galaxies are found to have high stellar masses in our SED-fitting, with a
median stellar mass of $\log_{10}M_{\star}/M_{\odot}$ = 10.30 in our fiducial
Bagpipes results, with five of them forming the most massive galaxies in our
fiducial sample. It is likely that at least some of these objects do contain
AGN, which would lower the inferred stellar masses and hence lower the high-
mass end of the GSMF. Further observations with NIRSpec or MIRI are essential
to ascertain the true nature of these sources. In Appendix A we recompute the
$z=7$ GSMF without any LRDs and show that these sources dominate the high mass
end of the GSMF. D’Silva et al. (2023) has shown that accounting for the
contribution of AGN lowers the cosmic star formation rate density by 0.4 dex
at $z\geq 9.5$, which will also lower the inferred stellar mass density, and
finds that a significant fraction of the LRDs are hidden AGN.

###  6.4 Impact of Modified IMF

Our implementation of a modified top-heavy IMF in Prospector has been shown to
reduce masses by up to $\sim$0.5 dex for galaxies at $z\geq$12 (HOT 60K) when
compared to a standard Kroupa (2001) IMF, as shown in Figure 5. The decrease
in mass seen is also found to be dependent on the star formation model, with a
larger decrease in stellar mass found when assuming a parametric “delayed” SFH
model (0.46 dex) than when assuming a non-parametric “continuity bursty” SFH
model (0.35 dex). Comparison of the best-fitting models has also shown that
the $\chi^{2}$ is almost unaffected by the change in IMF, suggesting the
modified IMF models can match the observed photometry of the galaxies as well
as the original data. We see similar results, with smaller decreases in
stellar mass ($\approx$0.3 dex), for the modified IMF model used at 8$\leq
z\leq 12$ (HOT 45K).

There are a number of comparable studies which also look at the impact of a
top-heavy IMF on the masses of high redshift galaxies. The most direct
comparison can be made to the results of Steinhardt et al. (2023) as we have
used the same top-heavy IMF model with a different SED fitting tool. Their
results are based on some of the earliest JWST observations and they only fit
a small number of galaxies. Direct comparison of their results to the default
EAZY-py fsps templates is also impacted by other differences between the two
template sets, including the re-parametrization of the IMF from Chabrier et
al. (2000) to Kroupa (2001) and different assumptions for SFH and nebular
emission. Our implementation changes only the IMF, with everything else
modelled in the same way. They observe decreases of $0.5-1$ dex in stellar
mass, whereas our implementation sees smaller decreases, typically from
$0.3-0.5$ dex. The reason for this discrepancy is unclear, but it may be
possible that the other differences between the standard EAZY-py templates and
their models have a larger effect on the stellar mass than expected, or that
the change in IMF in our modelling resulted in different star formation
histories that acted to somewhat decrease the impact of the modified IMF.

Woodrum et al. (2023) also look at other top-heavy modifications to commonly
used IMF parametrizations using Prospector. They look at a modification to the
Chabrier et al. (2000) IMF, rather than the Kroupa (2001) we use, but they
find comparable reductions in stellar mass (between 0.38 and 0.5 dex). They
also see no change in the goodness of fit when using a modified IMF.

Whilst we have shown that a modified top-heavy IMF can decrease the stellar
masses of high-$z$ galaxies, with no change in the simulated photometry, our
analysis of the $\Lambda$CDM limits on the growth of stellar mass in § 6.1
does not require a non-standard IMF in order to be compatible with
$\Lambda$CDM. In Figure 1 we have shown that the stellar masses can vary
significantly with other SED fitting assumptions (dust law, assumed SFH),
before any IMF change is considered. For the galaxies with high stellar masses
across all the models we consider, it is not possible to distinguish between a
modified IMF or simply a high star formation efficiency based on photometry
alone.

###  6.5 Comparing the measured GSMF with other observations and
theory/simulations

In Figure 8, we compare our GSMF estimates to a wide-range of observational
and theoretical/simulation based predictions of the GSMF. Here we briefly
discuss our GSMF estimates for each of the redshift bins. In order to make
direct comparisons where necessary all results have been converted to use a
Kroupa (2001) IMF.

The overall evolution of the derived Schechter parameters and a comparison to
the results derived by other studies can be seen in Figure 10. Whilst the
Schechter parameters are highly covariant, and our results typically have
large uncertainties, we observe an evolution in $\alpha$ and $\phi^{\star}$
compared, with both parameters decreasing compared to the results at $z\sim 4$
of Caputi et al. (2015), Duncan et al. (2014) and Grazian et al. (2015). We
see little evolution of M⋆ within the large range of uncertainties.

####  6.5.1 Redshift z=7 GSMF

We derive a mass function at $z\sim 7$ primarily as a proof of concept of our
method. We do not take advantage of galaxy lensing in this work, so any
reasonable mass completeness limit is higher than previous studies, nor do we
have the area of wide-field studies like Weaver et al. (2022) in order to
detect rare, bright and high mass galaxies. However with JWST we have seen a
surprising excess of UV-faint LRD like objects with high inferred stellar
masses, as discussed in § 6.3. The majority of these sources were previously
undetected with HST due to relatively weak Lyman-breaks, and so do not appear
in pre-JWST stellar mass estimates. Their inclusion in our GSMF has resulted
in an excess at the high mass end of the GSMF when compared to other
observational studies, and consequently a higher and poorly constrained
estimate of M⋆, as we see little evidence for any exponential turnover. The
highest mass GSMF data-points of Weaver et al. (2022) fall within our
1$-\sigma$ uncertainty region, but our results are significantly above the
measurements of Stefanon et al. (2021). At the low-mass end we fall below the
results of Kikuchihara et al. (2020) and Navarro-Carrera et al. (2023), but
agree within the uncertainties of Furtak et al. (2021) and Bhatawdekar et al.
(2019). Stefanon et al. (2021); Kikuchihara et al. (2020); Bhatawdekar et al.
(2019) and Furtak et al. (2021) are all based on HST+Spitzer observations of
the Hubble Frontier Fields, and incorporate lensing, which means they probe
the low mass end of the GSMF more accurately then this study. Our low mass
slope $\alpha=-1.94^{+0.1}_{-0.1}$ is in good agreement with Furtak et al.
(2021), but steeper than the results Stefanon et al. (2021) and Kikuchihara et
al. (2020). At the time of writing, Navarro-Carrera et al. (2023),
Gottumukkala et al. (2023) and Wang et al. (2024) are the only other studies
to incorporate JWST observations into their GSMF estimates. We see reasonable
agreement with the results of Navarro-Carrera et al. (2023) as our data is
within their GSMF uncertainties. This work relies almost entirely on JWST
observations, whereas they combine deep JWST observations of small volumes
($\leq$20 arcmin2) with HST and ground-based catalogues. This ground-based
data allows them to find more rare, high-mass galaxies than our study, but at
lower masses the small volumes probed with their JWST data are potentially
vulnerable to cosmic variance. Reliance on ground-based and HST data also
limits the maximum redshift they can probe to $z\leq 8$. Wang et al. (2024)
uses PRIMER observations with NIRCam and MIRI to measure the GSMF, notably
finding that the use of MIRI observations systematically reduces stellar
masses measured with SED fitting. We see good agreement in the measured GSMF
within the uncertainties of both studies, despite this work not incorporating
MIRI data or correcting for any systematic offset in mass arising from the
lack of restframe $>1\mu$m observations. Our GSMF does extend to higher
stellar mass than the result of Wang et al. (2024), resulting in a higher
value for $M^{\star}$, as can be seen in Figure 10.

Gottumukkala et al. (2023) examine the contribution of high-mass, dusty
galaxies at $3<z<8$ to the GSMF using data from the CEERS survey. Given that
our GSMF probes a wider galaxy population, we do not expect to see overlap at
all stellar masses. We see good overlap at the highest stellar masses
M${}_{\odot}\sim 10^{10.5}$, where our SMF estimate is dominated by dusty LRD
galaxies (as discussed in § 6.3).

When we compare to predictions from models and simulations, we see agreement
with the majority of models at the low-mass end but a significant excess at
higher masses that is not reproduced by any of the models. We find in
particular that the JAGUAR model we use for our completeness simulations shows
a more rapid decline at high stellar mass than the other models, but given
that we are not reliant on our completeness correction in this mass regime
this does not impact our estimate of the GSMF. We are closest to the
prediction of _Universe Machine_ (Behroozi et al., 2019) at the highest
stellar mass bin.

####  6.5.2 Redshift z=8 GSMF

Our fiducial GSMF estimate at $z\sim 8$ shows reasonable agreement with most
predictions. As we do not bootstrap in redshift when constructing the GSMF, we
do not account for galaxies scattering between redshift bins, and for example
a galaxy found to be at $z=7.49$ with Bagpipes would contribute only to the
$z=7$ GSMF, even in a significant fraction of the redshift PDF is above $z\geq
7.50$. This does not affect the majority of galaxies within our sample, but it
does explain some of the discrepancy between our results and the implied
results of Labbé et al. (2023), shown in purple in Figure 8 assuming 100%
completeness. The LRD galaxies of Labbé et al. (2023) which we also select are
all found to be at $z\leq 7.5$, meaning they do not contribute at all to our
estimate of the $z=8$ GSMF. The best-fitting redshift for these objects in
some cases is quite close to this boundary however, meaning that these objects
could theoretically contribute to the $z=8$ GSMF instead, which would boost
the high-mass end significantly. Our GSMF is also lower than the results of
Kikuchihara et al. (2020), which incorporates strong gravitational lensing in
order to probe to lower stellar mass. Our GSMF agrees with the results of Song
et al. (2016); Bhatawdekar et al. (2019); Stefanon et al. (2021), and appears
to validate the majority of pre-JWST GSMF estimates. We can also draw a
comparison to the results of Wang et al. (2024), whose GSMF results at $z=8$
are systematically above our results, but within the derived uncertainties of
both studies.

We see good agreement with most theoretical predictions of the GSMF at this
redshift, with _Universe Machine_ (Behroozi et al., 2019), _SC SAM GUREFT_
(Yung et al., 2023) and _FLARES_ (Lovell et al., 2021; Wilkins et al., 2023b)
having the most similar results.

####  6.5.3 Redshift z=9 GSMF

Our GSMF estimate at $z\sim 9$ is below the results of Bhatawdekar et al.
(2019) and Kikuchihara et al. (2020), but within the uncertainties of Stefanon
et al. (2021). We are below the implied result of Labbé et al. (2023), which
is derived from two galaxies in their sample in this redshift bin, but
assuming 100% completeness. We include one of these galaxies in our GSMF at
this redshift, as the other does not meet our selection criteria, as we do not
detect the Lyman break at 5$\sigma$. We additionally include one other
candidate from their sample in this GSMF, as our fiducial Bagpipes photo-$z$
places it within this redshift bin, rather than the $z\sim 8$ redshift bin
based on their photo-$z$. For both of their galaxies we do include in this
redshift bin we find $\sim 0.4$ dex lower stellar masses, meaning they
contribute to a lower stellar mass bin. Our reliance on the rest-frame UV to
robustly detect sources is one limitation of this work, although the increased
depth of JWST observations when compared to HST has reduced this in some
fields. We investigated less-stringent constraints on the Lyman-break, but
found that this dramatically increased rates of contamination within our
sample. When compared to simulations, _FLARES_ and _Universe Machine_ are
close to our GSMF estimate, but almost all the predictions are within our
posterior region. Interestingly, in this redshift bin we are below the
predictions of two recent JWST-era studies; Mauerhofer & Dayal (2023) and Li
et al. (2023), both of which incorporate higher star formation efficiencies
than typical models.

####  6.5.4 Redshift z=10.5 GSMF

At $z\sim 10.5$ observational comparisons can be made only to the pre-JWST
results of Stefanon et al. (2021). In comparison to their results we find a
significant excess of high mass galaxies in our observations. Our results show
that above $z\geq 10$ JWST observations are essential to accurately sample the
high-$z$ galaxy population. Our results are above the majority of theoretical
and simulation-derived predictions, but do show good agreement with _Universe
Machine_ (Behroozi et al., 2019) and _FLARES_ (Lovell et al., 2021; Wilkins et
al., 2023b).

####  6.5.5 Redshift z=12.5 GSMF

In our highest redshift bin, 11.5 $<z\leq 13.5$ which covers only $\approx$ 80
Myr, there are no published observationally-derived results and few
theoretical or simulation-based GSMF comparisons at this redshift. Pre-JWST
estimates of the GSMF were not possible at this redshift, and even with JWST
our GSMF estimate is also uncertain due to the difficulty in accurate stellar
mass estimates as well as the possible contribution of contaminants. At
$z=12.5$ the longest wavelength NIRCam filter falls within the rest-frame UV,
which is dominated by young stars, leading to highly uncertain star formation
histories and stellar masses. As we explored briefly in § 4.5, the more likely
possibility of a top-heavy IMF or exotic stellar populations in these early
galaxies further increases the systematic uncertainties in the stellar mass
estimates. An example of three galaxies within this redshift bin is shown in
the lower plot of Figure 3 and in Figure 4, and the range of stellar mass
estimates ($\sim 0.8-1dex$) for different Bagpipes and Prospector with very
little difference in the fitted rest-UV spectra shows the difficulty in
estimating stellar mass at these redshift. Previous studies at lower redshift
with HST and Spitzer have found that stellar masses estimated by HST alone,
with no measurement of the rest optical emission, typically underestimate
stellar masses by 0.62 dex, compared to measurements including HST and Spitzer
NIR observations (Furtak et al., 2021). At this redshift range, our JWST
NIRCam observations are probing comparable rest-frame UV wavelengths to HST
observations at $z\sim 6-7$, and it possible that our stellar masses are also
underestimated unless there is a significant change in stellar populations or
IMF. The possibility of more stochastic star formation histories at this
redshift compared to $z\sim 6-7$ may also lead to outshining, which further
increases the stellar mass discrepancy (Narayanan et al., 2023).

We note that several galaxies in this bin have been excluded from the GSMF in
this case due to our requirement that the contamination is less than 50%. The
inclusion of these possible contaminants would result in an $\approx$ 0.3 dex
increase in the lowest mass bin. The results of this are shown in Appendix C.
Whilst we do not attempt to fit the GSMF, we can make approximate comparisons
to the few available predictions. We see the closest agreement with _DELPHI_
(Mauerhofer & Dayal, 2023) and are within 1.5$\sigma$ of _FLARES_ (Lovell et
al., 2021; Wilkins et al., 2023b) in the higher mass bin, but have an excess
of $\approx 10^{8}$ M⊙ galaxies when comparing to _FLARES_ and _SC SAM GUREFT_
(Yung et al., 2023). Our results are significantly above the predictions of
_BLUETIDES_ (Feng et al., 2016; Wilkins et al., 2017) at all stellar masses.
Our fiducial GSMF prediction is slightly below the prediction of Li et al.
(2023) with maximum star formation efficiency $\epsilon_{\rm max}$ = 0.2, and
significantly below the upper limit of $\epsilon_{\rm max}$ = 1.

####  6.5.6 Alternative GSMF Estimates

Following on from the comparison of galaxy stellar mass estimates with
different choices of SFH and priors in § 4.3, it is possible to estimate the
GSMF for any of the different SED-fitting models. A full comparison of the
derived GSMF for every model, given the many possible combinations of possible
IMF and SFH models, is beyond the scope of this work, but we give a
representative example of the GSMF derived at z$\sim$10.5 for the models which
show the most variation in stellar mass when compared to our fiducial Bagpipes
fitting. Here we choose to investigate the GSMF dependence on the chosen SFH
model and SED fitting tool, rather than the choice of parameter prior or dust
law. This is because in § 4.3, there was larger variation in stellar mass with
little variation in $\chi^{2}$ for these alternative models. Additionally, as
discussed, the use of non-parametric SFHs is more common in the literature
(e.g. Tacchella et al., 2022; Giménez-Arteaga et al., 2023; Jain et al., 2024;
Giménez-Arteaga et al., 2024) for high-$z$ galaxies due to problems such as
outshining.

Figure 13: GSMF Schechter parametrization for the $z\sim$10.5 redshift bin
derived for our fiducial Bagpipes SED-fitting compared to alternative GSMF
estimates. We show the GSMF derived using Bagpipes with the non-parametric
“continuity bursty” SFH (labelled ‘bursty’) as well as for two GSMFs derived
using Prospector SED-fitting for a parametric and non-parametric SFH. We also
show for comparison the GSMF inferred with the alternative top-heavy IMF
model. The derived GSMF is clearly dependent on the choice of model and SED
fitting tool, with the “continuity bursty” SFH model typically shifting the
GSMF towards higher stellar mass. These Schechter functions are tabulated in
Appendix B.

We derive these GSMF estimates using the same method as described in § 5 for
the fiducial Bagpipes GSMF, replacing the stellar mass PDFs, best-fitting SEDs
and redshift estimates with those of the chosen model. Figure 13 shows a
comparison of the fiducial Bagpipes GSMF to a GSMF derived from the
“continuity bursty” non-parametric SFH model, which increases the stellar mass
estimates by 0.2 dex on average, but $\geq 1$ dex in some cases. This results
in the largest change in the overall GSMF when compared to the fiducial
Bagpipes result.

We also show mass functions derived from our Prospector SED-fitting, which are
offset above our fiducial Bagpipes GSMF for both the parametric and non-
parametric results. These results are somewhat comparable to the spread seen
in the Bagpipes results, although the low-mass end of the GSMF in the
“continuity bursty” SFH model produces a shallower slope than the other
models. Crucially we can see that the derived GSMFs are in tension with each
other, and do not typically fall within the confidence intervals across the
majority of the stellar mass range. This is consistent with Wang et al.
(2023b), who argue that the stellar mass uncertainties are typically
underestimated by SED fitting procedures.

The change in inferred stellar mass we observe with a modified IMF does not
appear to vary strongly with stellar mass, so the the impact on the GSMF can
generally be seen as a shift towards lower stellar mass of $0.3-0.4$ dex. This
is comparable in magnitude and opposite in direction to the shift seen when
moving from Bagpipes to Prospector when using a parametric SFH, which results
in little overall change in the resulting GSMF.

Resolved SED fitting from Giménez-Arteaga et al. (2023) has found that the
higher stellar masses inferred by the non-parametric SFHs can better account
for the out-shining of older stellar populations in bright, actively star
forming galaxies. How widespread the issue of outshining is among our sample
is unknown, but if it is common then the masses and resulting GSMF of the
“continuity bursty” SFH model may more accurately recover the true stellar
populations of these early galaxies.

These results demonstrate the overall systematic uncertainty different
assumptions cause in the GSMF which are not represented by the uncertainty
contours. Most GSMF estimates do not consider the overall uncertainty
introduced by the assumptions of their modelling, which often dwarfs the
statistical uncertainty in the fit itself. The variation in the derived GSMF
can also significantly impact the implied SMD, as discussed in the next
section.

###  6.6 Stellar Mass Density Evolution in the Early Universe

The growth of stellar mass density in the early Universe is highly uncertain.
Some observational studies (e.g. Oesch et al., 2014; Stefanon et al., 2021;
Willott et al., 2023a) have found a sharp decline in stellar mass density at
$z\geq$ 8, whereas others see a flatter evolution (Kikuchihara et al., 2020;
Bhatawdekar et al., 2019). On the theoretical side, Oesch et al. (2018) uses
dark matter halo evolutionary models to predict a deviation from the constant
star formation efficiency (CSFE) model of Madau & Haardt (2015), which follows
a significantly steeper slope at $z\geq$7.

Our results from our fiducial Bagpipes model fall between the predictions of
Madau & Haardt (2015) and Oesch et al. (2018). We see a flatter evolution with
redshift than predicted with (Oesch et al., 2018), but an overall lower
stellar mass density than the CFSE model of Madau & Haardt (2015). For our
other GSMF estimates at $z=10.5$, shown in Figure 13, we find that
$\rho_{\star}$ increases by up to 0.75 dex, which would bring it closer to the
constant star formation efficiency prediction of Madau & Haardt (2015). Whilst
we do not show the SMD scatter measured in other redshift bins, we typically
see the same behaviour at $z>7$, with our fiducial Bagpipes SMD result
producing lower $\rho_{\star}$ estimates than our alternative models. With our
fiducial Bagpipes results we see significant evolution of the GSMF between
$z=7$ and $z=8$, with $\rho_{\star}$ decreasing by $\sim 0.85$ dex. However we
see a significantly flatter evolution in the SMD derived from the “continuity
bursty” model GSMF, with a decrease of only $\sim 0.4$ dex across the same
redshift range. This is due partly to the overall increase in stellar mass
estimates observed with this SFH model when compared to our fiducial model, as
detailed in § 4.3, but is also due to the scattering of the high-mass LRD
galaxies scattering between the $z=7$ and $z=8$ redshift bins due to uncertain
photo-$z$ estimates, which significantly impacts the GSMF at higher stellar
masses.

We see a good agreement between the integration of the star formation rate
density of Adams et al. (2023), which uses the same sample, and our fiducial
SMD results. There are very few JWST-era GSMF estimates to directly compare
against, and so we have computed the inferred stellar mass density based on
the integral of the cosmic star formation rate density of other studies. We
note however the numerous works showing the increased scatter in mass-to-light
ratios observed due to bursty star formation (Santini et al., 2022; Asada et
al., 2023), which will impact the assumptions made to convert these UV
luminosity densities into stellar mass densities.

In Figure 11 we show the overall stellar mass density range we find when we
use a different SED fitting tool or star formation history model (dotted red
uncertainty). This is significantly larger than the statistical uncertainty in
the stellar mass density from our fiducial Bagpipes results. A change of up to
0.75 dex at $z\approx 10.5$ is possible, when only the SED fitting tool or SFH
model is varied and the overall sample is unchanged. More significant
variations are possible between the results of independent studies, which also
have to consider differences in reduction, source detection, photo-$z$
estimation, selection procedure, cosmic variance, and completeness
corrections. Not accounting for the contribution of AGN to the observed
photometry may cause overestimation of the stellar mass density at high-
redshift (D’Silva et al., 2023).

The range of stellar mass densities possible with our alternative GSMF
estimates at $z\sim 10.5$ is mostly above the 1$\sigma$ range of Adams et al.
(2023). A discrepancy between the integrated star formation rate density and
stellar mass density measured for the same sample could hint at a different
IMF, since the assumed return fraction is strongly dependent on the chosen
IMF, and the SMF and UVLF probe different stellar populations with different
characteristic stellar mass. However there are a number of other possible
issues with the conversion of the UVLF into a SMD estimate; the conversion of
UV flux to SFR ($\kappa_{\rm UV}$), is often assumed to be a constant factor
but actually dependent on the age and metallicity Madau & Dickinson (2014), as
well as the other assumptions used to calculate the return fraction (closed-
box model, constant IMF & metal yield and instantaneous recycling of metals)
which may not be valid at high-redshift.

As we show in § 4.3, in some cases discrimination between models or priors
based on the goodness of fit may be possible, but in others (e.g. assumed SFH
model), significant scatter in stellar mass estimates are possible with no
difference in $\chi^{2}$. Other studies which use only one method for
measuring stellar mass estimates will underestimate the overall uncertainty in
the derived GSMF and stellar mass density estimates.

##  7 Conclusions

In this paper we present an investigation into the properties of the EPOCHS v1
sample of 1120 high-redshift galaxies at 6.5 $\leq z\leq 13.5$ taken from a
uniform reduction of 187 arcmin2 of JWST data, including the GTO program
PEARLS as well as other public ERS/GO JWST programs.

We examine the consistency of galaxy properties, including stellar mass, under
different assumptions and using different SED fitting tools, including
Bagpipes and Prospector. In particular we examine the impact of different SFH
parametrizations as well as switching between a parametric and non-parametric
SFH models. We also also investigate the possible reduction in stellar mass
when assuming a top-heavy IMF. We then use this sample and our range of
stellar mass estimates to construct possible realisations of the galaxy
stellar mass function. Lastly we integrate our mass function estimates to
probe the buildup of stellar mass in the early Universe via the stellar mass
density.

The major conclusions from this study are as follows:

1\. We find that the stellar mass of high-redshift galaxies can depend
strongly on assumed models, their priors and the SED fitting package used. In
particular the estimated stellar mass can increase by $>$1 dex when a
parametric SFH is exchanged for a non-parametric SFH, with no change in the
goodness of fit. Higher stellar mass discrepancies are seen at $z>10$ due to a
lack of rest-optical emission.

2\. We find that the assumption of a modified top-heavy Kroupa (2001) IMF,
which may more accurately model the hot star-forming regions within high-$z$
galaxies, can reduce stellar mass estimates by up to 0.5 dex with no impact on
the goodness of fit.

3\. Whilst some of the stellar mass estimates imply a high star formation
efficiency, in our analysis of the most massive galaxies in our sample using
the Extreme-Value Statistics methodology of Lovell et al. (2023) we do not
find any galaxies which are incompatible with the existing $\Lambda$CDM
cosmology. The largest stellar mass estimates are typically found when fitting
the non-parametric SFH models, and often can be significantly reduced with an
alternative model. We not require a top-heavy IMF

4\. Across all of the fitted models, the highest mass galaxies in our sample
are ‘Little Red Dots’, with inferred masses of $>$1010 M⊙ at $z\approx 7$.
These galaxies dominate the highest mass bins of our galaxy stellar mass
function (GSMF) estimates, so understanding their true stellar populations and
accounting for the likely contribution of AGN (Greene et al., 2023; Furtak et
al., 2023, 2023) will be essential to more accurately constrain further GSMF
estimates.

5\. With the GSMF derived from our fiducial Bagpipes results, we typically see
good agreement with existing constraints on the GSMF at z$\leq$9.5. At the
limits of HST+Spitzer (z$\geq$10) we see an excess of galaxies when compared
to pre-JWST observations, but our GSMF results fall within predictions of
simulations and theory.

5\. The systematic variation in stellar mass estimates we find can
dramatically impact the inferred galaxy stellar mass function and therefore
the stellar mass density. We show that the choice of star formation history
model or SED fitting tool can cause up to a 0.75 dex shift in the overall
stellar mass density at z$\approx$10.5 with the same sample of galaxies. We
predict larger offsets between independent samples, where different
reductions, selection techniques and photo-$z$ estimates will increase the
uncertainties.

6\. We see a flatter evolution of the cumulative stellar mass density than
predicted by dark matter halo evolution models, whilst the slope of our
results is more consistent with constant star formation efficiency models. Our
results suggest that significant stellar mass had already formed at
$z\geq$11.5.

This is only the beginning of GSMF estimates at $z>10$, and the use of ultra-
deep observations (the second NGDEEP epoch (Bagley et al., 2023a), the JADES
Origins Field (Eisenstein et al., 2023b; Robertson et al., 2023a) and others)
and magnification by lensing clusters (PEARLS, UNCOVER, CANUCS; Windhorst et
al., 2023; Bezanson et al., 2022; Willott et al., 2023b) will help constrain
the GSMF at stellar masses below our completeness limit of $\sim 10^{8}$ M⊙,
whilst widefield surveys (e.g. PRIMER, UNCOVER, Cosmos-Webb; Dunlop et al.,
2021; Bezanson et al., 2022; Kartaltepe et al., 2021) will add area and rare
higher-mass sources. Deep MIRI F560W or F770W observations (e.g. the MIRI HUDF
survey, Norgaard-Nielsen & Perez-Gonzalez, 2017) will be crucial to provide
better constraints on stellar mass estimates at these redshifts by extending
the wavelength range further into the rest-frame optical, although the
sensitivity of MIRI decreases rapidly with increasing wavelength (e.g. Wang et
al., 2024). More complete NIRSpec coverage is also important to identify
interlopers, confirm photometric redshifts and distinguish between AGN
emission and star forming galaxies.

All of the raw JWST data used in this work are the same as used in Adams et
al. (2023) and can be accessed via this MAST DOI: DOI 10.17909/5h64-g193. All
proprietary data from the PEARLS program will all become accessible over 2024.
Catalogues for all high-$z$ galaxies will be published with the EPOCHS I paper
(Conselice et. al., in prep). The fiducial GSMF and SMD results from this work
are availale on GitHub, and results for our alternative models will be made
available upon request.

TH, CC, NA, DA, QL, JT, LW acknowledge support from the ERC Advanced
Investigator Grant EPOCHS (788113), as well as two studentships from the STFC.
AZ acknowledges support by Grant No. 2020750 from the United States-Israel
Binational Science Foundation (BSF) and Grant No. 2109066 from the United
States National Science Foundation (NSF); by the Ministry of Science &
Technology, Israel; and by the Israel Science Foundation Grant No. 864/23.
RAW, SHC, and RAJ acknowledge support from NASA JWST Interdisciplinary
Scientist grants NAG5-12460, NNX14AN10G and 80NSSC18K0200 from GSFC. CCL
acknowledges support from the Royal Society under grant RGF/EA/181016. The
Cosmic Dawn Center (DAWN) is funded by the Danish National Research Foundation
under grant No. 140. CNAW acknowledges funding from the JWST/NIRCam contract
NASS-0215 to the University of Arizona. M.N. acknowledges INAF-Mainstreams
1.05.01.86.20. CNAW acknowledges support from the NIRCam Development Contract
NAS5-02105 from NASA Goddard Space Flight Center to the University of Arizona.
This work is based on observations made with the NASA/ESA Hubble Space
Telescope (HST) and NASA/ESA/CSA James Webb Space Telescope (JWST) obtained
from the Mikulski Archive for Space Telescopes (MAST) at the Space Telescope
Science Institute (STScI), which is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS 5-03127
for JWST, and NAS 5–26555 for HST. The authors thank all involved with the
construction and operation of JWST, without whom this work would not be
possible. We also thank the PI’s and teams who designed and executed the ERS,
GTO and GO programs used within this work, including PEARLS (1176, 2738),
SMACS-0723 (2737), GLASS (1324), CEERS (1345), JADES (1180, 1210, 1895, 1963)
and NGDEEP (2079). This work makes use of astropy (Astropy Collaboration et
al., 2013, 2018, 2022b), matplotlib (Hunter, 2007), reproject, DrizzlePac
(Hoffmann et al., 2021), SciPy (Virtanen et al., 2020) and photutils (Bradley
et al., 2022).

##  Appendix A z = 7 Stellar Mass Function without ‘Little Red Dots’

As discussed in § 6.3, the ‘Little Red Dots’ (LRDs) dominate the high mass end
of our GSMF at $z=7$. As the contribution of AGN to their photometry is still
somewhat uncertain, and likely differs on an individual basis between
galaxies, in the main results of this paper we do not remove LRDs from the
GSMF estimates, or account for any possible AGN emission. In this Appendix we
briefly present the alternative case, where we remove all objects which meet
the color-color selection criteria of Kokorev et al. (2024) and reconstruct
the $z=7$ GSMF.

When we apply their ‘red2’ color selection, compactness criterion and SNR
requirements to our robust sample, we find 34 galaxies which meet these cuts.
There are 13 in the NEP-TDF, 17 in CEERS, 3 in the JADES DR1 field, and 1 in
the NGDEEP field. The median redshift is 7.16, with all candidates falling
between $z=6.5$ (our redshift cut) and $z=8.7$. The median fiducial Bagpipes
stellar mass is $\log_{10}M_{\star}/M_{\odot}=8.90$, with a maximum stellar
mass of $\log_{10}M_{\star}/M_{\odot}=10.70$.

Figure 14: Galaxy Stellar Mass Function at $6.5<z\leq 7.5$, excluding all
‘Little Red Dots’, compared to our fiducial Bagpipes results.

We exclude these 34 candidates from our sample and reconstruct the stellar
mass function at $z=7$. No other changes are made to our GSMF construction or
fitting procedures. Figure 14 shows the GSMF derived without including any
‘Little Red Dots’. Compared to the fiducial GSMF, this removes the two highest
mass bins entirely, which demonstrates our reliance on these galaxies at the
high mass regime. In terms of the derived Schechter parameters, the
exponential mass cutoff M⋆, which is not well constrained, decreases from
11.57${}^{+0.63}_{-0.85}$ to 10.64${}^{+1.25}_{-0.98}$ when we exclude the
LRDs. The median posterior $\phi^{\star}$ and $\alpha$ for the GSMF without
LRDs are $-5.40^{+1.34}_{-1.43}$ and $-2.04^{+0.18}_{-0.13}$ respectively.

##  Appendix B Tabulated Schechter parameters for alternative GSMF estimates
at z = 10.5

We give the Schechter function parameters for our alternative GSMF fits at
$z\sim 10.5$ in Table 6. These are the Schechter parameters representing the
fits shown in Figure 13. These GSMF estimates are equivalent to the fits given
in Table 3 for the fiducial GSMF, and are calculated using the same method,
simply replacing the redshift and stellar mass PDF used in constructing the
GSMF with those derived under the alternate SED fitting assumptions.

Table 6: Schechter function parameters for the GSMF at $9.5\leq z\leq 11.5$ for each of the alternative models shown in Figure 13. For $\alpha$, M⋆ and $\log_{10}\phi^{\star}$ we give both the median posterior and maximum likelihood values (in brackets). The details of the Bagpipes and Prospector configurations for each model are given in § 4.1 and § 4.2.  SED Fitting Tool | SFH Model | IMF | $\alpha$ | M⋆ | $\log_{10}\phi^{\star}$
---|---|---|---|---|---
Bagpipes | “continuity bursty” | Kroupa (2001) | $-1.93^{+0.21}_{-0.16}(-1.80)$ | $10.70^{+1.21}_{-1.06}(9.70)$ | $-5.94^{+1.33}_{-1.28}(-4.70)$
Prospector | “continuity bursty” | Kroupa (2001) | $-1.72^{+0.26}_{-0.18}(-1.51)$ | $10.51^{+1.34}_{-0.99}(9.54)$ | $-5.57^{+1.13}_{-1.19}(-4.46)$
Prospector | delayed-$\tau$ | Kroupa (2001) | $-1.98^{+0.22}_{-0.17}(-1.89)$ | $10.67^{+1.22}_{-1.07}(9.68)$ | $-6.22^{+1.37}_{-1.35}(-4.98)$
Prospector | “continuity bursty” | HOT 45K | $-1.93^{+0.25}_{-0.20}(-1.81)$ | $10.56^{+1.28}_{-1.10}(9.53)$ | $-6.09^{+1.39}_{-1.39}(-4.83)$
Prospector | delayed-$\tau$ | HOT 45K | $-2.19^{+0.25}_{-0.22}(-2.18)$ | $10.54^{+1.14}_{-1.10}(9.55)$ | $-6.67^{+1.63}_{-1.45}(-5.31)$

##  Appendix C z = 12.5 GSMF with no contamination limit

Our fiducial GSMF applies a 50% contamination limit on all galaxies. Given the
fields a galaxy is selected in and its stellar mass, our JAGUAR contamination
simulation computes a likelihood of contamination, based on simulated galaxies
with the same stellar mass. The highest contamination is seen in the $z=12.5$
GSMF, for the $10^{7.5}<$ M⋆/M${}_{\odot}\leq 10^{8.5}$ bin, and results in
several galaxies being removed from our fiducial GSMF estimate. As the
predictions of JAGUAR are uncertain at these redshifts, it is hard to judge
how accurate our contamination predictions are. In Figure 15 we have
recomputed the $z=12.5$ GSMF with no contamination limit, which boosts the
lower stellar mass bin by $\sim$0.3 dex. This brings it closer to the
predictions of Li et al. (2023)’s Feedback Free Model (FFB), which has higher
star formation efficiency than most models. The FFB model shown is for
$\epsilon_{\rm max}=0.20$ specifically, and the models with higher SFE
($\epsilon_{\rm max}=0.5-1$) overpredict the GSMF at this redshift compared to
our observations. If our contamination is over-estimated in this redshift bin,
then the FFB model of Li et al. (2023) or Mauerhofer & Dayal (2023)’s DELPHI
model provide the closest predictions, suggesting high but not extreme star
formation efficiency is required to produce the observed GSMF at this
redshift.

Figure 15: Galaxy Stellar Mass Function at $11.5<z\leq 13.5$ without a 50%
contamination limit, compared to our fiducial Bagpipes results.

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