Faint Galaxy Number Counts in the Durham and SDSS Catalogues
Marr John H.
Unit of Computational Science
Hundon, Suffolk, CO10 8HD, UK
<EMAIL_ADDRESS>

Abstract
Galaxy number counts in the $K$-, $H$-, $I$-, $R$-, $B$\- and $U$-bands from
the Durham Extragalactic Astronomy and Cosmology catalogue could be well-
fitted over their whole range using luminosity function (LF) parameters
derived from the SDSS at the bright region and required only modest luminosity
evolution with the steepening of the LF slope ($\alpha$), except for a sudden
steep increase in the $B$-band and a less steep increase in the $U$-band at
faint magnitudes that required a starburst evolutionary model to account for
the excess faint number counts. A cosmological model treating Hubble expansion
as an Einstein curvature required less correction at faint magnitudes than a
standard $\Lambda$CDM model, without requiring dark matter or dark energy.
Data from DR17 of the SDSS in the $g$, $i$, $r$, $u$ and $z$ bands over two
areas of the sky centred on the North Galactic Cap (NGC) and above the South
Galactic Cap (SGC), with areas of 5954 and 859 sq. deg., respectively, and a
combined count of 622,121 galaxies, were used to construct bright galaxy
number counts and galaxy redshift/density plots within the limits of redshift
$\leq 0.4$ and mag $\leq 20$. Their comparative densities confirmed an
extensive void in the Southern sky with a deficit of 26% out to a redshift
$z$$\leq$0.15. Although not included in the number count data set because of
its incompleteness at fainter magnitudes, extending the SDSS redshift-number
count survey to fainter and more distant galaxies with redshift $\leq 1.20$
showed a secondary peak in the number counts with many QSOs, bright X-ray and
radio sources, and evolving irregular galaxies with rapid star formation
rates. This sub-population at redshifts of 0.45–0.65 may account for the
excess counts observed in the $B$-band. Recent observations from the HST and
James Webb Space Telescope (JWST) have also begun to reveal a high density of
massive galaxies at high redshifts ($z>7$) with high UV and X-ray emissions,
and future observations by the JWST may reveal the assembly of galaxies in the
early universe going back to the first light in the universe.
Keywords
galaxies: number counts; galaxies: evolution; general relativity

##  1 Introduction

Galaxy number counts (GNCs) as a function of magnitude provided an early,
straightforward quantitative measurement in cosmology, and numerous surveys
have continued to increase the faint magnitude limit in six principal
observational photometric bands (K, H, I, R, B and U). The faint magnitude
depth has been regularly extended over the years for each spectral band, and
many individual series of observations have been collated and published by the
Extragalactic Astronomy and Cosmology Research Group of Durham University [1].
These observational data for GNCs have been accumulated over many years from a
variety of sources and across different sky fields, leading to much of the
data being widespread, with GNCs in the six principal optical bands
traditionally plotted as scatter plots to include each individual datum point.
One large source of error is that each bin contains galaxies with a range of
morphologies and redshifts and, hence, a range of ages and evolutionary
histories, so individual bins may require a range of corrections that have to
be averaged for each bin, but attempts to clarify these averaged counts by
quantifying them by morphology and redshift defeated the essential simplicity
of complete magnitude counts. For this paper, all counts were binned into
$0.5$ magnitude bins with error bars to indicate the range of observations
included in each bin, resulting in cleaner data points for each band.

The different rates of evolution in past epochs, which appear to have
gradually increased with decreasing observational wavelength from the $K$\- to
$U$-bands, and the excess numbers in the $B$-band and, to a lesser extent, in
the $U$-band were clearly evident. Metcalfe et al. [2] suggested that this may
be from a second peak at high redshift, possibly explained by a sub-population
of early-type galaxies with ongoing star formation. Arnouts et al. [3]
presented measurements of the galaxy LF at $1500$ Å in the range $0.2\leq
z\leq 1.2$ based on Galaxy Evolution Explorer VIMOS-VLT Deep Survey
observations for 1000 spectroscopic redshifts and at higher $z$ using existing
data sets. Their main results were: (a) luminosity evolution is observed with
$\Delta M^{*}$$\sim$$-$2.0 mag in the range $0\leq z\leq 1$ and $\Delta
M^{*}$$\sim$$-$1.0 mag in the range $1\leq z\leq 3$, confirming that star
formation activity was significantly higher in the past; (b) the LF slopes
vary in the range $-1.2\geq\alpha\geq-1.65$, with a possible further increase
at higher $z$; and (c) the analysis of three spectral-type classifications,
Sb-Sd, SdIrr and unobscured starbursts found that, although the bluest class
evolved less strongly in luminosity than the other two classes, their number
density increased sharply with $z$ from $\simeq 15\%$ in the local universe to
$\simeq$55% at $z\simeq 1$, while that of the reddest classes decreased.

Since the launch of the Hubble Space Telescope in 1990, there has been a
wealth of fresh data from newer deep-space telescopes and from the JWST, which
has begun to report observations of galaxies in the far-infrared at deep
magnitudes. The Hubble Space Telescope Medium Deep Survey (HST-MDS) found a
dwarf-rich population at $z$ = 0.3–0.5 [4], and a number of recent surveys
have identified galaxies with stellar masses as high as
$\sim$$10^{11}M_{\odot}$ out to redshifts $z\simeq 10$. This paper aims to
consider how some of the newly published deep redshift data fit into the
interpretation of the faint GNCs.

##  2 Calculating the Observed Volume

Number counts are determined in increments of apparent magnitude or half-
magnitude per square degree of observed sky; however, space is curved through
the expansion of the universe, and the intrinsic luminosity and density of
galaxies are not uniform. Because magnitude is a function of the luminosity
distance ($D_{L}$), whereas the observable area is a function of the diameter
distance ($D_{A}$), the volume increment must be parameterised in $z$,
producing a wineglass-shaped observable volume (Fig. 1). The volume
element/sterad is:

$\delta V=D_{H}^{3}D_{A}^{2}\delta D_{L}\,,$ (1)

where $D_{H}$ is the Hubble distance, $D_{A}$ is the angular diameter distance
and $D_{L}$ is the luminosity distance, and

$\begin{split}D_{H}&=c/H_{0}\\\ D_{A}&=D_{C}/(1+z)\\\
D_{L}&=D_{C}(1+z)\,.\end{split}$ (2)

$D_{C}$ is the line-of-sight comoving distance, generally obtained by
integrating the comoving equation for a $\Lambda$-Cold Dark Matter
($\Lambda$CDM) line-of-sight model derived from general relativity (GR) and
involving terms for dark matter and dark energy. The model for this paper
included a curvature term for Hubble expansion to incorporate the curvature of
light across an expanding universe [5]. This has been shown to closely
approximate a standard $\Lambda$CDM model for angular diameter distances
derived from baryonic acoustic oscillation (BAO) data [6] and for luminosity
distances derived from extensive supernova (SNe 1a) data [7]. By letting
$\Omega_{\Lambda}=0$ and assuming the intrinsic curvature term $\Omega_{K}=0$,
this has an analytical solution in $\Omega_{m}$ and $z$ (Equation (3)):

Figure 1: A graphic showing the observed volume/square degree with redshift in
terms of angular and luminosity distances ($D_{A}$ and $D_{L}$ respectively).

$D_{C}=\frac{1}{\sqrt{1-\Omega_{m}}}\log\left(\frac{(1+z)\left((1-0.5\Omega_{m})+\sqrt{1-\Omega_{m}}\,\right)}{1+0.5\Omega_{m}(z-1)+\sqrt{(1-\Omega_{m})(1+\Omega_{m}z)}}\right)$
(3)

$\delta V$ may now be expressed for the GR model in terms of $z$ as:

$\delta V=D_{H}^{3}\frac{D_{C}^{2}}{(1+z)^{2}}\frac{\\\ d}{\\\
dz}[D_{C}(1+z)]\delta
z~{}~{}~{}~{}[h^{3}\textnormal{~{}Mpc${}^{3}$~{}~{}sterad${}^{-1}$}]\,,$ (4)

while the density function must be multiplied by a further factor of
$(1+z)^{3}$ to allow for the contraction of space in three dimensions with
look-back time. Integrating to $z_{max}$, the redshift limit for the integral,
gives the total volume per square degree. The number of galaxies per square
degree per magnitude increment is then:

$n(m)=D_{H}^{3}h^{-3}(1/3283.8)\int_{0}^{z_{max}}\Phi_{z}(m)\delta
V\textnormal{\,d}z\textnormal{\,d}m\,.$ (5)

where $\Phi_{z}(m)$ is the luminosity function (Section 2.1). $z_{max}$ is
taken to be the volume limit for the integral, but, like much else in
magnitude-number count analysis, this limit is debated. Results from the
Wilkinson Microwave Anisotropy Probe (WMAP) suggest reionisation at $z$ =
11–30 [8], but an emission signature of broad and asymmetric Lyman-$\alpha$
emission may specify the redshift of the final stages of reionisation,
suggesting a redshift range $z$ = 6–9 [9]; however, counts of bright galaxies
at redshifts $z$$\sim$6 by Lyman-break selection find too few luminous
galaxies for their hot stellar populations to power reionisation [10]. This
makes the most likely culprit lower-luminosity galaxies, which may equate to
the small, low-metallicity objects found in significant numbers at
$z$$\sim$2–3 and sometimes termed subgalactic [11].

###  2.1 The Schechter Luminosity Function (LF)

For a specific sample of galaxies, $S$, it is usual to refer to the luminosity
distribution defined by Schechter [12] as $ns(L)$ galaxies per unit luminosity
for the sample. For a given sample size, the volume of the sample will vary
with the luminosity and can be defined as $V_{s}(L)$.

Then, the number of galaxies in $S$ in the luminosity interval $\delta L$
centred on $L$ is $n_{S}(L)\delta L$, and the luminosity function (LF)
$\Phi_{S}(L)$ of the sample $S$ has the units of number of galaxies per unit
luminosity per unit volume and is defined as $\Phi_{S}(L)\equiv\Phi(L)\delta
L~{}V_{S}(L)$, which is reached in the limit of $V_{S}(L)\rightarrow\infty$.
Schechter defined his luminosity function as

$\Phi(L)\textnormal{\,d}L=\Phi^{*}(L/L^{*})^{\alpha}\exp{(-L/L^{*})}\textnormal{\,d}(L/L^{*})$
(6)

but it is more useful to rewrite Equation (6) in terms of absolute magnitude
$M^{*}$:

$n_{S}(M)\textnormal{\,d}M=\Phi^{*}V_{C}K\textnormal{\,dex}[0.4(\alpha+1)(M^{*}-M)]\exp[-\textnormal{\,dex}[0.4(M^{*}-M)]]\textnormal{\,d}M$
(7)

Although there is considerable variation in the actual parameters, the
Schechter function has received good experimental confirmation, and there has
also been some theoretical justification from a model of the non-linear matter
distribution of rich galaxy clusters and their correlations, showing that the
luminosity function has the Schechter form [13]. Nevertheless, the actual form
of the Schechter function is an amalgam of galaxies of many types, ages and
distances and prone to wide variation, and it has been continuously refined
for several band-passes and galaxy types. Folkes et al defined the function in
terms of the spectral type from the Anglo-Australian Observatory 2dFGRS (Two-
degree-Field Galaxy Redshift Survey) [14], finding a range of values for the
three parameters, $M^{*},~{}\phi,$ and $\alpha$, and Metcalfe et al described
Schechter parameters for five different galaxy types (E/S0, Sab, Sbc, Scd and
Sdm) to produce a composite function [2].

###  2.2 Deriving the Number Count Curves

The absolute magnitude $M$ to produce each band of apparent magnitude $m$ over
the range of $D_{L}$ is derived from the definition of $D_{L}$ in the chosen
GR model:

$M=m-5\log_{10}(D_{L})-25$ (8)

where $D_{L}=D_{H}(1+z)D_{C}$ for the appropriate GR model. $n(m)$ is then
derived by substituting for $M$ in $\phi(z)$ in Equation (5) and integrating
Equation (5) to $z_{max}$ and over the range $m\pm 0.5$ per magnitude interval
(or $m\pm 0.25$ per half-magnitude interval). The resultant curve is an
asymmetric hyperbola (Fig. 2) with two distinct asymptotic slopes: the
classical $\textnormal{\,d}{}\log_{10}N/\textnormal{\,d}{m=0.6}$ Euclidean
slope at bright magnitudes and a slope of $-0.4(\alpha+1)$ at faint
magnitudes. Normalisation over the Euclidean region is determined by
$\phi^{*}$ and $M^{*}$. The asymmetric point of inflexion is a function of
$M^{*}$ and $\log(z_{max})$ that may be determined by numerical computation,
but approximates to:

$m_{inflexion}\simeq M^{*}+2.5\log(z_{max})+\textnormal{constant}\,.$ (9)

These contributions make no allowance for evolution. In practice, they will
show some change with redshift due to evolution from events such as
starbursts, luminosity evolution and mergers, making the true underlying cause
for any change in slope difficult to identify. Additionally, the observed
counts may tail off at the faintest magnitudes, which would depress the faint
end of the resultant curve. Blanton’s curves (Fig. 5) are essentially flat at
all bands at the faint end, corresponding to $\alpha\simeq-1.0$. Such flat LF
curves give rise to very shallow number count curves that do not reflect
reality; to account for the observations, there must be evolution of some or
all components of the LF with increasing redshift, as suggested in Fig. 2,
which combines redshift bands to form the single composite curve with
$\alpha=-1.5$.

Figure 2: Building a theoretical number count curve (red line) with composite
Schechter curves at increasing redshifts ($M^{*}=-23.0;\alpha=-1.5$). The
transition from a Euclidean slope of 0.6 to the $\alpha$-dependent slope is
clearly shown (dashed lines).

##  3 Observational Data

Since its first data release in 2003 until the most recent data release of
DR17, which became publicly available in December 2021 [15], the Sloan Digital
Sky Survey (SDSS) has provided increasing quantities of data with detailed
information on a large number of galaxies across five colour bands ($u$, $g$,
$r$, $i$ and $z$) out to apparent magnitude limits of $\sim$21–23. This has
produced clean, well-defined slopes for the bright galaxy counts. It should be
noted that the uppercase bands of the optical bands differ in their spectral
ranges from the lowercase designations of the SDSS bands, and their
correlation with earlier optical data used a standard conversion to the older
$B_{J}$-band to tie the deeper faint data to these more recent bright galaxy
magnitudes.

###  3.1 Observations from the SDSS

Region | RA (h) | Dec (deg) | Area (deg2) | Total Count | Density (gals deg-2)
---|---|---|---|---|---
NGC | 8:00 – 16:00 | 0 – 60 | 5,954 | 562,196 | 94.42
SGC | -2:00 – +2:00 | 0 – 30 | 859 | 59,925 | 69.76
Table 1: Parameters for the SDSS number count surveys (redshift$\leq 0.4$,
$B_{J}\leq 20$).
J2000 equatorial coordinates Figure 3: SDSS galaxy density in redshift bins of
0.0125. NGC (red bars) and SGC (blue bars). $B_{J}\leq 20$.

The Sloan Digital Sky Survey (SDSS) has provided a wealth of data over five
spectral bands, with spectra and usable redshifts for $\sim$2.6 million unique
galaxies out to limiting magnitudes of $u^{\prime}$ = 22.3, $g^{\prime}$ =
23.3, $r^{\prime}$ = 23.1, $i^{\prime}$ = 22.3 and $z^{\prime}$ = 20.8 [15].
Two principal areas are covered by the surveys: (a) the area centred on the
North Galactic Centre (NGC) and (b) an area above the South Galactic Centre
(SGC) (Table 1, J2000 equatorial coordinates). The counts for the smaller SGC
area suggest that it has a lower density of galaxies compared with the NGC
area, with a deficit of $26\%$ out to a redshift $z$$\sim$0.15 (Fig. 3), but
with an increased SGC density beyond $z$$\sim$0.1625 consistent with a local
fluctuation of $2\sigma$ in a large-scale structure. A model for an extensive
void was developed by Busswell et al [16], who noted from the 2dFGRS that the
Southern counts with $B<17$ mag were down by $\sim$30% out to $z$ = 0.1
relative to the Northern counts, and this appeared to be relatively
homogeneous over its whole range, suggesting it could be modelled by varying
the density normalisation $\Phi^{*}$. Beyond $z>0.15$, however, the over-
density in the Northern counts is reversed to become an under-density compared
with the Southern region. In using number count observations, it is therefore
helpful to average the data from several regions of the sky to compensate for
these variations in count density.

Extending the number count range to $z\leq 1.2$ with progressive increase in
$b_{J}\leq 24$ suggested a significant secondary peak between $0.48\leq z\leq
0.57$ with a maximum at $z\sim 0.51$ (Fig. 4). Many of these secondary sources
appear to be active galaxies with high star-forming activity and
correspondingly higher UV emission such as QSOs, AGNs, irregular dwarfs,
starburst galaxies or giant elliptical galaxies, leading to brighter absolute
magnitudes ($M^{*}$) in the past [17, 18]. However, it must be emphasised that
the SDSS is complete only up to the magnitude cutoff for each band and beyond
these limits the counts are progressively incomplete.

Figure 4: Extended SDSS Number counts (NGC, $z\leq 1.20$, progressive limit to
$b_{J}\leq 24$) Figure 5: Schechter curves for 5 bands at uniform redshift
($z=0.11$), after Blanton et al [19].

###  3.2 The LF from the SDSS

The work of Schechter and others to determine a luminosity function for
galaxies distributed randomly through space has been continuously refined.
Blanton et al used data from the extensive SDSS data to define Schechter
curves for five observational bands (0.1u, 0.1g, 0.1r, 0.1i and 0.1z) within a
tight band of redshift at $z=0.11$ for a number of cosmological models [19].
Their values for a Friedmann–Robertson–Walker (FRW) cosmological world model
with assumed matter density $\Omega_{0}=0.3$, vacuum pressure
$\Omega_{\Lambda}=0$ and Hubble constant $H_{0}=100$ h km s-1 Mpc-1 are
presented graphically (Fig. 5) and summarised in Table 2.

Band |  $\phi^{*}$ (10${}^{-2}~{}$h3 Mpc-3) |  $M^{*}-5\log_{10}$h | $\alpha$
---|---|---|---
0.1u | 3.26$\pm$0.40 | -17.89$\pm$0.04 | -0.94$\pm$0.09
0.1g | 2.42$\pm$0.10 | -19.34$\pm$0.02 | -0.92$\pm$0.04
0.1r | 1.69$\pm$0.06 | -20.37$\pm$0.02 | -1.03$\pm$0.03
0.1i | 1.62$\pm$0.06 | -20.78$\pm$0.03 | -1.02$\pm$0.04
0.1z | 1.47$\pm$0.05 | -21.12$\pm$0.02 | -1.07$\pm$0.03
Table 2: SDSS Schechter Function fits for $\Omega_{0}=0.3$ and
$\Omega_{\Lambda}=0$ cosmology at $z=0.11$ [19]

Blanton’s results provide consistent and detailed parameters for the colour
bands they selected and, although they do not provide a subdivision by galaxy
type, they have the advantage of uniformity in their selection by sampling a
well-defined shell at a fixed galactic distance.

##  4 Observations in the Optical Bands

Extensive number count data for six principal observational bands, ultraviolet
($U$-), blue ($B$-), red ($R$-, $I$-), and infrared ($H$-, $K$-), with the
Effective Wavelength Midpoint ($\lambda_{eff}$) for each standard filter at
365 nm, 445 nm, 658 nm, 806 nm, 1630 nm and 2190 nm respectively, have been
compiled over many years from many sources [2, 20, 21, 22, 23, 24] and are
available at the Durham Number Count Survey site [1]. Over small redshifts,
this volume is Euclidean, but as the counts probe deeper, the volume contained
in each interval of redshift begins to shrink with increasing look-back time
as observations come from earlier epochs of an expanding universe. Additional
corrections must be made for luminosity evolution and mergers and the
increasing density of galaxies within a reducing volume and composite curves
constructed for the six bands (Figs. 6-10).

The assumed initial values for $M^{*}$ and $\alpha$ for the model curves were
fitted to the GR model using the SDSS data of Blanton et al [19], assumed to
be correct at $z=0.1$, with $\Phi^{*}$ adjusted to normalise the curves to the
observational data (Table 3). The curves were fitted by pure luminosity
evolution (PLE) excepting the $K$-band which used a simple merger model, and
$B$-band which required a starburst addition to create the sharp rise above
$B_{J}$$\sim$22.

###  4.1 $K$-Band Magnitude Count

Figure 6: $K$-magnitude plots with no evolution (dashed blue line,
$\alpha=-1.03$) and evolution from $\alpha=-1.03$ to $\alpha=-1.70$ (solid
line) in the Hubble curvature GR model with $\Omega_{m}=0.035$, both with
$M^{*}=-23.0+5\log_{10}$ h. The error bars reflect the maximum range of each
bin. The GR model fully overlays a $\Lambda$CDM model with $\Omega_{m}=0.3$,
$\Omega_{\Lambda}=0.7$. The Euclidean 0.6 slope is also shown (dashed black
line). Adapted from composite means of 42 $K$-band surveys from Durham number
count archive [1].

A standard way to present all 42 published $K$-band series is to plot
scattered individual surveys overlapping along the curve. Figure 6 shows these
data with all series binned into bins of $0.5mag$ and plotting the mean of
each bin. The error bars show the range of counts in each bin from the many
surveys; the absence of error bars implies the bin has a single member. The
data for the $K$-magnitude-number counts demonstrate the theoretical
asymmetric hyperbola of the model with an inflexion at $K_{mag}\simeq 17.25$.
This binned representation gives a tight fit to the data points and is used to
show the remaining bands (Fig. 14).

There is still some controversy about the best parameters to use when building
these models. De Propris et al [25] suggested brighter $M^{*}$ but low alpha,
such as $M^{*}=-24.63$ to $-24.48$ and $\alpha=-0.21$ to $-0.81$, while Huang
et al [26] determined $M^{*}_{K}=-23.57\pm 0.08+5\log$ h, $\alpha=-1.33\pm
0.09$ and $\Phi^{*}=0.017\pm 0.002$ h3 Mpc-3 for a universe with
$\Omega_{0}=0.3$ and $\Omega_{\Lambda}=0$, and Kochanek et al [27] suggested
$M^{*}=-23.39-5\log$ h, $\alpha=-1.09$ and $\Phi^{*}=1.16\times 10^{-2}$ h3
Mpc-3.

The picture of a steepening $\alpha$ with redshift may be fundamental to
reconciling the steep observed slopes of the number counts at faint magnitudes
to the flatter LFs reported at the present epoch (or to $z=0.11$ for the SDSS
data). A pure galaxy merger model was therefore fitted to Fig. 6, where the
solid line shows the best fit for the Hubble curvature GR model using a single
Schechter model with modest evolution of $\alpha$, increasing with redshift
from $-1.03$ to $-1.70$ and integrating to $z=0.45$, with
$M^{*}=-23.0+5\log_{10}$h. The curve was normalised at $K_{mag}=10.5$ to the
Frith 2MASS data points [28]. For comparison, the flatter $\alpha=-1.03$ with
no evolution (blue dashed curve) and a Euclidean line with a slope of 0.6
(black dashed line) are also shown. At these low redshifts, integrating to
$z=0.45$, the Hubble curvature GR model with $\Omega_{m}=0.035$ overlay a
$\Lambda$CDM model with $\Omega_{m}=0.3$, $\Omega_{\Lambda}=0.7$ too closely
to be distinguished.

###  4.2 H-band

The H-band data (Fig. 7) are also plotted in bins of $0.5mag$. They show a
similar inflexion to the K-band, but with a flattening tail at faint magnitude
($H_{mag}\sim 25$).

Figure 7: H-band binned magnitude counts. GR model with PLE from $z=0.17$,
$\alpha=-1.07$ $M^{*}=-21.12$. (Adapted from composite means from 14 H-band
surveys, Durham number count archive, [1])

.

The GR-model in Fig. 7 was fitted using the SDSS data with $M^{*}=-21.12-\log
h$, $\alpha=-1.07$, and PLE beyond $z=0.17$. The curve was normalised to the
Frith 2MASS (2006) points. The tail comes from cutting the integration limit
at $z_{max}=1.0$. Further data beyond $H_{mag}>27$ may determine if the tail
is real, or if the true counts continue to rise.

Figure 8: I-band magnitude counts. Composite means from 26 I-band surveys. [1]

###  4.3 I-band

The I-band data (Fig. 8) show a gentle inflexion at $I_{mag}\sim 18$ with a
tailing off beyond $I_{mag}\sim 26$. The GR model mimics the data well (Fig.
8) using the SDSS parameters ($M^{*}=-20.78\log$ h; $\alpha=-1.02$;
$\Phi^{*}=0.0293h^{3}$ Mpc${-3}$ and PLE beyond $z=0.2$ ($\beta=-0.17$). As
with the H-band plots, a good fit is obtained to the tail by limiting the
integration to $z_{max}=2.0$.

###  4.4 R-band

Figure 9: R-band magnitude counts, composite means from 31 R-band surveys [1]

The R-band data show a weak inflexion at $R_{mag}\simeq 17$, with a slight
tail off at faint magnitude ($R_{mag}\gtrsim 27$) using $z_{max}=2.0$ and
$M^{*}=-20.37-5\log h$; $\alpha=-1.03$; $\Phi^{*}=0.028h^{3}$ Mpc-3 with a
modest PLE model at $z=0.15$ ($\beta=-0.18$) (Fig. 9).

###  4.5 B-band

Figure 10: $B$-band magnitude counts binned $\pm$0.25 m (composite means of 35
$B_{J}$-band surveys). Overlain are the GR starburst model (red line) with
starburst evolution from $0.3<z<1.2$, and the $\Lambda$CDM model (blue line).
Bars indicate the range of observations in each bin. (Adapted from Durham
galaxy number count archive [1]).

Ellis [29] described the $B$-band curves as an apparent excess of faint blue
galaxy counts over the number expected on the basis of local galaxy
properties. This has been referred to collectively as the faint blue galaxy
problem [30] and is evident in Fig. 10, constructed from 35 $B$-band surveys
again binned in half-magnitude bins, with the error bars showing the range of
data in each bin and the absence of error bars implying that the bin contains
only a single data point. The swan-neck rise associated with the $B_{J}$
counts contrasts with the other curves and is emphasised in Fig. 11 which
plots the rate of change (slope) of the counts compared with the more regular
$K$-band slope.

Metcalfe et al presented number count data with a blue magnitude limit of
$B_{mag}=27.5$, showing that it was not possible to reconcile the slope and
numbers of galaxies at $B>25$ with the slope of the local faint galaxy
luminosity function unless this was steeper in the past, or density evolution
had taken place [21]. Metcalfe et al [22] also performed a detailed analysis
of more than 110,500 galaxies from the 2dF Galaxy Redshift Survey to generate
an LF standardised to $z=0$; over the interval $-16.5>M_{B_{J}}-5\log$ h
$>-22$, their LF is accurately described by a Schechter function with
$M_{B_{J}}-5\log$ h=$-19.66\pm 0.07$, $\alpha=-1.21\pm 0.03$ and
$\Phi^{*}=1.61\pm 0.08\times 10^{-2}$ h3 Mpc-3, in broad agreement with
earlier calculations by Yasuda et al. [31].

Modelling the number count curves in the $B$-band is confounded in part
because of the ability to fit several different models to the same data by
varying the parameters, and alternative models with an extensive void, PLE or
a two-galaxy-type composite model can all be adjusted to a reasonable fit
[32]. The best and easiest fit is a starburst evolution model shown for the GR
model (red line) and for a $\Lambda$CDM model (blue line) in Fig. 10.
Parameters for the models were normalised to $B_{J}=16$ with
$M^{*}=-20.73+5\log_{10}$h and $\alpha$ increasing from $-0.92$ to $-1.45$
integrated out to $z=4.0$ for the GR model, and $M^{*}=-22.22+5\log_{10}$ h
for the $\Lambda$ CDM model. The starburst range was $0.6<z<1.2$, with the
inflexion beginning at magnitude $20.5$. This was in broad agreement with
ultra-deep counts from the Herschel and Hubble Deep Field observations [23],
although, at faint magnitudes, the $\Lambda$CDM model counts were too high,
requiring an increase in $M^{*}$ to $-22.22+5\log_{10}$h to tie it to the
faint $B_{J}$ data.

The need for faster rates of evolution in $b_{J}$ than in $r_{F}$ to fit the
counts may be explained by a model for evolution whereby some types of
galaxies radiated larger amounts of flux in the rest-U band relative to the
redder passbands [33]. This model has received support from McDonald et al who
reported X-ray, optical and infrared observations of the galaxy cluster SPT-
CLJ2344-4243 at redshift $z=0.596$ that reveal an exceptionally luminous
galaxy cluster that appears to be experiencing a massive starburst with a
formation rate of around 740 solar masses a year [34].

Figure 11: Change in slopes of $B$-band magnitude counts (blue) compared to
$K$-mag counts (red).

###  4.6 U-band

The U-band data, taken from 19 surveys, has a wide spread over much of its
range with an inflexion beyond $U_{mag}>22$ (Fig. 12). The best-fit model is a
PLE model using the $u_{0.1}$ parameters from the SDSS, with
$M^{*}=-17.89-5\log h$, $\alpha=-0.94$ ($\beta=-0.16$). Integrating out to
$z=1.0$ shows the slight tail-off beyond $U_{mag}>28$.

Figure 12: U-band magnitude, composite counts from 19 U-band surveys. [1] Param | K | H | I | R | B | U
---|---|---|---|---|---|---
$\alpha$ | -1.10 | -1.10 | -1.02 | -1.03 | -0.92 | -0.94
$M^{*}$ | -23.20 | -23.50 | -20.78 | -20.37 | -20.73 | -17.89
$z_{lim}$ | 1.00 | 1.20 | 2.00 | 2.00 | 2.30 | 1.00
$\phi^{*}$ | 0.018 | 0.006 | 0.0293 | 0.028 | 1.26$\times 10^{-4}$ | 3.91$\times 10^{-4}$
$m_{norm}$ | 12.00 | 12.00 | 14.00 | 16.00 | 16.00 | 18.00
Table 3: Schechter and PLE parameters for the six pass bands of Fig. 14,
adapted from the SDSS parameters of Blanton et al [19]
($H_{0}=100~{}h~{}km~{}s^{-1}~{}Mpc^{-1}$ with $h=1$).

###  4.7 Number Counts from the SDSS

The extinction-corrected magnitudes and redshifts of a total of 1,424,733
galaxies were downloaded for all surveyed galaxies in the five principal
filter colours of the surveys in the SDSS up to the survey cut-off magnitude
of each band, with the number densities counted into 0.5-magnitude bands over
a total sky area of 5954 square degrees (Fig. 13). The counts show a
systematic increase at all magnitudes from the ultraviolet towards the near-
infrared.

Figure 13: Galaxy number counts for the five SDSS observational bands
(crosses). Overlain are the bright $B$-band counts (circles $\pm$ bin ranges)
and the SDSS-derived $B$ curve (solid red line).

Cross-referencing historical magnitude systems with the more recent SDSS
filters is not a trivial problem [35], as the earlier B-filters were
calibrated on stars based on the Johnson–Cousins UBVRI System, generally
indicated by $B_{J}$. Empirical conversion to this system from the SDSS was
performed from the $g^{\prime}$ and $r^{\prime}$ bands using Equation (10)
[36], with appropriate de-reddening and k-corrections [37].

$B_{J}=g+0.155+0.152\times(g-r)$ (10)

This equivalence is shown in Fig. 13 over the range 14–18 mag. (red line),
where it overlays the $B_{J}$ observational number counts of Fig. 10 (circles)
and falls within the range errors of these counts, confirming this to be an
appropriate correction.

##  5 Discussion

The observational data for GNCs have been accumulated over many years from a
variety of sources and across different sky fields, causing much of the data
to be widespread, and GNCs in the six principal optical bands have
traditionally been plotted as scatter plots to include each individual data
point. In this paper, all counts were binned in $0.5$ magnitude bins producing
much cleaner data points for each band (Figs. 6–10 and 12). All curves were
well characterised by a Euclidean slope $N\varpropto 10^{0.6m}$ at bright
magnitudes, with counts increasing with wavelength from the UV to deep IR at
all magnitudes (Fig. 14) and an inflexion point whose apparent magnitude also
increased with wavelength.

Detailed GNCs were also plotted for the $u,g,r,i$ and $z$ colour bands of the
SDSS and showed good correlation with counts in the optical $B$-band up to the
SDSS cut-offs, using the conversion $B_{J}=g+0.155+0.152\times(g-r)$. Data
from the SDSS provide broad information on more than one million galaxies in
five colour bands, with detailed reference information for $\Phi^{*}$, $M^{*}$
and $\alpha$, the principal parameters of the Schechter function, at a well-
defined redshift. The presence of voids can bias GNCs in near-surveys,
confirmed by a deficiency in the count density/redshift between Northern and
Southern galactic hemispheres of 26% out to a redshift $z$$\sim$0.15 in the
SDSS counts, as noted in earlier papers [16]. The steepening number counts at
faint magnitudes reflect a much higher LF in the past. A simple merger model
has been shown to reproduce this decline with a derived merger rate at
$z=1.5$, close to the observed value based on the increase in number
densities, with most of these merging galaxies being of lower mass [38].

Figure 14: All six colour bands: GR model with PLE for $K$-, $H$-, $I$\- and
$R$-bands and starburst GR model for $B$\- and $U$-bands. Composite means of
all surveys (error bars omitted for clarity). (Data from Durham number count
archive [1].)

The need for faster rates of evolution in the $B_{J}$-band than the redder
$R$\- or $K$-bands may be explained by a model for evolution whereby some
types of galaxies radiated larger amounts of flux in the rest $U$-band
relative to the redder passbands [33]. Blanton et al found that most galaxies
in the SDSS with low redshift ($z<0.05$) at the faint end of the LF are blue,
and therefore, most dwarfs in the universe should be of the star-forming type
with the same luminosity evolution as the star-forming galaxies (SFGs) [39,
40]. Figure 15 illustrates the redshift at which each colour band begins to
pick up galaxies at increasingly shorter wavelengths. It will be noted that
the $B$-band moves into the UV by redshifts $z\simeq 0.2$, and the $B$-band
inflexion, beginning at $B_{mag}\simeq 20.5$, corresponds to redshift
$z$$\sim$0.5, where this band is picking up galaxies in the deep UV but does
not begin to pick up soft X-ray radiation until $z\simeq 10$.

Figure 15: Log–log plot of the observed emission wavelenth/redshift for the
six optical bands (solid lines) with reference lines for each band (dashed
lines). Diamonds show the cut-off redshifts used to generate the number count
plots for each band.

The giant ellipticals and spirals have been relatively stable since
$z$$\sim$0.7, whereas there has been rapid evolution of the irregular galaxy
population with an increasing number of irregular and peculiar systems with
increasing magnitude, with clear evolution of the blue population and little
or no evolution of the red population [41]. The Hubble Space Telescope Medium
Deep Survey (HST-MDS) found that the universe is dwarf-rich at $z=0.3-0.5$,
consistent with a dwarf luminosity function with a steep faint-end slope, and
a number of recent surveys have identified galaxies with stellar masses as
high as $\sim 10^{11}M_{\odot}$ out to redshifts $z\simeq 10$ [4]. The GOODS-
ALMA survey has detected a number of massive galaxies in the $H$-band with a
median stellar mass of $M^{*}=1.1\times 10^{11}M_{\odot}$ and median near-
infrared-based photometric redshifts of $z=2.92$, with two of them suggesting
redshifts $z>4$ ($z$$\sim$4.3 and $4.8$) [42]. The ALMA survey also noted that
$\geq 40\%$ of galaxies hosted an X-ray AGN, compared to $\sim$14% for other
galaxies of similar mass and redshift. One further possible cause for this
increase is an excess peak at $z$$\sim$0.69 from gravitational lensing, which
may inflate the true number count by as much as $\times 2.5$ out to $B\leq
27$, equating to an additional log count of $+0.4$ [43].

Starburst galaxies have an exceptionally high rate of star formation, up to
100 $M_{\odot}$ per year, compared with an average rate of star formation of
only $\sim 3M_{\odot}$ per year for a stable galaxy [17]. This is possibly
driven by an epoch-dependent galaxy merger rate causing a combination of
starbursts and weak AGN activity up to $z$$\sim$0.5–1.0, and some support for
such strong evolution comes from the microJansky radio source population,
thought to originate from a mixture of starburst, post-starburst and
elliptical galaxies [18]. Owen et al have suggested that there is also
extensive starburst activity at very high redshifts ($z\geq 7$), with extreme
X-ray luminosity to explain the rise in observable counts detected in the
$B$\- and $U$-bands, where the excess UV and X-ray emissions are red-shifted
into the blue and violet [44]. Sources making up the faint, sub-microJansky
radio sky include SFGs, radio-quiet active galactic nuclei, low-radio-power
ellipticals and dwarf galaxies. Kellermann et al selected low-redshift SDSS
quasi-stellar objects (QSOs) for an ultra-deep QSO survey and suggested that
the QSOs primarily comprise two components: (1) a small fraction of quasi-
stellar object (QSO) galaxies with radio-loud active galactic nuclei (AGNs)
accreting into a central, supermassive black hole, characterised by their
luminous, powerful radio emission, and (2) a large number of starburst-driven
QSO host galaxies with a radio-quiet AGN population but an appreciable
fraction of strong X-ray-emitting AGNs [45, 46].

##  6 Conclusions

Although originally seen as a possible cosmological test, GNCs are now
considered to be more sensitive to galactic evolution than to the cosmological
model [32]. Ellis has stated that, “although the original motivation was an
attempt to quantify the cosmological world model, [this is now] concerned with
the study of faint galaxies as a way of probing their evolutionary history”
[47], and Weinberg suggested that: “… in the interpretation of number counts,
I find it hard to believe that it is possible to disentangle the effects of
evolution” (Weinberg S., personal communication, 1993). The cosmological model
selected is, therefore, of less importance than changing the rate of evolution
to match the model chosen [47]. The general relativity (GR) model with Hubble
expansion as an Einstein curvature has accurately modelled the luminosity
distances derived from extensive supernovae (SNe 1a) data and the angular
diameter distances derived from baryonic acoustic oscillation (BAO) data [5],
and this was therefore selected as the cosmological model for the count data.
The Hubble Curvature GR model with $\Omega_{m}=0.035$ could be fitted to all
the GNC observational data with no intrinsic correction whereas, in
comparison, the $\Lambda$CDM model with $\Omega_{m}=0.3$,
$\Omega_{\Lambda}=0.7$ (Fig. 10) fitted the sudden rise in $B$ counts less
well.

The Schechter models used for this paper followed Blanton [19] for the SDSS,
using composite values for each colour band and, although each galaxy type may
have evolved with its own parameters and time scale, the curves described the
observations sufficiently well that additional discrimination of the
parameters for $M^{*}$ and $\alpha$ were not required. The four reddest
wavelengths ($K$-, $I$-, $R$\- and $H$-) required an evolutionary term in
$\alpha$ to produce steeper faint slopes to fit the observations, with
$\alpha$ evolving to reflect that there were more galaxies in past epochs,
reducing through mergers to the present epoch, although $M^{*}$ and $\Phi^{*}$
will also have evolved, and evolution rates probably varied over the lifetime
of these galaxies [23]. The two shorter wavelengths ($B$\- and $U$-) each
showed a sharp rise in their curves, particularly pronounced in the $B$-band,
and these required a modest starburst parameter with a rapid increase in
$M^{*}$ ($\leq 1$ mag) to give a good fit over the whole range of
observations.

The underlying physical cause for these rapidly evolving galaxies appears to
be related to their star formation rates. Galaxies with the strongest [O II]
emissions dominate the evolutionary trends, and their luminosity density has
fallen by a large factor since $z$$\sim$0.5 [48]. At the present epoch, such
systems all lie at the faint end of the LF, but even at modest redshifts, they
occupy a wide range of luminosities. Cohen [49] reported that $E$ galaxies
become more luminous by $z\simeq 1$, with the mean star formation rate (SFR)
increasing strongly in the range $0\leq z\leq 1$ by a factor of $\sim$10\. The
most luminous galaxies showed this trend very strongly, although Cohen
reported only a modest evolution of $M^{*}$ with $z$, consistent with passive
evolution.

Extending the SDSS count densities to higher redshifts and apparent magnitude
limits showed a secondary peak in the counts at redshift $\sim$0.5, with many
QSOs, AGNs, irregular dwarfs, starburst galaxies or giant elliptical galaxies.
These are all active galaxies with high star-forming activity and
correspondingly higher UV emission, leading to brighter absolute magnitudes
($M^{*}$) in the past [18, 17]. Further support for a second peak at high
redshifts has come from McDonald et al, who reported X-ray, optical and
infrared observations of the galaxy cluster SPT-CLJ2344-4243 at redshift
$z=0.596$ that revealed an exceptionally luminous galaxy cluster that appears
to be experiencing a massive starburst, with a formation rate of $\sim$740
solar masses per year [34]. Atek et al have probed the UV LF to $z$$\sim$7,
computing each field individually and comparing their results to the
compilation of the HST legacy fields, the LF fields and the UDF12 field [50].
One further important possible cause for this increase is an excess peak at
$z$$\sim$0.69 from gravitational lensing, which may inflate the true number
count by as much as $\times 2.5$ out to $B_{lim}\leq 27$, equating to an
additional log count of $+$0.4 [43].

The 1–5 $\upmu$m range from the JWST observations has allowed a search for
intrinsically red galaxies beyond the $\lambda_{rest}=1216$ Å Lyman break and
the $\lambda_{rest}$$\sim$3600 Å Balmer break in the first $\simeq 750$ Myrs
after the Big Bang, and six possible galaxies at $7.4\leq z\leq 9.1$ with
$M>10^{10}-10^{11}M_{\odot}$ have now been found, suggesting that the stellar
mass density in massive galaxies may be much greater than predicted by the
rest-frame UV-selected samples [51]. Finkelstein et al [52] have identified a
sample of 26 galaxies in the first 500 Myr of galaxy evolution at z$\sim$9–16
from the Cosmic Evolution Early Release Science (CEERS) JWST survey. These
objects were compact with a median half-light radius of $\sim$0.5 kpc and a
surprisingly high density. A high-redshift galaxy has also been observed at
$z$$\sim$11.09 that is extremely luminous in the UV, with a mass of
$\sim$$10^{9}M_{\odot}$[53]. An estimate of the $z$$\sim$11 rest-frame UV
luminosity function found that the number density of galaxies (arcmin-2)
showed little evolution from $z$$\sim$9 to $z$$\sim$11, and Mcgaugh has noted
that early, large galaxies at $z\geq 10$ may contribute to these very bright
number counts [54].

The new observational challenge is simultaneously to cover the low-mass and
high-redshift ranges of the galaxy population, especially at $1<z<4$ where
galaxies are most actively forming stars. As most spectral features move into
the optical rest-frame at these redshifts, deep near-infrared (NIR)- and
infrared (IR)-selected data will be essential for accurate photometric
redshifts and stellar masses, to probe differing galaxy environments, to
accurately trace the large-scale structure and to minimise the effect of
cosmic inhomogeneities [55]. For the future, kinetic Sunyaev Zel’dovich (kSZ)
tomography hopes to provide useful constraints on some bias parameters in
galaxy number counts and to enable a clearer distinction between standard
$\Lambda$CDM cosmology and relativistic Hubble expansion cosmology [56]. New
cosmological probes based on re-scattered CMB photons will enable huge galaxy
surveys to map vast volumes of the observable universe to the extent that
general relativistic corrections to the distribution of galaxies must be taken
into account, and the JWST, ranging from visible through the mid-infrared,
covering 0.6–28 $\upmu$m, may reveal the assembly of galaxies going back to
the first light in the early universe.

Acknowledgments I would like to thank the anonymous reviewers for their
constructive criticism; Tom Shanks and Nigel Metcalfe for their early support;
the Extragalactic Research Group of Durham University for collating and
publishing the many observations; the SDSS team for making their data fully
accessible; and the IoA at Cambridge for their facilities.

This research received no external funding.

## References

  * [1] Durham Galaxy Number-Counts. astro.dur.ac.uk/nm/pubhtml/counts/counts.html Accessed: 2022-08-04 (2010)
  * [2] Metcalfe, N., Shanks, T., Weilbacher, P.M., McCracken, H.J., Fong, R., Thompson, D.: Galaxy number counts - VI. An H-band survey of the Herschel Deep Field. Monthly Notices of the Royal Astronomical Society 370(3), 1257–1273 (2006). https://doi.org/10.1111/j.1365-2966.2006.10534.x
  * [3] Arnouts, S., Schiminovich, D., Ilbert, O., et al: The GALEX VIMOS-VLT Deep Survey Measurement of the Evolution of the 1500 Å Luminosity Function. ApJL 619(1), 43–46 (2005) arXiv:astro-ph/0411391 [astro-ph]. https://doi.org/10.1086/426733
  * [4] Oesch, P.A., Montes, M., Reddy, N., et al: HDUV: The Hubble Deep UV Legacy Survey. ApJS 237(1), 12 (2018) arXiv:1806.01853 [astro-ph.GA]. https://doi.org/10.3847/1538-4365/aacb30
  * [5] Marr, J.H.: Hubble Expansion as an Einstein Curvature. Journal of Modern Physics 13, 969–991 (2022) arXiv:1803.02198 [physics.gen-ph]
  * [6] Aubourg, É., Bailey, S., Bautista, J.E., et al, BOSS Collaboration: Cosmological implications of baryon acoustic oscillation measurements. Phys. Rev. D 92(12), 123516 (2015) arXiv:1411.1074 [astro-ph.CO]. https://doi.org/10.1103/PhysRevD.92.123516
  * [7] Betoule, M., Kessler, R., Guy, J., et al: Improved cosmological constraints from a joint analysis of the SDSS-II and SNLS supernova samples. A&Ap 568, 22 (2014) arXiv:1401.4064 [astro-ph.CO]. https://doi.org/10.1051/0004-6361/201423413
  * [8] Bennett, C.L., Halpern, M., Hinshaw, G., et. al: First-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results. ApJS 148(1), 1–27 (2003) [astro-ph]. https://doi.org/10.1086/377253
  * [9] Windhorst, R.A., Taylor, V.A., Jansen, R.A., et. al: A Hubble Space Telescope Survey of the Mid-Ultraviolet Morphology of Nearby Galaxies. ApJS 143(1), 113–158 (2002) [astro-ph]. https://doi.org/10.1086/341556
  * [10] Lehnert, M.D., Bremer, M.: Luminous Lyman Break Galaxies at $z>5$ and the Source of Reionization. ApJ 593(2), 630–639 (2003) [astro-ph]. https://doi.org/10.1086/376729
  * [11] Yan, H., Windhorst, R.A., Cohen, S.H.: Searching for z$\simeq$6 Objects with the Hubble Space Telescope Advanced Camera for Surveys: Preliminary Analysis of a Deep Parallel Field. ApJL 585(2), 93–96 (2003) [astro-ph]. https://doi.org/10.1086/374371
  * [12] Schechter, P.: An analytic expression for the luminosity function for galaxies. ApJ 203, 297–306 (1976). https://doi.org/10.1086/154079
  * [13] Schaeffer, R.: Galaxies, Clusters and Fluctuations. In: Audouze, J., Pelletan, M.-C., Szalay, A., Zel’dovich, Y.B., Peebles, P.J.E. (eds.) Large Scale Structures of the Universe, vol. 130, p. 215 (1988)
  * [14] Folkes, S., Ronen, S., Price, I., et. al: The 2dF Galaxy Redshift Survey: spectral types and luminosity functions. MNRAS 308(2), 459–472 (1999) [astro-ph]. https://doi.org/10.1046/j.1365-8711.1999.02721.x
  * [15] Abdurro’uf, Accetta, K., Aerts, C., Silva Aguirre, V., Ahumada, R., et. al.: The Seventeenth Data Release of the Sloan Digital Sky Surveys: Complete Release of MaNGA, MaStar, and APOGEE-2 Data. ApJS 259(2), 35 (2022) arXiv:2112.02026 [astro-ph.GA]. https://doi.org/10.3847/1538-4365/ac4414
  * [16] Busswell, G.S., Shanks, T., Frith, W.J., Outram, P.J., Metcalfe, N., Fong, R.: The local hole in the galaxy distribution: new optical evidence. MNRAS 354(4), 991–1004 (2004) arXiv:astro-ph/0302330 [astro-ph]. https://doi.org/10.1111/j.1365-2966.2004.08217.x
  * [17] Schneider, P.: Extragalactic Astronomy and Cosmology: An Introduction, pp. 1–459. Springer, Berlin, Heidelberg (2006). https://doi.org/10.1007/978-3-540-33175-9
  * [18] Windhorst, R.A.: The microJansky and nanoJansky population. New Astron. Rev. 47(4-5), 357–365 (2003). https://doi.org/10.1016/S1387-6473(03)00045-9
  * [19] Blanton, M.R., Hogg, D.W., Bahcall, N.A., et. al: The Galaxy Luminosity Function and Luminosity Density at Redshift z = 0.1. ApJ 592(2), 819–838 (2003) [astro-ph]. https://doi.org/10.1086/375776
  * [20] Jones, L.R., Fong, R., Shanks, T., Ellis, R.S., Peterson, B.A.: Galaxy number counts. - I. Photographic observations to B=23.5 mag. MNRAS 249, 481–497 (1991). https://doi.org/10.1093/mnras/249.3.481
  * [21] Metcalfe, N., Shanks, T., Fong, R., Jones, L.R.: Galaxy number counts - II. CCD observations to B = 25 mag. Monthly Notices of the Royal Astronomical Society 249(3), 498–522 (1991). https://doi.org/10.1093/mnras/249.3.498
  * [22] Metcalfe, N., Shanks, T., Fong, R., Roche, N.: Galaxy number counts - III. Deep CCD observations to B=27.5 mag. MNRAS 273(2), 257–276 (1995). https://doi.org/10.1093/mnras/273.2.257
  * [23] Metcalfe, N., Shanks, T., Campos, A., McCracken, H.J., Fong, R.: Galaxy number counts - V. Ultradeep counts: the Herschel and Hubble Deep Fields. Monthly Notices of the Royal Astronomical Society 323(4), 795–830 (2001). https://doi.org/10.1046/j.1365-8711.2001.04168.x
  * [24] McCracken, H.J., Metcalfe, N., Shanks, T., Campos, A., Gardner, J.P., Fong, R.: Galaxy number counts - IV. Surveying the Herschel Deep Field in the near-infrared. Monthly Notices of the Royal Astronomical Society 311(4), 707–718 (2000). https://doi.org/10.1046/j.1365-8711.2000.03096.x
  * [25] De Propris, R., Stanford, S.A., Eisenhardt, P.R., Holden, B.P., Rosati, P.: The rest-Frame K-band luminosity function of galaxies in clusters to z= 1.3. The Astronomical Journal 133(5), 2209–2215 (2007). https://doi.org/10.1086/513516
  * [26] Huang, J.-S., Glazebrook, K., Cowie, L.L., Tinney, C.: The Hawaii+Anglo-Australian Observatory K-Band Galaxy Redshift Survey. I. The Local K-Band Luminosity Function. ApJ 584(1), 203–209 (2003) arXiv:astro-ph/0209440 [astro-ph]. https://doi.org/10.1086/345619
  * [27] Kochanek, C.S., Pahre, M.A., Falco, E.E., Huchra, J.P., Mader, J., Jarrett, T.H., Chester, T., Cutri, R., Schneider, S.E.: The K-band galaxy luminosity function. The Astrophysical Journal 560(2), 566–579 (2001). https://doi.org/10.1086/322488
  * [28] Skrutskie, M.F., Cutri, R.M., Stiening, R., Weinberg, M.D., et al: The Two Micron All Sky Survey (2MASS). AJ 131(2), 1163–1183 (2006). https://doi.org/10.1086/498708
  * [29] Ellis, R.S.: Faint Blue Galaxies. ARA&A 35, 389–443 (1997) arXiv:astro-ph/9704019 [astro-ph]. https://doi.org/10.1146/annurev.astro.35.1.389
  * [30] Windhorst, R.A., Cohen, S.H., Hathi, N.P., et al: The Hubble Space Telescope Wide Field Camera 3 Early Release Science Data: Panchromatic Faint Object Counts for 0.2-2 $\mu$m Wavelength. ApJS 193(2), 27 (2011) arXiv:1005.2776 [astro-ph.CO]. https://doi.org/10.1088/0067-0049/193/2/27
  * [31] Yasuda, N., Fukugita, M., Narayanan, V.K., et al: Galaxy Number Counts from the Sloan Digital Sky Survey Commissioning Data. AJ 122(3), 1104–1124 (2001) arXiv:astro-ph/0105545 [astro-ph]. https://doi.org/10.1086/322093
  * [32] Marr, J.H.: Bright galaxy number counts. Master’s thesis, Durham University, Durham, UK (1995)
  * [33] Shanks, T., Stevenson, P.R.F., Fong, R., MacGillivray, H.T.: Galaxy number counts and cosmology. Monthly Notices of the Royal Astronomical Society 206(4), 767–800 (1984). https://doi.org/10.1093/mnras/206.4.767
  * [34] McDonald, M., Bayliss, M., Benson, B.A., et al: A massive, cooling-flow-induced starburst in the core of a luminous cluster of galaxies. Nature 488(7411), 349–352 (2012) arXiv:1208.2962 [astro-ph.CO]. https://doi.org/10.1038/nature11379
  * [35] Bessell, M.S.: Standard Photometric Systems. ARA&A 43(1), 293–336 (2005). https://doi.org/10.1146/annurev.astro.41.082801.100251
  * [36] Norberg, P., Cole, S., Baugh, C.M., et al: The 2dF Galaxy Redshift Survey: the bJ-band galaxy luminosity function and survey selection function. Monthly Notices of the Royal Astronomical Society 336(3), 907–931 (2002) https://academic.oup.com/mnras/article-pdf/336/3/907/2959521/336-3-907.pdf. https://doi.org/10.1046/j.1365-8711.2002.05831.x
  * [37] Blanton, M.R., Brinkmann, J., Csabai, I., Doi, M., Eisenstein, D., Fukugita, M., Gunn, J.E., Hogg, D.W., Schlegel, D.J.: Estimating Fixed-Frame Galaxy Magnitudes in the Sloan Digital Sky Survey. AJ 125(5), 2348–2360 (2003) arXiv:astro-ph/0205243 [astro-ph]. https://doi.org/10.1086/342935
  * [38] Conselice, C.J., Wilkinson, A., Duncan, K., Mortlock, A.: The Evolution of Galaxy Number Density at z ¡ 8 and Its Implications. ApJ 830(2), 83 (2016) arXiv:1607.03909 [astro-ph.GA]. https://doi.org/10.3847/0004-637X/830/2/83
  * [39] Blanton, M.R., Lupton, R.H., Schlegel, D.J., Strauss, M.A., Brinkmann, J., Fukugita, M., Loveday, J.: The Properties and Luminosity Function of Extremely Low Luminosity Galaxies. ApJ 631(1), 208–230 (2005) arXiv:astro-ph/0410164 [astro-ph]. https://doi.org/10.1086/431416
  * [40] Padovani, P.: The microjansky and nanojansky radio sky: source population and multiwavelength properties. MNRAS 411(3), 1547–1561 (2011) arXiv:1009.6116 [astro-ph.CO]. https://doi.org/10.1111/j.1365-2966.2010.17789.x
  * [41] Griffiths, R.E., Ratnatunga, K.U., Casertano, S., et al: The HST Medium Deep Survey: progress towards resolution of the faint blue galaxy problem. Astrophysical Letters and Communications 36(1-6), 355–361 (1997)
  * [42] Franco, M., Elbaz, D., Béthermin, M., et al: GOODS-ALMA: 1.1 mm galaxy survey. I. Source catalog and optically dark galaxies. A&Ap 620, 152 (2018) arXiv:1803.00157 [astro-ph.GA]. https://doi.org/10.1051/0004-6361/201832928
  * [43] Nistane, V., Jalilvand, M., Carron, J., Durrer, R., Kunz, M.: An estimator for the lensing potential from galaxy number counts. Journal of Cosmology and Astroparticle Physics 2022(06), 024 (2022). https://doi.org/10.1088/1475-7516/2022/06/024
  * [44] Owen, E.R., Wu, K., Jin, X., Surajbali, P., Kataoka, N.: Starburst and post-starburst high-redshift protogalaxies. The feedback impact of high energy cosmic rays. A&Ap 626, 85 (2019) arXiv:1905.00338 [astro-ph.GA]. https://doi.org/10.1051/0004-6361/201834350
  * [45] Kellermann, K.I., Condon, J.J., Kimball, A.E., Perley, R.A., Ivezić, Ž.: Radio-loud and Radio-quiet QSOs. ApJ 831(2), 168 (2016) arXiv:1608.04586 [astro-ph.GA]. https://doi.org/%****␣Counts.bbl␣Line␣825␣****10.3847/0004-637X/831/2/168
  * [46] Radcliffe, J.F., Barthel, P.D., Garrett, M.A., Beswick, R.J., Thomson, A.P., Muxlow, T.W.B.: The radio emission from active galactic nuclei. A&Ap 649, 9 (2021) arXiv:2104.04519 [astro-ph.GA]. https://doi.org/10.1051/0004-6361/202140791
  * [47] Ellis, R.S.: Unsolved Problems in Astrophysics. Princeton Univ. Press, Princeton (1996)
  * [48] Ellis, R.S., Colless, M., Broadhurst, T., Heyl, J., Glazebrook, K.: Autofib Redshift Survey - I. Evolution of the galaxy luminosity function. MNRAS 280(1), 235–251 (1996) [astro-ph]. https://doi.org/%****␣Counts.bbl␣Line␣875␣****10.1093/mnras/280.1.235
  * [49] Cohen, J.G.: A Progress Report on the Caltech Faint Galaxy Redshift Survey. In: Cristiani, S., Renzini, A., Williams, R.E. (eds.) Deep Fields, p. 49 (2001). https://doi.org/10.1007/10854354_10
  * [50] Atek, H., Richard, J., Jauzac, M., et al: Are ultra-faint galaxies at $z=6-8$ responsible for cosmic reionization? combined constraints from the hubble frontier fields, clusters and parallels. The Astrophysical Journal 814(1), 69 (2015). https://doi.org/10.1088/0004-637x/814/1/69
  * [51] Labbe, I., van Dokkum, P., Nelson, E., et al: A population of red candidate massive galaxies ~600 Myr after the Big Bang. Nature: Accelerated Article Preview. arXiv e-prints, 2207–12446 (2022) arXiv:2207.12446 [astro-ph.GA]. https://doi.org/10.48550/arXiv.2207.12446
  * [52] Finkelstein, S.L., Bagley, M.B., Ferguson, H.C., et al: CEERS Key Paper I: An Early Look into the First 500 Myr of Galaxy Formation with JWST. arXiv e-prints, 2211–05792 (2022) arXiv:2211.05792 [astro-ph.GA]. https://doi.org/10.48550/arXiv.2211.05792
  * [53] Oesch, P.A., Brammer, G., van Dokkum, P.G., et al: A Remarkably Luminous Galaxy at z=11.1 Measured with Hubble Space Telescope Grism Spectroscopy. ApJ 819(2), 129 (2016) arXiv:1603.00461 [astro-ph.GA]. https://doi.org/10.3847/0004-637X/819/2/129
  * [54] McGaugh, S.S.: Early galaxy formation and the hubble constant tension. Research Notes of the AAS 7(2), 20 (2023). https://doi.org/10.3847/2515-5172/acba9a
  * [55] Laigle, C., McCracken, H.J., Ilbert, O., et. al.: The COSMOS2015 Catalog: Exploring the 1 ¡ z ¡ 6 Universe with Half a Million Galaxies. ApJS 224(2), 24 (2016) arXiv:1604.02350 [astro-ph.GA]. https://doi.org/10.3847/0067-0049/224/2/24
  * [56] Contreras, D., Johnson, M.C., Mertens, J.B.: Towards detection of relativistic effects in galaxy number counts using kSZ Tomography. Journal of Cosmology and Astroparticle Physics 2019(10), 024 (2019). https://doi.org/10.1088/1475-7516/2019/10/024