Horizon thermodynamics and cosmological equations: a holographic-like connection between thermostatistical quantities on a cosmological horizon and in the bulk

Horizon thermodynamics is expected to be related to the effective energy based on the energy density calculated from the Friedmann equation for a Friedmann–Robertson–Walker (FRW) universe. In the present study, the effective energy and thermostatistical quantities on a cosmological horizon are examined to clarify the holographic-like connection between them, with a focus on a de Sitter universe. To this end, the Helmholtz free energy on the horizon is derived from horizon thermodynamics. The free energy is found to be equivalent to the effective energy calculated from the Friedmann equation. This consistency is interpreted as a kind of holographic-like connection. To examine this connection, Padmanabhan’s holographic equipartition law, which is related to the origin of spacetime dynamics, is applied to a de Sitter universe. It is found that the law should lead to a holographic-like connection. The holographic-like connection is considered to be a bridge between thermostatistical quantities on the horizon and in the bulk. For example, cosmological equations for a flat FRW universe can be derived from horizon thermodynamics by accepting the connection as a viable scenario. In addition, a thermal entropy equivalent to the Bekenstein–Hawking entropy is obtained from the Friedmann equation using the concept of a canonical ensemble in statistical physics. The present study should provide new insight into the discussion of horizon thermodynamics and cosmological equations.

Recently, we found that the effective energy calculated from the Friedman equation should be equivalent to the Helmholtz free energy on the horizon.This holographiclike connection may allow us to develop a deeper understanding of thermodynamic scenarios based on the holographic principle.For example, thermal entropy can be directly calculated from the free energy using the concept of a canonical ensemble in statistical physics [52].We expect that Padmanabhan's holographic equipartition law [39], which is related to the origin of spacetime dynamics, is suitable for examining the holographic-like connection because based on this law, cosmological equations are derived from the expansion of cosmic space due to the difference between the degrees of freedom on the surface and in the bulk [39].(This law has been used to examine the emergence of cosmological equations [40][41][42][43][44][45][46][47].)That is, Padmanabhan's holographic equipartition law is expected to be a bridge between thermostatistical quantities on the surface and in the bulk.An understanding of the relation between the effective energy calculated from the Friedman equation and thermostatistical quantities on the horizon should provide new insight into the discussion of cosmological equations and horizon thermodynamics.
In this context, we examine the effective energy calculated from the Friedman equation and the thermostatistical quantities on a cosmological horizon.In the present study, we consider a flat FRW universe, with a focus on a de Sitter universe whose horizon is considered to be in thermal equilibrium.
The remainder of the present article is organized as follows.In Sec.II, cosmological equations are reviewed.In addition, the effective energy based on the energy density is calculated from the Friedmann equation.In Sec.III, the Helmholtz free energy on the horizon is derived from horizon thermodynamics.In Sec.IV, a holographiclike connection between the effective energy calculated from the Friedmann equation and the free energy on the horizon is examined through Padmanabhan's holographic equipartition law.In Sec.V, thermostatistical quantities on the horizon are calculated from the Friedmann equation using the concept of a canonical ensemble in statistical physics.Finally, in Sec.VI, the conclusions of the study are presented.

II. COSMOLOGICAL EQUATIONS AND EFFECTIVE ENERGY
We consider a homogeneous, isotropic, and spatially flat universe, namely a flat FRW universe.In Sec.II A, a general formulation for cosmological equations is introduced.In Sec.II B, the standard cosmological equations used in this study are reviewed.In Sec.II C, the effective energy based on the energy density is calculated from the Friedmann equation.

A. General formulation
We introduce a general formulation for cosmological equations using the scale factor a(t) at time t according to previous works [20,21,[48][49][50].The general Friedmann and acceleration equations are written as where the Hubble parameter H(t) is defined by and w represents the equation-of-state parameter for a generic component of matter, which is given as Here, G, c, ρ(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 [20,21,[48][49][50].For a Λ-dominated universe, a matterdominated universe, and a radiation-dominated universe, w is −1, 0, and 1/3, respectively.The Λ-dominated universe corresponds to a de Sitter universe.Two extra driving terms, f Λ (t) and h B (t), are phenomenologically assumed for a Λ(t) cosmology and a bulk viscous cosmology, respectively [21,50].The general formulation is discussed in Sec.IV A.

B. Standard cosmological equations
We now consider the standard cosmological equations.To this end, we set f Λ (t) = h B (t) = 0. Consequently, Eqs. ( 1) and ( 2) are respectively rewritten as In addition, the continuity equation is given by The continuity equation is derived from the first law of thermodynamics under the assumption of adiabatic processes [3,18].(Alternatively, the continuity equation can be derived from the Friedmann and acceleration equations because two of the three equations are independent [3].) In the present paper, we focus on a de Sitter universe.The evolution of the scale factor is given by where the subscript 0 represents the present time.The properties of a de Sitter universe are characterized by H, which is constant during the evolution of the de Sitter universe [51].In addition, H is related to the temperature T H on the Hubble horizon, expressed as [23] T where k B and are the Boltzmann constant and the reduced Planck constant, respectively.The reduced Planck constant is defined as ≡ h/(2π), where h is the Planck constant [46,47].Equation (9) indicates that T H depends only on H. Accordingly, the horizon of a de Sitter universe is considered to be in thermal equilibrium because T H is constant.That is, the concept of a canonical ensemble in statistical physics can be applied to the horizon of a de Sitter universe [51].
C. Effective energy E eff based on energy density calculated from Friedmann equation In this subsection, we determine the effective energy based on the energy density calculated from the Friedmann equation, which corresponds to the energy equation.Using the Friedmann equation given by Eq. ( 5), the energy density ρc 2 is written as The energy density ρc 2 can be generally defined by where E eff is the effective energy and V is the volume of a sphere with the Hubble horizon (radius).The Hubble horizon r H and the Hubble volume V are respectively given by Note that the Hubble horizon is equivalent to an apparent horizon because of the flat universe.Solving Eq. ( 11) with respect to E eff and substituting Eqs. ( 10) and ( 13) into the resultant equation yields In this way, the effective energy E eff based on the energy density can be calculated from the Friedmann equation.The effective energy is expected to be related to the Helmholtz free energy F H on a cosmological horizon.We examine this in the next section.(The free energy corresponds to useful energy obtained from a closed thermodynamic system at a constant temperature.)

III. HORIZON THERMODYNAMICS
Horizon thermodynamics is closely related to the holographic principle [32], which assumes that the horizon of the universe has an associated entropy and an approximate temperature [14].In this section, based on horizon thermodynamics, the energy and the Helmholtz free energy on the horizon are derived.To this end, in Sec.III A, the entropy and the temperature on the horizon are introduced according to previous works [33][34][35][36][37][38][39][40][41][42]51].In Sec.III B, the energy and the free energy on the horizon are derived using the entropy and the temperature.In addition, the effective energy based on the energy density is discussed.Note that we consider a de Sitter universe whose horizon is in thermal equilibrium.
A. Definition of entropy SH and temperature TH We select a form of the Bekenstein-Hawking entropy as an associated entropy on the cosmological horizon because it is the most standard, as examined in previous works [33][34][35][36][37][38][39][40][41][42]51].In this subsection, we introduce the entropy on the horizon according to those works.
Based on the selected form of the Bekenstein-Hawking entropy, the entropy S H on the Hubble horizon is written as [51] where A H is the surface area of a sphere with the Hubble horizon.Substituting A H = 4πr 2 H into Eq.( 15) and applying Eq. ( 12) yields where K is a positive constant given by and L P is the Planck length, written as The standard form given by Eq. ( 16) is used for the entropy on the Hubble horizon.(Forms of the non-Gaussian black-hole entropy have been described elsewhere [53][54][55][56].)In addition, from Eq. ( 9), the temperature T H on the horizon is written as [23] T The horizon of a de Sitter universe is considered to be in thermal equilibrium because T H is constant.Throughout this study, Eq. ( 19) is used as an important thermostatistical quantity.

B. Derivation of energy EH and free energy FH
In this subsection, we derive the energy E H stored on the Hubble horizon from the thermodynamic relation dE H = T H dS H .In addition, we derive the Helmholtz free energy F H on the horizon.Hereafter, we refer to F H as the free energy (on the horizon).
Substituting Eq. (19) and given by Eq. ( 16) into Integrating this equation and using Eqs.( 16) and ( 19) yields where the integral constant E H,0 is set to zero so that E H = 0 is satisfied when T H = 0.That is, the energy E H stored on the Hubble horizon is given by Substituting Eqs. ( 16) and ( 19) into Eq.( 22) yields In this way, In addition, based on thermodynamics, the free energy F H on the horizon is defined as Substituting T H S H = E H /2 given by Eq. ( 22) into Eq.( 24) yields The free energy F H is half of E H . Substituting Eq. ( 23) into Eq.( 25) yields As expected, F H is equivalent to E eff given by Eq. ( 14).That is, the free energy F H on the Hubble horizon is equivalent to the effective energy E eff in the Hubble volume: This consistency, namely Eq. ( 27), is considered to be a kind of holographic-like connection.The interpretation of this connection is discussed in the next section using Padmanabhan's holographic equipartition law, which is related to the origin of spacetime dynamics [39].
When F H = E eff , the energy density ρc 2 for the Hubble volume should be written as The above equation is equivalent to Eq. ( 10), namely the Friedmann equation given by Eq. ( 5).That is, the Friedmann equation should be derived from the free energy on the horizon.Also, it is well-known that the continuity equation given by Eq. ( 7) is derived from the first law of thermodynamics [3].Accordingly, the acceleration equations can be derived using the Friedmann and continuity equations because two of the three equations are independent.In this way, the standard cosmological equations are derived from horizon thermodynamics when F H = E eff .(Note that another energy density obtained from E H is given by EH V = 3c 2 4πG H 2 , as previously reported [51].This is twice as large as Eq. ( 28), as described in Sec.I.) We expect that the holographic-like connection F H = E eff is a bridge between cosmological equations and horizon thermodynamics.In the next section, the holographic-like connection is examined through Padmanabhan's holographic equipartition law [39].
In this section, we derived E H and F H from the thermodynamic relation dE H = T H dS H based on horizon thermodynamics.The equipartition law of energy on the horizon was not used here.(The law is described in Sec.IV B.) It has been reported that the equipartition law of energy on the horizon can lead to E H = 2S H T H [34,37,38] and [48,51].The present result is consistent with previous works.However, the free energy F H and the holographic-like connection F H = E eff were not discussed in those works.

IV. HOLOGRAPHIC EQUIPARTITION LAW AND EFFECTIVE ENERGY
In this section, we examine the effective energy that can be obtained from Padmanabhan's holographic equipartition law.To this end, the law is introduced in Sec.IV A and its energy formula is discussed in Sec.IV B. In these two subsections, a flat FRW universe is considered.In Sec.IV C, the energy formula is applied to a de Sitter universe to examine the effective energy.

A. Padmanabhan's holographic equipartition law
Based on Padmanabhan's holographic equipartition law, cosmological equations can be derived from the expansion of cosmic space due to the difference between the degrees of freedom on the surface and in the bulk in a region of space [39].The law was reviewed in a previous study [46] based on Padmanabhan's work [39] and other related studies [40,41].In this subsection, we introduce Padmanabhan's holographic equipartition law based on those previous works.Note that the law has not yet been established in a cosmological spacetime, but is considered to be a viable scenario.
In an infinitesimal interval dt of cosmic time, the increase dV in the Hubble volume can be expressed as where N sur is the number of degrees of freedom on a spherical surface of the Hubble radius r H and N bulk is the number of degrees of freedom in the bulk [39].L p is the Planck length given by Eq. ( 18) and ǫ is a parameter defined as [39] ǫ ≡ +1 (ρc 2 + 3p < 0: an accelerating universe), −1 (ρc 2 + 3p > 0: a decelerating universe).
(30) We refer to Eq. ( 29) as Padmanabhan's holographic equipartition law.From this equation, the acceleration equation can be derived.The derivation used in this subsection is based on previous reports [46].(Equation ( 29) includes c because c is not set to 1.) To derive the acceleration equation, we first calculate the left-hand side of Eq. ( 29), namely dV /dt.Differentiating Eq. ( 13) with respect to t yields [46] where r is set to r H = c/H before the time derivative is calculated [39].Next, to calculate the right-hand side of Eq. ( 29), the number of degrees of freedom in the bulk N bulk is assumed to obey the equipartition law of energy [39]: where the Komar energy |E bulk | contained inside the Hubble volume V is assumed to be given by The Komar energy is different from the effective energy based on the energy density calculated from the Friedmann equation.The number of degrees of freedom on the spherical surface N sur is assumed to be given by where S sur is the entropy on the horizon [46].For generality, S sur is used here; S sur = S H is used later.We now derive an acceleration equation from Padmanabhan's holographic equipartition law.According to previous works [39,46], ρc 2 + 3p < 0 is selected and, therefore, ǫ = +1 from Eq. ( 30).This selection does not affect the following result [39].We first calculate N bulk on the right-hand side of Eq. ( 29).Substituting Eqs. ( 19) and (33) into Eq.( 32) and using Eqs.( 4) and ( 13) and ǫ = +1 yields [46] In addition, substituting ǫ = +1 and Eqs. ( 18), ( 31), (34), and (35) into Eq.( 29) and solving the resultant equation with respect to Ḣ yields [46] where K is given by Eq. ( 17).Substituting Eq. ( 36) into ä/a = Ḣ + H 2 and using S H = K/H 2 given by Eq. ( 16), we obtain the acceleration equation, written as [46] ä When S sur = S H , the second term H 2 (1 − S sur /S H ) on the right-hand side is non-zero [46].In this case, the obtained acceleration equation corresponds to the general form given by Eq. ( 2).
When S sur = S H , the second term on the right-hand side of Eq. ( 37) is zero and, therefore, the acceleration equation is written as This equation is equivalent to Eq. ( 6) in standard cosmology.For a specific case, we consider dV /dt = 0, corresponding to a de Sitter universe [39].That is, Ḣ = 0 and w = −1 are considered.Substituting Ḣ = 0 and w = −1 into Eq.( 38), we have the Friedmann equation, written as [39] The specific case is discussed again in Sec.IV C to examine the effective energy of a de Sitter universe.To this end, in the next subsection, we derive an energy formula of Padmanabhan's holographic equipartition law.
In addition, we set S sur = S H for standard cosmology.From Eq. ( 34), N sur is given by Substituting S H = EH 2TH given by Eq. ( 22) into Eq.( 44) yields Equation ( 45) represents the equipartition law of energy on the horizon [34,37].From Eq. ( 32), the number of degrees of freedom in the bulk N bulk is written as We note that the temperature T H on the horizon is included in Eqs. ( 45) and (46).Substituting Eqs. ( 45) and ( 46) into Eq.( 43) yields This is the energy formula of Padmanabhan's holographic equipartition law.In the next subsection, the energy formula is applied to a de Sitter universe.(The Komar energy |E bulk | is different from the effective energy E eff based on the energy density calculated from the Friedmann equation.) C. Effective energy Ẽeff in de Sitter universe calculated from Padmanabhan's holographic equipartition law We consider a de Sitter universe and examine the effective energy calculated from Padmanabhan's holographic equipartition law.A de Sitter universe corresponds to ṠH = 0 because H is constant.Accordingly, E H = |E bulk | is obtained from Eq. ( 47).That is, E H = |E bulk | is considered to be the so-called holographic duality based on the law.
We now discuss the effective energy.Substituting Eq. ( 33) into E H = |E bulk | and using w = p/(ρc 2 ) given by Eq. ( 4) yields (48) where ρ is assumed to be non-negative.The equation-ofstate parameter w is retained here and set to −1 later.(ǫ given by Eq. ( 30) is not used here.)Using Eq. ( 48), the energy density ρc 2 is written as where Ẽeff is the effective energy in the Hubble volume and is defined as a new parameter, which is given by Here, w = −1 is used.From Eq. ( 50), we can confirm that Ẽeff is equivalent to both F H given by Eq. ( 26) and E eff given by Eq. ( 14): This result implies that Padmanabhan's holographic equipartition law should lead to a holographic-like connection As mentioned above, E H = |E bulk | is considered to be the holographic duality.In addition, |E bulk | is equivalent to 2 Ẽeff .Consequently, a holographic-like connection F H = E eff (= Ẽeff ) given by Eq. ( 51) is obtained.Accordingly, the connection should be interpreted as a kind of reformulated holographic duality based on Padmanabhan's holographic equipartition law.
The holographic-like connection is expected to be a bridge between thermostatistical quantities on the horizon and in the bulk.This connection can lead to thermostatistical quantities such as a thermal entropy using a canonical ensemble in statistical physics.We examine this in the next section.
Note that an energy balance relation ρc 2 V = T H S H , similar to the holographic-like connection, has been derived by Padmanabhan [57].The energy balance relation was also described in, e.g., Ref. [58].However, the free energy F H and the holographic-like connection F H = E eff were not discussed in those previous works.

V. STATISTICAL PHYSICS AND FRIEDMANN EQUATION
So far, we have examined the free energy F H on the horizon and the effective energy E eff based on the energy density calculated from the Friedmann equation.
A holographic-like connection F H = E eff was obtained.This connection is related to Padmanabhan's holographic equipartition law and is expected to be a bridge between thermostatistical quantities on the horizon and in the bulk.To confirm this expectation, we calculate thermostatistical quantities on the horizon from the Friedmann equation by accepting the holographic-like connection as a viable scenario and using statistical physics.In Sec.V A, a standard formulation for a canonical ensemble in statistical physics is introduced.In Sec.V B, the formulation is applied to the horizon of a de Sitter universe to calculate the thermostatistical quantities on the horizon.

A. Canonical ensemble in statistical physics
A standard formulation for a canonical ensemble is introduced [52].We consider a canonical ensemble with a partition function Z(β) given by where H is the Hamiltonian for the system and β is the inverse temperature in the equilibrium state, written as where T represents the equilibrium temperature of the system.Based on statistical physics, the free energy F (β) is given by [52] Using F (β), the total energy E(β) is written as [52] E and the thermal entropy S(β) is given by Equations ( 55) and (56) indicate that the total energy E and the thermal entropy S can be calculated from the free energy F .In the next subsection, the formulations are applied to the horizon of a de Sitter universe.
B. Thermal entropy derived from effective energy calculated from Friedmann equation In this subsection, based on statistical physics, we calculate the thermal entropy on the horizon from the Friedmann equation.To this end, the formulations examined in Sec.V A are applied to a de Sitter universe whose horizon is considered to be in thermal equilibrium.The inverse temperature in the equilibrium state is written as where the temperature T H on the horizon is used.
In addition, we accept the holographic-like connection, namely F H = E eff , as a viable scenario.(E eff is the effective energy calculated from the Friedmann equation.)Substituting Eq. ( 14) into F H = E eff yields To discuss a canonical ensemble, the free energy F H is formulated as a function of the inverse temperature β.
Substituting 1 H = 2πkB TH given by Eq. ( 19) into Eq.( 58) and using Eq. ( 57) yields We now calculate the total energy E and the thermal entropy S. The symbols E H and S H are not used here because from now on, the total energy E and the thermal entropy S are calculated.Substituting Eq. ( 59) into Eq.( 55) yields From Eqs. ( 58) and ( 60), the total energy E is written as Substituting Eq. ( 59) into Eq.( 56) yields From this equation, the thermal entropy S is written as where Eq. ( 57) and 2πkB TH = 1 H given by Eq. ( 19) are used.Substituting Eq. ( 19) and the second line of Eq. ( 63) into 2ST H and using Eq. ( 61) yields (64) In this way, based on statistical physics, the total energy E and the thermal entropy S on the horizon are obtained from the effective energy E eff calculated from the Friedmann equation.We can confirm that the obtained equations, namely , and E = 2ST H , are equivalent to those examined in Sec.III.The interpretation of the holographic-like connection F H = E eff has not yet been established.However, when this connection is acceptable, the total energy and the thermal entropy on the horizon can be obtained from the Friedmann equation.Using an inverse operation, the Friedmann equation can be calculated from the thermostatistical quantities on the horizon.(A similar inverse operation was discussed in Sec.III.)In this sense, the holographic-like connection is considered to be a bridge between horizon thermodynamics and cosmological equations (i.e., a dictionary between thermostatistical quantities on the horizon and in the bulk).These results imply that thermostatistical quantities on the horizon and in the bulk can be examined through the holographiclike connection and Padmanabhan's holographic equipartition law.
In the present study, we assume a standard FRW cosmology, with a focus on a de Sitter universe in which H, r H , and T H are constant.The temperature T H given by Eq. ( 19) is obtained from field theory in the de Sitter space [23].In addition, we select a form of the Bekenstein-Hawking entropy as an associated entropy on the cosmological horizon.However, the selected from of the entropy on the horizon may modify the standard FRW cosmology.In fact, modified FRW cosmologies have been examined based on various forms of the entropy, see, e.g., the recent work of Nojiri et al. [59].Therefore, the question arises whether the holographiclike connection can be applied to the modified FRW cosmology.Probably, it should be difficult to apply this connection directly to modified FRW universes because horizons of these universes are generally dynamic, unlike for the de Sitter universe whose horizon is considered to be static.Instead of T H , we may need to use a dynamical temperature, to extend the holographic-like connection.
In fact, a dynamical temperature has been proposed to describe the temperature on a dynamic apparent horizon [60].The dynamical temperature should be suitable for discussing thermodynamics and statistical physics on dynamic horizons.Recently, we examined the dynamical temperature of a flat FRW universe, using a Λ(t) model similar to time-varying Λ(t) cosmologies [61].Consequently, we found a universe whose dynamical temperature is constant although H and r H vary with time [61].Such a universe at constant dynamical-temperature is expected to contribute to the study of the holographiclike connection in modified FRW cosmologies because the temperature on the horizon is considered to be constant.We expect that an extended holographic-like connection obtained from the modified Friedmann equation can be discussed, by focusing on a specific universe at constant dynamical-temperature in the modified FRW cosmology.Detailed studies are needed and are left for future research.
We note that the holographic entanglement entropy [62], which is related to the emergence of spacetime, has been extensively examined [63][64][65][66][67][68][69][70][71][72][73][74] based on anti-de Sitter/conformal field theory [75].The entanglement entropy can be treated as a thermal entropy; this has been examined in a de Sitter space [67][68][69].In particular, it has been reported that the formula for the entropy on a cosmological horizon in a de Sitter space should be equivalent to a holographic entanglement entropy [69].In addition, the relative entropy, corresponding to the free-energy difference, is expected to be related to gravity [71].We may be able to discuss a universe from a new viewpoint using these entropies and the holographic-like connection examined in the present paper.This task is left for future research.

VI. CONCLUSIONS
We examined the effective energy E eff based on the energy density calculated from the Friedmann equation and thermostatistical quantities on a cosmological horizon of a de Sitter universe to clarify the holographic-like connection between the two.
First, we derived the energy E H and the Helmholtz free energy F H on the horizon from horizon thermodynamics.The free energy F H was found to be half of the energy E H on the horizon and equivalent to the effective energy E eff obtained using the Friedmann equation, that is, E eff = F H = E H /2. The consistency is considered to be a kind of holographic-like connection.When this connection is acceptable, the Friedmann equation can be derived from the free energy F H on the horizon, that is, cosmological equations for a flat FRW universe can be derived from horizon thermodynamics.
Second, we examined the holographic-like connection through Padmanabhan's holographic equipartition law.Applying the law to a de Sitter universe, an effective energy Ẽeff in the Hubble volume was found to be given by E H /2, that is, Ẽeff = EH 2 = F H = E eff .This result implies that Padmanabhan's holographic equipartition law is related to the holographic-like connection F H = E eff and should lead to the connection.The holographic-like connection can be interpreted as a kind of reformulated holographic duality based on the law.
Finally, we examined thermostatistical quantities on the horizon by accepting the connection F H = E eff as a viable scenario and using the concept of a canonical ensemble in statistical physics.We confirmed that a thermal entropy equivalent to the form of the Bekenstein-Hawking entropy is obtained from the Friedmann equation.Using an inverse operation, the Friedmann equation can be calculated from thermostatistical quantities on the horizon.
The holographic-like connection is considered to be a bridge between horizon thermodynamics and cosmological equations (i.e., a dictionary between thermostatistical quantities on the horizon and in the bulk).The present study should provide new insight into the discussion of gravity and horizon thermodynamics.