Cosmology Due to Thermodynamics of Apparent Horizon

1. Introduction

It was proven that black holes obey thermodynamic laws where entropy is proportional to the horizon area [1,2] and temperature is connected with the surface gravity so that gravity is related to ordinary thermodynamics [3,4,5,6]. The Friedmann equations also can be obtained from the first law of apparent horizon thermodynamics [7,8,9,10,11,12,13,14,15,16,17]. Different forms of entropies were considered which lead to modified Friedmann’s equations [18,19,20,21,22,23,24,25]. Entropies are sources of holographic energy densities that can describe the dark energy of the universe [26,27]. Here, we propose new apparent horizon entropy $S_h=(1/\beta )arctan\left(\beta S_{BH}\right)$ with $S_{BH}$ being the Bekenstein–Hawking entropy. Our entropy, as well as other viable entropies, vanishes when the Bekenstein–Hawking entropy becomes zero. In addition, the entropy under consideration is the monotonically increasing function of the Bekenstein–Hawking entropy $S_{BH}$ and is positive which is the natural requirement. When parameter $\beta \to 0$ we have the Bekenstein–Hawking entropy, $S_h\to S_{BH}$. It is worth mentioning that the apparent horizon thermodynamics leads to the Friedmann equations within Einstein’s gravity only for a particular case when the matter is a perfect fluid with equation of state (EoS) given by $p=-\rho$, where p is the matter pressure and $\rho$ is the density energy of matter [16]. Here, we modify the Bekenstein–Hawking entropy $S_{BH}$ by apparent horizon entropy $S_h$ to consider the general case with arbitrary EoS state parameter for barotropic perfect fluid $w=p/\rho$ including the case $w=-1$. It is known that the long-range gravitational interactions are better described by generalized entropies.

We will show that our entropy results to modified Friedmann’s equations and the universe inflation. This approach corresponds to Einstein’s equations with dynamical cosmological constant. As a result, the universe inflation and dark energy are due to dynamical cosmological constant. It is worth mentioning that the inflation of the universe can be explained, for example, by coupling Einstein’s gravity with nonlinear electrodynamics (see [28] and references therein).

2. Thermodynamics of Apparent Horizon

In the following we consider the FLRW flat universe with the metric

$$d{s}^2=-d{t}^2+a{\left(t\right)}^2(d{r}^2+r^{2}d{\Omega }_2^2),$$

(1)

where $a\left(t\right)$ is a scale factor and $d{\Omega }_2^2$ denotes the line element of an 2-dimensional unit sphere. For the FLRW universe the radius of the apparent horizon $R_h=a\left(t\right)r$ reads

$$R_h={\displaystyle \frac{1}{H}},$$

(2)

with the Hubble parameter of the universe $H=\dot{a}\left(t\right)/a\left(t\right)$, where dot over $a\left(t\right)$ means the derivative with respect to the cosmological time t. The total energy inside the space is

$$E=\rho V_h={\displaystyle \frac{4\pi }{3}}\rho R_h^3.$$

(3)

while $-dE$ is the change of the energy inside the apparent horizon. Here, $\rho$ denotes the energy density of matter fields. The first law of apparent horizon thermodynamics is given by

$$dE=-T_{h}d{S}_h+Wd{V}_h,$$

(4)

where the work density in the cosmology is

$$W=-{\displaystyle \frac{1}{2}}Tr\left(T^{\mu \nu }\right)={\displaystyle \frac{1}{2}}(\rho -p).$$

where p being the matter pressure. The apparent horizon temperature is given by

$$T_h={\displaystyle \frac{H}{2\pi }}\left|1+{\displaystyle \frac{\dot{H}}{2H^2}}\right|.$$

(5)

From first law of apparent horizon thermodynamics (2.4), taking into account equations (2.2), (2.3) and (2.5), we obtain

$${\displaystyle \frac{H}{2\pi }}\left|1+{\displaystyle \frac{\dot{H}}{2H^2}}\right|d{S}_h=-{\displaystyle \frac{4\pi }{3H^3}}d\rho +{\displaystyle \frac{2\pi (\rho +p)}{H^4}}dH,$$

(6)

or

$${\displaystyle \frac{H}{2\pi }}\left|1+{\displaystyle \frac{\dot{H}}{2H^2}}\right|{\dot{S}}_h=-{\displaystyle \frac{4\pi \dot{\rho }}{3H^3}}\left(1+{\displaystyle \frac{\dot{H}}{2H^2}}\right),$$

(7)

where we have used the continuity equation (the energy momentum conservation)

$$\dot{\rho }=-3H(\rho +p).$$

(8)

3. Modified FLRW Equations

From Eqs. (2.7) and (2.8) we obtain

$${\displaystyle \frac{H}{2\pi }}{\dot{S}}_h={\displaystyle \frac{4\pi (\rho +p)}{H^2}}.$$

(9)

From our proposed entropy

$$S_h={\displaystyle \frac{1}{\beta }}arctan\left(\beta S_{BH}\right)$$

(10)

with $S_{BH}=\pi R_h^2/G=\pi /\left(G{H}^2\right)$, one finds from Eq. (3.1) the modified Friedmann equation

$${\displaystyle \frac{\dot{H}}{1+{\beta }^2{\pi }^2/\left(G^2{H}^4\right)}}=-4\pi G(\rho +p).$$

(11)

At $\beta \to 0$ equation (3.3) becomes the usual Friedmann equation for flat universe within general relativity. Utilizing Eq. (2.8) and after integrating Eq. (3.3), we obtain the second modified Friedmann equation

$$H^2-{\displaystyle \frac{\beta \pi }{G}}arctan\left({\displaystyle \frac{G{H}^2}{\beta \pi }}\right)={\displaystyle \frac{8\pi G}{3}}\rho .$$

(12)

At $\beta =0$ we comes to usual FLRW equation for flat universe within Einstein’s gravity. Equation (3.4) can be represented as

$$H^2={\displaystyle \frac{8\pi G}{3}}\rho +{\displaystyle \frac{{\Lambda }_{eff}}{3}}.$$

(13)

where ${\Lambda }_{eff}={\displaystyle \frac{3\beta \pi }{G}}arctan\left({\displaystyle \frac{G{H}^2}{\beta \pi }}\right)$ plays the role of the effective (a dynamical) cosmological constant. From Eq. (3.5) we obtain the dark energy density

$${\rho }_D={\displaystyle \frac{{\Lambda }_{eff}}{8\pi G}}={\displaystyle \frac{3\beta }{8G^2}}arctan\left({\displaystyle \frac{G{H}^2}{\beta \pi }}\right).$$

(14)

At small $\pi \beta /\left(G{H}^2\right)$ the effective cosmological constant becomes ${\Lambda }_{eff}\approx 3\beta {\pi }^2/2G$ while at small $G{H}^2/\left(\pi \beta \right)$ it reads ${\Lambda }_{eff}\approx 3H^2$. The plot of ${\Lambda }_{eff}$ versus H at different $b=\pi \beta /G$ is given in Figure 1.

Implying that dark substance obeys ordinary conservation law, where no mutual interaction between the cosmos components, we find the pressure

$$p_D=-{\displaystyle \frac{{\dot{\rho }}_D}{3H}}-{\rho }_D.$$

(15)

Making use of Eqs. (3.6) and (3.7) one finds the pressure

$$p_D=-{\displaystyle \frac{\pi {\beta }^2\dot{H}}{4G({\pi }^2{\beta }^2+G^2{H}^4)}}-{\displaystyle \frac{3\beta }{8G^2}}arctan\left({\displaystyle \frac{G{H}^2}{\beta \pi }}\right).$$

(16)

The second law of thermodynamics leads to the requirement ${\dot{S}}_h\ge 0$ and from Eq. (3.2) we obtain ${\dot{S}}_{BH}/(1+{\beta }^2{S}_{BH}^2)\ge 0$ or ${\dot{S}}_{BH}=-2\pi \dot{H}/\left(G{H}^3\right)\ge 0$. Thus, we have the same inequality as for the Bekenstein–Hawking entropy. This requirement for positive Hubble parameter gives $\dot{H}\le 0$. As a result, from Eq. (3.3) one finds $\rho +p\ge 0$ or for positive energy density we have for EoS parameter $w\ge -1$. One can use the redshift $z=a_0/a\left(t\right)-1$ instead of the scale factor $a\left(t\right)$, where $a_0$ is a constant corresponding to a scale factor at the current time. Then from the continuity equation (2.8) and EoS $p=w\rho$ we find the density energy of matter as

$$\rho ={\rho }_0{\left({\displaystyle \frac{1+z}{a_0}}\right)}^{3(1+w)}.$$

(17)

where ${\rho }_0$ is the density energy of matter at the present time. Making use of Eqs. (3.4) and (3.9) we obtain

$${\displaystyle \frac{1}{R_h^2}}-{\displaystyle \frac{\beta \pi }{G}}arctan\left({\displaystyle \frac{G}{\beta \pi R_h^2}}\right)={\displaystyle \frac{8\pi G}{3}}{\rho }_0{\left({\displaystyle \frac{1+z}{a_0}}\right)}^{3(1+w)}.$$

(18)

In Figure 2 we depicted the function of apparent horizon radius $R_h$ versus redshift z for $G=1$, ${\rho }_0=1$, $a_0=1$.

As redshift increases the apparent horizon radius decreases.

The deceleration parameter is defined as

$$q=-{\displaystyle \frac{\ddot{a}a}{{\dot{a}}^2}}=-1-{\displaystyle \frac{\dot{H}}{H^2}}.$$

(19)

When $q<0$ the acceleration phase takes place but as $q>0$ we have the universe deceleration. By virtue of Eqs. (3.3), (3.9) and (3.11) we obtain

$$q={\displaystyle \frac{4\pi {\rho }_0(1+w)(G^2{H}^4+{\pi }^2{\beta }^2)}{G{H}^6}}{\left({\displaystyle \frac{1+z}{a_0}}\right)}^{3(1+w)}-1.$$

(20)

Equations (3.4), (3.9) and (3.12) define the function of the deceleration parameter q versus redshift z. Making use of Eqs. (3.4) and (3.12) one finds also the deceleration parameter q as a function of H at fixed $\beta$ and EoS parameter w

$$q=$$

$${\displaystyle \frac{3(1+w)(G^2{H}^4+{\pi }^2{\beta }^2)}{2G^2{H}^6}}\left(H^2-{\displaystyle \frac{\beta \pi }{G}}arctan\left({\displaystyle \frac{G{H}^2}{\beta \pi }}\right)\right)-1.$$

(21)

In Figure 3 we plotted the function of the deceleration parameter q versus the Hubble parameter H for $G=1$, ${\rho }_0=1$, $a_0=1$.

For some parameters w and $\beta$ there are two eras: inflation and deceleration but in some w and $\beta$ we have only eternal university acceleration (inflation). From Eq. (3.13) we obtain the asymptotic

$$\underset{H\to \infty }{lim}q={\displaystyle \frac{3w+1}{2}}.$$

(22)

Thus, the asymptotic of the deceleration parameter does not depend on the entropy parameter $\beta$. At $\beta =0$ we obtain from Eq. (3.13) that $q=(3w+1)/2$. Figure 3 is in accordance with the formula (3.14). Making use of Eq. (3.14), we obtain the condition when two phases, acceleration and deceleration, take place: $w>-1/3$ ($q>0$). When $w<-1/3$ the eternal inflation is realised. With the help of Eq. (3.4) and (3.9) we obtain the redshift

$$z=$$

$$a_0{\left({\displaystyle \frac{3}{8\pi {\rho }_{0}G}}\left(H^2-{\displaystyle \frac{\beta \pi }{G}}arctan\left({\displaystyle \frac{G{H}^2}{\beta \pi }}\right)\right)\right)}^{1/\left(3\right(1+w\left)\right)}-1.$$

(23)

The approximate real and positive solutions to Eq. (3.13) for the transition redshifts $z_t$ when $q=0$, $G=1$, $w=-0.1$ are given in Table 1. Table I shows that when the entropy parameter $\beta$ increases the Hubble parameter H and reshift z also increase (at fixed w) for a divided point $q=0$ between two pases, universe acceleration and deceleration. One can calculate deceleration parameter q for matter dominated era ($w=0$) and for the current era ($z=0$), from Eqs. (3.13) and (3.15). We obtain from Eq. (3.15) for the current era, when $z=0$, solutions for the Habble parameter H and deceleration parameter q from Eq. (3.13) for different entropy parameters $\beta$, presented in Table II. Negative value of the deceleration parameter q in Table II indicates on the acceleration phase in the current time. According to [30] the deceleration parameter at the current time is $q_0\approx -0.6$. Table II shows that there is entropy parameter $\beta \approx 0.5$ which can produce that result. Making use of (3.15) we depicted the dependence of Habble parameter H on redshift z in Figure 4.

According to Figure 4, when z increases, H also increases. At fixed $\beta$, if EoS parameter w increases the Habble parameter H also increases. In accordance with figure, if parameter $\beta$ increases at fixed w the Habble parameter H also increases. With the help of Eqs. (3.13) and (3.15) we plotted the deceleration parameter q versus redshift z in Figure 5.

According to Figure 5, when z increases q also increases. At fixed $\beta$ (Left panel), if EoS parameter w increases the deceleration parameter q also increases. At $w=-0.1$ and $-0.3$ there are two phases, acceleration $q<0$ and deceleration $q>0$ but at $w=-0.5$ one has only the acceleration phase (eternal inflation). In accordance with Figure 5 (Right panel), if parameter $\beta$ increases at fixed w the deceleration parameter q also increases. Here we have two phases: acceleration and deceleration.

4. Summary

Thus, we have proposed entropy $S_h=(1/\beta )arctan\left(\beta S_{BH}\right)$ which shares similar property as the Bekenstein–Hawking entropy $S_{BH}$: it vanishes when the apparent horizon radius $R_h$ vanishes; $S_h$ monotonically increases as the apparent horizon radius $R_h$ increases and it is positive. We consider the barotropic perfect fluid and flat FLRW universe. From first law of apparent horizon thermodynamics we obtained the modified Friedmann’s equations. The addition term in the second Friedmann’s equation is treated as a dynamical cosmological constant. We showed that the universe inflation is due to holographic dark energy. It is worth noting that Barrow’s and Tsallis’s entropies also lead to Einsten’s equations with the dynamical cosmological constant [29]. By analysing the deceleration parameter we find that for some parameters our model can describe inflation and deceleration phases or only eternal inflation. We have calculated the transition redshifts when $q=0$, presented in Table I for some parameters w and $\beta$. Table II shows that at $\beta \approx 0.5$ and $w=-2/3$ we obtain the current deceleration parameter $q_0\approx -0.6$. Cosmology based on the modified Friedmann equations obtained may be of interest for a description of inflation and late time of the universe evolution.