Holographic thermodynamic relation for dissipative and non-dissipative universes
in a flat FLRW cosmology

We introduce a general formulation for cosmological equations in dissipative and non-dissipative universes, using the scale factor $a(t)$ at time $t$, based on previous works Koma6 ; Koma9 ; Koma14 ; Koma15 ; Koma16 . The general Friedmann, acceleration, and continuity equations are written as

with the Hubble parameter $H(t)$ defined as

where $G$, $c$, $\rho(t)$, and $p(t)$ are the gravitational constant, the speed of light, the mass density of cosmological fluids, and the pressure of cosmological fluids, respectively. Two extra driving terms, $f_{\Lambda}(t)$ and $h_{\textrm{B}}(t)$, are phenomenologically assumed Koma14 . Specifically, $f_{\Lambda}(t)$ is used for a $\Lambda(t)$ model, similar to $\Lambda(t)$CDM models, whereas $h_{\textrm{B}}(t)$ is used for a BV (bulk-viscous-cosmology-like) model, similar to bulk viscous models and CCDM models Koma14 ; Koma16 . That is, $f_{\Lambda}(t)$ is used for non-dissipative universes and $h_{\textrm{B}}(t)$ is used for dissipative universes. In this study, $f_{\Lambda}(t)$ and $h_{\textrm{B}}(t)$ are considered simultaneously. Only two of the three equations (the Friedmann, acceleration, and continuity equations) are independent Ryden1 . Therefore, the general continuity equation given by Eq. (3) can be derived from Eqs. (1) and (2). In addition, subtracting Eq. (1) from Eq. (2) yields

These equations are used in Sec. III.

Equation (3) indicates that the right-hand side of the general continuity equation is non-zero. A similar non-zero term appears in other cosmological models, such as energy exchange cosmology Barrow22 ; Wang0102 ; Dynamical20052013 and the bulk viscous and CCDM models Prigogine_1988-1989 ; Lima1992-1996 ; LimaOthers2023 ; Freaza2002Cardenas2020 ; BarrowLima ; BrevikNojiri ; EPJC2022 , as discussed in Refs. Koma9 ; Koma20 . For example, energy exchange cosmology assumes the transfer of energy between two fluids Barrow22 , such as the interaction between dark matter and dark energy Wang0102 . In this case, the two non-zero right-hand sides are totally cancelled because the total energy of the two fluids is conserved Koma20 . In the bulk viscous and CCDM models, an effective formulation can be obtained from an effective description for pressure, using a single fluid, as examined in the next paragraph. (When $f_{\Lambda}(t)=h_{\textrm{B}}(t)=0$, the general cosmological equations reduce to those for standard cosmology and, therefore, the continuity equation is given by $\dot{\rho}+3H[\rho+(p/c^{2})]=0$. The same continuity equation can be obtained when both $f_{\Lambda}(t)=\Lambda/3$ and $h_{\textrm{B}}(t)=0$ are considered, as for $\Lambda$CDM models.)

We now consider an effective formulation, using an effective pressure, $p_{e}=p+p^{\prime}$, which is given by Koma9

where $p^{\prime}(t)$ has been replaced by $-c^{2}h_{\textrm{B}}(t)/(4\pi G)$. Applying the effective pressure $p_{e}(t)$ to Eqs. (2), (3), and (5) yields Koma9

where $p_{e}(t)$ includes the $h_{\textrm{B}}(t)$ term. The effective pressure $p_{e}(t)$ can be related to irreversible entropies because the $h_{\textrm{B}}(t)$ term is considered to be related to an irreversible entropy arising from dissipative processes, such as the bulk viscosity Koma7 . The effective formulation is used in Sec. III. The right-hand side of Eq. (8) is non-zero except when $\dot{f}_{\Lambda}(t)=0$. This non-zero term affects the modified thermodynamics relation and is examined in Sec. III. The Friedmann equation for the effective formulation is given by Eq. (1), because the Friedmann equation does not include $h_{\textrm{B}}(t)$.

It should be noted that coupling Eq. (1) with Eq. (2) yields the cosmological equation Koma14 ; Koma15 ; Koma16 , given by

where $w$ represents the equation of the state parameter for a generic component of matter, which is given as $w=p/(\rho c^{2})$. For a matter-dominated universe and a radiation-dominated universe, the values of $w$ are $0$ and $1/3$, respectively. Instead of $p_{e}$, $p$ is used for $w$ because Eq. (10) includes $h_{\textrm{B}}(t)$. Equation (10) can describe background evolutions of the universe in various cosmological models Koma14 ; Koma15 ; Koma16 . Accordingly, Eq. (10) is used for the discussion of cosmological models, such as a constant $T_{\rm{KH}}$ model Koma19 , as examined later.

The horizon thermodynamics is closely related to the holographic principle Hooft-Bousso , which assumes that the horizon of the universe has an associated entropy and an approximate temperature EassonCai . The entropy $S_{H}$ and the temperature $T_{H}$ on the Hubble horizon are introduced according to previous works Koma11 ; Koma12 ; Koma17 ; Koma18 ; Koma19 ; Koma20 .

We select the Bekenstein–Hawking entropy $S_{\rm{BH}}$ as the associated entropy because it is the most standard. In general, the cosmological horizon is examined by replacing the event horizon of a black hole by the cosmological horizon Koma17 ; Koma18 . This replacement method has been widely accepted Jacob1995; Padma2010 ; Verlinde1 ; HDE ; Padma2012AB ; Cai2012 ; Moradpour ; Hashemi ; Wang ; Padmanabhan2004 ; ShuGong2011 ; Koma14 ; Koma15 ; Koma16 ; Koma17 ; Koma19 ; Koma20 and we use it here.

Based on the form of the Bekenstein–Hawking entropy, the entropy $S_{\rm{BH}}$ is written as Hawking1Bekenstein1

where $k_{B}$ and $\hbar$ are the Boltzmann constant and the reduced Planck constant, respectively. The reduced Planck constant is defined by $\hbar\equiv h/(2\pi)$, where $h$ is the Planck constant Koma11 ; Koma12 . $A_{H}$ is the surface area of the sphere with a Hubble horizon (radius) $r_{H}$ given by

Substituting $A_{H}=4\pi r_{H}^{2}$ into Eq. (11) and applying Eq. (12) yields

where $K$ is a positive constant given by

When a de Sitter universe is considered, $S_{\rm{BH}}$ is constant although the scale factor exponentially increases with time. Differentiating Eq. (13) with respect to $t$ yields the first derivative of $S_{\rm{BH}}$, given by Koma11 ; Koma12

The second law of thermodynamics on the horizon, $\dot{S}_{\rm{BH}}\geq 0$, is satisfied in favored cosmological models, such as $\Lambda$CDM models Koma14 . In the present study, the form of the Bekenstein–Hawking entropy, $S_{\rm{BH}}$, is typically used for the entropy $S_{H}$ on the cosmological horizon. (Various forms of black-hole entropy such as nonextensive entropy have been proposed Das2008 ; Radicella2010 ; LQG2004_123 ; Tsallis2012 ; Czinner1Czinner2 ; Barrow2020 ; Nojiri2022 ; Gohar2023 , as described in Refs. Koma18 ; Koma19 ; Koma20 . For a general form of entropy related to the Friedmann equation, see, e.g., Ref. Nojiri2024 .)

Next, we introduce an approximate temperature $T_{H}$ on the Hubble horizon. Before introducing a dynamical temperature, we review the Gibbons–Hawking temperature $T_{\rm{GH}}$, which is given by GibbonsHawking1977

This equation indicates that $T_{\rm{GH}}$ is proportional to $H$ and is constant during the evolution of de Sitter universes Koma17 ; Koma19 . In fact, $T_{\rm{GH}}$ is obtained from field theory in the de Sitter space GibbonsHawking1977 . However, most universes are not pure de Sitter universes in that their horizons are dynamic. That is, horizons of universes (including our Universe) are generally considered to be dynamic Koma19 ; Koma20 . Therefore, we introduce a dynamical Kodama–Hayward temperature, based on a previous work Koma19 .

In fact, a similar dynamic horizon for black holes has been examined in the works of Hayward Dynamical-T-1998 and Hayward et al. Dynamical-T-2008 , as described in Ref. Koma19 . Hayward suggested a dynamical temperature on a black hole horizon and clarified the relationship between the surface gravity and the temperature on a dynamic apparent horizon for the Kodama observer Dynamical-T-1998 . The Kodama–Hayward temperature on the cosmological horizon of an FLRW universe has been proposed Dynamical-T-2007 ; Dynamical-T-20092014 , based on the works of Hayward et al. Dynamical-T-1998 ; Dynamical-T-2008 .

The Kodama–Hayward temperature $T_{\rm{KH}}$ for a flat FLRW universe can be written as Tu2018 ; Tu2019

Here $H>0$ and $1+\frac{\dot{H}}{2H^{2}}\geq 0$ are assumed for a non-negative temperature in an expanding universe Koma19 ; Koma20 . When de Sitter universes are considered, $T_{\rm{KH}}$ reduces to $T_{\rm{GH}}$ because $\dot{H}=0$ and, therefore, $T_{\rm{KH}}$ is interpreted as an extended version of $T_{\rm{GH}}$ Koma19 ; Koma20 . In the present paper, the Kodama–Hayward temperature $T_{\rm{KH}}$ is typically used for the temperature $T_{H}$ on the horizon. Cosmological models used later satisfy a non-negative temperature.

As examined in Refs. Koma19 ; Koma20 , $T_{\rm{KH}}$ is constant when the following equation is satisfied:

where $\psi$ represents a dimensionless constant. Substituting Eq. (18) into Eq. (17) gives a constant temperature, $T_{\rm{KH}}=\hbar\psi H_{0}/(2\pi k_{B})$ Koma19 ; Koma20 . Accordingly, $T_{\rm{KH}}$ is non-negative when $\psi\geq 0$ is considered in an expanding universe. A universe at constant $T_{\rm{KH}}$ has been studied, using a cosmological model which includes a power-law term proportional to $H^{\alpha}$ (where $\alpha$ is a free variable) and the equation of state parameter $w$ Koma19 ; Koma20 . For example, substituting $w=1/3$, $f_{\Lambda}(t)=0$, and $h_{\textrm{B}}(t)=2\psi H_{0}H$ into Eq. (10) yields the cosmological equation equivalent to Eq. (18). (The $H$ term for $h_{\textrm{B}}(t)$ is obtained from $H^{\alpha}$ when $\alpha=1$.) In this universe, $T_{\rm{KH}}$ is constant though the Hubble parameter varies with time. Understanding the universe at constant $T_{\rm{KH}}$ should contribute to the study of horizon thermodynamics because systems at constant temperature play important roles in thermodynamics. In Sec. IV.2, we consider a constant $T_{\rm{KH}}$ universe, to simplify the modified thermodynamic relation. In Sec. V, the evolution of the constant $T_{\rm{KH}}$ universe is observed, using a constant $T_{\rm{KH}}$ model. To formulate the constant $T_{\rm{KH}}$ model, a power-law model with $H^{\alpha}$ terms is discussed in Sec. V.

We calculate $T_{\rm{GH}}\dot{S}_{\rm{BH}}$ and $T_{\rm{KH}}\dot{S}_{\rm{BH}}$, used to discuss the modified thermodynamic relation. Substituting Eqs. (16) and (15) into $T_{\rm{GH}}\dot{S}_{\rm{BH}}$ yields

where $K=\pi k_{B}c^{5}/(\hbar G)$ given by Eq. (14) has been used. Similarly, from Eqs. (17) and (15), $T_{\rm{KH}}\dot{S}_{\rm{BH}}$ is given by

The obtained $T_{\rm{GH}}\dot{S}_{\rm{BH}}$ and $T_{\rm{KH}}\dot{S}_{\rm{BH}}$ are used later.

In this subsection, we examine the $\dot{\rho}$ and $\dot{V}$ terms in the modified thermodynamic relation for constant $f_{\Lambda}(t)$. The $\dot{\rho}$ and $\dot{V}$ terms correspond to the left-hand side of Eq. (34), as confirmed later. The sum of the two terms is equivalent to $T_{\rm{KH}}\dot{S}_{\rm{BH}}$, from Eq. (34). Accordingly, we first discuss $T_{\rm{KH}}\dot{S}_{\rm{BH}}$ and then examine the $\dot{\rho}$ and $\dot{V}$ terms.

Using $T_{\rm{KH}}$ and $T_{\rm{GH}}$, $T_{\rm{KH}}\dot{S}_{\rm{BH}}$ can be written as

where $T_{\rm{GH}}$ is the Gibbons–Hawking temperature given by Eq. (16). The first term on the right-hand side of Eq. (39) is $T_{\rm{GH}}\dot{S}_{\rm{BH}}$. The second term is equivalent to the product of $T_{\rm{GH}}\dot{S}_{\rm{BH}}$ and a normalized deviation of $T_{\rm{KH}}$ from $T_{\rm{GH}}$. In de Sitter universes, this deviation reduces to $0$ because $T_{\rm{KH}}=T_{\rm{GH}}$. From Eq. (19), $T_{\rm{GH}}\dot{S}_{\rm{BH}}$ is given by

Using Eqs. (16), (17), and (40), we have the second term on the right-hand side of Eq. (39), written as

Coupling Eq. (40) with Eq. (41) yields

In addition, normalizing Eqs. (40) and (41) by $T_{\rm{GH},0}(S_{\rm{BH},0}H_{0})$ and coupling the two resulting equations yields

where $T_{\rm{GH},0}=\hbar H_{0}/(2\pi k_{B})$, $S_{\rm{BH},0}=K/H_{0}^{2}$, and $K=\pi k_{B}c^{5}/(\hbar G)$ have been used. $H_{0}$ represents the Hubble parameter at the present time. Equations (42) and (43) indicate that the magnitude of the second term is proportional to the square of the first term.

We now calculate $-\dot{E}_{\rm{bulk}}+W_{e}\dot{V}$, namely the left-hand side of Eq. (34). Using Eq. (27) and replacing $p$ and $W$ by $p_{e}$ and $W_{e}$, respectively, yields

where $p_{e}$ is an effective pressure given by Eq. (6) and $W_{e}$ is an effective work density $(\rho c^{2}-p_{e})/2$, given by Eq. (32). In this way, we confirm that $-\dot{E}_{\rm{bulk}}+W_{e}\dot{V}$ can be divided into $-\dot{\rho}c^{2}V$ and $-[(\rho c^{2}+p_{e})/2]\dot{V}$, namely the ‘$\dot{\rho}$ and $\dot{V}$ terms’. The $\dot{V}$ term is the sum of $-\rho c^{2}\dot{V}$ and $W_{e}\dot{V}$, where $-\rho c^{2}\dot{V}$ is implicitly included in $-\dot{E}_{\rm{bulk}}$. As examined previously, $\dot{\rho}$ can be replaced by applying the continuity equation $\dot{\rho}+3H[\rho+(p_{e}/c^{2})]=0$ given by Eq. (8) and $\dot{H}=-4\pi G[\rho+(p_{e}/c^{2})]$ given by Eq. (9), where $\dot{f}_{\Lambda}(t)=0$ has been used. From these equations, we obtain the first term on the right-hand side of Eq. (44), namely $-\dot{\rho}c^{2}V$, which can be written as

where $V=(4\pi/3)(c/H)^{3}$ given by Eq. (24) has been used. Equation (45) is equivalent to Eq. (40). In addition, substituting both $\dot{H}=-4\pi G[\rho+(p_{e}/c^{2})]$ and $\dot{V}=-4\pi c^{3}H^{-4}\dot{H}$ into the second term on the right-hand side of Eq. (44) yields

Equation (46) is equivalent to Eq. (41). From these two results, we have the following two equations:

Therefore, the modified thermodynamic relation given by Eq. (34) can be summarized as

where Eq. (44) is also used. Equation (49) indicates that the $\dot{\rho}$ term is equivalent to $T_{\rm{GH}}\dot{S}_{\rm{BH}}$, whereas the $\dot{V}$ term is equivalent to $[(T_{\rm{KH}}/T_{\rm{GH}})-1]T_{\rm{GH}}\dot{S}_{\rm{BH}}$. In this way, the modified thermodynamic relation can be interpreted as consisting of contributions from the $\dot{\rho}$ and $\dot{V}$ terms. We expect that Eq. (49) should provide a better understanding of the modified thermodynamic relation. For example, based on Eq. (49), Eq. (42) can be interpreted as showing that the magnitude of the $\dot{V}$ term is proportional to the square of the $\dot{\rho}$ term. This relationship is satisfied even when $W_{e}=W$, namely the first law of thermodynamics. In addition, solving $-\dot{\rho}c^{2}V=T_{\rm{GH}}\dot{S}_{\rm{BH}}$ given by Eq. (49) with respect to $\dot{\rho}$, substituting both $V=(4\pi/3)(c/H)^{3}$ and $T_{\rm{GH}}\dot{S}_{\rm{BH}}=(c^{5}/G)(-\dot{H}/H^{2})$ into the resulting equation and integrating yields the Friedmann equation, written as

where $C$ represents integral constants. That is, the $\dot{\rho}$ term in the thermodynamic relation corresponds to the Friedmann equation. Similarly, solving $-[(\rho c^{2}+p_{e})/2]\dot{V}=[(T_{\rm{KH}}/T_{\rm{GH}})-1]T_{\rm{GH}}\dot{S}_{\rm{BH}}$ with respect to $[\rho+(p_{e}/c^{2})]$ and substituting Eq. (41) and $\dot{V}=-4\pi c^{3}H^{-4}\dot{H}$ into the resulting equation gives $\dot{H}=-4\pi G[\rho+(p_{e}/c^{2})]$, which is equivalent to Eq. (9). In addition, adding this equation to the derived Friedmann equation yields an acceleration equation, written as

The derived Friedmann and acceleration equations are not unexpected because, in the previous section, the modified thermodynamic relation was similarly derived from general cosmological equations and the first law of thermodynamics. However, as examined above, Eq. (49) can clarify the roles of the $\dot{\rho}$ and $\dot{V}$ terms included in the modified thermodynamic relation. In Sec. V, we will study the evolution of the two terms, using cosmological models such as $\Lambda$CDM models.

In the previous subsection, we examined the $\dot{\rho}$ and $\dot{V}$ terms, corresponding to the left-hand side of the modified thermodynamic relation for constant $f_{\Lambda}(t)$. We expect that the right-hand side of this relation, namely $T_{H}\dot{S}_{H}$, reduces to $\dot{F}_{H}$ by applying the equipartition law of energy on the horizon, as if the holographic-like connection $E_{\rm{bulk}}=F_{H}$ is extended. For this, we first review the equipartition law, based on previous works Koma18 ; Koma20 . Then, we apply the equipartition law to the right-hand side of the modified thermodynamics relation. The equipartition law used here has not yet been established in a cosmological spacetime but is considered to be a viable scenario.