11footnotetext: Escuela de Ingeniería y Ciencias (EIC), Instituto Tecnológico
y de Estudios Superiores de Monterrey (ITESM), Monterrey, Nuevo León, México.
<EMAIL_ADDRESS>Instituto de Investigaciones en
Matemáticas Aplicadas y en Sistemas (IIMAS), Universidad Nacional Autónoma de
México (UNAM), Ciudad de México, México.
<EMAIL_ADDRESS>Primary 00A05, Secondary 00A66.

# Computing Fundamental Constants in the FLRW Universe using the Hawking
Radiation of the Cosmological Horizon

A. Meza1 and P. Padilla2

###### Abstract.

In this work, we compute the universe temperature for the cosmological horizon
for the FLRW metric. For this purpose, we consider a scalar field on the
cosmological horizon. This scalar field satisfies the Klein-Gordon equation in
a curved space-time. Recently, some authors like Barrow, Bekenstein and others
have proposed that the fundamental constants might vary with time. Along these
lines of thought, we derive the electromagnetic momentum and electromagnetic
field for the FLRW universe. This enables us to obtain a temporal dependence
of the Hubble parameter which, in turn, induces a time dependence of the
fundamental constants. In order to validate this model, the theoretical
predictions are then compared with observational data as a function of the
redshift, $z$, for the universe expansion. As a consequence, this time
dependence on the fundamental constants makes it possible to predict a change
in time of the informational content for the entropy of the universe surface
area, this is expressed as the bit number by using the holographic principle.

## 1\. Introduction

In recent years the possibility has been considered by various authors ([1],
[2], [3], [4], [5], [6], [7], [8] and [9]) that the fundamental constants
should change in time with the expansion of the universe. This problem about
the variation of fundamental constants has allowed multiple studies based on
the hypothesis of large Dirac numbers [10], [11], [12]. Even the theories of
quantum gravity and the attempts to unify the four fundamental forces take
into consideration the study of the temporal variation of the fundamental
constants [1], [2], [3], [4], [5], [6], [7], [8] and [9].

Different physical aspects that have been taken into account when analyzing
the variation of the fine structure constant are covariance, gauge invariance,
and causality (invariance in the face of temporal reversals). One of the few
ways to contrast unification theories is the study of the variation of the
fundamental constants. This is because the standard model of particle physics
and general relativity involve certain parameters that have the particularity
of remaining invariant in space-time and in a reference system, these
parameters are the fundamental constants [1], [2], [3], [4], [5], [6], [7],
[8] and [9].

In 1937 Dirac formulated the hypothesis that the fundamental constants are a
function of the age of the universe, that is, these constants can present
spatial and temporal variation with respect to the expansion of the universe
[10], [11], [12]. The standard model and general relativity have different
predictions for the variation of these parameters [1], [2], [3], [4], [5],
[6], [7], [8] and [9]. These parameters are not determined by theories but are
the result of experimental measurements [1], [2], [3], [4], [5], [6], [7], [8]
and [9].

Dirac’s observation to formulate his hypothesis of large numbers was that the
order of magnitude of the quotient between the electromagnetic and
gravitational forces between a proton and an electron coincided with the ratio
between the age of the universe and the time it takes for light to pass
through a hydrogen atom $(~{}10^{40})$ [10], [11], [12].

In this sense, quantum gravity theories such as superstring theories and M
theory predict the variation of fundamental constants in space-time; this can
be studied experimentally only at low energies [13] and [15].

Some of these theories are Bekenstein’s theories for the fine structure
constant $\alpha$ ([3], [4]). Also included are theories where the speed of
light $c$ is variable in time and space [16], [17], [18] and [19].

In these theories, the constant whose variation is to be studied is replaced
by a scalar field responsible for this variation. The prediction of the
variation of the constants is deduced from the dynamics of the field. In the
case of Bekenstein’s theory, the variation of the fine structure constant
$\alpha$ is studied based on the variation of an action that is a function of
the Planck mass $m_{pl}$ and a scalar field $\phi$ [3], [4].

On the other hand, it has recently been found in [14] that for the FLRW
metric, it is possible to consider the cosmological horizon. In this
cosmological horizon, the temperature belonging to Hawking radiation can be
measured. For this, it was considered that near the cosmological horizon,
there is the presence of a scalar field described by the Klein-Gordon equation
in curved space-time for the FLRW metric. The solution of this Klein-Gordon
equation is also the solution of the wave equation. From this wave equation,
we can understand that near the horizon there may be incoming or outgoing
waves that represent the creation and destruction of particles and
antiparticles. By having this, the strength of the incoming and outgoing waves
for the horizon was considered, with which the states of the particles and
antiparticles were obtained, this allowed, using a scalar product, to find a
thermal spectrum similar to that of a black body and with it find the
temperature belonging to the Hawking radiation for the cosmological horizon;
this temperature was found based on the Hubble parameter [14].

It is also possible thanks to the study of the holographic principle and the
$AdS/CFT$ correspondence that has been formulated by several authors: [20],
[21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33],
[34], [35], [36] and [37] among others. The temperature of black holes given
by Hawking radiation [25] can be associated with its entropy which is a
function of the surface area of the above mentioned black hole. In the
holographic principle, it is also stated that for every four Planck units
there is at least one degree of freedom (or a constant Boltzmann unit $k$ of
maximum entropy). This is known as the Bekenstein boundary $S<A/4$ [21], [22],
[23]. In a broader and more speculative sense, the theory suggests that the
entire universe can be viewed as a two-dimensional information structure
”painted” on the cosmological horizon, such that the three dimensions that are
observed would only be an effective description at macroscopic scales and low
energies; so the universe would actually be a hologram [30], [31]. In that
sense, Wheeler suggested that the number of bits for the universe is
$N=8\times 10^{88}$ [35], this quantity will be considered in our
calculations.

Now, the QED of the standard model is a gauge theory that allows us to
describe the electromagnetic interaction [38], [39], [40], [41], [42], [43],
[44], [45], [46] and [47]. Thanks to this theory, it is possible to compute
the Euler-Lagrange equations from the Klein-Gordon action in curved space-time
coupled to the electromagnetic field [48]. It has previously been possible to
find the Ward-Takahashi equation for the 4-momentum of the electromagnetic
field for the universe which must be solved numerically [49], [50] and [61].

In order to consider the contribution of the electromagnetic field to the
Hubble parameter, in this paper it will be seen that it is possible to take
into account this parameter as a function of the 4-momentum for the
electromagnetic field. This depends on the electromagnetic potential for the
radius of the universe (on the cosmological horizon). By obtaining this
factor, it is thus possible to use the expression for the temperature in the
cosmological horizon given by [14] as a function of the expansion of the
electromagnetic field given by the expansion of the universe.

Thanks to the above and considering the hypothesis of large Dirac numbers
[10], [11], [12], an expression will be obtained for the Hubble parameter as a
function of the temperature of the universe. Various fundamental constants
that arise from the electromagnetic interaction, such as the fine structure
constant $\alpha$, the electron charge $e$, the electron mass $m_{e}$, the
redshift $z$, the speed of light $c$ and the permeability of the vacuum
$\epsilon_{0}$ will be considered. From these values it will be possible to
address the change in the fundamental constants depending on the expansion of
the universe [10], [11], [12]. The latter can be analyzed as a case study
using astronomical data thanks to the concept of apparent magnitude.

In section 2, a brief review will be given about how Ya-Peng Hu [14]
calculated the temperature from the Hawking radiation for the cosmological
horizon for the Klein-Gordon equation in curved space-time. This will be
compared with the Bekenstein-Hawking entropy and the holographic principle to
give an expression for the number of bits in the universe [20], [21], [22],
[23], [24], [25], [26], [27], [35], [36] and [37].

In section 3, by considering the theory of QED [38], [39], [40], [41], [42],
[43], [44], [45], [46] and [47], the 4-momentum of the electromagnetic field
of the universe, the hypothesis of large Dirac numbers [10], [11], [12], the
expansion of the universe, the temperature of the cosmological horizon found
by Ya-Peng Hu [14], and the holographic principle [20], [21], [22], [23],
[24], [25], [26] and [27], it will be possible to find an expression for the
Hubble parameter [51] as a function of the temperature and radius of the
universe, the number of bits [35] contained in it and some fundamental
constants, taking special interest in the fine structure constant $\alpha$
[10], [11], [12]. This is done with the objective of understanding that the
expansion of the universe contributes to the variation of the fundamental
constants and, in turn, to the variation of the temperature and entropy of the
universe. This is reflected then in the fact that there may be a relationship
between the number of bits $N$ and the variation of the fundamental constants,
two important objects of study in quantum gravity theories that attempt to
unify all fundamental forces.

In section 4, a comparison is made between the change in the apparent
magnitude $\mu(z)$ with respect to the redshift $z$, making use of the Hubble
parameter found in section 3 with observational data. This will be contrasted
with Barrientos et al. work [52]. On the other hand, it is possible a
comparison between the Hubble parameter $H$ and the fine structure constant
$\alpha$.

In section 5 we discuss our results, while section 6 addresses the conclusions
and final comments.

## 2\. The Cosmological Horizon and its Hawking Radiation in an FLRW Universe

The observable universe, the horizon of the universe, and the cosmological
horizon constitute the visible part of the total universe, the global geometry
of the space-time of the universe appears to be flat. It has a radius of
$4.40\times 10^{26}$ m ($46.5$ billion light years), a volume of $1.08\times
10^{79}$ $m^{3}$ and a mass of $9.27\times 10^{52}$ kg, so the equivalent
mass-energy density is $8.58\times 10^{-27}$ $kg/m^{3}$. The word observable
in this sense does not refer to the ability of modern technology to detect
light or other information from an object, nor to whether there is anything to
detect. It refers to the physical limit created by the speed of light itself.
No signal can travel faster than light, so there is a maximum distance (called
the particle horizon) beyond which nothing can be detected, since the signals
could not have reached us yet [14].

A distinction is made between the visible universe, which includes only
signals emitted since recombination (when hydrogen atoms were formed from
protons and electrons and photons were emitted), and the observable universe,
which includes signals from the beginning of cosmological expansion (the Big
Bang in traditional physical cosmology, the end of the inflationary epoch in
modern cosmology) [14].

According to the calculations carried out, the current comoving distance to
the particles from which the Microwave Background Radiation $(CMBR)$ was
emitted, which represents the radius of the visible universe, is about $14$
billion parsecs (about $45.7$ billion light years). The comoving distance to
the edge of the observable universe is about $14.3$ billion parsecs (about
$46.6$ billion light years), approximately $2$ times greater. Therefore, the
radius of the observable universe is estimated to be about $46.5$ billion
light-years [14].

Currently, the limit of the observable universe is the cosmic microwave
radiation that is at a redshift distance of $z=1089$, which means that we can
see the universe since it was only $380$ thousand years old. If the cosmic
neutrino background or the stochastic gravitational wave background were
detected, the limit of the observable universe would extend to redshift
distances above $z>1010$ or to a few fractions of a second after the Big Bang
[14].

Also, the particle horizon (also called the cosmological horizon, the light
horizon, or the cosmic light horizon) is the maximum distance that particles
could have traveled toward the observer during the age of the universe. It
represents the boundary between the observable and unobservable regions of the
universe, so its distance at the present time defines the size of the
observable universe. The existence, properties, and importance of a
cosmological horizon depend on the particular cosmological model [14].

On the other hand, quantum field theory in curved space-time is an extension
of standard quantum field theory in which the possibility that the space-time
through which the field propagates is not necessarily flat is contemplated
(described by the Minkowski metric). A generic prediction of this theory is
that particles can be generated due to time-dependent gravitational fields, or
the presence of horizons [14].

One of the consequences of Heisenberg’s uncertainty principle is quantum
fluctuations in the vacuum. These consist of the creation, for very brief
moments, of particle-antiparticle pairs from a vacuum. These particles are
”virtual”, but the intense gravity of the black hole transforms them into real
ones. Such pairs quickly disintegrate, returning the energy borrowed for their
formation. However, at the edge of a black hole’s event horizon, the
probability that one member of the pair forms from the inside and the other
from the outside is not zero, so one of the components of the pair could
escape from the black hole. If the particle manages to escape, energy will
come from the black hole. That is, the black hole must lose energy to
compensate for the creation of the two particles separated. This phenomenon
has the consequences of the net emission of radiation by the black hole and
the decrease in its mass. Furthermore, a black hole emits thermalized Hawking
radiation, according to a distribution identical to that of the black body
corresponding to a temperature $T_{H}$, which expressed in terms of Planck
units, turns out to be $T_{H}=(\alpha_{H})/(2\pi)$ [14].

Ya-Peng Hu considered the FLRW metric as follows [14]:

(1)
$ds^{2}=-dt^{2}+a^{2}(t)\left(\frac{d\rho^{2}}{1-k\rho^{2}}+\rho^{2}d\Omega_{2}^{2}\right),$

where $t$ is the cosmic time, $\rho$ is the comoving radial coordinate, $a$ is
the scale factor, $d\Omega_{2}^{2}$ denotes the line element of a $2$
-dimensional sphere with unit radius, $k=1$, $0$ and $-1$ represent a closed,
flat and open FLRW universe respectively [14].

For convenience, Ya-Peng Hu defined $r=a\rho$. Therefore, the metric (1) can
be rewritten as [14]

(2)
$ds^{2}=-\frac{1-r^{2}/r_{A}^{2}}{1-kr^{2}/a^{2}}dt^{2}-\frac{2Hr}{1-kr^{2}/a^{2}}dtdr+\frac{1}{1-kr^{2}/a^{2}}dr^{2}+r^{2}d\Omega_{2}^{2},$

where $r_{A}=1/\sqrt{H^{2}+k/a^{2}}$ is the apparent horizon in an FLRW
universe [14].

The metric of the de Sitter spacetime used by Ya-Peng Hu is [14]

(3)
$ds^{2}=-\left(1-\frac{r^{2}}{l^{2}}\right)dt^{2}+\left(1-\frac{r^{2}}{l^{2}}\right)^{-1}dr^{2}+r^{2}d\Omega_{2}^{2}.$

Also the FLRW metric (2) could be put in the form

(4)
$ds^{2}=-\frac{1-r^{2}/r_{A}^{2}}{1-kr^{2}/a^{2}}(dt+\frac{Hr}{1-r^{2}/r_{A}^{2}}dr)^{2}+\frac{1}{1-r^{2}/r_{A}^{2}}dr^{2}+r^{2}d\Omega_{2}^{2}.$

For this reason, Ya-Peng Hu observed that the de Sitter spacetime is only a
particular case of the FLRW universe, in the previous equation $k=0$ and
$r_{A}=H^{-1}=l$ are defined is a constant in equation (4). Likewise, it is
known that $r=l$ is the de Sitter space-time horizon, so the existence of a
cosmological horizon in an FLRW universe is possible. By using the null
cosmological horizon property and considering the spherical symmetry in
equation (2), the corresponding cosmological horizon $r=r_{H}(t)$ satisfies
the constraint [14]

(5) $g^{\mu\nu}\frac{\partial f}{\partial x^{\mu}}\frac{\partial f}{\partial
x^{\nu}}=0,$

is

(6) $1-r_{H}^{2}/r_{A}^{2}=\dot{r}_{H}^{2}-2Hr_{H}\dot{r}_{H},$

here $f=r-r_{H}(t)$. From equation (6), it was possible for Ya-Peng Hu to
observe that the corresponding cosmological horizon $r_{H}(t)$ is just the
cosmological horizon of the de Sitter space-time when $k=0$ and
$\dot{r}_{H}=0$ [14].

Similarly, Ya Peng Hu considered the temperature of Hawking radiation coming
from the cosmological horizon for $r=r_{H}(t)$ in an FLRW universe. For
simplicity, in his work, the author considers the Klein-Gordon field in an
FLRW universe. The Klein-Gordon equation in curved space-time is [14]

(7) $(-m^{2})\psi=\frac{1}{\sqrt{-g}}\frac{\partial}{\partial
x^{\mu}}(\sqrt{-g}g^{\mu\nu}\frac{\partial}{\partial x^{v}})\psi-m^{2}\psi=0.$

In his paper, Ya-Peng Hu writes the Klein-Gordon equation (7) in terms of the
scalar field $\Phi$. But later it will be necessary to talk about the wave
function. For this reason, we will write the Klein-Gordon equation in terms of
the wave function $\psi$. This equation can be rewritten in the FLRW
coordinates (2) such that [14]

$\displaystyle-\frac{\partial}{\partial
t}(\frac{1}{\sqrt{1-\frac{k}{a^{2}}r^{2}}}\frac{\partial}{\partial
t})\frac{\rho(t,r)}{r}-\frac{\partial}{\partial
t}(\frac{Hr}{\sqrt{1-\frac{k}{a^{2}}r^{2}}}\frac{\partial}{\partial
r})\frac{\rho(t,r)}{r}-\frac{1}{r^{2}}\frac{\partial}{\partial
r}(\frac{r^{2}}{\sqrt{1-\frac{k}{a^{2}}r^{2}}}Hr\frac{\partial}{\partial
t})\frac{\rho(t,r)}{r}$ (8)
$\displaystyle+\frac{1}{r^{2}}\frac{\partial}{\partial
r}[\frac{r^{2}}{\sqrt{1-\frac{k}{a^{2}}r^{2}}}(1-r^{2}/r_{A}^{2})\frac{\partial}{\partial
r}]\frac{\rho(t,r)}{r}=[m^{2}+\frac{l(l+1)}{r^{2}}]\frac{1}{\sqrt{1-\frac{k}{a^{2}}r^{2}}}\frac{\rho(t,r)}{r}$

and

(9)
$\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}(\sin\theta\frac{\partial}{\partial\theta})Y_{lm}(\theta,\varphi)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\varphi^{2}}Y_{lm}(\theta,\varphi)+l(l+1)Y_{lm}(\theta,\varphi)=0,$

where $m$ is the rest mass of the Klein-Gordon particle,
$Y_{lm}(\theta,\varphi)$ are the usual spherical harmonics and $\psi$ has been
separated as [14]

(10) $\psi=\frac{1}{r}\rho(t,r)Y_{lm}(\theta,\varphi).$

The scalar field near the cosmological horizon behaves according to the
generalized comoving coordinate transformation [14]

(11) $r_{\ast}=r+\frac{1}{2\kappa}\ln[r_{H}(t)-r]$

and

(12) $t_{\ast}=t-t_{0},$

where $\kappa$ is the surface gravity, $r_{H}(t)$ is the location of the
cosmological horizon, and $t_{0}$ is a time constant that describes the time
at which particles radiate from the horizon. It is important to consider that
$\kappa$ here is the surface gravity of the event horizon or cosmological
horizon in stationary spacetime [14].

From (11) and (12), the radial equation (8) becomes [14]

(13)
$\begin{array}[]{ccc}\\{-\frac{2\kappa(r-r_{H})(\overset{.}{a}^{2}+k+a\overset{..}{a})}{a[r(2r\kappa-2r_{H}\kappa+1)\overset{.}{a}-a\overset{.}{r}_{H}]}+\frac{2\left(l^{2}+l+m^{2}r^{2}\right)\kappa
a(r-r_{H})}{r^{2}[r(2r\kappa-2r_{H}\kappa+1)\overset{.}{a}-a\overset{.}{r}_{H}]}\\}\rho\\\
\\\
+\\{\frac{-[\overset{.}{r}_{H}^{2}+(r-r_{H})\overset{..}{r}_{H}-1]a^{2}+[(r+r_{H})\overset{.}{a}\overset{.}{r}_{H}+r(r-r_{H})(2r\kappa-2r_{H}\kappa+1)\overset{..}{a}]a}{a(r-r_{H})[r(2r\kappa-2r_{H}\kappa+1)\overset{.}{a}-a\overset{.}{r}_{H}]}\\\
\\\ +\frac{r[2\kappa r^{2}+2\kappa
r_{H}^{2}-(4r\kappa+1)r_{H}](\overset{.}{a}^{2}+k)}{a(r-r_{H})[r(2r\kappa-2r_{H}\kappa+1)\overset{.}{a}-a\overset{.}{r}_{H}]}\\}\frac{\partial\rho}{\partial
r_{\ast}}+\\{\frac{(2r\kappa-2r_{H}\kappa+1)^{2}(\overset{.}{a}^{2}+k)r^{2}}{2\kappa
a(r-r_{H})[r(2r\kappa-2r_{H}\kappa+1)\overset{.}{a}-a\overset{.}{r}_{H}]}\\\
\\\
+\frac{-2(2r\kappa-2r_{H}\kappa+1)\overset{.}{a}\overset{.}{r}_{H}ar+a^{2}[-(2r\kappa+1)^{2}+4\kappa
r_{H}(2r\kappa+1)-4\kappa^{2}r_{H}^{2}+\overset{.}{r}_{H}^{2}]}{2\kappa
a(r-r_{H})[r(2r\kappa-2r_{H}\kappa+1)\overset{.}{a}-a\overset{.}{r}_{H}]}\\}\frac{\partial^{2}\rho}{\partial
r_{\ast}^{2}}\\\ \\\
+\frac{2\kappa(r-r_{H})\overset{.}{a}}{r(2r\kappa-2r_{H}\kappa+1)\overset{.}{a}-a\overset{.}{r}_{H}}\frac{\partial\rho}{\partial
t_{\ast}}+2\frac{\partial^{2}\rho}{\partial t_{\ast}\partial
r_{\ast}}+\frac{2\kappa
a(r-r_{H})}{r(2r\kappa-2r_{H}\kappa+1)\overset{.}{a}-a\overset{.}{r}_{H}}\frac{\partial^{2}\rho}{\partial
t_{\ast}^{2}}=0.\end{array}$

When $r\rightarrow r_{H}$ and $t\rightarrow t_{0}$, the radial equation (13)
is [14]

(14) $A\frac{\partial^{2}\rho}{\partial
r_{\ast}^{2}}+2\frac{\partial^{2}\rho}{\partial t_{\ast}\partial
r_{\ast}}+\alpha_{0}\frac{\partial\rho}{\partial r_{\ast}}=0.$

In [14], equation was used (13) and

(15)
$A=-\frac{H\overset{.}{r}_{H}-(H^{2}+k/a^{2})r_{H}}{\kappa(Hr_{H}-\overset{.}{r}_{H})}+2\overset{.}{r}_{H},~{}\alpha_{0}=\frac{(H^{2}+k/a^{2})r_{H}-H\overset{.}{r}_{H}+\overset{..}{r}_{H}-\frac{\overset{..}{a}}{a}r_{H}}{\overset{.}{r}_{H}-Hr_{H}}.$

The two linearly independent solutions of (14) are [14]

(16) $\rho_{out}=e^{-i\omega t_{\ast}},$

and

(17) $\rho_{in}=e^{-i\omega t_{\ast}+2i\omega
r_{\ast}/A}e^{-\alpha_{0}r_{\ast}/A}.$

These expressions represent the radial component for the scalar field within
the cosmological horizon ($r<r_{H}$). The Klein-Gordon equation in comoving
coordinates can be reduced to the standard form of the wave equation near the
horizon [14], [53], [54], [55] and [56].

(18) $\frac{\partial^{2}\rho}{\partial
r_{\ast}^{2}}+2\frac{\partial^{2}\rho}{\partial t_{\ast}\partial r_{\ast}}=0.$

It is possible to adjust the parameter $\kappa$ to make $A=1$, and obtain [14]

(19)
$\kappa=\frac{H\overset{.}{r}_{H}-(H^{2}+k/a^{2})r_{H}}{(Hr_{H}-\overset{.}{r}_{H})(2\overset{.}{r}_{H}-1)}.$

Here it is necessary to take into account that $A=1$ can be used in the
following special case for the de Sitter spacetime. Take into account that,
$k=0$ and $\dot{r}_{H}=0$ with $r_{A}=H^{-1}=l$, the $\kappa$ in (19) is
$\kappa=1/l$ which is simply the surface gravity of the cosmological horizon
in the de Sitter spacetime. If equation (19) is taken into account, it is
possible to find that $\kappa$ is in fact a constant related to the variable
$t_{0}$ [14].

Therefore, the ingoing wave of the Klein-Gordon field near the cosmological
horizon can be further rewritten as [14]

(20) $\rho_{in}=Ce^{-i\omega t_{\ast}+2i\omega
r_{\ast}}e^{-\alpha_{0}r_{\ast}}=Ce^{-i\omega t_{\ast}}e^{2i\omega
r-\alpha_{0}r}(r_{H}-r)^{i\omega/\kappa-\alpha_{0}/2\kappa},$

where [14] has used (11) and (12) and added the normalized factor $C$. Note
that, $\rho_{out}$ represents an outgoing wave and is well-behaved when
analytically extended outside $r>r_{H}$. However, we can find that the ingoing
wave $\rho_{in}$ (20) has a logarithmic singularity at the cosmological
horizon $r=r_{H}$ and is not analytical on the cosmological horizon. Thus we
can extend it by analytical continuation from the inside to the outside of the
cosmological horizon [14], [53], [54], [55] and [56]

(21) $(r_{H}-r)\rightarrow|r_{H}-r|e^{i\pi}=(r-r_{H})e^{i\pi}.$

Thanks to this, the incoming wave (20) becomes [14]

(22) $\begin{array}[]{cc}\rho_{in}\rightarrow{\tilde{\rho}}_{in}=Ce^{-i\omega
t_{\ast}}e^{2i\omega
r-\alpha_{0}r}(r-r_{H})^{i\omega/\kappa-\alpha_{0}/2\kappa}e^{-\frac{i\pi\alpha_{0}}{2\kappa}}e^{-\frac{\pi\omega}{\kappa}}&\\\
\\\ =Ce^{-i\omega t_{\ast}+2i\omega
r_{\ast}}e^{-\alpha_{0}r_{\ast}}e^{-\frac{i\pi\alpha_{0}}{2\kappa}}e^{-\frac{\pi\omega}{\kappa}},r>r_{H}.\end{array}$

Using the Heaviside function $Y$, in [14] it was found that

(23) $Y(x)=\Big{\\{}_{0,~{}~{}~{}x~{}<0}^{1,~{}~{}~{}x~{}\geq 0}$

the complete ingoing wave can be [14]

(24)
$\phi_{\omega}^{in}=N_{\omega}[Y(r_{H}-r)\rho_{in}+Y(r_{H}-r){\tilde{\rho}}_{in}],$

where $N_{w}$ is a normalization factor [14].

Here as in [14], we represent an incoming particle wave within the
cosmological horizon and an outgoing antiparticle wave of negative energy
outside the cosmological horizon (20) and (22) [53]. It has also been
mentioned previously that another interpretation is that an antiparticle of
positive energy incoming in the past disperses forward in time on the
cosmological horizon, and the incoming wave that describes this antiparticle
state is simply the equation (24), [14].

It is possible to construct an analogy with the WKB approximation in quantum
mechanics barrier penetration, $N_{\omega}^{2}$ can represent the force of a
particle wave entering or tunneling from the cosmological horizon, [14].

The explanation is as follows, the state of the antiparticle
$\phi_{\omega}^{in}$ is divided into two components. The first component is a
force particle wave $N_{\omega}^{2}$ entering from the horizon and the second
component is a negative antiparticle energy flow wave $N_{\omega}^{2}$
outgoing in the future towards the outside of the cosmological horizon. This
represents an antiparticle force wave $N_{\omega}^{2}e^{-2\pi\omega/\kappa}$
that has a flow of positive energy incoming in the past from outside the
cosmological horizon [14], [53], [54], [55] and [56].

We can use $\rho_{in}$, which is already normalized, and obtain the scalar
product of $\phi_{\omega}^{in}$ in (24), in order to obtain the temperature of
the Hawking radiation given in [14], [53], [54], [55] and [56]

(25)
$(\phi_{\omega_{1}}^{in},\phi_{\omega_{2}}^{in})=N_{\omega_{1}}N_{\omega_{2}}(\delta_{\omega_{1}\omega_{2}}-e^{-\pi(\omega_{1}+\omega_{2})/\kappa}\delta_{\omega_{1}\omega_{2}}),$

here the inner product of the wave function was used. To obtain negative
energy in equation (25) for bosonic particles, it is necessary to normalize
the negative $\delta$ function. Now, taking into account that, if $\kappa<0$
in (25), it is possible to rewrite the inner product as [14]

(26)
$(\phi_{w}^{in},\phi_{w}^{in})=-1=N_{\omega}^{2}(1-e^{-2\pi\omega/\kappa}).$

The latter can be associated with a thermal spectrum for a temperature
$T=-\kappa/2\pi$. Likewise, if we consider that $\kappa>0$ we have (see [14])

(27)
$(\phi_{\omega}^{in},\phi_{\omega}^{in})=1=N_{\omega}^{2}(1-e^{-2\pi\omega/\kappa}).$

Although strictly speaking it is not a thermal spectrum. In his work, Ya-Peng
Hu was able to redefine the complete incoming wave as (24) in [14]

(28)
$\phi_{\omega}^{in^{\prime}}=e^{\frac{\pi\omega}{\kappa}}N_{\omega}[Y(r_{H}-r)\rho_{in}+Y(r_{H}-r)\overset{\char
126\relax}{\rho}_{in}],$

from which we obtain [14]

(29)
$(\phi_{\omega}^{in^{\prime}},\phi_{\omega}^{in\prime})=1=N_{\omega}^{2}(e^{2\pi\omega/\kappa}-1).$

This is a thermal spectrum with the temperature $T=\kappa/2\pi$ [14].

Thus the thermal spectrum was obtained in [14] in both cases

(30) $N_{\omega}^{2}=\frac{1}{\exp(\omega/K_{B}T)-1},$

and the temperature $T$ is

(31)
$T=\frac{|\kappa|}{2\pi}=|\frac{(H^{2}+k/a^{2})r_{H}-H\overset{.}{r}_{H}}{2\pi(Hr_{H}-\overset{.}{r}_{H})(2\overset{.}{r}_{H}-1)}|.$

The entropy can be obtained considering [24], [25], [26] and [27]

(32) $dQ=TdS=\frac{\kappa}{8\pi}dA,$

and it is possible to understand that entropy can be expressed in the
following way

(33) $S=\frac{\kappa}{8\pi}\frac{A}{T}.$

This entropy can be expressed in terms of the temperature (31) as

(34) $S=\frac{\kappa}{8\pi
T}\frac{2\pi}{|\kappa|}=|\frac{\kappa}{8\pi}\frac{2\pi(Hr_{H}-\overset{.}{r}_{H})(2\overset{.}{r}_{H}-1)}{(H^{2}+k/a^{2})r_{H}-H\overset{.}{r}_{H}}|.$

The entropy $S$ is related to the number of bits $N$ by the equation

(35) $S=\frac{\kappa Nl_{p}^{2}}{4\pi},$

here $l_{p}$ is the Planck length and the Planck area is expressed as
$l_{p}^{2}=(\hbar G)/(c^{3})$. Also the number of bits is expressed as

(36)
$N=\frac{1}{2}|\frac{2\pi(Hr_{H}-\overset{.}{r}_{H})(2\overset{.}{r}_{H}-1)}{(H^{2}+k/a^{2})r_{H}-H\overset{.}{r}_{H}}|\frac{A}{l_{p}^{2}}.$

From here it can be seen that the entropy of the universe is a function of the
number of bits. This is known in some physical scenarios as a holographic
principle.

It can be seen, that there is a way to express the entropy of the universe
according to classical information theory.

In this sense, it is considered that the temperature of the universe
contributes to its entropy and consequently to the amount of information that
can be contained within the surface of the universe. Thus the information is
stored on the two-dimensional boundary of the universe and it is projected in
three dimensions, which gives shape to matter. One of the most studied aspects
of this principle is the AdS/CFT correspondence, in which it is considered
that the physics modeled on a 4-dimensional curved surface for an anti-de
Sitter spacetime (solution of Einstein’s equations with negative cosmological
constant). This presents an equivalence with conformal field theories (CFT)
which are quantum field theories defined on a boundary with a dimension
smaller than the AdS theory [34].

de Sitter spacetime may be a particular case of an FLRW universe; in the
cosmological horizon of de Sitter spacetime, there is Hawking radiation, this
was one of the conclusions of Ya-Peng Hu in [14].

Ya-Peng Hu mentions that “there are some clues that show that there is Hawking
radiation at the apparent horizon in an FLRW universe” [14]. This was done by
understanding how a Klein-Gordon field behaves near the cosmological horizon.
With this, it was understood that Hawking radiation comes from the
cosmological horizon of an FLRW Universe. It is important to emphasize here
that, when $\dot{r}_{H}=0$, the cosmological horizon at (6) is the same as the
apparent horizon, and the temperature in (31) is [14]

(37) $T=\frac{1}{2\pi Hr_{A}^{2}}.$

The previous temperature differs from the expression for the Hawking radiation
that other authors have previously obtained for the apparent horizon
$T=1/(2\pi r_{A})$ [57], [58]. As it can be seen, this expression does not
incorporate the Hubble parameter; intending to obtain a more coherent
cosmological model, this parameter must be considered. At that point the
temperature $T=1/(2\pi r_{A})$ is measured by a Kodama-type observer [14].

This Kodama-type observer is based on thermodynamic variables defined by a
Killing observer, which is absent in the time-dependent case, since our
universe is globally a non-static solution of the gravitational field
equations. The thermodynamic analogy of gravity in time-dependent spacetime
goes deeper. This difficulty could be somewhat solved by a geometrically
natural representation: a divergence-free preferred vector field proposed by
Kodama [59]. The so-called “Kodama vector” defines a natural time direction in
the general spherically symmetric spacetime, and induces an unexpected
conserved charge, which coincides with the Misner energy [14], [59] and [60].

The temperature measured by the observer

$(\partial/\partial t)^{a}$

in equation (2) is $T=1/(2\pi Hr_{A}^{2})$ [57]. Consider that the radial
velocity $\dot{r}_{H}=0$, ensures the observer in the coordinate system (11)
and (12) is the same as the observer $(\partial/\partial t)^{a}$ in (2) [14],
[59] and [60].

This condition $\dot{r}_{H}=0$ is consistent with the underlying integrability
condition in [57] and [58]. From $\dot{r}_{H}=0$, it was possible in [14] to
find that the cosmological horizon and the apparent horizon are equal. His
conclusion was that $\dot{R}_{H}=0$ can be reduced to $\dot{R}_{A}=0$ ([14]
and [60]).

The author also mentions that from equations (7) and (10) in [57], the
underlying integrability condition comes from
$\partial_{\tilde{r}}\partial_{t}S=\partial_{t}\partial_{\tilde{r}}S$ for this
reason it can also be deduced that $\dot{r}_{A}=0$. With this, the author
notes that his equation for temperature (31) is consistent with [14] and [60].

## 3\. Variation of the Fundamental Constants with the Expansion of the
Universe

In particle physics, quantum electrodynamics (QED) is the relativistic theory
of quantum electromagnetic fields. This describes how light interacts with
matter. Quantum electrodynamics mathematically describes all phenomena that
involve electrically charged particles through the exchange of photons.

The action for quantum electrodynamics is the following [61]

(38) ${S_{\text{QED}}=\int
d^{4}x\,\left[-{\frac{1}{4}}F^{\mu\nu}F_{\mu\nu}+{\bar{\psi}}\,(i\gamma^{\mu}D_{\mu}-m)\,\psi\right]},$

where $\gamma^{\mu}$ are Dirac matrices, $\psi$ a bispinor field of spin 1/2
particles, ${\bar{\psi}}\equiv\psi^{\dagger}\gamma^{0}$, called psi-bar, is
sometimes referred to as the Dirac adjoint [61].

The gauge covariant derivative is [61]

(39) ${\displaystyle D_{\mu}\equiv\partial_{\mu}+ieA_{\mu}+ieB_{\mu}},$

where $e$ is the coupling constant for the electric charge, equal to the
electric charge of the bispinor field, $A_{\mu}$ is the 4-covariant potential
of the electromagnetic field coming from the electron. This field have the
symmetry group ${\displaystyle{\text{U}}(1)}$, $B_{\mu}$ is the magnetic field
coming from an external source and $m$ is the mass of the electron or positron
[61].

Now, the electromagnetic field tensor is [61]

(40) ${\displaystyle F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}}.$

The covariant derivative is useful when developing a Lagrangian [61]

(41)
${\displaystyle{\mathcal{L}}=-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}+{\bar{\psi}}(i\gamma^{\mu}\partial_{\mu}-m)\psi-
ej^{\mu}A_{\mu}},$

where $j^{\mu}$ is the conserved current under the action of the
${\displaystyle{\text{U}}(1)}$ group arising from Noether’s theorem. It is
written ${\displaystyle j^{\mu}={\bar{\psi}}\gamma^{\mu}\psi}$ [61].

The covariant derivative can now be expanded in the Lagrangian to obtain [61]

(45)
$\displaystyle\begin{array}[]{cc}&\mathcal{L}=-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}+i{\bar{\psi}}\gamma^{\mu}\partial_{\mu}\psi-e{\bar{\psi}}\gamma^{\mu}A_{\mu}\psi-m{\bar{\psi}}\psi\\\
\\\
&=-{\frac{1}{4}}F_{\mu\nu}F^{\mu\nu}+i{\bar{\psi}}\gamma^{\mu}\partial_{\mu}\psi-m{\bar{\psi}}\psi-
ej^{\mu}A_{\mu}.\end{array}$

The external magnetic field $B_{\mu}$ has been set to zero, for simplicity.
Alternatively, we can absorb $B_{\mu}$ into a new gauge field ${\displaystyle
A^{\prime}_{\mu}=A_{\mu}+B_{\mu}}$ and relabel the new field as
${\displaystyle A_{\mu}}$ [61].

By developing the Euler-Lagrange equations for this Lagrangian and for
${\displaystyle{\bar{\psi}}}$, the equations of motion for the fields $\psi$
and $A_{\mu}$ can be obtained [61].

This Lagrangian does not contain terms
${\displaystyle\partial_{\mu}{\bar{\psi}}}$, so only the following term will
be present [61]

(46) ${\displaystyle{\frac{\partial{\mathcal{L}}}{\partial{\bar{\psi}}}}=0},$

so the equation of motion can be written [61]

(47)
${\displaystyle(i\gamma^{\mu}\partial_{\mu}-m)\psi=e\gamma^{\mu}A_{\mu}\psi}.$

Using the Euler–Lagrange equation for the $A_{\mu}$ field [61],

(48)
${\displaystyle\partial_{\nu}\left({\frac{\partial{\mathcal{L}}}{\partial(\partial_{\nu}A_{\mu})}}\right)-{\frac{\partial{\mathcal{L}}}{\partial
A_{\mu}}}=0,}$

the derivatives this time are [61]

(49)
${\displaystyle\partial_{\nu}\left({\frac{\partial{\mathcal{L}}}{\partial(\partial_{\nu}A_{\mu})}}\right)=\partial_{\nu}\left(\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}\right),}$

and

(50) ${\displaystyle{\frac{\partial{\mathcal{L}}}{\partial
A_{\mu}}}=-e{\bar{\psi}}\gamma^{\mu}\psi}.$

The above can be substituted into equation (42) to have the following result
[61]

(51) ${\displaystyle\partial_{\mu}F^{\mu\nu}=e{\bar{\psi}}\gamma^{\nu}\psi},$

which can be written in terms of the ${\displaystyle{\text{U}}(1)}$ current
$j^{\mu}$ as in [61]

(52) ${\displaystyle\partial_{\mu}F^{\mu\nu}=ej^{\nu}.}$

Here it is possible to impose a Lorentz gauge condition
${\displaystyle\partial_{\mu}A^{\mu}=0}$, and with this, the equation reduces
to ${A^{\mu}=ej^{\mu}}$, which is a wave equation for the 4-potential [61].

Now that all of these QED considerations have been made, and knowing that we
are working with the FLRW metric, it is more convenient to write the
4-electromagnetic momentum in a curved spacetime:

(53) $P_{\mu}=(E/c,P)=\hbar K^{\mu}=\hbar(\omega/c,\vec{k}),$

thus the 4-wavevector is related to the 4-momentum [62]. In this equation the
energy belonging to the electromagnetic field is

(54) $E=\int d^{3}\vec{k}|\vec{k}|\sqrt{g},$

here $\vec{k}$ is the electromagnetic wave vector.

With this, the electromagnetic linear momentum can be calculated using:

(55) $P_{\mu}=\int d^{3}K_{\mu}|\vec{k}|\sqrt{g},$

where $K\mu$ is the electromagnetic wave vector component.

Energy and momentum are related by the energy-momentum equation; in a curved
spacetime:

(56) $P_{\mu}P_{\mu}=-m^{2}g_{\mu\nu}.$

In spherical coordinates, the 4-electromagnetic momentum can be written as:

(57) $P_{\mu}=|Ep\theta p\phi|,$

where $E=-iK_{0}$, $P_{\theta}=-iK_{1}$ and $P_{\phi}=-iK_{2}$. It should be
taken into account that $K_{0}$ is the temporal component, $K_{1}$ the radial
component and $K_{2}$ the angular component.

Consider the metric for a photon traveling in a gravitational field in
spherical coordinates ($x=r\,sin\theta\,cos\phi$, $y=r\,sin\theta\,sin\phi$
and $z=r\,cos\theta$):

(58) $g_{\mu\nu}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&r^{2}&0\\\
0&0&r^{2}sin^{2}\theta\end{array}\right).$

In curved space-time we have the quantities $E=-iK_{0}/\sqrt{1}$,
$P_{\theta}=-iK_{1}/\sqrt{r^{2}}$ and
$P_{\phi}=-iK_{2}/\sqrt{r^{2}sin^{2}\theta}$.

The correct approximation is as the Ward-Takahashi equation for quantum field
theory in curved spacetime, in which the 4-momentum can be written in the form
[49], [50] and [61]:

(59) $P_{\mu}=-i\int
d^{3}\vec{k}K_{\mu}|\vec{k}|\sqrt{g}\langle\psi|K|\psi\rangle.$

In order to describe the scalar field introduced in the previous sections, the
Klein-Gordon equation coupled to the electromagnetic field will be used:

(60) $\frac{1}{\sqrt{-g}}\frac{\partial}{\partial
x^{\mu}}(\sqrt{-g}g^{\mu\nu}\frac{\partial}{\partial
x^{v}})\psi-m^{2}\psi=eA_{\mu}\frac{1}{\sqrt{-g}}\frac{\partial}{\partial
x^{\mu}}\psi.$

The wave function described in the above equation is used in spherical
coordinates

$\psi=\frac{1}{r}\rho(t,r)Y_{lm}(\theta,\varphi).$

The Klein-Gordon equation (57) can be rewritten as follows:

(61) $\begin{array}[]{cc}-\frac{\partial}{\partial
t}(\frac{1}{\sqrt{1-\frac{k}{a^{2}}r^{2}}}\frac{\partial}{\partial
t})\frac{\rho(t,r)}{r}-\frac{\partial}{\partial
t}(\frac{Hr}{\sqrt{1-\frac{k}{a^{2}}r^{2}}}\frac{\partial}{\partial
r})\frac{\rho(t,r)}{r}\\\ \\\ -\frac{1}{r^{2}}\frac{\partial}{\partial
r}(\frac{r^{2}}{\sqrt{1-\frac{k}{a^{2}}r^{2}}}Hr\frac{\partial}{\partial
t})\frac{\rho(t,r)}{r}\\\ \\\ +\frac{1}{r^{2}}\frac{\partial}{\partial
r}[\frac{r^{2}}{\sqrt{1-\frac{k}{a^{2}}r^{2}}}(1-r^{2}/r_{A}^{2})\frac{\partial}{\partial
r}]\frac{\rho(t,r)}{r}\\\ \\\
+\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}(\sin\theta\frac{\partial}{\partial\theta})Y_{lm}(\theta,\varphi)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial\varphi^{2}}Y_{lm}(\theta,\varphi)\\\
\\\ =eA_{r}\frac{1}{\sqrt{-g}}\frac{1}{r^{2}}\frac{\partial}{\partial
r}\rho(t,r)+eA_{\theta}\frac{1}{\sqrt{-g}}\frac{1}{\sin\theta}\frac{\partial}{\partial{\theta}}\rho(t,r)Y_{lm}(\theta,\varphi)\\\
\\\
+eA_{\varphi}\frac{1}{\sqrt{-g}}\frac{1}{\sin^{2}\theta}\frac{\partial^{2}}{\partial{\varphi^{2}}}\rho(t,r)Y_{lm}(\theta,\varphi).\end{array}$

The 4-electromagnetic momentum in curved spacetime (55) can be obtained from
the Klein-Gordon equation coupled to Maxwell’s equations, according to the
following relation:

(62) $\langle\psi(\vec{k})|A_{\mu}\partial_{\mu}|\psi(\vec{k})\rangle=-i\int
d^{3}\vec{k}|\vec{k}|\sqrt{g}\langle\psi(\vec{k})|\psi(\vec{k})\rangle.$

The above equation can be solved using the metric tensor given in (57). The
photon wave function on which the integral (57) must be evaluated is

(63) $\psi(\vec{k})=e^{-i\vec{k}\dot{\vec{x}}},$

which must be solved numerically.

Furthermore, because we want to work within the context of the hypothesis of
large Dirac numbers, which is proposed in the classical framework, to obtain
the 4-momentum value of the electromagnetic field of the universe,
semiclassical considerations must be made. In this work, we will limit
ourselves to mentioning the equation (54) and pointing out its importance for
finding relevant physical properties in cosmology that come from the presence
of the scalar field described by the Klein-Gordon equation. In a curved space-
time, the 4-momentum is written as:

(64) $P^{\mu}=\int(\int F^{\mu}_{\nu}J^{\nu}\sqrt{-g}dV)d\tau,$

in particular symmetric situations, this equation can be solved numerically.

The metric of the universe is the FLRW metric given in equation (1) and is
written in spherical coordinates. With this, we have the following Klein-
Gordon equation: the 4-momentum in spherical coordinates can be written as:

(65) $P^{\mu}=\int d^{3}x\sqrt{-g}T^{0\nu}.$

It is also possible to couple the potential and electromagnetic field to the
Klein-Gordon equation for a scalar field, this is reflected in the following
expression for the action:

(66) $S=-\int
d^{4}x\sqrt{-g}[\frac{1}{2}\Box_{\mu}\phi\Box^{\mu}\phi-V(\phi)-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-J^{\mu}A_{\mu}].$

With this, we have the following Klein-Gordon equation:

(67) $\Box_{\mu}\Box^{\mu}\phi-{\partial x^{\mu}V(\phi)}=J^{\mu}A_{\mu}.$

To simplify the calculations due to the mathematical complexity of expressions
(55), (58), (59) and (60) for the 4-momentum, it is possible to use the
integral:

(68) $P^{\mu}=\int A^{\mu}J^{\mu}dV,$

where $dV=\int\Phi c\rho r^{2}sin\theta drd\theta d\phi$.

For cosmological considerations, a relationship between the 4-momentum, the
Hubble parameter $H=\dot{a}(t)/(1+z)$ and the redshift $z$ can be obtained.
The 4-momentum in special relativity is $E^{2}=p^{2}c^{2}+m^{2}c^{4}$, and it
is also possible to consider Hubble’s law as $v=H*d$ (here $[H]=[1/t]$), where
the velocity is $v=c*z$. Taking into account the above, the 4-momentum for the
expansion of the universe as a function of the redshift $z$ can be written as:

(69)
$P^{\mu}=(mc,mc^{2}H\frac{z}{c}t,mc^{2}H\frac{z}{c}t,mc^{2}H\frac{z}{c}t).$

Another equation that relates the Hubble constant and the 4-electromagnetic
momentum is the following (for dimensional consistency, all equations used to
express the Hubble parameter will be in terms of Planck units):

(70) $H=\sqrt{(\frac{8\pi G\rho}{3})}.$

These Planck units are the following: $t_{p}$ is the Planck time, $l_{p}$ is
the Planck length, $\rho_{p}$ is the Planck density, $I_{p}$ is the Planck
electrical intensity, $q_{p}$ is the Planck charge, $m_{p}$ is the Planck mass
and $T_{p}$ is the Planck temperature. Taking into account the energy-momentum
tensor for the energy of the electromagnetic field [61]

(71) $T_{\sigma}^{\nu}=\frac{\partial\mathcal{L}_{m}}{\partial
A_{\mu,\nu}}A_{\mu,\sigma}-\delta_{\sigma}^{\nu}\mathcal{L}_{m}.$

With equation (68) it is possible to write this energy-momentum tensor for the
propagation of the electromagnetic field [61]

(72)
$\frac{1}{c}\int\left(T_{\sigma}^{\nu}+\delta_{\sigma}^{\nu}\mathcal{L}_{m}\right)dx^{\sigma}=0,$

using the Lagrangian density for the energy of the electromagnetic field

(73) $\mathcal{L}_{m}=\frac{1}{8\pi}(E^{2}-H^{2}),$

it is possible to find the energy density for the electromagnetic field [61]

(74) $T^{00}=\frac{1}{8\pi}(E^{2}+H^{2}).$

With this, it is possible to define the 4-momentum for the electromagnetic
field as [61]

(75) $P^{\mu}=\int T^{0\nu}dV.$

So by considering that the energy density $\rho=T^{0\nu}/V$ it is possible to
express the Hubble parameter (67) as

(76) $H=\sqrt{\frac{8\pi GP^{\mu}}{3V}}.$

This is consistent with what is seen in equation (66) in the following way

(77) $H=\sqrt{\frac{8\pi Gmcztv\rho_{p}l_{p}{t_{p}}^{2}}{3VD}}.$

In the previous equation, it was taken into account that the Hubble parameter
can be expressed as $H=v/D$. Because we are interested in understanding what
happens with the expansion of the universe, that is, when the radius $r_{U}$
of the universe varies, we use the radial component of the 4-momentum equation
(63):

(78) $P^{1}=\int A^{r_{U}}\rho cr^{2}_{U}sin\theta dr_{U}d\theta d\phi.$

This gives the following expression for the Hubble parameter $H$:

(79) $H=\frac{1}{c}\sqrt{\frac{8\pi G{t_{p}}^{4}l_{p}\int A^{r_{U}}\rho
cr^{2}_{U}sin\theta dr_{U}d\theta d\phi}{3V{l_{p}}^{3}t_{p}m_{p}}}.$

Now that we have calculated the expression for the Hubble parameter (74), it
is possible to find an expression for the temperature of the universe. We will
consider the expression for Hawking radiation given in equation (31) [14]

(80)
$T=\frac{|\kappa|}{2\pi}=|\frac{(H^{2}+k/a^{2})r_{H}-H\overset{.}{r}_{H}}{2\pi(Hr_{H}-\overset{.}{r}_{H})(2\overset{.}{r}_{H}-1)}|.$

Since we want to find the expression of the temperature for the universe and
thereby understand how the fundamental constants such as the fine structure
constant $\alpha=e^{2}/(4\pi\epsilon_{0}\hbar c)$ can vary with the expansion
of the entire universe, the cosmological parameters found in the expression
(74), which are the redshift $z$, the radius of the universe expressed as in
the Dirac large numbers hypothesis $(r_{U}/r_{e})=(r_{H}/r_{e})$. As it can be
seen in the Dirac large numbers hypothesis, the classical radius of the
electron $r_{e}=e^{2}/(4\pi\epsilon_{0}m_{e}c^{2})$ appears with an electron
mass $m_{e}$, an electrostatic radius
$r_{H}=e^{2}/(4\pi\epsilon_{0}m_{e}c^{2})$, with a mass for a hypothetical
particle $m_{H}=(Gm_{e}^{2})/(r_{e}c^{2})$.

Because we are also interested in the holographic principle, the number of
bits given for a spherical surface area will be considered
$N=A/(l^{2}_{p})=(c^{3}A)/(G\hbar)$, the critical density of the universe
$\rho_{c}=(3H^{2})/(8\pi G)$, $\rho(a)=\rho_{0}/a^{4}$ and the proportionality
constant of the Hubble parameter $H_{0}=(zc)/(r_{U})$ [10], [11] and [12].

If the charge distribution is approximately spherical but inhomogeneous, we
have:

(81)
$A^{r_{U}}=\frac{1}{4\pi\epsilon_{0}}\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{R_{U}}\frac{r^{\prime
2}_{U}\rho(r^{\prime},\theta,\phi)}{r^{\prime}_{U}}sin\theta
dr_{U}^{\prime}d\theta
d\phi=\frac{{l_{p}}^{2}q_{p}R^{3}_{U}\rho_{0}}{3\epsilon_{0}m_{p}{t_{p}}^{4}}.$

Due to the above, it is possible to solve the integral in the equation (76)
and thereby find a value for the temperature of the universe from the Hawking
radiation equation. To solve the equation (78) it is necessary to substitute
the value of the equation (75) given in the equation for the Hubble parameter
(76). Explicitly the solution to equation (76) is

(82)
$H=\sqrt{\frac{32\pi^{2}G\rho_{U}\rho_{0}{t_{p}}^{2}}{9\epsilon_{0}V{l_{p}}{q_{p}}^{2}}\frac{r_{U}^{6}}{6}}.$

Here it is necessary to consider that the critical density of the universe is
$\rho_{U}=\rho_{c}=3H^{2}/8\pi G$, where the number of bits $N$ has been taken
into account.

By substituting the Hubble parameter from equation (79) into the Hawking
radiation equation (77), a complicated integral is obtained. To simplify this,
an approximation is made using a Taylor expansion for the denominator of
equation (77), where the small parameter is $H$ as in equation (79):

(83)
$\begin{array}[]{ccc}T=\frac{|\kappa|}{2\pi}=|({(H^{2}+k/a^{2})r_{H}-H\overset{.}{r}_{H}})*({\frac{1}{2\pi\overset{.}{r}_{H}-4\pi\overset{.}{r}^{2}_{H}}}&&\\\
+{\frac{r_{H}H}{2\pi\overset{.}{r}^{2}_{H}-4\pi\overset{.}{r}^{3}_{H}}}+{\frac{{r^{2}}_{H}H^{2}}{2\pi\overset{.}{r}^{3}_{H}-4\pi\overset{.}{r}^{4}_{H}}}+{\frac{{r^{3}}_{H}H^{3}}{2\pi\overset{.}{r}^{4}_{H}-4\pi\overset{.}{r}^{5}_{H}}}+...)|.\end{array}$

By truncating this expansion to the first order, from equation (80) we can
solve for the Hubble parameter (For $r/\overset{.}{r}_{H}\neq 0$ and $T=0$):

(84)
$H=\frac{\pi\overset{.}{r}_{H}({-\frac{1}{2\pi}\pm\sqrt{\frac{1}{4\pi^{2}}-\frac{k{r^{2}}_{H}}{\pi^{2}a^{2}{t_{p}}^{2}\overset{.}{r^{2}}_{H}}}})}{r_{H}}.$

Likewise, it is now possible to truncate this expansion to second order and
obtain the following equations for the Hubble parameter:

a) With $r_{H}=0$, the constraint
$2a^{2}\overset{.}{r}^{2}_{H}-a\overset{.}{r}_{H}\neq 0$ and $T=0$

(85) $H=0.$

b) With $r_{H}=0$, the constraint
$2a^{2}\overset{.}{r}^{2}_{H}-a\overset{.}{r}_{H}\neq 0$ and $T=0$

(86)
$H=\frac{1}{t_{p}}\frac{(\frac{t_{p}}{l_{p}})\overset{.}{r}_{H}-2(\frac{{t_{p}}^{2}}{{l_{p}}^{2}})\overset{.}{r}^{2}_{H}}{\frac{t_{p}}{l_{p}}\overset{.}{r}_{H}}.$

c) With $ar_{H}\neq 0$, the constraint
$2\overset{.}{r}^{2}_{H}-\overset{.}{r}_{H}\neq 0$ and $T=0$

(87)
$H=\frac{l_{p}}{t_{p}}\frac{a^{2}(\frac{t_{p}}{l_{p}})\overset{.}{r}_{H}-\sqrt{a^{2}(a^{2}(\frac{{t_{p}}^{2}}{{l_{p}}^{2}})\overset{.}{r}^{2}_{H}-4k(\frac{{t_{p}}^{2}}{{l_{p}}^{2}})\overset{.}{r}^{2}_{H})}}{2a^{2}r_{H}}.$

d) With $ar_{H}\neq 0$, $2\overset{.}{r}^{2}_{H}-\overset{.}{r}_{H}\neq 0$ and
$T=0$

(88)
$H=\frac{l_{p}}{t_{p}}\frac{a^{2}(\frac{{t_{p}}}{{l_{p}}})\overset{.}{r}_{H}+(a^{2}(a^{2}(\frac{{t_{p}}^{2}}{{l_{p}}^{2}})\overset{.}{r}^{2}_{H}-4k(\frac{{t_{p}}^{2}}{{l_{p}}^{2}})\overset{.}{r}^{2}_{H}))}{2a^{2}r_{H}}.$

To use the expression for the Taylor series expansion to the first order the
Hawking radiation for temperature (77), it is necessary to consider that
$T\neq 0$, with this we have the following cases:

a) $r_{H}=0$ and $a\overset{.}{r}_{H}T\neq 0$, then we obtain of the following

(89) $H=\frac{2\pi T}{{t_{p}}{T_{p}}}.$

b) $ar_{H}=0$ and $a\overset{.}{r}_{H}T\neq 0$, then we obtain of the
following

(90)
$H=\frac{1}{t_{p}}\frac{a^{2}(\frac{{t_{p}}}{{l_{p}}})\overset{.}{r}_{H}-\sqrt{a^{2}(-8\pi
a^{2}(\frac{{t_{p}}}{{l_{p}}})\frac{r_{H}}{l_{p}}\overset{.}{r}_{H}\frac{T}{T_{p}}+a^{2}(\frac{{t_{p}}^{2}}{{l_{p}}^{2}})\overset{.}{r}^{2}_{H}-4k(\frac{{t_{p}}^{2}}{{l_{p}}^{2}})\overset{.}{r}^{2}_{H}})}{2a^{2}{l_{p}}\overset{.}{r}_{H}}.$

c) $ar_{H}=0$ and $a\overset{.}{r}_{H}T\neq 0$, then we have the following

(91) $H=\frac{1}{t_{p}}\frac{{(\sqrt{a^{2}(-8\pi
a^{2}(\frac{{t_{p}}}{{l_{p}}})\frac{r_{H}}{l_{p}}\overset{.}{r}_{H}\frac{T}{T_{p}}+a^{2}(\frac{{t_{p}}^{2}}{{l_{p}}^{2}})\overset{.}{r}^{2}_{H}-4k(\frac{{t_{p}}^{2}}{{l_{p}}^{2}})\overset{.}{r}^{2}_{H}}+a^{2}(\frac{{t_{p}}}{{l_{p}}})\overset{.}{r_{H}}}}{2a^{2}\overset{.}{r}_{H}}.$

At this point, the second-order expansion of the Hawking radiation is not
considered since it is no longer possible to find an exact mathematical
solution from the resulting equation for the Hubble parameter.

Considering that the scale parameter can be expressed as $a=1/(1+z)$ it is
possible to reexpress the previous equations b) and c) in terms of the
redshift $z$:

a) $ar_{H}=0$ and $a\overset{.}{r}_{H}T\neq 0$, we have

(92)
$\begin{array}[]{cc}H=\frac{1}{t_{p}}\frac{(\frac{1}{1+z})^{2}(\frac{{t_{p}}}{{l_{p}}})\overset{.}{r}_{H}-\sqrt{(\frac{1}{1+z})^{2}(-8\pi(\frac{1}{1+z})^{2}(\frac{{t_{p}}}{{l_{p}}})\frac{r_{H}}{l_{p}}\overset{.}{r}_{H}\frac{T}{T_{p}}+(\frac{1}{1+z})^{2}(\frac{{t_{p}}^{2}}{{l_{p}}^{2}})\overset{.}{r}^{2}_{H}-4k(\frac{{t_{p}}^{2}}{{l_{p}}^{2}})\overset{.}{r}^{2}_{H}})}{2(\frac{1}{1+z})^{2}{l_{p}}\overset{.}{r}_{H}}.\end{array}$

b) $ar_{H}=0$ and $a\overset{.}{r}_{H}T\neq 0$, the following holds

(93)
$\begin{array}[]{cc}H=\frac{1}{t_{p}}\frac{{(\sqrt{(\frac{1}{1+z})^{2}(-8\pi(\frac{1}{1+z})^{2}(\frac{{t_{p}}}{{l_{p}}})\frac{r_{H}}{l_{p}}\overset{.}{r}_{H}\frac{T}{T_{p}}+(\frac{1}{1+z})^{2}(\frac{{t_{p}}^{2}}{{l_{p}}^{2}})\overset{.}{r}^{2}_{H}-4k(\frac{{t_{p}}^{2}}{{l_{p}}^{2}})\overset{.}{r}^{2}_{H}}+(\frac{1}{1+z})^{2}(\frac{{t_{p}}}{{l_{p}}})\overset{.}{r_{H}}}}{2(\frac{1}{1+z})^{2}\overset{.}{r}_{H}}.\end{array}$

Now to use the hypothesis of Large Dirac Numbers it is necessary to replace
the expression of the Hubble parameter (79) in the expressions (89) and (90).
With this, it is possible to express the radius of the universe in the form
$r_{U}=e^{2}/(4\pi\epsilon_{0}m_{e}c^{2})=(\alpha\hbar)/(m_{e}c)$, that
$\rho_{U}=\rho_{c}=(3H^{2})/(8\pi G)$ and knowing that
$H=\overset{.}{a(t)}/a(t)$:

a) $ar_{H}=0$ and $a\overset{.}{r}_{H}T\neq 0$, we have

(94)
$\begin{array}[]{cc}H=\frac{1}{t_{p}}\sqrt{(\frac{432c^{4}{m_{e}}^{6}\epsilon_{0}\hbar
GN}{96\pi{m_{p}}^{5}l_{p}A_{H}\rho_{0}\alpha^{6}\hbar^{6}})}\frac{(\frac{1}{1+z})^{2}(\frac{{t_{p}}}{{l_{p}}})\overset{.}{r}_{H}-\sqrt{(\frac{1}{1+z})^{2}(-8\pi(\frac{1}{1+z})^{2}(\frac{{t_{p}}}{{l_{p}}})\frac{r_{H}}{l_{p}}\overset{.}{r}_{H}\frac{T}{T_{p}}+(\frac{1}{1+z})^{2}(\frac{{t_{p}}^{2}}{{l_{p}}^{2}})\overset{.}{r}^{2}_{H}-4k(\frac{{t_{p}}^{2}}{{l_{p}}^{2}})\overset{.}{r}^{2}_{H}})}{2(\frac{1}{1+z})^{2}{l_{p}}\overset{.}{r}_{H}}.\end{array}$

b) $ar_{H}=0$ and $a\overset{.}{r}_{H}T\neq 0$ is obtained

(95)
$\begin{array}[]{cc}H=\frac{1}{t_{p}}\sqrt{(\frac{432c^{4}{m_{e}}^{6}\epsilon_{0}\hbar
GN}{96\pi{m_{p}}^{5}l_{p}A_{H}\rho_{0}\alpha^{6}\hbar^{6}})}\frac{{(\sqrt{(\frac{1}{1+z})^{2}(-8\pi(\frac{1}{1+z})^{2}(\frac{{t_{p}}}{{l_{p}}})\frac{r_{H}}{l_{p}}\overset{.}{r}_{H}\frac{T}{T_{p}}+(\frac{1}{1+z})^{2}(\frac{{t_{p}}^{2}}{{l_{p}}^{2}})\overset{.}{r}^{2}_{H}-4k(\frac{{t_{p}}^{2}}{{l_{p}}^{2}})\overset{.}{r}^{2}_{H}}+(\frac{1}{1+z})^{2}(\frac{{t_{p}}}{{l_{p}}})\overset{.}{r_{H}}}}{2(\frac{1}{1+z})^{2}\overset{.}{r}_{H}}.\end{array}$

The fine structure constant $\alpha$ does not change in time, for this reason,
it is necessary to describe it as a parameter so that its variation with the
expansion of the universe makes sense. The value of the change in the fine
structure constant as a function of the expansion of the universe is taken as
$\overset{\cdot}{\alpha}/\alpha=-1.7\cdot 10^{-18}$ (per second).

This expression for $\delta\alpha$ coincides well with that obtained in [63].
In the Einstein-Cartan geometry framework, if we written it for the Riemannian
manifold, when the cosmological constant $\Lambda=0$, we have [63]

(96) $\alpha=\frac{c^{2}}{32\pi^{2}Gm}R.$

Varying this expression we obtain [64]

(97) $\delta\alpha=-\frac{H^{3}}{32\pi^{2}Gm}\delta t.$

This is a variation of the fine structure constant in time due to the change
in curvature of the Riemannian manifold.

Now with this equation, it is possible to express a variation of the Hubble
parameter $\delta H$ as a function of the variation of the fine structure
constant $\delta\alpha$ presented in equation (92). To do this, it is
necessary to use the variation of $\alpha$ that appears in the expressions
(89) and (90). To be able to observe how the variation of the fine structure
constant occurs as a function of the Hubble parameter:

a) $ar_{H}=0$ and $a\overset{.}{r}_{H}T\neq 0$, we have

(98) $\begin{array}[]{cc}\delta
H=\frac{1}{t_{p}}\sqrt{(\frac{432c^{4}{m_{e}}^{6}\epsilon_{0}\hbar
GN}{96\pi{m_{p}}^{5}l_{p}A_{H}\rho_{0}\alpha^{6}\hbar^{6}})}\frac{(\frac{1}{1+z})^{2}(\frac{\hbar{m_{p}}\delta\alpha}{{m_{e}}c})-\sqrt{(\frac{1}{1+z})^{2}(-8\pi(\frac{1}{1+z})^{2}(\frac{\hbar^{2}{m_{p}}^{2}\alpha}{{m_{e}}^{2}c^{2}})\delta\alpha(\frac{T}{T_{p}})+(\frac{\hbar^{2}{m_{p}}^{2}\delta\alpha^{2}}{{m_{e}}^{2}c^{2}})((\frac{1}{1+z})^{2}-4k))}}{2(\frac{1}{1+z})^{2}(\frac{\hbar{m_{p}}\delta\alpha}{{m_{e}}c})}\end{array}$

and

b) $ar_{H}=0$ and $a\overset{.}{r}_{H}T\neq 0$, the following holds

(99) $\begin{array}[]{cc}\delta
H=\frac{1}{t_{p}}\sqrt{(\frac{432c^{4}{m_{e}}^{6}\epsilon_{0}\hbar
GN}{96\pi{m_{p}}^{5}l_{p}A_{H}\rho_{0}\alpha^{6}\hbar^{6}})}\frac{\sqrt{(\frac{1}{1+z})^{2}(-8\pi(\frac{1}{1+z})^{2}(\frac{\hbar^{2}{m_{p}}^{2}\alpha}{{m_{e}}^{2}c^{2}})\delta\alpha(\frac{T}{T_{p}})+(\frac{\hbar^{2}\delta\alpha^{2}}{{m_{e}}^{2}c^{2}})((\frac{1}{1+z})^{2}-4k))}+(\frac{1}{1+z})^{2}(\frac{\hbar^{2}{m_{p}}^{2}\alpha}{{m_{e}}^{2}c^{2}})}{2(\frac{1}{1+z})^{2}(\frac{\hbar{m_{p}}\delta\alpha}{{m_{e}}c})}.\end{array}$

It should also be considered that the apparent magnitude $\mu(z)$ is a measure
of the brightness of a star or other astronomical objects. An object’s
apparent magnitude depends on its intrinsic luminosity, its distance, and any
extinction of the object’s light caused by interstellar dust along the line of
sight to the observer.

There is an independent relationship between the luminosity distance
$d_{L}(z)$ as a function of the redshift $z$ and the apparent magnitude $\mu$
is given by [52]

(100) $\mu(z)=5log[\frac{H_{0d_{L}(z)}}{c}]-5logH(z)+B.$

The light rays coming from the star usually go through an extinction
phenomenon; caused by particles (dust) located in the path of the beam.
Therefore, it is necessary to take into account the loss due to cosmic dust by
introducing a correction factor $B$ in the magnitude, this factor moves the
graph to the correct values, and in our calculations, such factor has the
value of $B=59$, where $H_{0}$ is the Hubble constant evaluated at the present
epoch $t_{0}$ and $h:=H_{0}/100$ $km/s/Mpc$ is the normalized Hubble constant,
with the luminosity distance for a flat universe given by [52]

(101)
$\begin{array}[]{cc}d_{L}(z)=\frac{c}{H_{0}}[z+\frac{1}{2}(1-q_{0})z^{2}-\frac{1}{6}(1-q_{0}-3q_{0}^{2}+j_{0})z^{3}&\\\
\\\
+\frac{1}{24}(2-2q_{0}-15q_{0}^{2}-15q_{0}^{3}+5j_{0}+10q_{0}j_{0}+s_{0})z^{4}+...].\end{array}$

Here $q_{0}$, $j_{0}$ and $s_{0}$ are the cosmographic parameters evaluated at
the present epoch [52]. The cosmographic parameters are obtained by performing
a Taylor series expansion for the scale factor $a(t)$ as a function of time,
thus

(102) $H(t)=\frac{1}{a}\frac{da}{dt},$

(103) $q(t)=-\frac{1}{a}\frac{d^{2}a}{dt^{2}}\frac{1}{h^{2}},$

(104) $j(t)=\frac{1}{a}\frac{d^{3}a}{dt^{3}}\frac{1}{h^{3}},$

(105) $s(t)=\frac{1}{a}\frac{d^{4}a}{dt^{4}}\frac{1}{h^{4}}$

and

(106) $l(t)=\frac{1}{a}\frac{d^{5}a}{dt^{5}}\frac{1}{h^{5}}.$

However, the Dirac large numbers hypothesis is an observation made by Paul
Dirac that relates the ratios of size scales in the Universe to force scales.
The proportions constitute very large, dimensionless numbers: about $40$
orders of magnitude in the current cosmological epoch. According to Dirac’s
hypothesis, the apparent similarity of these relationships could not be a mere
coincidence but could imply a cosmology with some unusual characteristics.
These characteristic features are that the force of gravity, represented by
the gravitational constant, is inversely proportional to the age of the
universe ${\displaystyle G\propto 1/t\,}$; the mass of the universe is
proportional to the square of the age of the universe ${\displaystyle M\propto
t^{2}}$; ”physical constants” are not constants at present. Their values
depend on the age of the Universe. Due to the above and the consideration that
the universe is expanding, our interest arises in the variation of the
fundamental constants with this cosmological expansion, this will be addressed
in more detail in the next section.

## 4\. Comparison with Observational Data

In this section, our interest is to compare the cosmological parameters such
as the redshift $z$ and the temperature of the universe $T_{U}$, that our
theoretical calculations show against astronomical data [66] and [67].

To achieve this objective, the following cosmological parameters are
considered: the temperature from the latest cosmic microwave background
scattering display $T_{FCMW}=2.725\pm 0.001$ $K$, the number of photons
$N=1/(e^{(h\nu/KT)}-1)\sim 10^{9}$ photons $\sim T\sim(1+z)$, the time-
dependent temperature of the universe $T(t)=T_{0}(1+z)$, the scale factor of
the FLRW metric $a(t)$ $\alpha$ $t^{1/2}$, the redshift is the variable to be
plotted, age of the universe $t_{U}=(R_{U}/c)=10-15\times 10^{9}$ years, the
entropy per unit mass for the average density of matter in the universe
$(S/M)=(1.1\times 10^{16})/(\Omega h^{2})$ $erg$ $K^{-}1$ $g^{-}1$ [68].

Using these data and equations (91) and (97), we wish to plot the value of the
redshift $z$ against that of the Hubble parameter $H$.

The graph generated from equation (97), where $H$ $vs.$ $z$ is plotted in the
Fig. $(1)$. For this, the following data reported in CODATA [65] are used
$c=299792458$ $m/s$, $\pi=3.141592654$, $N=8\times 10^{88}$ bits,
$h=6.626070040$ $J$ $\cdot$ $s$, $G=6.67408$ $\times$ $10^{-11}$ $m^{3}$
$kg^{-}1$ $s^{-}2$, $m_{e}=9.10938356\times 10^{-31}$ $kg$,
$\epsilon_{0}=8.854187187\times 10^{-12}$ $Fm^{-}1$, $R_{U}=4.4\times 10^{26}$
$m$, $|\vec{k}|=0.008975979$ $rad/m$, $r_{H}=1.5672\times 10^{27}$ $m$,
$T_{U}=2.725$ $K$, $\rho_{c}=\rho_{0}=1.8788*10^{-26}$ $h^{2}kgm^{-}3$,
$H_{0}=68.37$ $km/s/Mpc$, $z=20$, $\alpha=7.2973525693*10^{-3}$,
$A_{H}=7.716115\times 10^{54}$ $m^{2}$, $k=1$, $q_{0}=-0.55$ and $j_{0}=1$.

Fig. $1$. Apparent magnitude $\mu(z)$ vs. redshift $z$ diagram in comparison
with [52].

In this graph, it is possible to see how the apparent magnitude $\mu(z)$ grows
with the redshift $z$.

Likewise, considering equation (95) it is possible to observe the change in
the fine structure constant $\alpha$ as a function of the change in the Hubble
parameter $H$, this can be seen in Fig. (2).

Fig. $2$. Hubble parameter $H$ vs. fine structure constant $\alpha$, cf. [69].

In the following sections, a more detailed discussion of this result and the
conclusions will be given.

## 5\. Discussion

In this work, the FLRW metric was used for the Klein-Gordon equation. This
represents a scalar field from which the creation and destruction (entering
and leaving the cosmological horizon) of particles are observed, and which is
located on the surface of the observable universe. By using the Klein-Gordon
equation it is possible to find a wave equation, where the normalization of
the solution of this equation allows us to represent the temperature of the
universe. This temperature of the universe is dependent on the electromagnetic
field present throughout the universe. For this reason, it was considered the
4-electromagnetic momentum for such a field. Due to the expansion of the
universe, the electromagnetic field in it also expands. Therefore with this
temperature of the universe, it was possible to find the Hubble parameter. In
that sense, we could make a comparison of this Hubble parameter $H(z)$ vs. the
apparent magnitude $\mu(z)$ and plot it.

Now, in Fig. 2 we can see a curve that describes the variation of the fine
structure constant $\alpha$ as a function of the Hubble parameter $H$. This is
compared with Fig. 2 from the work of Kant and Chand [69]. This figure plots
the Hubble parameter $H$ as a function of time $t$. To analyze what was
mentioned in the previous paragraph, equation (52) described in [70] will be
used

(107) $(\textit{R}-4\Lambda)\frac{c^{4}}{16\pi G}=2\pi m_{e}c^{2}\alpha.$

Also in equation (30) described in [63]

(108)
$\delta\alpha=\frac{(1-\alpha^{2})^{3/2}Hc^{3}}{8\pi^{2}mcG}(\textit{R}-4\Lambda)\delta
t.$

In equation (104) the scalar curvature R can be written in terms of the Hubble
parameter $H$. It can be seen that the fine structure constant $\alpha$ has a
behavior directly proportional to the Hubble parameter $H$. In equation (105)
it can be seen that the fine structure constant $\alpha$ is directly
proportional to time $t$. For this reason, it is possible to compare the Fig.
2 in the present paper with Fig. 2 in [69].

In Fig. 1 of [52], the apparent magnitude $\mu$ is shown as a function of the
redshift $z$. This result is consistent with our Fig. 1 of that apparent
magnitude. In Barrientos et al [52] an expanding universe was considered in
which dark matter is not necessary and is consistent with astronomical
observations. This could even give us a way forward to enhance studies within
the framework of MOND.

Furthermore, the number of bits $N$ plays a fundamental role in our equation,
suggesting that the holographic principle is very important in the emergence
of the scalar field on the surface of the universe.

We also consider that it is possible to apply the Klein-Gordon equation in
curved space-time to the Schwarzschild and Reissner-Nordstrom metrics. In this
way, it would be possible to find the temperature for Hawking radiation and
the laws of black hole thermodynamics. This would represent a different path
to obtain these laws in a different way them Hawking’s. This would underline
the importance of the contribution of the scalar field given by the Klein-
Gordon equation to the temperature of both the universe and the black holes.

We conclude that the expansion of the universe contributes to the variation of
the fundamental constants and in turn to the variation of the temperature and
entropy of the universe. This is reflected in the fact that there may be a
relationship between the number of bits $N$ and the variation of the
fundamental constants, two important objects of study in quantum gravity
theories that attempt to unify all fundamental forces.

The wave function for the universe is given by the Wheeler-DeWitt wave
equation. In the Klein-Gordon equation in curved space-time given in (7) and
developed in equations (8) and (9); we have used the approximation for the
wave function corresponding to the wave equation of the scalar field. However,
although this equation is written in curved spacetime for the FLRW metric, the
wave function does not contain all the information for the universe. For this
reason, we consider that it is more appropriate to use the wave function given
by the Wheeler-DeWitt equation in the future. The use of the Wheeler-DeWitt
equation is an alternative formulation in comparison with the quantization of
the Klein-Gordon field, because with the wave function of the Wheeler-DeWitt
the quantization is for the universe, and with the Klein-Gordon equation it is
the quantization of a field.

We also consider that it is interesting to couple the Wheeler-DeWitt equation
and the Klein-Gordon equation in the FLRW metric, to find a value of the wave
function of the universe, which is a function of the scalar field in the
cosmological horizon. The quantum effects of the electromagnetic field are
described for the Wheeler-DeWitt equation $H|\psi\rangle=0$.

Due to what was commented on in the previous paragraph, when considering the
wave function explicitly, the semiclassical approximations for large Dirac
numbers and a relativistic classical electromagnetic field and momentum can no
longer be considered. In that case, we must start from the Ward-Takashi
equation given in equation (56). Furthermore, in a cosmological context, the
Ward-Takashi equation must be written in a curved space-time and its
corresponding wave function must contain all the information of the universe.
For this reason, it is more appropriate to use the wave function described by
Wheeler and Dewitt.

In various works on quantum gravity [71], the entropy for black holes has been
calculated in terms of qubits. Likewise, since the Klein-Gordon equation has
been used, we consider that the use of “quantum bits” or “qubits” is more
appropriate for describing the entropy of the universe in terms of information
theory.

## 6\. Conclusions

In this work, it was observed that it is possible to consider that a scalar
field given by the Klein-Gordon equation in curved space-time and the
electromagnetic field contribute to the temperature of the universe for the
cosmological horizon.

The presence of these scalar and electromagnetic fields in the expansion of
the universe has important consequences. This is done by incorporating the
hypothesis of large Dirac numbers, for this reason it was possible to find an
expression for the Hubble parameter as a function of the temperature of the
universe, the number of bits $N$ for the universe, and the fundamental
constants.

It could be observed that the expansion of the universe contributes to the
change of the fundamental constants, as well as its number of bits $N$.

As future work, we consider it interesting to further study the implications
and consequences of the number of bits in cosmology and the expansion of the
universe, in addition to what deeper aspects of the holographic principle can
be addressed.

On the other hand, in his article Duff [71] describes how the entropy of a
black hole can be expressed in terms of qubits instead of bits, we consider
that it is of interest to understand how the information expressed in qubits
for the entire universe can be considered a fundamental entity.

Furthermore, there is a formulation made by Frieden [72] where he derives the
laws of physics through information theory, thanks to constructing a
Lagrangian in informational terms, and through the Euler-Lagrange equations,
he obtains the most important equations of the laws of physics. It is natural
to address the question of whether there is a relationship between our work,
and Frieden’s approach, because a derivation of the laws of physics from the
laws of information theory seems to be fundamental.

We intend to derive from informational principle a Lagrangian that considers
the aforementioned coupling of the Klein-Gordon equation in the FLRW metric
and the Wheeler-DeWitt equation.

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