Varying constants entropic--$\Lambda$CDM cosmology

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.


I. INTRODUCTION
General Relativity is an established theory which explains the evolution of the universe on large scales [1]. 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 [2,3]. In order to obtain this accelerated expansion, one has to put an extra term, the cosmological constant Λ or dark energy into the Einstein-Friedmann equations. Resulted ΛCDM models [4][5][6] are consistent models to explain this accelerated expansion, but the observational value of Λ 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 [7,8].
The relation between the Einstein's gravity and thermodynamics is a puzzle. In the seventies of the twentieth century, Bekenstein and Hawking [9,10] derived the laws of black hole thermodynamics which emerged to have similar properties as in standard thermodynamics. Jacobson [11] 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. [12][13][14][15]. 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 [16], 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 [17].
Recently, the entropic cosmology based on the notion of the entropic force was developed in series of papers * mpdabfz@wmf.univ.szczecin.pl † hunzaie@wmf.univ.szczecin.pl [18][19][20][21][22] and especially it was compared with supernovae data in Ref. [23]. The authors added 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 gravity is still a fundamental force and that it includes extra driving force terms or boundary terms into the Einstein field equations. This is unlike Verlinde [12], who considers gravity as an entropic force, but not as a fundamental force (see also Refs. [24][25][26][27][28][29][30][31][32][33][34]). In this paper we expand entropic cosmology suggested in Refs. [18][19][20][21][22][23] for the theories with varying physical constants: the gravitational constant G and the speed of light c. 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 [35,36] 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 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 then modify the continuity and acceleration equations. We also determine Friedmann equation and give 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.
In Section VII we give our conclusions.

II. ENTROPIC FORCE FIELD EQUATIONS AND VARYING CONSTANTS
The main idea of our consideration is to follow Refs. [18,19,23] (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 In fact, the functions f (t) and g(t) play the role analogous to bulk viscosity (cf. Refs. [50-69] of the paper by Komatsu et. al. [18]) and this is why from (II.1) and (II.2) one obtains the modified continuity equatioṅ 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.

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 [24][25][26][27][28][29][30]37]. Defining the temperature and entropy on the cosmological horizons, one can use this law of thermodynamics for the whole universe 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 * = a(t)r (r is the comoving radius and a(t) is the scale factor) is We haveV where dot represents the derivative with respect to time and the Hubble function is H(t) =ȧ/a. The internal energy E and the energy density ε(t) of the universe are related by where ρ is the mass density of the universe. Now we generalize the Hawking temperature T [10] and Bekenstein entropy S [9] of the (time-dependent) Hubble horizon at r ≡ r h = r h (t) onto the varying c and G theories as follows Here A(t) = 4πr 2 h (t) is the horizon area, ℏ is the Planck constant, k B is the Boltzmann constant, and γ is an arbitrary and non-negative parameter of the order of unity O(1) which is usually taken to be 3 2π , 3 4π or 1 2 [18,19,23]. In fact, γ 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 which after applying (III.2) and (III.4) giveṡ (III.8) From (III.5) and (III.6) we have . (III.9) By using (III.7), (III.8), and (III.9) we get the modified continuity equation as followṡ where we have used the explicit definition of the Hubble horizon modified to varying speed of light models [23] r h (t) ≡ c(t) H(t) . (III.11) If we introduced the non-zero spatial curvature k = ±1, then we would have to apply the entropy and the temperature of the apparent horizon which reads . (III.12) Simple calculations give thaṫ which for k = 0 case reduces tȯ (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 increase (non-adiabatic expansion) In fact, (III.9) gives the condition which forċ =Ġ = 0 just says that the Hubble horizon must increaseṙ h ≥ 0. ForĠ(t) = 0, and by using (III.11) and (III.16), we have and forċ(t) = 0, we have where b 1 and b 2 are constants. It is worth noting that for the expanding universe (H > 0) in order to (III. 18) to hold, c should be growing and eventually when it is expanding forever, c → ∞ which is known as the Newtonian limit [38]. On the other hand, for the expanding universe H −2 is decreasing which means that G → 0 if the universe expands forever. Then, similarly as before, the universe reaches the Newtonian limit [38].

IV. GRAVITATIONAL THERMODYNAMICS -COSMOLOGICAL SOLUTIONS
Using the generalized continuity equation (III.10) 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 Having given f (t) and g(t) one is able to write down the equations (II.1) and (II.2) as follows which form a consistent set together with Eq. (III.10).
While fitting the functions f (t) and g(t) we set k = 0. If we were to investigate k = ±1 models then the the temperature T (III.5) and the entropy S (III.6) should be defined on the apparent horizon (III. 12). An alternative choice of f (t) and g(t) which is consistent with (III.10) is There  8) or (using the fact thatĠ/G = qH,ä/a =Ḣ + H 2 ) one hasḢ where A is constant. By using (IV.12), we can rewrite (IV.7) as (IV.13) The solution of (IV.12) is where t 0 is constant and without loss of generality we took A = 1. Bearing in mind the value of (IV.10), we can easily conclude that without entropic terms the solution (IV.14) corresponds to a standard barotropic fluid Friedmann evolution a(t) ∝ (t − t 0 ) (2/3(w+1)) . The scale factor for radiation, matter and vacuum (cosmological constant) dominated eras reads as 15) The solution (IV.15) shows that in varying G entropic cosmology even dust (w = 0) can drive acceleration of the universe provided (IV. 16) On the other hand, the solution which includes Λ−term (w = −1) drives acceleration for (2γ − 1) q ≤ 2. There is an interesting check of these formulas for the case when one takes the Hawking temperature parameter γ = 1, in all three cases (radiation, dust, vacuum) the conditions for accelerated expansion fall into one relation q ≤ 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 (̺ = 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.
B. c varying models only: c(t) = coa n ; c0, n = const., The solution of the continuity equation (III.10) for varying c is where ρ 0 is constant. Applying (IV.3) and (IV.4), we haveḢ The solution of (IV. 19) is where D is constant. By using H from (IV.21), we can write (IV.17) as Finally, the solution of (IV.21) for the scale factor gives where t 0 is a constant and we have taken D = 1 without loss of generality. For radiation, dust and vacuum we have, respectively: (IV.24) For these three cases, one derives inflation provided (4 + 2n) − (4 + 7n)γ ≤ 2, (radiation) 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 γ = 1, in all three cases (radiation, dust, vacuum) the conditions for accelerated expansion fall into one relation n ≥ −2/5, but this is also a special limit of Eq. (IV.3).
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 [39] but the standard approach nowadays picks up a 0 = 1 [40].

V. ENTROPIC PRESSURE MODIFIED EQUATIONS
In this section we start with the formal definition of the entropic force as given in [18,19,23]. We assume that the temperature and entropy are given by (III.5) and (III.6) and use the definition of the entropic force We calculate the entropic force on the horizon r = r h (t) by taking Forċ =Ġ = 0 this formula reduces to the value obtained in Ref. [18]: F = γ(c 4 /G). Now, we define the entropic pressure p E , as the entropic force per unit area A, and use (III.11) to get 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 and then write down the continuity equation (II.3) aṡ and the acceleration equation (II.2) as By using (V.9) and (V.10), we get for varying c(t) = c 0 a n and G(t) = G 0 a q : The cosmological solutions are obtained below. We consider two cases.
The Eq. (V.11) reduces to where K 1 is a constant of integration. Solving (V.18) for the scale factor a(t), one gets B. c varying models only:Ġ(t) = 0 andċ(t) = 0; q = 0, n = 0 From (V.11) we obtain Following the same procedure as in the subsection A, one can find the Hubble parameter and the scale factor for varying c as: and where, K 2 and t 0 are real constants and X is given by (V.23)

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 γ = 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 T dS as follows [31][32][33] dQ = T dS = −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 [32] dQ dt 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 The mass element flow is 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. 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 ∝ V 1/3 , then c = c(a) [41]. We have and besides by putting After integrating (VI.13) this gives generalized Friedmann equation For k = 0 (r A → r h = c/H) by taking the the ansatz of the form, c(t) = c 0 H m , c 0 = const., m = const. (or c(t) = c 0 (H/H 0 ) m , H 0 = const.; similar ansatz c(t) = a(t) was used in Ref. [42]), for varying c only, we have the following equations where K is the constant of integration which can be interpreted as the cosmological constant Λc 2 /3 [33]. By considering K=0, we can solve the above equations for the Hubble parameter, H and the scale factor a for varying c models as where C 2 is a constant.

VII. 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 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 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. In all these cases the entropic terms and the varying constants played the role of dark energy.

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