11institutetext: Univ Lyon, Univ Lyon1, Ens de Lyon, CNRS, Centre de Recherche
Astrophysique de Lyon UMR5574, F-69230, Saint-Genis-Laval, France
22institutetext: Leibniz-Institut fur Astrophysik Potsdam (AIP), An der
Sternwarte 16, 14482 Potsdam, Germany 33institutetext: Observatoire de Geneve,
Universite de Geneve, 51 Ch. des Maillettes, 1290 Versoix, Switzerland
44institutetext: Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA
Leiden, The Netherlands 55institutetext: Department of Astronomy, University
of Texas at Austin, 2515 Speedway, Austin, TX 78712, USA 66institutetext: Max
Planck Institute for Astronomy, Königstuhl 17, 69117, Heidelberg, Germany

# Median Surface Brightness Profiles of Lyman-$\alpha$ Haloes in the MUSE
Extremely Deep Field

Yucheng Guo Median Surface Brightness Profiles of Lyman-$\alpha$ Haloes in the
MUSE Extremely Deep Fielde-mail<EMAIL_ADDRESS>Surface
Brightness Profiles of Lyman-$\alpha$ Haloes in the MUSE Extremely Deep
Fielde-mail<EMAIL_ADDRESS>Roland Bacon Median Surface Brightness
Profiles of Lyman-$\alpha$ Haloes in the MUSE Extremely Deep FieldMedian
Surface Brightness Profiles of Lyman-$\alpha$ Haloes in the MUSE Extremely
Deep Field Lutz Wisotzki Median Surface Brightness Profiles of Lyman-$\alpha$
Haloes in the MUSE Extremely Deep FieldMedian Surface Brightness Profiles of
Lyman-$\alpha$ Haloes in the MUSE Extremely Deep Field Thibault Garel Median
Surface Brightness Profiles of Lyman-$\alpha$ Haloes in the MUSE Extremely
Deep FieldMedian Surface Brightness Profiles of Lyman-$\alpha$ Haloes in the
MUSE Extremely Deep Field Jérémy Blaizot Median Surface Brightness Profiles of
Lyman-$\alpha$ Haloes in the MUSE Extremely Deep FieldMedian Surface
Brightness Profiles of Lyman-$\alpha$ Haloes in the MUSE Extremely Deep Field
Joop Schaye Median Surface Brightness Profiles of Lyman-$\alpha$ Haloes in the
MUSE Extremely Deep FieldMedian Surface Brightness Profiles of Lyman-$\alpha$
Haloes in the MUSE Extremely Deep Field Johan Richard Median Surface
Brightness Profiles of Lyman-$\alpha$ Haloes in the MUSE Extremely Deep
FieldMedian Surface Brightness Profiles of Lyman-$\alpha$ Haloes in the MUSE
Extremely Deep Field Yohana Herrero Alonso Median Surface Brightness Profiles
of Lyman-$\alpha$ Haloes in the MUSE Extremely Deep FieldMedian Surface
Brightness Profiles of Lyman-$\alpha$ Haloes in the MUSE Extremely Deep Field
Floriane Leclercq Median Surface Brightness Profiles of Lyman-$\alpha$ Haloes
in the MUSE Extremely Deep FieldMedian Surface Brightness Profiles of
Lyman-$\alpha$ Haloes in the MUSE Extremely Deep Field Leindert Boogaard
Median Surface Brightness Profiles of Lyman-$\alpha$ Haloes in the MUSE
Extremely Deep FieldMedian Surface Brightness Profiles of Lyman-$\alpha$
Haloes in the MUSE Extremely Deep Field Haruka Kusakabe Median Surface
Brightness Profiles of Lyman-$\alpha$ Haloes in the MUSE Extremely Deep
FieldMedian Surface Brightness Profiles of Lyman-$\alpha$ Haloes in the MUSE
Extremely Deep Field John Pharo Median Surface Brightness Profiles of
Lyman-$\alpha$ Haloes in the MUSE Extremely Deep FieldMedian Surface
Brightness Profiles of Lyman-$\alpha$ Haloes in the MUSE Extremely Deep Field

(Submitted 2023)

We present the median surface brightness profiles of diffuse Ly$\alpha$ haloes
(LAHs) around star-forming galaxies by stacking 155 spectroscopically
confirmed Ly$\alpha$ emitters (LAEs) at $3<z<4$ in the MUSE Extremely Deep
Field (MXDF), with median Ly$\alpha$ luminosity $\mathrm{L_{Ly\alpha}\approx
10^{41.1}erg\,s^{-1}}$. After correcting for a systematic surface brightness
offset we identified in the datacube, we detect extended Ly$\alpha$ emission
out to a distance of $\approx$270 kpc. The median Ly$\alpha$ surface
brightness profile shows a power-law decrease in the inner 20 kpc, and a
possible flattening trend at larger distance. This shape is similar for LAEs
with different Ly$\alpha$ luminosities, but the normalisation of the surface
brightness profile increases with luminosity. At distances larger than 50 kpc,
we observe strong overlap of adjacent LAHs, and the Ly$\alpha$ surface
brightness is dominated by the LAHs of nearby LAEs. We find no clear evidence
of redshift evolution of the observed Ly$\alpha$ profiles when comparing with
samples at $4<z<5$ and $5<z<6$. Our results are consistent with a scenario in
which the inner 20 kpc of the LAH is powered by star formation in the central
galaxy, while the LAH beyond a radius of 50 kpc is dominated by photons from
surrounding galaxies.

###### Key Words.:

galaxies: high-redshift – galaxies: formation – galaxies: evolution –
intergalactic medium– cosmology: observations

##  1 Introduction

Galaxies are born and bred in their gaseous haloes (e.g. Tumlinson et al.,
2017). Within the circumgalactic medium (CGM), gas and metals can be ejected
from galaxies by feedback processes or stripping, or they can be (re-)accreted
to fuel star formation. On larger scales, the intergalactic medium (IGM)
traces the cosmic web of matter that connects, forms and fuels galaxies, and
its properties may be changed by feedback processes. Mapping the gas
distribution in the IGM and the CGM, and investigating their relationship with
the interstellar medium (ISM) of the galaxies are crucial for understanding
the evolution of galaxies.

The hydrogen Ly$\alpha$ line is a powerful tool for mapping the CGM and IGM at
high redshift (e.g. Steidel et al., 2011; Ouchi et al., 2020; Bacon et al.,
2021). Theoretical studies predict substantial amounts of neutral hydrogen in
the CGM and IGM (e.g. Fumagalli et al., 2011; van de Voort et al., 2012), and
this has been observed through absorption measurements (e.g. Lee et al., 2018;
Newman et al., 2020). The Ly$\alpha$ photons are produced by hydrogen
recombination following ionization, or by collisional excitation following
photo- or shock- heating. As they propagate through the ISM, CGM and IGM, they
can be repeatedly absorbed and re-emitted by the neutral hydrogen gas. This
resonant nature of Ly$\alpha$ makes it a unique tracer of neutral gas
distribution, though the likelihood of it being absorbed by galactic dust is
non-negligible. In addition to its utility in mapping the hydrogen gas, the
Ly$\alpha$ emission line is also a highly informative probe of high-redshift
star-forming galaxies (e.g. Ouchi et al., 2008; Shibuya et al., 2012; Guo et
al., 2020a; Bacon et al., 2023). It is intrinsically the strongest emission
line in the rest-frame UV/optical spectrum (e.g. Partridge & Peebles, 1967).
Therefore, observing the large-scale spatial distribution of Ly$\alpha$
emission can provide a map of the large-scale structure of the early Universe,
e.g., the pristine gas distribution as well as the distribution of high-
redshift galaxies.

In the past decades, extended Ly$\alpha$ emissions, known as Ly$\alpha$ haloes
(LAHs), have been observed around Ly$\alpha$ emitting galaxies (LAEs) at
$\mathrm{z\gtrsim 2}$ using the narrow band (NB) technique (e.g. Matsuda et
al., 2012; Feldmeier et al., 2013; Momose et al., 2014, 2016; Xue et al.,
2017; Wu et al., 2020; Kikuta et al., 2023) and integral field unit (IFU)
facilities (e.g. Wisotzki et al., 2016; Leclercq et al., 2017; Daddi et al.,
2021; Kusakabe et al., 2022). The LAHs have also been observed in the local
Universe in the vicinity of galaxies (e.g. Hayes et al., 2013; Duval et al.,
2016). The LAHs have physical scales of several tens of kpc, where most
important processes regulating galaxy formation and evolution occur. By
combining deep exposures and stacking, recent studies have expanded the
detection of extended Ly$\alpha$ emission to hundreds of kpc (e.g. Kakuma et
al., 2021; Kikuchihara et al., 2022; Lujan Niemeyer et al., 2022b, a). Despite
the substantial observational work on the spatial content of the LAHs, the
physical mechanisms of the production and propagation of Ly$\alpha$ photons
are still under debate (e.g. Cantalupo et al., 2005; Zheng et al., 2011;
Rosdahl & Blaizot, 2012; Byrohl et al., 2021).

The ESO-VLT instrument MUSE (the Multi Unit Spectroscopic Explorer, Bacon et
al., 2010) revolutionized the observation of the CGM and IGM. This IFU
facility with a large field of view is highly efficient in mapping the
extended emission from the CGM, or even the IGM, using multiple tracers, e.g.,
Ly$\alpha$ (e.g. Wisotzki et al., 2016, 2018; Cai et al., 2017, 2019; Leclercq
et al., 2017, 2020; Gallego et al., 2018, 2021; Claeyssens et al., 2019, 2022;
Bacon et al., 2021; Kusakabe et al., 2022), C IV, He II, C III] (e.g. Borisova
et al., 2016; Guo et al., 2020b), Mg II (e.g. Zabl et al., 2021; Leclercq et
al., 2022), and [O II] (e.g. Johnson et al., 2022). At $z\gtrsim 3$, MUSE
observations have identified large samples of individual LAEs, and allowed for
the analysis of the surface brightness profiles of the LAHs individually or by
stacking. Wisotzki et al. (2018) find that the LAHs on average extend to tens
to hundreds kpc, thus covering nearly all of the sky by projection. Bacon et
al. (2021) detect the cosmic web in Ly$\alpha$ emission on scales of several
comoving Mpc. Kusakabe et al. (2022) study the LAHs around UV-selected
galaxies instead of objects selected from their Ly$\alpha$ emission. The
existence of extended Ly$\alpha$ emission around non-LAEs implies a
significant amount of neutral hydrogen in the CGM of the normal star-forming
galaxies. The deep MUSE observations have also allowed for the spatially
resolved spectroscopic analysis of a sample of bright individual LAHs (e.g.
Claeyssens et al., 2019; Leclercq et al., 2020).

In this work, we measure the median Ly$\alpha$ surface brightness profile of
369 spectroscopically confirmed LAEs at $3<z<6$, with our primary focus being
$3<z<4$. The current dataset is 5-14 times deeper than that of Wisotzki et al.
(2018). We therefore expect to expand the detection of extended Ly$\alpha$
emission out to hundreds of kpc from the LAE center. Previous studies have
suggested that the dominant origin of the LAHs differs at different radii
relative to the galaxies (e.g. Lake et al., 2015; Mitchell et al., 2021;
Byrohl et al., 2021). By measuring the Ly$\alpha$ profiles out to large radii,
we expect to provide more constraints on the origin of the LAHs at different
distances.

The paper is organized as follows: we describe the definition of the LAE
sample and our data reduction in Section 2. Section 3 provides the results,
highlights the median Ly$\alpha$ surface brightness profile at $3<z<4$, and
presents the profiles of different subsamples and datasets. In Section 4 we
discuss our results by comparing with previous observations and theoretical
predictions, and also interpret the Ly$\alpha$ profiles from the point of view
of cross-correlation functions. We summarize in Section 5.

We adopt the standard $\mathrm{\Lambda CDM}$ cosmology with
$H_{0}\mathrm{=70\,km\,s^{-1}\,Mpc^{-1}}$, $\mathrm{\Omega_{m}=0.3}$ and
$\mathrm{\Omega_{\lambda}=0.7}$. All distances are proper, unless noted
otherwise.

##  2 The galaxy sample and data analysis

This work is based on the data release 2 (DR2) of the MUSE Hubble Ultra Deep
Field surveys (Bacon et al., 2023). The DR2 data consists of 3 datasets, a
$\mathrm{3\times 3}$ arcmin2 mosaic of 9 MUSE fields at 10-hour depth
(hereafter MOSAIC), a $\mathrm{1\times 1}$ arcmin2 field at 31-hour depth
(hereafter UDF-10), and the MUSE eXtremely Deep Field (MXDF), with a diameter
of 1 arcmin and the deepest achieved exposure of 141 hours. In this work, we
mainly use the MXDF. It is the deepest spectroscopic survey ever performed,
reaching an unresolved emission line median $\mathrm{1\sigma}$ surface
brightness limit of $\mathrm{<10^{-19}\,erg\,s^{-1}\,cm^{-2}\,arcsec^{-2}}$.
The MOSAIC dataset, which covers an area approximately nine times larger than
the MXDF but has an exposure time approximately 14 times shorter, is a useful
comparison dataset. In Section 3.4, we describe our efforts to detect more
extended emission in MOSAIC. For further details on the data reduction of the
MXDF and MOSAIC, we refer the reader to Bacon et al. (2023).

###  2.1 The LAE sample

Figure 1: Distribution of the redshift (first panel), Ly$\alpha$ luminosity
(second panel), Ly$\alpha$ line rest-frame FWHM (third panel) and S/N of
Ly$\alpha$ line (fourth panel) of the LAE sample. The thick lines present the
distributions of the whole galaxy samples in MXDF (blue lines) or MOSAIC (grey
lines). The narrow lines show the LAEs at $3<z<4$, the redshift range of most
interest for this work.

Bacon et al. (2023) perform a meticulous work that involves multiple cross-
checked procedures to achieve a high-quality and homogeneous source detection.
Their final catalogue provides the redshifts, multi-band photometries,
morphological and spectral properties, as well as measurements of stellar mass
and star formation rate of all the galaxies discovered in the MOSAIC, UDF-10
and MXDF fields. The MXDF field contains a total of 369 LAEs at $3<z<6$, while
MOSAIC contains 693 detected LAEs. In this work, we focus on the redshift
range of $3<z<4$. In this redshift range, the efficiency of MUSE is high, the
redshift dimming is low, there are no strong OH sky line, enabling us to
acquire a highly representative LAE sample that extends to very low Ly$\alpha$
luminosity. In the redshift range $3<z<4$, there are 155 LAEs detected in MXDF
and 329 in MOSAIC.

Figure 1 shows the distribution of redshift, Ly$\alpha$ luminosity
($\mathrm{L_{Ly\alpha}}$), rest-frame Ly$\alpha$ line FWHM and Ly$\alpha$ line
S/N for all LAEs and for the sample at $3<z<4$. All these quantities are
derived from Bacon et al. (2023). The median Ly$\alpha$ luminosity of LAEs at
$3<z<4$ in the MXDF is $\mathrm{L_{Ly\alpha,median}\approx
10^{41.1}erg\,s^{-1}}$. Their median stellar mass is $\mathrm{M_{*}\approx
10^{7.6}M_{\odot}}$. The median Ly$\alpha$ S/N for LAEs at $3<z<4$ in the MXDF
is 7.7, with the 95th percentile of 53.4 and the 5th percentile of 2.9.
According to Herrero Alonso et al. (2023), the typical halo mass for MXDF LAEs
over the whole redshift range of is
$\mathrm{log(M_{h}/[h^{-1}M_{\odot}])=10.77^{+0.13}_{-0.15}}$. The number
should be smaller for MXDF LAEs at $3<z<4$, as their typical Ly$\alpha$
luminosity is lower than their parent sample (second panel of Figure 1).

We try to avoid contamination from the bright AGNs that can be dominant in
powering the extended Ly$\alpha$ emission. Given the sky coverage of MXDF and
MOSAIC (1-3 arcmin), the possibility of detecting a substantial number of AGNs
is very low. There is one type-I AGN at $z$=3.2. There is another type-II AGN
at $z$=3.06 that is cross-matched by the Chandra Chandra 7Ms catalog (Luo et
al., 2017). Both AGNs are found in MOSAIC (Bacon et al., 2023), and are
excluded from our analysis. Considering that the LAEs in our sample have
Ly$\alpha$ luminosity $\mathrm{L_{Ly\alpha}<10^{43}erg\,s^{-1}}$, the AGN
contamination should be very low (e.g. Sobral et al., 2018; Calhau et al.,
2020; Zhang et al., 2021). According to the estimation of Zhang et al. (2021),
the AGN fraction at $\mathrm{L_{Ly\alpha}<10^{43}erg\,s^{-1}}$ is $\lesssim$
0.05.

###  2.2 Masking and continuum subtraction

The exposure of the MXDF field is not uniform, with the integration time
decreasing from 141 hours at the field centre to several hours at the edge of
the field (Bacon et al., 2023). We mask out the area with exposure time
smaller than 110 hours to keep a high S/N and a more homogeneous sample
throughout the selected FoV. We also mask the wavelength slices affected by
bright sky lines to avoid strong sky line residuals. We remove the continuum
by performing a spectral median filtering using a wide spectral window of 200
Å. This approach provides a rapid and effective way to remove continuum
sources in the search for extended line emission.

We focus on detecting the extended Ly$\alpha$ line at large distances from the
LAEs. To achieve this focus, we need to remove the contamination from emission
and absorption lines from foreground galaxies and from the LAEs themselves. We
mask out all the detected emission and absorption lines from all the galaxies
in the continuum-subtracted datacube, with the exception of the Ly$\alpha$
line. The mask is based on the composite segmentation images provided by Bacon
et al. (2023). The segmentation images are provided by SExtractor (Bertin &
Arnouts, 1996), with a S/N limit of 2.

Note that in case that more than one LAE exists in one NB, we keep all the
objects. In Section 3.2, we will demonstrate that as the exposure becomes
deeper and the distance increases, neighboring LAEs inevitably appear within
the field of view. Therefore, in Section 3.2, we will also present an
additional measurement after removing the neighboring LAEs.

Figure 2:  The stacked exposure time in each radial bin for LAEs at $3<z<4$.
The black and blue symbols show the MXDF and MOSAIC sample, respectively

###  2.3 Extraction of Ly$\alpha$ surface brightness

We construct the pseudo-NB image for each LAE based on the MUSE datacube. Each
NB is centered on the peak wavelength of the Ly$\alpha$ line. In the case of
Ly$\alpha$ lines with two peaks, we center the NB on the red peak. The
spectral bandwidth of the Ly$\alpha$ NB is fixed at 12 cMpc, which corresponds
to approximately 15.0 Å (in the observational frame) or 924.4 km/s at
$z\approx 3$, and 34.3 Å or 1206.4 km/s at $z\approx 6$. This bandwidth is
chosen to include most of the Ly$\alpha$ flux, providing a reasonable
tolerance for the variety of Ly$\alpha$ line widths (Figure 1), uncertainty in
the measurement of the Ly$\alpha$ peak wavelength, and the spatial variation
of the Ly$\alpha$ line at large radii (Guo et al. 2023 in preparation).

In each NB image, we measure the surface brightness of the Ly$\alpha$ emission
in radial bins. We average the flux inside successive concentric annuli
centered on each LAE, with the central radii of annuli ranging from 10 to 270
kpc. The final annulus includes signal as distant as 470 kpc. We ignore the
possible spatial offset of the Ly$\alpha$ centroid and the UV centroid, as our
point of interest is the extended emission at large radius. For bright
$2.9<z<6.7$ LAEs that can be detected both in UV and Ly$\alpha$, the median
spatial offset between UV and Ly$\alpha$ emission is reported to be
0.58$\pm$0.14 kpc (Claeyssens et al., 2022). Therefore, most of the possible
offsets between the UV and Ly$\alpha$ should reside in the first radial bin
($\approx$1.4”), and have a negligible impact on the detection at large
distances.

Finally, we stack all individual radial profiles to obtain the median
Ly$\alpha$ surface brightness profile of the LAH. In order to retain physical
units, we do not normalize the surface brightness profile of each LAE. As is
shown in Section 3, we present the results for different redshift intervals
and different $\mathrm{L_{Ly\alpha}}$ bins.

Note that in this method, we make use of the full potential of the MXDF’s
field of view. As we extend to the furthest radial bin, it approaches the edge
of the field. Consequently, the limited area of the field introduces
uncertainties associated with reduced exposure time. In Figure 2, we sum up
the exposure time of all pixels within each radial bin, and plot it against
radial distance. Indeed, at larger distances ($\gtrsim$300 kpc for the MXDF,
and $\gtrsim$800 kpc for the MOSAIC), the stacked exposure time start to
decrease due to field limitations. However, for the distances of primary
interest in this study (within $\approx$100 kpc), this effect is negligible.

Our methods for signal extraction and stacking are not exactly the same as
used in previous works of the same instrument (e.g. Wisotzki et al., 2018). We
first extract the individual radial profiles and then median-stack those
profiles, while Wisotzki et al. (2018) measure the radial profile from the
median-stacked NB image. In this work, we define the annuli in physical
distance, whereas Wisotzki et al. (2018) stack pseudo-NBs of LAEs at different
redshifts in the MUSE observed frame. Our method has the advantage that we
include the average of any anisotropic (or even filamentary) emission at large
distance. The method of Wisotzki et al. (2018) has the advantage that the
median-stacking of the images screens out the signals from the neighbors,
while those contributions are inevitable in our profiles. In Section 3.2 we
further quantify the contribution of the neighbors detected in the field.

###  2.4 Estimation of the surface brightness systematic offset

Although Bacon et al. (2023) provide datacubes that underwent very high-
quality data reduction and sky subtraction, a residual background may still
exist, including the sky residuals or other unknown systematic effects. In
this work, we refer to this residual background as systematic surface
brightness offset. At low surface brightness levels, the uncertainties
introduced by the systematic offset may not be negligible. Here we estimate
this offset using two independent methods.

$Random\;Wavelength$. We randomly select positions in the datacube at the same
sky coordinate as a real LAE, but at different wavelength layers. These
positions are adjacent to the real Ly$\alpha$ line within a wavelength range
of 5 Å to 50 Å (rest-frame). We extract the surface brightness profiles from
these randomly selected positions, using the same procedures employed as the
real LAEs. As the datacube is continuum-subtracted, the surface brightness
profiles should be zero on average. If the stacked profile is not zero, we can
infer that a systematic offset is present. We generate 100 these random
measurements for each object. We estimate the potential systematic offset by
taking the median of these random surface brightness profiles.

$Random\;Position$. In the pseudo-NB image, we pick up random positions on the
sky. For each real LAE, we perform 100 such measurements at random positions
that are at least 8” away, but within the same image. We then extract the
surface brightness profiles from these randomly positions. The median of these
random profiles is then taken as another measurement of the systematic offset.

The systematic surface brightness offsets are finally subtracted from the
surface brightness profile of each individual LAE before we stack all the
profiles. As illustrated in the lower panels of Figure 3, the two measurements
of systematic surface brightness offsets exhibit good agreement with each
other. The median systematic offset is very small ($\mathrm{\approx
1.67\times\ 10^{-21}\,erg\,s^{-1}\,cm^{-2}\,arcsec^{-2}}$). To estimate the
stacking uncertainty, we employ a bootstrap algorithm, randomly resampling 70%
of the sample for 10,000 iterations.

##  3 Results

###  3.1 The median Ly$\alpha$ surface brightness profiles

Figure 3: The median Ly$\alpha$ surface brightness profiles of our MXDF LAE
sample after subtracting the systematic surface brightness offset. The left
and right columns present the two methods for measuring the systematic offset
(Section 2.4), by inserting fake objects at the same sky position as the real
LAEs but in different wavelength layers (left column), and by inserting fake
objects in the same wavelength layers but at different sky positions (right
column). The blue dashed lines show the power-law fit. The small panels at the
bottom show the systematic offset on a linear $y$-scale. The final radial bin
is centered at 270 kpc, but it includes signal out to $\approx$470 kpc.
Different colors show different redshift ranges. All the symbols are slightly
shifted horizontally for better visualization. We plot the PSFs in the MXDF at
3000Å and 9000Å for comparison (dotted lines).

Figure 3 presents the median Ly$\alpha$ surface brightness profiles around
LAEs in different redshift ranges. The left and right panels show the two
measurements of the systematic surface brightness offset. The PSFs of the MXDF
at 5000Å and 9000Å are also plotted for comparison. For all redshift ranges,
the median Ly$\alpha$ surface brightness profiles are clearly more extended
than the PSF.

In the left and right columns of Figure 3, we compare the surface brightness
profiles correcting from the systematic offset estimated using the two
independent methods presented in Section 2.4. These two independent
measurements give consistent results. In the following sections we only show
the results of the $Random\;Wavelength$ method. However, we verified that the
results of $Random\;Position$ or the average of these two methods does not
change our conclusions.

As shown by the blue dots in Figure 3, at $3<z<4$, we detect the extended
Ly$\alpha$ emission out to hundreds of kpc. Within approximately 20 kpc, the
Ly$\alpha$ surface brightness exhibits a gradual decrease. If we consider
solely the scattering of Ly$\alpha$ photons from the central galaxy to the
CGM, a power-law surface brightness profile with a slope of -2.4 has been
predicted (Kakiichi & Dijkstra, 2018). We apply a power-law model to our
stacked profile, specifically targeting the inner $\approx$20 kpc while also
accounting for the PSF effect. The resulting power-law index,
$\mathrm{-2.47\pm 0.11}$ agrees with the theoretical prediction, suggesting a
notable influence of the scattering effect within this distance.

At radii larger than 20 kpc, a noticeable flattening trend emerges, which is
more plausible within the range of 20 to 50 kpc. Within this radial span, the
Ly$\alpha$ surface brightness stays at a level of $\mathrm{4.73\pm 1.95\times\
10^{-21}\,erg\,s^{-1}\,cm^{-2}\,arcsec^{-2}}$. The detections within the 20–50
kpc range have a slightly above 2$\sigma$ significance, which means we cannot
entirely rule out variations attributed to errors. Nevertheless, as
demonstrated in Section 3.4, this change of slope in the Ly$\alpha$ surface
brightness profile at large radius is consistently observed across different
data samples and facilities (e.g. Wisotzki et al., 2018; Lujan Niemeyer et
al., 2022b).

The furthest radial bin includes signals as far out as 470 kpc, with its
center located approximately at 270 kpc. Within the 50–270 kpc range, we
detect a tentative signal, with the combined surface brightness
$\mathrm{1.20\pm 0.49\times\ 10^{-21}\,erg\,s^{-1}\,cm^{-2}\,arcsec^{-2}}$.
Despite the presence of errors, a comparison with the 20–50 kpc range reveals
a break in the Ly$\alpha$ surface brightness at 50–270 kpc, with a drop
occurring after approximately 50 kpc.

At higher redshift, we do not detect extended emission at as large distance as
at $3<z<4$. We achieve a robust detection within approximately 30 kpc and 15
kpc for LAEs at $4<z<5$ and $5<z<6$, respectively. Comparing the three
redshift intervals, we do not detect any obvious evolution of the observed
Ly$\alpha$ surface brightness profiles. This agrees with previous works (e.g.
Wisotzki et al., 2018). When we compare the Ly$\alpha$ surface brightness
profiles at different redshift, we cannot ignore the observational effects.
Due to the stronger cosmological surface brightness dimming at higher
redshift, lower efficiency of MUSE at the red end and the stronger sky
background contamination, the high-redshift LAEs in our sample are more
luminous than those at $3<z<4$ (see the second panel of Figure 1). As we will
show in Section 3.3, there is a strong correlation between Ly$\alpha$
luminosity of the galaxy and the brightness of the LAH. Therefore,
observational incompleteness may complicate the comparison of the Ly$\alpha$
surface brightness profiles at different redshift.

Because we detect the extended Ly$\alpha$ haloes out to hundreds of kpc, the
neighboring galaxies within the field of view, if they exist, may contribute
to the surface brightness profile and are more likely to do so with increasing
distance. In Section 3.2, we mask all nearby LAEs to provide a “cleaner”
Ly$\alpha$ surface brightness profile. However, since these neighbors are
quite common, and we are not able to distinguish two adjacent neighboring
LAHs, we take Figure 3 as the more valuable result, that contains all the
averaged environmental information at large distance.

###  3.2 The influence of the nearby galaxies

Figure 4:  The distribution in the upper panel presents the number of
neighbors for each LAE within the NB. The middle panel displays the projected
distance to the nearest neighbor. The lower panel displays the projected
distance to the brightest neighbor. All three panels are based on the
catalogue of the MXDF.  Figure 5:  The Ly$\alpha$ surface brightness profiles
after masking the neighboring LAEs (green symbols) at $3<z<4$. For comparison,
we also plot the profiles before masking the neighboring LAEs (blue symbols).
The symbols are slightly shifted horizontally for better visualization.

As we have already mentioned in Section 2.2, since we are detecting the
extended Ly$\alpha$ emission at hundreds of kpc, the influence of nearby LAEs
(or the nearby LAHs) is inevitable. This is demonstrated by our statistics in
Figure 4. In the upper panel of Figure 4, we calculate the number of neighbors
for each LAE in the MXDF. Here the neighbor of a LAE is defined as any LAE at
the field of view and its Ly$\alpha$ peak wavelength $\pm$ 0.5$\times$FWHM at
the wavelength range of the same NB. The LAE and its neighbor(s) are within a
projected distance of 1’, but not necessarily physically or gravitationally
related. The distribution in Figure 4 is highly observationally-biased, as the
sample incompleteness changes with redshift and because neighbors can also
exist outside the field of view. Despite this incompleteness, we find that out
of 369 LAEs in the MXDF, only 36 LAEs do not have any detected neighbor within
the same NB. Although it is not shown in Figure 4, we also take into account
the LAEs identified in the MOSAIC, that includes the brighter neighbors at
larger distance. After including the MOSAIC LAEs, only 1 out of the 369 LAEs
is totally isolated. The number of the neighbors can exceed 30.

In the middle panel of Figure 4, we present the distance of a MXDF LAE to its
nearest neighbor. In the lower panel of Figure 4, we present the distance of a
MXDF LAE to the neighbor with the highest Ly$\alpha$ luminosity. The median
distance of a LAE to its nearest neighbor is 110 kpc, the 10th and 90th
percentiles are 29 and 274 kpc, respectively. The median distance to the
brightest neighbor is 226 kpc, the 10th and 90th percentiles are 97 and 377
kpc, respectively. In this case, at the scale of hundreds of kpc, the overlap
of the LAHs cannot be ignored. Compared to previous work, this LAH-overlapping
problem is more evident for us, because our extremely deep observations
identify lots of faint LAEs that could not be detected in previous
observations. In fact, the previous work, restricted by observational depth or
spatial resolution, cannot remove the effect of the neighboring LAEs to get a
“cleaner” radial profile. In other words, the previous observations with
shorter integration time or lower spatial resolution already included lots of
faint neighbors in their Ly$\alpha$ surface brightness profiles.

We provide another measurement of the Ly$\alpha$ surface brightness profile by
excluding the nearby LAEs. In each NB, except the target LAE, we mask all the
nearby galaxies based on the segmentation images given by Bacon et al. (2023).
Then we compute the median profiles following the same procedures as in
Section 2.3. This operation only removes the bright area of the nearby LAHs,
because we are not able to decompose each galaxy from its LAH. The final
surface brightness profile may still contain the diffuse content of the nearby
LAHs.

The median Ly$\alpha$ surface brightness profile after masking the neighboring
LAEs is shown by the green dots in Figure 5. For comparison, we also show the
Ly$\alpha$ surface brightness profile before masking the neighbors (blue dots
in Figure 5, same as Figure 3). Within the inner 50 kpc, the two profiles are
very similar, though there are inconspicuous differences at 20-50 kpc. Beyond
50 kpc we do not get a robust detection after masking the neighbors. The upper
limits in the last two radial bins further show the break of the Ly$\alpha$
surface brightness profile at $\approx$ 50 kpc. Limited by the radial sampling
and the S/N, we are not able to provide better constraints on this break, in
particular its exact distance. However, the downward trend of the profile is
quite robust. After stacking the outer two radial bins, we provide a 2-sigma
upper limit of Ly$\alpha$ surface brightness at 50-270 kpc of
$\mathrm{0.93\times\ 10^{-21}\,erg\,s^{-1}\,cm^{-2}\,arcsec^{-2}}$, about more
than half an order-of-magnitude lower than the surface brightness at 20-50
kpc.

The potential flattening of the Ly$\alpha$ surface brightness profile beyond
20 kpc is intriguing. This observation suggests a change in the dominant
mechanism(s) responsible for producing Ly$\alpha$ photons at varying
distances. Moreover, the possible break at around 50 kpc likely indicates that
the mechanisms governing the production and propagation of Ly$\alpha$ photons
at smaller radii become less influential at larger distances.

###  3.3 The LAHs of bright and faint LAEs

Figure 6: The Ly$\alpha$ surface brightness profiles at $3<z<4$. We split the
LAE sample into two subsamples based on $\mathrm{L_{Ly\alpha}}$. The profiles
before masking the neighboring LAEs are shown by small symbols. The profiles
after masking are shown by light-colored large symbols. For comparison, we
also plot the result of Wisotzki et al. (2018). The symbols are slightly
shifted horizontally for better visualization.

To study the possible correlations between Ly$\alpha$ surface brightness
profiles and LAE properties, we divide the LAE sample at $3<z<4$ into two
subsamples based on the median Ly$\alpha$ luminosity $\mathrm{L_{Ly\alpha}}$.
The corresponding Ly$\alpha$ surface brightness profiles are shown in Figure
6. The green and blue dots present the high- and low-$\mathrm{L_{Ly\alpha}}$
subsamples, respectively. For each subsample, we show the profiles before and
after masking the neighboring galaxies.

For all subsamples, the Ly$\alpha$ surface brightness profiles decrease within
approximately 20 kpc. The Ly$\alpha$ profiles of high- and
low-$\mathrm{L_{Ly\alpha}}$ LAEs have similar slopes, but the normalisation is
$\approx$1-dex fainter for the low-$\mathrm{L_{Ly\alpha}}$ subsample. This
suggests that the production of Ly$\alpha$ photons in the central galaxies
plays an important role in boosting the LAHs within $\approx$20 kpc, and that
the mechanisms that produce and propagate the Ly$\alpha$ photons outward may
scale with $\mathrm{L_{Ly\alpha}}$.

At larger radii ($\gtrsim$20 kpc), the two radial profiles show evidence of
flattening, despite the large errors (or upper limits). The flattening trend
appears for both subsamples, but the change occurs at a shorter distance for
the low-$\mathrm{L_{Ly\alpha}}$ subsample.

As we have discussed in Section 3.2, nearby LAEs are contributors to the LAHs
at distances larger than 50 kpc. The trend is not very clear for the two
subsamples, because the S/N is low. Still, the measured values (or upper
limits) before masking the neighboring galaxies are slightly higher than those
after masking the neighboring galaxies. Nonetheless, the contribution from
neighboring galaxies may not be the key reason for the flattening of the
profiles at 20-50 kpc, which has already been demonstrated by Section 3.2 and
Figure 5.

The Ly$\alpha$ surface brightness profile of higher-$\mathrm{L_{Ly\alpha}}$
LAEs agrees with Wisotzki et al. (2018). The sample of Wisotzki et al. (2018)
is built from brighter LAEs in the UDF-10 and MOSAIC fields. The good
agreement between Wisotzki et al. (2018) and our high-$\mathrm{L_{Ly\alpha}}$
profile further demonstrates the robustness of our measurement. Similar to our
work here, Wisotzki et al. (2018) split their LAE sample by
$\mathrm{L_{Ly\alpha}}$. They also reach a similar conclusion that the radial
profiles follow the same trend but with different normalisations of the
surface brightness.

###  3.4 Comparison of different datasets

We have shown the median Ly$\alpha$ surface brightness profiles of LAEs at
$3<z<4$ out to $\approx$270 kpc using the MXDF data. In this section, we
firstly try to detect the Ly$\alpha$ signal at larger distance. The MOSAIC is
a good supplement to the MXDF, with $\approx$9 times larger sky coverage but
at the expense of 10-14 times shorter integration time. We would like to use
MOSAIC to constrain the median Ly$\alpha$ surface brightness as far as 1000
kpc. We present the radial profile in Figure 7. We do not achieve a robust
detection at the distance of 60-1000 kpc. Instead, we provide a 2-sigma upper
limit of $\mathrm{2.82\times\ 10^{-21}\,erg\,s^{-1}\,cm^{-2}\,arcsec^{-2}}$.

In Figure 7 we compare with the Ly$\alpha$ surface brightness profiles of
Wisotzki et al. (2018). The profile of MOSAIC agrees well with Wisotzki et al.
(2018) within approximately 60 kpc. The median Ly$\alpha$ profile of MXDF is
about 0.5 dex fainter than that of the MOSAIC and Wisotzki et al. (2018).
Considering that the shallower observation of the MOSAIC can only identify the
bright LAEs (Figure 1), this comparison of Ly$\alpha$ surface brightness
profiles in the MOSAIC and the MXDF reflects again the trend with
$\mathrm{L_{Ly\alpha}}$ (see Figure 6).

We also show the observational result of the Hobby-Eberly Telescope Dark
Energy Experiment (HETDEX, Gebhardt et al., 2021). The HETDEX Ly$\alpha$
surface brightness profile around LAEs at $z\approx 2.5$ is provided by Lujan
Niemeyer et al. (2022b), and shown by yellow dots in Figure 7. The Ly$\alpha$
surface brightness profile is scaled by $(1+z)^{4}$. Our observation and
HETDEX have different PSF size, spatial resolution and observational strategy.
HETDEX performs shallow observations over a large sky area, while our MXDF and
MOSAIC concentrate on small sky areas with extremely deep exposures. The LAE
sample of Lujan Niemeyer et al. (2022b) is 1 order-of-magnitude brighter than
our sample. Their typical $\mathrm{L_{Ly\alpha}}$ is
$\mathrm{10^{42.8}erg\,s^{-1}}$. For comparison, the median
$\mathrm{L_{Ly\alpha}}$ of the MXDF and MOSAIC sample at $3<z<4$ are
$\mathrm{L_{Ly\alpha,MXDF}\approx 10^{41.1}erg\,s^{-1}}$ and
$\mathrm{L_{Ly\alpha,MOSAIC}\approx 10^{41.5}erg\,s^{-1}}$ respectively. In
Figure 7, the profiles are strongly influenced by the different PSF. Lujan
Niemeyer et al. (2022b) tried to re-scale the profile of Wisotzki et al.
(2018) to their PSF size and LAE luminosity, and get very good agreement.

Despite the different luminosities, observational facilities and noise levels,
the Ly$\alpha$ surface brightness profiles of the MXDF, MOSAIC, and Lujan
Niemeyer et al. (2022b) show similar patterns. Among these datasets, the MXDF
obviously achieves a deeper detection limit. Therefore, here we take the
radial profile of the MXDF as an example. At small distances, the profile
shows a power law decrease. The normalisation of the profile depends strongly
on $\mathrm{L_{Ly\alpha}}$. Then the profile likely reaches a plateau. The
transition between the decrease and plateau appears at $\approx$20 kpc for
MXDF. The transition distance increases with higher $\mathrm{L_{Ly\alpha}}$.
Beyond the plateau ($\approx$50 kpc for MXDF), the profile likely drops to a
very low surface brightness level, and becomes dominated by neighboring
galaxies (Section 3.2). The distance of this break also increases with
$\mathrm{L_{Ly\alpha}}$.

Note that a similar trend is also observed in the Ly$\alpha$ surface
brightness profile of quasars at $2<z<4$, though the physical mechanisms in
behind may be different. Several works find that the Ly$\alpha$ surface
brightness profile within the distance of approximately 100 kpc to the quasar
decrease by a power with power law index of approximately -1.8 (e.g. Borisova
et al., 2016; Cai et al., 2019). According to Lin et al. (2022), the profile
within 1 cMpc decreases by a similar power law. At larger radii (1-100 cMpc),
the profile shows a flatter slope but with large variation, which can be well
explained by a two halo-term of the clustered Ly$\alpha$ sources.

Figure 7: The Ly$\alpha$ surface brightness profiles of LAEs at $3<z<4$ in the
MOSAIC field (blue symbols) and in the MXDF field (red symbols). Both profiles
include nearby LAEs. For comparison, we also plot the result of Wisotzki et
al. (2018) and Lujan Niemeyer et al. (2022b). The Lujan Niemeyer et al.
(2022b) profile is scaled by $(1+z)^{4}$.

##  4 Discussion

###  4.1 Physical origins of the LAHs

While the existence of LAHs around star-forming galaxies at $2<z<6$ has been
confirmed by previous works (e.g. Steidel et al., 2011; Matsuda et al., 2012;
Momose et al., 2014; Wisotzki et al., 2016, 2018; Leclercq et al., 2017;
Kusakabe et al., 2022), there is no consensus yet on the dominant physical
mechanism giving rise to the LAHs. Several physical origins are on the list
(e.g. Momose et al., 2016; Ouchi et al., 2020), including the resonant
scattering of Ly$\alpha$ photons produced by recombination of gas photoionized
by star formation or an AGN (in the central galaxy and also in unresolved
satellites), the collisional excitation and recombination in cooling gas
flowing into a galaxy, as well as the fluorescent emission from the UV
background. The dominant origin of the Ly$\alpha$ photons may change with the
distance to the galaxies (e.g. Lake et al., 2015; Mitchell et al., 2021;
Byrohl et al., 2021).

In this work, we detect Ly$\alpha$ emission around LAEs extending out to
$\approx$270 kpc by median-stacking a representative sample of LAEs (Figure
3), re-demonstrating that LAHs are a common property of LAEs. Our measurement
provides the typical Ly$\alpha$ surface brightness profile around the LAE with
median Ly$\alpha$ luminosity $\mathrm{\approx 10^{41.1}erg\,s^{-1}}$. In this
section, we will discuss the dominant physical origin of the Ly$\alpha$
photons at different radial ranges.

Before discussing the physics dominating at different distances, we firstly
clarify the difference between the extent of the LAHs and the size of dark
matter haloes. Following the method of Leclercq et al. (2017), we estimate the
virial radius based on the $r_{vir}$ \- UV magnitude relation predicted by the
semi-analytic model of Garel et al. (2015). The typical virial radius
$r_{vir}$ of the MXDF LAEs is approximately 20 kpc. The estimation of
$r_{vir}$ is always model-dependent and we expect a large amount of scatter.
Our detection of extended Ly$\alpha$ emission out to $\approx$270 kpc implies
that the Ly$\alpha$ emission can extend to $\gtrsim$10 times larger radii than
$r_{vir}$.

####  4.1.1 $\mathrm{r\lesssim 20\leavevmode\nobreak\ kpc}$

The production of Ly$\alpha$ photons in the inner region of the LAH is closely
related to the “central engine”, e.g., the star formation in the galaxy. The
Ly$\alpha$ photons are produced by recombination in the star-forming regions
of the host galaxy, part of them succeed in escaping the dusty ISM. The
existence of outflows may help the escape of Ly$\alpha$ photons. These escaped
Ly$\alpha$ photons can be scattered by the neutral hydrogen in the CGM and
produce the observed LAH. Spectroscopic observations find that the majority of
Ly$\alpha$ lines are red-shifted with respect to the systemic redshift and
have red-asymmetric profiles, which is interpreted as a signature of galactic
outflows (e.g. Verhamme et al., 2006; Schaerer et al., 2011; Dijkstra &
Kramer, 2012; Chang et al., 2022). Such a model of Ly$\alpha$ scattering
through outflowing media has been successful in reproducing the observed
Ly$\alpha$ spectral and surface brightness profiles (e.g. Yang et al., 2016;
Song et al., 2020; Li et al., 2022).

In our analysis, within the central $\approx$20 kpc ($r_{vir}$), the
Ly$\alpha$ surface brightness profile drops down as a power law with
increasing radius (Figure 3). As is shown in Figure 6, dividing the sample
into high-$\mathrm{L_{Ly\alpha}}$ and low-$\mathrm{L_{Ly\alpha}}$ subsamples
results in similarly shaped Ly$\alpha$ surface brightness profiles, but the
normalisation increases with $\mathrm{L_{Ly\alpha}}$. This is also seen in
Figure 7 by comparing datasets spanning a larger range of
$\mathrm{L_{Ly\alpha}}$. If we assume that the Ly$\alpha$ photons within this
radius are primarily produced by star formation, the similar slopes of the
high-$\mathrm{L_{Ly\alpha}}$ and low-$\mathrm{L_{Ly\alpha}}$ subsamples
suggest that the production and propagation mechanism of the Ly$\alpha$
photons in the inner CGM are similar for LAEs with different
$\mathrm{L_{Ly\alpha}}$.

####  4.1.2 $\mathrm{20\lesssim r\lesssim 270\leavevmode\nobreak\ kpc}$

At this distance of $\mathrm{20\lesssim r\lesssim 270\leavevmode\nobreak\
kpc}$, the Ly$\alpha$ surface brightness profile exhibits a trend of
flattening (Figure 3), despite the variation of errors. This trend is more
robustly observed at 20-50 kpc ($\approx$1-3 $r_{vir}$). At this radial span,
the Ly$\alpha$ surface brightness profile only exhibits marginal inconsistency
after masking the nearby LAEs (Figure 5). As is shown in Figure 4, neighboring
LAEs start to emerge at tens of kpc, but the majority is at hundreds of kpc.
These findings suggest that the flattening trend observed cannot be solely
attributed to the presence of nearby detected LAEs. The interpretation of the
Ly$\alpha$ surface brightness profile at this distance (20-50 kpc) is
complicated, because multiple factors are at play. At this distance, the
neighbors start to have an effect, and the “central engine” begins to lose its
dominance. The flattening of the Ly$\alpha$ profile likely indicates a change
in the dominant power source, which can be explained as signs of cooling
radiation or undetected satellites, or a mixture of all (e.g. Haiman et al.,
2000; Rosdahl & Blaizot, 2012; Mitchell et al., 2021). The Ly$\alpha$ emission
at large distance to the central galaxy can also be explained as the
fluorescence emission from gas illuminated by the UV background (e.g.
Cantalupo et al., 2005; Mas-Ribas & Dijkstra, 2016). For example, Gallego et
al. (2021) use the Ly$\alpha$ emission at large distance to constrain the
photoionization rate of hydrogen and the covering fraction of Lyman limit
systems, based on the assumption that all the Ly$\alpha$ emission at large
distances originates from Ly$\alpha$ fluorescence in optically thick H I
clouds.

Compared to 20-50 kpc, the Ly$\alpha$ surface brightness profile is likely to
drop at approximately 50 kpc, and then stays at a lower surface brightness
level out to 270 kpc (Figure 3). This potential break at approximately 50 kpc
is clearer after we remove the nearby LAEs. In Figure 5, after masking the
nearby LAEs within the same NB, the Ly$\alpha$ emission at 50-270 kpc is below
the detection limit. The comparison of the two profiles in Figure 5 thus shows
that bright nearby LAHs make a major contribution to the observed Ly$\alpha$
surface brightness profile at 50-270 kpc ($\gtrsim 3r_{vir}$).

In the simulations of Mitchell et al. (2021), the LAHs at $3<z<4$ at tens of
kpc are mainly powered by satellite galaxies. The neighbors defined in this
work (Section 3.2) are not necessarily bounded within the gravitational
potential of a more massive galaxy. The clustering analysis of Herrero Alonso
et al. (2023) suggests that only $\lesssim$10% of the MXDF LAEs (same as our
LAE sample) are satellites. Therefore, in our case, the neighbors that
contribute to the LAHs at 50-270 kpc are mostly bright central galaxies
instead of satellites. Byrohl et al. (2021) study a similar case. They apply
Monte Carlo radiative transfer of Ly$\alpha$ to TNG50 cosmological
magnetohydrodynamical simulations. Their simulations find that most Ly$\alpha$
photons at large radii originate from nearby bright haloes and are then
scattered in the CGM of the target halo and also in the IGM. In our work,
taking advantage of deep exposures and high spatial resolution, we directly
observe the LAHs being enhanced by nearby bright LAEs (Figure 6).

Therefore, at radii of 50-270 kpc, we can confirm that the star formation in
nearby bright LAEs provides a major contribution to the extended Ly$\alpha$
emission. The contribution of other Ly$\alpha$ emission mechanisms at these
radii, including cooling radiation, flourescence, and satellite galaxies,
cannot be robustly measured. Instead, we provide an upper limit on the total
of these mechanisms (Section 3.2).

The potential break observed in the Ly$\alpha$ surface brightness profile at
approximately 50 kpc is intriguing. Limited by the S/N, we cannot rule out the
possibility of error variations. Here we provide a few speculative
considerations. It seems to indicate a quick change in the powering mechanism
of the LAH or physical state of the CGM at this distance. This may suggests a
typical clustering pattern of satellites, or enhancement of the Ly$\alpha$
flux due to collisional excitation of gas inflows at $\mathrm{20\lesssim
r\lesssim 50\leavevmode\nobreak\ kpc}$. Another possibility is that the
neutral hydrogen density decreases at approximately 50 kpc. According to the
simulations of Mitchell et al. (2021), the hydrogen number density shows a
hint of decrease with radius, but their simulations stops at 1 $r_{vir}$. The
gas state and physical processes at large distance to the galaxy (around 3
$r_{vir}$ in our case) is not very clear. To further address this question, we
conduct detailed simulations as will be described in Guo et al. in
preparation.

###  4.2 The Ly$\alpha$ surface brightness profile as a cross-correlation
function

Intensity mapping of the cumulative Ly$\alpha$ and other emission lines (e.g.,
H I, CO) has been proposed as a a powerful tool for understanding the line
emissivity and the large-scale matter distribution in the Universe (e.g.
Bernal & Kovetz, 2022). In particular, by stacking the cross-correlation
functions (CCFs) between the source redshift catalogue and maps expected to
include Ly$\alpha$ emission from the same sources, efforts have been made to
constrain the integrated Ly$\alpha$ emission over large cosmological volumes
(e.g. Croft et al., 2016, 2018; Kakuma et al., 2021; Kikuchihara et al., 2022;
Lin et al., 2022). Based on the Sloan Digital Sky Survey (DR12) spectra, Croft
et al. (2016, 2018) measure the cross-correlation between quasar positions and
Ly$\alpha$ emission imprinted in the residual spectra of luminous red
galaxies. They detect positive signal of extended Ly$\alpha$ emission around
quasars on scales as far as 15 cMpc. This measurement is renewed by Lin et al.
(2022), using DR16 of the Sloan Digital Sky Survey. The extended Ly$\alpha$
emission around normal galaxies is orders-of-magnitude fainter than the
quasars and thus more difficult to detect. Recently, positive detections of
the Ly$\alpha$-LAE CCF as far as $\approx$1 cMpc have been obtained by
stacking deep NB images around bright LAEs (Kakuma et al., 2021; Kikuchihara
et al., 2022).

In the 2 dimensional case, the CCF between Ly$\alpha$ emission intensity and
the position of a given LAE mathematically equals the Ly$\alpha$ surface
brightness at different radii (e.g. Kakuma et al., 2021). Restricted by the
sky area of the MXDF and MOSAIC, our observations are not ideal for mapping
the extended emission at a scale of several cMpc, but the extremely deep
exposure and good spatial resolution provide a unique experiment that
identifies very faint galaxies and foreground contaminators. This could be a
good supplement to Ly$\alpha$ intensity mapping experiments. Our work is more
efficient in removing line and continuum contaminators. Our dedicated pseudo-
NBs include all the potential signal but not extra noise. More importantly, we
quantify the contribution of the faint LAEs that are not detected in previous
works, and anonymously contribute to their Ly$\alpha$ CCFs.

Our precise and reliable measurement of the Ly$\alpha$ surface brightness
profile will be helpful for the design of future intensity mapping
experiments, which usually have lower spatial resolution and perform much
shallower observations. It would also be very meaningful to compare our CCFs
with cosmological simulations.

##  5 Summary

Thanks to the extremely deep MUSE observations of the Hubble Ultra Deep Field,
we unveil the typical Ly$\alpha$ surface brightness profile around LAEs down
to unprecedented depth and distance. Our major results are summarized below.

Based on the MXDF data, we present the median Ly$\alpha$ surface brightness
profiles of LAEs with Ly$\alpha$ luminosity $\mathrm{\approx
10^{41.1}erg\,s^{-1}}$ at $3<z<4$. After carefully correcting for the
systematic surface brightness offsets of the MUSE datacube, we detect extended
Ly$\alpha$ emission out to 270 kpc. The Ly$\alpha$ surface brightness profile
decreases as a power law within a radius of 20 kpc, followed by a flattened
profile at 20-50 kpc, and a potential drop to a lower level at 50-270 kpc. We
observe a possible break in the Ly$\alpha$ profile at a radius of
approximately 50 kpc.

We find that the nearby LAEs make a major contribution to the Ly$\alpha$
surface brightness profile at 50-270 kpc, at the level of $\mathrm{1.20\pm
0.49\times\ 10^{-21}\,erg\,s^{-1}\,cm^{-2}\,arcsec^{-2}}$. In addition, we
provide an upper limit on the contribution of other physical mechanisms to the
profile at this distance, which is estimated to be $\mathrm{0.93\times\
10^{-21}\,erg\,s^{-1}\,cm^{-2}\,arcsec^{-2}}$.

We divide our LAE sample in subsamples by $\mathrm{L_{Ly\alpha}}$. Within
approximately 20 kpc, the high-$\mathrm{L_{Ly\alpha}}$ sample exhibits a
higher Ly$\alpha$ surface brightness profile, but the profiles of different
subsamples have similar slopes. This indicates that star formation in the
central galaxy may dominate the Ly$\alpha$ surface brightness at small
distance. The production and propagation of Ly$\alpha$ photons likely follows
similar mechanisms in different subsamples.

We attempt to detect the extended Ly$\alpha$ emission at larger radii using
MOSAIC. We are not able to provide a robust detection from 60 kpc to 1 Mpc ,
but provide an upper limit of $\mathrm{2.82\times\
10^{-21}\,erg\,s^{-1}\,cm^{-2}\,arcsec^{-2}}$.

Although this work mainly focuses on the LAHs at $3<z<4$, we find that there
is no significant evolution in the observed Ly$\alpha$ surface brightness
profiles when we compare with $4<z<5$ and $5<z<6$.

Our results support a scenario in which star formation in the central galaxy
dominates the LAHs at small radii (within 20 kpc), while Ly$\alpha$ photons
from nearby galaxies dominate the Ly$\alpha$ surface brightness at large radii
(50-270 kpc). The Ly$\alpha$ surface brightness profile at the distance range
of 20-50 kpc is more difficult to interpret. Deeper observations of the
Ly$\alpha$ line profiles and further simulations are needed.

###### Acknowledgements.

YG, RB acknowledge support from the ANR L-INTENSE (ANR-20-CE92-0015). LW
acknowledges support by the ERC Advanced Grant SPECMAG-CGM (GA101020943).

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