Anisotropic Multiverse with Varying c and G and Study of Its Thermodynamics

Abstract

We assume the anisotropic model of the Universe in the framework of a varying speed of light c and a varying gravitational constant G theories and study different types of singularities. We write the scale factors for the singularity models in terms of cosmic time and find some conditions for possible singularities. For future singularities, we assume the forms of a varying speed of light and varying gravitational constant. For regularizing the Big Bang singularity, we assume two forms of scale factors: the sine model and the tangent model. For both models, we examine the validity of null and strong energy conditions. Starting from the first law of thermodynamics, we study the thermodynamic behaviors of a number n of universes (i.e., multiverse) for (i) varying c, (ii) varying G and (iii) varying both c and G models. We find the total entropies for all the cases in the anisotropic multiverse model. We also find the nature of the multiverse if the total entropy is constant.

1. Introduction

The multiverse theory states that there are multiple versions of the Universe, and each of them is slightly different from the other. Each universe is known as a parallel universe, and the whole collection of these parallel universes is known as the multiverse (for review, see [1]). The notion of a cosmic multiverse has been introduced in the context of eternal inflation [2,3] caused by an inflation field. Thus, due to the inflationary expansion, a continuously growing number of subuniverses is produced which is surrounded by rapidly growing regions of inflating space [4]. In ref. [4], the model of multiverse has been discussed by taking the scalar field model of dark energy. The observational consequences of an interacting multiverse have been studied in [5]. Marosek et al. [6] analyzed the isotropic cyclic multiverse model and studied several kinds of singularities for a varying speed of light c and a varying gravitational constant G. Furthermore, they studied the thermodynamics of the doubleverse model. The scenarios of parallel cyclic multiverses with varying G were studied in [7]. They described the formalism of quantum entanglement in the context of the multiverse and the thermodynamics of entanglement for the cosmological models. The preinflation scenario from the multiverse was studied in [8]. A canonically quantized multiverse was studied in [9]. The vacuum decay in an interacting multiverse was studied in [10]. Marosek et al. [6] considered the isotropic cyclic multiverse model and studied several kinds of singularities for a varying speed of light c and a varying gravitational constant G and regularizing the singularities in the cyclic Universe. Furthermore, they studied the thermodynamics of the doubleverse model and found the nature of the doubleverse when the total entropy of the doubleverse is constant. Furthermore, the regularizing cosmological singularities were studied in [6,11,12] by varying physical constant (say, c and G) theories. Motivated by their works, here we study the singularities in the anisotropic model of the Universe for a varying c and G. We study the regularizing singularities for varying constants in the anisotropic model of the cyclic Universe. We discuss the thermodynamic behaviors in the anisotropic multiverse (n universes) model and investigate the nature of the multiverse if the total entropy of the multiverse is constant. The paper is organized as follows: In Section 2, we study the anisotropic space–time model with a varying c and G. In Section 3, we describe some kinds of singularities. In Section 4, we discuss regularizing the Big Bang and Big Rip singularities for the cyclic model. In Section 5, we study thermodynamics in a multiverse with a varying c and G. The discussions and some concluding remarks are presented in the Section 6.

2. Anisotropic Model of the Universe

Marosek et al. [6] studied an isotropic model of the Universe for a varying speed of light and a varying gravitational constant [13,14,15]. In this section, we study the anisotropic model of the Universe, and we write Einstein’s field equations with the continuity equation with a varying speed of light and a varying gravitational constant. Now, we consider the homogeneous and anisotropic space–time model of the Universe described by the line element [16,17,18,19].

$$d{s}^2=-c^2\left(t\right)d{t}^2+a^2\left(t\right)d{x}^2+b^2\left(t\right)d{\Omega }_k^2$$

(1)

where a and b are functions of time t only and

$$\begin{array}{c}\hfill d{\Omega }_k^2=\left\{\begin{array}{c}d{y}^2+d{z}^2,\mathrm{when}k=0\hfill \\ \left(\mathrm{Bianchi}\mathrm{I}\mathrm{model}\right)\hfill \\ d{\theta }^2+si{n}^2\theta d{\varphi }^2,\mathrm{when}k=+1\hfill \\ \left(\mathrm{Kantowaski}-\mathrm{Sachs}\mathrm{model}\right)\hfill \\ d{\theta }^2+sin{h}^2\theta d{\varphi }^2,\mathrm{when}k=-1\hfill \\ \left(\mathrm{Bianchi}\mathrm{III}\mathrm{model}\right)\hfill \end{array}\right.\end{array}$$

(2)

Here, $k(=0,+1,-1)$ is the curvature index of the corresponding 2-space, so that the above three types are described by Thorne [20] as flat, closed and open, respectively, and $c=c\left(t\right)$ is the time-varying speed of light.

As per the proposal of a varying speed of light [13], the Friedman equations remain valid even when $\dot{c}\ne 0$. Due to the same prescription, the Einstein’s field equations in the framework of varying speed of light and varying gravitational constant theories in an anisotropic universe are [16,17,18]

$$\frac{{\dot{b}}^2}{b^2}+2\frac{\dot{a}}{a}\frac{\dot{b}}{b}+\frac{k{c}^2\left(t\right)}{b^2}=8\pi G\left(t\right)\rho ,$$

(3)

$$\frac{2\ddot{b}}{b}+\frac{{\dot{b}}^2}{b^2}+\frac{k{c}^2\left(t\right)}{b^2}=-\frac{8\pi G\left(t\right)p}{c^2\left(t\right)},$$

(4)

$$\frac{\ddot{a}}{a}+\frac{\ddot{b}}{b}+\frac{\dot{a}}{a}\frac{\dot{b}}{b}=-\frac{8\pi G\left(t\right)p}{c^2\left(t\right)}$$

(5)

where $\rho$ and p are, respectively, the mass density and pressure. Consequently, $\rho c^2$ is the energy density. Here, $G=G\left(t\right)$ is the time-varying gravitational constant.

From Equations (3)–(5), we obtain

$$\frac{\ddot{a}}{a}+2\frac{\ddot{b}}{b}=-4\pi G\left(t\right)\left(\rho +\frac{3p}{c^2\left(t\right)}\right)$$

(6)

For a varying c and a varying G, the modified continuity equation is obtained as

$$\dot{\rho }+\left(\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}\right)\left(\rho +\frac{p}{c^2}\right)=-\rho \frac{\dot{G}}{G}+\frac{kc\dot{c}}{4\pi G{b}^2}$$

(7)

In the next sections, we study the singularities for varying constants in an anisotropic space–time model.

3. Singularities

In this section, we study several singularities in an anisotropic universe. For varying c and varying G models, we find the conditions so that singularities can occur. For a two-scale-factor model (anisotropic), the Big Bang initial singularity [21] occurs at $t\to 0$, and at this time, $a\to 0$, $b\to 0$, $\rho \to \infty$ and $\left|p\right|\to \infty$. The future singularities can be classified in the following ways:

• Type I (Big Rip) [22]: For $t\to t_s$, $a\to \infty$, $b\to \infty$, $\rho \to \infty$ and $\left|p\right|\to \infty$.
• Type II (sudden future) [23]: For $t\to t_s$, $a\to a_s$, $b\to b_s$, $\rho \to {\rho }_s$ and $\left|p\right|\to \infty$.
• Type III (finite scale factors) [24]: $t\to t_s$, $a\to a_s$, $b\to b_s$, $\rho \to \infty$ and $\left|p\right|\to \infty$.
• Type IV (Big Separation) [24]: For $t\to t_s$, $a\to a_s$, $b\to b_s$, $\rho \to 0$ and $\left|p\right|\to 0$.
• Type V (w-singularity) [25]: For $t\to t_s$, $a\to a_s$, $b\to b_s$, $\rho \to 0$, $\left|p\right|\to 0$, $w=p/\rho \to \infty$.
• Type VI (Little Rip) [26]: For $t\to \infty$, $a\to \infty$, $b\to \infty$, $\rho \to \infty$ and $\left|p\right|\to \infty$.

where $t_s$, $a_s$, $b_s$ and ${\rho }_s$ are constants with $a_s\ne 0$, $b_s\ne 0$. The Big Bang, Big Rip and Little Rip are strong singularities while the other singularities mentioned above are weak singularities. Since $a\left(t\right)$ and $b\left(t\right)$ are unknown functions of t, to keep the anisotropy of the universe, we must have $a\left(t\right)/b\left(t\right)\ne$ constant. Therefore, for simplicity of calculation, we may assume $b\left(t\right)$ is related to the power law form of $a\left(t\right)$ i.e., $b\left(t\right)=b_0{a}^{\alpha }\left(t\right)$ [16,17,18,19,27,28,29,30,31], where $b_0$ and $\alpha$ are positive constants. The physical importance of this assumption is that it gives a constant ratio of shear and expansion scalar [32]. Here, we use the two scale factors, which, after appropriate choice of parameters, admit Big Bang, Big Rip, sudden-future, finite-scale-factor and w-singularities and read as [6,11]

$$a\left(t\right)=a_s{\left(\frac{t}{t_s}\right)}^{m}exp{\left|1-\frac{t}{t_s}\right|}^n$$

(8)

where $a_s,t_s,m$ and n are positive constants. Then, $b\left(t\right)$ can be written as

$$b\left(t\right)=b_0{a}_s^{\alpha }{\left(\frac{t}{t_s}\right)}^{m\alpha }exp\left(\alpha {\left|1-\frac{t}{t_s}\right|}^n\right)$$

(9)

Since the first powers of Equations (8) and (9) give the Big Bang singularity, the second parts of (8) and (9) provide the future singularity. Now we want to analyze the future singularity for our anisotropic model. For the future singularity part, taking the expressions of a and b from Equations (8) and (9) and putting their values in Equations (3) and (4), we directly obtain the energy density and pressure for the future singularity as [6,11]

$$\begin{array}{c}\hfill \rho =\frac{1}{8\pi G\left(t\right)}\left[\frac{({\alpha }^2+2\alpha )n^2}{t_s^2}{\left|1-\frac{t}{t_s}\right|}^{2n-2}\right.\\ \hfill \left.+\frac{k{c}^2\left(t\right)}{a_s^{2\alpha }}exp\left(-\alpha {\left|1-\frac{t}{t_s}\right|}^n\right)\right]\end{array}$$

(10)

and

$$\begin{array}{c}\hfill p=-\frac{c^2\left(t\right)}{12\pi G\left(t\right)}\left[\frac{(5{\alpha }^2+2\alpha +2)n^2}{2t_s^2}{\left|1-\frac{t}{t_s}\right|}^{2n-2}\right.\\ \hfill +\frac{(1+2\alpha )n(n-1)}{t_s^2}{\left|1-\frac{t}{t_s}\right|}^{n-2}\\ \hfill \left.+\frac{k{c}^2\left(t\right)}{2a_s^{2\alpha }}exp\left(-\alpha {\left|1-\frac{t}{t_s}\right|}^n\right)\right]\end{array}$$

(11)

We observe that for $0\le n\le 1$, there is a finite-scale-factor singularity, for $1\le n\le 2$, there is a sudden-future singularity and for $n\ge 2$, there is a generalized sudden-future singularity. Since a future singularity occurs at $t=t_s$, we may assume the variable G is in the power law form [6,11]:

$$G\left(t\right)=G_0{\left|1-\frac{t}{t_s}\right|}^{-r}$$

(12)

where $r>0$ and $G_0>0$ are constants. For this form of $G\left(t\right)$, at $t=t_s$, we have $G\to \infty$. At $t=t_s$, the density and pressure are finite for $r>2-n$. Therefore, a sudden singularity is regularized due to the strong gravitational coupling ($G\to \infty$ at $t=t_s$). However, for $t\to \infty$, the scale factors $a\to \infty$ and $b\to \infty$ and both the density $\rho \to \infty$ and pressure $\left|p\right|\to \infty$, which yields to a Little Rip singularity. On the other hand, similar to the choice of $G\left(t\right)$, we may choose the varying speed of light c in the power law form [6,11]:

$$c\left(t\right)=c_0{\left|1-\frac{t}{t_s}\right|}^{\frac{\beta }{2}}$$

(13)

where $\beta >0$ and $c_0>0$ are constants. For this form of $c\left(t\right)$, at $t=t_s$, we have $c\to 0$. However, for $\beta >2-n$ at $t=t_s$, both the density $\rho \to \infty$ and pressure $p\to \infty$, which regularizes to a sudden singularity. Furthermore, for $t\to \infty$, we obtain $a\to \infty$, $b\to \infty$, $\rho \to \infty$ and $\left|p\right|\to \infty$, which yields at Little Rip singularity. Thus, due to the above choices of the varying c and G, a sudden singularity can be regularized.

4. Regularizing Strong Singularity with Varying $\mathit{G}$

In this section, we study regularizing strong singularities such as the Big Bang and Big Rip singularities for varying G models in an anisotropic cyclic universe. Therefore, we assume a constant c in order to construct a regularized anisotropic universe [6]. In the following subsection, we discuss the sine model to study regularizing a Big Bang singularity, while in Section 4.2, we discuss the tangent model to study regularizing Big Bang and Big Rip singularities. Furthermore, we study the validity of the energy conditions for both models.

4.1. Regularizing Big Bang Singularity: Sine Model

Here, for a cyclic universe, we assume the “sine” model to describe regularizing the Big Bang singularity. Therefore, the scale factors $a\left(t\right)$ and $b\left(t\right)$ can be chosen in the forms [6]:

$$a\left(t\right)=a_0\left|sin\left(\pi \frac{t}{t_c}\right)\right|$$

(14)

and

$$b\left(t\right)=b_0{a}^{\alpha }\left(t\right)=b_0{a}_0^{\alpha }\left|si{n}^{\alpha }\left(\pi \frac{t}{t_c}\right)\right|$$

(15)

where $a_0,b_0$ and $\alpha$ are positive constants. Here, $t_c$ is the turning point for the cyclic universe where scale factors a and b are zero. For $t\to 0$, we get $a\to 0$ and $b\to 0$, which follows a Big Bang singularity. For constant c, we assume

$$G\left(t\right)=\frac{G_0}{ab}$$

(16)

where $G_0$ is a positive constant. Thus, at the Big Bang singularity (at $t=0$), $G\to \infty$. However, at the turning point ($t=t_c$), we have $a\to 0$, $b\to 0$ and $G\to \infty$. That means at the turning point of the cyclic universe, the curvature singularity (Big-Bang-like singularity) can be regularized due to the strong gravitational coupling. Here, “like” is used because the density and pressure are regular for this singularity but they are not regular at the Big Bang. Using Einstein’s field equations, we obtain the density and pressure in the forms:

$$\begin{array}{c}\hfill \rho =\frac{a_0^{2\alpha +1}{b}_0^2}{8\pi G_0}si{n}^{2\alpha +1}\left(\pi \frac{t}{t_c}\right)\left[({\alpha }^2+2\alpha )co{t}^2\left(\pi \frac{t}{t_c}\right)\right.\\ \hfill \left.+\frac{k{c}^2}{a_0^{2\alpha }{b}_0^2}cose{c}^{2\alpha }\left(\pi \frac{t}{t_c}\right)\right]\end{array}$$

(17)

and

$$\begin{array}{c}\hfill p=\frac{\pi c^2}{12G_0{t}_c^2}\left[(1+2\alpha )-3{\alpha }^{2}co{t}^2\left(\pi \frac{t}{t_c}\right)\right]\\ \hfill -\frac{k}{24\pi G{a}_0^{2\alpha }{b}_0^2}cose{c}^{2\alpha }\left(\pi \frac{t}{t_c}\right)\end{array}$$

(18)

Now, we study the null energy condition and strong energy condition for this model. From the above expressions (17) and (18), we obtain

$$\begin{array}{c}\hfill p+\rho c^2=\frac{\pi c^2{a}_0^{\alpha +1}{b}_0}{12G_0{t}_c^2}\left|si{n}^{\alpha +1}\left(\pi \frac{t}{t_c}\right)\right|\times \\ \hfill \left[(1+2\alpha )+(4\alpha -{\alpha }^2)co{t}^2\left(\pi \frac{t}{t_c}\right)\right]\\ \hfill +\frac{k{c}^4}{12a_0^{\alpha -1}{b}_0}cose{c}^{\alpha }\left(\pi \frac{t}{t_c}\right)\end{array}$$

(19)

and

$$\begin{array}{c}\hfill 3p+\rho c^2=\frac{\pi c^2{a}_0^{\alpha +1}{b}_0}{4G_0{t}_c^2}\left|si{n}^{\alpha +1}\left(\pi \frac{t}{t_c}\right)\right|\times \\ \hfill \left[(1+2\alpha )+2(\alpha -{\alpha }^2)co{t}^2\left(\pi \frac{t}{t_c}\right)\right]\end{array}$$

(20)

The null energy condition $p+\rho c^2>0$ is satisfied for $0<\alpha \le 4$ with $k\ge 0$. The strong energy condition $3p+\rho c^2>0$ is satisfied for $0<\alpha \le 1$.

At the turning points, we have

$$\begin{array}{c}\hfill a\left(n{t}_c\right)=a_0,b\left(n{t}_c\right)=b_0{a}_0^{\alpha },G\left(n{t}_c\right)=\frac{G_0}{b_0{a}_0^{\alpha +1}},\\ \hfill \rho \left(n{t}_c\right)=\frac{k{c}^2}{8\pi G_0{b}_0{a}_0^{\alpha -1}}\end{array}$$

(21)

and

$$\begin{array}{c}\hfill p\left(n{t}_c\right)=\frac{\pi c^2{b}_0{a}_0^{\alpha +1}}{12G_0{t}_c^2}(1+2\alpha )-\frac{k{c}^4}{24\pi G_0{b}_0{a}_0^{\alpha -1}}\\ \hfill =-\frac{1}{3}{c}^2\rho \left(n{t}_c\right)+\frac{\pi c^2{b}_0{a}_0^{\alpha +1}}{12G_0{t}_c^2}(1+2\alpha )\end{array}$$

(22)

where $n=\frac{1}{2},\frac{3}{2},\frac{5}{2},\dots$. However, at $t=m{t}_c$ ($m=0,1,2,\dots$), $a\to 0$, $b\to 0$, $\rho =$ constant, $p=$ constant and $G\to \infty$. So the curvature singularity (Big-Bang-like singularity) is regularized in each cycle due to the strong gravitational coupling.

Now, the Hubble parameter is defined by

$$H=\frac{1}{3}\left(\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}\right)=\frac{\pi }{3t_c}(1+2\alpha )cot\left(\pi \frac{t}{t_c}\right)$$

(23)

and its derivative is obtained by

$$\dot{H}=-\frac{{\pi }^2}{3t_c^2}(1+2\alpha )cose{c}^2\left(\pi \frac{t}{t_c}\right)$$

(24)

We observe that for $m{t}_c\le t\le (m+1)t_c$, $H>0$, which represents the expansion and for $(m+1)t_c\le t\le (m+2)t_c$, $H<0$, which represents the contraction. Furthermore, since $\dot{H}<0$, there is no proper bounce of the universe.

4.2. Regularizing Big Bang and Big Rip Singularities: Tangent Model

Here, we consider a “tangent” model to describe regularizing Big Bang and Big Rip singularities. The forms of the scale factors are [6]

$$a\left(t\right)=a_0\left|tan\left(\frac{\pi t}{t_s}\right)\right|,$$

(25)

and

$$b\left(t\right)=b_0{a}^{\alpha }\left(t\right)=b_0{a}_0^{\alpha }\left|ta{n}^{\alpha }\left(\frac{\pi t}{t_s}\right)\right|$$

(26)

and the gravitational constant is

$$G\left(t\right)=\frac{G_s}{b^{\beta }\left(t\right)}=\frac{G_s}{b_0^{\beta }{a}_0^{\alpha \beta }\left|ta{n}^{\alpha \beta }\left(\frac{\pi t}{t_s}\right)\right|}$$

(27)

The mass density and pressure are obtained by

$$\begin{array}{c}\hfill \rho =\frac{b_0^{\beta }{a}_0^{\alpha \beta }}{8\pi G_s{t}_s^2}\left|ta{n}^{\alpha \beta }\left(\frac{\pi t}{t_s}\right)\right|\left[4{\pi }^2({\alpha }^2+2\alpha )cose{c}^2\left(2\frac{\pi t}{t_s}\right)\right.\\ \hfill \left.+\frac{k{c}^2{t}_s^2}{b_0^2{a}_0^{2\alpha }}\left|co{t}^{2\alpha }\left(\frac{\pi t}{t_s}\right)\right|\right]\end{array}$$

(28)

and

$$\begin{array}{c}\hfill p=\frac{\pi c^2{b}_0^{\beta }{a}_0^{\alpha \beta }}{6G_s{t}_s^2}\left|ta{n}^{\alpha \beta }\left(\frac{\pi t}{t_s}\right)\right|\times \\ \hfill \left[-(5{\alpha }^2+2\alpha +2)cose{c}^2\left(2\frac{\pi t}{t_s}\right)\right.\\ \hfill +2(1+2\alpha )cosec\left(2\frac{\pi t}{t_s}\right)cot\left(2\frac{\pi t}{t_s}\right)\\ \hfill \left.-\frac{k{c}^2{t}_s^2}{4{\pi }^2{b}_0^2{a}_0^{2\alpha }}\left|co{t}^{2\alpha }\left(\frac{\pi t}{t_s}\right)\right|\right]\end{array}$$

(29)

The minimum values of the mass density and pressure are given by

$${\rho }_{min}=\rho \left(\kappa t_s\right)=\frac{b_0^{\beta }{a}_0^{\alpha \beta }}{8\pi G_s{t}_s^2}\left[4{\pi }^2({\alpha }^2+2\alpha )+\frac{k{c}^2{t}_s^2}{b_0^2{a}_0^{2\alpha }}\right]$$

(30)

and

$$p_{min}=p\left(\kappa t_s\right)=-\frac{c^2{b}_0^{\beta }{a}_0^{\alpha \beta }}{24\pi G_s{t}_s^2}\left[4{\pi }^2(5{\alpha }^2+2\alpha +2)+\frac{k{c}^2{t}_s^2}{b_0^2{a}_0^{2\alpha }}\right]$$

(31)

where $\kappa =\frac{2m+1}{4}(m=0,1,2,\dots )$.

We see that the scale factors $a\to \infty$, $b\to \infty$, $G\to 0$, density $\rho =$ constant and pressure $p=$ constant for $t=n{t}_s$ with $n=1/2,3/2,5/2,\dots$. Therefore, the Big-Rip-like singularity is regularized. However, the scale factors $a\to 0$, $b\to 0$, $G\to \infty$, density $\rho =$ constant and pressure $p=$ constant for $t=m{t}_s$ with $m=0,1,2,\dots$. As a result, the curvature singularity (Big-Bang-like singularity) is also regularized.

Now, we study the null energy condition and strong energy condition for this model. From Equations (28) and (29), we get

$$\begin{array}{c}\hfill p+\rho c^2=\frac{\pi c^2{b}_0^{\beta }{a}_0^{\alpha \beta }}{3G_s{t}_s^2}\left|ta{n}^{\alpha \beta }\left(\frac{\pi t}{t_s}\right)\right|\times \\ \hfill \left[-({\alpha }^2+1)cose{c}^2\left(2\frac{\pi t}{t_s}\right)\right.\\ \hfill +(1+2\alpha )cosec\left(2\frac{\pi t}{t_s}\right)cot\left(2\frac{\pi t}{t_s}\right)\\ \hfill \left.+\frac{k{c}^2{t}_s^2}{4{\pi }^2{b}_0^2{a}_0^{2\alpha }}\left|co{t}^{2\alpha }\left(\frac{\pi t}{t_s}\right)\right|\right]\end{array}$$

(32)

At $t=m{t}_s$, we get: (i) $p+\rho c^2>0$, i.e., the null energy condition is satisfied for $0<\alpha \le 2$ and $k\ge 0$; (ii) $p+\rho c^2<0$, i.e., the null energy condition is violated for $\alpha >2$ and $k<0$. However, at $t=n{t}_s$, we get: $p+\rho c^2<0$ i.e., the null energy condition is always violated.

Furthermore, from Equations (28) and (29), we have

$$\begin{array}{c}\hfill 3p+\rho c^2=\frac{\pi c^2{b}_0^{\beta }{a}_0^{\alpha \beta }}{G_s{t}_s^2}\left|ta{n}^{\alpha \beta }\left(\frac{\pi t}{t_s}\right)\right|\times \\ \hfill \left[-(2{\alpha }^2+1)cose{c}^2\left(2\frac{\pi t}{t_s}\right)\right.\\ \hfill \left.+(1+2\alpha )cosec\left(2\frac{\pi t}{t_s}\right)cot\left(2\frac{\pi t}{t_s}\right)\right]\end{array}$$

(33)

At $t=m{t}_s$, we get: (i) $3p+\rho c^2>0$, i.e., the strong energy condition is satisfied for $0<\alpha \le 1$; (ii) $3p+\rho c^2<0$ i.e., the strong energy condition is violated for $\alpha >1$. However, at $t=n{t}_s$, we get: $3p+\rho c^2<0$ i.e., the strong energy condition is always violated.

5. Thermodynamics in the Multiverse

In ref. [6], the authors studied the classical thermodynamics of two universes (doubleverse) and their consideration that the entropy of the two universes was changing in such a correlated way that the total entropy was always the same (constant). Motivated by their work, in this section, we study the thermodynamics of the multiverse (n universes) model. For this purpose, we use the first law of thermodynamics and calculate the total entropy of the multiverse for (i) a varying c with a constant G model, (ii) a varying G with a constant c model and (iii) a model with both c and G varying. Then, we show that the total entropy is always the same (constant), provided there are some relations between the n universes.

5.1. Thermodynamics for Varying c

First, we consider the varying c model with a constant G and analyze the thermodynamic nature of n universes, i.e., a multiverse model. The first law of thermodynamics is given by [6]

$$dE=TdS-pdV$$

(34)

where $E,T,S,p$ and V are the internal energy, temperature, entropy, pressure and volume, respectively. From Einstein’s mass–energy equation we have

$$E=m{c}^2=\rho V{c}^2$$

(35)

where $\rho$ is the mass density and $V=a{b}^2$. From the above two equations, we obtain [6]

$$\dot{\rho }+\frac{\dot{V}}{V}\left(\rho +\frac{p}{c^2}\right)+2\rho \frac{\dot{c}}{c}-\frac{T}{V{c}^2}\dot{S}=0$$

(36)

Using the continuity equation

$$\dot{\rho }+\left(\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}\right)\left(\rho +\frac{p}{c^2}\right)=\frac{kc\dot{c}}{4\pi G{b}^2}$$

(37)

and the above Equation (36), we have

$$\frac{T}{V{c}^2}\dot{S}-2\rho \frac{\dot{c}}{c}=\frac{kc\dot{c}}{4\pi G{b}^2}$$

(38)

Defining [6]

$$\tilde{\rho }=\frac{1}{8\pi G}\left(\frac{{\dot{b}}^2}{b^2}+2\frac{\dot{a}}{a}\frac{\dot{b}}{b}+\frac{2k{c}^2}{b^2}\right)=\rho +\frac{k{c}^2}{8\pi G{b}^2}$$

(39)

and

$$\tilde{p}=p-\frac{k{c}^2}{24\pi G{b}^2}$$

(40)

we obtain

$$\dot{S}=2\tilde{\rho }\frac{V{c}^2}{T}\frac{\dot{c}}{c}$$

(41)

Here, we consider the density and pressure contributions from the nonflat ($k\ne 0$) effect, so $\tilde{\rho }$ and $\tilde{p}$ represent the effective density and effective pressure. For the equation of state of an ideal gas with respect to the effective density and pressure, we can take [6]

$$\tilde{\rho }\frac{V{c}^2}{T}=constant=\frac{N{k}_B}{\tilde{w}}$$

(42)

where N is the number of particles, $k_B$ is the Boltzmann constant and the parameter $\tilde{w}=\frac{\tilde{p}}{\tilde{\rho }{c}^2}$ (here $\tilde{w}$ is constant). Using Equations (41) and (42), and after integrating the above Equation (41), we get the entropy in the following form:

$$S\left(t\right)=\frac{2N{k}_B}{\tilde{w}}log\left[A_{0}c\left(t\right)\right]$$

(43)

where $A_0$ is constant of integration. It should be mentioned that the entropy is in a form analogous to the above standard Gibbs’s relation (which comes from the ideal gas equation of state) where the volume of the phase space is proportional to the speed of light. For a flat ($k=0$) model, we have $\tilde{\rho }=\rho$. Now, using the barotropic equation of state $p=w\rho c^2$ with the equation of state parameter $w=$ constant, we have

$$S\left(t\right)=\frac{2N{k}_B}{w}log\left[A_{0}c\left(t\right)\right]$$

(44)

The nature of the entropy depends on the nature of $c\left(t\right)$.

Now, we want to study the nature of the entropy in the multiverse model. The total entropy of the n universes (i.e., multiverse) is given by

$$\dot{S}=\sum _{i=1}^n{\dot{S}}_i\left(t\right)$$

(45)

where the entropy of the ith universe is

$$S_i\left(t\right)=\frac{2N_i{k}_B}{\tilde{w}}log\left[A_0{c}_i\left(t\right)\right],i=1,2,\dots ,n$$

(46)

We assume the following form for $c\left(t\right)$ [6]:

$$A_0{c}_i\left(t\right)=e^{{\lambda }_i{\varphi }_i\left(t\right)},i=1,2,\dots ,n$$

(47)

where the ${\lambda }_i$’s are arbitrary constants. Therefore, the total entropy of the multiverse is [6]

$$S=\sum _{i=1}^n{S}_i\left(t\right)=\sum _{i=1}^n\frac{2k_B{N}_i{\lambda }_i}{\tilde{w}}{\varphi }_i\left(t\right)$$

(48)

Now, we can study the relation between the natures of the n universes if we assume the total entropy of the multiverse to be constant. In ref. [6], the authors considered the doubleverse system and they studied the nature of two universes when the total entropy of the two universes was constant. For this purpose, here we may assume in the multiverse there are an even number of universes (i.e., n is even). To get a constant total entropy of the multiverse (i.e., $S=$ constant), we may assume [6]

$${\varphi }_i\left(t\right)=\left\{\begin{array}{c}si{n}^2\left(\frac{\pi t}{t_s}\right),i=1,2,\dots ,\frac{n}{2}\hfill \\ \\ co{s}^2\left(\frac{\pi t}{t_s}\right),i=\frac{n}{2}+1,\frac{n}{2}+2,\dots ,n\hfill \end{array}\right.$$

(49)

with

$$N_1{\lambda }_1=N_2{\lambda }_2=\dots =N_n{\lambda }_n$$

(50)

From these, we may conclude that if the entropies of $n/2$ number of universes are growing/diminishing and the entropies of the other $n/2$ number of universes are diminishing/growing then the total entropy of the multiverse may be constant. In this case, the speed of light of $n/2$ number of universes is growing/diminishing and its value for the other $n/2$ number of universes is diminishing/growing.

5.2. Thermodynamics for Varying G

Now, we consider the varying G model with a constant c and analyze the thermodynamic nature of n universes (i.e., multiverse). For a varying gravitational constant G with a constant c, the continuity equation is

$$\dot{\rho }+\left(\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}\right)\left(\rho +\frac{p}{c^2}\right)=-\rho \frac{\dot{G}}{G}$$

(51)

Then, using Equation (36), we get

$$\dot{S}=-\frac{\rho V{c}^2}{T}\frac{\dot{G}}{G}$$

(52)

Using the equation of state of an ideal gas

$$\frac{\rho V{c}^2}{T}=constant=\frac{N{k}_B}{w}$$

(53)

we get

$$S=\frac{N{k}_B}{w}log\left[\frac{B_0}{G\left(t\right)}\right]$$

(54)

where $B_0$ is an integration constant.

Thus, the entropy of the ith universe is

$$S_i\left(t\right)=\frac{N_i{k}_B}{w}log\left(\frac{B_0}{G_i\left(t\right)}\right),i=1,2,\dots ,n$$

(55)

Therefore, the total entropy of the multiverse is

$$S=\sum _{i=1}^n{S}_i=\sum _{i=1}^n\frac{N_i{k}_B}{w}log\left(\frac{B_i}{G_i\left(t\right)}\right)$$

(56)

Similar to the previous subsection, we assume that the multiverse contains an even number of universes (i.e., n is even). Now, we assume the gravitational constant $G\left(t\right)$ in terms of the scale factors is in the following form:

$$G_i\left(t\right)=\left\{\begin{array}{c}\frac{G_{0i}}{a_i\left(t\right)b_i\left(t\right)},i=1,2,\dots ,\frac{n}{2}\hfill \\ \\ G_{0i}{a}_i\left(t\right)b_i\left(t\right),i=\frac{n}{2}+1,\frac{n}{2}+2,\dots ,n\hfill \end{array}\right.$$

(57)

If all the scale factors of all the universes are equal, i.e., $a_i\left(t\right)=a\left(t\right)$ and $b_i\left(t\right)=b\left(t\right),i=1,2,\dots ,n$, then all the universes have the same type of evolution. To get the constant total entropy, we need the following conditions:

$$\begin{array}{c}\hfill N_1=N_2=\dots =N_n,\mathrm{with}\frac{B_1}{G_{01}}=\frac{B_2}{G_{02}}=\dots =\frac{B_n}{G_{0n}}\end{array}$$

(58)

We observe that the entropies and gravitational constant of $n/2$ number of universes are growing (or diminishing) and that of the other $n/2$ number of universes are diminishing (or growing) but the total entropy of the multiverse will be constant.

5.3. Thermodynamics for Varying c and G

Finally, we consider a model with both c and G varying and analyze the thermodynamic nature of n universes, i.e., multiverse model. When c and G both vary, the continuity equation becomes

$$\dot{\rho }+\left(\frac{\dot{a}}{a}+2\frac{\dot{b}}{b}\right)\left(\rho +\frac{p}{c^2}\right)=-\rho \frac{\dot{G}}{G}+\frac{kc\dot{c}}{4\pi G{b}^2}$$

(59)

Using this equation and Equation (36), we get

$$\frac{T}{V{c}^2}\dot{S}-2\rho \frac{\dot{c}}{c}=-\rho \frac{\dot{G}}{G}+\frac{kc\dot{c}}{4\pi G{b}^2}$$

(60)

Now, we obtain

$$\dot{S}=2\frac{\tilde{\rho }V{c}^2}{T}\frac{\dot{c}}{c}-\frac{\rho V{c}^2}{T}\frac{\dot{G}}{G}$$

(61)

where $\tilde{\rho }$ is defined in Equation (39). Due to an ideal gas’ equation of states, we may take

$$\begin{array}{c}\hfill \tilde{\rho }\frac{V{c}^2}{T}=constant=\frac{N{k}_B}{\tilde{w}},\\ \hfill and\rho \frac{V{c}^2}{T}=constant=\frac{N{k}_B}{w}\end{array}$$

(62)

Using the above relations, integrating (61), we get the entropy

$$S\left(t\right)=\frac{2N{k}_B}{\tilde{w}}log\left[c\left(t\right)\right]-\frac{N{k}_B}{w}log\left[G\left(t\right)\right]+log\left[D_0\right]$$

(63)

where $D_0$ is an integration constant. The entropy of the ith universe is

$$\begin{array}{c}\hfill S_i\left(t\right)=\frac{2N_i{k}_B}{\tilde{w}}log\left[c_i\left(t\right)\right]-\frac{N_i{k}_B}{w}log\left[G_i\left(t\right)\right]+log\left[D_i\right],\\ \hfill i=1,2,\dots ,n\end{array}$$

(64)

As a result, the total entropy of the multiverse is obtained by

$$\begin{array}{cc}\hfill S=\sum _{i=1}^n{S}_i=& \sum _{i=1}^n\left[\frac{2N_i{k}_B}{\tilde{w}}log\left[c_i\left(t\right)\right]\right.\hfill \\ & \left.-\frac{N_i{k}_B}{w}log\left[G_i\left(t\right)\right]+log\left[D_i\right]\right]\hfill \end{array}$$

(65)

Since here, both $c_i\left(t\right)$ and $G_i\left(t\right)$ are varying with t, to get the constant total entropy of the n universes (where n is considered as even), we can choose $c_i\left(t\right)$ from Equation (47) where ${\varphi }_i\left(t\right)$ is satisfying Equation (49) with the condition in (50) and $G_i\left(t\right)$ is satisfying Equation (57) with the condition in (58). In this case, the speed of light and gravitational constant of $n/2$ number of universes are growing/diminishing and their values for the other $n/2$ number of universes are diminishing/growing. Thus, for both a varying speed of light and a varying gravitational constant, we may conclude that the total entropy of the multiverse is constant.

6. Discussions and Concluding Remarks

We assumed an anisotropic model of the Universe in the framework of a varying speed of light c and a varying gravitational constant G theories. We mentioned different types of weak and strong singularities for the anisotropic model of the Universe. To study the strong and weak singularity models, we wrote the scale factors $a\left(t\right)$ and $b\left(t\right)$ in terms of cosmic time and found some conditions for possible weak and strong singularities. For the future singularity, the density and the pressure were obtained. We observed that there was a finite-scale-factor singularity if $0\le n\le 1$. However, there was a sudden-future singularity if $1\le n\le 2$. On the other hand, there was a generalized sudden-future singularity if $n\ge 2$. For the future singularity, we assumed particular power law forms of $c\left(t\right)$ and $G\left(t\right)$. For the choice of $G\left(t\right)$, at $t=t_s$ (the future singularity time), we obtained $G\to \infty$. At $t=t_s$, the density and pressure were finite for $r>2-n$, so the sudden singularity was regularized due to the strong gravitational coupling ($G\to \infty$ at $t=t_s$). However, for $t\to \infty$, the scale factors $a\to \infty$ and $b\to \infty$ and both the density $\rho \to \infty$ and pressure $\left|p\right|\to \infty$, which yielded a Little Rip singularity. For the choice of $c\left(t\right)$, at $t=t_s$, we obtained $c\to 0$. However, for $\beta >2-n$ at $t=t_s$, $\rho \to \infty$ and $p\to \infty$, which regularized to a sudden singularity. Furthermore, for $t\to \infty$, we obtained $a\to \infty$, $b\to \infty$, $\rho \to \infty$ and $\left|p\right|\to \infty$, which yielded at Little Rip singularity.

For a strong singularity with varying G, we assumed two forms of scale factors: sine model and tangent model. For the sine model, at the Big Bang singularity (at $t=0$), we obtained $G\to \infty$. Furthermore, at the turning point $t=m{t}_c$ ($m=0,1,2,\dots$), we obtained $a\to 0$, $b\to 0$, $\rho =$ constant, $p=$ constant and $G\to \infty$. That means at the turning point of the cyclic universe, the Big-Bang-like singularity was regularized due to the strong gravitational coupling. However, for the tangent model, the scale factors $a\to \infty$, $b\to \infty$, $G\to 0$, $\rho =$ constant and $p=$ constant for $t=n{t}_s$ with $n=1/2,3/2,5/2,\dots$. Therefore, the Big-Rip-like singularity was regularized. However, the scale factors $a\to 0$, $b\to 0$, $G\to \infty$, density $\rho =$ constant and pressure $p=$ constant for $t=m{t}_s$ with $m=0,1,2,\dots$. Thus, the Big-Bang-like singularity was also regularized. For both models, we examined the validity of the null and strong energy conditions. For the sine model, the null energy condition was satisfied for $0<\alpha \le 4$ with $k\ge 0$ and strong energy condition was satisfied for $0<\alpha \le 1$. However, for the tangent model, the null energy condition was satisfied for $0<\alpha \le 2$ and $k\ge 0$ and the strong energy condition was satisfied for $0<\alpha \le 1$.

Finally, we studied the thermodynamic nature of the multiverse (n universes) model. Using the first law of thermodynamics, we obtained the total entropy of the multiverse for (i) a varying c with a constant G model, (ii) a varying G with a constant c model and (iii) a model with both c and G varying. Then, we showed that the total entropy was always the same (constant), provided there were some relations between the n universes, where n was considered an even number. We assumed the speed of light and gravitational constant of the $n/2$ number of universes were growing/diminishing, and their values in the other $n/2$ number of universes were diminishing/growing. As a result, for both a varying speed of light and varying gravitational constant, we concluded that the total entropy of the multiverse was constant if the entropies of the $n/2$ number of universes were growing/diminishing and the entropies of the other $n/2$ number of universes were diminishing/growing.