Cosmological scenario based on the first and second laws of thermodynamics: Thermodynamic constraints on a generalized cosmological model

Lambda cold dark matter ($\Lambda$CDM) models, which assume a cosmological constant $\Lambda$ and dark energy, can explain an accelerated expansion of the late Universe PERL1998_Riess1998 ; Planck2018 ; Hubble2017 . However, the $\Lambda$CDM model suffers from several theoretical problems Weinberg1989 . For example, it is well-known that $\Lambda$ measured by observations is approximately $60\sim 120$ orders of magnitude smaller than the theoretical value obtained from quantum field theory Weinberg1989 ; Pad2003 ; Barrow2011 ; Bao2017 . To resolve those problems, astrophysicists have proposed various cosmological models, such as time-varying $\Lambda(t)$ cosmology FreeseOverduin ; Nojiri2006etc ; Valent2015Sola2019 ; Sola_2009-2022 , bulk viscous cosmology BarrowLima ; BrevikNojiri ; EPJC2022 , and creation of CDM (CCDM) models Prigogine_1988-1989 ; Lima1992-1996 ; LimaOthers2023 ; Freaza2002Cardenas2020 . In addition, thermodynamic scenarios based on the holographic principle Hooft-Bousso , such as entropic cosmology EassonCai ; Basilakos1 ; Koma45 ; Koma6 ; Koma78 ; Koma9 ; Neto2022 ; Gohar2024 and holographic cosmology Padmanabhan2004 ; ShuGong2011 ; Padma2012AB ; Cai2012 ; Hashemi ; Moradpour ; Koma10 ; Koma11 ; Koma12 ; Koma18 ; Pad2017 ; Tu2018 ; Tu2019 ; HDE ; Krishna20172019 ; Mathew2022 ; Chen2022 ; Luciano ; Mathew2023 ; Mathew2023b ; Koma14 ; Koma15 ; Koma16 ; Koma17 ; Koma19 ; Koma20 , have been examined extensively Koma21 ; Cai2005 ; Cai2011 ; Dynamical-T-2007 ; Dynamical-T-20092014 ; Sheykhi1 ; Sheykhi2Karami ; Mirza2015 ; Silva2015 ; Santos2022 ; Sheykhia2018 ; ApparentHorizon2022 ; Cai2007 ; Cai2007B ; Cai2008 ; Sanchez2023 ; Nojiri2024 ; Odintsov2023ab ; Odintsov2024 ; Mohammadi2023 ; Odintsov2024B ; Nojiri2024B .

In the thermodynamic scenarios, black hole thermodynamics Hawking1Bekenstein1 is applied to a cosmological horizon, which is assumed to have an associated entropy and an approximate temperature EassonCai . In those models Cai2007 ; Cai2007B ; Cai2008 ; Sanchez2023 ; Nojiri2024 ; Odintsov2023ab ; Odintsov2024 ; Mohammadi2023 ; Odintsov2024B ; Nojiri2024B , cosmological equations are derived from the first law of thermodynamics using a dynamical Kodama–Hayward temperature Dynamical-T-2007 ; Dynamical-T-20092014 and various forms of black hole entropy (including Bekenstein–Hawking entropy Hawking1Bekenstein1 ). For example, modified cosmological equations, which include extra driving terms, can be formulated by applying a power-law-corrected entropy Das2008 ; Radicella2010 , Tsallis–Cirto entropy Tsallis2012 , Tsallis–Rényi entropy Czinner1Czinner2 , Barrow entropy Barrow2020 , and a generalized six-parameter entropy Nojiri2022 . In addition, an arbitrary entropy on the horizon can be used to derive a generalized cosmological model from the first law, as examined by Odintsov et al. Odintsov2024 . We expect that the second law of thermodynamics constrains driving terms in the generalized model, as if the cosmological constant problem could be discussed from a thermodynamics viewpoint. (For the first law, see, e.g., the previous works of Akbar and Cai Cai2007 ; Cai2007B and Cai et al. Cai2008 and the recent works of Sánchez and Quevedo Sanchez2023 , Nojiri et al. Nojiri2024 , Odintsov et al. Odintsov2023ab ; Odintsov2024 ; Odintsov2024B , and the present author Koma21 . See also a recent review Nojiri2024B .)

In fact, the present author has examined thermodynamic constraints on an extra driving term in holographic equipartition models, similar to a time-varying $\Lambda(t)$ cosmology Koma11 ; Koma12 . However, theoretical backgrounds and cosmological equations for the model are different from those for the generalized cosmological model derived from the first law of thermodynamics. In addition, the generalized cosmological model should include two different driving terms, such as $f_{\Lambda}(t)$ and $h_{\textrm{B}}(t)$, unlike for the holographic equipartition model. (Usually, $f_{\Lambda}(t)$ and $h_{\textrm{B}}(t)$ are used for, e.g., time-varying $\Lambda(t)$ cosmology and bulk viscous cosmology, respectively, based on a general formulation Koma16 ; Koma21 .) The two terms should be related to a deviation of horizon entropy from the Bekenstein–Hawking entropy. We expect that the generalized cosmological model provides a consistent thermodynamic scenario; that is, the generalized model is derived from the first law, whereas the second law constrains the two terms in the model. The generalized cosmological model and thermodynamic constraints on the two driving terms have not yet been examined from those viewpoints. (Note that the second law itself has been examined in, e.g., Refs. Odintsov2024 ; Nojiri2024B .)

In this context, we examine thermodynamic constraints on the two terms in the generalized cosmological model, extending previous work Koma11 ; Koma12 . In the present study, cosmological equations are derived from the first law of thermodynamics using an arbitrary entropy on the cosmological horizon, in accordance with Ref. Odintsov2024 . The original cosmological equations implicitly include extra driving terms. Therefore, the cosmological equations are systematically formulated again, based on a general formulation that explicitly includes two extra driving terms, $f_{\Lambda}(t)$ and $h_{\textrm{B}}(t)$. In addition, we universally examine the thermodynamic constraints on the two driving terms using the second law of thermodynamics. The present study should contribute to a better understanding of thermodynamic scenarios and may provide new insights into the discussion of the cosmological constant problem. Inflation of the early universe is not discussed here because we focus on the late universe.

The remainder of the present article is organized as follows. In Sec. II, a general formulation for cosmological equations is reviewed. In Sec. III, an associated entropy and an approximate temperature on a cosmological horizon are introduced. In Sec. IV, cosmological equations are derived from the first law of thermodynamics using an arbitrary entropy on the horizon. In addition, based on the general formulation, the cosmological equations are systematically formulated. In Sec. V, thermodynamic constraints on the two terms in the present model are examined based on the second law of thermodynamics. Finally, in Sec. VI, the conclusions of this study are presented.

The present study considers a flat Friedmann–Lemaître–Robertson–Walker (FLRW) universe. In addition, an expanding universe is assumed from observations PERL1998_Riess1998 ; Hubble2017 ; Planck2018 . A general formulation for cosmological equations was previously examined in Refs. Koma6 ; Koma9 ; Koma14 ; Koma15 ; Koma16 and recently examined in Ref. Koma21 . In this section, we introduce a general formulation using the scale factor $a(t)$ at time $t$, in accordance with those works.

The general Friedmann, acceleration, and continuity equations are written as Koma21

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. Also, $f_{\Lambda}(t)$ and $h_{\textrm{B}}(t)$ are extra driving terms Koma14 ; Koma21 . Usually, $f_{\Lambda}(t)$ is used for a $\Lambda(t)$ model, similar to $\Lambda(t)$CDM models, whereas $h_{\textrm{B}}(t)$ is used for a bulk-viscous-cosmology-like model, similar to bulk viscous models and CCDM models Koma14 ; Koma16 ; Koma21 . In this paper, the two terms are phenomenologically assumed and are considered to be related to an associated entropy on a cosmological horizon, as examined later. The general continuity given by Eq. (3) can be derived from Eqs. (1) and (2), because only two of the three equations are independent Ryden1 . In addition, subtracting Eq. (1) from Eq. (2) yields Koma21

These equations are used in Sec. IV to examine cosmological equations derived from the first law of thermodynamics.

Equation (3) indicates that the right side of the general continuity equation is usually non-zero, as discussed in Refs. Koma9 ; Koma20 ; Koma21 . However, when both $f_{\Lambda}(t)=\Lambda/3$ and $h_{\textrm{B}}(t)=0$ are considered, the continuity equation reduces to the standard continuity equation, namely $\dot{\rho}+3H[\rho+(p/c^{2})]=0$. Exactly speaking, Eq. (3) reduces to the standard continuity equation when the following relation is satisfied:

In the present study, we derive a generalized cosmological model from the first law of thermodynamics, by applying the standard continuity equation, as examined in previous works Cai2007 ; Cai2007B ; Cai2008 ; Sanchez2023 ; Nojiri2024 ; Odintsov2023ab ; Odintsov2024 ; Odintsov2024B ; Nojiri2024B . That is, we assume that Eq. (6) holds. We can confirm that substituting Eq. (6) into Eq. (3) gives the standard continuity equation, written as

The right side of this equation is zero but includes $f_{\Lambda}(t)$ and $h_{\textrm{B}}(t)$ implicitly.

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 $w=p/(\rho c^{2})$ Koma21 . For a $\Lambda$-dominated universe and a matter-dominated universe, the values of $w$ are $-1$ and $0$, respectively. In this paper, $w>-1$ is considered because $f_{\Lambda}(t)$ can behave as a varying cosmological-constant-like term instead of $w=-1$. (Note that $w$ is retained for generality.) For example, when both $f_{\Lambda}(t)=\Lambda/3$ and $h_{\textrm{B}}(t)=0$ are considered, the general formulation reduces to that for $\Lambda$CDM models. The order of the density parameter $\Omega_{\Lambda}$ for $\Lambda$ is $1$, based on the Planck 2018 results Planck2018 . Here $\Omega_{\Lambda}$ is defined by $\Lambda/(3H_{0}^{2})$, and $H_{0}$ is the current Hubble parameter. Therefore, the order of the cosmological constant term measured by observations, namely $O(\Lambda_{\textrm{obs}}/3)$ should be written as Koma10 ; Koma11 ; Koma12

The orders of two extra driving terms are discussed later.

In thermodynamic scenarios, a cosmological horizon is assumed to have an associated entropy and an approximate temperature EassonCai . In this section, the entropy $S_{H}$ and the temperature $T_{H}$ on the horizon are introduced in accordance with previous works Koma11 ; Koma12 ; Koma17 ; Koma18 ; Koma19 ; Koma20 ; Koma21 .

First, we review a form of the Bekenstein–Hawking entropy as an associated entropy on the cosmological horizon because it is the most standard approach Koma19 ; Koma20 ; Koma21 . In fact, a deviation from the Bekenstein–Hawking entropy plays an important role in extra driving terms, as discussed later. In general, the cosmological horizon is examined by replacing the event horizon of a black hole with the cosmological horizon Jacob1995 ; Padma2010 ; Verlinde1 ; Padma2012AB ; Cai2012 ; Moradpour ; Hashemi ; Padmanabhan2004 ; ShuGong2011 ; Koma10 ; Koma11 ; Koma12 ; Koma14 ; Koma15 ; Koma16 ; Koma17 ; Koma18 ; Koma19 ; Koma20 ; Koma21 . We use this replacement method. In the present paper, the Hubble horizon is equivalent to the apparent horizon of the universe because a flat FLRW universe is considered.

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. (10) and applying Eq. (11) yields

where $K$ is a positive constant given by

Differentiating Eq. (12) with regard to $t$ yields Koma11 ; Koma12

Cosmological observations indicate that $H>0$ and $\dot{H}<0$ (see, e.g., Ref. Hubble2017 ). Accordingly, in our Universe, $\dot{S}_{\rm{BH}}$ should be positive as follows Koma11 ; Koma12 :

Of course, various forms of black hole entropy have been proposed Tsallis2012 ; Czinner1Czinner2 ; Barrow2020 ; Nojiri2022 , as described in Refs. Koma18 ; Koma19 ; Koma20 . These entropies can be interpreted as an extended version of $S_{\rm{BH}}$. In this study, we consider an arbitrary form of entropy $S_{H}$ on the Hubble horizon and derive a generalized cosmological model from the first law of thermodynamics, as examined in Sec. IV.

Next, we introduce an approximate temperature $T_{H}$ on the Hubble horizon, in accordance with previous works Koma19 ; Koma20 ; Koma21 . We first review the Gibbons–Hawking temperature $T_{\rm{GH}}$ because a dynamical Kodama–Hayward temperature is interpreted as an extended version of $T_{\rm{GH}}$. The Gibbons–Hawking temperature GibbonsHawking1977 is given by $T_{\rm{GH}}=\frac{\hbar H}{2\pi k_{B}}$. Accordingly, $T_{\rm{GH}}$ 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 (including our Universe) are not pure de Sitter universes in that their horizons are dynamic Koma19 ; Koma20 ; Koma21 . Therefore, we introduce a dynamical Kodama–Hayward temperature Dynamical-T-1998 ; Dynamical-T-2008 . The Kodama–Hayward temperature for 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 $\dot{H}\geq-2H^{2}$ are assumed for a non-negative temperature in an expanding universe Koma19 ; Koma20 ; Koma21 . When de Sitter universes are considered, $T_{\rm{KH}}$ reduces to the Gibbons–Hawking temperature $T_{\rm{GH}}$. In the present paper, the Kodama–Hayward temperature $T_{\rm{KH}}$ is used for the temperature $T_{H}$ on the horizon, to discuss the first law of thermodynamics in accordance with Refs. Cai2007 ; Cai2007B ; Cai2008 ; Sanchez2023 ; Nojiri2024 ; Odintsov2023ab ; Odintsov2024 ; Odintsov2024B ; Nojiri2024B .

In this section, we review the first law of thermodynamics and introduce cosmological equations derived from the first law, in accordance with Refs. Nojiri2024 ; Odintsov2024 . Then, we reformulate the cosmological equations and determine the two extra driving terms, $f_{\Lambda}(t)$ and $h_{\textrm{B}}(t)$, based on the general formulation introduced in Sec. II. The first law of thermodynamics Cai2007 ; Cai2007B ; Cai2008 ; Sanchez2023 ; Nojiri2024 ; Odintsov2023ab ; Odintsov2024 ; Odintsov2024B ; Nojiri2024B was recently examined in a previous work Koma21 and, therefore, the first law is reviewed based on that work and the references therein. Note that the Hubble horizon is equivalent to an apparent horizon because a flat FLRW universe is considered.

The first law of thermodynamics is written as Cai2007 ; Cai2007B ; Cai2008 ; Sanchez2023 ; Nojiri2024 ; Odintsov2023ab ; Odintsov2024 ; Odintsov2024B ; Nojiri2024B

where $E_{\rm{bulk}}$ is the total internal energy of the matter fields inside the horizon, given by

$W$ represents the work density done by the matter fields Nojiri2024 , which is written as

and $V$ is the Hubble volume, written as

where $r_{H}=c/H$ is given by Eq. (11) Koma21 . Equation (17) indicates that the entropy on the horizon is generated based on both the decreasing total internal energy of the bulk ($-dE_{\rm{bulk}}$) and the work done by the matter fields ($WdV$) Nojiri2024 .

In addition, Eq. (17) can be written as Koma21

or equivalently,

Here the Kodama–Hayward temperature $T_{\rm{KH}}$ is used for the temperature $T_{H}$ on the horizon Cai2007 ; Cai2007B ; Cai2008 ; Sanchez2023 ; Nojiri2024 ; Odintsov2023ab ; Odintsov2024 ; Odintsov2024B ; Nojiri2024B . In this study, an arbitrary entropy $S_{H}$ on the horizon is considered, as examined in Ref. Odintsov2024 . Note that a general form of entropy is discussed in Ref. Nojiri2024 .

To examine Eq. (22), we calculate the left side of this equation Koma21 . Substituting Eqs. (18) and (19) into $-\dot{E}_{\rm{bulk}}+W\dot{V}$ yields Nojiri2024

Equation (23) corresponds to the left side of Eq. (22). Therefore, Eq. (22) can be written as Odintsov2024

To derive cosmological equations, the standard continuity equation is applied Cai2007 ; Cai2007B ; Cai2008 ; Sanchez2023 ; Nojiri2024 ; Odintsov2023ab ; Odintsov2024 ; Odintsov2024B ; Nojiri2024B . From Eq. (7), the continuity equation is written as

and $h_{\textrm{B}}(t)$ given by Eq. (6) is written as

Based on the above preparations, we are able to derive cosmological equations from the first law of thermodynamics. In fact, Odintsov et al. derived cosmological equations from the first law using an arbitrary entropy $S_{H}$ on the horizon Odintsov2024 . The cosmological equations are considered to be a generalized cosmological model. The derivation is summarized in Appendix A, and the results are used here. From Eq. (71), the Friedmann equation from the first law is written as

where $C$ is an integral constant and should be given by $\Lambda/3$. The above equation corresponds to the Friedmann equation given by Eq. (1). In accordance with Ref. Odintsov2024 , we use a symbol with brackets, $(\partial S_{H}/\partial S_{\rm{BH}})$. Note that the integral constant $C$ is retained here, to avoid confusion with $\Lambda$ measured by observations. In addition, from Eq. (67), a cosmological equation corresponding to Eq. (5) is written as

The acceleration equation is obtained from Eqs. (27) and (28), as examined later. In fact, Eqs. (27) and (28) implicitly include two extra driving terms, namely $f_{\Lambda}(t)$ and $h_{\textrm{B}}(t)$, which are used for the general formulation introduced in Sec. II. However, it is difficult to find the two terms from the above two equations directly. Therefore, in the next subsection, the cosmological equations are systematically formulated again, based on the general formulation.

In the previous section, we formulated a generalized cosmological model from the first law of thermodynamics and determined two extra driving terms, $f_{\Lambda}(t)$ and $h_{\textrm{B}}(t)$. From Eqs. (39) and (40), the two terms for the present model are written as

where the deviation $S_{\Delta}$ is $S_{H}-S_{\rm{BH}}$, which is given by Eq. (29). Note that $S_{H}$ is an arbitrary entropy on the horizon and $S_{\rm{BH}}$ is the Bekenstein–Hawking entropy.

In this section, based on the second law of thermodynamics, we universally examine thermodynamic constraints on the two terms in the present model, extending the method used in previous works Koma10 ; Koma11 ; Koma12 . Similar constraints are discussed in those works, using holographic equipartition models that are different from the present model. (The second law itself has been examined: for example, to discuss the second law, the total entropy, namely the sum of the horizon entropy and the entropy of the matter fields, was considered Odintsov2024 .) In our Universe, the horizon entropy is extremely larger than the sum of the other entropies Egan1 . In addition, the horizon entropy is included in the two terms. Therefore, we focus on the horizon entropy. Consequently, the second law of thermodynamics is written as

As examined in Sec. III, we consider $\dot{S}_{\rm{BH}}>0$ given by Eq. (15), because $H>0$ and $\dot{H}<0$ in our Universe Hubble2017 . Also, $\dot{H}\geq-2H^{2}$ is assumed because the horizon temperature given by Eq. (16) is considered to be non-negative. In addition, $S_{\Delta}\neq 0$ is considered. Note that various nonextensive entropies satisfy $S_{\Delta}\neq 0$, except when $S_{H}=S_{\rm{BH}}$ Das2008 ; Radicella2010 ; Tsallis2012 ; Czinner1Czinner2 ; Barrow2020 ; Nojiri2022 .

We now examine thermodynamic constraints on the present model. Substituting Eq. (30) into Eq. (42) yields

where $(\partial S_{H}/\partial S_{\rm{BH}})=\dot{S}_{H}/\dot{S}_{\rm{BH}}$ is used from Eq. (65). Solving Eq. (44) with regard to $\dot{S}_{H}$ gives

From Eq. (45), to satisfy $\dot{S}_{H}\geq 0$, we require

where $\dot{S}_{\rm{BH}}>0$ is used. This inequality corresponds to the thermodynamic constraint on $h_{\textrm{B}}(t)$. Applying $\dot{H}<0$ to Eq. (46) gives

The above inequality implies a lower limit of $h_{\textrm{B}}(t)$. The lower limit is negative because $\dot{H}<0$. Substituting Eq. (5) into Eq. (47) yields

In addition, substituting $w=p/(\rho c^{2})$ into Eq. (48) gives

where $\rho$ is positive. In this way, the constraint on $w$ can be obtained from the constraint on $h_{\textrm{B}}(t)$. We note that $w>-1$ considered in Sec. II satisfies Eq. (49). An upper limit of $h_{\textrm{B}}(t)$ and the order of $h_{\textrm{B}}(t)$ are discussed later.

In the present study, $h_{\textrm{B}}(t)$ is given by Eq. (26), namely $\dot{f}_{\Lambda}(t)/(2H)$, because the standard continuity equation is considered. Substituting Eq. (26) into Eq. (46) yields a constraint on $\dot{f}_{\Lambda}(t)$:

Also, from Eq. (41), we can obtain a similar constraint on $df_{\Lambda}(t)$:

The two inequalities correspond to the constraints on $\dot{f}_{\Lambda}(t)$ and $df_{\Lambda}(t)$. However, the two inequalities do not constrain the extent of $f_{\Lambda}(t)$, although the two should help to examine cosmological models from a thermodynamics viewpoint.

In fact, thermodynamic constraints on $f_{\Lambda}(t)$ can be discussed using Eq. (8), which is written as

where $w>-1$ is considered, as examined in Sec. II. Solving Eq. (52) with regard to $h_{\textrm{B}}(t)$ and substituting the resultant equation into Eq. (45) yields

Using $\dot{S}_{\rm{BH}}>0$, $\dot{H}<0$, and $w>-1$ to satisfy $\dot{S}_{H}\geq 0$, we require

or equivalently,

Equations (54) and (55) imply an upper limit of $f_{\Lambda}(t)$. When $0<H$ and $H_{0}\leq H$ (obtained from $\dot{H}<0$), the strictest constraint from the past to the present is given by

and the order of $f_{\Lambda}(t)$ can be written as

In addition, we can examine an upper limit of $h_{\textrm{B}}(t)$. Applying $\dot{H}<0$ to Eq. (52) and using Eq. (55) yields

where $f_{\Lambda}(t)\geq 0$ is also used. (Such a non-negative $f_{\Lambda}(t)$ can be obtained from, e.g., a power-law-corrected entropy, as examined in Appendix B.) Equation (58) implies the upper limit of $h_{\textrm{B}}(t)$. Coupling Eq. (47) with Eq. (58) and applying $\dot{H}\geq-2H^{2}$ yields

This inequality corresponds to an extended thermodynamic constraint on $h_{\textrm{B}}(t)$. Applying $0<H_{0}\leq H$ to Eq. (59) gives the order of $h_{\textrm{B}}(t)$, written as

These results indicate that the second law of thermodynamics should constrain $f_{\Lambda}(t)$ and $h_{\textrm{B}}(t)$. From Eqs. (57) and (60), the orders of the upper limits of $f_{\Lambda}(t)$ and $h_{\textrm{B}}(t)$ are likely consistent with the order of the cosmological constant measured by observations, namely $\Lambda_{\textrm{obs}}$. We note that $O(\Lambda_{\textrm{obs}}/3)\approx O(H_{0}^{2})$ is given by Eq. (9), which is based on the Planck 2018 results Planck2018 . Also, from Eq. (60), the absolute value of the order of the lower limit of $h_{\textrm{B}}(t)$ is the same as the order of the upper limit.

Figure 1 shows the orders of the upper limits of the two terms, the order of $\Lambda_{\textrm{obs}}$ Planck2018 , and the order of the theoretical value from quantum field theory Weinberg1989 ; Pad2003 ; Barrow2011 ; Bao2017 . A region for the lower limit of $h_{\textrm{B}}(t)$ is not shown in this figure because the lower limit is negative and the order is the same as the order of the upper limit. We can confirm that the orders of the upper limits of the two terms are consistent with the order of $\Lambda_{\textrm{obs}}$. The discrepancy of $60\sim 120$ orders of magnitude appears to be avoided, as if a consistent scenario could be obtained from thermodynamics.

Finally, we use the present model to discuss an important model similar to the $\Lambda$CDM models ($f_{\Lambda}(t)\rightarrow C$ and $h_{\textrm{B}}(t)\rightarrow 0$). To this end, a deviation $S_{\Delta}$ from the Bekenstein–Hawking entropy is considered to be close to zero but not zero. Accordingly, Eqs. (41) and (42) should reduce to $f_{\Lambda}(t)\approx C+\epsilon_{1}$ and $h_{\textrm{B}}(t)\approx\epsilon_{2}$, respectively. Here, two parameters, $\epsilon_{1}$ and $\epsilon_{2}$, should be close to zero because $S_{\Delta}$ is close to zero. Therefore, the constraint on $h_{\textrm{B}}(t)$ given by Eq. (46) should be satisfied, and the order of $h_{\textrm{B}}(t)$ can be given by

In addition, the constraint on $f_{\Lambda}(t)$ given by Eq. (57) can be written as

Equation (62) implies that the order of $C$ is consistent with the order of $\Lambda_{\textrm{obs}}$. In this sense, we can use the present model to discuss the cosmological constant problem from a thermodynamic viewpoint when $S_{\Delta}$ is close to zero but not zero. The small deviation $S_{\Delta}$ from the Bekenstein–Hawking entropy may play an important role in the accelerated expansion of our late Universe.

An arbitrary entropy $S_{H}$ on the horizon is considered here and, therefore, we can use various forms of entropy on the horizon; see, for example, Appendix B and Refs. Odintsov2024 ; Nojiri2024B . Before fine-tuning, we should be able to discuss thermodynamic constraints on driving terms calculated from those entropies. We expect that effective entropies, which deviate slightly from the Bekenstein–Hawking entropy, are favored in our Universe. Further studies are needed, and those tasks are left for future research.

It should be noted that, in this section, we consider $S_{\Delta}\neq 0$ and discuss the thermodynamic constraints. When $S_{\Delta}=0$ (i.e., $S_{H}=S_{\rm{BH}}$), the two terms reduce to $f_{\Lambda}(t)=C$ and $h_{\textrm{B}}(t)=0$, respectively, as for $\Lambda$CDM models. In this case, Eq. (45) reduces to $\dot{S}_{H}=\dot{S}_{\rm{BH}}$. In fact, Eq. (45) is used in the calculation of the constraint on $f_{\Lambda}(t)$, through Eq. (53). Accordingly, when $S_{\Delta}=0$, it is difficult to discuss the thermodynamic constraints. To avoid this difficulty, we consider $S_{\Delta}\neq 0$. Various nonextensive entropies satisfy $S_{\Delta}\neq 0$, except when $S_{H}=S_{\rm{BH}}$. Note that we also consider $\dot{S}_{\rm{BH}}>0$, $H>0$, $-2H^{2}\leq\dot{H}<0$, and $w>-1$.

In this paper, we examine the thermodynamic constraints on the two driving terms in a generalized cosmological model derived from the first law of thermodynamics, using an arbitrary entropy on the horizon. The thermodynamic constraints imply that the orders of the two terms are consistent with the order of the cosmological constant measured by observations. Of course, we cannot exclude all the other contributions, such as quantum field theory, because these contributions have not been examined in this study. In addition, the assumptions used here have not yet been established but are considered to be viable. The present study should contribute to a better understanding of thermodynamic scenarios, which may provide new insights into the discussion of the cosmological constant problem.

The first and second laws of thermodynamics should lead to a consistent scenario for discussing the cosmological constant problem. To establish such a thermodynamic scenario, we have derived cosmological equations in a flat FLRW universe from the first law, using an arbitrary entropy $S_{H}$ on the horizon. The derived cosmological equations implicitly include extra driving terms. Therefore, we have systematically reformulated the cosmological equations using a general formulation that includes two extra driving terms, $f_{\Lambda}(t)$ and $h_{\textrm{B}}(t)$, explicitly. The present model is essentially equivalent to that derived in Ref. Odintsov2024 , but is suitable for the discussion of thermodynamic constraints.

Based on the second law of thermodynamics, we have universally examined the thermodynamic constraints on the two terms in the present model. It is found that a variation in the deviation $S_{\Delta}$ of $S_{H}$ from the Bekenstein–Hawking entropy plays an important role in the thermodynamic constraints. In our late Universe, the second law should constrain both the upper limit of $f_{\Lambda}(t)$ and the upper and lower limits of $h_{\textrm{B}}(t)$. The upper limits imply that the orders of the two terms are consistent with the order of the cosmological constant $\Lambda_{\textrm{obs}}$ measured by observations. The lower limit of $h_{\textrm{B}}(t)$ leads to a constraint on $w$, and the absolute value of the order of the lower limit is likely consistent with the order of $\Lambda_{\textrm{obs}}$ as well. In particular, when the deviation $S_{\Delta}$ is close to zero but not zero, $h_{\textrm{B}}(t)$ and $f_{\Lambda}(t)$ should reduce to zero and a constant (consistent with the order of $\Lambda_{\textrm{obs}}$), respectively, as if a consistent and viable scenario could be obtained from thermodynamics. The thermodynamic scenario may imply that the accelerated expansion of our Universe is related to the small deviation from the Bekenstein–Hawking entropy.