics, National Tsing Hua University, Kuang Fu Rd. Sec. 2, Hsinchu 30013, Taiwan
    Center for Informatics and Computation in Astronomy, National Tsing Hua University, Kuang Fu Rd. Sec. 2, Hsinchu 30013, Taiwan
    Lund Observatory, Division of Astrophysics, Department of Physics, Lund University, Box 43, SE-221 00 Lund, Sweden
    Max Planck Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg, Germany
    Department of Physics, University of Surrey, Guildford GU2 7XH, UK
   Universität Heidelberg, Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Im Neuenheimer Feld 205, D-69120 Heidelberg, Germany
    Universität Heidelberg, Zentrum für Astronomie, Institut für Theoretische Astrophysik, Albert-Ueberle-Straße 2, D-69120 Heidelberg, Germany
    Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotnik ow 32/46, 02-668 Warsaw, Poland
    Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH13LE, United Kingdom

Extragalactic Stellar Tidal Streams: Observations meet Simulation
Miró-Carretero et al.



 
  
  
   According to the well established hierarchical framework for galaxy evolution, galaxies grow through mergers with other galaxies and the ΛCDM cosmological model predicts that the stellar halos of galaxies are rich in remnants from minor mergers. The Stellar Streams Legacy Survey has provided a first release of a catalogue with a statistically significant sample of stellar streams in the Local Universe that can be used to study minor mergers and test the cosmological models. 
  
   The main objective is to compare the results of the observations of stellar tidal streams with the predictions of state-of-the-art cosmological simulations regarding the formation of stellar streams up to a redshift z < 0.02, according to the ΛCDM model.
  
   We use the predictions of the cosmological simulations Copernicus Complexio, TNG50 of the IllustrisTNG project and Auriga to generate 225 mock-images of nearby halos at a distance of 70 Mpc, and search for stellar streams. We compare the obtained stream frequency and characteristics with those obtained from the Stellar Streams Legacy Survey.
  
   We find  good agreement between the results of analysing real images from the Dark Energy Survey and mock-images from cosmological simulations. We obtained predictions for the detection rate of stellar streams to a surface brightness limit of 35 mag arcsec^-2
  
  The cosmological simulations predict that for a surface brightness limit of 32 mag arcsec^-2 a frequency of almost 70% in the detection of streams around galaxies can be achieved.
   

   Extragalactic Stellar Tidal Streams: Observations meet Simulation
    Juan Miró-Carretero 1,2, Maria A. Gómez-Flechoso 1,3, David Martínez-Delgado4,5,6ARAID Fellow, Andrew P. Cooper 7,8, Santi Roca-Fàbrega 9, Mohammad Akhlaghi 4, Annalisa Pillepich 10, Konrad Kuijken 2, Denis Erkal 11, Tobias Buck 12,13, Wojciech A. Hellwing 14, Sownak Bose 15
   
    
===========================================================================================================================================================================================================================================================================================





§ INTRODUCTION



According to the well established hierarchical framework for galaxy evolution, galaxies grow through mergers with other galaxies. These mergers can be major mergers, when the merging galaxies are of similar stellar mass; a mass ratio > 1/3 is a generally accepted threshold, see e.g. <cit.>, and minor mergers, when the host galaxy accretes a dwarf galaxy in its halo. 
The Lambda Cold Dark Matter (ΛCDM) cosmological model predicts that the stellar halos of galaxies are rich in remnants from minor mergers that, in the Local Universe, on the basis of observations and simulations, are expected to be more frequent than major mergers <cit.>. 
 




 The detection and study of extragalactic stellar tidal streams contributes to augment the stream census, mostly built so far from streams detected in the Milky Way and Local Volume <cit.>, and helps us to contrast their frequency and characteristics with the predictions of the ΛCDM model on a statistically sound basis (the Local Volume is considered here to be a spherical region with a radius of 11 Mpc around the Milky Way  or up to a radial velocity of redshift of z < 0.002). 
 
 This motivates the search
 for streams beyond the Local Volume, and up to a distance for which surveys are available with the required depth <cit.>. 
 Due to their low surface brightness  
 
 much less data can be gathered for each individual stream in a survey of distant hosts, in comparison to the 
 streams in our galaxy or in the Local Volume. 
 
 
 In particular, star-by-star photometric and kinematic measurements are not possible for
 streams at these distances.
 

A number of relevant surveys of tidal features beyond the Local Volume have been reported in the last decade: <cit.>. <cit.> present the results of visual inspection of a sample of 838 edge-on  galaxies using images from three surveys: SDSS Strip-82, Subaru HSC and DESI (DECals, MzLS, BASS). This study, like our present work, was motivated to construct a deep photometric sample and obtain better statistics of tidal structures in the Local Universe in order to compare with cosmological simulations. The definition of tidal features used in that study includes also disc deformations and tidal tails, typical of major mergers. Their results will be discussed further in Section <ref>.


Simulations are required to interpret the observations and to infer the 
physics that determines the
origin and evolution of 
streams. 
 The commonly used analytical/semi-empirical methods to study stellar streams formation in the Local Universe cannot be used when the observational data is scarce and when the central system’s mass distribution evolves with time. So, to understand the formation of the unresolved stellar streams at large distances, accounting for the full temporal evolution, cosmological simulations are needed. However, 
 
 cosmological simulations 
 also have disadvantages.
 They cannot reach the high spatial and mass resolution of bespoke models. Furthermore, the impact of uncertainties in sub-grid models of baryonic astrophysics (e.g. star formation and supernova feedback) is poorly known on the very small scales probed by tidal streams.
 
 
 
 


 
 
 Cosmological simulations nevertheless provide a powerful means  not only to understand the origin and evolution of streams as observed in the surveys above, but also 
 to predict their photometric characteristics.
 State-of-the-art models are now detailed enough to be constrained by stream observations, and can also inform assessments of the design and completeness of the observations themselves.
 
 We already know that future surveys will need to be able to produce deeper images than available today, if new extra galactic streams, presumed to exist in great numbers, are to be discovered. 
 
 However, it remains unclear what 
 critical image depth
 is
 needed to significantly increase 
 the number of known streams.
 
 This is important 
 
 to motivate and plan for
 surveys such as ESA's space mission Euclid <cit.> and the Vera C. Rubin Observatory's Legacy Survey of Space and Time <cit.>. 
 In the context of minor mergers, predictions for streams and their progenitor galaxies are close to the limit of the capabilities of current large-volume cosmological simulations, and the robustness of those predictions has not yet been explored in detail. The use of cosmological simulations to plan for and interpret new surveys therefore has to proceed in tandem with their validation against existing extragalactic stream data, mostly at surface brightness limits brighter than ∼ 29 mag arcsec^-2 <cit.>.
 

The use of cosmological simulations to study tidal features has also increased significantly in the recent times <cit.>
Relevant work on stream detectability using mock-images from cosmological simulations is reported in <cit.>. The authors have inspected surface brightness maps generated from 30 Auriga project simulations <cit.> of Milky Way-like galaxies looking for the brightest streams. They report that no streams have been detected in images with a surface brightness limit brighter than 25 mag arcsec^-2. Their stream detection frequency increases significantly between 28 and 29 mag arcsec^-2. They 

find a correlation between infall time and infall mass of the 

stream progenitors, such that more massive progenitors tend to be accreted at later times.

 
<cit.> report on a theoretical investigation of the extended diffuse light around galaxies and galaxy groups by visually inspecting mock-images produced using the NEWHORIZON cosmological simulations. This is carried out on a sample of 37 simulated objects at redshifts z = 0.2, 0.4, 0.6 and 0.8, spanning a stellar mass range of 10^9.5 < M_⋆ < 10^11.5M_⊙.
Through production of surface brightness maps at different surface brightness limits, they predict the fraction of tidal features that can be expected to be detected at different limiting surface brightnesses. 

















<cit.>

identified and classified

tidal features in LSST-like mock-images from 


four sets of hydrodynamical cosmological simulations:

EAGLE, IllustrisTNG and Magneticum). 

These features
comprise streams/tails, shells, plumes or asymmetric stellar halos and double nuclei, and as such do not distinguish between minor and major mergers as the origin of the such features. The results of this previous work and the one presented in the preceding paragraphs will be discussed in more detail in Section <ref>.

As in this work, the works by <cit.> and <cit.> rely on visual inspection of mock-images from cosmological simulations. However 

they
focus on the detection of tidal features and tidal tails, while we focus our analysis on the detection and characterisation of remnants of minor mergers, low surface brightness features that are of an accreted origin. 

One important conclusion of <cit.>, with which we concur, is that a higher level of domain knowledge is required to perform robust visual classifications of tidal features (more so than to separate spiral and elliptical galaxies, for example). This work follows 

our previous surveys  
to detect stellar streams in 
images from the DESI Legacy Surveys <cit.>, thus, the inspection of mock images in this paper 

benefits from 

our experience gathered 
from working with
comparable
observational images. 
 
In this work we use the term stellar tidal streams to refer to the remnants of minor mergers, in line with the nomenclature used in <cit.>[great circles, hereinafter referred to as circles, are streams that result from satellites along mildly eccentric orbits, with an arc-like shape, sometimes featuring complete loops around the host, but in most cases (in our sample) seen as covering only a small part of a loop; umbrellas, structures often appearing on both sides of the host galaxy, displaying an elongated shaft ending in the form of a shell (sometimes only the shells are visible) resulting from satellites that were on more eccentric, radial orbits; giant plumes, hereinafter referred to as plumes, structures appearing to shoot out of the host, generally for quite a long distance]. 
Our work focuses on stellar tidal streams in the Local Universe up to a distance of 100 Mpc (redshift z < 0.02). 
We consider as stellar tidal streams only those low surface
brightness (LSB) features that are of an accreted origin, whatever their apparent morphology (shells, circles, plumes etc.); as we will discuss later in the paper, the apparent morphology is strongly dependent on the line of sight of the observation. We can broadly characterise stellar tidal streams as LSB structures in the halo of galaxies, at distances between ∼ 20 and 120 kpc from the host centre and with surface brightness fainter than ∼ 25 mag arcsec^-2.
Stellar tidal streams are thus a particular case of LSB structures and are significantly (several mag arcsec^-2) fainter than tidal tails, 

another type of LSB feature
resulting primarily from major mergers <cit.>.

The results from observations of the Dark energy Survey (DES) presented in <cit.> allow for a direct, quantitative comparison of the abundance and characteristics of stellar tidal streams in the Local Universe with the predictions from state-of-the-art cosmological simulations based on the ΛCDM paradigm. 

In particular, we can compare statistics derived from the observed stream population (for example, the number of stream detections at at given surface brightness limit, or the distribution of photometric observables for detected streams)
with those 

predicted by
cosmological simulations.

To do this,

we obtain predictions of stream formation from three cosmological simulations: Copernicus Complexio <cit.>, Illustris TNG50 <cit.> and Auriga <cit.>. 


We have carried out this work in the 

context 
of the Stellar Stream Legacy Survey <cit.>, whose main objective is to perform a systematic survey of stellar tidal streams in a parent galaxy sample of ∼ 3200 nearby galaxies using images from the recently completed DESI Legacy Survey imaging surveys. Examples of stellar streams detected in the SSLS can be seen in Figure <ref>. 
A catalog of streams from the first batch of galaxies in this survey is presented in <cit.>. 
 
In this paper, we

compare the stellar streams in a sample of galaxies observed by the DES survey
with 

streams in mock images 

derived from the simulations listed above, for a matched sample of hosts.
We compare the detection frequency and photometric characteristics measured in both samples

and discuss the results. 

In Section <ref> we introduce the main characteristics of the 

cosmological simulations. The selection of the halos from the simulations to be analysed is presented in Section <ref>. Section <ref> is devoted to the process of generating mock images. In Section <ref> we present the predicted detectability of streams at different surface brightness limits. 

The results of the comparison are discussed in Section <ref> and the summary, conclusions and outlook are given in Section <ref>.




§ COSMOLOGICAL SIMULATIONS




This work makes use of several different cosmological simulations. An overview of the available cosmological simulations, as well as the underlying tools and modelling paradigms, can be found in <cit.> and as part of the AGORA collaboration <cit.>. For our work, we have selected three simulation sets, each belonging to one of the broad categories in which the cosmological simulations are classified at the highest level:



 
  * Volume simulations produce large, statistically complete samples of galaxies 
 
 
 but typically do not resolve spatial scales smaller than ∼100 pc. Physical processes on scales smaller than the explicit hydrodynamical scheme, such as star formation and feedback, are incorporated via semi-analytical `sub-grid' models.

 
  * Zoom-in simulations produce smaller samples of galaxies with a higher spatial and mass resolution and thereby model baryonic processes on smaller scales.

 
  * Semi-analytical simulations are the result of a combination of numerical dark matter-only simulations, and analytic models for the prescription of baryonic physics. They are computationally much more efficient than the above categories, at the cost of self-consistency in the dynamics of the baryonic component.
 


For our comparison with the observational data we have chosen one state-of-the-art simulation suite of each of the types listed above.



 §.§ Copernicus Complexio



The Copernicus Complexio <cit.> is a ΛCDM cosmological N-body simulation post-processed with a semi-analytic galaxy formation model and the `Stellar Tags in N-Body Galaxy Simulations' (STINGS) particle tagging technique <cit.>.

COCO provides both high mass resolution 
and an approximate analogue of the Local Volume (a high-resolution spherical region of radius ∼25 Mpc, with density slightly lower than the cosmic mean, embedded in a lower-resolution box of 100 Mpc/side). 

The Galform semi-analytic model of <cit.> was used to predict the evolution of the baryonic component in each dark matter halo. This model is calibrated to a range of low and high-redshift observables, including optical and near-IR luminosity functions, the HI mass function, and the relationship between the masses of bulges mass and central supermassive black holes. The 6-dimensional phase space of each single-age stellar population formed in the model is mapped to an individual subset of dark matter particles using the STINGS technique. These models will be presented in detail in a future publication (Cooper et al. in prep.). The specific characteristics of the COCO simulations are listed in <ref>.



















 §.§ Illustris TNG50



IllustrisTNG is a suite of large volume, cosmological, magnetohydrodynamical simulations run with the moving-mesh code AREPO (Springel 2010). The Illustris TNG50 <cit.> simulations characteristics are listed in Table <ref>:










The IllustrisTNG50 simulation includes a comprehensive model for galaxy formation physics, which is able to realistically follow the formation and evolution of galaxies across cosmic time <cit.>. Each IllustrisTNG50 simulation self-consistently solves the coupled evolution of dark matter, cosmic gas, luminous stars, and supermassive blackholes from a starting redshift of z=127 to the present day, z=0. We select the TNG50-1 simulation run and the snapshots at z < 0.02.



 §.§ Auriga


 
The Auriga simulations <cit.> are a set of cosmological zoom-in magnetohydrodynamical simulations

also carried out with the AREPO code.
Auriga

re-simulates at higher resolution a sample of halos selected by the
EAGLE project <cit.>.

From the available simulations in the Auriga portal[<https://wwwmpa.mpa-garching.mpg.de/auriga/data.html>] we 
use all 30 halos in the Original/4 series.


























§ HALO SELECTION

 

In order to allow for a consistent comparison between the cosmological simulations and the DES image sample, we selected samples of simulated halos in a comparable range of both stellar mass and halo mass.

To determine the range of host stellar mass in the DES sample, we 
construct an empirical relation between the absolute magnitude of the 

galaxies in the B-band, M_B, available from the HyperLeda database[<http://leda.univ-lyon1.fr/>] <cit.>, and 



their stellar mass, M_⋆, as determined by
the Spitzer Survey of Stellar Structure in Galaxies <cit.>. 
We find the following relation:

    log_10 (M_* / M_⊙) = - 0.396  M_B + 2.21






As shown in Figure <ref>, this empirical relation is linear over the magnitude range of the DES host sample, with small scatter (generally less than 0.2 log_10 (M_* / M_⊙) bar a few outliers).




The histogram in Figure <ref> shows the stellar mass distribution of the DES sample and of the selected halos in the COCO, IllustrisTNG50 and Auriga simulations. 
The average stellar mass of the DES sample is log_10 M_⋆/M_⊙ = 10.00 and the standard deviation σ = 0.38.
While the average of the distributions is not the same, there is sufficient overlap between the DES sample stellar mass range and the stellar mass range of the simulated halos selected for the comparison.
Figure <ref> shows the halo mass distribution of the selected halos in the COCO, IllustrisTNG50 and Auriga simulations. Here all the simulations overlap and the average values are close to one another.

In this stellar mass range, COCO galaxies occupy a broader range of halo masses than IllustrisTNG50 galaxies.








§ MOCK IMAGES

 



To compare the predictions of cosmological simulations with the observations from the DES sample,

we have generated mock images from
snapshots of the simulations described in Section <ref> in the redshift range 0 < z < 0.02. 

The continuous stellar mass density field in all these simulations is represented by discrete tracers, called star particles[In the COCO simulation, stellar mass is associated with a subset of the collisionless particles in post-processing, rather than an independent particle species in the original simulation, but the principle is the same from the point of view of our analysis; more details are given in <cit.>.]. The stellar mass associated with each star particle corresponds to a stellar population with a single age and metallicity. As in any N-body realization of a density field, each particle notionally corresponds to an irregular volume of phase space centered on the location of the particle.




We have applied 

the following
transformations 
to the properties of the star particles from the simulation snapshots in order to recover the observables that can be identified in real images: 



  * Expansion of the discrete star particles into an approximation of the implied continuous 3-dimensional of stellar mass distribution, by convolution with an adaptive smoothing kernel;

  * Projection of the continuous distribution of stellar mass into a 2-dimensional plane. The orientation of the central galaxy relative to the observer's line of sight is a parameter of our method, and can be either random or specific (for example, to view the galaxy face-on or edge-on);


  * Conversion of stellar mass density to luminosity density, by convolution of an SED appropriate to the age and metallicity of each particle with a specific photometric bandpass.






We use
the open source tool pNbody [<https://obswww.unige.ch/ revaz/pNbody/>] <cit.> to produce mock images by implementing the transformations above.




Figure <ref> shows, as an example, the r-band surface brightness map produced by processing one of the COCO galaxies with pNbody. 

The contour lines identify the isophotes for an intuitive view of the possible low surface brightness (LSB) structures present in the image. 





We have carried out tests to asses the 
robustness 

of the resulting mock-images 

to variations in
the pNbody configuration parameters for the generation of 
mock images with the 
DECam instrument. 
We have generated surface brightness maps for a halo at a distance of 70 Mpc from the Sun, as a representative distance for the DES galaxy sample that ranges from 40 to 100 Mpc (changing the distance within the DES galaxy sample range does not noticeably influence the detectability of streams, as explained in Section  <ref>). 


For the expansion (smoothing) step, we distribute the luminosity (flux) of each particle over the image pixels by convolution with a Gaussian kernel. The scale of this kernel, h, which we refer to as the smoothing scale, is set to a parameterized multiple to the root-mean-square average distance to the 16^th-nearest star particle neighbour, h_16. It therefore adapts to the local star particle density. The logic for setting h is broadly similar to that used to determine the kernel scale in a smoothed particle hydrodynamics calculation. However, the smoothing in our case only serves to interpolate between the original particles and does not have any physical significance. It is therefore somewhat arbitrary; our method reflects a balance between smoothing sufficiently to reduce the visual impression of a discrete particle distribution, while preserving the small-scale features.



Figure <ref> shows 


mock images of the same three galaxies (left to right) with kernels of scale h=0.6 h_16 (top row) and h=0.3 h_16 (bottom row). Over this range (with the DESI pixel scale) the bulk of the particle granularity in the image is removed, but nature, extent and surface brightness of tidal features relevant to our subsequent visual inspection (see Section <ref>) are not noticeably different. 















We use the mock images to assess the detectability of streams over a range of surface brightness limits. 

To follow the same process of stellar stream detection by visual inspection as with real images, we have transformed the surface brightness maps into counts images. Then we have added background noise to the images, which depends on the type of analysis to be carried out.

In order to assess the detectability of the streams under different image depths, a simple realization of background noise was added to the images, as flat Gaussian noise with variable amplitude. This gives us the flexibility to emulate different surface brightness limits in an easy way; we choose limits between 25 and 34 mag arcsec^-2 in intervals of 1 mag arcsec^-2. 
For this purpose we used the state-of-the-art  GNU Astronomy Utilities (Gnuastro)[<http://www.gnu.org/software/gnuastro>] software.

To asses the impact of the smoothing length on the detection of a possible stream 

in the mock images by visual inspection, 

we have examined count images with different choices of smoothing length, after adding 
background noise corresponding to a surface brightness limit of 29 mag arcsec^-2.
Figure <ref> shows 

this test for smoothing lengths h=0.6 h_16, h=0.3 h_16 and an alternative smoothing scheme in which h is instead set to the absolute distance to the 5^th nearest neighbour.

Again, regarding stream detection by visual inspection, we find no significant difference between the images.
We therefore adopt a smoothing scale of h=0.6 h_16.



    

To measure 

photometric properties of the mock images 
and 

compare the results with those of the DES sample, we add a more realistic sky background to the mock images. We first extract the sky background from selected real DES sample images, 

removing the central galaxy and replacing it by real sky background from an area of the same image without significant point sources. 
The image from which we extracted this fiducial background image was
selected 

according to
the following criteria: i) the central galaxy should be edge-on, in order to minimise its area of influence on the image; ii) 
the image
should have no stream detection, in order not to interfere with the synthetic halo image; and iii) the image should have an r-band surface brightness limit representative of the DES sample. 

The selected image had a surface brightness limit of 28.65 [mag arcsec^-2].  
The fiducial sky background extracted from this image, as above, was
then 

superimposed onto 

each mock image
with the central 

galaxy
and no sky background, creating an image with a synthetic central galaxy and a real DES background. 

We use Gnuastro to mask the central galaxy with an ellipsoidal aperture, and place instead an ellipsoidal 'patch' of the same dimensions extracted from a region of the image without bright sources. The size and orientation of the ellipsoidal mask is such that covers the area of influence of the central galaxy in its surroundings, i.e. it reaches up to the point where the surface brightness profile flattens. 




Figure <ref> shows an example of a COCO
galaxy

to which we have added the sky background of a real DES image with a surface brightness limit of 28.65 mag arcsec^-2. We compare this to

the same mock image combined with a simple Gaussian noise background equivalent to the same surface brightness limit.

A stream with a umbrella /shell morphology can be clearly appreciated in the image with the real DES background and its appearance is consistent with that in the image with an artificial background. Adding a synthetic background to the mock image does not seem to impact significantly the detection of streams by means of visual inspection with respect to the real background in this surface brightness limit regime. This method is very efficient to emulate different levels of surface brightness limit and we adopt it in this work to assess the detectability to those levels currently not achievable in available surveys. However we acknowledge that for much fainter surface brightness limits, this method provides only an approximation as the confusion of sources and possibly cirri will become much more significant and the difference between the synthetic background and the real background will increase.
 

 





§ STREAM DETECTION



In this section, the detectability of tidal streams under different image depths is analysed on the basis of mock-images. At the present time, there is no automatic method for detecting tidal streams, though this is an important subject of current research.

Therefore detectability is based here on the method combining visual inspection and image analysis tools, as presented in Section <ref>. This is the method applied to the detection of streams in real images, as presented in <cit.> and that has been the basis for generating the stream catalogue presented there.



We made all the photometry measurements by applying Gnuastro's MakeCatalog subroutine <cit.> on the sky-subtracted images generated by Gnuastro's NoiseChisel <cit.>. The method is identical to the one applied to the analysis of the DES sample images and explained in detail in <cit.>. 



The analysis has been done on the basis of the r-band images. This band has been chosen as it 

has been used in other relevant observational studies on tidal features <cit.>, thus allowing for comparison of results across different studies. In the mock-images, the SDSS r-filter produces brighter measurements of the stream than the SDSS g-filter, in agreement with the observations. Depending on the region in the image this difference can be up to ∼ 0.6 mag arcsec^-2.

In order to assess the impact of the host distance on the detectability of streams, mock images have been produced at different distances (50, 70 and 90 Mpc), covering the distance range of the observed hosts with their streams, 

which ranges
from 40 to 100 Mpc. Although the surface brightness is independent of the distance, the distance has an impact on the S/N at pixel level that may in turn impact the observability. However, as expected, this  small relative difference in the distance does not show a noticeable difference in the visual perception of the images.






























 §.§ Stellar Streams in the COCO Simulation



For the comparison between the DES sample and the COCO simulations, in the first step we selected a sample of simulated halos containing central galaxies with

stellar mass 

in a range around
the DES stellar sample average ± 1σ, that is, between  4.17 × 10^9 M_⊙ 
(log_10 M_⋆/M_⊙ = 9.62) and 2.4 × 10^10 M_⊙ 
(log_10 M_⋆/M_⊙ = 10.38). 
In a second step, we selected a sample of simulated 

halos containing central galaxies with
stellar mass 

corresponding to the DES host sample average stellar mass value +1σ and +2σ (log_10 M_⋆/M_⊙ = 10.76). 



For the COCO simulations, 108 halos have been selected with central galaxy stellar masses overlapping with 70% of the DES sample with streams.
A comparison of the host stellar mass distribution of the DES sample with the COCO simulation sample, within the selected stellar mass range, 

is shown in Figure <ref>



 We have generated mock images with surface brightness limit  between 25 and 34 mag arcsec^-2 in intervals of 1 mag arcsec^-2. As the depth of the image increases, that is, the fainter the surface brightness limit is, the clearer the streams appears to the observer, as can be seen in Figure <ref>.















Then the resulting ∼ 1000 images were visually inspected and for each one, the surface brightness limit at which a streams could be visually detected for the first time was identified and noted. The inspection was carried out on the image 

FITS-format files, displayed with the SAO DS9 tool. As for the inspection of the observational images, suitable colour, scale and analysis block level options were selected to improve the perception. 
The DES observational images have a ‘real’ sky background while for the mock-images we have a sky background that while corresponding to a similar surface brightness limit, is much smoother, allowing for an easier detection of the streams. This has been taken into account in assessing the detectability of streams by reporting one level deeper when the stream was not clearly distinguishable by visual inspection.

The result is a curve indicating the percentage of streams detected within the COCO halo sample as a function of the image depth, that is, as a function of the image surface brightness limit. The result is depicted in Figure <ref>. The curve follows 

an approximately
linear trend between 
surface brightness limits of
26 and 

34 mag  arcsec^-2, with a 

12.5% 
increase in streams detected per 1 mag arcsec^-2 increase in surface brightness limit. In 97% of the mock-images, we detect by visual inspection what we consider to be (part of) a stream up to a SB-limit
of 34 mag arcsec^-2.
However,

we estimate that
that only 89% of such faint structures could actually be measured with a reasonable level of error in real images.
   
Regarding the stream morphology, 

in the COCO 

sample, at SB-limit 28-29 arcsec^-2, around 70-80% of the detected streams are shells (a segment of the wider morphology class known as umbrella, see stream morphology classification in <cit.> while at SB-limit 34 arcsec^-2 around 80-90% of the detected streams are shells, with only 2-3% of the streams displaying a circular morphology. 




We have investigated a possible correlation between the surface brightness limit at which streams are detected and the host galaxy stellar mass. In particular, we have analysed whether more massive host galaxies show streams at a lower (brighter) surface brightness limit. Figure <ref> shows the stellar mass distribution for the galaxies identified as hosting streams versus the surface brightness limit at which those streams have been detected. 

No correlation is 

evident.







 §.§ Stellar Streams in the IllustrisTNG50 Simulation



We have generated surface brightness maps from the IllustrisTNG50 simulation for 60 halos.
40 of these


have central galaxies in the
stellar mass range between
between  3.02 × 10^10 M_⊙ 
(log_10 M_⋆/M_⊙ = 10.48) and 5.73 × 10^10 M_⊙ 
(log_10 M_⋆/M_⊙ = 10.76),
corresponding to the range of 

stellar mass between the average value of the DES sample +1σ and +2σ. 
In order 

to compare with Milky Way-like galaxies (such as 

those in the Auriga simulations) we have selected 20 additional halos in an extension of the stellar mass range to 8.0 × 10^10 M_⊙ ( log_10 M_⋆ / M_⊙ = 10.9), see Section <ref> for details.
 
We have transformed the surface brightness maps into images with counts and added sky background corresponding to 10 levels of surface brightness limit between 25 and 34 mag arcsec^-2 (

see
Section <ref>).  As we increase the depth of the image (apply a higher background noise level to the image corresponding to a fainter surface brightness limit), streams become more visible and can be detected by visual inspection, as can be seen in Figure  <ref>.
We have visually inspected the resulting ∼ 600 images and for each halo / host galaxy, the surface brightness limit at which a streams could first be visually appreciated was identified and noted. 

As with our analysis of COCO, FITS images were inspected with DS9.



















Figure <ref> shows the resulting curve indicating the percentage of streams detected within the the IllustrisTNG50 halo sample as a function of the image surface brightness limit.
The curve shows a steep gradient between SB-limit = 30 and SB-limit=32 mag / arcsec^2, of about 25 % increase in streams detected per 1 mag  arcsec^-2 increase in surface brightness limit.
In ∼ 70% of the mock-images a (part of a) stream can be detected by visual inspection at a SB-limit of 34 mag / arcsec^2.   
The detection percentage level resulting from inspection of the IllustrisTNG50 mock-images at a SB-limit of 28.65 mag arcsec^-2 (the SB limit value corresponding to the average SB-limit for the DES sample in the r band) is between 3% and 10% (corresponding to the SB-limit values for 28 and 29 mag arcsec^-2 in the curve, respectively). 

To assess the dependency of the detection rate for a certain SB-limit on the host stellar mass, we have compared the detectability curve obtained for 30 halos in the stellar mass range between log_10 M_⋆/M_⊙ = 10.48 and  log_10 M_⋆/M_⊙ = 10.76 with the one obtained for 60 halos in an extended stellar mass range up to log_10 M_⋆/M_⊙ = 10.9. As can be seen in Figure <ref>, both curves are very similar, the one for the extended mass range appearing smoother, due to the increased population of galaxies included. This seems to reinforce the speculation made in Section <ref> using COCO simulations that the stellar mass of the host galaxy does not noticeably influence the brightness of the streams around it. 

We have taken into account the fact that the smooth background added to the image makes 
streams easier to detect, by reporting one level fainter of the surface brightness limit, when the stream cannot be clearly distinguished by visual inspection at a certain level.

The morphology analysis shows 20-43% shells (part of the cosmological class umbrella/shell) and 10-16% circular shapes, some showing a clear loop around the host. 






 §.§ Stellar Streams in the Auriga Simulation



We have available 30 Auriga zoom simulations of Milky Way-mass halos

at z < 0.02.


Since 

this sample
may not be statistically representative, and in order to compare with the COCO and IllustrisTNG50 simulations, three surface brightness maps have been generated from each halo, taking the three 

orthogonal axes of the simulation coordinate system as lines of sight. None of the axes are aligned with face-on or edge-on directions so that overall the orientation of the galaxies is random. This process results in 

90 images, 

each of which we combine with a range of sky backgrounds
between the SB limit of 25 and 34 mag arcsec^-2, at intervals of 1 mag arcsec^-2, thus yielding 900 mock images in total. Examples of halos with streams of different morphology can be seen in Figure <ref>.

 



















Figure <ref> shows the detection rate curve obtained for Auriga from visual inspection of the mock images, together with those obtained for COCO and IllustrisTNG50, and the observational results of the DES sample. The figure shows the percentage of the sample for which at least one stream is detected for each of the 10 surface brightness limit levels analysed. The curve indicates detections of streams with a reasonable level of confidence. When we detect a LSB feature but have doubts 

about
whether that LSB feature constitutes a stream, we take 

the conservative approach of not counting it towards the results included in the figure
Two of the halos (Au20 and Au30) seem to show an ongoing major merger between two galaxies of similar apparent size, clearly visible at a surface brightness limit of 27 mag arcsec^-2.

These two halos have been discarded, 

because
we are looking for remnants of minor mergers only. LSB structures that could appear similar to streams at fainter surface brightness limits are not accounted for in the detectability curve.

The curve shows that there are no stream detections 
at a SB limit 26 mag arcsec^-2 or brighter. Between 26 and 29 mag arcsec^-2 there is an increase in detections with a gradient of around 4% detection rate per 
mag arcsec^-2 the SB limit, reaching ∼ 12% detection rate at 29 mag arcsec^-2. Between 29 and 31 mag arcsec^-2 the gradient is about 4 times steeper, reaching ∼ 34% detection rate at 31 mag arcsec^-2. The gradient increases again to ∼ 25% per mag arcsec^-2 increase in the SB limit between 31 and 32 mag arcsec^-2 reaching ∼ 65% detection rate at this SB limit. Beyond this value, the gradient flattens.
We detect streams by visual inspection with a reasonable level of confidence in 95% of the mock images up to a SB-limit of 34 mag arcsec^-2.

The percentage of streams detected in the AURIGA sample at a SB-limit corresponding to the average SB-limit for the DES sample in the r band (28.65 mag arcsec^-2) is between 8% and 13% (corresponding to the SB-limit values for 28 and 29 mag arcsec^-2 in the curve, respectively).

The morphology analysis yields 41-62% shells and 11-24% circular shapes. However, the observed stream morphology is strongly dependent on the line-of-sight from which the streams is observed (Youdong et al. in prep.). As an example, Figure <ref> shows halos Au14 and Au29 seen from two different lines of sight. As can be appreciated in the images, looking at the same halo from these two different perspectives would suggest a different morphological classification.








§ STREAM PHOTOMETRY



The results are presented in this Section and compared with the results obtained from the photometric characterisation and analysis of the streams detected in the DES observations sample (see <cit.>). This comparison between the observations and simulations includes also the results of the statistical analysis.
The mock images generated from cosmological simulations though the process described in Section <ref> have been subject to photometric characterisation and analysis. In order to be able to measure photometry parameters on these mock images, and compare with observations, the corresponding surface brightness maps have been generated for the SDSS r, SDSS g and SDSS z filters. 


The line of sight in the z coordinate direction has been chosen, as the halos are not aligned with the volume coordinates, the chosen line of sight represents a random orientation of the galaxy and therefore it does not introduce an orientation bias of the system.  For the photometry analysis, the same smoothing method and length have been selected as for the stream detectability analysis in Section <ref>. We have generated mock images from the surface brightness maps by transforming surface brightness readings to count readings, applying a zero point ZP=22.5 mag as per DECam images. 















The comparison between the mock-images photometry and the observations from the DES galaxy sample is carried out on the basis of the average stream surface brightness, average stream (g-r)_0 colour as well as the average distance of stream to the host centre. The resulting histograms are shown in Figures <ref>, <ref> and <ref>. 










to be completed for TNG50 and Auriga







§ DISCUSSION



We now compare the predictions of the cosmological simulations with one another and with the observational data regarding frequency, detectability and morphology
of streams.



Looking at Figure <ref>, the cosmological simulations all seem to predict that, for a surface brightness limit of 32 mag  arcsec^-2,

almost 70% of galaxies in the mass range we study have one or more detectable streams.





However, the simulations show discrepancies with one another regarding 

detection rates 

for
surface brightness 
limits brighter than
 32 mag  arcsec^-2. 
In particular, the 
two hydrodynamical simulations,

IllustrisTNG50 and Auriga, 
agree well with each other but predict a lower detectability rate than COCO.


For limits fainter than
32 mag  arcsec^-2, however, IllustrisTNG50

has a lower detection rate than COCO and Auriga (which agree well with each other, in this regime).

The simulations also show discrepancies with one another regarding the morphology of the streams detected, as will be discussed further down in this section.


The small differences in the stellar mass ranges

of the samples of galaxies drawn from the three simulations
(see Figure <ref>) 
do not seem to play an important role in stream detectability for the range of stellar masses analysed in this work, as discussed in Section <ref>. We therefore speculate that these differences  

can be attributed mainly to the treatment of baryon physics in

the simulations (all the simulations use the same N-body treatment of gravitational dynamics and very similar cosmological parameters).  
Pinpointing specific explanations for these differences 


will require an in-depth analysis, outside of the scope of this work.
However, since the most striking difference concerns the apparently greater number of relatively brighter streams detectable in the COCO particle tagging models, compared to the two hydrodynamical simulations, we speculate on why differences in the dynamical treatment of the baryons may give rise to this result. 

As discussed in <cit.>, when comparing particle tagging simulations with hydrodynamical methods, even with identical initial conditions, it can be difficult to separate effects due to the dynamical approximation of particle tagging from the effects of different star formation models. In our case, the IllustrisTNG and COCO models have both been calibrated to observations of the galaxy mass and luminosity functions at z=0 <cit.>, as well as other low-redshift data, notably the galaxy size–mass relation. The fundamental relationship between stellar mass and virial mass in both simulations agrees well with (for example) that inferred from galaxy abundance matching. Although the typical star formation histories of host galaxies in our sample, and their stream progenitors, may still differ in detail between the two simulations, we expect that they are broadly similar. This makes it more likely (although by no means certain) that the differences we observe are related to dynamical factors, rather than differences in how stars populate dark matter halos.

Two particularly relevant dynamical factors are neglected (by construction) in particle tagging models. First, the gravitational potentials of stream progenitors can be altered by the inflow and outflow of baryons associated with cooling and feedback; this may make satellites either more or less resilient to tidal stripping, depending on whether baryonic processes produce density cores or density cusps. Second, stars (and gas) could make a significant contribution to the central potential of the host halo, in particular through the formation of a massive baryonic disk.

<cit.> explore the consequences of neglecting these factors when predicting satellite disruption (and hence stream and stellar halo formation) in particle tagging models. In the hydrodynamical model used in that work, massive satellites were found to disrupt somewhat earlier than realizations of the same systems in a particle tagging model[This of course depends on the detail of the hydrodynamical scheme and its subgrid recipes for star formation and feedback; the specific scheme used in IllustrisTNG was not examined by <cit.>. A detailed case study of a single bright stream relevant to this discussion is given in Appendix A of that paper.].









This could explain why more streams are visible at surface brightness limits of ≲ 28 mag arcsec^-2

in COCO compared to IllustrisTNG; if satellites are disrupted earlier in IllustrisTNG, their streams may have more time to phase mix, lowering their surface brightness. 


A similar argument could also explain the greater abundance of shells detected in COCO. Shells originate from satellites on radial orbits <cit.>.  

The relative fraction of radial and circular orbits may differ between IllustrisTNG and COCO, because the evolution of the progenitor's orbit during pericentric passages (due to exchange of angular momentum with the host) may be significantly different with and without a massive baryonic disk.


A related effect is explored by <cit.>,
who study the relation between the rotation of the hosts and the presence and morphology of tidal features

One of the conclusions of that work is that shells appear more frequently in slow rotating hosts than in fast rotating hosts. This hypothesis could be tested in future work. More generally, further exploration of these differences particle tagging and hydrodynamical simulations could help to understand how observations of streams can constrain the nature of the gravitational potentials of stream progenitors and their host galaxies.



Turning now to comparison between the simulations and the observational data, 8.7±1.1% of galaxies in the DES sample have detectable streams at an average r-band surface brightness limit of 28.65 mag  arcsec^-2.


of 28.65 mag  arcsec^-2 is 8.7 ± 1.1 %. 
This seems to match very well with the predictions of IllustrisTNG50 and Auriga, 
and is lower than the prediction from COCO.





 

Stream morphology is not an observable that can be used reliably to constrain simulations, because it is strongly dependent on the line of sight along
which the stream is observed. Nevertheless, comparison of the 

fraction of streams with different morphologies is interesting because it could be related to the 

same dynamical differences between simulations methods that give rise to differences in stream abundance.

20-43% of streams in the IllustrisTNG50 sample are shells, which is within the range found in the DES sample, and 10-16% have circular morphology, which is also not too far from the observations. 
As noted above, there is a significant discrepancy in the relative fractions of different stream morphologies 
between observations and the COCO simulations. In the COCO mock images, 70-90% of streams are identified as having Umbrella/Shell morphology, while in the DES sample,  
only 27-38% of streams are identified as such. The COCO mock images show only 2-3 % of streams with circular morphologies, compared to 21-35 % in the DES images.
Auriga predicts 41-62% of streams with Umbrella/Shell 
morphologies, significantly above the fraction in the DES sample, and 10-24% of stream with circular morphology, somewhat below the DES figures.
Table <ref> summarises the stream morphology findings for the different samples derived from the total number of streams detected. Note that more than one stream is detected in some of the halos. 
It is beyond the scope of this work to analyse the reasons for these discrepancies in 
stream morphology; as noted above, further work would be a valuable contribution to understanding the relationship between the observable properties of streams and the galaxy formation process as a whole.



    


Regarding the stream photometry measurements, the stream average surface brightness range in the COCO simulation matches generally well the observations in the DES stream sample, though the distribution is skewed towards the brighter end of the range (Figure <ref>). This is consistent with the abundance of shell-shaped streams, typically brighter than other stream morphologies, as confirmed by the observations. 

The stream (g-r)_0 colour distribution of the COCO simulations also matches generally well with the observations. The mean value of the COCO simulation distribution is 0.54±0.12 mag , versus 0.57±0.14 mag for the DES observations sample. However, the simulations are slightly skewed towards the blue end as can be seen in Figure <ref>.

Consistent with the overabundance of shell stream morphology in the COCO simulations, the average distance of the stream to the host galaxy matches the observations, if the comparison is done with the shell-shape streams in the DES observation sample, see Figure <ref>.

photometry analysis to be completed for TNG50 and Auriga



 

 §.§ Comparison with Previous Work



We present the comparison with other observations of streams as far as reported in the literature:




  * <cit.> reports a frequency of 6-19% at a SB-limit of ∼ 27 mag arcsec^-2. We consider the lowest value of 6% as the most reliable for maximising purity in the detection of streams (see <ref>.


  * <cit.> reports a LSB feature (a superset of tidal streams) detection frequency of 12-18% at an average SB-limit of ∼ 27.7 mag arcsec^-2 for the g-band. Note that the g-band in the DES sample is fainter in average than the r-band by ∼ 0.5 mag arcsec^-2. It is also to be noted that the calculation of the SB limit in that paper is based on 1σ of the noise variation in apertures of 1.2 arcsec^2 placed on empty regions of the images and therefore differs from the one applied in this work [3σ, 100 arcsec^2] that follows the standard proposed in <cit.>. 


  * <cit.> reports a frequency of ∼ 10% at a SB limit of 28 mag arcsec^-2.


  * <cit.> reports 15% of tidal features, including 5±2% for streams and 5±2% shells (which we consider together in our curve) at a SB limit of 28.5 - 29  mag arcsec^-2 for the g-band. Note that the g-band in the DES sample is fainter in average as the r-band by ∼ 0.5 mag arcsec^-2.


  * <cit.> have characterised the morphology of more than 350 low surface brightness structures up to a distance of 42 Mpc through annotation of images from the Canada–France Imaging Survey (CFIS2) and the Mass Assembly of early-Type gaLAxies with their fine Structures survey (MATLAS3). They obtained 84 annotations for streams and 260 for shells, but out of these figures alone it is not possible to derive detectability figures for the brightest streams and shells, as there could be several of them in one image.


  * <cit.> reports a frequency of 12.2 ± 2.4% for Milky Way-like galaxies at a SB limit of 28.40 mag arcsec^-2.


  * <cit.> use deep imaging from the Subaru-Hyper Suprime Cam Wide data to search for tidal features in massive [log_10 M_⋆ / M_⊙ > 10] early-type galaxies (ETGs) in the SAMI Galaxy Survey. They report a tidal feature detection rate of 31 ±2 % at a surface brightness limit of 27 ±0.5 mag arcsec^-2 for the r-band. They calculate the surface brightness limit following the same approach as <cit.>, thus different to our method [3σ, 100 arcsec^2]. For comparison, our method applied to a test Subaru HSC image yields a surface brightness limit of 29.79 mag arcsec^-2.


  * In <cit.> the results of visual inspection of a sample of 838 edge-on  galaxies using images from three surveys: SDSS Strip-82, Subaru HSC and DESI (DECals, MzLS,BASS) are presented. In total, 49 tidal features out of 838 images are reported, equivalent to a frequency of 5.8% at a SB limit of 28.60 mag arcsec^-2. In that study, tidal features include also disc deformations and tidal tails, typical of major mergers.


 




Also a number of papers in the literature report the frequency of tidal features (including streams and shells) in cosmological simulations:




  * <cit.> reports on detection of tidal features inspecting mock-images produced using the NEWHORIZON cosmological simulations. Through production of surface brightness maps at different surface brightness limits, they predict the fraction of tidal features that can be expected to be detected at different limiting surface brightnesses.

In this study, tidal features comprise: (i) Stellar streams, (ii) Tidal tails, (iii)
Asymmetric stellar halos, (iv) Shells  (v) Tidal bridges, (vi) Merger remnants (vii)
Double nuclei, of which only (i) and (iv) are clearly of accreted nature.
For a surface brightness limit of 35 mag arcsec^-2), expert classifiers were able to identify specific tidal features in close to 100 per cent of galaxies (M_⋆ > 10^9.5 M_⊙), in agreement with our results.
For the range of stellar mass of the DES sample, log_10 M_⋆ = 10, they report ∼ 25% detection rate for streams and shells together at a surface brightness limit of 30 mag arcsec^-2 in the r-band. The detection rate becomes 35% for a surface brightness limit of 31 mag arcsec^-2. These figures match very well our predictions with Auriga and IllustrisTNG50.





 


  * In <cit.>, the results of identification and classification of tidal featurest in LSST-like mock-images from cosmological simulations are reported. Four sets of hydrodynamical cosmological simulations are used (NewHorizons, EAGLE, IllustrisTNG and Magneticum). The frequency of tidal features, expressed in fractions of the total number of images, is between 0.32 and 0.40 showing consistency across the different simulations. Tidal Features comprise streams/tails, shells, Plumes or Asymmetric Stellar Halos and Double Nuclei. Looking only at streams and shells, the percentage of detections varies between 5-15% depending on the level of confidence, for a SB limit of 30.3 mag arcsec^-2 in the r-band.


  * In <cit.>, the authors report the results of inspecting surface brightness maps generated from 30 Auriga cosmological simulations <cit.> of Milky Way-like galaxies looking for the brightest streams. These simulations are the same we have applied in this work. They report that no streams have been detected in images with a surface brightness limit brighter than 25 mag arcsec^-2 and that the stream detection frequency increases significantly between 28 and 29 mag arcsec^-2. For a surface brightness limit of 28.50 mag arcsec^-2 in the r-band they achieve a detection rate of ∼ 18 - 30%. This is far higher than our results show with the same Auriga simulations and we believe the reason is the different pixel size of the mock-images. While we use the pixel size of the DECam instrument, 0.262 arcsec, equivalent to 0.08 kpc at 70 Mpc distance, in <cit.> they use a pixel size of 1 kpc. This is relevant for the visual inspection of count images, as it has an impact on the S/N per pixel and could account for a significant difference in the surface brightness limit at which the streams are detected. 



  * <cit.> use the Magneticum Box4 hydrodynamical cosmological simulations to detect tidal features (streams, shells and tidal tails) and connect their morphology to the internal kinematics of their host galaxies. In Table 1 they present the fraction of galaxies with the different types of tidal features for host galaxies with M_⋆≥ 10^11 M_⊙. Looking at their Figure 3, the fraction of shells and streams together is ∼ 10% for galaxies with 10^10 M_⊙ < M_⋆ < 10^11 M_⊙, comparable with the stellar mass range of our simulations, at a surface brightness limit of 28.5 - 29 mag arcsec^-2. This results are similar to the ones obtained in this work.
 

    




 §.§ Caveats




The comparison of our stream observability results with those of other surveys and cosmological simulations is not always straightforward, due to: i) the range of halo mass and stellar mass is not always the same, although our analysis of stream observability does not reveal a significant correlation of the stream observation rate with the host stellar mass within the range of stellar masses considered in this work; ii) the method applied for the  visual inspection of the images, some of the previous works seem to be based on the inspection of surface brightness maps with cut-offs of the surface brightness limit, while we inspect count images, where the pixel size bears an influence on the S/N per pixel and thereby on the detectability of streams;  iii) the different ways of calculating the surface brightness limits of the images by the different authors, as pointed out in the previous Section and iv) the different classification schemes used for the LSB features and their meaning, e.g. streams, shells or tidal tails used by the different authors.

The observability result obtained with mock-images can only be seen as an indicative reference for predicting the stream frequency to be met by present and future surveys for a number of reasons. First, this result has been obtained with an idealised flat image background, while in real-life observations the background is often not sufficiently flat, preventing the detection to reach the theoretical surface brightness limit, as explained in <cit.> regarding DES images. As mentioned earlier, for surface brightness limits much fainter than those of the DES sample, the confusion of sources and possibly cirri will become more significant and the synthetic background will be less representative of the real one, making detection more difficult. Second, this prediction is dependent on the modelling assumptions underlying the cosmological simulations. Nevertheless, the fact that hydrodynamic simulations provide very similar results with one another and match also the results from observations of the DES sample, as well as previous surveys and simulation analysis, provides confidence in the predictions.

Regarding the method of visual inspection of images, it is clear that the human factor plays a role in the results of the detection. However, the confidence in the method can be increased by having a team with experience in searching specifically for streams in real observations, and a systematic and rigorous method to proceed. While different scientists may come to different conclusions in specific cases, overall, in a large survey, these differences may not have a significant influence in the global results. Furthermore the inspection of mock-images with a flat background (as seen in Figures) does not leave a large margin of interpretation regarding the presence of tidal features. It is in the classification of these features where the divergences are more likely to appear. In any case, as discussed above, the morphology is a weak observable  because its perception is dependent 

on
the line of sight. In the absence of a mature automatic detection method for extragalactic streams, visual inspection remains the state-of-the-art used by all the works in this domain reported in the literature.



§ CONCLUSIONS AND OUTLOOK



From the results obtained from comparing the stream frequency, characteristics

of the DES galaxy sample observations with 

cosmological simulations, we can conclude that overall the methods applied here work well and provide a valid reference for the analysis of stellar streams.

Generally the predictions of the simulations are in agreement with the results of the 
analysis
carried out on 

the DES sample, following the same approach to visual inspection as the present work, used as a reference <cit.>, and with previous work reported in the literature, as presented in Section <ref>.
This provides a degree of confidence in the simulation predictions regarding detection of streams in future surveys at surface brightness limits for which we do not have observations today. The cosmological simulations we have have analysed here predict that, in the absence of a confusion limit due to background and foreground sources, and with a pixel size similar to the one of the DECam instrument, a frequency of almost 70% in the detection of streams around galaxies can be achieved for a surface brightness limit of 32 mag arcsec^-2. This prediction can be extrapolated to other observations taking into account the effect of the pixel size on the S/N.
Nevertheless, there are some noticeable differences in the stream morphologies between observations and simulations and between simulations themselves, that should be further analysed in order to understand their origin and 

be able to evolve the specific simulation resolution and physics modelling required for stream analysis where needed.





 
We present a method for comparison of stream observations with cosmological simulations based on novel tools for generation of mock images. 

When inspecting the mock-images, we follow exactly the same approach and apply the same criteria, with the same team, as we did for inspecting the DES observational image sample, and reported in <cit.>.

In this work, we only compare with the output of the simulations at z=0

Exploring the evolution of streams from high redshift to the current time in the simulations could provide more insight into the history, mass ratio and kinematics of the preceding mergers,
 allowing to compare the simulated reality with the conclusions of the visual inspection. 
This may also help to understand the differences in stream morphology that have been highlighted in this work.

Overall, the results of our work indicate that surveys reaching a surface brightness limit fainter than 31 mag arcsec^-2 would be able to reach a stellar tidal stream detection rate of at least 50%, and thereby test the predictions of the ΛCDM model as implemented by state-of-the-art cosmological simulations.










         JMC wants thank the Leiden Observatory for hosting and providing computer infrastructure and facilities for carrying out part of this work, as well as the Universidad Complutense de Madrid for providing computer infrastructure used in this work.
         JMC wants to thank Yves Revaz for support in the use of the pNbody tool.

         DMD acknowledges the grant CNS2022-136017 funding by MICIU/AEI /10.13039/501100011033 and the European Union NextGenerationEU/PRTR and finantial support from the Severo Ochoa Grant CEX2021-001131-S funded by MCIN/AEI/10.13039/501100011033 and project (PDI2020-114581GB-C21/ AEI / 10.13039/501100011033). 

         


        JMC and MAGF acknowledge financial support from the Spanish Ministry of Science and Innovation through the project PID2022-138896NB-C55

        APC acknowledges support from a Taiwan Ministry of Education Yushan Fellowship and Taiwan National Science and Technology Council grants 112-2112-M-007-017 and 113-2112-M-007-009. 
        
        SRF acknowledge support from the Knut and Alice Wallenberg Foundation, the Swedish Research Council (grant 2019-04659), and the Swedish National Space Agency (SNSA Dnr 2023-00164). 
        SRF also acknowledges financial support from the Spanish Ministry of Science and Innovation through the project PID2021-123417ob-i00. 
        
        For this work we have used GNU Astronomy Utilities (Gnuastro, ascl.net/1801.009) versions 0.17, 0.18 and 0.20. Work on Gnuastro has been funded by the Japanese MEXT scholarship and its Grant-in-Aid for Scientific Research (21244012, 24253003), the European Research Council (ERC) advanced grant 339659-MUSICOS, and from the Spanish Ministry of Economy and Competitiveness (MINECO) under grant number AYA2016-76219-P.

        M.A acknowledges the financial support from the Spanish Ministry of Science and Innovation and the European Union - NextGenerationEU through the Recovery and Resilience Facility project ICTS-MRR-2021-03-CEFCA and the grant PID2021-124918NA-C43.

This work used high-performance computing facilities operated by the
Center for Informatics and Computation in Astronomy (CICA) at National
Tsing Hua University. This equipment was funded by the Ministry of
Education of Taiwan, the Ministry of Science and Technology of Taiwan,
and National Tsing Hua University.

This work is supported by the National Science Center, Poland under Agreement No.  2020/39/B/ST9/03494.

SB is supported by the UK Research and Innovation (UKRI) Future Leaders Fellowship (grant number MR/V023381/1).

        




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§ MOCK-IMAGES PHOTOMETRY






























