# Can Early Dark Energy be Probed by the High-Redshift Galaxy Abundance?

Liu, Weiyang,1,2 Zhan, Hu,1,3 Gong, Yan,1,2 and Wang, Xin1,2
1Key Laboratory of Space Astronomy and Technology, National Astronomical
Observatories, Chinese Academy of Sciences, Beijing, 100101, China P.R.
2School of Astronomy and Space Science, University of Chinese Academy of
Sciences, Beijing, 100049, China P.R.
3The Kavli Institute for Astronomy and Astrophysics, Peking University,
Beijing, 100871, China P.R E-mail<EMAIL_ADDRESS>(NAOC)

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

The Hubble tension and $\sigma_{8}$ tension are two of the major issues of the
standard $\Lambda$ Cold Dark Matter ($\Lambda$CDM) model. The analysis of the
Cosmic Microwave Background (CMB) data acquired by the Atacama Cosmology
Telescope (ACT) and the large-scale ($\ell\lesssim 1300$) Planck Telescope
manifest their preference for the Early Dark Energy (EDE) theory, which was
set to alleviate the Hubble tension by decreasing the sound horizon $r_{s}$,
and gives $H_{0}\approx 72$ km s-1 Mpc-1. However, the EDE model is commonly
questioned for exacerbating the $\sigma_{8}$ tension on top of the
$\Lambda$CDM model, and its lack of preference from the late-time matter power
spectrum observations, e.g., Baryon Oscillation Spectroscopic Survey (BOSS).
In light of the current obscurities, we inspect if the high redshift galaxy
abundance, i.e., Stellar Mass Function/Density (SMF/SMD) and Luminosity
Function (LF), can independently probe the EDE model and ameliorate the
challenges it encounters. Our result shows that the EDE model produces more
observable galaxies than $\Lambda$CDM at $z>8$. The LF and SMD, in particular,
are consistent with the recent unexpectedly high results observed by the James
Webb Space Telescope (JWST), which may posit another observational hint of
EDE. This result also implies an efficient suppression mechanism of the galaxy
formation rate that leads to the conversion between EDE- and $\Lambda$CDM-like
Universe around $z\sim 7-10$, and that $\sigma_{8}$ tension could be more of a
tension of galaxy evolution than cosmology, hence diminishing its impact on
EDE or $\Lambda$CDM theory.

###### keywords:

cosmology – dark energy – galaxies: high-redshift – galaxies: abundances

††pubyear: 2024††pagerange: Can Early Dark Energy be Probed by the High-
Redshift Galaxy Abundance?–References

##  1 Introduction

The discovery of the accelerating expansion of the Universe and the resulting
cosmological constant dark energy (Riess et al. (1998); Perlmutter et al.
(1999) ) led to the transition of the cosmological model from Einstein de
Sitter ($\Omega_{m}=1$) to $\Lambda$CDM Universe. This was later recognised as
the standard model of cosmology for it was endorsed by numerous follow-up
observations. However, despite the success, the last decade witnessed growing
doubts regarding this model, due to the appearance of the (third) Hubble
tension that implies the values of the Hubble constant measured from the early
and late Universe are inherently inconsistent. For instance, when assuming
$\Lambda$CDM, Planck Collaboration et al. (2020a) derived $H_{0}=67.36\pm
0.54$ km s-1 Mpc-1 from CMB, while the direct local observational result from
Riess et al. (2022) showed that $H_{0}=73.30\pm 1.04$ km s-1 Mpc-1. Although
many efforts have been made to check whether the inconsistency between the
local direct and early indirect measurement of $H_{0}$ is caused by some
statistical flukes or overlooked observational errors, the consensus still has
not been reached as yet. Instead, the rising concern is that the modification
of the cosmological model itself is required to alleviate the tension.

On top of the Hubble tension, the $\Lambda$CDM model is also challenged by the
anomaly on the matter clustering, dubbed as the $S_{8}$ tension. The factor
$S_{8}\equiv\sigma_{8}\sqrt{\Omega_{m}/0.3}$ measures the amplitude of the
matter clustering and is closely related to the growth rate of the large-scale
structure via $f\sigma_{8}$, where $f=[\Omega_{m}(z)]^{0.55}$. The observed
result of this factor from the Weak Lensing (WL), e.g., Kilo-Degree Survey
(KiDS), is $S_{8}\approx 0.759$ (Asgari et al. (2021)), about $2-3\sigma$
lower than the expected value from CMB that gives $S_{8}\approx 0.834$ (Planck
Collaboration et al. (2020b)). Likewise, there are two possible approaches
towards the alleviation of the $S_{8}$ tension, the observational/statistical
errors, and the alteration of the cosmological model. The consensus, however,
has not yet been reached either. More information about the aforementioned
tensions can be found in Abdalla et al. (2022).

Schöneberg et al. (2022) summarised and compared the most common 17 theories
proposed to resolve the Hubble tension and among them, the EDE theory is one
of the few that carries both theoretical capability and observational
evidence. The basic idea of EDE is to introduce a scalar field that behaves
like dark energy before the recombination and dilutes faster than the
radiation such that the post-recombination evolution remains to be
$\Lambda$CDM. This extra component decreases the physical sound horizon
$r_{s}$ by slightly increasing the cooling rate of the Universe. As a result,
the corresponding $H_{0}$ inferred from CMB is increased accordingly.

The most direct observational evidence of EDE by far comes from Poulin et al.
(2021); Hill et al. (2022), in which they analysed the CMB temperature power
spectrum data from the Wilkinson Microwave Anisotropy Probe (WMAP), ACT and
Planck Telescope. The results show that although EDE is not favoured by the
full-scale Planck data, the preference of the model is shown in WMAP+ACT. The
main difference between Planck and ACT is their small scale, e.g. $\ell\sim
1000$ and $\ell\sim 1500$. Smith et al. (2022b) further indicated that the
truncated Planck data limited to $\ell<1300$ also shows a preference for EDE.
On the other hand, the observational result of the late-time matter power
spectrum made by BOSS, e.g., Hill et al. (2020), does not reveal clear signs
of EDE. However, other studies such as Simon et al. (2023); Herold et al.
(2022) argued that the conclusion drawn from BOSS power spectrum could be
affected by the other non-cosmological factors, e.g., volume effects or the
normalisation of the window function. More research attempting to search for
the trace of EDE can be found in Jiang & Piao (2022); Lin et al. (2020);
Goldstein et al. (2023); Gómez-Valent et al. (2022); Reeves et al. (2023);
Smith et al. (2022a); Murgia et al. (2021), in which various probes (e.g.,
weak lensing and Lyman-$\alpha$ forest) and alterations of the model (e.g.,
massive neutrinos) were applied. Generally speaking, at this stage, it is
still far from concluding the debate over the observational evidence of EDE.
In light of the ambiguity of the current observational results, it is
indispensable to verify the model with another probe and break the stalemate
on the small-scale and post-recombination epoch.

In this paper, we aim to study if the abundance of the galaxies at high
redshift is capable of probing the EDE. This idea was first proposed by Klypin
et al. (2021), in which they calculated the Halo Mass Function (HMF) in the
EDE scenario and found that there are more halos formed than in the
$\Lambda$CDM case. Interestingly, this phenomenon is more prominent at higher
redshift. Hence, the galaxy abundance, such as HMF, satisfies what is required
to study the EDE model in both ways: 1. The observation of the galaxy
abundance can be conducted on small-scale surveys, which makes it possible to
extract information from the galaxy distribution at ultra-high redshift. Note
that the typical redshift of the BOSS matter power spectrum is below $z\sim 1$
(see e.g., Simon et al. (2023); Hill et al. (2020)); 2. The expectation of a
predominant outnumbered halo number density at ultra-high redshift in the EDE
model could make it distinguishable from $\Lambda$CDM. Thanks to the deep
field observation archives from e.g., Hubble Space Telescope (HST) and Spitzer
Infrared Array Camera (Spitzer/IRAC), as well as the advent of the James Webb
Space Telescope data, the detection of the EDE on the high redshift galaxy
abundance is now ready.

An unexpected feature observed by JWST is that it appears to suggest that the
number density of the massive galaxies at redshift $z>7$ is considerably
higher than the extrapolation from the previous studies (Labbé et al. (2023)).
Specifically, Boylan-Kolchin (2023) argued that these massive galaxies lie on
the upper limit of what is allowed by physics, which requires all baryons to
be converted into the stellar mass of these objects. Although there are also
other speculations regarding the adjustment of the Initial Mass Function
(IMF), e.g., Steinhardt et al. (2023), or the uncertainty on the photometric
redshift (Arrabal Haro et al. (2023)), this surprising discovery may imply
that the current galaxy formation theory or the $\Lambda$CDM itself require
further contemplation. As we shall see, the predicted galaxy abundance under
the EDE model is consistent with the unexpectedly high number density of the
massive galaxies found by JWST.

On top of all of the concerns above, we also notice that the change in
cosmology itself can simultaneously affect the estimation of the stellar mass
through the change of the galaxy formation history. Alternatively, one might
stick with the luminosity of the galaxies to avoid the uncertainty of the
stellar mass estimation, which involves the composite of cosmology and various
baryonic interaction processes. However, the observational result of the
luminosity depends on the knowledge of the distance modulus and the dust
attenuation (e.g., Meurer et al. (1999)) that may behave differently at the
ultra-high redshifts. Bearing all in mind, we will show that the luminosity is
not noticeably affected by EDE and leave the full-scale estimation of the
stellar mass under EDE for future work since it is beyond the topic of this
paper.

The structure of this paper is as follows. In Section 2, we introduce the
theory of EDE. Then we present the way to predict the high redshift galaxy
abundance in Section 3. The comparison of our predictions with the
observational data is described in Section 4. Finally, we conclude this paper
in Section 5.

##  2 Early Dark Energy Model

The angular size of the sound horizon $\theta_{s}\equiv r_{s}/D_{A}$ at the
epoch of recombination is precisely measured by the CMB observations. Here
$r_{s}$ and $D_{A}$ are the physical size of the sound horizon and the
distance towards the surface of the last scattering, respectively. Starting
from a fixed $\theta_{s}$ and a given cosmological model as the preconditions,
we can derive the corresponding $H_{0}$ as follows (Kamionkowski & Riess
(2023)),

$\displaystyle
H_{0}=\sqrt{3}H_{ls}\theta_{s}\frac{\int_{0}^{\infty}dz[\rho(z)/\rho_{0}]^{-1/2}}{\int_{z_{ls}}^{\infty}dz[\rho(z)/\rho(z_{ls})]^{-1/2}(1+R)^{-1/2}},$
(1)

where $H_{ls}$ is the Hubble parameter at the last scattering, $z_{ls}\simeq
1080$. $\rho(z)$ and $R=(3/4)(\omega_{b}/\omega_{\gamma})/(1+z)$ represent the
total energy density and the density ratio of the baryons to photons at $z$,
respectively. By decreasing the integral of the denominator, i.e., increasing
the energy density $\rho(z)$ before recombination, the inferred $H_{0}$ can be
elevated accordingly, and the corresponding physical size of the sound horizon

$\displaystyle r_{s}=\int_{z_{ls}}^{\infty}\frac{c_{s}(z)dz}{H(z)}$ (2)

is decreased. Here $c_{s}(z)=c[3(1+R)]^{-1/2}$ is the sound speed of the
baryon-photon fluid. The collection of the methods following this type of
mechanism is normally dubbed as the Early-time solution, of which a
representative example is the EDE model we apply in this paper.

The EDE model by far is the collective name of an ad hoc postulation with
various alterations of dynamics. In this paper, we focus on the most studied
Axion-like EDE model derived from the string theory (McDonough & Scalisi
(2022)). As elaborated by Poulin et al. (2018, 2019); Karwal & Kamionkowski
(2016); Kamionkowski & Riess (2023), the EDE model introduces an oscillating
axion scalar field $\phi$ with a potential

$\displaystyle V_{n}(\phi)=m^{2}f^{2}[1-\cos{(\theta)}]^{n}$ (3)

to accelerate the cooling rate of the Universe before the recombination takes
place. Here $m$ denotes the effective mass of the axion particle, $f$
represents the decay constant, and $\theta\equiv\phi/f$ is a re-normalisation
factor so that $-\pi\leq\theta\leq\pi$. The decay rate of the axion field is
determined by the index $n$ via the equation of state $w_{n}=(n-1)/(n+1)$.
When $n=1$, it behaves like the normal dark matter axion field. However, if
$n>2$ the field decays faster than radiation so that the post-recombination
epoch remains to be the $\Lambda$CDM-like Universe. The EDE field is initially
frozen at a fixed value until the critical redshift $z_{c}$, after which it
starts to oscillate and decay. The fractional energy density of EDE at this
point is denoted by $f_{\text{EDE}}(z_{c})$. The first observational evidence
showed by Poulin et al. (2021) indicated that if $n=3$, the best-fit EDE
parameters reconstructed from the ACT data are
$f_{\text{EDE}}=0.158_{-0.094}^{+0.051}$ and
$\log_{10}(z_{c})=3.326_{-0.093}^{+0.2}$, which give
$H_{0}=73.43_{-3.4}^{+2.6}$ km s-1 Mpc-1.

There has been a growing number of studies, following the first observational
hint, attempting to search for more evidence of EDE from CMB. Smith et al.
(2022b) took the South Pole Telescope (SPT) data into account and found that
ACT+SPT+Planck polarisation gives a similar conclusion that favours the EDE
model. They also found that the Planck temperature power spectrum also shows a
preference for EDE if the small-scale data ($\ell>1300$) are discarded. More
works, e.g., Jiang & Piao (2022); Hill et al. (2022), also found pro-EDE
evidence from the observations of CMB.

However, applying the EDE-like mechanism to resolve the Hubble tension comes
with the cost of an exacerbated $S_{8}$ tension since it results in a denser
Universe. Generally speaking, the locally observed values of $S_{8}$ from
e.g., Weak Lensing, are lower than those inferred from $\Lambda$CDM. For
instance, Asgari et al. (2021) analysed the data from KiDS and obtained
$S_{8}\simeq 0.759$, while the result from Planck Collaboration et al. (2020b)
is about $2-3\sigma$ higher and reaches $S_{8}\simeq 0.834$. In comparison,
the result from Poulin et al. (2021) showed that $S_{8}\simeq 0.862$ in the
EDE model, which is slightly higher than the $\Lambda$CDM case. As a
consequence, most of the post-recombination observations do not show any
preferences on EDE over $\Lambda$CDM. For instance, Goldstein et al. (2023)
attempted to use Lyman-$\alpha$ forest to constrain the EDE parameters and
found that $f_{\text{EDE}}<0.08$ with $H_{0}=67.9_{-0.4}^{+0.4}$ km s-1 Mpc-1,
$>4\sigma$ away from the Supernovae and $H_{0}$ for the Equation of State of
dark energy (SH0ES) result (Riess et al. (2022)). On top of that, the signal
of EDE does not emerge in the late-time large-scale structure (LSS)
observations, either. Hill et al. (2020) reanalysed the EDE scenario with the
LSS data, including CMB lensing, Baryon Acoustic Oscillation (BAO), Redshift
Distortion (RSD), Dark Energy Survey Year 1 (DES-Y1), Hubble Source Catalog
(HSC) and KiDS, and found no clear signal of EDE. This is further indicated in
Herold et al. (2022) that gives $f_{\text{EDE}}=0.072_{-0.036}^{+0.036}$, a
value too small to resolve Hubble tension. Although Simon et al. (2023)
reassessed the BOSS data by applying a new normalisation method of the window
function and slightly ameliorated its disfavour over EDE, the challenge of
$S_{8}$ tension remains intact for the increment of the matter density is an
inevitable consequence of EDE.

Briefly speaking, despite the “spark of hope” to resolve the Hubble tension
lit by the EDE theory, the mechanism behind it is still under challenges from
the small-scale Planck temperature power spectrum and the low-redshift
observations. The detailed reviews of the current dilemma of EDE in 2023 can
be found in McDonough et al. (2023); Poulin et al. (2023). Be that as it may,
one notices that the current analyses focus on the recombination and late-time
Universe while omitting the observations of the post-recombination high
redshifts, e.g., the Cosmic Dawn and the reionisation. We argue that the study
of EDE in these epochs can complement what is needed for resolving the current
stalemate: 1. It provides an alternative way to study the small-scale problem
of EDE; 2. It is relatively less complicated by the late-time non-linear
process of the cosmic structure/galaxy formation history which is one of the
speculations of the reason for the $S_{8}$ tension. A new probe from the realm
of the small-scale and post-recombination high redshift is yearned for the
verification of EDE. As we shall see below, the galaxy abundance might indeed
bear the probability to deliver the just cause for it.

##  3 Galaxy Abundance in the EDE Scenario

The basic idea of the galaxy formation theory in the modern perspective is
that the dark matter halo distribution and merger history form the skeleton of
the galaxy distribution. Then, the baryons gravitationally bound by these
systems start to form stars, during which various kinds of feedback (such as
stellar and AGN feedback) take place and suppress or quench the further star
formation process. The basic empirical approach towards galaxy formation
theory is the Abundance matching, which implies that the most massive galaxies
reside in the most massive halos (Wechsler & Tinker (2018)). Consequently, the
predicted abundance of the galaxies, normally expressed in terms of the
stellar mass function or luminosity function, depends on the halo mass
function and the scaling relation between the halo mass $M_{h}$, stellar mass
$M_{\ast}$, and luminosity at the ultraviolet band $L_{\text{UV}}$ (or the
absolute magnitude $M_{\text{UV}}$). In Section 3.1, we compare the halo mass
functions in the EDE and $\Lambda$CDM models. In particular, We adopt the EDE
and $\Lambda$CDM parameters inferred by Smith et al. (2022b) and Planck
Collaboration et al. (2020a), respectively. The exact values of these
parameters are listed in Table 1. Then, we apply the empirical scaling
relations to derive the corresponding stellar mass functions in Section 3.2
and luminosity functions in Section 3.3.

Parameter | EDE |  $\Lambda$CDM
---|---|---
$h$ | 0.7420 | 0.6736
$\omega_{c}$ | 0.1356 | 0.1200
$n_{s}$ | 1.0010 | 0.9649
$\sigma_{8}$ | 0.8446 | 0.8111
$n$ | 3 |
$\lg({z_{c}})$ | 3.526 |
$f_{c}$ | 0.163 |
Table 1: The cosmological parameters of the EDE and $\Lambda$CDM models to
generate the matter power spectrum. The values of EDE are the best fit of
ACT+SPT+Planck TT650TEEE from Smith et al. (2022b) and the $\Lambda$CDM case
is the result of Planck Collaboration et al. (2020a).

###  3.1 Halo Mass Function

Broadly speaking, the derivation of the halo mass function depends on two
aspects: the cosmology that mainly affects the matter power spectrum and the
properties of the dark matter that determine the formation and evolution of
the halos. In EDE theory, the properties of the dark matter stay unchanged,
such that we only need to consider the change of the matter power spectrum for
the approximation of the halo mass function in the EDE/$\Lambda$CDM model. In
order to stay consistent with previous studies, we make use of the AxiCLASS
(Smith et al. (2020); Poulin et al. (2018)), a modified version of the
Einstein-Boltzmann code CLASS (Blas et al. (2011)), to compute the matter
power spectra in the scenarios of EDE and $\Lambda$CDM.

Figure 1: (Left) The matter power spectra of the EDE (solid lines) and
$\Lambda$CDM (dashed lines) model. The EDE generates more power at all scales
and most prominently at the small scale, due to its larger primordial power
spectrum index $n_{s}$; (Right) The ratio of the matter power spectra between
the EDE and $\Lambda$CDM model. Their difference decreases as the Universe
evolves over redshift.

The left side of Figure 1 shows the matter power spectra of EDE (solid lines)
and $\Lambda$CDM model (dashed lines) in 5 different redshifts. It is clear
that the EDE generates larger power than $\Lambda$CDM at all scales and most
prominently, at the small scale. This is caused by the slightly increased
value of the primordial power spectrum index $n_{s}$ from the best-fit EDE
parameters shown in Table 1. Consequently, the EDE Universe is expected to be
denser than the $\Lambda$CDM case (Klypin et al. (2021)). On top of that, if
we compare the ratio of the two scenarios which is shown on the right side of
Figure 1, we see that the difference between them decreases with redshift,
i.e., the Universe becomes more $\Lambda$CDM-like in an EDE Universe. This
phenomenon is significantly more prominent in the halo mass function as we
shall see below. Finally, with the matter power spectra prepared, we can
calculate the corresponding theoretical halo mass function.

We use the halo mass function approximation from Sheth & Tormen (1999) to
estimate the number of halos in the EDE model (also see Klypin et al., 2021;
Despali et al., 2016, for more information). The halo mass function describes
the number density of the halo with mass $M$ at redshift $z$. It is formulated
by the smoothed matter power spectrum $\sigma$ as follows,

$\displaystyle\frac{dn}{dM}=f(\sigma)\left[\frac{\Omega_{m}\rho_{\text{crit}}}{M^{2}}\right]\frac{d\ln{\sigma}}{d\ln{M}}.$
(4)

Here

$\displaystyle\sigma^{2}(M,z)\equiv\int\frac{dk}{k}\frac{k^{3}P(k,z)}{2\pi^{2}}W^{2}[kR(M)]$
(5)

denotes the extent of the overdensity within the region confined by the window
function $W[KR(M)]$, where $M$ is the mass enclosed inside the radius $R$,
$M=(4\pi/3)\rho_{m}R^{3}$. The $P(k,z)$ is the matter power spectrum
extrapolated to redshift $z$ as is shown in Figure 1. Note that the window
functions are normally top-hat or Gaussian. Next, it is commonly accepted to
assume that the threshold of the overdensity is $\delta_{\text{crit}}\approx
1.68$ at which the density of the overdense region is high enough to form a
halo. Define the relative height of the density peak by

$\displaystyle\nu=\frac{\delta_{\text{crit}}}{\sigma(M,z)},$ (6)

so that the halo mass function can be rewritten as

$\displaystyle\nu f(\nu)=A\left(1+\frac{1}{\nu^{\prime
p}}\right)\left(\frac{\nu^{\prime}}{2\pi}\right)e^{-\nu^{\prime}/2},$ (7)

where $\nu^{\prime}=a\nu$ and the simulated coefficients, $(A,a,p)$, determine
the normalisation, high mass cut-off, and low mass shape of the function,
respectively.

In practice, we use hmf (Murray et al. (2021, 2013)) and adopt the
coefficients from Despali et al. (2016), i.e., $(A,a,p)=(0.3295\pm
0.0003,0.7689\pm 0.0011,0.2536\pm 0.0026)$ to calculate the halo mass
functions for the EDE model. In their work, they found that the halo mass
function remains universal for Planck 2013 $\Lambda$CDM Universe and when
$\sigma_{8}=0.9$, which is higher than the values of both EDE and Planck 2018
$\Lambda$CDM listed in Table 1. We therefore argue that their result is a
legitimate choice for our purpose, as the elevated value of $\sigma_{8}$ is
the main consequence of EDE in the post-recombination epoch. Figure 2 shows
the redshift and mass evolution of the halo mass function. Interestingly, the
uncertainty gets larger at the massive end as the redshift goes higher. This
phenomenon is consistent with the result in Despali et al. (2016) since the
halo mass function is more sensitive to the normalisation factor $A$ and the
high mass cut-off parameter $a$, as $\nu$ gets larger with a higher redshift.

Figure 2: The halo mass function under EDE model in different redshifts. The
error shade corresponds to the uncertainties on the coefficients $(A,a,p)$
from Despali et al. (2016) in Equation 7 as well as the uncertainties of the
cosmological parameters provided by Smith et al. (2022b) and Planck
Collaboration et al. (2020a). The uncertainty gets larger at the massive end
as the redshift goes higher. We also plot the $\Lambda$CDM cases (dashed
lines) for comparison.

To see the difference between the EDE and $\Lambda$CDM predictions more
clearly, we neglect the uncertainty of the coefficients and re-plot the halo
mass functions of both models and their ratio in Figure 3, where the EDE is
denoted by the solid lines and the dashed lines are the $\Lambda$CDM cases. As
is expected, the EDE model produces more halos in all redshifts. When $z<3$,
both cases are almost identical, while the difference of them gets larger at
higher redshift. This phenomenon is caused by the fact that the exponential
part of Equation 7 dominates the halo mass function when $\nu$ is large, which
is equivalent to the function becomes more sensitive to $\sigma(M,z)$ as
redshift goes higher (Klypin et al. (2021)). Hence, through Equation 5, the
effect of EDE on the power spectrum is revealed via a denser distribution of
the dark matter halo. In this case, the feature of the EDE halo mass function
is therefore manifested in two aspects: 1. It is consistent with $\Lambda$CDM
in the local Universe; 2. It shows an observable deviation from $\Lambda$CDM
in the early Universe.

Therefore, bearing the tight bond between the halos and galaxies in mind,
according to the abundance matching, we expect that the feature of the
outnumbered massive halo density in the EDE model when $z\geq 4$ should also
be propagated to the high redshift galaxy abundance, albeit the number density
difference of the galaxies is not expected to be as high as the halos due to
the baryonic feedback processes of the galaxy formation history and, as was
demonstrated in both semi-analytical studies and simulations (Efstathiou
(1992); Bullock et al. (2000); Simpson et al. (2018)), that most of the low
mass halos are unable to form galaxies, both of which suppress or quench the
further merging and enrichment of the galaxies.

Figure 3: (Left) The halo mass functions in different redshifts. The solid
lines are the EDE case and the dashed lines denote the $\Lambda$CDM. The
redshift evolution patterns of both models are akin, albeit in the EDE case
the denser initial matter power spectrum results in more halos produced in
each redshift; (Right) The halo mass function ratio between the EDE and
$\Lambda$CDM cases. The ratio goes up as the mass and redshift get higher,
which implies that EDE can produce more massive galaxies in the early Universe
than $\Lambda$CDM. The evolution of their ratio suggests that the observation
of the massive galaxies at high redshift should be able to distinguish the two
models.

###  3.2 Stellar Mass Function

The mass of the dark matter halo determines the maximum scale that a galactic
system can reach. Empirically speaking, the principle of their relationship
follows the abundance matching as we mentioned above. However, their
relationship is not linear but depends on a composite of various dynamic
processes, such as the feedback from star formation or AGN. In particular, the
Stellar-to-Halo Mass Ratio (SHMR, or $\epsilon$) at high redshift may differ
from the low redshift case, for the stars and AGN need time to form.
Notwithstanding the previous simulations being implemented to study the
galaxies with high precision, it remains an open question to simulate the
formation history of them accurately. Therefore, we apply the empirical SHMR
based on known data to establish our prediction of the galaxy abundance, for
it contains the authentic information required to reconstruct the relation
between halos and galaxies, despite the ambiguity of the specific mechanisms
concealed within the black box of the galaxy formation process.

Stefanon et al. (2021) analysed the stellar mass of the galaxy candidates
observed by HST and Spitzer at $z\sim 6-10$ and found that the SHMR relation
does not strongly depend on redshift. Namely,

$\displaystyle\frac{M_{\ast}}{M_{h}}=2N\left[\left(\frac{M_{h}}{M_{c}}\right)^{-\beta}+\left(\frac{M_{h}}{M_{c}}\right)^{\gamma}\right]^{-1}.$
(8)

Here $N=0.0297\pm 0.0065$ is the normalisation factor, $M_{c}=10^{11.5\pm
0.2}$ M⊙ is the characteristic halo mass at which the star formation
efficiency reaches its maximum. The slope $\gamma=0.4$ (Tacchella et al.
(2018)) and $\beta=1.35\pm 0.26$ denote the shape of the relation at the high-
mass and low-mass regimes, respectively.

The SHMR expression of Equation 8 also indicates that the galaxy formation
efficiency is not a constant or a monotonically increasing function, but peaks
at around $M_{h}\sim 10^{11.5}$ M⊙, as is shown in Figure 4 (the blue solid
line). In comparison, we also draw the common assumption that $5-30$ per cent
of the baryonic mass is converted into stars (the grey area), i.e.,
$\epsilon=(0.05,0.3)\times\Omega_{b}/\Omega_{m}$. It is clear that the low-
mass halos cannot efficiently form galaxies due to their limited capacity to
bind the masses contained in them. On the other hand, the turnover of the
massive end indicates the potential strong feedback at the high redshift that
quenches further enrichment of the stellar mass in these halos. In the low
redshift regime, the suppression of the massive end is normally considered the
result of the AGN feedback (Wechsler & Tinker (2018)). However, considering
only very few numbers of AGN are detected in the galaxies at $z>8$ (e.g., the
rare and most distant AGN up-to-date, CEERS_1019 at $z\approx 8.72$, Larson et
al. (2023) and GN-z11 at $z\approx 10.603$, Scholtz et al. (2023)), the
mechanism behind the turnover above $z\sim 8$ might be different from their
local counterparts, for AGN also needs time to form. Alternatively, since the
SHMR at $z\geq 8$ is poorly constrained by the pre-JWST data, we can also
speculate that the turnover does not exist at the early stage of the Universe.
Therefore, we also extrapolate an AGN-free SHMR by letting $\gamma=0.01$ and
remove the turnover (blue dashed line). Instead, if future observations indeed
find the turning point, the mechanism behind the suppression at the massive
end might be the collective result of the insufficient merging process between
the galaxy progenitors within the given time and the early-formed AGN.
Nevertheless, considering the redshift-free relation of Equation 8 stands as
early as $z=10$, we assume that the potential AGN feedback is not a major
concern for applying the SHMR at a higher redshift.

Meanwhile, the redshift-free empirical scaling relation suggests that the
early formation process of the galaxies is dominated by the dark matter halos
rather than the baryonic compositions which will lead to the evolving feedback
mechanisms that alter the formation process over time. According to Tacchella
et al. (2018), the redshift-free SHMR does not stand below $z\sim 4$.
Considering the purpose of our work is to investigate the galaxies at high
redshift, we therefore cautiously suggest not to extend the usage of this
scaling relation for redshift $z<6$, for the late-time galaxies are formed in
a much more dusty and complicated environment than their early-time
progenitors.

Figure 4: The galaxy formation efficiency proposed by Stefanon et al. (2021)
changes over $M_{h}$ (blue solid line). The grey area marks the constant
assumption that 5 to 30 per cent of the baryons are converted into $M_{\ast}$.
In particular, the efficiency of the low-mass halos could be as low as $\leq
5$ per cent of the total baryonic mass, while the massive halos appear to have
a turning point around $M_{h}^{11.5}$ M⊙. It is known that the low-mass halos
are highly inefficient in forming stars, while the turnover is presumably
caused by the feedback from the low-mass AGN or the insufficient merging
process between the galaxy progenitors. Alternatively, since the pre-JWST data
at $z\geq 8$, upon which this relation is built, does not put a strong
constraint on the turning point, we also extrapolate the relation to an AGN-
free version by letting $\gamma=0.01$ in Equation 8 so that the massive end is
flattened (blue dashed line).

Finally, by applying the Jacobian $dM_{h}/dM_{\ast}$ derived from Equation 8
to the halo mass function, Equation 4, we obtain the corresponding stellar
mass function as is shown in Figure 5,

$\displaystyle\frac{dn}{dM_{\ast}}=\frac{dn}{dM_{h}}\frac{dM_{h}}{dM_{\ast}}.$
(9)

The black solid lines on the figure are the Schechter best-fitting which is
the same as Figure 7 in Stefanon et al. (2021), and the blue solid and red
dashed lines are the EDE and $\Lambda$CDM predictions, respectively. The error
shade of the EDE lines is propagated from the uncertainties of the halo mass
function (Figure 2) and the SHMR scaling relation parameters
$(N,M_{c},\beta)$. We also plot the HST/Spitzer IRAC data from Stefanon et al.
(2021); Song et al. (2016); Kikuchihara et al. (2020); Bhatawdekar et al.
(2019) for comparison. The result shows that the predictions from $\Lambda$CDM
appear to fit better with the existing data. However, it is yet too soon to
conclude that EDE is excluded by the high redshift galaxy abundance already.
As we shall see in the next two sections, the advent of the JWST data may have
been a “game-changer” in our investigation.

The application of the EDE model may also affect the estimation of $M_{\ast}$
due to e.g., the change of the distance measurement, the potentially earlier
starting time of the galaxy formation process and a denser cool gas
distribution. The distance change may slightly decrease the measured value of
the luminosity, as we shall see in Section 3.3, and thus the estimation of
$M_{\ast}$. In contrast, the potential changes in the galaxy formation process
itself are expected to lead to an increment in the measurement. In this paper,
we apply the vanilla results of the stellar mass data and neglect the possible
shift of it under the EDE model, for the scale at which the galaxy formation
takes place is way smaller than the context of cosmology, and the properties
of dark matter remain unchanged. Nevertheless, a detailed evaluation of the
impact that the EDE model may inflict on the stellar mass estimation is
required for a more solid argumentation.

Figure 5: The stellar mass function from the best-fitting Schechter (black
solid lines, Stefanon et al. (2021)), EDE (blue solid lines) and $\Lambda$CDM
(red dashed lines). The error shade of the EDE lines is propagated from the
uncertainties of the halo mass function (Figure 2) and the $M_{h}-M_{\ast}$
scaling relation parameters $(N,M_{c},\beta)$ given by Stefanon et al. (2021).
The data points are from the HST/Spitzer IRAC obtained by Stefanon et al.
(2021); Song et al. (2016); Kikuchihara et al. (2020); Bhatawdekar et al.
(2019). Note that the stellar mass data are not calibrated by the EDE model.
The result of the comparison shows that the prediction of the $\Lambda$CDM
model fits the data better than that of EDE. However, it is too early to veto
EDE as yet, for the advent of JWST appears to have changed the situation.

###  3.3 Luminosity Function

In observation, the luminosity function directly counts the number density of
the galaxies in terms of the surface brightness with limited hypotheses of
cosmology and galaxy formation. The stellar mass function, on the other hand,
requires the mass estimation of the galaxies that depends on the theories of
both cosmology and the baryonic interactions in the galaxy formation history.
Therefore, the detected luminosity function is considered a more direct tool
to test the mechanisms of galaxy formation.

The prediction of the luminosity function, however, is unable to be inferred
in a model-free way. One of the greatest challenges of predicting the
luminosity function from the halo mass function is the difficulty of finding
the straightforward scaling relation between the halo mass $M_{h}$ and the
luminosity of a galaxy $L_{\ast}$. The $L_{\ast}$ is a direct representative
of the properties and distribution of the stars, while the $M_{h}$ does not
directly reflect the stellar distribution of the system, i.e., a massive halo
may form a galaxy with different compositions of stars according to the age,
metallicity and other factors, which correspond to different values of
luminosity at a given band. There are attempts to find the luminosity from the
halo mass, e.g., Mason et al. (2023); Sabti et al. (2022), while some
essential assumptions of the galaxies, such as the Star-Formation Rate (SFR),
are required to obtain the corresponding $L_{\ast}$, which is equivalent to
using the stellar mass $M_{\ast}$ as a proxy to mediate the $M_{h}-L_{\ast}$
relation in between. Following the same logic, we first fix the scaling
relation between the stellar mass and luminosity and then derive the
corresponding luminosity function as is done in Equation 9, i.e.,

$\displaystyle\frac{dn}{dM_{\text{UV}}}=\frac{dn}{dM_{h}}\frac{dM_{h}}{dM_{\ast}}\frac{dM_{\ast}}{dM_{\text{UV}}}.$
(10)

Note that we replace the luminosity $L_{\ast}$ by the absolute magnitude at
the UV band, $M_{\text{UV}}$, to align with the observations.

There have been a few number of works discussing the form of the scaling
relation between $M_{\text{UV}}$ and $M_{\ast}$, e.g., Stefanon et al. (2021);
Kikuchihara et al. (2020); Bhatawdekar et al. (2019); Duncan et al. (2014). In
their works, the scaling relation is approximated by a linear relation,

$\displaystyle M_{\text{UV}}=[\lg({M_{\ast})-a_{0}}]/a_{1},$ (11)

where $(a_{0},a_{1})$ are the slope and intercept of the relation fitted from
the pre-JWST data. Here we keep up with the relation provided by Stefanon et
al. (2021) both to keep the consistency and to cover a higher redshift limit
which was not reached by the other works. We notice that in their work,
$a_{0}$ is not directly given but calculated by
$a_{0}=\lg{M_{\ast}}-a_{1}\times-20.5$ where $M_{\ast}$ is the stellar mass
corresponds to $M_{\text{UV}}=-20.5$ in each redshift bin (see Table 3 in
their work for more information).

The $M_{\text{UV}}$ converted by Equation 11 is the intrinsic absolute
magnitude of a galaxy. In real observations, however, the UV light emitted
from the galaxy is absorbed by dust and re-emitted in the Infrared (IR) band.
Meurer et al. (1999) proposed a simple dust attenuation rule to calibrate the
absolute magnitude by using the UV spectral slope $\beta$, which is equivalent
to the colour of the galaxy,

$\displaystyle A_{1600}=4.43+1.99\beta,$ (12)

where $A_{1600}$ is the dust attenuation factor at 1600 $\mathring{\text{A}}$.
Note that the relation is built upon the assumption that the high redshift
galaxies are scaled-up starburst galaxies in the local Universe (Meurer et al.
(1999); Sun et al. (2023)). Given this, one can obtain the calibration of the
dust attenuation in different redshifts with the measurement of $\beta$.

With the intrinsic luminosity function and the dust attenuation law prepared,
we can finally derive a realistic luminosity function prediction in the EDE
Universe to compare with the observational results.

##  4 Results

Figure 6: The predicted luminosity functions at different redshifts with dust
attenuation calibration from Cullen et al. (2023). The black solid and blue
dashed lines correspond to the EDE and $\Lambda$CDM predictions, respectively.
We also plot the black dashed lines for the AGN-free scenario (Figure 4) and
the light lines for the dust uncalibrated results. For comparison, we consider
both JWST and pre-JWST data from Castellano et al. (2023); Finkelstein et al.
(2022b); Harikane et al. (2023a, b); Morishita & Stiavelli (2023); Naidu et
al. (2022); Pérez-González et al. (2023); Donnan et al. (2023); Bouwens et al.
(2023a, b); Bagley et al. (2022); Bouwens et al. (2021); Finkelstein et al.
(2022a); Morishita et al. (2018); Oesch et al. (2018); Bowler et al. (2020);
Stefanon et al. (2019). The black data points are collected from pre-JWST
observations, while the red data points are from JWST. We also modify the
luminosity distance of the JWST data to make the corresponding $M_{\text{UV}}$
consistent with EDE cosmology. The alteration of the distance brings about
$0.15$ right-shift of the $M_{\text{UV}}$ to the fainter end. The error shade
collects the uncertainties from both stellar mass function (Figure 5) and the
$M_{\ast}-M_{\text{UV}}$ scaling relation from Stefanon et al. (2021). The
results show that when $z\leq 9$ the $\Lambda$CDM predictions are consistent
with the observational results, while the EDE fits better with the JWST data
when $z>10$. The $z\sim 10$ appears to be the boundary of the two models,
which implies an efficient suppression mechanism of the galaxy formation
around this era.

The predicted luminosity functions under the EDE and $\Lambda$CDM models as
well as the observed results of the pre-JWST and JWST data from Castellano et
al. (2023); Finkelstein et al. (2022b); Harikane et al. (2023a, b); Morishita
& Stiavelli (2023); Naidu et al. (2022); Pérez-González et al. (2023); Donnan
et al. (2023); Bouwens et al. (2023a, b); Bagley et al. (2022); Bouwens et al.
(2021); Finkelstein et al. (2022a); Morishita et al. (2018); Oesch et al.
(2018); Bowler et al. (2020); Stefanon et al. (2019), are plotted in Figure 6.
The black data points are from pre-JWST (e.g., HST and Spitzer/IRAC)
observations and the red data points are from JWST. We also plot both dust
attenuation uncalibrated and calibrated cases of the two models in light and
bold lines, respectively. Here we apply the dust attenuation calibration
adopted from the JWST $\beta-M_{\text{UV}}$ relation obtained by Cullen et al.
(2023), i.e., $\beta=-0.17M_{\text{UV}}-5.40$. This relation is roughly
consistent with the result obtained from the lower redshift while mildly
pointing towards the bluer end, implying that the galaxies at high redshift
are slightly more active than their local counterpart starbursts. The
deviations of the two cases show that the influence of the dust attenuation
calibration is less significant at higher redshifts, as the dust also needs
time to form. Also, if we assume the SHMR is free from AGN suppression (Figure
4), the EDE luminosity function will be slightly lifted at the luminous end
(black dashed lines). Its effect, however, is limited as the luminosity of the
most observed galaxy candidates is below the turning point extrapolated from
the SHMR under the given $M_{\ast}-M_{\text{UV}}$ relation. We also draw the
uncertainty of this case that collects the errors from both stellar mass
function (Figure 5) and the $M_{\ast}-M_{\text{UV}}$ scaling relation given by
Stefanon et al. (2021).

In addition, since the observational results of $M_{\text{UV}}$ are affected
by the background cosmology via the change of the luminosity distance $D_{L}$,
we also adjust the data points accordingly. Assuming $(m,M)$ are the apparent
and absolute magnitudes of the observed galaxy, they are correlated by

$\displaystyle m=M+DM+K_{QR},$ (13)

where the distance modulus $DM=5\lg[D_{L}/10\text{ pc}]$ and $K_{QR}$ is the
K-correction factor determined by the observed flux and the properties of the
detector (Hogg et al. (2002)). Therefore, the only factor affected by the
background cosmology is $DM$. Apply the EDE model to the calibration of the
data and we find that the change of cosmology will slightly shift the
corresponding $M_{\text{UV}}$ to the right (dimmer) side by $\sim 0.15$ mag.
This is not surprising since $D_{L}=(1+z)\times
c\int_{0}^{z}1/H(z^{\prime})dz^{\prime}$ decreases with a larger $H_{0}$. Note
that in observation it is common to take $H_{0}=70$ km s-1 Mpc-1.

In the case of $z\sim 8$ in Figure 6, we see that the number density of the
galaxies obtained by pre-JWST observations is roughly consistent with the JWST
data. As the redshift goes up to $z\sim 9-10$, the pre-JWST results begin to
be lower than the JWST cases. The results from Bagley et al. (2022);
Finkelstein et al. (2022a), however, are exceptionally higher than other pre-
JWST data. Generally speaking, the data from JWST does not show strong
evolution among the redshift range $z\sim 6-12$ and the constancy may extend
to up to $z\sim 14$. Note that the future JWST spectroscopic redshift result
may considerably affect the current data at $z>12$ due to their limited number
of objects and the uncertainties brought by the highly redshifted UV light
signals. If we constrain ourselves to the redshift $z\leq 12$, the comparison
between the data and our prediction clearly shows that the $\Lambda$CDM fits
better with the $z\leq 9$ cases, while the EDE case demonstrates more
consistency with the data at $z>9$, and particularly at $z\sim 11-12$. We
speculate that there are three possible explanations for the outnumbered
galaxy abundance at $z>9$ as follows,

  1. 1.

The pollution of the data that leads to errors in the luminosity measurements;

  2. 2.

The dust attenuation calibration method at that redshift range should be
different from the lower redshifts;

  3. 3.

The efficient feedback mechanisms that intrinsically suppress the galaxy
formation over redshift, so that the luminosity function does not grow as fast
as the EDE model expected.

The calibration of the pollution is one of the major challenges of the high
redshift galaxy observations. According to Equation (13), there are in
principle three possible causes that lead to the inaccuracy of the luminosity:
the disturbance on the flux that pollutes the apparent magnitude $m$, the
instrumental error of the telescope itself, and the uncertainty of the
redshift $z$, Among them, the most commonly discussed error source is the
uncertainty of the redshift from JWST. By the time our analysis was finished,
the spectroscopic redshift $z_{\text{spec}}$ of the JWST galaxies were not yet
available, while there were some reservations over the accuracy of their
photometric redshift $z_{\text{photo}}$, e.g., Arrabal Haro et al. (2023).
However, considering the luminosity function contains the information of
multiple galaxies, rather than the single. The chance is low that all of their
$z_{\text{photo}}$ are systematically overestimated. The pollution from e.g.,
the selection effect or observational bias, on the other hand, is possible, as
JWST brings unprecedented resolution of the low surface brightness objects.
The detailed discussion on this matter 1 and an elaborate study on the high
redshift dust attenuation calibration method 2, however, are beyond the scope
of this paper.

In this paper, we are most interested in the possibility 3. Should the data
sets themselves be accurate, and the dust attenuation not significantly change
at higher redshifts, we argue that the dark matter-dominated early Universe is
better described by the EDE model until the baryonic feedback mechanism begins
to considerably affect the galaxy formation process and suppress the growth of
the galaxy abundance, as was indicated by Bowler et al. (2020). Consequently,
the pure dark matter-dominated EDE Universe at $z\sim 10-12$ is transformed
into the baryonic/dark matter-dominated $\Lambda$CDM Universe at lower
redshifts.

The comparison of the predicted and observed luminosity function, however, is
not as direct as it looks. As mentioned above, the prediction of the
luminosity function strongly depends on the stellar mass function, for there
is no one-to-one bond between the halo mass and the luminosity of the
corresponding galaxy. Consequently, although the scaling relation of
$M_{\ast}-M_{\text{UV}}$ can correctly convert the stellar mass and the
luminosity data upon which the scaling relation is built, it may not reliably
reflect the properties of the data from other observations, for the estimation
of the stellar mass by itself varies with the hypotheses of the Initial Mass
Function (IMF), SFR and other critical conditions each work applies. As is
shown in Figure 16 in Stefanon et al. (2021), the scaling relations built upon
various works can still be noticeably diverse. Be that as it may, now that we
see tantalising evidence for EDE from the JWST observation, it becomes
necessary to investigate further and directly compare our prediction with the
recent results of the JWST galaxy abundance in terms of $M_{\ast}$, which
fermented a “turmoil” on our previous understanding of the galaxy number
density based on the pre-JWST observations.

Labbé et al. (2023) estimated the stellar mass of six massive galaxies at
redshift $z\sim 7.4-9.1$ from the early release of the JWST data and found
that the stellar mass density, the integral of the stellar mass function above
a given mass $M_{\ast}$, is about an order of magnitude larger than the
extrapolation of the previous best-fitting from HST+Spitzer. As is mentioned
in Meurer et al. (1999), the high redshift galaxies are treated as the scaled-
up starburst galaxies. Hence it is common to estimate the mass of the high
redshift galaxies by the locally confirmed IMF and other assumptions. Boylan-
Kolchin (2023) found that according to the current understanding of the galaxy
formation theory, the unexpectedly high abundance of these massive galaxies
would require all of the baryons to form stars, which lies on the very edge of
what is allowed by $\Lambda$CDM and way beyond the empirical limit of the star
formation efficiency shown in Figure 4, thus put the current theory of the
galaxy formation at the early time under the $\Lambda$CDM model into
challenge.

Should the pipeline to approach the stellar mass of the JWST galaxy candidates
be appropriate and the cosmology be allowed to vary, we find that this
phenomenon can be reproduced by the EDE model. In Figure.7, we demonstrate the
stellar mass density that spans the range of $z\sim 7-10$ in both EDE (blue)
and $\Lambda$CDM (red) models. On top of that, considering the possible AGN-
free scaling relation shown in Figure 4, we also plot the stellar mass density
without the suppression at the massive end (blue dashed). The JWST data are
marked by orange ($z\sim 7.5$) and red ($z\sim 9.1$) points, and the black
data points are the pre-JWST data observed by Stark et al. (2013); Stefanon et
al. (2021); Kikuchihara et al. (2020); Oesch et al. (2014); Song et al.
(2016); Bhatawdekar et al. (2019). It is clearly shown that the pre-JWST data
are covered by the $\Lambda$CDM region, while the EDE region, particularly the
AGN-free case, covers the JWST data better. In comparison, we also plot the
$\Lambda$CDM case with $z=7.5$ in the grey shade area, assuming
$\epsilon=(0.05-0.3)\times\Omega_{b}/\Omega_{m}$. In this scenario, however,
neither pre-JWST nor JWST data points can be nicely explained. The reason is
that the star formation efficiency of the massive galaxy-forming regions is
normally higher, while the low-mass halos normally share a lower value. As a
consequence, the empirical $M_{h}-M_{\ast}$ in Figure 4 determines the
flattened and sharpened shape of the low mass and high mass region,
respectively, in the stellar mass density shown in Figure 7.

Furthermore, the JWST stellar mass density we refer to may also imply that the
JWST luminosity observations with $z\leq 10$ are incomplete. In Figure 6, the
$\Phi_{\text{UV}}$ from JWST in these redshifts almost show no inconsistency
against the pre-JWST results, and the consistency is likely to remain in their
corresponding stellar mass estimation as well, which would contradict the
results from Labbé et al. (2023). Therefore, we expect more observations of
the luminosity function at these redshifts in the coming years may tell us if
the data in the redshift range $z\sim 7-10$ also follows the prediction of the
EDE model.

Through the comparison between the theoretical prediction of the stellar mass
function (Figure 5), luminosity function (Figure 6) and stellar mass density
(Figure 7), we argue that the galaxy abundance at high redshift ($z\geq 8$)
indeed shows a promising prospect to verify the EDE model (or other models, as
long as they can put a similar effect on the matter power spectrum), for it
demonstrates considerable deviation under the EDE and $\Lambda$CDM model. From
the luminosity function comparison, we find that the JWST data at $z\sim
10-12$ shows the best consistency with EDE, which suggests a possible
suppression mechanism that slows down the growth of the galaxy number density
as the Universe evolves to the lower redshift. However, the observation of the
stellar mass density from JWST appears to shed light on a different path, that
is even the $z\sim 7-10$ JWST data shows a preference for EDE over
$\Lambda$CDM. The intrinsic discrepancy between them may lie in the technical
differences between the observation projects. The answer to this issue,
however, relies on more JWST data to come in the future.

The intrinsic concern over the stellar mass density measured by JWST is that
the scaling relation between luminosity and stellar mass at high redshift may
not follow the empirical rule constructed from the previous observations.
I.e., if there are two galaxies sharing the same luminosity. One is a local
starburst and the other is an early galaxy with $z\geq 7.5$. Should the
stellar mass of the high redshift candidate be smaller than its local
counterpart, due to the possible IMF variation or high redshift low mass AGN?
If otherwise, one will find that the extrapolation of the pre-JWST best-fit
stellar mass density and the $\Lambda$CDM prediction are unable to explain the
JWST data. The result of our calculation, however, provides an alternative
interpretation that the abundance of the massive galaxies could indeed be
higher than previously expected because of the EDE model. Consequently, both
the luminosity function and stellar mass density under this model are elevated
and fit better with the JWST data. The scaling relation, on the other hand,
remains unchanged for the scale of the galaxy formation is too small to be
strongly affected by the presence of EDE. Bearing all the discussions above in
mind, albeit it is yet too soon to claim that another observational evidence
of EDE has been found, we argue that it is safe to regard the high redshift
abundance of the galaxies as another probe of the EDE model. We also suggest
that further investigations should focus on two aspects: 1. The detailed re-
evaluation of the stellar mass under the EDE model; 2. More luminosity
function data with $z\geq 7$. Should it come to pass that the studies of days
to come accord with our results, we might find it needful to contemplate the
possibility that EDE be a fairer model to portray the early Universe, in the
stead of $\Lambda$CDM.

Figure 7: The stellar mass density under EDE (blue) and $\Lambda$CDM (red)
cosmology. The blue dashed lines are the EDE with AGN-free scaling relation as
is shown in Figure 4. The blue and red shades span the redshift from $7$ to
$10$ to match the range of the corresponding data. The grey shade area
corresponds to the fixed $5-30$ per cent star formation efficiency at $z=7.5$.
The black data are from the pre-JWST observations from Stark et al. (2013);
Stefanon et al. (2021); Kikuchihara et al. (2020); Oesch et al. (2014); Song
et al. (2016); Bhatawdekar et al. (2019) and the coloured data are what Labbé
et al. (2023) estimated from JWST. The pre-JWST data is consistent with the
$\Lambda$CDM scenario, while the prediction under EDE, particularly the AGN-
free case, matches the JWST results better.

##  5 Conclusions

The early dark energy model is one of the few theories to resolve the Hubble
tension that carries both theoretical capability and observational evidence.
At present, the signal of EDE has been detected in some of the CMB
observations. However, the verification of EDE suffers severe challenges on
the small-scale (e.g., full-scale Planck CMB) and low-redshift (e.g., BOSS
matter power spectrum) observations. Also, applying the EDE-like mechanism to
resolve the Hubble tension will inevitably exacerbate the $S_{8}$ tension, for
it incubates a denser Universe than the $\Lambda$CDM case, which already
showed discordance with the weak lensing observational results. Considering
the complications brought by the baryonic matter in the observations of the
low redshift galaxies, we instead attempted to inspect if the EDE model can be
probed by the high redshift galaxy abundance, for it provides an alternative
scope to detect the small-scale density evolution of the Universe when it was
less affected by the baryons.

Our investigation found that despite the complications of the galaxy formation
processes, the current number density of the observed high redshift galaxies
expressed in terms of the luminosity function, stellar mass function, and
stellar mass density, can indeed detect the difference of the corresponding
predictions under the EDE and $\Lambda$CDM model. This is particularly true
for the JWST data, as it discovered a surprisingly high number density of
luminous/massive galaxies which is concordant with the EDE prediction.
Although the current data is not enough to put an unarguable Aye or Nay to the
EDE model as yet, the upcoming JWST luminosity function data within redshift
$z\sim 8-12$ as well as the corresponding stellar mass estimation can be
expected to verify EDE independent of CMB and low redshift galaxy survey.

The way we applied in this paper to predict the galaxy abundance depends on
the theoretical prediction of the halo mass function and the empirical scaling
relations between $(M_{h},M_{\ast},M_{\text{UV}})$. Assuming the properties of
the dark matter are unaffected by EDE, we found that the EDE and $\Lambda$CDM
halo mass functions are almost identical at lower redshifts while
significantly distinguishable when $z>4$, which is consistent with Klypin et
al. (2021) (see Figure 2 and 3). This result is ideal for our study because
the abundance matching method indicates that the corresponding observable
stellar mass or luminosity function should share a similar detectability.

Next, we estimated the corresponding stellar mass function under the two
models. As shown in Figure 5, the existing pre-JWST data show a clear
preference towards $\Lambda$CDM rather than the EDE model up to $z\sim 10$.
The redshift-free scaling relation between $M_{h}$ and $M_{\ast}$ (Equation 8)
indicates that the low mass halos are extremely inefficient in forming stars,
while the turnover at $M_{h}\sim 10^{11.5}\ M_{\odot}$ may need to be treated
with extra care. At lower redshift, the turning point of the massive galaxy
number density is normally caused by the AGN feedback, while the situation at
higher redshift could be different due to the rareness of AGN in these
extremely young galaxies. Considering the data to form the scaling relation in
Stefanon et al. (2021) does not constrain the turning point at redshift $z>8$,
it is reasonable to assume that $z\sim 8$ is too early for most AGN to form.
Therefore, we also extrapolated the scaling relation by making $\gamma=0.01$
rather than the original fit $0.4$ in Equation 8 to flatten the turnover, such
that more massive galaxies can form due to the lack of AGN feedback (Figure
4). The accurate fitting of the scaling relation, however, depends on whether
the future high redshift observations that cover the stellar mass
$M_{\ast}\sim 10^{11.5}\ M_{\odot}$ detect a pervasive presence of AGN in the
young galaxies and the turning point in the scaling relation.

The first possible evidence of EDE in this paper was found in the luminosity
function. The prediction of it depends on the stellar mass function, the
scaling relation between $M_{\ast}$ and $M_{\text{UV}}$, and the dust
attenuation calibration rule. We applied the scaling relation obtained by
Stefanon et al. (2021) for the consistency with the stellar mass function and
the upper limit of the redshift. The dust attenuation calibration was adapted
from the recent JWST observation. In addition, the change in cosmology also
affects the distance modulus and consequently, the observational result of
$M_{\text{UV}}$. Our calculation showed that EDE will cause a slightly fainter
result by $\sim 0.15$ mag. Finally, Our result in Figure 6 suggests that the
$\Lambda$CDM model is favoured by the observational results at $z\sim 8-9$,
while the $z\sim 11-12$ shows a preference for EDE. The $z\sim 10$ appears to
be a transition era of the two models. We also found that the AGN-free SHMR in
Figure 4 does not have a strong effect on the luminosity function because most
of the data are below the turning point extrapolated from the stellar mass
function, according to the given $M_{\ast}-M_{\text{UV}}$ relation. Given that
$z\sim 10$ is indeed the boundary between the preference of the EDE and
$\Lambda$CDM model, we speculate that it may imply an efficient baryonic
suppression mechanism of the galaxy formation around this epoch, such that the
early galaxy formation rate that follows the EDE model is slowed down and
converted into the $\Lambda$CDM Universe.

The other plausible evidence of EDE appears in the stellar mass density shown
in Figure 7. We found that the denser Universe produced by EDE can fit the
JWST result observed by Labbé et al. (2023) better than the $\Lambda$CDM case.
In particular, the AGN-free SHMR relation can further promote the consistency
between the EDE prediction and the JWST data. Also, considering that the
luminosity function data from JWST at $z\sim 7-9$ does not prefer EDE while
the stellar mass density shows otherwise. It may also imply that the current
luminosity function observation from JWST is incomplete. However, the
estimation of the stellar mass itself may also be affected by EDE via, e.g.,
the lower luminosity or a more efficient gas cooling process. In our
calculation, we apply the vanilla results of the stellar mass since the impact
that the EDE model may inflict on the stellar mass estimation will (partly)
cancel out each other. Nevertheless, a detailed evaluation of it is
recommended for a more solid argumentation.

To conclude, should the luminosity function and stellar mass density obtained
by JWST be reliable, what it discovered may imply a revolutionary change in
our understanding of the galaxy formation history and/or cosmology at high
redshift. Specifically, we found that the $\Lambda$CDM model is preferred by
the stellar mass function and luminosity function when $z<10$, regardless of
the data source. In contrast, the $z\sim 10-12$ data (mainly JWST and 2
reported by pre-JWST observations) show a preference for EDE. The JWST stellar
mass density data provides extra plausible evidence of EDE at $z\sim 7-9$,
which may indicate the incompleteness of the luminosity function observation
at this redshift range. The net consequence of our comparison suggests that
the galaxy formation rate around $z\sim 7-10$ might be suppressed by an
efficient suppression mechanism which leads to the transition of the best-fit
cosmological model from EDE to $\Lambda$CDM. On top of that, considering the
consistency between the galaxy abundance and our prediction for EDE that
essentially points out a denser early stage of the Universe, it may imply that
the $S_{8}$ tension is more of the tension for the galaxy formation history
than it is for cosmology, thus the impact it may inflict on the EDE-like
models could be diminished, and a larger $S_{8}$ alone is insufficient to veto
the modifications of the cosmological theory. The hope to alleviate the
$S_{8}$ tension, therefore, lies on the observations of the Universe at
redshift $z\geq 7$.

## Acknowledgements

LW would like to thank Mauro Stefanon and Jiang, Junqian for their
constructive and timely suggestions, and Roland Timmerman for organising the
talk in Leiden where this work was first presented. LW, ZH and GY are
supported by the National Key R&D Program of China grant No. 2022YFF0503400,
2022YFF0503404 and China Manned Space Project grant No. CMS-CSST-2021-B01. GY
is also supported by the National Key R&D Program of China grant No.
2020SKA0110402, CAS Project for Young Scientists in Basic Research grant No.
YSBR-092 and China Manned Space Project grant No. CMS-CSST-2021-A01. WX is
supported by the Fundamental Research Funds for the Central Universities and
the CAS Project for Young Scientists in Basic Research grant No. YSBR-062.

## References

  * Abdalla et al. (2022) Abdalla E., et al., 2022, Journal of High Energy Astrophysics, 34, 49
  * Arrabal Haro et al. (2023) Arrabal Haro P., et al., 2023, arXiv e-prints, p. arXiv:2303.15431
  * Asgari et al. (2021) Asgari M., et al., 2021, A&A, 645, A104
  * Bagley et al. (2022) Bagley M. B., et al., 2022, arXiv e-prints, p. arXiv:2205.12980
  * Bhatawdekar et al. (2019) Bhatawdekar R., Conselice C. J., Margalef-Bentabol B., Duncan K., 2019, MNRAS, 486, 3805
  * Blas et al. (2011) Blas D., Lesgourgues J., Tram T., 2011, J. Cosmology Astropart. Phys., 2011, 034
  * Bouwens et al. (2021) Bouwens R. J., et al., 2021, AJ, 162, 47
  * Bouwens et al. (2023a) Bouwens R., Illingworth G., Oesch P., Stefanon M., Naidu R., van Leeuwen I., Magee D., 2023a, MNRAS, 523, 1009
  * Bouwens et al. (2023b) Bouwens R. J., et al., 2023b, MNRAS, 523, 1036
  * Bowler et al. (2020) Bowler R. A. A., Jarvis M. J., Dunlop J. S., McLure R. J., McLeod D. J., Adams N. J., Milvang-Jensen B., McCracken H. J., 2020, MNRAS, 493, 2059
  * Boylan-Kolchin (2023) Boylan-Kolchin M., 2023, Nature Astronomy, 7, 731
  * Bullock et al. (2000) Bullock J. S., Kravtsov A. V., Weinberg D. H., 2000, ApJ, 539, 517
  * Castellano et al. (2023) Castellano M., et al., 2023, ApJ, 948, L14
  * Cullen et al. (2023) Cullen F., et al., 2023, MNRAS, 520, 14
  * Despali et al. (2016) Despali G., Giocoli C., Angulo R. E., Tormen G., Sheth R. K., Baso G., Moscardini L., 2016, MNRAS, 456, 2486
  * Donnan et al. (2023) Donnan C. T., et al., 2023, MNRAS, 518, 6011
  * Duncan et al. (2014) Duncan K., et al., 2014, MNRAS, 444, 2960
  * Efstathiou (1992) Efstathiou G., 1992, MNRAS, 256, 43P
  * Finkelstein et al. (2022a) Finkelstein S. L., et al., 2022a, ApJ, 928, 52
  * Finkelstein et al. (2022b) Finkelstein S. L., et al., 2022b, ApJ, 940, L55
  * Goldstein et al. (2023) Goldstein S., Hill J. C., Iršič V., Sherwin B. D., 2023, arXiv e-prints, p. arXiv:2303.00746
  * Gómez-Valent et al. (2022) Gómez-Valent A., Zheng Z., Amendola L., Wetterich C., Pettorino V., 2022, Phys. Rev. D, 106, 103522
  * Harikane et al. (2023a) Harikane Y., Nakajima K., Ouchi M., Umeda H., Isobe Y., Ono Y., Xu Y., Zhang Y., 2023a, arXiv e-prints, p. arXiv:2304.06658
  * Harikane et al. (2023b) Harikane Y., et al., 2023b, ApJS, 265, 5
  * Herold et al. (2022) Herold L., Ferreira E. G. M., Komatsu E., 2022, ApJ, 929, L16
  * Hill et al. (2020) Hill J. C., McDonough E., Toomey M. W., Alexander S., 2020, Phys. Rev. D, 102, 043507
  * Hill et al. (2022) Hill J. C., et al., 2022, Phys. Rev. D, 105, 123536
  * Hogg et al. (2002) Hogg D. W., Baldry I. K., Blanton M. R., Eisenstein D. J., 2002, arXiv e-prints, pp astro–ph/0210394
  * Jiang & Piao (2022) Jiang J.-Q., Piao Y.-S., 2022, Phys. Rev. D, 105, 103514
  * Kamionkowski & Riess (2023) Kamionkowski M., Riess A. G., 2023, Annual Review of Nuclear and Particle Science, 73, 153
  * Karwal & Kamionkowski (2016) Karwal T., Kamionkowski M., 2016, Phys. Rev. D, 94, 103523
  * Kikuchihara et al. (2020) Kikuchihara S., et al., 2020, ApJ, 893, 60
  * Klypin et al. (2021) Klypin A., et al., 2021, MNRAS, 504, 769
  * Labbé et al. (2023) Labbé I., et al., 2023, Nature, 616, 266
  * Larson et al. (2023) Larson R. L., et al., 2023, ApJ, 953, L29
  * Lin et al. (2020) Lin M.-X., Hu W., Raveri M., 2020, Phys. Rev. D, 102, 123523
  * Mason et al. (2023) Mason C. A., Trenti M., Treu T., 2023, MNRAS, 521, 497
  * McDonough & Scalisi (2022) McDonough E., Scalisi M., 2022, arXiv e-prints, p. arXiv:2209.00011
  * McDonough et al. (2023) McDonough E., Hill J. C., Ivanov M. M., La Posta A., Toomey M. W., 2023, arXiv e-prints, p. arXiv:2310.19899
  * Meurer et al. (1999) Meurer G. R., Heckman T. M., Calzetti D., 1999, ApJ, 521, 64
  * Morishita & Stiavelli (2023) Morishita T., Stiavelli M., 2023, ApJ, 946, L35
  * Morishita et al. (2018) Morishita T., et al., 2018, ApJ, 867, 150
  * Murgia et al. (2021) Murgia R., Abellán G. F., Poulin V., 2021, Phys. Rev. D, 103, 063502
  * Murray et al. (2013) Murray S. G., Power C., Robotham A. S. G., 2013, Astronomy and Computing, 3, 23
  * Murray et al. (2021) Murray S. G., Diemer B., Chen Z., Neuhold A. G., Schnapp M. A., Peruzzi T., Blevins D., Engelman T., 2021, Astronomy and Computing, 36, 100487
  * Naidu et al. (2022) Naidu R. P., et al., 2022, ApJ, 940, L14
  * Oesch et al. (2014) Oesch P. A., et al., 2014, ApJ, 786, 108
  * Oesch et al. (2018) Oesch P. A., Bouwens R. J., Illingworth G. D., Labbé I., Stefanon M., 2018, ApJ, 855, 105
  * Pérez-González et al. (2023) Pérez-González P. G., et al., 2023, ApJ, 951, L1
  * Perlmutter et al. (1999) Perlmutter S., et al., 1999, ApJ, 517, 565
  * Planck Collaboration et al. (2020a) Planck Collaboration et al., 2020a, A&A, 641, A6
  * Planck Collaboration et al. (2020b) Planck Collaboration et al., 2020b, A&A, 641, A8
  * Poulin et al. (2018) Poulin V., Smith T. L., Grin D., Karwal T., Kamionkowski M., 2018, Phys. Rev. D, 98, 083525
  * Poulin et al. (2019) Poulin V., Smith T. L., Karwal T., Kamionkowski M., 2019, Phys. Rev. Lett., 122, 221301
  * Poulin et al. (2021) Poulin V., Smith T. L., Bartlett A., 2021, Phys. Rev. D, 104, 123550
  * Poulin et al. (2023) Poulin V., Smith T. L., Karwal T., 2023, arXiv e-prints, p. arXiv:2302.09032
  * Reeves et al. (2023) Reeves A., Herold L., Vagnozzi S., Sherwin B. D., Ferreira E. G. M., 2023, MNRAS, 520, 3688
  * Riess et al. (1998) Riess A. G., et al., 1998, AJ, 116, 1009
  * Riess et al. (2022) Riess A. G., et al., 2022, ApJ, 934, L7
  * Sabti et al. (2022) Sabti N., Muñoz J. B., Blas D., 2022, Phys. Rev. D, 105, 043518
  * Scholtz et al. (2023) Scholtz J., et al., 2023, arXiv e-prints, p. arXiv:2306.09142
  * Schöneberg et al. (2022) Schöneberg N., Abellán G. F., Sánchez A. P., Witte S. J., Poulin V., Lesgourgues J., 2022, Phys. Rep., 984, 1
  * Sheth & Tormen (1999) Sheth R. K., Tormen G., 1999, MNRAS, 308, 119
  * Simon et al. (2023) Simon T., Zhang P., Poulin V., Smith T. L., 2023, Phys. Rev. D, 107, 063505
  * Simpson et al. (2018) Simpson C. M., Grand R. J. J., Gómez F. A., Marinacci F., Pakmor R., Springel V., Campbell D. J. R., Frenk C. S., 2018, MNRAS, 478, 548
  * Smith et al. (2020) Smith T. L., Poulin V., Amin M. A., 2020, Phys. Rev. D, 101, 063523
  * Smith et al. (2022a) Smith T. L., Poulin V., Simon T., 2022a, arXiv e-prints, p. arXiv:2208.12992
  * Smith et al. (2022b) Smith T. L., Lucca M., Poulin V., Abellan G. F., Balkenhol L., Benabed K., Galli S., Murgia R., 2022b, Phys. Rev. D, 106, 043526
  * Song et al. (2016) Song M., et al., 2016, ApJ, 825, 5
  * Stark et al. (2013) Stark D. P., Schenker M. A., Ellis R., Robertson B., McLure R., Dunlop J., 2013, ApJ, 763, 129
  * Stefanon et al. (2019) Stefanon M., et al., 2019, ApJ, 883, 99
  * Stefanon et al. (2021) Stefanon M., Bouwens R. J., Labbé I., Illingworth G. D., Gonzalez V., Oesch P. A., 2021, ApJ, 922, 29
  * Steinhardt et al. (2023) Steinhardt C. L., Kokorev V., Rusakov V., Garcia E., Sneppen A., 2023, ApJ, 951, L40
  * Sun et al. (2023) Sun L., et al., 2023, arXiv e-prints, p. arXiv:2311.15664
  * Tacchella et al. (2018) Tacchella S., Bose S., Conroy C., Eisenstein D. J., Johnson B. D., 2018, ApJ, 868, 92
  * Wechsler & Tinker (2018) Wechsler R. H., Tinker J. L., 2018, ARA&A, 56, 435