# Mock Observations: Formation and Evolution of diffuse light in Galaxy Groups
and Clusters in the IllustrisTNG Simulations

Lin Tang School of Physics and Astronomy, China West Normal University, ShiDa
Road 1, 637002, Nanchong, China  CSST Science Center for the Guangdong-
Hongkong-Macau Greater Bay Area, DaXue Road 2, 519082, Zhuhai, China  Weipeng
Lin School of Physics and Astronomy, Sun Yat-sen University, DaXue Road 2,
519082, Zhuhai, China  CSST Science Center for the Guangdong-Hongkong-Macau
Greater Bay Area, DaXue Road 2, 519082, Zhuhai, China  Yang Wang Department of
Mathematics and Theories, Peng Cheng Laboratory, No. 2, Xingke 1st Street,
Nanshan District, Shenzhen, 518000, China  CSST Science Center for the
Guangdong-Hongkong-Macau Greater Bay Area, DaXue Road 2, 519082, Zhuhai, China
Jing Li School of Physics and Astronomy, China West Normal University, ShiDa
Road 1, 637002, Nanchong, China  Yanyao Lan  School of Physics and Astronomy,
Sun Yat-sen University, DaXue Road 2, 519082, Zhuhai, China  CSST Science
Center for the Guangdong-Hongkong-Macau Greater Bay Area, DaXue Road 2,
519082, Zhuhai, China Lin Tang, Weipen Lin<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Received XXX; Revised YYY; Accepted ZZZ)

###### Abstract

In this paper, by analyzing mock images from the IllustrisTNG100-1 simulation,
we examine the properties of the diffuse light and compare them to those of
central and satellite galaxies. Our findings suggest that the majority of the
diffuse light originates from satellites. This claim is supported by the
similarity between the age and metallicity distributions of the diffuse light
and those of the satellites. Notably, the color distribution of the diffuse
light gradually evolves to resemble that of the centrals at lower redshifts,
suggesting a coevolution or passive process. The radial profiles of the
diffuse light reveal distinct trends, with the inner regions displaying a
relatively flat distribution and the outer regions showing a descending
pattern. This finding suggests that the formation of the diffuse light is
influenced by both major mergers and stellar tidal stripping. Moreover, strong
correlations are found between the stellar mass of the diffuse light and the
overall stellar mass of the satellites, as well as between the stellar mass of
the diffuse light and the number of satellites within groups or clusters.
These relationships can be described by power-law and logarithmic functions.
Overall, the diffuse light components predominantly originate from satellites
with intermediate ages and metallicities. These satellites typically fall
within the stellar mass range of $\rm 8<\log_{10}M_{star}/M_{\odot}<10$ and
the color range of $\rm-1<[g-r]^{0.1}<0$. As the redshift decreases, the
growth of the diffuse light is primarily influenced by the redder satellites,
while the most massive and reddest satellites have minimal roles in its
growth.

galaxies: clusters: general – galaxies: clusters: intracluster medium –
galaxies: evolution – galaxies: statistical – method: numerical

††journal: ApJ

##  1 Introduction

In the past two decades, there has been extensive research on intracluster
light (also known as diffuse light), which is a widespread and diffuse stellar
component found in galaxy clusters. Typically located at the cluster center,
encompassing the brightest cluster galaxies (BCGs), the intracluster light is
associated with the overall gravitational potential well of the cluster,
rather than any individual galaxy. The study of the intracluster light is
crucial because it contributes significantly to our understanding of galaxy
clusters and their host dark matter halos, potentially shedding light on their
assembly histories (Montes, 2019, 2022; Contini, 2021, for reviews).

Note that the intracluster light is not only present in clusters but is also
observed in groups of galaxies, as supported by simulations and observations
(e.g., Gonzalez et al., 2005, 2007; Krick & Bernstein, 2007; Contini et al.,
2019). Considering that we treat both groups and clusters in this work, we use
the general term “diffuse light” when describing the diffuse stellar component
associated with the overall gravitational potential well of the groups or
clusters, rather than any individual galaxy. In contrast to the typically
observed luminous galaxies, the diffuse light represents a fainter aspect of
the universe. Exploring these faint components can provide valuable insights
that enhance our understanding of galaxy formation and evolution.

The properties, formation, and evolution of the diffuse light have been
extensively investigated and discussed in the literature. For instance, the
stellar mass fractions of the diffuse light exhibit a wide range across
different observed filters, masses of groups and clusters, and redshifts.
Jiménez-Teja et al. (2018) discovered that the fractions of the diffuse light
in 11 massive clusters of galaxies, as detected by the Hubble Space Telescope,
vary from $2\%$ to $23\%$ in three broadband optical filters: F435W, F606, and
F814W. Building upon the analysis of Jiménez-Teja et al. (2018), de Oliveira
et al. (2022) found that the diffuse light percentages for both A370 and
AS1063 range from $7\%$ to $25\%$ and $3\%$ to $22\%$ of the total light of
the respective clusters. When detecting the diffuse light at different
redshifts, researchers have found that the fraction of the diffuse light
increases as the redshift decreases (e.g., Rudick et al., 2011; Burke et al.,
2015; Jiménez-Teja et al., 2018; Montes & Trujillo, 2018; Furnell et al.,
2021; Kluge et al., 2021; Montes et al., 2021) and grows alongside the
assembly of groups and clusters of galaxies (e.g., Mihos, 2016; Zhang et al.,
2019; Cañas et al., 2020; Kluge et al., 2021; Sampaio-Santos et al., 2021).
However, Joo & Jee (2023) recently found that the diffuse light is already
highly abundant in galaxy clusters with redshifts of $z\geq 1$, and that its
fraction is about twice the theoretical prediction of Rudick et al. (2011) at
high redshifts. A high fraction of the diffuse light at high redshifts is also
reported in more recent work (Werner et al., 2023; Jiménez-Teja et al., 2023).

Furthermore, the fraction of the diffuse light is proportional to the mass of
the host dark matter halos in simulations (e.g., Pillepich et al., 2018a; Tang
et al., 2021; Sampaio-Santos et al., 2021), while the observations show a
scattered distribution of the diffuse light fraction within halos greater than
$\rm 10^{12}\ M_{\odot}$ (e.g., Gonzalez et al., 2005, 2007; Krick &
Bernstein, 2007). The results of several studies also indicate that the
definition of the diffuse light significantly influences its fraction (e.g.,
Tang et al., 2018; Contini, 2021; Montes, 2022, for reviews).

Considering that the majority of the diffuse light is located in the center of
dark matter halos, we can reasonably assume a significant correlation between
the evolution of the diffuse light and the BCGs (e.g., Zibetti et al., 2005;
Rudick et al., 2009). However, numerous investigations have demonstrated that
the distribution of the diffuse light is broader than that of the BCGs (e.g.,
Burke et al., 2012). This finding indicates a clear distinction between the
diffuse light and BCGs, suggesting two primary formation mechanisms for the
diffuse light: galaxy mergers and tidal stripping (e.g., Murante et al., 2007;
Purcell et al., 2007; Puchwein et al., 2010; Rudick et al., 2011; Burke et
al., 2015; DeMaio et al., 2015, 2018; Groenewald et al., 2017; Morishita et
al., 2017; Montes & Trujillo, 2018; Contini et al., 2018, 2019; Contini, 2021;
Montes & Trujillo, 2022, and the references therein). These mechanisms are
differentiated because one would not expect color/metallicity gradients in the
BCG + diffuse light system if mergers are primarily responsible for the
diffuse light formation. Importantly, note that only a small amount of the
diffuse light is formed in situ from cold gas (e.g., Jordán et al., 2009;
Takamiya et al., 2009; Puchwein et al., 2010).

The question of which mechanism plays a key role in the formation of the
diffuse light is crucial. To answer this question, researchers have conducted
additional studies. For instance, some groups and clusters of galaxies with
flat color gradients have been found and identified (e.g., Krick & Bernstein,
2007). However, distinct radial gradients of the color and metallicity of BCG
+ diffuse light systems have been discovered, with the BCG being redder and
richer in metals than the diffuse light (e.g., Montes & Trujillo, 2014;
Morishita et al., 2017; DeMaio et al., 2018; Montes & Trujillo, 2018). This
finding suggests that stellar stripping from satellite galaxies is the primary
mechanism of the diffuse light formation, as confirmed by semi-analytic models
in Contini et al. (2019). Yoo et al. (2021) extracted the diffuse light from a
fossil cluster and found that the color of the diffuse light is comparable to
that of the BCG out to $\sim 70{\rm\ {kpc}}$, becoming somewhat bluer toward
the outskirts. In addition, Montes & Trujillo (2022) derived the color profile
of a cluster in the James Webb Space Telescope (JWST) Early Release
Observations and discovered a nearly constant color profile in the inner
portions and a significant dip of 0.6 magnitudes in the outer parts. These
findings can be explained by a combination of major mergers in the inner
portions and tidal stripping in the outer parts.

The identification of the galaxies responsible for the generation of the
diffuse light is another crucial aspect for understanding its formation.
Numerous studies have concluded that the diffuse light primarily originates
from massive galaxies with stellar masses comparable to that of the Milky Way
(e.g., Contini et al., 2014; DeMaio et al., 2018; Contini et al., 2018).
Further, Montes & Trujillo (2018) investigated the diffuse light from six
massive clusters in the Hubble Frontier Fields survey. It is found that the
average color and metallicity of the diffuse light are similar to those of the
Milky Way’s outskirts. However, Morishita et al. (2017) determined that the
diffuse light more likely originates from satellites with stellar masses
$\rm\log M_{star}/M_{\odot}<10$. Contini et al. (2019) anticipated a negative
radial metallicity and color gradient in the BCG + diffuse light system based
on a semi-analytic model of galaxy formation. By comparing the typical colors
of the diffuse light with those of satellite galaxies, they discovered that
low-mass galaxies with stellar masses in the range of $\rm 9<\log
M_{star}/M_{\odot}<10$ are the primary contributors to the early stages of the
diffuse light formation. In contrast, intermediate/massive galaxies with
stellar masses in the range of $\rm 10<\log M_{star}/M_{\odot}<11$ contribute
to the growth of the diffuse light content at lower redshifts.

Only the diffuse light from a few groups and clusters of galaxies have been
extensively observed, as these observations require expensive wide-field
observations with high spatial resolution and very deep exposure. Conversely,
simulations can provide a large sample of groups and clusters of galaxies
spanning a wide range of masses. As a result, simulations are a valuable tool
for statistically studying the properties and formation of the diffuse light.
In this paper, we aim to investigate the relationships between the diffuse
light properties and those of central galaxies (henceforth referred to as
centrals), satellite galaxies (henceforth referred to as satellites), and dark
matter halos, as well as to determine the formation and evolution processes of
the diffuse light. We utilize the TNG100-1 simulation of IllustrisTNG, the
state-of-the-art N-body/hydrodynamical simulations, to determine which factor
has the most significant influence on the diffuse light and understand the
relationship between them.

The rest of this paper is organized as follows. In Sections 2 and 3, we
briefly describe the simulation used and the method employed to define the
diffuse light. In Section 4, we present the age, metallicity, and color
statistics of the diffuse light, as well as the comparisons with centrals and
satellites. The radial profiles of the diffuse light properties are presented
in Section 5. In Section 6, we examine the relationships between the
satellites and diffuse light. Finally, we briefly discuss and further
summarize our main results in Section 7.

##  2 Simulation

In this study, the utilized simulation is the TNG100-1 simulation of
IllustrisTNG suite 111https://www.tng-project.org. This is a cosmological
hydrodynamical simulation with $1820^{3}$ dark matter and $1820^{3}$ gas
particles in a cubic box of $(110.7{\rm\ {Mpc}})^{3}$. The mass resolution of
the simulation is $1.4\times 10^{6}\>{\rm M_{\odot}}$ for baryon particles
while the spatial resolution is set by the Plummer softening length, which is
$0.74{\rm\ {kpc}}$. The following cosmological parameters are obtained from
Planck (Planck Collaboration et al., 2016): $\Omega_{m}=0.3089$,
$\Omega_{\Lambda}=0.6911$, $\Omega_{b}=0.0486$, $n_{s}=0.9667$,
$\sigma_{8}=0.8159$, and $h=0.6774$. The detailed description of the database
can be found in Nelson et al. (2019). Refer to the paper series on
TNG50/100/300 for a more detailed description of the stellar content
(Pillepich et al., 2018b), galaxy clustering (Springel et al., 2018), galaxy
colors (Nelson et al., 2018), chemical enrichment (Naiman et al., 2018), and
magnetic fields (Marinacci et al., 2018). Compared to the previous Illustris
simulations (Vogelsberger et al., 2014), the TNG100-1 simulation includes a
revised active galactic nucleus (AGN) feedback model, which controls the star
formation efficiency of massive galaxies (Weinberger et al., 2017), and a
galactic wind model, whose feedback inhibits the efficient star formation in
low- and intermediate-mass galaxies (Pillepich et al., 2018b).

To reduce the mass-resolution effect of the simulation, we select the friend-
of-friend (FoF) groups with masses $\rm M_{200}>10^{12}\>{\rm M_{\odot}}$ and
member galaxies with stellar masses $\rm M_{star}>10^{8}\>{\rm M_{\odot}}$.
$\rm M_{200}$ is the mass within the virial radius $\rm R_{200}$, which is the
radius within which the average density is 200 times the critical density of
the universe. The most massive FoF group in our sample has a mass
$\bm{M}_{200}\sim 10^{14.64}\>{\rm M_{\odot}}$. To determine the evolution
processes of the diffuse light, we extract FoF groups from simulation
snapshots 91, 78, 67 and 50 ($z=0.1,0.3,0.5$, and $1.0$).

Figure 1:  Illustration of luminosity profile of a dark matter halo with a
virial radius of around $1.15\>h^{-1}{\rm{Mpc}}$ and $\rm M_{200}$ mass of
$\rm 2.96\times 10^{14}M_{\odot}$ at redshift $z=0.1$. The projection of this
halo is shown on the x-y plane, with each side of the square being
approximately $1.0\>h^{-1}{\rm{Mpc}}$ and centered at the center of mass.
$Left$: total luminosity profile of the halo; $Right$: luminosity of the halo
after applying a mask that excludes the galaxies defined by the SBLSP method.
The color bar represents the surface brightness in the V band.

Figure 2:  Relationships between intrinsic properties (age, metallicity, and
color in the top, middle, and bottom panels, respectively) of the centrals (or
satellites, diffuse light) and dark matter halo mass $\rm M_{200}$. Each panel
shows the results for the centrals, satellites, and diffuse light, represented
by the red, green, and blue open circles, respectively. The panels are
arranged from left to right to show the results at redshifts $z=0.1,0.3,0.5$,
and $1.0$. The error bars represent the standard deviations.

Figure 3:  Number distributions of galaxy properties are shown in each panel,
with the fraction normalized by dividing the maximum number in the value bins.
The solid lines represent Gaussian fits to the distributions. In each panel,
the red, green, and blue open circles represent the results for the centrals,
satellites, and diffuse light, respectively. The panels from left to right
correspond to the results at redshifts $z=0.1$, $0.3$, $0.5$, and $1.0$,
respectively.

##  3 Diffuse light definition

Multiple definitions of the diffuse light exist (Arnaboldi, 2004; Tang et al.,
2018; Montes, 2022, for review). In observational studies, the diffuse light
is commonly defined as the excess of light beyond the profile of a BCG when
removing the sky background and satellite galaxies (e.g., Gonzalez et al.,
2000, 2005; Zibetti et al., 2005; Seigar et al., 2007; Jee, 2010; Melnick et
al., 2012; Giallongo et al., 2014; Spavone et al., 2018, 2020). However, the
choices of the fitting method and outer fitting radius can lead to variations
in the results. Another approach is the surface brightness (SB) threshold
method, where the stellar component with a surface brightness fainter than a
certain threshold, typically the observational surface brightness limit, is
defined as the diffuse light (e.g., Zibetti et al., 2005; Puchwein et al.,
2010; Presotto et al., 2014).

In Tang et al. (2018), the evolution of the observational light fraction of
the diffuse light was reproduced by the SB threshold method in simulations. In
Tang et al. (2021), it was found that the stellar mass in the mock galaxy
sample resembles the observations more closely than approaches based on sub-
finder structures (Springel et al., 2001) and aperture-selected galaxies
(Pillepich et al., 2018a). To obtain more accurate results, we define the
diffuse light based on mock galaxy images (Tang et al., 2018), effectively
eliminating the influences of the components of the central and field galaxies
extracted by the surface brightness level segmentation procedure (SBLSP)
introduced in Tang et al. (2020, 2021), which is based on SB thresholds to
distinguish between the member galaxies and the diffuse light in the TNG100-1
simulation of IllustrisTNG.

Initially, we generate mock images of the simulated FoF groups on three
projecting planes ($xy$, $yz$, $yz$) by considering various factors, such as
the point-spread function (PSF) characterized by a Moffat function (Moffat,
1969; Trujillo et al., 2001), the pixel size (D), surface brightness limit
(SBL) of the image, spatial resolution, dust extinction, and redshift dimming
effects. During this process, the following astrophysical quantities can be
calculated for each projecting mesh: (1) the total stellar mass, obtained by
summing the masses of all star particles within each mesh; (2) the total
luminosity, obtained by summing the luminosities of all the star particles in
four filters ($B$, $V$, SDSS $g$, and SDSS $r$) within each mesh; (3)
$\rm[g-r]^{0.1}$ color, calculated by applying the transformation
$\rm[g-r]^{0.1}=0.7188+1.3197[(g-r)-0.6102]$ Blanton & Roweis (2007) to
correct the color at $z=0.1$; (4) the mean age and metallicity, calculated
with weights based on the stellar luminosity; and (5) the sum of the
individual star formation rates of all gas cells within the mesh. Note that
there is no age of the stellar particles in the IllustrisTNG datasets, which
can be defined as the cosmic time at the snapshot redshift minus the cosmic
time of the stellar particle formation listed in the IllustrisTNG datasets.
The cosmic time is calculated using the functions (Mo et al., 2010),

$t(z)=\int_{0}^{a(z)}\frac{{\rm
d}a}{\dot{a}}=\frac{1}{H_{0}}\int_{z}^{\infty}\frac{{\rm d}z}{(1+z)E(z)},$ (1)
$\begin{split}E(z)=[\Omega_{\Lambda,0}+(1-\Omega_{0})(1+z)^{2}+\\\
\Omega_{m,0}(1+z)^{3}+\Omega_{r,0}(1+z)^{4}]^{1/2},\end{split}$ (2)

where $a$ is the scale factor and $z$ is the redshift, $\Omega_{m,0}=0.3089$,
$\Omega_{\Lambda,0}=0.6911$, $\Omega_{r,0}=0$, $\Omega_{0}=1$, and
$H_{0}=100h$. The stellar age is defined as $t(z_{\rm snap})-t(z_{\rm sft})$,
where $t(z_{\rm snap})$ and $t(z_{\rm sft})$ are the cosmic time of star
formation and the snapshot, respectively.

The surface brightness for each pixel is given by

$\mu_{x}=-2.5\log\frac{I_{x}}{L_{\odot,x}\cdot pc^{-2}}+21.572+M_{\odot,x},$
(3)

where $I_{x}$ is

$I_{x}=\frac{L_{x}}{\pi^{2}D^{2}}(1+z)^{-4}.$ (4)

Here, $x$ indicates different filters in the rest frame, and $\rm M_{\odot,x}$
is the absolute solar magnitude, with values of $5.36,\ 4.80,\ 5.12,$ and $\
4.64\ \rm mag/arcsec^{-2}$ (Blanton & Roweis, 2007) for the $B$, $V$, SDSS
$g$, and SDSS $r$ filters in the AB system, respectively.

Next, we separate the diffuse light components from the member galaxies by
applying a given SB threshold. To clearly delineate the boundaries of the
galaxies with different SB levels, as shown in Figure 6 of Tang et al. (2020),
we repeat this step with a series of SB thresholds to obtain sample templates.
Subsequently, we utilize a segmentation procedure, the SBLSP, which is
introduced in detail in Tang et al. (2020, 2021), to obtain the final galaxy
sample from the sample templates defined by various SB thresholds. Finally,
the diffuse light refers to the stellar particles that do not belong to
galaxies. All the galaxies in our samples are chosen with stellar masses
greater than $10^{8}\>{\rm M_{\odot}}$.

Note that the softening length is set as the fiducial ‘simulation-CCD’ pixel
size of the projected images. For the charge-coupled diode (CCD) pixel, the
variance of the Poisson distribution is set to 30 with units of $\rm
mag/arcsec^{-2}$ and the magnitude of the CCD pixel, representing the sky
background and CCD noise, respectively. Additionally, we convert the softening
length into an angular resolution to serve as a ‘simulation-PSF’. In the SBLSP
method, we employ the radiative transfer code SKIRT (Baes et al., 2011) to add
a dust component to the star-forming mock galaxies. The observational limit is
set at 26.5 $\rm mag/arcsec^{-2}$. The series of SB thresholds used in the
second procedure range from 18 to 26.5 $\rm mag/arcsec^{-2}$, with surface-
brightness intervals of 0.1 $\rm mag/arcsec^{-2}$.

In Figure 1, we present the diffuse light within a dark matter halo at
redshift $z=0.1$, as defined by the SBLSP method. The halo has an
$\bm{R}_{200}$ of $\rm 1.15{\rm\ {Mpc}}$ and an $\rm M_{200}$ mass of
$2.96\times 10^{14}\>{\rm M_{\odot}}$. The x-y projected plane has a central
region of $1.0\ \>h^{-1}{\rm{Mpc}}\times 1.0\ \>h^{-1}{\rm{Mpc}}$. The left
panel of Figure 1 illustrates the total luminosity profile of the halo, while
the right panel shows the luminosity profile after applying a mask to exclude
member galaxies. All the peaks in the total luminosity profile presented in
the left panel are eliminated due to the masking of member galaxies defined by
the SBLSP method. Consequently, the remaining luminosity is significantly
faint and primarily exhibits a surface brightness in the V band of
$\rm\mu_{V}\lesssim 25\ mag/arcsec^{-2}$. As shown in the right panel of
Figure 1, the majority of the diffuse light is concentrated in the center of
the dark matter halo. Additionally, a large satellite object is situated close
to the halo center, surrounded by the diffuse light. In the top-right corner
of the right panel, a considerable amount of the diffuse light is found, which
is believed to represent remnants from a falling halo. Notably, dark stellar
streams can be identified between the centers of dark matter halos and
neighboring member galaxies.

##  4 Diffuse light properties and comparison with centrals and satellites

In this section, we present a statistical analysis of the diffuse light
properties, including the age, metallicity, color, and stellar mass. We also
examine the centrals and satellites for comparison. Those properties of the
three components are calculated in the same way for each mesh, as described in
Section 3. Note that we define the most massive galaxies located in the dark
matter halos as the central galaxies and the remaining galaxies as the
satellite galaxies.

Figure 2 illustrates the relationships between the dark matter halo mass $\rm
M_{200}$ and the age, metallicity, and color for the diffuse light, centrals,
and satellites. Figure 3 displays the normalized number distributions of these
intrinsic properties (age, metallicity, and color). To demonstrate the
redshift evolution, we present the results of four redshifts ($z=0.1$, $0.3$,
$0.5$, and $1.0$) from left to right in Figure 2 and 3. Additionally, Figure 4
shows the relationships between the dark matter halo mass and the stellar
masses of the diffuse light, centrals, and satellites. Due to their
similarity, only the results at $z=0.1$ are shown in Figure 4.

Figure 4:  Relationships between the dark halo mass $M_{500}$ and the stellar
masses of the centrals (left panel), total satellites (middle panel), and
diffuse light (right panel) within dark matter halos at redshift $z=0.1$. The
gray points represent the results for each component in the dark matter halos.
The open circles with error bars indicate the average values within the mass
bins, and the solid lines represent the fitting results. The error bars show
the standard deviations of the data points.

###  4.1 Age, metallicity, and color

The top panels of Figure 2 reveal that the age of the centrals remains nearly
constant as the mass of the dark matter halos increases, but slightly
increases at redshift $z=1.0$. However, the older diffuse light and satellites
tend to be found in more massive halos at lower redshifts. In the middle
panels of Figure 2, a positive correlation is found between the metallicities
of the centrals, satellites, and diffuse light and the masses of the dark
matter halos. In the bottom panels of the same figure, it can be found that
the color of the centrals remains fairly constant, while the redder diffuse
light and satellites are more prevalent in the more massive dark matter halos.

From the top panels of Figure 2, note that on average, the diffuse light
components are younger than those of the centrals and close to those of the
satellites. This trend continues from redshift 1 to 0.1. This distinction
between the age of the diffuse light and centrals is further highlighted in
the top panel of Figure 3, where the ages of the diffuse light resemble those
of the satellites more than those of the centrals. As shown in the middle
panels of Figure 2, the diffuse light components exhibit significantly lower
metallicity than the satellites and centrals. Also, note that while the
metallicities of the diffuse light and centrals show little evolution,
satellites exhibit a stronger trend of decreasing average metallicity, with
their metallicities falling between those of the diffuse light and centrals,
as shown in the middle panels of Figures 2 and 3. This finding can be
explained by the fact that early infalling satellites, which are similar in
metallicity to the centrals during the major merger stage, have either fully
merged or their remaining cores are too faint to be detected. However, late
infalling satellites represent younger, metal-poor galaxies. This finding is
consistent with those of previous studies (e.g., Engler et al., 2018), which
also indicates that early infalling satellites are relatively more metal rich
than late infalling satellites.

The bottom panels in Figures 2 and 3 reveal a different trend than those in
the top and middle panels. In contrast to the age and metallicity results,
where satellites fall between the diffuse light and centrals, the color of the
diffuse light falls between the satellites and centrals, while all three
components become redder with decreasing redshift. Importantly, the color
distribution of the diffuse light is closer to that of the satellites at
higher redshift and becomes closer to that of the centrals at lower redshift ,
which can be found in the bottom panels of Figure 3.

The results of the color distribution can be explained as follows. At
redshifts $z>1$, the stars in the diffuse light components are newly formed
(e.g., Burke et al., 2012; Ko & Jee, 2018; Joo & Jee, 2023), either stripped
from satellites or formed in situ, resulting in a bluer color compared to that
of the satellites. However, as the diffuse light components evolve toward
lower redshifts ($z\leq 1$), the formation of the diffuse light decreases
significantly, with only a few stars still forming (possibly in situ), while
the majority of stars in the diffuse light are dying. As a result, the diffuse
light rapidly undergoes quenching and shifts toward a red color due to the
strong physical mechanisms, in the densest regions of the dark matter halos.
In comparison, even though satellites are quenched, the presence of ongoing
star formations leads to a relatively blue color. Also, note that the color
distribution for the diffuse light and satellite appears broad in Figure 3,
whereas the majority of the centrals have very red colors due to significant
quenching caused by the AGN and stellar feedback, which is determined by the
simulation settings. However, the long tail in the color distribution of
centrals, along with the presence of a second peak or tail in the age
distribution, indicates that a portion of the centrals still exhibit a blue
color, suggesting the ongoing formation of young stars from the cold gas.

Figure 5:  Radial profiles of intrinsic properties for the diffuse light, with the age, metallicity, and color shown from left to right. The lines in different colors (blue, green, and red) represent the profiles within halos of different mass ranges ($12<{\rm log_{10}M_{200}/\>{\rm M_{\odot}}}<13,13<{\rm log_{10}M_{200}/\>{\rm M_{\odot}}}<14,14<{\rm log_{10}M_{200}/\>{\rm M_{\odot}}}<15$). The error—indicated by sky blue, light cyan, and papaya—represent the standard deviations.  Redshift | Central–Halo | Satellite–Halo | Diffuse light–Halo
---|---|---|---
a | b | a | b | a | b
CombinationA | 0.51 | 11.52 | 1.30 | 12.21 | 1.08 | 12.17
0.1 | 0.67 | 11.91 | 1.38 | 12.21 | 1.02 | 11.99
0.3 | 0.66 | 11.82 | 1.34 | 12.18 | 1.00 | 12.07
0.5 | 0.69 | 11.80 | 1.35 | 12.16 | 1.00 | 12.13
1.0 | 0.70 | 11.71 | 1.33 | 12.10 | 1.00 | 12.26
Table 1:  Best-fitting parameters of stellar mass–halo mass relationships for
the centrals, satellites, and diffuse light.
A: combination results at redshifts $z=0.01,0.1,0.2$ in Tang et al. (2021).

In conclusion, our analysis reveals that the diffuse light components differ
from those of the centrals but resemble those of the satellites in terms of
their relatively young ages, poor metallicities, and red colors. This finding
suggests that the diffuse light components are primarily formed through merger
events or tidal stripping of infalling satellites but evolve passively with
the centrals. Furthermore, the ages, metallicities and colors for the diffuse
light, satellites, and centrals show positive correlations with the halo mass,
with the diffuse light components displaying the highest sensitivity. This
finding indicates that more massive halos tend to host older, metal-richer and
redder centrals, satellites, and diffuse light. Note that the dips in the most
massive bins are likely a result of limited statistical data.

###  4.2 Stellar mass

Tang et al. (2021) investigated the relationships between the dark matter halo
mass ($\rm M_{500}$) and the stellar mass of the diffuse light, satellites,
and centrals, and found that the results are consistent with the observational
findings. $\rm M_{500}$ here denotes the mass within the virial radius $\rm
R_{500}$, which is the radius within which the average density is 500 times
the critical density of the universe. As reported in Tang et al. (2021), the
correlation between the dark matter halo mass and stellar mass is weaker for
the centrals than for the satellites. The fitting parameters for the diffuse
light and satellites show similar results, with slopes of 1.08 and 1.30 and
intercepts of 12.17 and 12.21, respectively. However, the fitting parameters
for the centrals are notably different, with a slope of approximately 0.5 and
an intercept lower than 12.00. Previous results show significant scatter due
to the combination of data from three redshifts ($z=0.01,0.1,$ and $0.2$). In
this work, we specifically examine the stellar mass–dark matter halo mass
relationships for a set of redshifts—namely $z=0.1,0.3,0.5$, and $1.0$—to
investigate the redshift evolution.

The results are presented in Figure 4, and the parameters of best-fit are
listed in Table 1. Due to the similarity, only the results at $z=0.1$ are
displayed in Figure 4 for viewing. The fitting equation is the same as that in
Pillepich et al. (2018a), which is given as follows,

$y=am+b,$ (5)

where $m={\rm log}_{10}({\rm M}_{\rm 500}/\>{\rm M_{\odot}})-14$, $y={\rm
log}_{10}(\rm M_{\rm stars}/\>{\rm M_{\odot}})$, $a$ is the slope and $b$ is
the intercept.

The results obtained from our analysis demonstrate that the stellar mass–dark
matter halo mass relationships for the centrals, satellites, and diffuse light
exhibit strong correlations, and that these relationships remain relatively
unchanged across the examined redshifts. The slopes $a$ and intercepts $b$ for
the central stellar mass–dark matter halo mass relationship are consistently
around 0.6 and 11.8 at all four redshifts. In comparison, the satellites and
diffuse light have larger slopes and intercepts than the centrals, despite
having similar fitting parameters. The satellites have a slope of
approximately 1.3, while the diffuse light has a slope of approximately 1.0.
Both the satellites and diffuse light have intercepts of approximately 12.1.
These findings are consistent with the results presented in Tang et al.
(2021).

Figure 6:  $Left$: relationship between the stellar mass of the diffuse light
and the total stellar mass of satellites that are more massive than
$10^{8}\>{\rm M_{\odot}}$. $Right$: relationship between the stellar mass of
the diffuse light and the number of satellites that are more massive than
$10^{8}\>{\rm M_{\odot}}$. These relationships are analyzed at different
redshifts—$z=0.1,0.3,0.5$, and $1.0$—represented by the colors black, red,
green, and blue, respectively. Our results are depicted by the open circles
with error bars, where the error bars represent the standard deviation within
each mass bin. The solid lines represent fitting lines using different
functions. The fitting functions and parameters can be found in Table 2. Note
that the inset in the right panel provides the same results but with a linear
axis for the satellite number, instead of a logarithmic scale.  Figure 7:
Dispersion of the best-fitting Equation 6 in left panel of Figure 6 as a
function of $\gamma$. Specifically, we focus on dispersions in $\log_{10}({\rm
M_{star}\ of\ diffuse\ light})$, as the dispersions in $\rm\log_{10}M_{star}\
of\ satellites$ are nearly 0 for $\gamma>2$.

##  5 Radial profiles of diffuse light

Several studies have reported an apparent radial trend in the metallicities
and colors of the diffuse light, showing an outward tilt (e.g., Montes &
Trujillo, 2014, 2018; Morishita et al., 2017; Contini et al., 2019; Edwards et
al., 2020). However, Yoo et al. (2021) found a similarity in the color between
the diffuse light and BCG out to a radius of $70{\rm\ {kpc}}$ for a massive
fossil cluster at an intermediate redshift based on deep imaging with
Gemini/GMOS-N. Furthermore, Montes & Trujillo (2022) conducted an analysis of
the BCG + diffuse light color profile using the JWST Early Release
Observations, and found an almost constant color profile within the inner
approximately $100{\rm\ {kpc}}$.

As presented in Figure 2, our previous analysis in Section 2 reveals that the
centrals exhibit a redder, older, and more metal-rich nature than the diffuse
light. Consequently, when considering both locations, we find that the
centrals + diffuse light property profiles show a decline toward the outer
regions. In this section, we aim to validate the observed constant profile in
the inner region by examining the radial profiles of the intrinsic properties
of the diffuse light in the simulations. Figure 2 also indicates that a slight
correlation exists between the properties of the diffuse light and dark matter
halo mass. To further investigate this, we divide the dark matter halos into
three subsamples based on the $\rm M_{200}$ values of the dark matter halos:
$12<{\rm log_{10}M_{200}/\>{\rm M_{\odot}}}<13,13<{\rm log_{10}M_{200}/\>{\rm
M_{\odot}}}<14$, and $14<{\rm log_{10}M_{200}/\>{\rm M_{\odot}}}<15$. For the
radial profiles of the diffuse light, we calculate the average age,
metallicity, and color within redial bins from $R$ to $R+\Delta R$, where
$\Delta R$ is equal to $\bm{0}.05R_{200}$. In Figure 5, we present the radial
profiles of the diffuse light properties, specifically at redshift $z=0.1$.
Note that the radial profiles at higher redshifts exhibit similar trends to
those at $z=0.1$, but they are explicitly discussed in the text.

$z$ | $\alpha_{1}$ | $\beta_{1}$ | $\gamma$ | $\alpha_{2}$ | $\beta_{2}$
---|---|---|---|---|---
0.1 | 0.54 | 9.70 | 7.2 | 1.08 | 9.55
0.3 | 0.50 | 9.83 | 7.6 | 1.10 | 9.68
0.5 | 0.45 | 9.96 | 8.5 | 1.08 | 9.82
1.0 | 0.42 | 10.14 | 9.1 | 1.07 | 10.13
Table 2:  Best-fitting parameters in Figure 6. $\alpha_{1}$, $\beta_{1}$, and
$\gamma$ correspond to the left panel, while $\alpha_{2}$ and $\beta_{2}$
correspond to the right panel. Fits are in the form of Equation 6 for
$\alpha_{1}$, $\beta_{1}$, and $\gamma$ in the left panel and Equation 7 for
$\alpha_{2}$ and $\beta_{2}$ in the right panel.

Figure 5 demonstrates clear gradients in the radial profiles of the color,
metallicity, and age, which are consistent with the results in previous works
(e.g., Contini et al., 2019; Edwards et al., 2020). Furthermore, these
gradients have pronounced dark matter halos with lower masses. Notably, the
radial profiles of the diffuse light exhibit distinct flattened distributions
within approximately $\bm{0}.1R_{200}$, in general agreement with the
observations in Yoo et al. (2021) and Montes & Trujillo (2022). Moreover, the
flattening of the color profile is more pronounced for the halos with lower
masses.

The differences between the diffuse light and centrals, as well as the
similarity between the diffuse light and satellites, as shown in Figure 2, can
be readily explained by satellite stripping, as indicated by the gradients
shown in Figure 5. In the hierarchical clustering diagram, smaller halos form
first and subsequently undergo infalling or merging to form bigger halos. In
this scenario, the substructures (or satellites) within the halos act as
centrals prior to infalling, with declines in color, age, and metallicity
outward along the radial profiles, corresponding to an inside-out picture for
the galaxy formation (e.g., Chiappini et al., 1997; van Dokkum et al., 2010;
Patel et al., 2013; González Delgado et al., 2014; Tacchella et al., 2015;
Carrillo et al., 2023). During satellite stripping caused by the tidal forces
of the dark matter halo, the outer components of the satellites become
separated from their central components, resulting in the formation of the
diffuse light. This process explains why the diffuse light is younger and
metal poorer than that of satellites, providing an explanation for the results
presented in the top and middle panels of Figure 2.

##  6 Relationship between diffuse light and satellites

Here, we examine the relationship between the diffuse light and satellites.
First, we investigate the dependence of the stellar mass of the diffuse light
on the total stellar mass and the number of satellites within the same dark
matter halos. Subsequently, we take advantage of the age–metallicity planes of
the satellites to confirm the types of satellites that are most closely
associated with the diffuse light formation.

Figure 8:  Satellite age–metallicity planes, compared to those of the diffuse
light. $Open\ diamonds$ and $open\ squares$ represent the average values of
the satellite and diffuse light properties located in the same dark matter
halo, respectively. The black, red, green, and blue symbols represent the
results at redshifts $z=0.1$, $0.3$, $0.5$, and $1.0$, respectively. The error
bars represent the standard deviations. We divide the stellar masses $\rm
M_{star}$ of the satellites or color $\rm[g-r]^{0.1}$ into four bins, $\rm
8<\log_{10}M_{star}/M_{\odot}<9$, $\rm 9<\log_{10}M_{star}/M_{\odot}<10$, $\rm
10<\log_{10}M_{star}/M_{\odot}<11$, and $\rm 11<\log_{10}M_{star}/M_{\odot}$,
from left to right, and $\rm-1<[g-r]^{0.1}<0$, $\rm 0<[g-r]^{0.1}<0.4$, $\rm
0.4<[g-r]^{0.1}<0.8$, and $\rm 0.8<[g-r]^{0.1}$, from top to bottom.

###  6.1 Dependence of diffuse light on the stellar mass and number of
satellites

Figure 6 presents the relationship between the stellar mass of the diffuse
light and two variables: the total stellar mass of the satellites (left panel)
and the number of satellites (right panel) within their host halos. The open
circles represent our results at four different redshifts, distinguished by
various colors, as indicated in the labels. Note that the stellar masses of
all the satellites shown in Figure 6 are greater than $10^{8}\>{\rm
M_{\odot}}$.

As shown in the left panel of Figure 6, we fit the relationship between the
stellar mass of the diffuse light and satellites by a power-law equation, as
follow:

$y=\alpha_{1}m^{\gamma}+\beta_{1},$ (6)

where $m=(\log_{10}(\rm M_{Sat}/\>{\rm M_{\odot}}))/10)$, $y=\log_{10}(\rm
M_{\rm diffuse\ light}/\>{\rm M_{\odot}})$, $\alpha_{1}$ is the slope,
$\beta_{1}$ is the intercept, and $\gamma$ is the index.

In the right panel, the fitted equation is given by

$y=\alpha_{2}\log_{10}N+\beta_{2},$ (7)

Where $y=\log_{10}(\rm M_{\rm diffuse\ light}/\>{\rm M_{\odot}})$, $N$ is the
number of satellites in a cluster, slope $\alpha_{2}$ is the slope, and
$\beta_{2}$ is the intercept. $\rm M_{diffuse\ light}$ in Equations 6 and 7 is
the stellar mass of the diffuse light.

The parameter $\gamma$ is the best fit index. Here, we examine the dispersion
of $\gamma$. The dispersion is calculated by the standard deviation of the
chi-squared error in the $Linfit\ Routine$ of the IDL software. In Figure 7,
we illustrate the estimated dispersion of $\log_{10}{\rm M_{star}}$ of the
diffuse light as a function of $\gamma$, which allows us to determine the
best-fitting index in Equation 6. We find that the minimum dispersion occurs
at $\gamma=7.2,7.6,8.5$, and $9.1$ at redshifts $z=0.1,0.3,0.5$, and $1.0$,
respectively. Additionally, note that higher redshifts correspond to larger
$\gamma$ values. We do not present the dispersion of $\rm\log_{10}M_{star}$ of
the satellites here, as it steadily decreases with increasing $\gamma$,
eventually dropping to nearly 0 (around $10^{-4}$) at $\gamma=2$. Table 2
provides the best-fitting parameters for Equations 6 and 7. Notably, the
slopes $\alpha_{1}$ and $\alpha_{2}$ exhibit a decreasing trend as the
redshift increases. Conversely, $\beta_{1}$ and $\beta_{2}$ show an increase
with higher redshifts.

At a given redshift (e.g., $z=1.0$, represented by the blue line in the left
panel of Figure 6), the increasing trend of the diffuse light stellar mass
with a higher satellite mass within the range of $\rm\log_{10}M_{star}/\>{\rm
M_{\odot}}\lesssim 10$ is less steep than that within the range of
$\rm\log_{10}M_{star}/\>{\rm M_{\odot}}\gtrsim 10$. The different colored
lines in the left panel of Figure 6 demonstrate that the slopes of the
increasing trend become steeper at lower redshifts within the range of
$\rm\log_{10}M_{star}/\>{\rm M_{\odot}}\lesssim 10$, and vice versa within the
range of $\rm\log_{10}M_{star}/\>{\rm M_{\odot}}\gtrsim 10$. In the right
panel of Figure 6, the data points at each redshift exhibit a close-to-linear
relationship. As shown in the inset of the right panel of Figure 6, the
stellar mass of the diffuse light exhibits a steep increase with the number of
satellites (N) up to approximately 50 satellites, after which the trend
becomes nearly flat for N $\gtrsim$ 100\. In conclusion, the diffuse light
components tend to have higher average masses within host halos that contain a
greater number of satellites or more massive satellites.

From the left panel of Figure 6, It seems that the mass of the diffuse light
increases with increasing redshift. We have checked the relative mass or light
fractions of the diffuse light compared to the total stellar masses within the
host halos, and find that these fractions are about 21%, 27%, 33%, and 46% at
redshifts z = 0.1, 0.3, 0.5, and 1.0 respectively. This finding agrees with
the results presented in Figure 6 of Tang et al. (2018) but contradicts with
the diffuse light evolution described in Burke et al. (2015) who introduced an
evolving factor for the SBL, i.e., $\bm{2}.5\log(1+z)^{4}$, meaning that the
SBL at higher redshift will be fainter. Similarly, as has been done in Tang et
al. (2018), if we also made such a cosmological correction on the SBL, the
stellar mass fractions are about 17%, 16%, 13%, and 12% at redshift z = 0.1,
0.3, 0.5, and 1.0 respectively, in good agrees with those of previous works
(e.g., Rudick et al., 2011; Burke et al., 2015; Tang et al., 2018; Montes &
Trujillo, 2018), with decreasing diffuse light fraction to higher redshifts.
Otherwise, if not applying such SBL correction, the predictions of increasing
diffuse light fraction with increasing redshift are primarily caused by
observational effects. At high redshift, many satellites with low surface
brightness are smoothing out and missed in the projecting image by the SBLSP
method, as the galaxies must be brighter than the SBL ($\bm{2}6.5\
mag/arcsec^{2}$ in this paper). Therefore, it turns out that the number of
satellites will be smaller at higher redshift, as shown in the right panel of
Figure 6. The stellar mass of these missing satellites will be then classified
as the diffuse light, causing the larger total mass of the diffuse light at
higher redshift. Nevertheless, the contradiction with traditional expectation
should be resolved and further investigations are required, although it is not
the primary focus of this paper.

###  6.2 Age–metallicity planes of diffuse light and satellites

Informed by the study in Contini et al. (2019), we aim to gain further
insights into the crucial types of satellites that contribute significantly to
the diffuse light formation. To that end, we investigate the relationship
between the intrinsic properties of the diffuse light and satellites,
considering their redshift evolution. In Figure 8, we present the
age–metallicity planes of the satellites, represented by open diamonds, along
with the corresponding average properties of the diffuse light, indicated by
open squares, allowing for direct comparisons.

Note that we employ the age–metallicity plane, which differs from the
color–metallicity plane utilized in Contini et al. (2019). As shown in Figures
2 and 3, the color ($\rm[g-r]^{0.1}$) distribution of the diffuse light
exhibits a closer resemblance to that of the centrals than that of the
satellites, particularly at lower redshifts. This indicates a potential
coevolution of the diffuse light and centrals in terms of color. However,
discernible variations are found in the age and metallicity distributions of
the diffuse light. The peaks of the age distribution of the diffuse light
assemble with those of the satellites, while the peaks of the metallicity
distribution of the diffuse light are lower than those of the satellites.
Consequently, by employing the age–metallicity plane of the satellites, we
effectively mitigate scattering effects and gain a clearer understanding of
the intrinsic properties. Similarly to Contini et al. (2019), we classify the
satellites into four stellar mass bins to identify which types of satellites
exhibit the most similar properties of the diffuse light. Additionally, we
categorize the satellites into four color bins, enabling a comprehensive
exploration of the satellite–diffuse light relationship.

From the overall view of Figure 8, it is found that the properties of the
satellites and diffuse light are located in close regions in the
age–metallicity planes only in panels (a) and (b) for all four redshifts. From
the right panels to the left panels, the plotted points for the diffuse light
and satellites move closer to each other in the age–metallicity planes at four
redshifts, for all the color bins. From the top panel to the bottom panel, the
bluer diffuse light and satellites become more similar. However, in panels
(d), (h), (l), and (p) (most massive mass bins) and in panel (o), no clear
correlation is found between the diffuse light and satellites.

It is important to determine which kinds of satellites are related to the
formation of the diffuse light at high or low redshifts. In our results, at
redshift $z=1.0$, the satellites with stellar masses within the range of $\rm
8<\log_{10}M_{star}/M_{\odot}<10$ and colors within the range of
$\rm-1<[g-r]^{0.1}<0$ (panels (a) and (b)) are the primary sources of the
formation of the diffuse light. At lower redshifts, the redder satellites
(panels (e), (f), and (i)) are related to the build-up of the diffuse light.
However, we cannot confirm the relationships between the smallest and reddest
satellite (panel (m)) and the formation of the diffuse light.

##  7 Discussion and Conclusions

In this study, we utilize the TNG-100 simulation of IllustrisTNG to
investigate the age, metallicity, and color $\rm[g-r]^{0.1}$ of the diffuse
light, as well as their relationships with the centrals and satellites. To
distinguish between the diffuse light and galaxies, we employ an improved
surface brightness level segmentation procedure called SBLSP. This procedure
is based on an iterative surface brightness level cutting in projected mock
images generated by converting the stellar particles from TNG100-1 snapshots
into seeing-convolved pixelated images. To obtain realistic observational
conditions and reproduce imaging-like data, we consider Moffat PSF, dust
extinction, and redshift dimming effects. Furthermore, to reduce resolution
effects in the simulations, we focus on the dark matter halos with total mass
$\rm M_{200}$ greater than $\rm 10^{12}M_{\odot}$ and the stellar mass of
galaxies $\rm M_{star}$ greater than $\rm 10^{8}M_{\odot}$.

We perform a comprehensive analysis of various properties of the diffuse
light, including the age, metallicity, color $\rm[g-r]^{0.1}$, and stellar
mass. We then examine the number distributions of these properties and
investigate their dependence on the dark matter halo mass $\rm M_{200}$.
Additionally, we compare these distributions with those of the centrals and
satellites to gain a better understanding of the formation processes. To
further explore the nature of the diffuse light, we analyze their radial
profiles, which allows us to validate the presence of gradients of properties
such as the color, metallicity, and age, which are consistent with previous
studies. Furthermore, we examine the relationship between the stellar mass of
the diffuse light and the stellar mass or number of satellites within the same
dark matter halos. Finally, we explore the age–metallicity planes for the
satellites and diffuse light to gain insights into their intrinsic properties
and evolution. The results are summarized as follows.

  * 1.

The age and metallicity of the diffuse light components resemble those of the
satellites, and both are distinct from those of the centrals. However, as the
redshift decreases, the color of the diffuse light approaches that of the
centrals. These findings indicate that the diffuse light components are
associated with satellite galaxies, and they gradually turn red by passive
evolution with the centrals (coevolution).

  * 2.

Consistent with recent observational findings, the age, metallicity, and color
radial profiles of the diffuse light appear flat in the inner region but drop
sharply in the outer region. This finding provides a clue for the build-up of
the diffuse light by mixed mechanisms of the major merger and tidal stripping.

  * 3.

The stellar mass of the diffuse light correlates strongly with the total
stellar mass and number of satellites in the host halo, and these
relationships can be modeled using a power-law and logarithmic function,
respectively.

  * 4.

Satellites with stellar masses within the range of $\rm
8<\log_{10}M_{star}/M_{\odot}<10$ and colors within the range of
$\rm-1<[g-r]^{0.1}<0$ are found to be the primary source of the diffuse light.
From high to low redshifts, an increasing number of redder satellites
participate in the formation of the diffuse light, whereas the most massive
and reddest satellites contribute little to the diffuse light formation.

Previous studies (e.g., Montes & Trujillo, 2014; Morishita et al., 2017;
Contini et al., 2019) have commonly utilized diffuse light color diagrams to
investigate the satellites contributing to the diffuse light. However, our
study reveals a coevolutionary relationship between the color of the diffuse
light and centrals, as well as a wide distribution of the diffuse light color.
We caution readers to avoid relying solely on diffuse light color diagrams, as
they may introduce scatter and potentially lead to spurious conclusions. In
Figure 6, we also suggest a novel approach for determining the stellar mass or
number of satellites in dark mark halos.

Notably, our result slightly differs from that in Contini et al. (2019), which
is that the low-mass galaxies, with $\rm 9<log_{10}M_{star}M_{\odot}<10$, are
the most important contributors of the diffuse light at high redshifts, but
that massive contents are contributed by intermediate/massive galaxies, with
$\rm 10<log_{10}M_{star}/M_{\odot}<11$, at lower redshifts. The stellar masses
of the satellites relative to the diffuse light formation in Contini et al.
(2019) are roughly one order of magnitude higher than our results. The main
reason is that the mock galaxies defined by our method are typically less
massive than those found by the traditional substructure extraction algorithm
(Tang et al., 2021), because the method distinguishes the diffuse light
located around the galaxies. Otherwise, our results confirm that the
satellites that formed the diffuse light are blue galaxies and show
color–redshift dependence.

By applying cosmological hydrodynamical simulations with high resolution, we
can improve our theoretical understanding of the spatial distribution of the
diffuse light, as discussed in Section 5. We find a flattened distribution of
the diffuse properties in the innermost region and a significant drop in the
outer region. However, only color and metallicity gradients have been found in
previous observational and theoretical works (e.g., Montes & Trujillo, 2018;
Contini et al., 2019). The platform in the innermost region of the diffuse
light properties, such as the color, has been confirmed in recent observations
(e.g., Yoo et al., 2021; Montes & Trujillo, 2022).

Furthermore, the relationship between the stellar mass of the diffuse light
and the number of satellites serves as an effective indicator of the
interaction efficiency within groups and clusters of galaxies, considering
that the diffuse light arises from merging and tidal stripping processes
(e.g., Montes, 2019). Consistent with previous studies (e.g., Rudick et al.,
2006; Montes & Trujillo, 2018), our findings demonstrate that groups and
clusters of galaxies with a higher abundance of the diffuse light tend to
include a greater number of satellites, enabling more frequent satellite
interactions. Conversely, groups and clusters of galaxies with smaller numbers
of satellites are less efficient in terms of interactions, resulting in slower
increases in the stellar mass of the diffuse light.

Overall, our investigations provide a crucial insight into the formation and
evolution of the diffuse light. To gain a deeper understanding of the specific
mechanisms involved in the formation and evolution of the diffuse light, in
the future, we plan to trace the spatial distribution of stellar particles to
elucidate their origins and the processes through which they become diffuse
light components.

## Acknowledgments

The authors thank the anonymous referee for the useful suggestions and the
Illustris and IllustrisTNG projects for providing simulation data. We
acknowledge support from the NSFC grant (No. 12073089, No. 12003079) and the
National Key Program for Science and Technology Research and Development (No.
2017YFB0203300). L.T. is supported by Natural Science Foundation of Sichuan
Province (No. 2022NSFSC1842), the Fundamental Research Funds of China West
Normal University (No. 21E029), and the Sichuan Youth Science and Technology
Innovation Research Team (21CXTD0038). Y.W. is supported by The Major Key
Project of PCL. J. L. is supported by an NSFC grant (No. 12273027). The
analysis carried out in this work is done at the Kunlun HPC facility of the
School of Physics and Astronomy, Sun Yat-Sen University. We thank LetPub
(www.letpub.com) for its linguistic assistance during the preparation of this
manuscript.

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