Entropic cosmology for a generalized black-hole entropy

Entropic cosmology for a generalized black-hole entropy

Abstract

An entropic-force scenario, i.e., entropic cosmology, assumes that the horizon of the universe has an entropy and a temperature. In the present study, in order to examine entropic cosmology, we derive entropic-force terms not only from the Bekenstein entropy but also from a generalized black-hole entropy proposed by C. Tsallis and L.J.L. Cirto [Eur. Phys. J. C 73, 2487 (2013)]. Unlike the Bekenstein entropy, which is proportional to area, the generalized entropy is proportional to volume because of appropriate nonadditive generalizations. The entropic-force term derived from the generalized entropy is found to behave as if it were an extra driving term for bulk viscous cosmology, in which a bulk viscosity of cosmological fluids is assumed. Using an effective description similar to bulk viscous cosmology, we formulate the modified Friedmann, acceleration, and continuity equations for entropic cosmology. Based on this formulation, we propose two entropic-force models derived from the Bekenstein and generalized entropies. In order to examine the properties of the two models, we consider a homogeneous, isotropic, and spatially flat universe, focusing on a single-fluid-dominated universe. The two entropic-force models agree well with the observed supernova data. Interestingly, the entropic-force model derived from the generalized entropy predicts a decelerating and accelerating universe, as for a fine-tuned standard CDM (lambda cold dark matter) model, whereas the entropic-force model derived from the Bekenstein entropy predicts a uniformly accelerating universe.

pacs:
98.80.-k, 98.80.Es, 95.30.Tg

I Introduction

Since the late 1990s, an accelerated expansion of the late universe has gradually been accepted as a new paradigm (1); (2); (3); (4); (5); (6). In order to explain this accelerated expansion, various cosmological models have been suggested (7); (8); (9); (10); (11). (See, e.g., Refs. (8); (9); (10); (11) and the references therein.) As one such model, Easson, Frampton, and Smoot (12); (13) recently proposed an entropic-force scenario called ‘entropic cosmology’. In entropic cosmology, an extra driving term, i.e., an entropic-force term, should be added to the Friedmann–Lemaître acceleration equation, without introducing new fields (14). The entropic-force term is derived from the usually neglected surface terms on the horizon of the universe, assuming that the horizon has an entropy and a temperature due to the information holographically stored there (12).

Entropic cosmology has been extensively examined from various viewpoints (15); (16); (17); (18); (19); (20); (21). As an entropy on the horizon, i.e., the Bekenstein entropy (22) is always used, substituting the horizon of the universe for the event horizon of a black hole. In fact, the Bekenstein entropy is proportional to area (or horizon) and is additive. However, self-gravitating systems exhibit peculiar features (23); (24), such as nonequilibrium thermodynamics and nonextensive statistical mechanics (25); (27); (26); (28); (29); (30); (31); (32); (33); (34); (35); (36); (37); (38); (39); (40); (41). Accordingly, for example, the Tsallis entropy (26) and the Renyi entropy (27) have been proposed for nonadditive (nonextensive) generalized entropies and have been investigated from astrophysical viewpoints (28); (29); (30); (31); (32); (33); (34); (35); (36); (37); (38); (39); (40); (41). In particular, Tsallis and Cirto recently suggested a generalized black-hole entropy proportional to its volume based on appropriate nonadditive generalizations (41). Using the generalized entropy instead of the Bekenstein entropy will provide new insight into entropic cosmology. Therefore, it is important to derive entropic-force models from the two entropies and examine the properties of the two models in order to understand entropic cosmology more deeply. Note that power-law and logarithmic entropic-corrections have been discussed (13); (14); (17); (18); (19); (42); (43).

In addition, in entropic cosmology, the entropy on the horizon of the universe can increase during the evolution of the universe (15), even if we consider a homogeneous and isotropic universe. However, a bulk viscosity of cosmological fluids (48); (44); (45); (46); (47); (49); (50); (51); (53); (54); (55); (56); (58); (59); (60); (61); (62); (63); (64); (66); (67); (68); (52); (57); (65); (69) is usually the only thing that can generate an entropy in the homogeneous and isotropic universe (49). (Such a cosmological model is referred to as bulk viscous cosmology.) Through the study of entropic cosmology, we may be able to discuss the classical entropy generated by bulk viscous stresses.

In this context, we examine entropic cosmology using a generalized black-hole entropy (proportional to its volume) proposed by Tsallis and Cirto (41). In the present study, we derive entropic-force terms not only from the Bekenstein entropy but also from the generalized entropy. Moreover, using an effective description similar to bulk viscous cosmology, we formulate the modified Friedmann, acceleration, and continuity (conservation) equations for entropic cosmology. Note that the entropic-force considered here is different from the idea that gravity itself is an entropic force (70); (71). Since we focus on background evolutions of the late universe, we do not discuss the inflation of the early universe.

The remainder of the present paper is organized as follows. In Sec. II, we present a brief review of three cosmological models, i.e., CDM (lambda cold dark matter) cosmology, bulk viscous cosmology, and entropic cosmology. In Sec. III, we derive entropic-force terms from both the Bekenstein entropy and a generalized black-hole entropy. We also formulate the modified Friedmann, acceleration, and continuity equations for two entropic-force models based on the obtained entropic-force terms. In Sec. IV, we examine a model that combines the two entropic-force models. In Sec. V, we discuss the evolution of the universe in the two entropic-force models using solutions of the combined model. Finally, in Sec. VI, we present a discussion and our conclusions.

Ii Cosmological models

In the present paper, we consider a homogeneous, isotropic, and spatially flat universe and examine the scale factor at time in the Friedmann–Lemaître–Robertson–Walker metric. In this section, we present a brief review of three cosmological models, i.e., CDM, bulk viscous, and entropic cosmologies, focusing on the Friedmann, acceleration, and continuity equations. (A spatially non-flat universe is discussed in the last paragraph of in Sec. IV.)

ii.1 CDM cosmology

We first introduce the well-known CDM models (8); (7); (9). In the standard CDM model, the Friedmann equation is given as

(1)

and the acceleration equation is

(2)

In addition, the continuity equation is given by

(3)

where the Hubble parameter is defined by

(4)

and , , , , and are the gravitational constant, a cosmological constant, the speed of light, the mass density of cosmological fluids, and the pressure of cosmological fluids, respectively. Equations (1) and (2) include the extra driving terms, i.e., , which can explain the accelerated expansion of the late universe.

The continuity equation is consistent with the Friedmann and acceleration equations, because two of the three equations are independent (7). In other words, if the Friedmann and acceleration equations are used, Eq. (3) is derived from these two equations. Note that Eq. (3) can be derived from the first law of thermodynamics as well, assuming adiabatic (isentropic) processes, without using the Friedmann and acceleration equations (7).

ii.2 Bulk viscous cosmology

In bulk viscous cosmology, a bulk viscosity of cosmological fluids is assumed (48); (44); (45); (46); (47); (49); (50); (51); (53); (54); (55); (56); (58); (59); (60); (61); (62); (63); (64); (66); (67); (68); (52); (57); (65); (69). Usually, the bulk viscosity is the only thing that can generate an entropy in the homogeneous and isotropic universe (49). Such a cosmological model is referred to as bulk viscous cosmology. For example, in the 1980s, Barrow (46); (47), Davies (48), and Lima et al. (49) investigated bulk viscous cosmology in order to discuss the inflation of the early universe. Recently, a number of studies have examined not only the inflation but also the accelerated expansion of the late universe, based on bulk viscous cosmology (50); (51); (53); (54); (55); (56); (58); (59); (60); (61); (62); (63); (64); (66); (67); (68); (52); (57); (65); (69).

Generally, an effective pressure for bulk viscous cosmology is given by

(5)

where, for simplicity, is a constant. (It is possible to assume a variable bulk-viscosity. For instance, see Refs. (64); (67) and the references therein.) In bulk viscous cosmology, the Friedmann equation is given by

(6)

Unlike the CDM model [Eq. (1)], Eq. (6) does not include an extra term such as a cosmological constant. Using the effective pressure , the acceleration equation for bulk viscous cosmology is given by

(7)

Substituting Eq. (5) into Eq. (7) and rearranging, we obtain the acceleration equation as

(8)

The last term, , corresponds to the extra driving term (15). Accordingly, instead of the cosmological constant, the extra term due to the bulk viscosity can explain the accelerated expansion of the universe. Note that the extra term considered here is proportional to the Hubble parameter . The continuity equation for bulk viscous cosmology is given by

(9)

Substituting Eq. (5) into Eq. (9) and rearranging, we have

(10)

Equation (10) has a non-zero right-hand side related to a classical entropy generated by bulk viscous stresses (46); (47); (15).

We now derive the continuity equation using a different approach. To this end, we use the generalized continuity equation obtained from the general Friedmann and acceleration equations (15), because two of the three equations are independent. The general Friedmann and acceleration equations are given, respectively, by

(11)

and

(12)

where and are general functions. Using Eqs. (11) and (12), we obtain the generalized continuity equation (15) given by

(13)

Comparing [Eqs. (11) and (12)] with [Eqs. (6) and (8)], we can set the general functions as and . Substituting these functions into Eq. (13), we obtain

(14)

Equation (14) is the same as Eq. (10). This indicates that Eq. (10) obtained from the effective pressure is consistent with Eq. (14) obtained from the Friedmann and acceleration equations. The consistency plays an important role in formulating the entropic-force models discussed in Sec. III.

ii.3 Entropic cosmology

In the entropic cosmology suggested by Easson et al. (12); (13), the horizon of the universe is assumed to have an associated entropy and an approximate temperature. The entropy considered here is the Bekenstein entropy. In the present study, we refer to this cosmology as the standard entropic cosmology. (The Bekenstein entropy is discussed in detail in Sec. III.1.)

In a study by Koivisto et al. (14), the modified Friedmann and acceleration equations are summarized as

(15)

and

(16)

The four coefficients , , , and are dimensionless constants. The and terms with the dimensionless constants correspond to the extra driving terms, i.e., entropic-force terms. The entropic-force terms are derived, taking into account the entropy and temperature on the horizon of the universe due to the information holographically stored there (12). We can solve the two equations assuming the single-fluid-dominated universe (14); (15). Note that we neglect high-order terms for quantum corrections, because we do not discuss the inflation of the early universe in the present paper. (The dimensionless constants were expected to be bounded by and , and typical values for a better fitting were and (12). It was argued that the extrinsic curvature at the surface was likely to result in something like and (13); (14).)

As examined in Ref. (15), we can simplify the two modified Friedmann equations, assuming a non-adiabatic-like expansion of the universe. The simple modified Friedmann and acceleration equations are summarized as

(17)
(18)

Equations (17) and (18) do not include terms. In fact, Easson et al. first proposed that the entropic-force terms of the modified acceleration equation are or ; i.e., terms are not included in the entropic-force terms (12). In other words, Eq. (18) is consistent with their original acceleration equation, as discussed in Sec. III.1. In the present paper, we select Eqs. (17) and (18) as standard entropic-force models. (Note that it may be possible to neglect the entropic-force terms of the modified Friedmann equation, i.e., . In the standard entropic cosmology (14), the entropic-force terms are added not only with the acceleration equation but also with the Friedmann equation, as in the case of the CDM models.)

In Ref. (12), Easson et al. considered that the continuity equation was given by , assuming an adiabatic (isentropic) expansion. However, in entropic cosmology, since the entropy on the horizon is assumed, the entropy can increase during the evolution of the universe. Therefore, in a previous study, we derived the continuity equation from the first law of thermodynamics, taking into account a non-adiabatic-like process caused by the entropy and the temperature on the horizon (15). Consequently, the modified continuity equation was written as

(19)

Because of the non-zero right-hand side, Eq. (19) was likely consistent with the continuity equation obtained from the modified Friedmann and acceleration equations (15). In the present study, we do not use Eq. (19) as the modified continuity equation. Instead, in the next section, we derive the modified continuity equation from an effective pressure for entropic cosmology.

We point out that a similar non-zero right-hand side of the continuity equation appears not only in bulk viscous cosmology but also in ‘energy exchange cosmology’, in which the transfer of energy between two fluids is assumed (15); (72), e.g., the interaction between matter and radiation (73), matter creation (74), interacting quintessence (75), the interaction between dark energy and dark matter (76), and dynamical vacuum energy (77); (78); (79); (80); (81); (82); (83); (84); (85); (86); (87). In particular, cosmological equations for dynamical vacuum energy models are very similar to those for entropic-force models. Therefore, in the following paragraph, we discuss a similarity between the two models.

Instead of a cosmological constant, a variable cosmological term is assumed in dynamical vacuum energy models, i.e., CDM models (77); (78); (79); (80); (81); (82); (83); (84); (85); (86); (87). Various terms (e.g., , , and constant terms) have been examined in those works. For example, see Refs. (79); (80); (81); (82); (83). Recently, and terms have been investigated in Refs. (21); (87). [The and terms are the same as the entropic-force terms shown in Eqs. (15) and (16).] Consequently, the influence of terms was found to be similar to that of terms. In other words, essentially new properties of the late universe were not obtained from terms. This implies that the terms can be neglected as if Eqs. (17) and (18) were the approximated Friedmann and acceleration equations, respectively. In addition, in CDM models, the continuity equation for matter ’’ is arranged as (82). The continuity equation is equivalent to Eq. (19), when is proportional to . In this way, cosmological equations for CDM models are similar to those for entropic-force models. Accordingly, the cosmological equations and their solutions discussed in the present paper are similar to those for CDM models, especially in the work of Basilakos, Plionis, and Solà (82). However, a theoretical background of the CDM model is different from that of the entropic-force model.

Finally, we discuss a fundamental problem of the standard entropic cosmology which includes and terms [Eqs. (15) and (16)]. In fact, and terms of entropic-force terms cannot describe a decelerating and accelerating universe predicted by the standard CDM model. This problem has been discussed in Refs. (21); (85) and our previous paper (15). In particular, Basilakos, Polarski, and Solà have shown that not the and terms but extra constant terms are important for describing the decelerating and accelerating universe (21). We emphasize that the standard entropic cosmology has such a fundamental problem, since entropic-force terms are usually considered to be and terms. However, a generalized black-hole entropy (41) is expected to solve the problem, as discussed later. (It should be noted that the above problem does not occur in CDM models, as examined in Refs. (21); (85). This is because the extra constant term is naturally obtained from an integral constant of the renormalization group equation for the vacuum energy density. For details, see a summarized review (84) and a recent thorough review (86).)

Iii Derivation of entropic-force for entropic cosmology

In this section, in order to discuss entropic cosmology, we derive entropic-force terms from the Bekenstein entropy and a generalized black-hole entropy proposed by Tsallis and Cirto (41). In Sec. III.1, we derive the standard entropic-force term from the Bekenstein entropy, which is proportional to the surface area of a sphere with the Hubble radius. Moreover, using an effective description for pressure, we reformulate the entropic-force model. In Sec. III.2, we assume the generalized entropy proportional to the volume. We derive an entropic-force term from the generalized entropy and propose a new entropic-force model. In the present paper, we use the Hubble radius as the preferred screen, because the apparent horizon coincides with the Hubble radius in the spatially flat universe (12).

iii.1 Entropic-force from the Bekenstein entropy

In the standard entropic cosmology, the modified Friedmann and acceleration equations include terms [Eqs. (15)–(18)], as entropic-force terms. We derive the entropic-force term from the Bekenstein entropy, according to the work of Easson et al. (12). We also reformulate the entropic-force model, using an effective pressure for entropic cosmology.

Derivation of entropic-force from the Bekenstein entropy

For entropic cosmology, we assume that the Hubble horizon has an approximate temperature and an associated entropy , where the Hubble horizon (radius) is given by

(20)

The temperature on the Hubble horizon is given by

(21)

where and are the Boltzmann constant and the reduced Planck constant, respectively. The reduced Planck constant is defined by , where is the Planck constant. The temperature considered here is obtained by multiplying the horizon temperature, , by (15). In the present study, is a non-negative free parameter on the order of , typically or . A similar parameter for the screen temperature has been discussed in Refs. (12); (17); (18); (19). (Note that the temperature on the horizon can be evaluated as K, which is slightly lower than the temperature of our cosmic microwave background radiation, K. We use the temperature on the horizon, assuming thermal equilibrium states based on a single holographic screen (12); (13).)

As an associated entropy on the Hubble horizon, the Bekenstein entropy is given as

(22)

where is the surface area of the sphere with the Hubble radius . As shown in Eq. (22), the Bekenstein entropy is proportional to . Substituting into Eq. (22) and using as given in Eq. (20), we obtain

(23)

where is a positive constant (15) given by

(24)

The entropic-force can be given by

(25)

where the minus sign indicates the direction of increasing entropy or the screen corresponding to the horizon (12). Substituting Eqs. (21) and (22) into Eq. (25) and using , we have the entropic-force as

(26)

Therefore, the pressure derived from the Bekenstein entropy is given by

(27)

The above derivation is based on the original idea of Easson et al. (12). Note that Eq. (27) includes used in this study, where is a free parameter for the temperature.

In the following, in order to reformulate entropic cosmology, we consider an effective description similar to bulk viscous cosmology. In other words, we assume an effective pressure for entropic cosmology. We will discuss the reason for this in Sec. III.2. We assume that the effective pressure based on the Bekenstein entropy is given by

(28)

Using , the acceleration equation can be written as

(29)

Accordingly, substituting Eq. (28) into Eq. (29), we have

(30)

The term is the entropic-force term, which can explain the accelerated expansion of the universe. When , Eq. (30) corresponds to the modified acceleration equation derived by Easson et al. (12).

We now examine the continuity equation for entropic cosmology. Using the effective pressure , the continuity equation is expected to be given by

(31)

Substituting Eq. (28) into Eq. (31) and rearranging, we have

(32)

This is the modified continuity equation obtained from the effective pressure for entropic cosmology. In the present paper, we consider Eq. (32) as the modified continuity equation for entropic cosmology derived from the Bekenstein entropy. [Note that Eqs. (32) and (19) are consistent with each other when . The universe for corresponds to the empty universe, as discussed in Sec. IV.]

We can determine two dimensionless constants and included in the modified Friedmann and acceleration equations [Eqs. (17) and (18)] using two continuity equations. The first continuity equation is Eq. (32), whereas the second continuity equation can be derived from the modified Friedmann and acceleration equations. In order to obtain the second continuity equation, we use the generalized continuity equation, i.e., Eq. (13). Comparing [Eqs. (11) and (12)] with [Eqs. (17) and (18)], we can set general functions as and . Substituting these functions into Eq. (13) and rearranging, we have the second continuity equation, which is given by

(33)

We expect that the two modified continuity equations, i.e., Eqs. (32) and (33), are consistent with each other. Consequently, we obtain and as

(34)

Therefore, we can neglect the entropic-force term of the modified Friedmann equation shown in Eq. (17). This is because, in the present paper, we assume the effective pressure for entropic cosmology.

As mentioned previously, we consider as a free parameter for the temperature. Moreover, a free parameter for the entropy may be required for calculating . However, we do not use the free parameter for the entropy, because we assume that the Bekenstein entropy is given by [Eq. 22].

Entropic-force model for the Bekenstein entropy

In Sec. III.1.1, we derived an entropic-force term from the Bekenstein entropy and obtained an entropic-force model. Consequently, the modified Friedmann, acceleration, and continuity equations are summarized as

(35)
(36)

and

(37)

where and are given by

(38)

Note that we leave in Eq. (35) in order to discuss a combined model later.

iii.2 Entropic-force from a generalized entropy

Thus far, we have considered the Bekenstein entropy to discuss the standard entropic cosmology. Instead of the Bekenstein entropy, we now assume a generalized black-hole entropy proposed by Tsallis and Cirto (41).

Derivation of entropic-force from a generalized entropy

Recently, Tsallis and Cirto (41) examined a black-hole entropy using appropriate nonadditive generalizations for -dimensional systems and suggested a generalized black-hole entropy. In the following, we introduce the generalized black-hole entropy, according to the work of Tsallis and Cirto (41). In their study, a nonadditive entropy (for a set of discrete states) is defined by

(39)

where is a probability distribution (24); (41). When , recovers the Boltzmann–Gibbs entropy given by

(40)

If we compose two probabilistically independent subsystems and , the Boltzmann–Gibbs entropy is additive:

(41)

However, when , is nonadditive:

(42)

This is because, for , is given by

(43)

In Ref. (41), Tsallis and Cirto demonstrated that a generalized black-hole entropy can be written as

(44)

where the Bekenstein entropy is given by

(45)

and is the event horizon area of a black hole. For details, see Ref. (41).

We now apply the generalized black-hole entropy to an entropy for entropic cosmology. To this end, substituting Eq. (45) into Eq. (44), replacing with , and rearranging, the entropy on the Hubble horizon can be evaluated as

(46)

where is the surface area of the sphere with the Hubble radius , and, therefore, . Because of nonadditive generalizations, the generalized entropy is proportional to , i.e., volume. Accordingly, from Eqs. (45) and (46), we assume a generalized entropy given by

(47)

where is a non-negative free-parameter. Note that is a dimensional constant. We hereinafter refer to as the generalized entropy on the Hubble horizon. Substituting [Eq. (20)] into Eq. (47), we obtain

(48)

where is [Eq. (24)].

In order to calculate an entropic-force, we use the generalized entropy [Eq. (47)] and the temperature on the horizon [Eq. (21)]. Substituting Eqs. (21) and (47) into Eq. (25) and using , we obtain the entropic-force as

(49)

Therefore, the pressure derived from the generalized entropy is given as

(50)

In this equation, is shown separately in order to clarify the difference between Eqs. (27) and (50). Similarly, is shown separately, in the following.

In the present study, we assume an effective pressure for entropic cosmology. Using Eq. (50), the effective pressure based on the generalized entropy is given by

(51)

Substituting Eq. (51) into Eq. (29), we have

(52)

This equation is the modified acceleration equation derived from the generalized entropy. The term on the right-hand side corresponds to an extra driving term to explain the accelerated expansion of the universe. The extra driving term is found not to be the term but rather to be the term, unlike in the case of the standard entropic cosmology. The continuity equation is expected to be given by

(53)

Substituting Eq. (51) into Eq. (53) and rearranging, we have

(54)

In this paper, we consider Eq. (54) as the modified continuity equation based on the generalized entropy (88).

Interestingly, the effective pressure, Eq. (51), is similar to Eq. (5), i.e., , for bulk viscous cosmology. Therefore, Eqs. (52) and (54) are also similar to Eqs. (8) and (10) for bulk viscous cosmology. This is probably because the generalized entropy assumed here is proportional not to the surface area but to the volume. However, the similarity may be interpreted as a sign that the generalized entropy behaves as if it were a classical entropy generated by bulk viscous stresses. In other words, the generalized entropy may be related to a bulk viscosity of cosmological fluids through a holographic screen. In fact, the bulk viscosity is usually the only thing that can generate an entropy in the homogeneous and isotropic universe. Accordingly, this interpretation may help to explain the origin of the bulk viscosity of cosmological fluids. As an alternative interpretation, the bulk viscosity may be derived from an extra entropy that is proportional to the volume of the universe.

Entropic-force model for a generalized entropy

In Sec. III.2.1, terms are derived from a generalized entropy on the horizon. Accordingly, we expect that the modified Friedmann and acceleration equations are summarized, respectively, as

(55)

and

(56)

where and are defined by

(57)

The two coefficients and are dimensionless constants, and is the present value of the Hubble parameter. As shown in Eq. (54), the modified continuity equation is given by

(58)

We can determine and , which are included in Eqs. (55) and (56), from two continuity equations. In this case, the first continuity equation is Eq. (58). The second continuity equation (derived from the modified Friedmann and acceleration equations) is calculated from the generalized continuity equation, i.e., Eq. (13). Comparing [Eqs. (11) and (12)] with [Eqs. (55) and (56)], we can set general functions as and . Therefore, substituting these functions into Eq. (13) and rearranging, we have the second continuity equation given by

(59)

Since Eqs. (58) and (59) are expected to be consistent with each other, we obtain and as

(60)

Accordingly, the dimensionless constants and are given as

(61)

As expected, we can neglect the entropic-force term of the modified Friedmann equation, Eq. (55). This is because, in the above discussion and in the present study, we assume an effective pressure for entropic cosmology. Note that we leave in Eq. (55) in order to discuss a combined model in the next section.

Parameter Bekenstein Generalized

Table 1: Entropic-force terms for the modified acceleration equation. The Bekenstein and Generalized columns indicate the information for the entropic-force models derived from the Bekenstein and generalized entropies, respectively.

In this section, we derive entropic-force terms from the Bekenstein and generalized entropies, assuming that the temperature on the horizon is given by Eq. (21), i.e., . Consequently, terms are derived from the Bekenstein entropy, whereas terms are derived from the generalized entropy, which is proportional to volume. The two entropic-force terms for the modified acceleration equation are summarized in Table 1. Interestingly, the term is similar to an extra driving term for bulk viscous cosmology. Therefore, we assume an effective pressure for entropic cosmology. The modified acceleration equation is found to include the entropic-force terms, whereas the Friedmann equation does not include the entropic-force terms. Based on these results, we propose two entropic-force models. In the next section, we discuss solutions of the two entropic-force models.

Iv Combined model

In the previous section, the Friedmann equation was found not to include entropic-force terms, because we use an effective pressure for entropic cosmology. Moreover, two entropic-force terms, i.e., and terms, are discussed separately, as shown in Secs. III.1 and III.2. This is because the term is derived from the Bekenstein entropy, whereas the term is derived from a generalized entropy. However, in this section, in order to obtain general solutions that are widely used, we consider a combined model in which the Friedmann and acceleration equations include both and terms as the entropic-force terms. In order to solve the equations, we extend our solution method, which is discussed in Ref. (15). Note that the two extra driving terms for the acceleration equation, i.e., and terms, have been examined in bulk viscous cosmology. For example, see the work of Avelino and Nucamendi (64). Of course, similar extra driving terms have been discussed using a variable cosmological term. For instance, Basilakos, Plionis, and Solà (82) have examined and terms in detail. Several types of variable cosmological terms, which include constant terms, are closely investigated in Ref. (82). For variable cosmologies, see Ref. (78), a recent review (86), and the references therein. We point out that cosmological equations and their solutions discussed in this section are similar to those examined in the above previous works.

iv.1 Formulations for the combined model

For a combined model, we consider both and terms as extra driving terms. Accordingly, the modified Friedmann equation is given as

(62)

and the modified acceleration equation is given as

(63)

where is given by

(64)

and are defined by

(65)

and represents the equation of state parameter for a generic component of matter. For non-relativistic matter (or the matter-dominated universe) is , and for relativistic matter (or the radiation-dominated universe) is . The four coefficients , , , and are dimensionless constants. For entropic-force models for the Bekenstein and generalized entropies, three dimensionless constants are set as , and , respectively.

Coupling [ Eq. (62)] with [ Eq. (63)] and rearranging, we obtain

(66)

where

(67)

and

(68)

Equation (66) includes not only the term but also the term. (In our previous study (15), the term was not considered.) Here, we point out that is a dimensionless parameter, whereas is a dimensional parameter. Substituting Eq. (65) into Eq. (68), we obtain a dimensionless parameter given by

(69)

From Eq. (66), is calculated as

(70)

We can rearrange Eq. (70) as

(71)

where is defined by

(72)

Because of the term, Eq. (71) is slightly more complicated than the equation examined in Ref. (15). We can solve Eq. (71), as discussed in the next subsection.

iv.2 Solutions for the combined model in the single-fluid-dominated universe

In the previous subsection, we obtain Eq. (71) for the combined model. We can solve Eq. (71) when and are constant. In fact, and are constant when five parameters, i.e., , , , , and , are constant. Therefore, we assume that these five parameters are constant. This indicates that we assume a single-fluid-dominated universe. For example, is for the matter-dominated universe and for the radiation-dominated universe. In the following, we consider , , and as non-negative free parameters. For simplicity, we also assume .

When and are constant, Eq. (71) is integrated as

(73)

Solving this integral, and using , we have

(74)

and dividing this equation by gives

(75)

where is an integral constant. Dividing Eq. (75) by , we have

(76)

where is the present value of the scale factor. Rearranging Eq. (76), and substituting [Eq. (69)] into the resulting equation, we obtain

(77)

Equation (77) indicates that and play an important role in the combined model. We can determine and from Eqs. (67) and (69), respectively. Since is related to the terms, Eq. (77) is somewhat complicated. For the case in which , typical results have been discussed in our previous study (15). For example, when , for , , , and are consistent with for the radiation-dominated, matter-dominated, empty, and -dominated universes, respectively (15).

Scale factor

We examine the scale factor , using Eq. (77). For this purpose, Eq. (77) is rearranged as

(78)

where , , and are defined by

(79)

Multiplying Eq. (78) by , we obtain

(80)

On the other hand, we can calculate as