Testing cosmological models with the brightness profile of distant
galaxies–References

# Testing cosmological models with the brightness profile of distant galaxies

I. Olivares-Salaverri1 and M. B. Ribeiro2

1Observatório do Valongo  Universidade Federal do Rio de Janeiro  Brazil
2Instituto de Física  Universidade Federal do Rio de Janeiro  Brazil

(2013)

###### Abstract

The goal of this work is to use observed galaxy surface brightness profiles at
high redshifts to determine, among a few candidates, the cosmological model
best suited to interpret these observations. Theoretical predictions of
galactic surface brightness profiles are compared to observational data in two
cosmological models, $\Lambda$CDM and Einstein-de Sitter, to calculate the
evolutionary effects of different spacetime geometries in these profiles in
order to try to find out if the available data is capable of indicating the
cosmology that most adequately represents actual galactic brightness profiles
observations. Starting from the connection between the angular diameter
distance and the galactic surface brightness as advanced by Ellis and Perry
(1979), we derived scaling relations using data from the Virgo galactic
cluster in order to obtain theoretical predictions of the galactic surface
brightness modeled by the Sérsic profile at redshift values equal to a sample
of galaxies in the range $1.5\lesssim z\lesssim 2.3$ composed by a subset of
Szomoru’s et al. (2012) observations. We then calculated the difference
between theory and observation in order to determine the changes required in
the effective radius and effective surface brightness so that the observed
galaxies may evolve to have features similar to the Virgo cluster ones. Our
results show that within the data uncertainties of this particular subset of
galaxies it is not possible to distinguish which of the two cosmological
models used here predicts theoretical curves in better agreement with the
observed ones, that is, one cannot identify a clear and detectable difference
in galactic evolution incurred by the galaxies of our sample when applying
each cosmology. We also concluded that the Sérsic index $n$ does not seem to
play a significant effect in the evolution of these galaxies. Further
developments of the methodology employed here to test cosmological models are
also discussed.

###### keywords:

cosmology: theory - galaxies: distances and redshifts, structure, evolution

##  1 Introduction

Observational cosmology attempts to understand the large-scale matter
distribution in the universe and its geometry by basically following two
different methodologies. The first, known as the direct-manner, or data-
driven, approach, seeks to describe what is actually observed without
addressing the question of why we observe such an universe the way it is,
whereas the theory-based, or model-based, one interprets the observations
based on explanations that can produce the observed patterns (Ellis 2006).

The theory-based approach consists of assuming a model based on a spacetime
geometry and then determines the values of the free parameters by comparing
the theoretical predictions with astronomical observations of distant objects.
Currently, the most accepted cosmological model, the $\Lambda$CDM cosmology,
is based on this theory-based approach. It concludes that the universe is
almost entirely made up of dark matter and dark energy, whose compositions are
presently unknown.

The data-driven approach of observational cosmology claims that we are in
principle capable of determining the spacetime geometry on the null cone by
means of astrophysical observations, that is, using data available on the past
null cone (Kristian & Sachs 1966; Ellis et al. 1985). One specific study that
follows this direct manner methodology was advanced by Ellis & Perry (1979),
who developed a very detailed discussion connecting the galactic brightness
profiles with cosmological models. Their aim was to determine the spacetime
geometry of the universe by measuring the angular diameter distance
$d_{\scriptscriptstyle A}$, also known as area distance, of distant galaxies
through their surface brightness photometric data. Such a task was, however,
made very difficult due to lack of detailed knowledge about the structure and
evolution of galaxies. To this day this difficulty still remains.

Here we propose a method for testing cosmological models partially based on
Ellis & Perry (1979) methodology, but less ambitious than theirs. Our approach
differs from these authors in the sense that we do not aim to determine the
entire spacetime geometry from observations. Our goal is to seek consistency
between detailed astronomical observations of quantities describing actual
galactic structure as compared to their predictions made by a specific
cosmological model. To be precise, we start by assuming a cosmological model
and then discuss the consistency between the model’s predictions and actual
observations of the surface brightness of distant galaxies and other related
quantities. The idea is to obtain a glimpse of the structure and evolution of
galaxies by means of observations of our local universe, identify observing
parameters which, in principle, are cosmological-model independent, that is,
independent of the spacetime geometry, and then assume that galactic scaling
relations do not change significantly, say within $3\sigma$, with the
redshift. In this way we could select distant galaxies that form a homogeneous
class of objects, defined as a set of similar galactic properties which can be
found at different galactic evolutionary stages, such as morphology, so that
one can compare objects at different redshift values. This implies in assuming
that the variations in the scaling relations are related to the variations in
the intrinsic structural parameters of a previously selected homogeneous class
of objects (Ellis et al. 1984). By starting with a cosmological model we are
able to assess to what extent the assumed cosmology affects actual galaxy
evolution modeling carried out in extragalactic astrophysics where some
cosmological model, nowadays the $\Lambda$CDM, is implicitly assumed.

In this paper we address the first step in this approach of cosmological model
testing using actual galactic data, whose basic methodology was briefly
advanced elsewhere (Olivares-Salaverri & Ribeiro 2009, 2010). The purpose here
is to verify if sample galactic brightness profiles vary with predicted
theoretical ones when one changes the cosmological model, that is, if surface
brightness profiles are affected by the spacetime geometry. We adopt two
distinct cosmologies, the $\Lambda$CDM and, for simplicity in this initial
approach, Einstein-de Sitter (EdS) model. We then calculate photometric
scaling relations using Virgo cluster galaxies by means of the Kormendy et al.
(2009) data and use our theory to predict the galactic surface brightness at
high redshift values in the two cosmological models, assuming that these
scaling relations do not change with the redshift. We then compare these
predictions with a subsample of high redshift galactic surface brightness data
of Szomoru et al. (2012; from now on S12).

Our results show that the observed high-redshift galactic brightness profiles
differ from the theoretically predicted ones obtained by assuming that they
follow scaling relations derived from the Virgo cluster. Such a difference
occurs when the theoretical results are obtained by using both cosmological
models studied here. Therefore, for these galaxies to change their features to
the ones found in the Virgo cluster, an intrinsic evolution must take place.
Such an evolution is similar in both cosmological models. Consequently, our
results do not allow us to conclude which cosmology produces more or less
evolution or is more suitable to represent the process in which high redshift
galaxies develop into local galaxies having scaling relations similar to the
ones observed in the Virgo galactic cluster, at least as far the chosen
particular subset of S12 galaxies is concerned.

The outline of the paper is as follow. In §2 we discuss cosmological distance
measures and their connections to astrophysical observables and §3 shows how
the surface brightness of cosmological sources are connected to those distance
measures. We present in §4 the received surface brightness using the profile
due to Sérsic (1968). In §5 we calculate two photometric scaling relations of
the Virgo cluster so that the next section (§6) shows our comparison, using
the two cosmologies assumed here, between the prediction of the surface
brightness obtained by means of these scaling relations and the observations
of S12 high redshift galaxies. In §7 we calculate in both cosmological models
how galaxies whose redshift values are equal to the ones in our chosen
subsample of S12 observations would have to evolve to end up with features
similar to the galaxies in the Virgo cluster. Finally, in §8 we summarize the
results and present our conclusions.

##  2 Cosmological distances

We start by considering that source and observer are at relative motion to one
another. From the point of view of the source, the light beams that travel
along future null geodesics define a solid angle
$\mathrm{d}\Omega_{\scriptscriptstyle G}$ with the origin at the source and
have a transverse section area $\mathrm{d}\sigma_{\scriptscriptstyle G}$ at
the observer (Ellis 1971; see also Fig. 2 of Ribeiro 2005 where cosmological
distances are also discussed in some detail).

The specific radiative intensity $F_{em}$ is the emitted, or intrinsic,
radiation measured at the source in a unit 2-sphere $S_{unit}$ lying in the
locally Euclidean space at rest with the source and centered at it, also
assumed to radiate locally with spherical symmetry. It is related to the
intrinsic source luminosity $L$ by,

$L=\int_{S_{unit}}F_{em}{\mathrm{d}}\Omega_{\scriptscriptstyle G}=4\pi
F_{em}.$ (1)

Let us now define $F_{re}$ as the flux radiated by the source, but measured by
an observer located at some future time $t_{0}$ relative to the source. This
is, of course, the received flux in the area
$\mathrm{d}\sigma_{\scriptscriptstyle G}$ at rest with the observer and
implies a certain distance between source and observer, distance which is
geometrically defined along a null curve in an expanding spacetime where both
source and observer are located. Thus, the source luminosity is given by,

$L=\int_{S_{\\!\\!phys}}(1+z)^{2}F_{re}{\mathrm{d}}\sigma_{\scriptscriptstyle
G},$ (2)

where $z$ is the redshift and $S_{\\!\\!phys}$ is the physical surface
receiving the flux, e.g., a detector. The factor $(1+z)^{2}$ appears here
because both source and observer have their geometrical locus in a curved and
expanding spacetime (Ellis 1971). Now, it has long been known that the area
law establishes that the source intrinsic luminosity is independent from the
observer (Ellis 1971). Therefore, these two equations are equal, yielding,

$L=\int_{S}F_{em}\;{\mathrm{d}}\Omega_{\scriptscriptstyle
G}=\int_{S}(1+z)^{2}F_{re}\;{\mathrm{d}}\sigma_{\scriptscriptstyle G},$ (3)
$F_{em}\;{\mathrm{d}}\Omega_{\scriptscriptstyle
G}=const=(1+z)^{2}F_{re}\;{\mathrm{d}}\sigma_{\scriptscriptstyle G}.$ (4)

Considering the source’s viewpoint, we may define the galaxy area distance
$d_{\scriptscriptstyle G}$ as (Ellis 1971; see also Fig. 2 of Ribeiro 2005),

${\mathrm{d}}\sigma_{\scriptscriptstyle G}={d_{\scriptscriptstyle
G}}^{2}{\mathrm{d}}\Omega_{\scriptscriptstyle G}.$ (5)

Thus, equation (4) becomes,

$F_{re}=\frac{F_{em}}{{d_{\scriptscriptstyle
G}}^{2}(1+z)^{2}}=\frac{L}{4\pi}\frac{1}{{d_{\scriptscriptstyle
G}}^{2}(1+z)^{2}}.$ (6)

The factor $(1+z)^{2}$ can be understood as arising from (i) the energy loss
of each photon due to the redshift $z$, and (ii) the lower measured rate of
incoming photons due to time dilation (Ellis 1971). Since the galaxy area
distance $d_{\scriptscriptstyle G}$ appearing in equation (6) cannot be
measured as ${\mathrm{d}}\Omega_{\scriptscriptstyle G}$ is defined at the
source, we need to change this equation into another one containing measurable
quantities. This can be done as follows.

From the point of view of the observer, the light beams that travel along its
past null geodesics leave the source and converge to the observer, defining a
solid angle $d\Omega_{\scriptscriptstyle A}$ with the origin at the observer
and having transverse section area $d\sigma_{\scriptscriptstyle A}$ at the
source (Ellis 1971; see also Fig. 1 of Ribeiro 2005). Then we can define the
angular diameter distance $d_{\scriptscriptstyle A}$ as being given by,

${\mathrm{d}}\sigma_{\scriptscriptstyle A}={d_{\scriptscriptstyle
A}}^{2}{\mathrm{d}}\Omega_{\scriptscriptstyle A}.$ (7)

Now we can use the reciprocity theorem, due to Etherington (1933; see also
Ellis 1971, 2007), to relate $d_{\scriptscriptstyle G}$ to
$d_{\scriptscriptstyle A}$. This theorem is written as follows,

${d_{\scriptscriptstyle G}}^{2}=(1+z)^{2}{d_{\scriptscriptstyle A}}^{2}.$ (8)

Thus, it is now possible to connect the flux received by the observer and the
angular diameter distance by combining equations (6) and (8), yielding

$F_{re}=\frac{F_{em}}{{d_{\scriptscriptstyle A}}^{2}(1+z)^{4}}.$ (9)

The received flux $F_{re}$ and the redshift $z$ are astronomically measurable
quantities. So, if the angular diameter distance $d_{\scriptscriptstyle A}$ is
somehow determined astronomically, or obtained from theory as a function of
$z$, then the intrinsic flux $F_{em}$ and, therefore, the intrinsic luminosity
$L$ are both determined for all redshifts.

##  3 Connection with the surface photometry of cosmological sources

As discussed in §2, equation (9) connects the received and emitted fluxes of
sources located in a curved spacetime, but that expression is valid for point
like sources. Galaxies, however, form extended sources of light and their
characterization requires defining another quantity, the surface brightness,
better suited to describe them. The received surface brightness $B_{re}$ is
defined as the ratio between the received flux and the observed solid angle of
the galaxy,

$B_{re}\equiv\frac{F_{re}}{{\mathrm{d}}\Omega_{\scriptscriptstyle A}}.$ (10)

Considering equations (7) and (9), this expression can be rewritten as,

$B_{re}=\frac{F_{em}}{{\mathrm{d}}\sigma_{\scriptscriptstyle
A}}\frac{1}{(1+z)^{4}}.$ (11)

If we define the emitted surface brightness $B_{em}$ as the intrinsic flux of
the source $F_{em}$ per area unit in the rest frame of the source, we have
that

$B_{em}\equiv\frac{F_{em}}{{\mathrm{d}}\sigma{\scriptscriptstyle A}}.$ (12)

Thus, equation (11) in fact connects the received and emitted surface
brightness, as follows,

$B_{re}=\frac{B_{em}}{(1+z)^{4}}.$ (13)

This expression is simply the so-called Tolman surface brightness test for
cosmological sources, showing that galactic surface brightness does not depend
on the distance. This equation also shows that if there is no significant
cosmological effects, that is, if source and observer are close enough to be
considered at rest with one another and Newtonian approximation is valid, then
the source redshift is not significant. In this case there is no cosmological
contribution ($z\sim 0$) and $B_{re}=B_{em}$. This can be considered as a
consequence of the Liouville theorem (Bradt 2004). It is also worth mentioning
that several authors name the radiation measured by the observer as intensity
$I$, and the radiation emitted by the source as surface brightness $B$,
instead of terms adopted here, respectively, received surface brightness
$B_{re}$ and emitted surface brightness $B_{em}$. This is often the case in
texts where General Relativity is not considered.

Actual astronomical observations are carried out in observational bandwidths
and, therefore, the equations discussed so far should take this fact into
account. The specific received surface brightness $B_{re,\nu_{re}}$ gives the
amount of radiation received by the observer per unit solid angle measured at
the observer in the frequency range $\nu_{re}$ and
$\nu_{re}+{\mathrm{d}}\nu_{re}$. Clearly
$B_{re}=\int_{0}^{\infty}B_{re,\nu_{re}}{\mathrm{d}}\nu_{re}$. Considering
equation (13), we have that,

$B_{re,\nu_{re}}{\mathrm{d}}\nu_{re}=\frac{B_{em,\nu_{em}}}{(1+z)^{4}}\,{\mathrm{d}}\nu_{em},$
(14)

where we had defined the specific emitted surface brightness as follows,

$B_{em,\nu_{em}}=B_{em}J(\nu_{em}).$ (15)

Here $J(\nu)$ is the galactic spectral energy distribution (SED) giving the
proportion of radiation at each frequency, being normalized by the condition

$\int_{0}^{\infty}J(\nu)\,{\mathrm{d}}\nu=1.$ (16)

From our definitions it also follows that
$B_{em}=\int_{0}^{\infty}B_{em,\nu_{em}}{\mathrm{d}}\nu_{em}$. We need now to
relate the received and emitted frequencies. This is accomplished by the
definition of the redshift,

$\nu_{em}=\nu_{re}(1+z),$ (17)

implying that the SED of the source is observed according to
$J(\nu_{em})=J\left[\nu_{re}(1+z)\right]$ and equation (14) can be rewritten
as follows,

$B_{re,\nu_{re}}{\mathrm{d}}\nu_{re}=\frac{B_{em,\nu_{em}}}{(1+z)^{3}}\,{\mathrm{d}}\nu_{re}.$
(18)

The variables in the equation above depend on some implicit parameters. In
order to reveal these dependencies, let us start by assuming our galaxy as
having spherical symmetry with space points defined by the radius $R$.
Furthermore, if this galaxy has a circular projection in the celestial sphere,
any angle $\alpha$ measured by the observer corresponds to the radius $R$ in
the source by means of the angular diameter distance at a given redshift.
Hence, we have that (see Ellis & Perry 1979, Fig. 1),

$R(z)=d_{\scriptscriptstyle A}(z)\;\alpha.$ (19)

This expression is in fact a simplification of equation (7) where area and
solid angle are respectively approximated to length and angle so that
$d_{\scriptscriptstyle A}$ can be estimated observationally (Ellis 1971,
Ribeiro 2005). Indeed, it is used in observational cosmology tests under the
name “angular diameter redshift relation” since the angular diameter distance
has all cosmological information. So, different values of the angular diameter
distance are related to different cosmological models. As mentioned above, in
this paper the two chosen cosmological models are EdS and $\Lambda$CDM.

The EdS cosmology has zero curvature and no cosmological constant so, in this
model the angular diameter distance may be written as below,

$d_{\scriptscriptstyle
A,EdS}(z)=\frac{2c}{H_{0}(1+z)}\biggl{[}1-\frac{1}{\sqrt{(1+z)}}\biggr{]},$
(20)

where $c$ is the light speed and $H_{0}$ is the Hubble constant. This
expression shows that $d_{\scriptscriptstyle A,EdS}$ reaches a maximum value
at $z=1.25$ and then starts decreasing, asymptotically vanishing at the big
bang singularity hypersurface.

In the case of the $\Lambda$CDM model, several tests have been carried out in
the last few years to measure the parameter values of the model leading to
such a degree of accuracy that it became known as the concordance model, being
the most accepted cosmology nowadays. Those tests involve studies of the
cosmic microwave background radiation, baryonic acoustic oscillations and type
Ia supernovae. Komatsu et al. (2009) presented values for several parameters
in this cosmology to a high degree of accuracy, such as $H_{0}=71.8$ km s-1
Mpc-1, $\Omega_{m_{0}}=0.273$ and $\Omega_{\Lambda}=0.727$. We used these
values to calculate numerically the angular diameter distance in both
cosmologies, that is, $d_{\scriptscriptstyle A,\Lambda CDM}$ and
$d_{\scriptscriptstyle A,EdS}$, where the latter is evaluated directly from
equation (20).

Returning to equation (18), other implicit parameter dependence also occurs in
both specific surface brightness. The received one depends on the observed
parameters $\alpha$, $z$ and $\nu_{re}$, so that we should write it as
$B_{re,\nu_{re}}=B_{re,\nu_{re}}(\alpha,z)$. The specific emitted surface
brightness depends on the source parameters $R$, $\nu_{em}$ and, if allowed
for the intrinsic evolution of the source, also in the $z$. Thus,
$B_{em,\nu_{em}}=B_{em}(R,z)\,J[\nu_{re}(1+z),R,z]$. With these dependencies,
equation (18) turns out to be written as the expression below (Ellis & Perry
1979),

$B_{re,\nu_{re}}(\alpha,z)=\frac{B_{em}(R,z)}{(1+z)^{3}}\,J\left[\nu_{re}(1+z),R,z\right].$
(21)

Note that this equation is completely general, i.e., valid for any
cosmological model.

Next we shall show how the surface brightness $B_{em}(R,z)$ can be
characterized by means of the Sérsic (1968) profile and obtain an explicit
expression for the received surface brightness.

##  4 Received Sérsic surface brightness

The Sérsic profile was not commonly considered among astronomers after its
proposal. Gradually, however, some authors started to claim that the Sérsic
index is not simply a parameter capable of providing a better mathematical
fit, but that it does have a physical meaning (Ciotti 1991; Caon et al. 1993;
D’Onofrio et al. 1994). Nowadays, this profile seems to be more accepted since
one can find in the recent literature several papers using it as well as
relating its parameters to other astrophysical quantities (Davies et al. 1988;
Prugniel & Simien 1997; Ciotti & Bertin 1999; Trujillo et al. 2001; Mazure &
Capelato 2002; Graham 2001, 2002; Graham & Driver 2005; La Barbera et al.
2005; Coppola et al. 2009; Chakrabarty & Jackson 2009; Laurikainen et al.
2010). In view of this, we believe that this profile is the most suitable for
the purposes of this paper.

The Sérsic profile can be presented in two slightly different parametric
formats, although both of them characterize the same brightness profile. The
difference lies in the interpretation of the parameters. The first one can be
written as,

$B_{\mathrm{S_{1}},em}(R,z)=B_{0}(z)\,\exp\left\\{-\left[\frac{R(z)}{a(z)}\right]^{1/n}\right\\},$
(22)

where $B_{0}$ is the brightness amplitude, $a$ is the scalar radius and $n$ is
the Sérsic index. Ellis & Perry (1979) implicitly used this form. The second
way of writing the Sérsic profile is given by the following expression,

$B_{\mathrm{S_{2}},em}(R,z)=B_{eff}(z)\,\exp\left\\{\displaystyle-
b_{n}\left[\left(\frac{R(z)}{R_{eff}(z)}\right)^{1/n}-1\right]\right\\},$ (23)

where $R_{eff}(z)$ is the effective radius, $B_{eff}(z)$ is the brightness at
the effective radius and $b_{n}$ is a parameter dependent on the value of $n$.

The main difference between these two equations is the nature of their
parameters. In equation (22) $B_{0}$ and $a$ do not have a clear physical
meaning, whereas the parameters $B_{\scriptscriptstyle{eff}}$ and
$R_{\scriptscriptstyle{eff}}$ appearing in equation (23) are more easily
interpreted. $R_{\scriptscriptstyle{eff}}$ is defined as the isophote that
contains half of the total luminosity and $B_{\scriptscriptstyle{eff}}$ is the
value of the brightness in that radius (Ciotti 1991; Caon et al. 1993). For
this reason we believe that the second form above is more appropriate for our
analysis and we shall use it from now on.

A relationship between the different parameters in equations (22) and (23) can
be obtained by equating these two expressions, yielding,

$B_{0}(z)=B_{\scriptscriptstyle eff}(z)\,e^{b_{n}},$ (24)
$a(z)=\frac{R_{\scriptscriptstyle eff}(z)}{{b_{n}}^{n}}.$ (25)

The evolution of the galactic structure is implicit in this profile in view of
the fact that both $B_{eff}(z)$ and $R_{eff}(z)$ are redshift dependent. In
addition, the connection of this profile to a cosmological model and,
therefore, to the underlying curved spacetime geometry and its evolution
occurs in the intrinsic radius $R(z)$ by means of equation (19). Thus,
galaxies can possibly appear to experience two simultaneous evolutionary
effects, intrinsic source evolution and cosmological, or geometrical,
evolution. Since at our current knowledge of galactic structure both effects
cannot be easily separated, if they can be separated at all, from now on we
shall assume that $R(z)$ depends only on the underlying spacetime geometry
given by a chosen cosmological model. Furthermore, at first we shall not
consider a possible intrinsic evolution of the Sérsic index in the form
$n=n(z)$, because we assume that its change is produced via galactic merger
processes (Naab & Trujillo 2006). So, the way that the evolutionary dependency
is defined in the emitted surface brightness (eq. 23) means that we are
implicitly considering galaxies belonging to a set with similar properties, or
a homogeneous class of objects. Departures from this class occur by smooth
dependency in the evolution of the intrinsic parameters such as $B_{eff}(z)$
and $R_{eff}(z)$ (Ellis et al. 1984).

Let us now return to the properties of the Sérsic profile. There are
analytical and exact expressions relating to $b_{n}$ and $n$. Analytical
expressions for $b_{n}$ were given by several authors (Capaccioli 1989; Ciotti
1991; Prugniel & Simien 1997), whereas others worked out exact values for this
parameter (Ciotti 1991; Graham & Driver 2005; Mazure & Capelato 2002). Ciotti
& Bertin (1999) analyzed the exact value form and concluded that a fourth
order expansion is enough to obtain good results, which are even better than
the values obtained from the analytical expressions. Such an expansion is
enough for the purposes of this paper and may be written as below,

$b_{n}=2n-\frac{1}{3}+\frac{4}{405n}+\frac{46}{25515n^{2}}.$ (26)

Having expressed $B_{em}$ in terms of the Sérsic profile, we can now obtain
the equation for the received surface brightness $B_{re}(\alpha,z)$ since
equation (21) gives the relationship between the emitted and received surface
brightness. Considering the emitted brightness as modeled by the second form
of the Sérsic profile (eq. 23), we then substitute the latter equation into
the former and obtain the following expression,

$\displaystyle B_{re,\nu_{re}}(\alpha,z)$ $\displaystyle=$
$\displaystyle\frac{B_{eff}(z)}{(1+z)^{3}}\;J[\nu_{re}(1+z),R,z]\times$ (27)
$\displaystyle\times\exp{\left\\{-b_{n}\left[{\left(\frac{R(z)}{R_{eff}(z)}\right)}^{1/n}-1\right]\right\\}}.$

Let us now define two auxiliary quantities required to study this problem from
an extragalactic point of view (Caon et al. 1993; Graham & Driver 2005),

$\mu_{re,\nu_{re}}(\alpha,z)\equiv-2.5\log\left(B_{re,\nu_{re}}\right),$ (28)
$\mu_{eff}(z)\equiv-2.5\log\left[B_{eff}(z)\right].$ (29)

Equation (28) is given in units of $[\mu_{re,\nu_{re}}]$ = [mag/arc sec2].
Considering these definitions, equation (27) can be rewritten as,

$\displaystyle\mu_{re,\nu_{re}}(\alpha,z)$ $\displaystyle=$
$\displaystyle\mu_{eff}(z)+7.5\log(1+z)$ (30)
$\displaystyle-2.5\log\left\\{J\left[\nu_{re}(1+z),R,z\right]\right\\}$
$\displaystyle+\left[\frac{2.5}{\ln(10)}\right]b_{n}\left\\{\left[\frac{R(z)}{R_{eff}(z)}\right]^{1/n}-1\right\\}.$

The equation above allows us to predict the galactic surface brightness in a
given redshift and in a specific cosmological model. Nevertheless, in order to
relate this expression with actual observations, we still need further
assumptions regarding galactic structure. Next we shall make use of Kormendy
et al. (2009) data to derive photometric scaling relations of the Virgo galaxy
cluster and suppose that these relations do not change drastically out to the
redshift range under study here.

##  5 Photometric scaling relations of the Virgo Cluster

In order to compare theoretical predictions of the surface brightness profiles
with observational data we require information regarding the galactic
structure, information which can be obtained after investigating scaling
relations that different kinds of galaxies follow in the local universe. The
most well-known of those relations are the ones that relate luminosity with
velocity dispersion, such as the Faber-Jackson relation (Faber & Jackson 1976)
for elliptical galaxies, and the Tully-Fisher relation (Tully & Fisher 1977)
for spiral galaxies. Another, more general, scaling relation is the
fundamental plane (Djorgovski & Davis 1987) which correlates velocity
dispersion with effective brightness and effective radius instead of the
luminosity. It is worth mentioning that the fundamental plane is a
generalization of the Kormendy relation (Kormendy 1977).

The parameters involved in these scaling relations are measured through
spectroscopic and photometric techniques. After knowing that the Sérsic index
and the velocity dispersion correlate, a fundamental plane that only uses
photometric parameters was proposed by Graham (2002). This variant, called
photometric plane, uses parameters which appear in the definition of the
Sérsic profile, i.e., the Sérsic index $n$, the effective surface brightness
$\mu_{eff}$ and the effective radius $R_{eff}$. Since the study presented in
this paper deals with photometric parameters, we shall use the photometric
plane to obtain the galactic scaling relations required in our analysis.
Specifically, our focus will be on the galaxies belonging to the Virgo cluster
as their data (Kormendy et al. 2009) form the most exhaustively studied
galactic surface brightness dataset in the local Universe.

###  5.1 Virgo cluster data

Kormendy et al. (2009) studied 42 galaxies of the Virgo cluster on the V band,
$\lambda_{V,eff}=(5450\pm 880)$ Å, whose morphological types are elliptic (E),
lenticular (S0) and spheroid (Sph). They used the Sérsic profile to fit the
observed surface brightness and, in the authors’ words, “the Sérsic functions
fit the main parts of the profiles of both elliptical and spheroidal galaxies
astonishingly well on large ranges in surface brightness.”

As the observed data of the Virgo cluster is well fitted by the Sérsic
profile, the parameters obtained through these fittings can be considered as
reliable. Thus, using the parameters involved in the photometric plane, $n$,
$\mu_{eff}$ and $R_{eff}$, we carried out linear fittings relating $\mu_{eff}$
to $n$ and $R_{eff}$ to $n$. Plots showing these fittings are presented in
Fig. 1 and the fitted parameters are as follows,

$\mu_{eff,V}=(0.38\pm 0.09)n+(20.5\pm 0.5),$ (31) $\log R_{eff,V}=(0.21\pm
0.03)n-(0.5\pm 0.1).$ (32)

The index $\scriptstyle V$ stands for the Virgo cluster.

Figure 1: Left: Plot of the logarithm of the effective radius $R_{eff,V}$ vs.
the Sérsic index $n$ of galaxies from the Virgo cluster. The straight line
represents the linear fit whose values are given in eq. (32). Right: Plot of
the effective brightness $\mu_{eff,V}$ and the Sérsic index $n$ of galaxies
belonging to the Virgo cluster. The straight line represents the linear fit
whose values are given in eq. (31). Both: Data taken from Kormendy et al.
(2009).

Analyzing the plots we can observe that most galaxies with low Sérsic index do
not correlate. These galaxies are mostly of spheroidal types and are the main
contributors to the large errors in the Y-axis. As our aim is to compare the
prediction of the surface brightness with high-redshift observational data, if
we knew the morphological types of these high-redshift galaxies we could
select specific galaxy types to obtain the scaling relation. But, this is
usually not possible because galactic morphology at high redshifts is not as
well established as in the local universe. Considering such constraint, we
resorted on using galaxies of the Virgo cluster whose scaling relations were
just derived in order to predict the surface brightness at any redshift and
compare it with high redshift galaxy observations of S12. On top of the
scaling relations, this is done by employing the two cosmological models
adopted here.

##  6 High-redshift vs. predicted galactic surface brightness profiles in
$\Lambda$CDM and EdS cosmologies

In order to analyze the possible effects that different cosmological models
can have in the estimated galactic evolution, we shall compare theoretical
predictions of the surface brightness data with high-redshift galactic
brightness profiles. An important issue concerning high redshift galactic
surface brightness is the depth in radius of these images. Observing the
universe at high redshift involves lower resolution images, so obtaining a
good sample of the brightness profiles as complete as possible in radius is
not an easy task.

###  6.1 The high redshift galactic data of Szomoru et al. (2012)

S12 were mainly interested in investigating if the interpretation of the
compactness of high redshift galaxies was due to a lack of deep images in
radius which would lead to a misinterpretation of the compactness pattern.
They observed a stellar mass limited sample of 21 quiescent galaxies in the
redshift range $1.5<z<2.5$ having ${\cal{M}}_{*}>5.10^{10}{\cal{M}}_{\odot}$
and obtained their surface brightness using the Sérsic profile with high
values in radius. We chose S12 data for our purposes mainly because of these
features.

S12 used NIR data taken with HST WFC3 as part of the CANDELS survey.
Specifically, the surface brightness profiles were obtained in the $H_{160}$
band, $\lambda_{H,eff}=(13923\pm 3840)$ Å, that is, comparable with the $V$
band rest-frame in the redshift range under consideration. From the 21
galaxies we have selected a subsample having Sérsic indexes grouped in three
sets: $n\sim 1$, $n\sim 4$ and $n\sim 5$. The last two sets are more numerous
and are located in several redshift values, especially the group with $n\sim
5$. Table LABEL:table1 shows our subsample with the identification number in
the S12 catalog and their respective Sérsic indexes and the redshifts.

Table 1: Identification (ID) number, Sérsic index $n$ and redshift of the galaxy subsample selected from S12 and used in this paper. ID | $n$ | $z$
---|---|---
2.856 | 1.20 $\pm$ 0.08 | 1.759
3.548 | 3.75 $\pm$ 0.48 | 1.500
2.531 | 4.08 $\pm$ 0.30 | 1.598
3.242 | 4.17 $\pm$ 0.45 | 1.910
3.829 | 4.24 $\pm$ 1.15 | 1.924
1.971 | 5.07 $\pm$ 0.31 | 1.608
3.119 | 5.09 $\pm$ 0.60 | 2.349
6.097 | 5.26 $\pm$ 0.56 | 1.903
1.088 | 5.50 $\pm$ 0.67 | 1.752

###  6.2 Theoretical predictions of the surface brightness

As discussed above, equation (30) allows us to calculate the surface
brightness theoretical prediction of a hypothetical galaxy in a given
redshift. We have already calculated scaling relations for local galaxies
which we assume to be maintained out to the observed S12 redshift values.
However, to see if the observed high redshift galaxies behave like the Virgo
cluster ones, i.e., if they obey the assumed scaling relations, we must
calculate the errors of the theoretical predictions in eq. (30).

Errors for $\Delta\mu_{re,\nu_{re}}$ are estimated quadratically as follows,

$\displaystyle\Delta\mu_{re,\nu_{re}}$ $\displaystyle=$
$\displaystyle\biggl{[}\biggl{(}\frac{\partial\mu_{re,\nu_{re}}}{\partial\mu_{eff}}\Delta\mu_{eff}\biggr{)}^{2}+\biggr{(}\frac{\partial\mu_{re,\nu_{re}}}{\partial
z}\Delta z\biggr{)}^{2}+\biggr{(}\frac{\partial\mu_{re,\nu_{re}}}{\partial
n}\Delta n\biggr{)}^{2}$ (33)
$\displaystyle+\biggr{(}\frac{\partial\mu_{re,\nu_{re}}}{\partial R}\Delta
R\biggr{)}^{2}+\biggr{(}\frac{\partial\mu_{re,\nu_{re}}}{\partial\log(R_{eff})}\Delta\log(R_{eff})\biggr{)}^{2}\biggl{]}^{1/2}.$

Let us analyze individually each of the uncertainty terms in this expression.

####  6.2.1 $\Delta\mu_{eff}$

The effective surface brightness is calculated from the linear fit of the
Virgo cluster galaxy scaling relation,

$\mu_{eff}=A_{\mu_{eff}}n+B_{\mu_{eff}},$ (34)

where $A$ and $B$ are the linear fit parameters given in equation (31). Its
uncertainty yields,

$\Delta\mu_{eff}=\biggl{[}\biggl{(}\frac{\partial\mu_{eff}}{\partial
A_{\mu_{eff}}}\Delta
A_{\mu_{eff}}\biggr{)}^{2}+\biggl{(}\frac{\partial\mu_{eff}}{\partial n}\Delta
n\biggr{)}^{2}+\biggl{(}\frac{\partial\mu_{eff}}{\partial B_{\mu_{eff}}}\Delta
B_{\mu_{eff}}\biggr{)}^{2}\biggr{]}^{1/2}.$ (35)

This expression can can be rewritten as below,

$\Delta\mu_{eff}=\biggl{[}(n\Delta A_{\mu_{eff}})^{2}+(A_{\mu_{eff}}\Delta
n)^{2}+(\Delta B_{\mu_{eff}})^{2}\biggr{]}^{1/2},$ (36)

where $\Delta A$ and $\Delta B$ come from the linear fit and $\Delta n$ comes
from the observation.

####  6.2.2 $\Delta z$

Redshift uncertainty was given by S12 only if $z$ was measured
photometrically. However, some of S12’s galaxies had their redshifts measured
spectroscopically, then the errors were not made available. Since there is a
mixture of photometric and spectroscopic redshifts in S12’s sample, we decided
to avoid including them in our calculations and have effectively assumed
$\Delta z\sim 0$.

####  6.2.3 $\Delta n$

The value of Sérsic index and its error came directly from the observations
shown in S12 (see Table LABEL:table1).

####  6.2.4 $\Delta R$

The projected radius is obtained by means of equation (19), whose quadratic
uncertainty may be written as follows,

$\Delta R=\biggl{[}\biggl{(}\frac{\partial
R}{\partial\alpha}\Delta\alpha\biggr{)}^{2}+\biggl{(}\frac{\partial
R}{\partial d_{A}}\Delta d_{A}\biggr{)}^{2}\biggr{]}^{1/2}.$ (37)

The angle uncertainty $\Delta\alpha$ was neglected by S12, so
$\Delta\alpha\sim 0$. Regarding the angular diameter distance, the small
uncertainties in the parameters of the cosmological models are such that once
propagated they change very little the value of $d_{A}$. So, we effectively
have $\Delta d_{A}\sim 0$.

####  6.2.5 $\Delta\log(R_{eff})$

Similarly to $\Delta\mu_{eff}$, the uncertainty $\Delta\log(R_{eff})$ is
derived from the linear fit of the Virgo galaxy cluster,

$\log(R_{eff})=A_{\log(R_{eff})}n+B_{\log(R_{eff})},$ (38)

where $A$ and $B$ are the parameters given by equation (32). Therefore,

$\displaystyle\Delta\log(R_{eff})$ $\displaystyle=$
$\displaystyle\biggl{\\{}\biggl{[}\frac{\partial\log(R_{eff})}{\partial
A_{\log(R_{eff})}}\Delta
A_{\log(R_{eff})}\biggr{]}^{2}+\biggl{[}\frac{\partial\log(R_{eff})}{\partial
n}\Delta n\biggr{]}^{2}$ (39)
$\displaystyle+\biggl{[}\frac{\partial\log(R_{eff})}{\partial
B_{\log(R_{eff})}}\Delta B_{\log(R_{eff})}\biggr{]}^{2}\biggr{\\}}^{1/2},$

which may be rewritten as,

$\Delta\log(R_{eff})=\biggl{\\{}\bigl{[}n\Delta
A_{\log(R_{eff})}\bigr{]}^{2}+(A\Delta n)^{2}+\bigl{[}\Delta
B_{\log(R_{eff})}\bigr{]}^{2}\biggr{\\}}^{1/2}.$ (40)

Just like in $\Delta\mu_{eff}$, $\Delta A_{\log(R_{eff})}$ and $\Delta
B_{\log(R_{eff})}$ come from the linear fit and $\Delta n$ is given by
observations.

####  6.2.6 $\Delta\mu_{re,\nu_{re}}$

Putting together all these expressions for uncertainties in equation (33) and
remembering the relationship between $b_{n}$ and $n$ (eq. 26), the uncertainty
in the theoretical prediction of the surface brightness yields,

$\displaystyle\Delta\mu_{re,\nu_{re}}$ $\displaystyle=$
$\displaystyle\Biggl{(}(n\Delta A_{\mu_{eff}})^{2}+(A_{\mu_{eff}}\Delta
n)^{2}+(\Delta B_{\mu_{eff}})^{2}+$ (41)
$\displaystyle+\biggl{[}\biggl{(}\frac{2.5}{\ln
10}\frac{{\mathrm{d}}b_{n}}{{\mathrm{d}}n}\biggl{\\{}\bigg{[}\frac{R}{10^{\log(R_{eff})}}\biggr{]}^{1/n}-1\biggr{\\}}-$
$\displaystyle-\frac{2.5b_{n}}{n^{2}\ln
10}\biggl{[}\frac{R}{10^{\log(R_{eff})}}\biggr{]}^{1/n}\log\biggl{\\{}\frac{R}{10^{\log(R_{eff})}}\biggr{\\}}\biggr{)}\Delta
n\biggr{]}^{2}+$
$\displaystyle+\biggl{\\{}\frac{2.5b_{n}}{n}\biggl{[}\frac{R}{10^{\log(R_{eff})}}\biggr{]}^{1/n}\biggr{\\}}^{2}\biggl{\\{}[n\Delta
A_{\log(R_{eff})}]^{2}+$ $\displaystyle+[A_{\log(R_{eff})}\Delta
n]^{2}+[\Delta B_{\log(R_{eff})}]^{2}\biggr{\\}}\Biggr{)}^{1/2}.$

####  6.2.7 Comparing theory and observation

Before we can actually compare our S12 galaxy subsample with the theoretical
predictions of the surface brightness profile, we still need to estimate the
spectral energy distribution (SED) $J$. We proceed on this point from the very
simple working assumption of a constant value for the SED, since using the
overall energy distribution of the galaxies in the Virgo cluster in different
bandwidths is beyond the aims of this paper. Henceforth, we assume the working
value of $J=0.5$.

The scaling relations obtained from the galaxies of the Virgo cluster, eqs.
(31) and (32), allow us to calculate $\mu_{eff}$ and $R_{eff}$ for a given
Sérsic index value. So, these two equations may be rewritten as,

$\mu_{eff,V}=0.38n+20.5,$ (42)

where its error is given by,

$\Delta\mu_{eff,V}=0.09n+0.5,$ (43)

and

$\log R_{eff,V}=0.21n-0.5,$ (44)

whose uncertainty is,

$\Delta\log R_{eff,V}=0.03n+0.1.$ (45)

Graphs showing the S12 data, labeled as “Obs”, and the theoretical predictions
of the surface brightness profile, labeled as “Pre”, in each of the two
cosmological models adopted in this work, $\Lambda$CDM and Einstein-de Sitter,
are shown in Figs. 2 to 11. One can clearly see that the observed and
predicted results are very different, a result which shows that the Virgo
cluster galaxies do not behave as our S12 subsample. So, for our high redshift
galaxies to evolve into the Virgo cluster ones the scaling relations
parameters have to change in order to reflect such an evolution. One can also
see that the difference in the results when employing each of the two
cosmological models is not at all significant. Thus, our next step is to work
out the changes in the parameter to find out if the evolution these galaxies
have to sustain is strongly dependent on the assumed underlying cosmology.

Figure 2: Both: galaxy 2.856 from S12 labeled as “Obs” at $z=1.759$ with
$n=1.2\pm 0.08$. Theoretical prediction using eqs. (30) and (41) assuming the
scaling relations of Virgo cluster galaxies, labeled as “Pre”. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.

Figure 3: Both: galaxy 2.531 galaxy from S12 labeled as “Obs” at $z=1.598$
with $n=4.08\pm 0.3$. Theoretical prediction using eqs. (30) and (41) assuming
the scaling relations of Virgo cluster galaxies, labeled as “Pre”. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.

Figure 4: Both: galaxy 3.242 from S12 labeled as “Obs” at $z=2.47$ with
$n=4.17\pm 0.45$. Theoretical prediction using eqs. (30) and (41) assuming the
scaling relations of Virgo cluster galaxies, labeled as “Pre”. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.

Figure 5: Both: galaxy 3.548 from S12 labeled as “Obs” at $z=1.5$ with
$n=3.75\pm 0.48$. Theoretical prediction using eqs. (30) and (41) assuming the
scaling relations of Virgo cluster galaxies, labeled as “Pre”. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.

Figure 6: Both: galaxy 3.829 from S12 labeled as “Obs” at $z=1.924$ with
$n=4.24\pm 1.15$. Theoretical prediction using eqs. (30) and (41) assuming the
scaling relations of Virgo cluster galaxies, labeled as “Pre”. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.

Figure 7: Both: galaxy 1.088 from S12 labeled as “Obs” at $z=1.752$ with
$n=5.5\pm 0.67$. Theoretical prediction using eqs. (30) and (41) assuming the
scaling relations of Virgo cluster galaxies, labeled as “Pre” points. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.

Figure 8: Both: galaxy 1.971 from S12 labeled as “Obs” at $z=1.608$ with
$n=5.07\pm 0.31$. Theoretical prediction using eqs. (30) and (41) assuming the
scaling relations of Virgo cluster galaxies, labeled as “Pre”. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.

Figure 9: Both: galaxy 2.514 from S12 labeled as “Obs” at $z=1.548$ with
$n=5.73\pm 0.93$. Theoretical prediction using eqs. (30) and (41) assuming the
scaling relations of Virgo cluster galaxies, labeled as “Pre”. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.

Figure 10: Both: galaxy 3.119 from S12 labeled as “Obs” at $z=2.349$ with
$n=5.09\pm 0.6$. Theoretical prediction using eqs. (30) and (41) assuming the
scaling relations of Virgo cluster galaxies, labeled as “Pre”. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.

Figure 11: Both: galaxy 6.097 from S12 labeled as “Obs” at $z=1.903$ with
$n=5.26\pm 0.56$. Theoretical prediction using eqs. (30) and (41) assuming the
scaling relations of Virgo cluster galaxies, labeled as “Pre”. Left:
Considering $\Lambda$CDM cosmological model. Right: Considering Einstein-de
Sitter cosmological model.

##  7 Evolution to the Virgo cluster scaling parameters in $\Lambda$CDM and
EdS cosmologies

As seen above, the theoretical prediction of the surface brightness profiles
obtained through scaling relations derived from data of the Virgo galactic
cluster are very different from the observed surface brightness of S12
galaxies in both cosmological models studied here. Hence, in order to
ascertain the possible evolution that these galaxies would have to experience
such that they end up with surface brightness equal to the ones in the Virgo
cluster, we need to look carefully at the parameter evolution of the scaling
relations. Assuming that they do not evolve through the Sérsic index $n$ (see
above), we can study such an evolution via the effective brightness and the
effective radius.

Let $\mu_{eff,evo}$ and $\log R_{eff,evo}$ be, respectively, the evolution of
the effective brightness and the effective radius. We can estimate these two
quantities similarly to our previous calculation of the scaling relations. For
values of the Sérsic index of our S12 galactic subsample, we can obtain their
respective Virgo cluster galaxies effective brightness and effective radius
$\mu_{eff,V}$ and $\log R_{eff,V}$ by means of the expressions (42) to (45).
We then modify the linear fit parameters $A$ and $B$ in these expressions
($y=Ax+B$), adjusting them so that the results approximate the points of the
theoretical predictions with S12’s observed values in order to find
$\mu_{eff,Szo}$ and $\log R_{eff,Szo}$, that is, the values of effective
brightness and effective radius of our subsample of S12’s galaxies. Therefore,
both $\mu_{eff,evo}$ and $\log R_{eff,evo}$ are given as follows,

$\mu_{eff,evo}=\mu_{eff,Szo}-\mu_{eff,V},$ (46) $\log R_{eff,evo}=\log
R_{eff,Szo}-\log R_{eff,V}.$ (47)

The respective uncertainties yield,

$\Delta\mu_{eff,evo}=\Delta\mu_{eff,Szo}+\Delta\mu_{eff,V},$ (48) $\Delta\log
R_{eff,evo}=\Delta\log R_{eff,Szo}+\Delta\log R_{eff,V}.$ (49)

Table LABEL:table2 shows the effective brightness and effective radius for the
Virgo cluster galaxies obtained with Sérsic indexes $n$ equal to those in our
S12 subsample presented in Table LABEL:table1. These quantities were obtained
using the Virgo scaling relations.

Table 2: Effective brightness $\mu_{eff,V}$ and effective radius $\log R_{eff,V}$ obtained with Virgo cluster galaxies scaling relations for values of $n$ equal to those in our subsample of S12 galaxies. ID | $\mu_{eff,V}$ | $\log R_{eff,V}$
---|---|---
2.856 | 21.0 $\pm$ 0.7 | -0.3 $\pm$ 0.2
3.548 | 21.9 $\pm$ 0.9 | 0.3 $\pm$ 0.3
2.531 | 22.1 $\pm$ 0.9 | 0.4 $\pm$ 0.3
3.242 | 22.1 $\pm$ 0.9 | 0.4 $\pm$ 0.3
3.829 | 22.1 $\pm$ 0.9 | 0.4 $\pm$ 0.3
1.971 | 22 $\pm$ 1 | 0.6 $\pm$ 0.3
3.119 | 22 $\pm$ 1 | 0.6 $\pm$ 0.3
6.097 | 22 $\pm$ 1 | 0.6 $\pm$ 0.3
1.088 | 23 $\pm$ 1 | 0.6 $\pm$ 0.3

Table LABEL:table3 shows the same quantities for the adjusted parameters of
our subsample of S12 galaxies using the two cosmological models considered
here and Table LABEL:table4 presents the evolution of the effective brightness
and effective radius calculated using eqs. (46) to (49) in both cosmological
models in the V band.

Table 3: Effective brightness $\mu_{eff,Szo}$ and effective radius $\log R_{eff,Szo}$ of the adjusted parameters from the galaxies selected from S12 for values of $n$ equal to the ID galaxies in the V band in the two cosmological models considered in this paper. ID | $\mu_{eff,Szo,\Lambda CDM}$ | $\log R_{eff,Szo,\Lambda CDM}$ | $\mu_{eff,Szo,EdS}$ | $\log R_{eff,Szo,EdS}$
---|---|---|---|---
2.856 |  16.5 $\pm$ 0.6 |  -0.1 $\pm$ 0.1 |  16.7 $\pm$ 0.6 |  -0.1 $\pm$ 0.1
3.548 |  16.5 $\pm$ 0.8 |  -0.2 $\pm$ 0.3 |  16.4 $\pm$ 0.8 |  -0.3 $\pm$ 0.2
2.531 |  16.7 $\pm$ 0.9 |  0.0 $\pm$ 0.3 |  17.1 $\pm$ 0.9 |  -0.1 $\pm$ 0.3
3.242 |  14.9 $\pm$ 0.9 |  -0.4 $\pm$ 0.3 |  14.3 $\pm$ 0.9 |  -0.6 $\pm$ 0.3
3.829 |  18.1 $\pm$ 0.9 |  0.2 $\pm$ 0.2 |  17.1 $\pm$ 0.9 |  -0.1 $\pm$ 0.3
1.971 |  19.3 $\pm$ 0.9 |  0.5 $\pm$ 0.3 |  20 $\pm$ 1 |  0.5 $\pm$ 0.3
3.119 |  16 $\pm$ 1 |  -0.1 $\pm$ 0.3 |  16 $\pm$ 1 |  -0.3 $\pm$ 0.3
6.097 |  17 $\pm$ 1 |  0.3 $\pm$ 0.3 |  18 $\pm$ 1 |  0.3 $\pm$ 0.3
1.088 |  16 $\pm$ 1 |  17 $\pm$ 1 |  17 $\pm$ 1 |  -0.2 $\pm$ 0.3
Table 4: Evolution of the effective brightness $\mu_{eff,evo}$ and effective radius $\log R_{eff,evo}$ of the galaxies selected from S12 for values of $n$ equal to the ID galaxies in the V band in the two cosmological models considered here. ID | $\mu_{eff,evo,\Lambda CDM}$ | $\mu_{eff,evo,EdS}$ | $\log R_{eff,evo,\Lambda CDM}$ | $\log R_{eff,evo,EdS}$
---|---|---|---|---
2.856 | -5 $\pm$ 2 | -4 $\pm$ 2 | 0.2 $\pm$ 0.3 | 0.2 $\pm$ 0.3
3.548 | -6 $\pm$ 2 | -6 $\pm$ 2 | -0.4 $\pm$ 0.4 | -0.6 $\pm$ 0.5
2.531 | -5 $\pm$ 2 | -5 $\pm$ 2 | -0.4 $\pm$ 0.5 | -0.5 $\pm$ 0.5
3.242 | -7 $\pm$ 2 | -7 $\pm$ 2 | -0.7 $\pm$ 0.5 | -1.0 $\pm$ 0.4
3.829 | -4 $\pm$ 2 | -5 $\pm$ 2 | -0.2 $\pm$ 0.3 | -0.5 $\pm$ 0.5
1.971 | -3 $\pm$ 2 | -2 $\pm$ 2 | 0.0 $\pm$ 0.5 | 0.0 $\pm$ 0.5
3.119 | -6 $\pm$ 2 | -6 $\pm$ 2 | -0.7 $\pm$ 0.5 | -0.8 $\pm$ 0.5
6.097 | -5 $\pm$ 2 | -4 $\pm$ 2 | -0.3 $\pm$ 0.5 | -0.2 $\pm$ 0.5
1.088 | -6 $\pm$ 2 | -6 $\pm$ 2 | -0.7 $\pm$ 0.6 | -0.8 $\pm$ 0.6

The results show that the evolution that will have to occur so that the S12
high redshift galaxies have effective brightness and effective radius equal to
the ones in the Virgo cluster is similar in both cosmological models.
Specifically, it seems that in EdS model the effective radius evolution is
higher than the one occurred in the $\Lambda$CDM model. Nevertheless, the
evolution of the effective surface brightness is almost the same in both
models. We also note that the difference in the Sérsic index values do not
appear to affect our results, which also clearly show that the uncertainties
in the measurements of both quantities we deal with here are just too high to
allow us to distinguish the underlying cosmological model that best represents
the data. Basically our methodology is limited by the uncertainties, at least
as far as the S12 subset data is concerned.

As final words, this work is based on the assumption that the galaxies we
compare belong to a group whose members share at least one common feature
regardless of the redshift, otherwise comparison among them becomes
impossible. In other words, our basic assumption is that our selected galaxies
belong to a homogeneous class of galaxies. In the analysis we carried out
above we grouped our galaxies using only the Sérsic indexes, this therefore
being our defining criterion of a homogeneous class of objects. So, in a sense
we followed Ellis & Perry (1979) and adopted morphology as our definition of a
homogeneous class.

##  8 Summary and conclusions

In this paper we have compared high redshift surface brightness observational
data with theoretical surface brightness predictions for two cosmological
models, namely the $\Lambda$CDM and Einstein-de Sitter, in order to test if
such comparison allows us to distinguish the cosmology that best fits the
observational data. We started by reviewing the expressions for the emitted
and received bolometric source brightness and then obtained their respective
specific expressions in the context of galactic surface brightness (Ellis
1971; Ellis & Perry 1979).

Using the Sérsic profile, we have obtained scaling relations between the
surface effective brightness $\mu_{eff}$ and Sérsic index, as well as between
the effective radius $R_{eff}$ and $n$, for the Virgo cluster galaxies using
Kormendy et al. (2009) data. Assuming this scaling relation, we have
calculated theoretical predictions of the surface brightness and compared them
with some of the observed surface brightness profiles of high-redshift
galaxies in a subsample of Szomoru et al. (2012) galaxies in the two
cosmological models considered here. Our results showed that although the
Sérsic profile fits well the observed brightness, the results for surface
brightness is different from the theoretical predictions. Such difference was
used to calculate the amount of evolution that the high redshift galaxies
would have to experience in order to achieve the Virgo cluster structure once
they arrive at $z\sim 0$. We concluded from our results that the cosmological
evolution is quite similar in the two models considered in this paper. We also
noted that galaxies having different Sérsic indexes do not seem to follow a
different evolutionary path.

Overall, with the data and errors available for the chosen subset of galactic
profiles used here we cannot distinguish between the two different
cosmological models assumed in this work. That is, assuming that the high
redshift galaxies will evolve to have features similar to the ones found in
the Virgo cluster, is it not possible to conclude which cosmological model
will predict theoretical surface brightness curves similar to the observed
ones due to the high uncertainties in the data used here. We also noted that
the Sérsic index does not seem to play any significant evolutionary role, as
the evolution we discussed is apparently not affected by the value of $n$.
Nevertheless, this work used only the Sérsic index to define a homogeneous
class of objects.

Summing up those results, it seems reasonable that future studies of this kind
should also select galaxies based on other features besides morphology in
order to increase the number of common properties between high and low
redshift galaxies, and, of course, using different data samples than those
adopted here. More common features such as the Sérsic index are essential for
a better definition of a homogeneous class of cosmological objects whose
observational features are possibly able to distinguish among different
cosmological scenarios. However, care should be taken to avoid features which
can possibly suffer dramatic cosmological evolution.

## Acknowledgments

I.O.-S. is grateful to the Brazilian agency CAPES for financial support.

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