# Varying constants entropic–$\Lambda$CDM cosmology

Mariusz P. Da̧browski<EMAIL_ADDRESS>Institute of Physics,
University of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland, National
Centre for Nuclear Research, Andrzeja Sołtana 7, 05-400 Otwock, Poland,
Copernicus Center for Interdisciplinary Studies, Sławkowska 17, 31-016 Kraków,
Poland H. Gohar<EMAIL_ADDRESS>Institute of Physics, University
of Szczecin, Wielkopolska 15, 70-451 Szczecin, Poland Vincenzo Salzano
<EMAIL_ADDRESS>Institute of Physics, University of Szczecin,
Wielkopolska 15, 70-451 Szczecin, Poland

###### Abstract

We formulate the basic framework of thermodynamical entropic force cosmology
which allows variation of the gravitational constant $G$ and the speed of
light $c$. Three different approaches to the formulation of the field
equations are presented. Some cosmological solutions for each framework are
given and one of them is tested against combined observational data
(supernovae, BAO, and CMB). From the fit of the data it is found that the
Hawking temperature numerical coefficient $\gamma$ is two to four orders of
magnitude less than usually assumed on the geometrical ground theoretical
value of $O(1)$ and that it is also compatible with zero. Besides, in the
entropic scenario we observationally test that the fit of the data is allowed
for the speed of light $c$ growing and the gravitational constant $G$
diminishing during the evolution of the universe. We also obtain a bound on
the variation of $c$ to be $\Delta c/c\propto 10^{-5}>0$ which is at least one
order of magnitude weaker than the quasar spectra observational bound.

###### pacs:

98.80.Jk; 95.36.+x; 04.50.Kd; 04.70.Dy

##  I Introduction

General Relativity is an established theory which explains the evolution of
the universe on large scales EMM . Although it is not complete because it
contains singularities, it explains the dynamics of the universe in a
consistent way. Besides, the current phase of accelerated evolution of the
universe has been discovered perlmutter ; riess . In order to obtain this
accelerated expansion, one has to put an extra term, the cosmological constant
$\Lambda$ or dark energy into the Einstein-Friedmann equations. Resulted
$\Lambda$CDM models Peebles84 ; KofmanStar85 ; Ann86 ; Weinberg89  are
consistent models to explain this accelerated expansion, but the observational
value of $\Lambda$ is over 120 orders of magnitude smaller than the value
calculated in quantum field theory, where it is interpreted as vacuum energy.
This motivates cosmologists to look for alternative models which can explain
the effect c2 ; c3 .

The relation between the Einstein’s gravity and thermodynamics is a puzzle. In
the seventies of the twentieth century, Bekenstein and Hawking BekensteinS ;
HawkingT  derived the laws of black hole thermodynamics which emerged to have
similar properties as in standard thermodynamics. Jacobson e  derived Einstein
field equations from the first law of thermodynamics by assuming the
proportionality of the entropy and the horizon area. A more extensive work in
this direction was made by Verlinde and Padmanabhan in Refs. f ; f1 ; f2 ; f3
. Verlinde derived gravity as an entropic force, which originated in a system
by the statistical tendency to increase its entropy. He assumed the
holographic principle g , which stated that the microscopic degrees of freedom
can be represented holographically on the horizons, and this piece of
information (or degrees of freedom) can be measured in terms of entropy. The
approach got criticized on the base of neutron experiments though Archil .

Recently, the entropic cosmology based on the notion of the entropic force was
developed in series of papers komatsu ; komatsu_proc ; i1 ; i2 ; i3  and
especially it was compared with supernovae data in Ref. easson . However,
supernovae test is not very strong and so the Ref. easson  got criticized on
the base of galaxy formation problem (e.g Koivisto11 ; Basilakos12 ).
Basically, the idea of entropic cosmology is to add extra entropic force terms
into the Friedmann equation and the acceleration equation. This force is
supposed to be responsible for the current acceleration as well as for an
early exponential expansion of the universe. It is pertinent to mention that
the entropic cosmology suggested in these references assumes that gravity is
still a fundamental force and that it includes extra driving force terms or
boundary terms in the Einstein field equations. This is unlike Verlinde f ,
who considers gravity as an entropic force, but not as a fundamental force
(see also Refs. Cai05 ; Cai07 ; Cai08 ; Akbar06 ; Akbar071 ; Akbar072 ;
Eling06 ; Hayward99 ; BakRay2000 ; Danielsson ; Wang10 ). All frameworks were
discussed in detail in Ref. Visser11  by Visser. Entropic cosmology is also
related to dynamical vacuum energy models which have been discussed and
confronted with data in Refs. Basilakos09 ; Grande10 ; Grande11 ; Gomez141 ;
Sola15 .

In this paper we expand entropic cosmology suggested in Refs. komatsu ;
komatsu_proc ; i1 ; i2 ; i3 ; easson  for the theories with varying physical
constants: the gravitational constant $G$ and the speed of light $c$. Although
easson  is problematic in the context galaxy formation test, we use it as a
starting point for further discussion. We discuss possible consequences of
such variability onto the entropic force terms and the boundary terms. As it
has been known for the last fifteen years, varying constants cosmology VG ; Vc
was proposed as an alternative to inflationary cosmology, because it can to
solve all the cosmological problems (horizon, flatness, and monopole). In the
paper we try three different approaches to formulate the entropic cosmology
with varying constants. In section II we present a consistent set of the field
equations which describes varying constants entropic cosmology with general
entropic force terms. In Section III we derive the continuity equation from
the first law of thermodynamics and fit general entropic terms to the field
equations derived in Section II using explitic definitions of Bekenstein
entropy and Hawking temperature. We also discuss the constraints on the models
which come from the second law of thermodynamics. In section IV we study
single-fluid accelerating cosmological solutions to the field equations
derived in Section III. In Section V we derive the entropic force for varying
constants, define appropriate entropic pressure, and modify the continuity and
acceleration equations. We also determine Friedmann equation and give single-
fluid accelerating cosmological solutions. In Section VI we derive
gravitational Einstein field equations using the heat flow through the horizon
to which Bekenstein entropy and the Hawking temperature is assigned. Sections
VII and VIII are devoted to observationally testing the many-fluid entropic
force models with varying constants. For this sake the data from supernovae,
Baryon Acoustic Oscillations (BAO), and Cosmic Microwave Background (CMB) is
used. In Section IX we give our conclusions.

##  II Entropic force field equations and varying constants

The main idea of our consideration is to follow Refs. easson ; komatsu ;
komatsu_proc  (assuming homogeneous Friedmann geometry) and generalize field
equations which contain the entropic force terms $f(t)$ and $g(t)$ onto the
case of varying speed of light $c$ and varying Newton gravitational constant
$G$ theories. It is easy to realize that the modified Einstein equations can
be written down as follows

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi
G(t)}{3}\rho-\frac{kc^{2}(t)}{a^{2}}+f\left(t\right),$ (II.1)
$\frac{\ddot{a}}{a}=-\frac{4\pi
G(t)}{3}\left[\rho+\frac{3p}{c^{2}(t)}\right]+g(t).$ (II.2)

In fact, the functions $f(t)$ and $g(t)$ in general play the role analogous to
bulk viscosity (cf. Refs. [50-69] of the paper by Komatsu et. al. komatsu )
and this is why from (II.1) and (II.2) one obtains the modified continuity
equation

$\displaystyle\dot{\rho}$ $\displaystyle+$ $\displaystyle
3H\left[\rho+\frac{p}{c^{2}(t)}\right]+\rho\frac{\dot{G(t)}}{G(t)}-3\frac{kc(t)\dot{c}(t)}{4\pi
G(t)a^{2}(t)}$ (II.3) $\displaystyle=$ $\displaystyle\frac{3H}{4\pi
G(t)}\left[g(t)-f(t)-\frac{\dot{f}(t)}{2H}\right],$

which will further be used in our paper to various thermodynamical scenarios
of the evolution of the universe. It is clear that (II.3) has dissipative
terms in full analogy to bulk viscosity models. However, if the functions
$f(t)$ and $g(t)$ are equal and have the value of the $\Lambda$-term modified
by varying speed of light $c(t)$ i.e.

$f(t)=g(t)=\frac{\Lambda c^{2}(t)}{3},$ (II.4)

then they give modified varying $c$ and $G$ Einstein field equations with the
continuity equation as JCAP15

$\dot{\varrho}+3\frac{\dot{a}}{a}\left(\varrho+\frac{p}{c^{2}(t)}\right)+\varrho\frac{\dot{G}(t)}{G(t)}=\frac{\left(3k-\Lambda
a^{2}\right)}{4\pi G(t)a^{2}}c(t)\dot{c}(t),$ (II.5)

which finally reduce to the standard $\Lambda$-CDM equations for $c$ and $G$
constant. Another point is the $f(t)$ and $g(t)$ terms can also be considered
as time-dependent (dynamical) vacuum energy Basilakos09 ; Grande10 ; Grande11
; Gomez141 ; Sola15 .

##  III Gravitational thermodynamics and varying constants

In this section we start with basic thermodynamics in order to get entropic
force varying constants field equations. We remind that the first law of
thermodynamics has been widely used to interlink different gravity theories
with thermodynamics GW77 ; Cai05 ; Cai07 ; Cai08 ; Akbar06 ; Akbar071 ;
Akbar072 ; Eling06 . Defining the temperature and entropy on the cosmological
horizons, one can use this law of thermodynamics for the whole universe

$dE+pdV=TdS,$ (III.1)

where $dE$, $dV$, and $dS$ describe changes in the internal energy $E$, the
volume $V$, and the entropy $S$, while $T$ is the temperature, and $p$ is the
pressure. The volume of the universe contained in a sphere of the proper
radius $r_{\ast}=a(t)r$ ($r$ is the comoving radius and $a(t)$ is the scale
factor) is

$V(t)=\frac{4}{3}\pi a^{3}r^{3}\text{ .}$ (III.2)

We have

$\dot{V}(t)=3V(t)\frac{\dot{a}}{a}=3V(t)H(t)$ (III.3)

where dot represents the derivative with respect to time and the Hubble
function is $H(t)=\dot{a}/a$. The internal energy $E$ and the energy density
$\varepsilon(t)$ of the universe are related by

$E(t)=\varepsilon(t)V(t),\hskip 8.5359pt\varepsilon(t)=\rho(t)c^{2}(t),$
(III.4)

where $\rho$ is the mass density of the universe.

Now we generalize the Hawking temperature $T$ HawkingT  and Bekenstein entropy
$S$ BekensteinS  of the (time-dependent) Hubble horizon at $r\equiv
r_{h}=r_{h}(t)$ onto the varying $c$ and $G$ theories as follows

$\displaystyle T$ $\displaystyle=$ $\displaystyle\frac{\gamma\hslash
c(t)}{2\pi k_{B}r_{h}(t)},$ (III.5) $\displaystyle S$ $\displaystyle=$
$\displaystyle\frac{k_{B}}{4\hslash}\left[\frac{c^{3}(t)A(t)}{G\left(t\right)}\right].$
(III.6)

Here $A(t)=4\pi r_{h}^{2}(t)$ is the horizon area, $\hslash$ is the Planck
constant, $k_{B}$ is the Boltzmann constant, and $\gamma$ is an arbitrary,
dimensionless, and non-negative theoretical parameter of the order of unity
$O(1)$ which is usually taken to be $\frac{3}{2\pi}$, $\frac{3}{4\pi}$ or
$\frac{1}{2}$ easson ; komatsu ; komatsu_proc . In fact, $\gamma$ can be
related to a corresponding screen or boundary of the universe to define the
temperature and the entropy on that preferred screen. Here the screen will be
the Hubble horizon i.e. the sphere of the radius $r_{h}$. Dividing (III.1) by
time differential $dt$, we have

$\frac{dE}{dt}+p\frac{dV}{dt}=T\frac{dS}{dt},$ (III.7)

which after applying (III.2) and (III.4) gives

$\dot{E}+p\dot{V}=\left[\dot{\rho}+2\frac{\dot{c}(t)}{c(t)}\rho+3\frac{\dot{a}}{a}\left(\rho+\frac{p}{c^{2}(t)}\right)\right]Vc^{2}(t).$
(III.8)

From (III.5) and (III.6) we have

$T\dot{S}=\frac{\gamma
c^{4}(t)}{2G(t)}r_{h}\left[3\frac{\dot{c}(t)}{c(t)}+2\frac{\dot{r}_{h}}{r_{h}}-\frac{\dot{G}(t)}{G(t)}\right].$
(III.9)

By using (III.7), (III.8), and (III.9) we get the modified continuity equation
as follows

$\displaystyle\dot{\rho}$ $\displaystyle+$ $\displaystyle
3H\left[\rho+\frac{p}{c^{2}(t)}\right]=-2\frac{\dot{c}(t)}{c(t)}\rho$
$\displaystyle+$ $\displaystyle\frac{3\gamma H^{2}}{8\pi
G(t)}\left[\left(5\frac{\dot{c}(t)}{c(t)}-\frac{\dot{G}(t)}{G(t)}\right)-2\frac{\dot{H}}{H}\right]$

where we have used the explicit definition of the Hubble horizon modified to
varying speed of light models easson

$r_{h}(t)\equiv\frac{c(t)}{H(t)}.$ (III.11)

If we introduced the non-zero spatial curvature $k=\pm 1$, then we would have
to apply the entropy and the temperature of the apparent horizon which reads

$r_{A}=\frac{c(t)}{\sqrt{{H^{2}}+\frac{kc^{2}(t)}{a^{2}(t)}}}.$ (III.12)

Simple calculations give that

$\frac{\dot{r}_{A}}{r_{A}}=-\frac{Hr_{A}^{2}}{c^{2}}\left(\dot{H}-\frac{kc^{2}}{a^{2}}\right)+\frac{\dot{c}}{c}\left(1-\frac{k}{a^{2}}r_{A}^{2}\right),$
(III.13)

which for $k=0$ case reduces to

$\frac{\dot{r}_{h}}{r_{h}}=\frac{\dot{c}(t)}{c(t)}-\frac{\dot{H}}{H}.$
(III.14)

In this section we restrict ourselves to $k=0$ case in order to get the
general functions $f(t)$ and $g(t)$.

In order to constrain possible sets of varying constant models we can apply
the second law of thermodynamics according to which the entropy of the
universe remains constant (adiabatic expansion) or increases (non-adiabatic
expansion)

$\frac{dS}{dt}\geq 0.$ (III.15)

In fact, (III.9) gives the condition

$3\frac{\dot{c}(t)}{c(t)}-\frac{\dot{G}(t)}{G(t)}\geq-2\frac{\dot{r}_{h}}{r_{h}}=-2\left(\frac{\dot{c}(t)}{c(t)}-\frac{\dot{H}}{H}\right).$
(III.16)

or

$5\frac{\dot{c}(t)}{c(t)}-\frac{\dot{G}(t)}{G(t)}\geq
2\frac{\dot{H}}{H}=2\left(\frac{\ddot{a}}{\dot{a}}-\frac{\dot{a}}{a}\right)$
(III.17)

which for $\dot{c}=\dot{G}=0$ just says that the Hubble horizon must increase
$\dot{r}_{h}\geq 0$. For $\dot{G}(t)=0,$ and by using (III.11) and (III.16),
we have

$c\left(t\right)\geq b_{1}H^{\frac{2}{5}},$ (III.18)

and for $\dot{c}(t)=0,$ we have

$G(t)\leq b_{2}H^{-2},$ (III.19)

where $b_{1}$ and $b_{2}$ are constants.

##  IV Gravitational thermodynamics – cosmological solutions

Using the generalized continuity equation (III) one is able to fit the
functions $f(t)$ and $g(t)$ from a general varying constants entropic force
continuity equation (II.3) as follows

$\displaystyle f(t)$ $\displaystyle=$ $\displaystyle\gamma H^{2}$ (IV.1)
$\displaystyle g(t)$ $\displaystyle=$ $\displaystyle\gamma
H^{2}+\frac{\gamma}{2}\left(5\frac{\dot{c}(t)}{c(t)}-\frac{\dot{G}(t)}{G(t)}\right)H$
(IV.2) $\displaystyle+$ $\displaystyle\frac{4\pi
G(t)}{3H}\left(\frac{\dot{G}(t)}{G(t)}-2\frac{\dot{c}(t)}{c(t)}\right)\rho.$

Having given $f(t)$ and $g(t)$ one is able to write down the equations (II.1)
and (II.2) as follows

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi G(t)}{3}\rho+\gamma H^{2},$
(IV.3) $\displaystyle\frac{\ddot{a}}{a}$ $\displaystyle=$ $\displaystyle\gamma
H^{2}-\frac{4\pi G(t)}{3}\left(\rho+\frac{3p}{c^{2}(t)}\right)$ (IV.4)
$\displaystyle+$
$\displaystyle\left(\frac{7\gamma-2}{2}\right)\frac{\dot{c}(t)}{c(t)}H+\left(\frac{1-2\gamma}{2}\right)\frac{\dot{G}(t)}{G(t)}H,$

which form a consistent set together with Eq. (III). While fitting the
functions $f(t)$ and $g(t)$ we set $k=0$. If we were to investigate $k=\pm 1$
models then the the temperature $T$ (III.5) and the entropy $S$ (III.6) should
be defined on the apparent horizon (III.12). A different choice of $f(t)$ and
$g(t)$ which is consistent with (III) would be for example as follows

$\displaystyle f(t)$ $\displaystyle=$ $\displaystyle 0,$ (IV.5) $\displaystyle
g(t)$ $\displaystyle=$
$\displaystyle\gamma\dot{H}+\frac{\gamma}{2}\left(5\frac{\dot{c}(t)}{c(t)}-\frac{\dot{G}(t)}{G(t)}\right)H$
(IV.6) $\displaystyle+$ $\displaystyle\frac{4\pi
G(t)}{3H}\left(\frac{\dot{G}(t)}{G(t)}-2\frac{\dot{c}(t)}{c(t)}\right)\rho.$

However, both choices (IV.1)-(IV.2) and (IV.5) and (IV.6) do not allow for a
constant term like the cosmological constant (unless one fine-tunes $H=$
const.) so that an alternative choice which fulfills this requirement would be

$\displaystyle f(t)$ $\displaystyle=$ $\displaystyle\gamma H^{2}+K_{1},$
(IV.7) $\displaystyle g(t)$ $\displaystyle=$ $\displaystyle\gamma
H^{2}+K_{1}+\frac{\gamma}{2}\left(5\frac{\dot{c}(t)}{c(t)}-\frac{\dot{G}(t)}{G(t)}\right)H$
(IV.8) $\displaystyle+$ $\displaystyle\frac{4\pi
G(t)}{3H}\left(\frac{\dot{G}(t)}{G(t)}-2\frac{\dot{c}(t)}{c(t)}\right)\rho,$

where $K_{1}$ is a constant acting on the same footing as the cosmological
$\Lambda$-term in standard $\Lambda$-CDM cosmology securing the model with
respect to structure formation tests (cf. komatsu ; Koivisto11 ; Basilakos12 ;
Basilakos14 ; Gomez15 ).

There is a full analogy of varying constants generalized equations (IV.3),
(IV.4), and (III) with the entropic force equation given in easson  when one
applies the specific ansätze for varying $c$ and $G$:

$c(t)=c_{o}a^{n}\hskip 14.22636pt{\rm and}\hskip 14.22636ptG(t)=G_{o}a^{q}$
(IV.9)

with $n,q=$ const. which gives $\dot{c}(t)/c(t)=nH$ or $\dot{G}(t)/G(t)=qH$.
It is worth emphasizing that our ansätze should be $c(t)=c_{o}(a/a_{0})^{n}$
and $G(t)=G_{o}(a/a_{0})^{q}$ PLB14  but the standard approach nowadays picks
up $a_{0}=1$ amendola .

We note that the application of the ansätze (IV.9) to the growing entropy
requirement (III.17) gives the bound that

$n\geq-2/5\hskip 22.76228pt{\rm and}\hskip 22.76228ptq\leq 2.$ (IV.10)

As we shall see in Section VIII (or Table I) these limits are in agreement
with the observational values we have obtained. They also allow the Newtonian
limit $c\to\infty$ ($n\to\infty$) or $G\to 0$ ($q\to-\infty$) okun .

The cosmological solutions of the set of varying constants Eqs. (IV.3), (IV.4)
and (III) are given below.

###  IV.1 $G$ varying models only: $G(t)=G_{o}a^{q};$ $q,G_{o}=const.$,
$\dot{c}(t)=0$.

Defining the barotropic index equation of state parameter $w$ by using the
barotropic equation of state, $p=w\rho c^{2}$ for varying $G=G_{o}a^{q}$, we
can integrate the continuity equation (III) to get

$\rho=\rho_{0}a^{3\left(1+w\right)}\left[\left(\frac{G(t)}{G_{o}}\right)\left(\frac{H}{H_{0}}\right)^{2}\right]^{\frac{\gamma}{\gamma-1}},$
(IV.11)

where $\rho_{0}$ is a constant with the dimension of mass density, $G_{o}$ the
gravitational constant, and $H_{0}$ the Hubble parameter. Using (IV.3) and
(IV.4) and then multiplying (IV.3) by $\left(1+3w\right)$ and (IV.4) by $2$ we
get

$\dot{H}=\left(\frac{1-2\gamma}{2}\right)\frac{\dot{G}}{G}H-\frac{3}{2}\left(1+w\right)\left(1-\gamma\right)H^{2}$
(IV.12)

or (using the fact that $\dot{G}/{G}=qH$, $\ddot{a}/a=\dot{H}+H^{2}$) one has

$\dot{H}=-\bar{w}H^{2},$ (IV.13)

where

$\bar{w}=\frac{1}{2}\left[3(w+1)\left(1-\gamma\right)-(1-2\gamma)q\right].$
(IV.14)

The Eq. (IV.13) solves easily using a new variable $N=\ln a$ komatsu  i.e.

$\left(\frac{dH}{da}\right)a=\frac{dH}{dN}=\left(\frac{dH}{dt}\right)\frac{dt}{da}a=-\bar{w}H,$
(IV.15)

which integrates to give

$H=H_{0}a^{-\bar{w}},$ (IV.16)

where $H_{0}$ is constant. The solution of (IV.16) is

$a\left(t\right)=\bar{w}^{\frac{1}{\bar{w}}}\left[H_{0}\left(t-t_{0}\right)\right]^{\frac{1}{\bar{w}}},$
(IV.17)

where $t_{0}$ is constant. Bearing in mind the value of (IV.14), we can easily
conclude that without entropic terms the solution (IV.17) corresponds to a
standard barotropic fluid Friedmann evolution
$a(t)\propto(t-t_{0})^{(2/3(w+1))}$. The scale factor for radiation, matter
and vacuum (cosmological constant) dominated eras reads as

$a(t)\propto\left\\{\begin{array}[]{c}\left[H_{0}(t-t_{0})\right]^{\frac{2}{\left(4-q\right)+2\gamma\left(q-2\right)}};w=\frac{1}{3},\text{
(radiation)}\\\
\left[H_{0}(t-t_{0})\right]^{\frac{2}{\left(3-q\right)+\left(2q-3\right)\gamma}};w=0,\text{
(dust)}\\\
\left[H_{0}(t-t_{0})\right]^{\frac{2}{\left(2\gamma-1\right)q}};w=-1.\text{
(vacuum)}\end{array}\right.$ (IV.18)

The solution (IV.18) shows that in varying $G$ entropic cosmology even dust
($w=0$) can drive acceleration of the universe provided

$\left(3-q\right)+\left(2q-3\right)\gamma\leq 2~{}.$ (IV.19)

On the other hand, the solution which includes $\Lambda-$term $(w=-1)$ drives
acceleration for $\left(2\gamma-1\right)q\leq 2$. There is an interesting
check of these formulas for the case when one takes the Hawking temperature
parameter $\gamma=1$, in all three cases (radiation, dust, vacuum) the
conditions for accelerated expansion fall into one relation $q\leq 2$. In
fact, this limit is very special which can be seen from Eq. (IV.3) in which
the terms involving $H^{2}$ cancel and lead to empty universe ($\varrho=0$) so
that it is no wonder that the acceleration does not depend on the barotropic
index parameter $w$. Finally, we conclude that in all these cases the entropic
terms and the varying constants can play the role of dark energy.

One may also consider a more than one component models i.e. the models which
allow matter, radiation as well as other cosmological fluids of negative
pressure like the cosmological constant which give a turning point of the
evolution compatible with current observational data (early-time deceleration
and late-time acceleration). We will consider such models numerically in
Section VIII, where we test these models with observational data.

###  IV.2 $c$ varying models only: $c(t)=c_{o}a^{n};$ $c_{0},n=const.$,
$\dot{G}(t)=0$

The solution of the continuity equation (III) for varying $c$ is

$\rho=\rho_{0}a^{3\left(1+w\right)}\left[\left(\frac{c(t)}{c_{0}}\right)^{7\gamma-2}\left(\frac{H}{H_{0}}\right)^{-2\gamma}\right]^{\frac{1}{1-\gamma}},$
(IV.20)

where again $\rho_{0}$ is a constant with the dimension of mass density,
$c_{o}$ the velocity of light, and $H_{0}$ the Hubble parameter. Applying
(IV.3) and (IV.4), we have

$\dot{H}=\frac{7\gamma-2}{2}\frac{\dot{c}}{c}H-\frac{3}{2}\left(1-\gamma\right)\left(1+w\right)H^{2},$
(IV.21)

or

$\dot{H}=-\tilde{w}H^{2},$ (IV.22)

where

$\tilde{w}=\frac{1}{2}\left[3(1+w)\left(1-\gamma\right)-n\left(7\gamma-2\right)\right].$
(IV.23)

The solution of (IV.22) is

$H=H_{0}a^{-\tilde{w}},$ (IV.24)

where $H_{0}$ is constant. Finally, the solution of (IV.24) for the scale
factor gives

$a(t)=\tilde{w}^{\frac{1}{\tilde{w}}}\left[H_{0}\left(t-t_{0}\right)\right]^{\frac{1}{\tilde{w}}},$
(IV.25)

where $t_{0}$ is constant. For radiation, dust and vacuum we have,
respectively

$a(t)\propto\left\\{\begin{array}[]{c}\left[H_{0}(t-t_{0})\right]^{\frac{2}{\left(4+2n\right)-\left(4+7n\right)\gamma}};w=\frac{1}{3},\text{
(radiation)}\\\
\left[H_{0}(t-t_{0})\right]^{\frac{2}{\left(3+2n\right)-\left(3+7n\right)\gamma}};w=0,\text{
(dust)}\\\
\left[H_{0}(t-t_{0})\right]^{\frac{2}{\left(2-7\gamma\right)n}};w=-1.\text{
(vacuum)}\end{array}\right.$ (IV.26)

For these three cases, one derives inflation provided

$(4+2n)-(4+7n)\gamma\leq 2,\text{ (radiation)}$ $(3+2n)-(3+7n)\gamma\leq
2,\text{ (dust)}$ $(2-7\gamma)n\leq 2,\text{ (vacuum)}$

and the entropic force terms play the role of dark energy which can be
responsible for the current acceleration of the universe. As in the previous
subsection, here also after taking the Hawking temperature parameter
$\gamma=1$, in all three cases (radiation, dust, vacuum) the conditions for
accelerated expansion fall into one relation $n\geq-2/5$, but this is also a
special empty universe limit of Eq. (IV.3).

As in the previous subsection one may also consider a more than one component
models – the matter we deal with numerically in Section VIII. We would like to
emphasize again that here we have presented one-component solutions only while
in Section VII we will be studying multi-component models which allow the
transition from deceleration to acceleration.

##  V Entropic Pressure Modified Equations

In this section we start with the formal definition of the entropic force as
given in komatsu ; komatsu_proc ; easson . We assume that the temperature and
entropy are given by (III.5) and (III.6) and use the definition of the
entropic force

$F=-T\frac{dS}{dr}.$ (V.1)

We calculate the entropic force on the horizon $r=r_{h}(t)$ by taking

$dS/dr_{h}=\dot{S}/\dot{r}_{h}$ (V.2)

to obtain

$F=-\frac{\gamma
c^{4}(t)}{2G(t)}\left[\frac{5\frac{\dot{c}(t)}{c(t)}-\frac{\dot{G}(t)}{G(t)}-2\frac{\dot{H}}{H}}{\frac{\dot{c}(t)}{c(t)}-\frac{\dot{H}}{H}}\right].$
(V.3)

For $\dot{c}=\dot{G}=0$ this formula reduces to the value obtained in Ref.
komatsu : $F=\gamma(c^{4}/G)$ which is presumably the value of maximum tension
in general relativity Gibbons2002 ; schiller ; BG2014 . It has been shown in
Ref. MaxTen  that (V.3) may recover infinite tension thus violating the so-
called Maximum Tension Principle Gibbons2002  in the framework of varying
constants theories.

Now, we define the entropic pressure $p_{E}$, as the entropic force per unit
area $A$, and use (III.11) to get

$p_{E}=-\frac{\gamma c^{2}(t)H^{2}}{8\pi
G(t)}\left[\frac{5\frac{\dot{c}(t)}{c(t)}-\frac{\dot{G}(t)}{G(t)}-2\frac{\dot{H}}{H}}{\frac{\dot{c}(t)}{c(t)}-\frac{\dot{H}}{H}}\right].$
(V.4)

Out of the set of initial equations (II.1)-(II.3) only two of them are
independent. On the other hand, only (II.2) (acceleration equation) and (II.3)
(continuity equation) contain the pressure. This is why while having (V.4) we
will define the effective pressure

$p_{eff}=p+p_{E}$ (V.5)

and then write down the continuity equation (II.3) as

$\dot{\rho}+3H\left(\rho+\frac{p_{eff}}{c^{2}(t)}\right)+\frac{\dot{G}(t)}{G(t)}\rho=0,$
(V.6)

or

$\displaystyle\dot{\rho}$ $\displaystyle+$ $\displaystyle
3H\left(\rho+\frac{p}{c^{2}(t)}\right)+\frac{\dot{G}(t)}{G(t)}\rho$
$\displaystyle=$ $\displaystyle\frac{3\gamma H^{3}}{8\pi
G(t)}\left[\frac{5\frac{\dot{c}(t)}{c(t)}-\frac{\dot{G}(t)}{G(t)}-2\frac{\dot{H}}{H}}{\frac{\dot{c}(t)}{c(t)}-\frac{\dot{H}}{H}}\right],$

and the acceleration equation (II.2) as

$\frac{\ddot{a}}{a}=-\frac{4\pi
G(t)}{3}\left(\rho+\frac{3p_{eff}}{c^{2}(t)}\right)$ (V.8)

or

$\frac{\ddot{a}}{a}=-\frac{4\pi
G(t)}{3}\left(\rho+\frac{3p}{c^{2}(t)}\right)+\frac{\gamma
H^{2}}{2}\left[\frac{5\frac{\dot{c}(t)}{c(t)}-\frac{\dot{G}(t)}{G(t)}-2\frac{\dot{H}}{H}}{\frac{\dot{c}(t)}{c(t)}-\frac{\dot{H}}{H}}\right]$
(V.9)

In order to solve the continuity equation (V) we have to put $f(t)=0$ in the
Friedmann equation. Alternatively, we see by comparing (V) and (II.3) for
$k=0$, that we need to put $f(t)=0$. We then obtain the simplest form of the
Friedmann equation to use

$\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi G(t)}{3}\rho,$ (V.10)

By using (V.9) and (V.10), we get for varying $c(t)=c_{0}a^{n}$ and
$G(t)=G_{0}a^{q}$:

$\left(\dot{H}\right)^{2}-\left(\frac{B_{1}+2n}{2}\right)\dot{H}H^{2}-\left(\frac{nB_{2}+q\gamma}{2}\right)H^{4}=0,$
(V.11)

where

$B_{1}=-3(1+w)+2\gamma$ (V.12) $B_{2}=3(1+w)-5\gamma$ (V.13)

The cosmological solutions are obtained below. We consider two cases.

###  V.1 $G$ varying models only: $\dot{G}(t)\neq 0$ and $\dot{c}(t)=0$;
$q\neq 0,n=0$.

The Eq. (V.11) reduces to

$\left(\dot{H}\right)^{2}-\left(\frac{B_{1}}{2}\right)\dot{H}H^{2}-\left(\frac{q\gamma}{2}\right)H^{4}=0,$
(V.14)

or we can write

$\dot{H}-\frac{B_{1}H^{2}}{4}=\pm\sqrt{\left(\frac{q\gamma}{2}+\frac{B^{2}_{1}}{16}\right)}H^{2}$
(V.15)

or

$\dot{H}=-WH^{2},$ (V.16)

where

$W=\mp\sqrt{\left(\frac{q\gamma}{2}+\frac{B^{2}_{1}}{16}\right)}+\frac{B_{1}}{4}.$
(V.17)

Solving (V.16) for the Hubble parameter, we have

$H=H_{0}a^{-W},$ (V.18)

where $H_{0}$ is a constant of integration. Solving (V.18) for the scale
factor $a(t)$, one gets

$a(t)=W^{\frac{1}{W}}\left[H_{0}(t-t_{0})\right]^{\frac{1}{W}}.$ (V.19)

###  V.2 $c$ varying models only: $\dot{G}(t)=0$ and $\dot{c}(t)\neq 0$;
$q=0,n\neq 0$

From (V.11) we obtain

$\left(\dot{H}\right)^{2}-\left(\frac{B_{1}+2n}{2}\right)\dot{H}H^{2}-\left(\frac{nB_{2}}{2}\right)H^{4}=0,$
(V.20)

Following the same procedure as in the subsection A, one can find the Hubble
parameter and the scale factor for varying $c$ as:

$H=H_{0}a^{-X}$ (V.21)

and

$a(t)=X^{\frac{1}{X}}\left[H_{0}(t-t_{0})\right]^{\frac{1}{X}},$ (V.22)

where, $H_{0}$ and $t_{0}$ are real constants and $X$ is given by

$X=-\left(\pm\sqrt{\left(\frac{nB_{2}}{2}+\frac{(B_{1}+2n)^{2}}{16}\right)}+\frac{B_{1}+2n}{4}\right).$
(V.23)

Both of the above cases have the same non-varying constants limit ($n\to 0$ or
$G\to 0$) of $W=B_{1}/2$.

##  VI Gravitational thermodynamics - horizon heat flow

In this section we use yet another approach to derive entropic cosmology which
is based on the application of the idea that one can get gravitational
Einstein field equations using the heat flow through the horizon to which
Bekenstein entropy (III.6) and Hawking temperature (III.5) (with $\gamma=1$)
are assigned.

The heat flow $dQ$ out through the horizon is given by the change of energy
$dE$ inside the apparent horizon and relates to the flow of entropy $TdS$ as
follows Danielsson ; BakRay2000 ; Hayward99

$dQ=TdS=-dE.$ (VI.1)

If the matter inside the horizon has the form of a perfect fluid and $c$ is
not varying, then the heat flow through the horizon over the period of time
$dt$ is BakRay2000

$\frac{dQ}{dt}=T\frac{dS}{dt}=A(\varrho+\frac{p}{c^{2}})=4\pi
r_{A}^{2}(\varrho+\frac{p}{c^{2}})$ (VI.2)

However, in our case $c$ is varying in time and we have to take this into
account while calculating the flow so that bearing in mind that the mass
element is $dM$ we have the energy through the horizon as

$-dE=c^{2}dM+2Mcdc+pdV.$ (VI.3)

The mass element flow is

$dM=A(vdt)\varrho=dV\varrho,$ (VI.4)

where $vdt=s$ is the distance travelled by the fluid element, $v$ is the
velocity of the volume element, and $dV$ is the volume element. The velocity
of a fluid element can be related to the Hubble law of expansion

$v=Hr_{A}$ (VI.5)

so that (VI.4) can be written down as

$dM=AHr_{A}\varrho dt.$ (VI.6)

We assume that the speed of light is the function of the volume through the
scale factor i.e. $c=c(V)$ and since $a\propto V^{1/3}$, then $c=c(a)$ youm .
We have

$\frac{dc}{dV}=\frac{1}{3}\frac{1}{V^{2/3}}\frac{dc}{da}$ (VI.7)

and besides by putting $M=V\varrho$ in (VI.3) we get

$-dE=c^{2}dV\left(\varrho+\frac{p}{c^{2}}+\frac{2}{3}\varrho\frac{a}{c}\frac{dc}{da}\right).$
(VI.8)

Using (VI.6), (VI.8) and (III.9) (replacing $r_{h}$ by $r_{A}$) one has from
(VI.1)

$4\pi
r_{A}^{2}H\left(\varrho+\frac{p}{c^{2}}+\frac{2}{3}\varrho\frac{a}{c}\frac{dc}{da}\right)=\frac{c^{2}}{2G}\left(3\frac{\dot{c}}{c}+2\frac{\dot{r}_{A}}{r_{A}}-\frac{\dot{G}}{G}\right),$
(VI.9)

or after explicitly using (III.12) we get a generalized acceleration equation

$\displaystyle\dot{H}$ $\displaystyle=$ $\displaystyle-4\pi
G(\rho+\frac{p}{c^{2}})+\frac{1}{2}\left(5\frac{\dot{c}}{c}-\frac{\dot{G}}{G}\right)H$
(VI.10) $\displaystyle-$ $\displaystyle\frac{8\pi
G}{3}\frac{\dot{c}}{c}\frac{\rho}{H}+\frac{1}{2}\frac{kc^{2}}{a^{2}H}\left(\frac{\dot{c}}{c}-\frac{\dot{G}}{G}+2H\right),$

which for $\dot{c}=\dot{G}=0$ gives the Eq. (A6) of Ref. easson .

In order to get the Friedmann equation, we have to use the continuity equation
(VII) but for adiabatic expansion ($dS=0$) to obtain

$H\left(\rho+\frac{p}{c^{2}}\right)=-\frac{\dot{\rho}}{3}-\frac{2}{3}\frac{\dot{c}}{c}\rho.$
(VI.11)

By using Eq. (VI.10) in (VI.11), we have

$\displaystyle H\dot{H}$ $\displaystyle=$ $\displaystyle\frac{4\pi
G(t)}{3}\dot{\rho}+\frac{1}{2}\left(5\frac{\dot{c}}{c}-\frac{\dot{G}}{G}\right)H^{2}$
(VI.12) $\displaystyle+$
$\displaystyle\frac{1}{2}\frac{kc^{2}}{a^{2}H}\left(\frac{\dot{c}}{c}-\frac{\dot{G}}{G}+2H\right).$

After integrating (VI.12) one obtains a generalized Friedmann equation

$\displaystyle H^{2}$ $\displaystyle=$ $\displaystyle\frac{8\pi}{3}\int
G(t)\dot{\rho}dt+\int\left(5\frac{\dot{c}}{c}-\frac{\dot{G}}{G}\right)H^{2}dt$
(VI.13) $\displaystyle+$
$\displaystyle\frac{1}{2}\int\frac{kc^{2}}{a^{2}H}\left(\frac{\dot{c}}{c}-\frac{\dot{G}}{G}+2H\right)$

For $k=0$ ($r_{A}\to r_{h}=c/H$) by taking the ansatz of the form

$c(t)=c_{0}[H(t)]^{m},$ (VI.14)

$c_{0}=$ const., $m=$ const. (or $c(t)=c_{0}(H/H_{0})^{m}$, $H_{0}=$ const.;
similar ansatz $c(t)=\dot{a}(t)$ was used in Ref. buchalter ), for varying $c$
only (i.e. for $\dot{G}=0$), we have the following equations

$H^{2}=\frac{8\pi G}{3}\rho+\frac{5m}{2}H^{2}+K,$ (VI.15)
$\frac{\ddot{a}}{a}=-\frac{4\pi
G}{3}(\rho+\frac{3p}{c^{2}})+\frac{5m}{2}H^{2}+(\frac{3m}{2}+\frac{5m^{2}}{2})\dot{H}+K+mK\frac{\dot{H}}{H^{2}},$
(VI.16)
$\dot{\rho}+3H\left(\rho+\frac{p}{c^{2}(t)}\right)+2m\frac{\dot{H}}{H}\rho=0,$
(VI.17)

where solely $K$ is the constant of integration which can be interpreted as
the cosmological constant $\Lambda c^{2}/3$ Danielsson  provided $m=0$ (cf.
the discussion in Section II and the formula (II.5)). For small $m$ one may
expand $c(t)$ given by (VI.14) in Taylor series

$\displaystyle c(t)$ $\displaystyle=$ $\displaystyle
c_{0}\left[H(t)\right]^{m}$ $\displaystyle=$ $\displaystyle
c_{0}\left[1+m\ln{H(t)}+\frac{m^{2}}{2}\left(\ln{H(t)}\right)^{2}+\ldots\right]$

and use numerical procedures to calculate the consequences of variability of
the speed of light but we keep this beyond the scope of the paper.

Our set of equations (VI.15)-(VI.17) contains two effects: the entropic force
contribution $K$ as well as many new terms related to variability of $c$ (all
the terms which involve the parameter $m$). In fact, there are as many as four
such latter terms in the equation (VI.16) (including a cross-term with $K$)
and each of them may play the role in accelerating the universe instead of
$K$-term.

For $K=0$ case one can easily solve for the Hubble parameter $H$ and the scale
factor $a$ for varying $c$ models as

$H=H_{0}a^{-\frac{3(1+w)}{2(1+m)}},$ (VI.19)

and

$a(t)\propto\left[H_{0}(t-t_{0})\right]^{\frac{2(1+m)}{3(1+w)}}.$ (VI.20)

where $H_{0}$ is the constant of integration. Besides, the continuity equation
solves by

$\rho=\rho_{0}a^{-\frac{3(1+w)}{1+m}},$ (VI.21)

where $\rho_{0}$ is a constant with the dimension of mass density. The
solutions for $K\neq 0$ can be found numerically, but we do not present them
here.

##  VII Observational Parameters

In this section we will try to give some more quantitative information about
our approach, by applying our model to observational data. We will leave the
single-fluid approach we have considered in past section, to move to the more
realistic case of a multi-fluid scenario. We will take into account the
components which make up the total mass density $\rho$, i.e., radiation
$\rho_{r}$, matter $\rho_{m}$, and some unknown vacuum energy component
$\rho_{v}$ (which can also be the cosmological constant). We take the model
with $f(t)$ and $g(t)$ given by (IV.1) and (IV.2) which do not contain the
constant $K_{1}$-term as in (IV.7) and (IV.8). However, we will get this
constant term effectively as the energy density of vacuum
$\rho_{\Lambda}=(\Lambda c^{2})/(8\pi G)$. With these assumptions we can write
the continuity equation (III) as

$\displaystyle\sum_{i}\dot{\rho_{i}}$ $\displaystyle+$ $\displaystyle
3H\left[\sum_{i}\rho_{i}+\frac{\sum_{i}p_{i}}{c^{2}(t)}\right]=-2\frac{\dot{c}}{c}\sum_{i}\rho_{i}$
$\displaystyle+$
$\displaystyle\frac{\gamma}{1-\gamma}\sum_{i}\rho_{i}\left[\left(5\frac{\dot{c}(t)}{c(t)}-\frac{\dot{G}(t)}{G(t)}\right)-2\frac{\dot{H}}{H}\right]$

where summation on $i$ runs for radiation, matter and dark energy. From (VII)
one can easily check that one can separate the contribution of the three
fluids obtaining three separate continuity equations; if we introduce the
equation of state parameter $w_{i}$ from $p_{i}=w_{i}\;\rho_{i}$, then the
continuity equations have the form:

$\displaystyle\dot{\rho_{i}}$ $\displaystyle+$ $\displaystyle
3H\rho_{i}\left(1+w_{i}\right)=-2\frac{\dot{c}}{c}\rho_{i}$ $\displaystyle+$
$\displaystyle\frac{\gamma}{1-\gamma}\rho_{i}\left[\left(5\frac{\dot{c}(t)}{c(t)}-\frac{\dot{G}(t)}{G(t)}\right)-2\frac{\dot{H}}{H}\right]\;,$

where, of course, $w_{i}=0$ for matter, $w_{i}=1/3$ for radiation, and
$w_{i}=-1$ for vacuum. From them, it is easy to check that, on one hand, no
interaction term is present, in the way of exchanging energy among the fluids;
but, on the other, entropic forces and the varying constants influence the
behaviour of the fluids, by the same amount and separately. Thus, for each of
them, a separate continuity equation holds, and we never have any violation of
the mass-energy conservation law.

The solution for each fluid from (VII) can be easily found; once we use our
ansatze, $c=c_{0}a^{n}$ and $G=G_{0}a^{q}$, we have:

$\rho_{i}=\frac{\rho_{0}}{H_{0}^{\frac{2\gamma}{1-\gamma}}}H^{\frac{2\gamma}{1-\gamma}}a^{f^{X}_{i}(\gamma,n,q)}\;$
(VII.24)

where, as usual, $H$ is the Hubble function, $H_{0}$ the Hubble constant, $a$
the scale factor, and $f_{i}(\gamma,n,q)$ are general functions obtained by
solving (VII). When considering only a varying $c$, these functions are:

$f^{c}_{i}(\gamma,n)=-3\left[1+w_{i}+\frac{n(2-7\gamma)}{3(1-\gamma)}\right]\;,$
(VII.25)

while for a varying $G$ we have:

$f^{G}_{i}(\gamma,q)=-3\left[1+w_{i}+\frac{q\gamma}{3(1-\gamma)}\right]\;.$
(VII.26)

We can note down that the main changes to the equation of state parameters
come for the varying constant assumptions: in the limit of $n\rightarrow 0$
and $q\rightarrow 0$, we recover the usual behaviours, $a^{-3}$ for matter,
$a^{-4}$ for radiation, and constancy for the vacuum. But still we have some
dynamical effects on the densities from the entropic forces, through the
$H^{\frac{2\gamma}{1-\gamma}}$ term. Thus, even in the case of no-varying
constant, the entropic forces make the vacuum dynamical.

Starting from the Friedmann equation (IV.3), after some simple algebra, we can
write the Hubble function $H$, which we explicitly need for observational
fitting. In the case of varying $c$ it will be

$E^{2}=\left(\frac{H}{H_{0}}\right)^{2}=\left[\sum_{i}\frac{\Omega_{i,0}}{1-\gamma}a^{f^{c}_{i}}(\gamma,n)\right]^{\frac{\gamma-1}{2\gamma-1}}$
(VII.27)

while for varying $G$ it will be

$E^{2}=\left(\frac{H}{H_{0}}\right)^{2}=\left[a^{q}\sum_{i}\frac{\Omega_{i,0}}{1-\gamma}a^{f^{G}_{i}}(\gamma,q)\right]^{\frac{\gamma-1}{2\gamma-1}}$
(VII.28)

We have defined the dimensionless density parameters as

$\Omega_{i}=\frac{8\pi G_{0}\rho_{i,0}}{3H^{2}_{0}},$ (VII.29)

where, $G_{0}$ is the current value of Newton’s gravitational constant.
Finally, in order to check if our model (VII) allows a transition from
deceleration to acceleration during the evolution of the universe at some
redshift $z$ in a similar way to a “pure” $\Lambda$CDM model, we have looked
at the deceleration parameter, defined as:

$q(z)=\frac{(1+z)}{2\,H^{2}(z)}\frac{dH^{2}(z)}{dz}-1\;,$ (VII.30)

where the cosmological redshift is given by $1+z=1/a$.

Figure 1: Transition from deceleration to acceleration for the model (VII).
Solid line is for standard $\Lambda$CDM model; dashed line for
varying–$G$–entropic–$\Lambda$CDM model; dotted line is for
varying–$c$–entropic–$\Lambda$CDM model. The values of the parameters are
taken from Table I and from the Planck data ($\Lambda$CDM).

##  VIII Data Analysis

The analysis has involved the largest updated set of cosmological data
available so far, and includes: Type Ia Supernovae (SNeIa); Baryon Acoustic
Oscillations (BAO); Cosmic Microwave Background (CMB); and a prior on the
Hubble constant parameter, $H_{0}$.

###  VIII.1 Type Ia Supernovae

We used the SNeIa (Supernovae Type Ia) data from the JLA (Joint-Light-curve
Analysis) compilation (JLA, ). This set is made of $740$ SNeIa obtained by the
SDSS-II (Sloan Digital Sky Survey) and SNLS (Supenovae Legacy Survey)
collaboration, covering a redshift range $0.01<z<1.39$. The $\chi^{2}_{SN}$ in
this case is defined as

$\chi^{2}_{SN}=\Delta\boldsymbol{\mathcal{F}}^{SN}\;\cdot\;\mathbf{C}^{-1}_{SN}\;\cdot\;\Delta\boldsymbol{\mathcal{F}}^{SN}\;,$
(VIII.31)

with
$\Delta\boldsymbol{\mathcal{F}}^{SN}=\mathcal{F}^{SN}_{theo}-\mathcal{F}^{SN}_{obs}$,
the difference between the observed and the theoretical value of the
observable quantity $\mathcal{F}^{SN}$; and $\mathbf{C}_{SN}$ the total
covariance matrix (for a discussion about all the terms involved in its
derivation, see (JLA, )). For JLA, the observed quantity will be the predicted
distance modulus of the SNeIa, $\mu$, given the cosmological model and two
other quantities, the stretch (a measure of the shape of the SNeIa light-
curve) and the color. It will read

$\mu(\boldsymbol{\theta})=5\log_{10}[D_{L}(z,\boldsymbol{\theta_{c}})]-\alpha
X_{1}+\beta\mathcal{C}+\mathcal{M}_{B}\;,$ (VIII.32)

where $D_{L}$ is the luminosity distance

$D_{L}(z,\boldsymbol{\theta_{c}})=\frac{c_{0}}{H_{0}}(1+z)\
\int_{0}^{z}\frac{\mathrm{d}z^{\prime}}{E(z^{\prime},\boldsymbol{\theta_{c}})}\;,$
(VIII.33)

with $H(z)\equiv H_{0}E(z)$ (following (JLA, ), we assume $H_{0}=70$ km/s
Mpc-1), $c_{0}$ the speed of light here and now, and $\boldsymbol{\theta_{c}}$
the vector of cosmological parameters. The total vector $\boldsymbol{\theta}$
will include $\boldsymbol{\theta_{c}}$ and the other fitting parameters, which
in this case are: $\alpha$ and $\beta$, which characterize the stretch-
luminosity and color-luminosity relationships; and the nuisance parameter
$\mathcal{M}_{B}$, expressed as a step function of two more parameters,
$\mathcal{M}^{1}_{B}$ and $\Delta_{m}$:

$\mathcal{M}_{B}=\begin{cases}\mathcal{M}^{1}_{B}&\mbox{if}\quad
M_{stellar}<10^{10}M_{\odot},\\\
\mathcal{M}^{1}_{B}+\Delta_{m}&\mbox{otherwise}.\end{cases}$ (VIII.34)

Further details about this choice are given in Ref. (JLA, ). The formula
(VIII.35) stands for the constant $c$ cases; when $c$ is varying according to
(IV.9), it is modified into PLB14 ; JCAP14

$D_{L}(z,\boldsymbol{\theta_{c}})=\frac{c_{0}}{H_{0}}(1+z)\
\int_{0}^{z}\frac{(1+z^{\prime})^{-n}}{E(z^{\prime},\boldsymbol{\theta_{c}})}\mathrm{d}z^{\prime}\;.$
(VIII.35)

###  VIII.2 Baryon Acoustic Oscillations

The $\chi^{2}_{BAO}$ for Baryon Acoustic Oscillations (BAO) is defined as

$\chi^{2}_{BAO}=\Delta\boldsymbol{\mathcal{F}}^{BAO}\;\cdot\;\mathbf{C}^{-1}_{BAO}\;\cdot\;\Delta\boldsymbol{\mathcal{F}}^{BAO}\;,$
(VIII.36)

where the quantity $\mathcal{F}^{BAO}$ can be different depending on the
considered survey. We used data from the WiggleZ Dark Energy Survey
(WiggleZ_0, ), evaluated at redshifts $z=\\{0.44,0.6,0.73\\}$, and given in
Table 1 of (WiggleZ, ); in this case the quantities to be considered are the
acoustic parameter

$A(z,\boldsymbol{\theta_{c}})\equiv
100\sqrt{\Omega_{m}\,h^{2}}\frac{D_{V}(z,\boldsymbol{\theta_{c}})}{c_{0}\,z}\,,$
(VIII.37)

and the Alcock-Paczynski distortion parameter

$F(z,\boldsymbol{\theta_{c}})\equiv(1+z)\frac{D_{A}(z,\boldsymbol{\theta_{c}})\,H(z,\boldsymbol{\theta_{c}})}{c_{0}}\,,$
(VIII.38)

where, $D_{A}$ is the angular diameter distance

$D_{A}(z,\boldsymbol{\theta_{c}})=\frac{c_{0}}{H_{0}}\frac{1}{1+z}\
\int_{0}^{z}\frac{\mathrm{d}z^{\prime}}{E(z^{\prime},\boldsymbol{\theta_{c}})}\;,$
(VIII.39)

and $D_{V}$ is a combination of the physical angular-diameter distance $D_{A}$
(tangential separation) and Hubble parameter $H(z)$ (radial separation)
defined as

$D_{V}(z,\boldsymbol{\theta_{c}})=\left[(1+z)^{2}D^{2}_{A}(z,\boldsymbol{\theta_{c}})\frac{c_{0}\,z}{H(z,\boldsymbol{\theta_{c}})}\right]^{1/3}.$
(VIII.40)

When dealing with varying $c$, Eqs. (VIII.37) (VIII.38) (VIII.39) (VIII.40)
have to be changed into PLB14 ; JCAP14 :

$A(z,\boldsymbol{\theta_{c}})\equiv
100\sqrt{\Omega_{m}\,h^{2}}\frac{D_{V}(z,\boldsymbol{\theta_{c}})}{c_{0}(1+z)^{-n}\,z}\,,$
(VIII.41)
$F(z,\boldsymbol{\theta_{c}})\equiv(1+z)\frac{D_{A}(z,\boldsymbol{\theta_{c}})\,H(z,\boldsymbol{\theta_{c}})}{c_{0}(1+z)^{-n}}\,,$
(VIII.42) $D_{A}(z,\boldsymbol{\theta_{c}})=\frac{c_{0}}{H_{0}}\frac{1}{1+z}\
\int_{0}^{z}\frac{(1+z^{\prime})^{-n}}{E(z^{\prime},\boldsymbol{\theta_{c}})}\mathrm{d}z^{\prime}\;,$
(VIII.43)
$D_{V}(z,\boldsymbol{\theta_{c}})=\left[(1+z)^{2}D^{2}_{A}(z,\boldsymbol{\theta_{c}})\frac{c_{0}(1+z)^{-n}\,z}{H(z,\boldsymbol{\theta_{c}})}\right]^{1/3}.$
(VIII.44)

We have also considered the data from SDSS-III Baryon Oscillation
Spectroscopic Survey (BOSS) DR$10$-$11$, described in (BOSS1, ; BOSS2, ). Data
are expressed as

$D_{V}(z=0.32)=(1264\pm 25)\frac{r_{s}(z_{d})}{r^{fid}_{s}(z_{d})}\,,$
(VIII.45)

and

$D_{V}(z=0.57)=(2056\pm 20)\frac{r_{s}(z_{d})}{r^{fid}_{s}(z_{d})}\,,$
(VIII.46)

where $r_{s}(z_{d})$ is the sound horizon evaluated at the dragging redshift
$z_{d}$, and $r^{fid}_{s}(z_{d})$ is the same sound horizon but calculated for
a given fiducial cosmological model used, being equal to $149.28$ Mpc (BOSS1,
; BOSS2, ). The redshift of the drag epoch is well approximated by
(Eisenstein, )

$z_{d}=\frac{1291(\Omega_{m}\,h^{2})^{0.251}}{1+0.659(\Omega_{m}\,h^{2})^{0.828}}\left[1+b_{1}(\Omega_{b}\,h^{2})^{b2}\right]$
(VIII.47)

where

$\displaystyle b_{1}$ $\displaystyle=$ $\displaystyle
0.313(\Omega_{m}\,h^{2})^{-0.419}\left[1+0.607(\Omega_{m}\,h^{2})^{0.6748}\right],$
$\displaystyle b_{2}$ $\displaystyle=$ $\displaystyle
0.238(\Omega_{m}\,h^{2})^{0.223}.$ (VIII.48)

And the sound horizon is defined as:

$r_{s}(z)=\int^{\infty}_{z}\frac{c_{s}(z^{\prime})}{H(z^{\prime},\boldsymbol{\theta_{c}})}\mathrm{d}z^{\prime}\,,$
(VIII.49)

with the sound speed

$c_{s}(z)=\frac{c_{0}}{\sqrt{3(1+\overline{R}_{b}\,(1+z)^{-1})}}\,$ (VIII.50)

and

$\overline{R}_{b}=31500\Omega_{b}\,h^{2}\left(T_{CMB}/2.7\right)^{-4}\,,$
(VIII.51)

with $T_{CMB}=2.726$ K.

We have also added data points from Quasar-Lyman $\alpha$ Forest from SDSS-III
BOSS DR$11$ (Lyman, ):

$\displaystyle\frac{D_{A}(z=2.36)}{r_{s}(z_{d})}$ $\displaystyle=$
$\displaystyle 10.8\pm 0.4$ $\displaystyle\frac{c_{0}}{H(z=2.36)r_{s}(z_{d})}$
$\displaystyle=$ $\displaystyle 9.0\pm 0.3.$ (VIII.52)

When working with varying $c$ models, of course, we will have to change
$D_{A}$ and $D_{V}$ as described above, and also the sound horizon, through
the definition of the sound speed, Eq. (VIII.50), which now will be PLB14 ;
JCAP14

$c_{s}(z)=\frac{c_{0}(1+z)^{-n}}{\sqrt{3(1+\overline{R}_{b}\,(1+z)^{-1})}}.$
(VIII.53)

Thus, we will have three different contributions to $\chi^{2}_{BAO}$, e.g.,
$\chi^{2}_{WiggleZ},\chi^{2}_{BOSS},\chi^{2}_{Lyman}$, depending on the data
sets we consider.

###  VIII.3 Cosmic Microwave Background

The $\chi^{2}_{CMB}$ for Cosmic Microwave Background (CMB) is defined as

$\chi^{2}_{CMB}=\Delta\boldsymbol{\mathcal{F}}^{CMB}\;\cdot\;\mathbf{C}^{-1}_{CMB}\;\cdot\;\Delta\boldsymbol{\mathcal{F}}^{CMB}\;,$
(VIII.54)

where $\mathcal{F}^{CMB}$ is a vector of quantities taken from (WangWang, ),
where Planck first data release is analyzed in order to give a set of
quantities which efficiently summarize the information contained in the full
power spectrum (at least, for the cosmological background), and can thus be
used in alternative to the latter (WangMukherjee, ). The quantities are the
CMB shift parameters:

$\displaystyle R(\boldsymbol{\theta_{c}})$ $\displaystyle\equiv$
$\displaystyle\sqrt{\Omega_{m}H^{2}_{0}}\frac{r(z_{\ast},\boldsymbol{\theta_{c}})}{c_{0}}$
$\displaystyle l_{a}(\boldsymbol{\theta_{c}})$ $\displaystyle\equiv$
$\displaystyle\pi\frac{r(z_{\ast},\boldsymbol{\theta_{c}})}{r_{s}(z_{\ast},\boldsymbol{\theta_{c}})}\,,$
(VIII.55)

and the baryonic density parameter, $\Omega_{b}\,h^{2}$. Again, $r_{s}$ is the
comoving sound horizon, but evaluated at the photon-decoupling redshift
$z_{\ast}$, given by the fitting formula (Hu, ):

${}z_{\ast}=1048\left[1+0.00124(\Omega_{b}h^{2})^{-0.738}\right]\left(1+g_{1}(\Omega_{m}h^{2})^{g_{2}}\right)\,,$
(VIII.56)

with

$\displaystyle g_{1}$ $\displaystyle=$
$\displaystyle\frac{0.0783(\Omega_{b}h^{2})^{-0.238}}{1+39.5(\Omega_{b}h^{2})^{-0.763}}$
$\displaystyle g_{2}$ $\displaystyle=$
$\displaystyle\frac{0.560}{1+21.1(\Omega_{b}h^{2})^{1.81}}\,;$ (VIII.57)

while $r$ is the comoving distance defined as:

$r(z,\boldsymbol{\theta_{c}})=\frac{c_{0}}{H_{0}}\int_{0}^{z}\frac{\mathrm{d}z^{\prime}}{E(z^{\prime},\boldsymbol{\theta_{c}})}\mathrm{d}z^{\prime}\;.$
(VIII.58)

When considering varying $c$ models, again, the sound horizon will change as
described above, and the comoving distance will be PLB14 ; JCAP14

$r(z,\boldsymbol{\theta_{c}})=\frac{c_{0}}{H_{0}}\int_{0}^{z}\frac{(1+z^{\prime})^{-n}}{E(z^{\prime},\boldsymbol{\theta_{c}})}\mathrm{d}z^{\prime}\;,$
(VIII.59)

and the shift parameter $R$ will become

$R(\boldsymbol{\theta_{c}})\equiv\sqrt{\Omega_{m}H^{2}_{0}}\frac{r(z_{\ast},\boldsymbol{\theta_{c}})}{c_{0}(1+z_{\ast})^{-n}}.$
(VIII.60)

Moreover, we have added a gaussian prior on the Hubble constant, $H_{0}$

$\chi^{2}_{H_{0}}=\frac{(H_{0}-69.6)^{2}}{0.07^{2}}$ (VIII.61)

derived from (Bennett, ).

Thus, the total $\chi^{2}_{Tot}$ will be the sum of:
$\chi^{2}_{SN},\chi^{2}_{WiggleZ},\chi^{2}_{BOSS},\chi^{2}_{Lyman},\chi^{2}_{CMB},\chi^{2}_{H_{0}}$.
We minimize $\chi^{2}_{Tot}$ using the Markov Chain Monte Carlo (MCMC) method.

Finally, we should make a few comments about the parameters which will be
constrained. The total parameters vector $\boldsymbol{\theta_{c}}$ will be
equal to
$\\{\Omega_{m},\Omega_{b},h,q,\gamma,\alpha\,\beta\,M^{1}_{B},\Delta_{M}\\}$
when considering the varying $G$ cases, and
$\\{\Omega_{m},\Omega_{b},h,n,\gamma,\alpha\,\beta\,M^{1}_{B},\Delta_{M}\\}$
when considering the varying $c$ ones. The actual observationally fitted
components of this vector are given in Table 1.

The parameter $h$ is defined in a standard way by $H_{0}\equiv 100\,h$. The
density parameters entering $H(z)$ are $\Omega_{m},\Omega_{r},\Omega_{v}$;
assuming zero spatial curvature, we can express
$\Omega_{v}=1-\gamma-\Omega_{m}-\Omega_{r}$, in order to ensure the condition
$E(z=0)=1$. Moreover, the radiation density parameter $\Omega_{r}$ will be
defined (WMAP, ) as the sum of photons and relativistic neutrinos

$\Omega_{r}=\Omega_{\gamma}(1+0.2271\mathcal{N}_{eff})\,,$ (VIII.62)

where $\Omega_{\gamma}=2.469\times 10^{-5}\,h^{-2}$ for $T_{CMB}=2.726$ K; and
the number of relativistic neutrinos is assumed to be
$\mathcal{N}_{eff}=3.046$.

###  VIII.4 Results

Our main result is presented in Fig. 2. First novelty is that we have found
the observational bounds on the Hawking temperature coefficient $\gamma$ which
(on the theoretical basis) was usually taken to of order of unity $O(1)$. Our
evaluation gives that it should be of the order of $10^{-2}-10^{-4}$. This
difference is not unexpected, because the $O(1)$ estimation was based on
purely theoretical considerations, with no previous connection to data. Now,
we show that observations are not consistent with such large values of
$\gamma$. Instead, it is at least two orders of magnitude less. Thus, the
entropic force in the model we have considered gives only a small
contribution. Similar results were obtained in (Basilakos09, ; Gomez141, ).
Another novelty is the bound on the variability of the speed of light $c$ and
the gravitational constant $G$. According to them, in the entropic scenario we
have investigated both $G$ (Fig. 2, left panel) and $c$ (Fig. 2, right panel)
should be increasing with the evolution of the universe. Bearing in mind that
the speed of light is related to the inverse of the fine structure constant
defined as

$\alpha=\frac{e^{2}}{\hbar c},$ (VIII.63)

where $e$ is the electron charge and $\hbar$ is the Planck constant, by using
(IV.9) and (VIII.63) one has

$\frac{\Delta c}{c}=-\frac{\Delta\alpha}{\alpha}=n\frac{\Delta
a}{a}\sim\frac{n}{10},$ (VIII.64)

then one can derive from Table 1 and Fig. 2 that the change in $c$ and so in
$\alpha\;(\Delta\alpha/\alpha_{0})$ from our fit is $\sim 10^{-5}$ in a
redshift range $[1;2]$ ($n=4.9\cdot 10^{-4}>0$), while other observational
bounds, in the same range (see table II of Ref. carlos  which is based on LP1
; alphaMolaro ; alphaChand ; alphaAgafonova ), give
$\Delta\alpha/\alpha_{0}\sim 10^{-6}$. But still our estimation is compatible
with other cosmological constraints, as the ones derived from CMB Planck first
release, see OBrian . Moreover, recent observations show that both positive
and negative values of $n$ are possible (the so-called $\alpha-$dipole webb ).

Finally, we can enumerate some general conclusions as follows:

  * •

the entropic scenario plus varying $c$ and/or $G$ is quite indistinguishable
from a pure-$\Lambda$CDM model, that is why we call it an
entropic-$\Lambda$CDM model. Present data is still unable to differentiate
between the two scenarios;

  * •

the model obtained (entropic-$\Lambda$CDM cosmology) is a variation of the
exchange of energy between vacuum and matter model studied in Refs.
Basilakos09 ; Grande11 ; Gomez141 .

  * •

the best fit for the value of the Hawking temperature coefficient $\gamma$ is
quite different from the theoretical values used in literature, i.e.
$\gamma=3/(2\pi)$ or $1/2$; it should be pointed out that other considered
entropic scenarios have the values of $O(1)$ (e.g. i1 );

  * •

the model with small values of the parameter $\gamma$ is equivalent to a
dynamical vacuum model with small variation of the vacuum energy studied in
Refs. Basilakos09 ; Gomez141 ;

  * •

the value for $\gamma$ is compatible with zero since we were able to put only
an upper limit to it. This would mean that the Hawking temperature were zero
for the models under study;

  * •

it is also clear that we still have the deceleration-acceleration transition,
as we show in the plot of the relation for $q(z)$ and also for $q(a)$ in Fig.
1, where our models are compared with a standard $\Lambda$CDM resulting, as
said above, barely distinguishable.

The models we have studied here involve a mixture of matter and the dark
energy fluid which is typically the energy of vacuum with small modifications
due to the variability of $c$ and $G$. This means that the discussion of the
structure formation problem (perturbation equations, the formation of the
structures, linear growth rate) is similar to those of dynamical vacuum models
given in Ref. Grande11  with $\gamma$ parameter here being analogous to $\nu$
parameter of that reference. In fact, the models the models $III$ and $IV$ of
Ref. Grande11  are indistinguishable from $\Lambda$CDM while the models $I$
and $II$ exhibit some difference what can be seen from Fig. 1 of Grande11 ,
where the density contrast and the linear growth rate of clustering are shown.

Table 1: Observational parameters of the entropic models under study.

$id.$ | $\Omega_{m}$ | $\Omega_{b}$ | $h$ | $q/n$ | $\gamma$ | $\alpha$ | $\beta$ | $\mathcal{M}^{1}_{B}$ | $\Delta_{m}$
---|---|---|---|---|---|---|---|---|---
$G=G_{0}\;a^{q}$ | $0.314^{+0.009}_{-0.008}$ | $0.0453^{+0.0009}_{-0.0009}$ | $0.698^{+0.007}_{-0.007}$ | $0.048^{+0.042}_{-0.033}$ | $<0.022$ | $0.141^{+0.007}_{-0.006}$ | $3.106^{+0.077}_{-0.087}$ | $-19.044^{+0.018}_{-0.019}$ | $-0.071^{+0.023}_{-0.023}$
$c=c_{0}\;a^{n}$ | $0.311^{+0.007}_{-0.007}$ | $0.046^{+0.001}_{-0.001}$ | $0.696^{+0.007}_{-0.007}$ | $0.00049^{+0.00049}_{-0.00053}$ | $<0.0007$ | $0.141^{+0.007}_{-0.007}$ | $3.100^{+0.080}_{-0.080}$ | $-19.043^{+0.018}_{-0.018}$ | $-0.070^{+0.023}_{-0.022}$

Figure 2: (Left.) Varying $G$ scenario: $68\%$ and $95\%$ confidence levels
for $q$ and $\gamma$. (Right.) Varying $c$ scenario: $68\%$ and $95\%$
confidence levels for $n$ and $\gamma$.

##  IX Conclusions

In this paper we extended the entropic cosmology onto the framework of the
theories with varying gravitational constant $G$ and varying speed of light
$c$. We discussed the consequences of such variability onto the entropic force
terms and the boundary terms using three different approaches which possibly
relate thermodynamics, cosmological horizons and gravity. We started with a
general set of the field equations which described varying constants entropic
cosmology with a general form of the entropic terms. In the first approach we
derived the continuity equation from the first law of thermodynamics,
Bekenstein entropy as well as Hawking temperature to fit the general entropic
terms to this continuity equation. We found appropriate single-fluid
accelerating cosmological solutions to these field equations. We also
discussed the constraints on the models which come from the second law of
thermodynamics. In the second approach we derived the entropic force for
varying constants, defined the entropic pressure, and finally modified the
continuity and the acceleration equations. Then, we determined the Friedmann
equation and gave single-fluid accelerating cosmological solutions as well.
Finally, in the third approach we got gravitational Einstein field equations
using the heat flow through the horizon to which Bekenstein entropy and
Hawking temperature were assigned.

We have also examined some of the many-fluid (first accelerating and then
decelerating) entropic models against observational data (supernovae, BAO, and
CMB). We have used data from JLA compilation of SDSS-II and SNLS collboration
(supernovae), WiggleZ Dark Energy Survey and SDSS-III Baryon Oscillation
Spectroscopic Survey (BOSS) as well as Planck data (CMB). We found that the
observational bound on the Hawking temperature coefficient $\gamma$ was much
smaller ($10^{-2}-10^{-4}$) than it is usually assumed on the theoretical
basis to be of order of unity $O(1)$. We have also found that in our entropic
models $G$ should be diminishing while $c$ should be increasing with the
evolution of the universe. Our bound on the variation of $c$ being $\Delta
c/c\sim 10^{-5}$ is at least one order of magnitude weaker than observational
bound obtained from analysis of the quasar spectra.

##  X Acknowledgements

This project was financed by the Polish National Science Center Grant
DEC-2012/06/A/ST2/00395.

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