Different aspects of entropic cosmology

Citation: Nojiri, S.; Odintsov, S.D.; Paul, T. Different Aspects of Entropic Cosmology. Universe 2024,10, 352. https://doi.org/10.3390/ universe10090352 Academic Editor: Jean-Pierre Gazeau Received: 24 July 2024 Revised: 22 August 2024 Accepted: 29 August 2024 Published: 3 September 2024 Copyright: © 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). universe Review Different Aspects of Entropic Cosmology Shin’ichi Nojiri 1,2 , Sergei D. Odintsov 3,4 and Tanmoy Paul 5,* 1 KEK Theory Center, Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK), Oho 1-1, Tsukuba 305-0801, Ibaraki, Japan; [email protected] 2Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University, Nagoya 464-8602, Japan 3ICREA, Passeig Lluis Companys, 23, 08010 Barcelona, Spain; [email protected] 4Institute of Space Sciences (ICE-CSIC), C. Can Magrans s/n, 08193 Barcelona, Spain 5Department of Physics, Visva-Bharati University, Santiniketan 731235, India *Correspondence: tanmoy[email protected] Abstract: We provide a short review of the recent developments in entropic cosmology based on two thermodynamic laws of the apparent horizon, namely the first and the second laws of thermodynamics. The first law essentially provides the change in entropy of the apparent horizon during the cosmic evolution of the universe; in particular, it is expressed by TdS =−d(ρV) + WdV (where W is the work density and other quantities have their usual meanings). In this way, the first law actually links various theories of gravity with the entropy of the apparent horizon. This leads to a natural question—“What is the form of the horizon entropy corresponding to a general modified theory of gravity?”. The second law of horizon thermodynamics states that the change in total entropy (the sum of horizon entropy + matter fields’ entropy) with respect to cosmic time must be positive, where the matter fields behave like an open system characterised by a non-zero chemical potential. The second law of horizon thermodynamics importantly provides model-independent constraints on entropic parameters. Finally, we discuss the standpoint of entropic cosmology on inflation (or bounce), reheating and primordial gravitational waves from the perspective of a generalised entropy function. Keywords: entropic cosmology; inflation; reheating; primordial gravitational waves; bounce 1. Introduction The benchmark of Bekenstein–Hawking entropy connects two apparently different sectors, namely gravity and thermodynamics, on equal footing. In particular, the black hole horizon exhibits thermal behaviour whereby the entropy of the horizon scales by the area of the horizon and the surface gravity fixes the corresponding temperature [1–4]. In the cosmological context, there exists an apparent horizon that is a marginally trapped surface with vanishing expansion that divides the observable universe from the unobservable universe. Similar to black hole thermodynamics, the cosmic horizon is generally considered to exhibit thermal behaviour [ 5 – 25 ], and one can motivate it using the following arguments: • During the cosmic evolution of the universe, the matter fields inside of the horizon show a flux from inside to outside the horizon (the flux is outward in nature during the accelerating stage), which results in a decrease in the matter fields’ entropy. This violates the second law of thermodynamics, which states that the change in total entropy must be positive. Therefore, the cosmic horizon should be incorporated with entropy to manage the increase in total entropy (the sum of horizon entropy + matter fields’ entropy). • The cosmological field equations are time-reversal-symmetric; thus, they always come with a contracting solution, along with an expanding one. However, our observational data indicate that the universe is expanding. Therefore, the natural question that comes to mind is “Why does the universe always choose the expanding solution?”. Universe 2024,10, 352. https://doi.org/10.3390/universe10090352 https://www.mdpi.com/journal/universe Universe 2024,10, 352 2 of 19 In order to answer this question, we need to associate thermal behaviour with the cosmic horizon. Then, the second law of horizon thermodynamics actually disagrees the contracting solution in order to achieve a positive change in total entropy. Consequently, the subject of “entropic cosmology” has gained a lot of interest, in which the cosmic horizon is associated with entropy that follows the following thermodynamic law [5–8,19]: ThdSh=−d(ρV)+1 2(ρ−p)dV (1) where V is the volume enclosed by the apparent horizon expressed by Rh= 1 /H (with H being the Hubble parameter of the universe); moreover, Th and Sh are the temperature and entropy of the horizon, respectively (and the other quantities have their usual meanings). We now assume that the universe is homogeneous and isotropic, owing to which the total energy inside the apparent horizon is expressed as U=ρV ; otherwise, if ρ depends on spatial coordinates, then the energy inside the horizon should be expressed by the integral (over the volume of the horizon) of U=RρdV . Here, Th is fixed by the surface gravity of the apparent horizon, which, in the case of the spatially flat Friedmann–Lemaître– Robertson–Walker (FLRW) metric, turns out to be Th=H 2π 1+˙ H 2H2 . (2) At this stage, we would like to mention that the microscopic origin of horizon entropy (and its associated temperature) is still a debatable topic and needs further investigation ( see [26,27] for some progress in this regard). Together with Equation (1) (which is the first law of horizon thermodynamics in cosmology), the second law states that [20] d(Sh+Sm)>0, (3) where Sm represents the entropy of the matter fields inside of the horizon. With a specific form of horizon entropy, the first law of horizon thermodynamics (along with the local conservation law of the matter fields) leads to the cosmological field equation; for instance, the Bekenstein–Hawking-like entropy of the cosmic horizon provides the usual FLRW equations of Einstein gravity from Equation (1). However, for a different form of horizon entropy (compared to Bekenstein–Hawking-like horizon entropy), we end up with the following modified cosmological field equations: ˙ H∂Sh ∂S=−4πG(ρ+p)(4) and Z∂Sh ∂SdH2=8πG 3ρ+Λ 3, (5) where S=π/(GH2) is the Bekenstein–Hawking entropy (clearly, for Sh≡S , one obtains the usual FLRW equations). Such a modified cosmic scenario has some interesting cosmological consequences starting from inflation (or bounce) to the dark energy era [5–25,28–39]. Some variants of Bekenstein–Hawking entropy are Tsallis entropy [ 40 ], Rényi entropy [ 41 ], Barrow entropy [ 42 ], Sharma–Mittal entropy [ 43 ], Kaniadakis entropy [ 44 ], loop quantum gravity entropy [ 45 ], etc. The important point is that all of these entropies share some common features; for example, they all vanish at the limit of S→ 0, and they show monotonically increasing behaviour with respect to the Bekenstein–Hawking entropy variable ( S ). One more reason for interest in entropic cosmology is that it is related to the holographic cosmology initiated by Witten and Susskind [ 46 – 48 ]. In particular, entropic cosmology proves to be equivalent to the generalised holographic scenario with suitable Universe 2024,10, 352 3 of 19 holographic cut-offs [ 49 , 50 ]. The significant contributions of holographic cosmology (or, equivalently, entropic cosmology) corresponding to the aforementioned entropies explain the dark energy era of our universe, namely holographic dark energy (HDE) [51–69]. Based on the above arguments, some immediate questions that arise include the following: 1. What is the form of the horizon entropy that leads to the cosmological field equations for a general modified theory of gravity from Equation (1)? 2. Does there exist any generalised entropy that can generalise all the known entropies proposed so far (like Tsallis entropy, Rényi entropy, Barrow entropy, Sharma–Mittal entropy, Kaniadakis entropy, etc.)? This question is well motivated, as all these entropies share some common properties, as mentioned above. 3. If a generalised form of entropy exists, then what are the constraints on the generalised entropic parameters coming from the second law of horizon thermodynamics? Furthermore, what is the standpoint of generalised entropy on primordial gravitational waves? Does the constraint coming from the primordial gravitational waves match that based on the second law of horizon thermodynamics? The present article, based on some of our previous works [ 19 , 20 , 24 , 25 , 28 , 33 ], provides a brief review, answering the above questions. We follow the (−,+,+,+,....)signature of a spatially flat (n+ 1 ) dimensional spacetime metric, and we take G=c=¯h= 1, where G is the Newton’s constant, cis the speed of light and ¯his Planck’s constant. 2. First Law of Horizon Thermodynamics: Consistent Entropy for a General Modified Theory of Gravity The question that we encounter in this section is the following: What is the form of entropy that, based on the thermodynamic law (1), can produce the cosmological field equations for a general modified theory of gravity [19]? The FLRW equations for a general modified theory of gravity in (n+ 1 ) dimensional spacetime can be expressed as H2=16π n(n−1)(ρ+ρc)and ˙ H=−8π (n−1)(ρ+ρc+p+pc), (6) where ρc and pc represent the modifications compared to Einstein gravity. Owing to such modifications, we can expect that the corresponding horizon entropy for a general modified gravity theory will take the following form: Sh=A 4+Sc(A), (7) where the suffix ‘h’ represents horizon’ entropy. Here, A=nΩnRhn−1 represents the area of the apparent horizon in n+ 1 dimensional spacetime, and Sc is a function of A . In particular, the horizon entropy for a general theory of gravity is considered to be corrected over that in the case of Einstein gravity, namely Equation (7). The correction ( Sc ) explicitly depends on the modification of gravitational action, which we need to find in such a way that the following thermodynamic law holds with E=ρVand W=1 2(ρ−p): ThdSh=−dE +WdV. (8) By using Sh=A 4+Sc, the above equation can be equivalently written as ThdSc dt =d dt(ρcV)−WcdV dt +−d dt(ρV+ρcV)+1 2(ρ+ρc−p−pc)dV dt −Td dtA 4, (9) Universe 2024,10, 352 4 of 19 where Wc=1 2(ρc−pc). With Rh=1 Hin combination with Equation (2), we obtain dSc dt =−2πnΩn(ρc+pc)Rhn, (10) upon the integration of which, we obtain Sc=2πnΩnZRhn−2ρc+pc ˙ HdRh. (11) Equation (11) argues that Sc should be a function of only Rh (or, equivalently, the area of the horizon), as dRh is the only differential present in the rhsof the above expression. In general, the integration into Equation (11) should be realized by specifying the scale factor ( a=a(t) ) as a function of the cosmological time ( t ). In particular, if we consider a specific scale factor, then H(t) and, consequently, Rh are also expressed as a function of t , and as a result, the integrand in Equation (11) can be expressed in terms of Rh . Then, Equation (11) can be integrated to yield Sc=Sc(Rh) . With the above form of Sc , Equation (7) provides the full entropy corresponding to a general modified theory of gravity as Sh=A 4+2πnΩnZRhn−2ρc+pc ˙ HdRh, (12) It can be observed that for ρc=pc= 0, i.e., for Einstein’s theory of gravity, the entropy from Equation (12) becomes Sh=A/ 4, as per our expectation. Moreover, for ρ=p= 0, i.e., without any matter fields, Equation (6) yields (ρc+pc)/˙ H=− 8 π/(n− 1 ) , which immediately yields to Sh= 0 from Equation (12). However, this is expected, as there is no flux of matter fields from inside to outside of the horizon for ρ=p= 0 or, equivalently, there is no information loss associated with the horizon. Below, we present some specific examples of gravity theories and determine the respective entropy from Equation (12). • For (n+1)dimensional GB gravity, where the FLRW equations are expressed as H2+λ(n−2)(n−3)H4=16π n(n−1)ρ, h1+2λ(n−2)(n−3)H2i˙ H=−8π (n−1)(ρ+p), (13) with λ being the GB parameter, the corresponding horizon entropy from Equation (12) is expressed as Sh=A 41+2λ(n−1)(n−2) Rh2. (14) • For (3+1)dimensional f(Q)gravity theory, the FLRW equations are expressed as H2=1 6fQρ+f 2, ˙ H=−1 2fQρ 2+f 2+12H2fQQ ˙ H−f 4+p 2−˙ H fQ, (15) where fQ and fQQ represent the first and second derivatives of f(Q) (with respect to the variable ( Q )), respectively. Clearly, in this case, one requires a certain form of f(Q) , which is taken to be a power law type, i.e., f(Q) = Qn (with n being a constant). Such a form of f(Q) , along with Equation (12), leads to the following entropy corresponding to the (3+1)dimensional f(Q)gravity: Universe 2024,10, 352 5 of 19 Sh=A 4"1−32π(1+n(1−2n) (2−n)6π An−1)#. (16) It can be noted that in both GB gravity and f(Q) gravity theories, the integration of Equation (12) can be performed without specifying the scale factor ( a=a(t) ). This may not be the case when the gravitational field equations contain higher derivatives of the Hubble parameter; for instance, in the F(R) gravity theory, where the FLRW equations contain ... H , one needs to specify the scale factor ( a(t) ) to perform the integration and to determine Sh (corresponding to F(R) gravity) from Equation (12) [ 19 ]. On other hand, it is well known that a F(R) theory can be recast as a scalar-tensor theory through the conformal transformation of the spacetime metric, where the scalar potential depends on the form of F(R) under consideration. However, it was pointed out in [ 19 ] that such a mathematical equivalence between F(R) and scalar-tensor theory is spoiled from the perspective of the entropy of the apparent horizon. At this stage it deserves mentioning that the determination of the horizon entropy from the thermodynamic law (1) encounters a problem. In particular, it requires the quantity 1+˙ H 2H2 to be positive; otherwise, the entropy (particularly for Einstein gravity) turns out to be negative, which is impossible. The condition, namely 1 +˙ H 2H2< 0, may occur during the reheating process, where the EoS parameter of the matter field ( ω=p/ρ ) is larger than 1 / 3. Then, one may argue that for such a reheating era, where ω> 1 / 3, there exists no such entropy (of the horizon) that connects the FLRW Equation (6) with the thermodynamic law (1). In order to resolve this issue, the following modified thermodynamic law was proposed in the context of cosmology [19]: ThdS(m) h=−dE +ρdV, (17) where Th is shown in Equation (2) and the superfix ‘ m ’ stands for the ‘modified’ thermodynamic law. Such a modified thermodynamic law, indeed, resolves the aforementioned problem [ 19 ] and, thus, is considered to be more general than the previous law (1) which is a limiting case of the modified thermodynamics for p=−ρ. This modified thermodynamic law surely affects the horizon entropy compared to that of the previous law. For instance, • In the case of (n+ 1 ) dimensional Einstein gravity, the modified thermodynamic law (17) leads to the corresponding horizon entropy, which is expressed as S(m) h=n(n−1) 4ΩnZRhn−2 1+˙ H 2H2 dRh, (18) which, for a constant EoS parameter for the matter field, i.e., for a constant of ω=p/ρ , results in the following form: S(m) h(constant ω) = A |4−n−nω|. (19) Equation (19) clearly indicates that S(m) h explicitly depends on the value of ω . Therefore, in this modified thermodynamic law, the form of entropy corresponding to Einstein’s gravity changes with the evolution era of the universe. This is unlike the previous case, where the horizon entropy for Einstein’s gravity is given by A/ 4, which does not change in its form with the evolution of the universe. Universe 2024,10, 352 6 of 19 S(m) h(constant ω) = A 4during inflation when ω=−1, S(m) h(constant ω) = A |4−n|, during matter-dominated era when ω=0, S(m) h(constant ω) = A 4|1−n/3|, during radiation era when ω=1/3, • For (n+ 1 ) dimensional GB gravity theory, the required entropy corresponding to (17) is expressed as S(m) h=A 41+2α Rh2n−1 n−3, (20) when H=constant, and S(m) h=1 |1−1/(2h0)|A 41+2α Rh2n−1 n−3 (21) when H=h0/t, with A=nΩnRhn−1representing the area of the apparent horizon. • For a general modified theory of gravity, the corresponding horizon entropy coming from the modified thermodynamic law (17) is obtained as S(m) h=n(n−1)Ωn 4ZRhn−2 1+˙ H 2H2 dRh+2πnΩnZRhn−2  ρc+pc ˙ H1+˙ H 2H2  dRh, (22) The above expressions of horizon entropy (for different gravity theories) arising from the modified thermodynamic law (17) are proven to exist, irrespective of whether 1+˙ H 2H2is positive or negative. 3. Second Law of Horizon Thermodynamics Until now, we have used only the first law of thermodynamics of the apparent horizon. However, in the context of horizon thermodynamics, a consistent cosmology also demands the validity of the second law of thermodynamics, i.e., the change in total entropy (which is the sum of the horizon entropy and the entropy of the matter fields) with cosmic time should be positive [20]. ˙ Sh+˙ Sm>0. (23) Equation (1) immediately yields the change in horizon entropy as ˙ Sh=8π H3(ρ+p). (24) Besides the thermodynamics of the apparent horizon governed by Equation (1), we also need to consider the thermodynamics of the matter fields. In particular, the matter fields inside the apparent horizon obey the following thermodynamic law: TmdSm=d(ρV)+pdV −µdN, (25) where Tm and Sm represent the temperature and the entropy of the matter fields, respectively; note that Tm , in general, is different than the horizon temperature. The matter fields exhibit a flux through the horizon, which is either outward or inward, depending on the background cosmic era of the universe. Owing to this flux, the matter fields behave like an open system, where µ (in Equation (25)) is the chemical potential and dN represents the change in particle number (within time dt ) inside of the horizon. Due to V=4π 3H3 , the above expression takes the following form: Universe 2024,10, 352 7 of 19 Tm˙ Sm=−4π H2(ρ+p)1+˙ H H2−µ˙ N. (26) For the purpose of ˙ N , we need to understand that the speed of the formation of the apparent horizon is different than the comoving expansion speed of the universe. Actually, the speed of the formation of the apparent horizon turns out to be vh=−˙ H/H2 , while the comoving speed of the universe at a physical distance of d from an observer is expressed as vc=Hd . Therefore, vc= 1 at the apparent horizon (i.e., at d= 1 /H ). Therefore, vc>vh when the universe undergoes an accelerating era, while for a decelerating era, we have vc<vh. Hence, we calculate Vc(t+dt)−V(t+dt) = 4π 31 H−˙ H H2dt3 −4π 31 H+dt3 =4π H2(1−ϵ)dt, (27) with ϵ=−˙ H/H2 , which represents the gap between the comoving volume and the volume enclosed by the horizon. Therefore, we can write dN dt =−ρ u d dt[Vc(t+dt)−V(t+dt)], (28) where u is the energy per particle. The above expression, along with Equation (27), immediately results in µ˙ N=−4πρ H2(1−ϵ). (29) To arrive at Equation (29), we used µ≡∂ ∂N(total energy)=u . Plugging this into Equation (26) yields Tm˙ Sm=−4π H2(ρ+p)1+˙ H H2+4πρ H2(1−ϵ). (30) Equations (24) and (30) provide the change in horizon entropy, as well the change in matter fields’ entropy (with respect to the cosmic time). These immediately determine the change in total entropy as ThdSh dt +TmdSm dt =−2π(ρ+p)˙ H H4+4πρ H2(1−ϵ), (31) We can eliminate ρ and p from the above expression by using the Friedmann Equations (4) and (5). As a result, we obtain ThdSh dt +TmdSm dt =ϵ2 2G∂Sh ∂S−3(ϵ−1) 2G 1 H2Z∂Sh ∂SdH2. (32) Equation (32) indicates that the total change in entropy depends on the following two factors: (a) the background cosmic evolution of the universe (through the Hubble parameter) and (b) the form of the horizon entropy under consideration (through Sh ). Below, we consider some specific forms of horizon entropy and establish the constraints on the corresponding entropic parameters in order to validate the second law of horizon thermodynamics during a wide range of cosmic eras. Universe 2024,10, 352 8 of 19 • For Tsallis entropy ( Sh≡ST=Sδ , where the suffix ‘T’ stands for Tsallis entropy and S=π GH2 is the Bekenstein–Hawking entropy), the change in total entropy from Equation (32) is expressed as ThdST dt +TmdSm dt =δ 2Gπ GH2δ−1ϵ2−3(ϵ−1) (2−δ). (33) 1. During inflation , ϵ≃ 0; thus, in order to have in order for Th˙ ST+Tm˙ Sm> 0 from Equation (33), the Tsallis exponent has to fulfill the following condition: 0<δ<2. (34) 2. During the reheating stage , ϵ=3 2(1+ω0) , where ω0 is the effective EoS parameter; thus, Equation (33) leads to the following constraint on δ to satisfy Th˙ ST+Tm˙ Sm> 0, as the EoS parameter may vary within the range of ω0= [ 0,1 ] : 0<δ<5 4. (35) 3. During the radiation era , the changes in the matter fields’ entropy and the horizon entropy are expressed as ˙ Sm∝3 a3H2(ϵ−1), (36) and ˙ ST=4π GH δ 2−δπ GH2δ−1, (37) respectively. Clearly, ˙ Sm> 0, as during the radiation era, ϵ is larger than unity; moreover, the positivity of ˙ STleads to 0<δ<2. (38) Because δ remains constant with the cosmic expansion of the universe, all the above constraints on δ during different cosmic eras are simultaneously fulfilled if the following condition is met: 0<δ<Min 2, 5 4,2=5 4. (39) Here, it can be noted that such a range of δ also covers the case of Bekenstein–Hawking entropy, where δ= 1, i.e., Bekenstein–Hawking entropy also fulfills the requirement of the second law of horizon thermodynamics. Following the above procedure, one can determine the constraints on entropic parameters for other forms of the horizon entropy. Here, we present a list for several forms of Sh[20]: • For Rényi entropy, Sh≡SR=1 αln (1+αS) (with α being the parameter), and the constraint on the Rényi exponent, from an inflationto radiation-dominated era, followed by a reheating stage, is expressed as follows: α>GH2 I π, (40) where HIis the Hubble scale during inflation. Universe 2024,10, 352 9 of 19 • For Kaniadakis entropy, Sh≡SK=1 Ksinh(KS) ,and the second law of horizon thermodynamics is fulfilled from the inflation → reheating → radiation era if the Kaniadakis exponent obeys the following constraint: −1.4 GH2 I π!≲K≲1.4 GH2 I π!. (41) • The four-parameter generalised entropy given by Sh≡Sg[α+,α−,β,γ]=1 γ"1+α+ βSβ −1+α− βS−β#, (42) is the minimal version of generalised entropy that is able to generalise all the known entropies proposed so far. The parameters should lie within the following constraints in order to validate the second law of horizon thermodynamics: α± β>GH2 I π, 0 <β<5 4and γ>0. (43) Importantly, the above ranges provide model-independent constraints on entropic parameters (for different entropy functions of the apparent horizon) directly from the second law of horizon thermodynamics during a wide range of cosmic eras of the universe. 4. Generalised Entropy Functions As mentioned in Equations (4) and (5), different forms of horizon entropy ( Sh ) lead to different cosmological scenarios. In this regard, several entropies have been proposed, like Tsallis entropy [ 40 ], Rényi entropy [ 41 ], Barrow entropy [ 42 ], Sharma–Mittal entropy [ 43 ], Kaniadakis entropy [ 44 ], loop quantum gravity entropy [ 45 ], etc. However, irrespective of the form, Shmaintains some common properties, including the following: •Sh is a monotonically increasing function of the Bekenstein–Hawking entropy variable (S=A/(4G), where A=4πR2 hdenotes the area of the apparent horizon); •Sh goes to zero in the limit of S→ 0, which can be thought of as equivalent to the third law of thermodynamics. Such common properties indicate that there should exist some generalised form of entropy (with few parameters) that can generalise all the known entropies proposed so far at suitable representatives of the entropic parameters. Motivated by this idea, a few parameter-dependent generalised entropy functions (both singular and on-singular) have been proposed, which are able to generalise known entropies, like Tsallis entropy, Rényi entropy, Barrow entropy, Sharma–Mittal entropy, Kaniadakis entropy and loop quantum gravity entropy. Initially, six-parameter and three-parameter generalised entropies were proposed in [23], taking the forms of S6(α±,β±,γ±) = 1 α++α−"1+α+ β+Sγ+β+ −1+α− β−Sγ−−β−#, (44) and S3(α,β,γ) = 1 γ"1+α βSβ −1#, (45) respectively, where the respective entropic parameters are shown in brackets. However, after the introduction of this proposal, it was soon realized that the minimum number of parameters required in a generalised entropy function that can generalise all the aforemen- Universe 2024,10, 352 16 of 19 2. Quasi-matter bounce is described by the following scale factor: a(t) = "1+a0t t02#n , (74) where n , a0 and t0 are connected to the entropic parameters. In particular, one may take n=√α , a0=π 4β and t0=√G/ϵ , with G being the gravitational constant. Consequently, γ(N)that leads to such quasi-matter bounce is expressed as γ(N) = 1+1 ϵtanhe−N/√αeN/√α−11 2β −1+1 ϵtanhe−N/√αeN/√α−11 2−β . (75) At this stage, it deserves mentioning that the comoving Hubble radius in the case of exponential bounce monotonically decreases with time and asymptotically goes to zero at both sides of the bounce. This results in the fact that the primordial perturbation modes are generated near the bounce where all the modes lie in the sub-Hubble regime; moreover, the perturbation modes in the distant past remain outside of the Hubble radius. As a result, exponential bounce suffers from a horizon problem. This is unlike quasi-matter bounce, where the comoving Hubble radius monotonically increases with cosmic time and eventually diverges in the asymptotic regime. Thus, the perturbation modes generate and lie in the sub-Hubble regime in the distant past, far away from the bounce. This resolves the horizon problem in this case. Moreover, quasi-matter bounce also leads to viable observable quantities consistent with the Planck data [25]. 7. Brief Discussion on Future Perspectives In Section 2, we discussed how a gravity theory is linked with a specific form of horizon entropy through the first law of horizon thermodynamics. Therefore, it is important to understand the gravity theory corresponding to four-parameter generalised entropy. This is important because four-parameter generalised entropy is able to generalise all the known entropies; thus, the theory of gravity equivalent to such a generalised entropy must have rich consequences in the context of cosmology, as well as in black hole physics. It is evident from Section 5that the theoretical expectation of GW spectra based on four-parameter generalised entropy does not intersect with the sensitivity curve of NANOGrav. This may indicate that the standard inflationary evolution may not be the full story of the early universe. Thus, a modified inflationary evolution—for instance, a short deceleration epoch inside inflation—may be required to corroborate the theoretical GW spectra with NANOGrav data. The aspect of the generalised entropy in such a modified cosmic evolution and its consistency with NANOGrav data have significance in their own right. Apart from the early universe scenario, there are few unsolvable issues of entropic cosmology yet to be solved in the context of the dark energy era. For instance, (a) Can we obtain a viable reason from entropic cosmology regarding the cosmic transition of the universe from standard deceleration to late-time acceleration?; (b) What is the view of entropic cosmology on the Hubble tension issue, as well as on the LCDM epoch?, etc. Moreover, with development of entropic cosmology, black holes and compact objects remain to be studied, which are obtained with entropic modification of FLRW equations. These issues are timely and will, perhaps, be studied in the future. Author Contributions: Conceptualization, S.D.O. and T.P.; methodology, T.P.; validation, S.D.O., T.P. and S.N.; formal analysis, T.P.; investigation, S.D.O. and T.P.; writing—original draft preparation, T.P.; writing—review and editing, S.D.O., T.P. and S.N.; supervision, S.D.O. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. 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