Modified Friedmann Equations from Tsallis Entropy

Physics Letters B 785 (2018) 118–126

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Physics Letters B

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Modiﬁed Friedmann equations from Tsallis entropy ∗ Ahmad Sheykhi a,b, a Physics Department and Biruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran b Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran a r t i c l e i n f o a b s t r a c t

Article history: It was shown by Tsallis and Cirto that thermodynamical entropy of a gravitational system such as black Received 8 June 2018 β hole must be generalized to the non-additive entropy, which is given by Sh = γ A , where A is the Received in revised form 2 August 2018 horizon area and β is the nonextensive parameter [1]. In this paper, by taking the entropy associated Accepted 21 August 2018 with the apparent horizon of the Friedmann–Robertson–Walker (FRW) Universe in the form of Tsallis Available online 24 August 2018 entropy, and assuming the first law of thermodynamics, dE = T dS + WdV, holds on the apparent Editor: N. Lambert h h horizon, we are able to derive the corresponding Friedmann equations describing the dynamics of the universe with any spatial curvature. We also examine the time evolution of the total entropy and show that the generalized second law of thermodynamics is fulfilled in a region enclosed by the apparent horizon. Then, modifying the emergence proposal of gravity proposed by Padmanabhan and calculating the difference between the surface degrees of freedom and the bulk degrees of freedom in a region of space, we again arrive at the modified Friedmann equation of the FRW Universe with any spatial curvature which is the same as one obtained from the first law of thermodynamics. We also study the cosmological consequences of Tsallis cosmology. Interestingly enough, we find that this model can explain simultaneously the late time acceleration in the universe filled with pressureless matter without invoking dark energy, as well as the early deceleration. Besides, the age problem can be circumvented automatically for an accelerated universe and is estimated larger than 3/2age of the universe in standard cosmology. Taking β = 2/5, we find the age of the universe ranges as 13.12 Gyr < t0 < 16.32 Gyr, which is consistent with recent observations. Finally, using the Jeans’s analysis, we comment, in brief, on the density perturbation in the context of Tsallis cosmology and found that the growth of energy differs compared to the standard cosmology. © 2018 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction portance because it confirms that the Einstein field equations is nothing but the first law of thermodynamics for the spacetime. Although gravity is the most universal forces of nature, under- Following Jacobson, a lot of studies have been carried out to dis- standing its origin has been a mystery for a long time. Einstein close the deep connection between gravity and thermodynamics believed that gravity is just the spacetime curvature and regarded [3–5]. The studies were also generalized to the cosmological se- it as an emergent phenomenon which describes the dynamics of tups [6–14], where it has been shown that the Friedmann equation spacetime. In the past decades a lot of attempts have been done to of Friedmann–Robertson–Walker (FRW) universe can be written in disclose the nature of gravity. A great step in this direction put for- the form of the first law of thermodynamics on the apparent hori- warded by Jacobson [2] who studied thermodynamics of spacetime zon. Although Jacobson’s derivation is logically clear and theoreti- and showed explicitly that Einstein’s equation of general relativ- cally sound, the statistical mechanical origin of the thermodynamic ity is just an equation of state for the spacetime. Combining the nature of general relativity remains obscure. Clausius relation δ Q = T δS, together with the entropy expression, In 2010 Verlinde [15]put forwarded the next great step to- he derived Einstein field equations. This derivation is of great im- ward understanding the nature of gravity who claimed that gravity is not a fundamental force and can be interpreted as an entropic force caused by changes of entropy associated with the informa- tion on the holographic screen. Verlinde’s proposal is based on two * Correspondence to: Physics Department and Biruni Observatory, College of Sci- ences, Shiraz University, Shiraz 71454, Iran. principles, namely the holographic principle and the equipartition E-mail address: [email protected]. law of energy. Using these principles he derived the Newton’s law https://doi.org/10.1016/j.physletb.2018.08.036 0370-2693/© 2018 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. A. Sheykhi / Physics Letters B 785 (2018) 118–126 119 of gravitation, the Poisson equation and in the relativistic regime not applicable, and large-scale gravitational systems are known to the Einstein field equations [15](see also [16]). The investigation fall within this class. Tsallis generalized standard thermodynam- on the entropic origin of gravity have been extended in different ics (which arises from the hypothesis of weak probabilistic cor- setups [17–24] and references therein). Although Verlinde’s pro- relations and their connection to ergodicity) to nonextensive one, posal has changed our understanding on the origin and nature which can be applied in all cases, and still possessing standard of gravity, but it considers the gravitational field equations as the Boltzmann–Gibbs theory as a limit. Hence, the usual Boltzmann– equations of emergent phenomenon and leave the spacetime as a Gibbs additive entropy must be generalized to the nonextensive, pre-existed background geometric manifold. A new perspective to- i.e. non-additive entropy (the entropy of the whole system is not wards emergence of spacetime dynamics was suggested in 2012 necessarily the sum of the entropies of its sub-systems), which is by Padmanabhan [25]. He argued that the spatial expansion of named Tsallis entropy [45]. our Universe can be regarded as the consequence of emergence of It is worth mentioning that in deriving Friedmann equations space and the cosmic space is emergent as the cosmic time progresses. from the first law of thermodynamics, the entropy expression as- By calculating the difference between the number of degrees of sociated with the horizon plays a crucial role [12]. Thus, it is in- freedom in the bulk and on the boundary, Padmanabhan [25]de- teresting to see how the Friedmann equation get modified if the rived the Friedmann equation of the flat FRW Universe. This pro- entropy-area relation gets corrections by some reasons. Starting posal was also applied for deriving the Friedmann equations of a from the first law of thermodynamics at apparent horizon of a higher dimensional FRW Universe in Einstein, Gauss–Bonnet and FRW universe, and assuming that the associated entropy with ap- more general Lovelock cosmology [26,27]. By modification the Pad- parent horizon has a logarithmic quantum corrected relation, the manabhan’s proposal, one may extract the Friedmann equation of modified Friedmann equations were derived in [46]. Also, taking FRW Universe with any spatial curvature not only in Einstein grav- the associated entropy with apparent horizon as the power-law- ity, but also in Gauss–Bonnet and more general Lovelock gravity corrected relation, one is able to obtain the corrected Friedmann [28]. The novel idea of Padmanabhan has got a lot of attentions in equation by using the first law of thermodynamics at the apparent the literatures [29–35]. horizon [47]. Besides, if thermodynamical interpretation of gravity It is important to note that in order to rewrite the Friedmann near apparent horizon is generic feature, one should also not only equations, in any gravity theory, in the form of the first law of check the first law of thermodynamics but also the generalized thermodynamics, dE = ThdSh + WdV, on the apparent horizon second law of thermodynamics. The latter is a universal princi- and vice versa, one should consider the entropy expression of the ple governing the evolution of the total entropy of the Universe. black hole in each gravity theory. The only change one should be In the context of the accelerating Universe, the generalized second law of thermodynamics has been explored in [48–51]. It should done is replacing the black hole horizon radius r+ by the appar- be noted that dark energy and a modified Friedmann equations ent horizon radius rÃ . However, the entropy expression associated in the context of Tsallis entropy and from different perspective, with the black hole horizon get modified from the inclusion of were first investigated in [45,52]. It was argued that Tsallis entropy quantum effects. Several types of quantum corrections to the area parameter change the strength of the gravitational constant and law have been introduced in the literatures, among them are loga- consequently the energy density of the dark components of the rithmic and power-law corrections. Logarithmic corrections, arises universe, requiring more (less) dark energy to provide the observed from the loop quantum gravity due to thermal equilibrium fluctua- late time universe acceleration [52]. They also explored some phe- tions and quantum fluctuations [36–38]. The logarithmic term also nomenological aspects as well as some observational constraints appears in a model of entropic cosmology which unifies the in- from a modified Friedmann equations induced by Tsallis entropy flation and late time acceleration [39]. Another form of correction [45,52]. In the context of the nonextensive Kaniadakis statistics to area law, namely the power-law correction, appears in dealing [53], the Jeans length was investigated and the results were com- with the entanglement of quantum fields inside and outside the pared with the Jeans length obtained in the non-extensive Tsallis horizon [40–43]. Another modification for the area law of entropy statistics [54]. Recently, modified cosmology through nonextensive comes from the Gibbs arguments who pointed out that in systems Tsallis entropy have been investigated in [55]. It was shown that with divergency in the partition function, like gravitational system, the universe exhibits the usual thermal history, with the sequence the Boltzmann–Gibbs (BG) theory cannot be applied. As a result of matter and dark energy eras, and depending on the value of thermodynamical entropy of such nonstandard systems is not de- nonextensive parameter β the equation of state of dark energy can scribed by an additive entropy but must be generalized to the even cross the phantom-line [55]. In this work, we shall derive the modified Friedmann equation in a universe which its entropy non-additive entropy [44]. Based on this, and using the statistical is given by the nonextensive Tsallis entropy. Then, we investigate arguments, Tsallis and Cirto argued that the microscopic mathe- the cosmological implications of the obtained modified Friedmann matical expression of the thermodynamical entropy of a black hole equations in the matter and radiation dominated era. In our study, does not obey the area law and can be modified as [1], we do not need to invoke the dark energy component and the early deceleration as well as the late time acceleration of the uni- S = γ Aβ , (1) h verse expansion can be achieved in the presence of radiation and where A is the black hole horizon area, γ is an unknown con- pressureless matter. Besides, we shall show that our model can stant and β known as Tsallis parameter or nonextensive parameter, solve the age problem of the universe. Our approach and the ob- which is a real parameter which quantifies the degree of nonexten- tained Friedmann equations, from the first law of thermodynamics sivity [1]. It is obvious that the area law of entropy is restored for completely differ from those investigated in [45,52,55]. = = 2 = = = ¯ β 1 and γ 1/(4L p ). Through this paper we set kB 1 c h This paper is organized as follows. In the next section, we de- for simplicity. In fact, at this limit, the power-law distribution of rive the modified Friedmann equations by applying the first law probability becomes useless, and the system is describable by the of thermodynamics, dE = ThdSh + WdV, at apparent horizon of ordinary distribution of probability [1]. a FRW universe and assuming the entropy associated with appar- Let us briefly review the concept of nonextensive, or Tsallis ent horizon is in the form of Tsallis entropy (1). In section 3, we entropy. In 1902 Gibbs pointed out that, in systems where the par- examine the generalized second law of thermodynamics for the to- tition function diverges, the standard Boltzmann–Gibbs theory is tal entropy including the corrected entropy-area relation together 120 A. Sheykhi / Physics Letters B 785 (2018) 118–126 with the matter field entropy inside the apparent horizon. In sec- horizon radius. We assume the first law of thermodynamics on the tion 4, we adopt the modified version of Padmanabhan’s proposal apparent horizon is satisfied and has the form [25]for deriving the Friedmann equation corresponding to Tsal- = + lis entropy (1). By assuming the difference between the number dE ThdSh WdV. (9) of degrees of freedom in the bulk and on the boundary is propor- It is clear that this equation is similar to the standard first law tional to the volume change of the spacetime, we find the modified of thermodynamics, unless the work term −pdV is replaced by Friedmann equation which coincides with the one obtained from WdV. For a pure de Sitter space where ρ =−p, the work term the first law of thermodynamics. In section 5, we investigate the reduces to the standard −pdV and one arrives at the standard first cosmological consequences of the modified Friedmann equations law of thermodynamics. derived from the non-extensive Tsallis entropy. The last section is We suppose the total energy content of the universe inside a devoted to conclusion and discussion. ˜ = = 4π ˜3 3-sphere of radius r A , is E ρV where V 3 r A is the volume enveloped by 3-dimensional sphere with the area of apparent hori- 2. Modified Friedman equation from the first law of zon A = 4πr˜2 . Taking differential form of the total matter and thermodynamics A energy inside the apparent horizon, we find 4 We assume the background spacetime is spatially homogeneous = ˜2 ˜ + π ˜3 ˙ dE 4πr A ρdr A r A ρdt. (10) and isotropic which is described by the line element 3 Using the continuity equation (6), we obtain 2 μ ν 2 2 2 2 ds = hμνdx dx + r˜ (dθ + sin θdφ ), (2) 2 3 dE = 4πr˜ ρdrÃ − 4π Hr˜ (ρ + p)dt. (11) 0 1 2 2 A A where r˜ = a(t)r, x = t, x = r, and hμν = diag(−1, a /(1 − kr )) represents the two dimensional metric. The open, flat, and closed We further assume the entropy associated with the apparent hori- universes corresponds to k = 0, 1, −1, respectively. We also assume zon is in the form of Tsallis entropy (1), where now A is the the physical boundary of the Universe, which is consistent with apparent horizon area of the Universe. Differentiating the modified entropy-area relation (1), we get laws of thermodynamics, is the apparent horizon with radius = β−1 = 2 β−1˜ ˜ 1 dSh γ β A dA 8πγβ(4πr A) r Adr A . (12) rÃ = . (3) H2 + k/a2 Substituting Eqs. (8), (11) and (12)in the first law (9) and using relation (5)we get the differential form of the Friedmann equation The associated temperature with the apparent horizon can be de- as fined as [6] β β−1 γ ˜2 ˜ = + ˙ 4πr A dr A H(ρ p)dt. (13) κ 1 r˜ πr˜3 = =− − A A Th ˜ 1 ˜ , (4) 2π 2πr A 2Hr A Using the continuity equation (6), we can rewrite it as β−1 ˜˙ ≤ ˜ 2 2 2π where κ is the surface gravity. For r A 2Hr A , the temperature − 4πr˜ drÃ = dρ. (14) ≤ ˜3 A 3 β becomes T 0. To avoid the negative temperature one may define r A γ T =|κ|/2π . Also, within an infinitesimal internal of time dt one ˙ Next, we integrate Eq. (14), may assume rÃ 2HrÃ , which physically means that the apparent − 2 horizon radius is kept fixed. Thus there is no volume change in it − β−1 ˜2β 5 ˜ = π 2 (4π) r A dr A ρ, (15) and one may define T = 1/(2πrÃ ) [8]. The profound connection 3γ β between temperature on the apparent horizon and the Hawking which can be written as radiation has been considered in [56], which further confirms the − 1 2π(2 β) 1−β existence of the temperature associated with the apparent horizon. = (4π) ρ, (16) ˜4−2β 3γ β We assume the matter and energy content of the Universe is in r A the form of perfect fluid with stress-energy tensor where we have set the integration constant equal to zero. Substi- tuting r˜ from Eq. (3)we obtain = + + A Tμν (ρ p)uμuν pgμν, (5) − 2 β 8 L2 2 k π p where ρ and p are the energy density and pressure, respectively. H + = ρ, (17) a2 3 The conservation of the energy-stress tensor in the FRW back- μν provided we define ground, ∇μT = 0, leads to the continuity equation as 2 − β − ˙ + + = ≡ (4 )1 β . (18) ρ 3H(ρ p) 0, (6) γ 2 π 4βL p = ˙ where H a/a is the Hubble parameter. Following [57], we define Equation (17)is nothing but the modified Friedmann equation ob- the work density as tained through the non-additive Tsallis entropy. In this way we have not only derived the modified Friedmann equation by start- 1 μν W =− T hμν. (7) ing from the first law of thermodynamics at apparent horizon of a 2 FRW universe, and assuming that the associated entropy with ap- A simple calculation gives parent horizon has a corrected relation (1), but also find a general 1 definition for the unknown constant γ in the entropy expression W = (ρ − p). (8) (1). In the limiting case where β = 1, we recover the standard 2 Friedmann equation in Einstein gravity. Besides, from Eq. (18), we The work density term is indeed the work done by the volume have always β<2 which puts an upper bound on the non-additive change of the Universe, which is due to the change in the apparent parameter of the Tsallis entropy. A. Sheykhi / Physics Letters B 785 (2018) 118–126 121

3. Generalized second law of thermodynamics entropy the total entropy namely the entropy of the boundary together with the matter entropy inside the bulk is a non decreasing Now we want to examine the validity of the generalized sec- function of time and hence the generalized second law of thermo- ond law of thermodynamics in a region enclosed by the apparent dynamics is fulfilled. horizon. Combining Eq. (14)with Eq. (6) and using relation (18), we arrive at 4. Modified Friedmann equation from emergence of cosmic space 2 ˙ 2β−2 (2 − β)r˜ r˜ = 8π L2 H(ρ + p). (19) ˜3 A A p r A According to the Padmanabhan’s proposal [25], the basic equa- ˙ tion governing the evolution of the Universe can be derived by Solving for r˜ we find A relating the emergence of space to the difference between the 4π L2 H number of degrees of freedom in the holographic surface and the ˜˙ p ˜5−2β r A = r (ρ + p). (20) one in the emerged bulk. In this viewpoint, the spatial expansion 2 − β A of our Universe can be regarded as the consequence of emergence ˙ Since β<2, thus the sign of rÃ depends on the sign of ρ + p. For of space and the cosmic space is emergent, following the progress- ρ + p > 0, which physically means the dominant energy condition ing in the cosmic time. Thus, he argued that in an infinitesimal ˜˙ ˙ holds, we have r A > 0. Let us now turn to find out Th Sh: interval dt of cosmic time, the increase dV of the cosmic volume, ˙ is given by [25] 1 rÃ d T S˙ = 1 − γ (4πr˜2 )β . (21) h h ˜ ˜ A dV 2 2πr A 2Hr A dt = L (Nsur − Nbulk), (27) dt p After some simplifications and using Eq. (20)we obtain where N is the number of degrees of freedom on the bound- sur ˙ ary and Nbulk is the number of degrees of freedom in the bulk. rÃ T S˙ = 4 Hr˜3 ( + p) 1 − . (22) Using this new idea, Padmanabhan derived the Friedmann equa- h h π A ρ ˜ 2Hr A tion of a flat FRW Universe [25]. It was argued that this proposal failed to derive the Firedmann equations of a nonflat FRW universe At present, our Universe is undergoing an acceleration phase and in other gravity theories such as Gauss–Bonnet and Lovelock grav- thus its equation of state parameter can cross the phantom line ity [26]. The modification of Padmanabhan’s proposal was done by (w D = p/ρ < −1). This implies that in an accelerating universe the dominant energy condition may violate, ρ + p < 0, implying that the present author [28], who argued that in a nonflat Universe the ˙ proposal should be generalized as the second law of thermodynamics, Sh ≥ 0, does not hold. There- fore, we should examine the validity of the generalized second law ˜ ˙ ˙ dV 2 r A of thermodynamics, namely S + S ≥ 0. = L (Nsur − Nbulk) . (28) h m dt p H−1 From the Gibbs equation we have [58] This implies that the volume increase, in a nonflat universe, is still TmdSm = d(ρV ) + pdV = Vdρ + (ρ + p)dV, (23) proportional to the difference between the number of degrees of freedom on the apparent horizon and in the bulk, but the function where Tm and Sm are, respectively, the temperature and the en- of proportionality is not just a constant, and is equal to the ratio of tropy of the matter fields inside the apparent horizon. We assume the apparent horizon and Hubble radius. Clearly, for spatially flat −1 the local equilibrium hypothesis holds. Therefore, the thermal sys- universe, rÃ = H , and one recovers the proposal (27). tem bounded by the apparent horizon remains in equilibrium so Now, we would like to see whether one can derive the modi- that the temperature of the system must be uniform and the same fied Friedmann equation from the relation (28), when the entropy as the temperature of its boundary, Tm = Th [58]. Note that if the associated with the apparent horizon get modified to Tsallis en- temperature of the fluid differs much from that of the horizon, tropy. First of all, we define the effective area of the holographic there will be spontaneous heat flow between the horizon and the surface corresponding to the entropy (1)as bulk fluid and the local equilibrium hypothesis will no longer hold. β Thus, from the Gibbs equation (23)wecan obtain = β = ˜2 A A 4πr A . (29) ˙ = ˜2 ˜˙ + − ˜3 + Th Sm 4πr Ar A (ρ p) 4πr A H(ρ p). (24) Next, we calculate the increasing in the effective volume as In order to examine the generalized second law of thermodynam- dV r˜ dA = A = ˜2 β ˜˙ + β(4πr ) r A ics, we should study the evolution of the total entropy Sh Sm. dt 2 dt A Adding equations (22) and (24), we get ˜5 r d 2β−4 = β(4π)β A r˜ . (30) A − A ˙ + ˙ = ˜2 + ˜˙ = + ˜˙ (2β 4) dt Th(Sh Sm) 2πr A (ρ p)r A (ρ p)r A , (25) 2 Inspired by (30), we propose that the number of degrees of free- ˙ where A is the apparent horizon area. Inserting rÃ from Eq. (20) dom on the apparent horizon with Tsallis entropy, is given by into (25)we find 4 β = γ ˜2 β Nsur (4πr A) . (31) 8 2 − 2 − β ˙ ˙ π 2 ˜7 2β 2 Th(Sh + Sm) = L Hr (ρ + p) . (26) 2 − β p A We also assume the temperature associated with the apparent horizon is the Hawking temperature, which is given by [8] Since β<2, the right hand side of the above equation is always a non-negative function during the universe history, which means 1 ˙ ˙ T = , (32) that Sh + Sm ≥ 0. This implies that for a universe with Tsallis 2πrÃ 122 A. Sheykhi / Physics Letters B 785 (2018) 118–126 1−β 2−β and the energy contained inside the sphere with volume V = a¨ k k (4 − 2β) H2 + + (2β − 1) H2 + ˜3 2 2 4πr A /3is the Komar energy a a a =−8π L2 p. (42) EKomar =|(ρ + 3p)|V . (33) p According to the equipartition law of energy, the bulk degrees of This is indeed the second modified Friedmann equation govern- freedom obey ing the evolution of the Universe in Tsallis cosmology which is based on the nonextensive Tsallis entropy. When β = 1, the above 2|E | Komar equation reduces to the second Friedmann equation in standard Nbulk = . (34) T cosmology,

In order to have Nbulk > 0, we take ρ + 3p < 0[25]. Thus the a¨ k number of degrees of freedom in the bulk is given by 2 + H2 + =−8π L2 p. (43) a a2 p 16 2 =− π ˜4 + Nbulk r (ρ 3p). (35) We can also obtain the equation for the second time derivative 3 A of the scale factor. To this aim, we combine the first and second 2 We also replace L p with 1/(4γ ) and V with V in proposal (28) modified Friedmann equations (39) and (42). We find and write it down as ¨ 1−β 2 ˜ a k 4π L p dV r A H2 + =− [(2β − 1)ρ + 3p] . (44) 4γ = (Nsur − Nbulk). (36) 2 − dt H−1 a a 3(2 β) Substituting relations (30), (31) and (35)in Eq. (36), after simpli- Since our Universe is currently undergoing an acceleration phase fying, we arrive at (a¨ > 0), thus from the above equation we should have ˜˙ 2 1 − 2β 2β−5 r A 2β−4 4π L p (2 − β)r˜ − r˜ = (ρ + 3p), (37) (2β − 1)ρ + 3p < 0 −→ ω < , (45) A H A 3 3 where we have also used definition (18). Multiplying the both where ω = p/ρ is the equation of state parameter. Again, for hand side of Eq. (37)by factor 2aa˙ , after using the continuity equa- β = 1the above condition for the accelerated universe leads to − tion (6), we reach the well-known inequality ω < 1/3for the equation of state in standard cosmology. A close look on the inequality (45)show that 8 L2 ≥ d 2 2β−4 π p d 2 for β 1/2, we should always have ω < 0. However, for β<1/2, a r˜ = (ρa ). (38) ≥ dt A 3 dt we can have ω 0in an accelerated universe. For example, for β = 1/3the above inequality implies ω < 1/9. This is an interest- Integrating, yields ing result, which indicates that in Tsallis cosmology, the late time 2−β 2 accelerated universe can be achieved even in the presence of the k 8π L p H2 + = ρ, (39) ordinary matter with positive equation of state parameter. Thus, a2 3 if we assume our Universe is now dominated with pressureless where we have set the integration constant equal to zero. This matter with (ω = 0), then it can undergo an accelerated expan- is nothing but the modified Friedmann equation motivated from sion provided we choose the nonextensive parameter β less than Tsallis entropy-area relation (1). We see that our result from the 1/2, without needing to an additional component of energy such emergence approach coincides with the modified Friedmann equa- as dark energy. tion derived from the first law of thermodynamics in section 2. Next, we are going to find the scalae factor as a function of time Our study indicates that the approach presented here is enough in Tsallis cosmology. For simplicity we only consider the flat uni- powerful and further supports the viability of the Padmanabhan’s verse with k = 0, although the calculations can be easily extended perspective of emergence gravity and its modification given by to the nonflat universe (k = 0). We shall consider the radiation and Eq. (36). matter dominated era, separately.

5. Tsallis cosmology 5.1. Matter-dominated era

In this section, we would like to investigate cosmological con- It is well-known that the pressure caused by random motions sequences of the modified Friedmann equation derived in Eqs. (17) of galaxies and clusters of galaxies is negligible. This is due to the and (39). Let us begin by deriving the second modified Friedmann fact that the random velocities of the galaxies are of the order of − equation in Tsallis cosmology. Taking the time derivative of the 108 cm s 1 or less [59]. Besides, the average energy density of the − − first Friedmann equation (39), we get universe is of the order of 10 30 gcm 3. Thus, we can write 1−β 2 k k 8π L p 2 −5 2 2 2H(2 − β) H˙ − H2 + = ρ˙. (40) p ∼ ρ < v >∼ 10 ρc ρc . (46) a2 a2 3 This implies that the pressure is negligible compared with ρc2. Using the continuity equation (6), we arrive at Thus if the energy density of the universe is dominated by the − k k 1 β nonrelativistic matter with negligible pressure, we can write p = 0. − ˙ − 2 + =− 2 + (2 β) H H 4π L p(ρ p) (41) In this case, the continuity equation (6)canbe written ˙(t) + a2 a2 ρ 3Hρ(t) = 0, which can be easily integrated to yield where we have also used Eq. (39)in the last step. Now using the − fact that H˙ = a¨/a − H2, after some calculations, we can rewrite the 3 ρ(t) = a(t) above equation as , (47) ρ(t0) a(t0) A. Sheykhi / Physics Letters B 785 (2018) 118–126 123

3 where t0 is the present time of the universe. Therefore, ρa = Note that 2/(3H0) is the age of the universe in standard cosmol- constant., which is exactly the same as in standard cosmology in ogy. Let us see how the above relation can alleviate the problem of the matter dominated era. Substituting ρ from (47)in the first age in standard cosmology. The Hubble constant is usually written Friedmann equation (39), we arrive at as − ˙ 4 2β 2 3 − − a 8π L pρ0a0 = 1 = × 42 = , (48) H0 100h km sec Mpc 2.1332 h 10 GeV, (60) a 3a3 where h describes the uncertainty on the value H0. The obser- which can be rewritten as vations of the Hubble Key Project constrain this value to be h = − ± = = × da 4 2β 0.72 0.08 [60]. Thus the Hubble time is tH 1/H0 9.78 1−2β 9 −1 = C1a , (49) 10 h years. Using this value for h, the age of the Universe in dt standard cosmology is in the range 8.2Gyr< t0 < 10.2Gyr [61]. where we have defined Carretta et al. [62]estimated the age of globular clusters in the Milky Way to be 12.9 ± 2.9Gyr, whereas Jimenez et al. [63]ob- 8 L2 a3 π pρ0 0 tained the value 13.5 ± 2Gyr. Hansen et al. [64] constrained the C1 ≡ . (50) 3 age of the globular cluster M4to be 12.7 ± 0.7Gyr by using the Eq. (49) has the following solution, method of the white dwarf cooling sequence. In most cases the ages of globular clusters are larger than 11 Gyr which indicates (4−2β)/3 a(t) = (C2t) , (51) that the cosmic age estimated in standard cosmology is inconsis- tent with the ages of the oldest globular clusters. It was argued where that, in standard model of cosmology, this problem cannot be cir- 1/(4−2β) cumvented unless the cosmological constant (dark energy) is taken 3 C C ≡ 1 > 0. (52) into account [61]. 2 − 2 (2 β) However, in the framework of Tsallis cosmology, the problem of In the limit of standard cosmology where β = 1, the above solution age can be circumvented automatically for an accelerated universe. restores More precisely, from relation (59)the age of the universe in Tsal- 2/3 lis cosmology is larger than 3/2age of the universe in standard 3 2/3 a(t) = C1 t . (53) cosmology, namely 2 3 Let us now calculate the second time derivative of the scale factor. t0|T > t0|S , (61) We find 2 (4−2β)/3 provided we choose β<1/2. Here subscript “T” and “S” stand for C − + a¨(t) = 2 (4 − 2β)(1 − 2β) t (2 2β)/3. (54) Tsallis and Standard cosmology, respectively. As an example, taking 9 β = 2/5, from relation (58)we find t0|T = 1.6 t0|S . Consequently, Again, we see that in the matter dominated universe, the accel- the age of the accelerated universe in Tsallis cosmology is in the | erated expansion (a¨(t) > 0) can be achieved provided we take range 13.12 Gyr < t0 T < 16.32 Gyr, which is larger than the age β<1/2, which is consistent with our previous discussion. Thus, of the oldest globular clusters. Therefore, the cosmic age problem in the framework of Tsallis cosmology, without invoking any kind can be properly alleviated in an accelerated universe in the frame- of dark energy and only with pressureless matter, the late time work of Tsallis cosmology without invoking additional component acceleration of the universe expansion can be understood. of energy. We can also calculate the evolution of the energy density, the Hubble and the deceleration parameters as 5.2. Radiation-dominated era 1 ρ(t) ∝ , (55) It is well-known that, in an expanding universe, the proper mo- t4−2β menta of freely moving particles decreases in term of scale factor a˙ 4 − 2β H(t) = = , (56) as 1/a(t). Thus, small random velocities of particles seen today a 3t should have been large in the past when the scale factor was much aa¨ 2β − 1 q(t) =− = . (57) smaller than its present value [59]. As a result, the pressureless a˙2 4 − 2β approximation is break down in the early universe. In this subsec- Again, all above parameters are reduced to the one in standard tion, we are going to find the solution for the modified Friedmann cosmology for β = 1. From the deceleration parameter, we see that equation in the framework of Tsallis cosmology by assuming that for β<1/2, we have q < 0 and a¨ > 0, a result which confirm the universe is filled with a highly relativistic gas (radiation) with = the accelerated universe. Now, we look at the Hubble parameter equation of state p ρ/3. In this case from the continuity equa- ˙ + = which can be employed for estimating the age of the universe at tion, ρ(t) 4Hρ(t) 0, we can get the present time (t = t0). We have − ρ(t) a(t) 4 4 − 2β = , (62) t0 = , (58) ρ(t1) a(t1) 3H0 where t1 is an arbitrary reference time in the radiation dominated where H = H(t ) is the Hubble constant. In an accelerated uni- 0 0 era. Thus, in this case ρa4 = constant., which is similar to the verse we have β<1/2, which implies 4 − 2β>3, and thus result obtained in the radiation dominated era in standard cosmology. Substituting ρ from (62)in the first Friedmann equation (39), 1 = 3 2 t0 > . (59) we get H0 2 3H0 124 A. Sheykhi / Physics Letters B 785 (2018) 118–126 4−2β 2 4 2 2 a˙ 8π L pρ1a vs κ = 1 , (63) f (t) = − 4π Gρ(t). (73) a 3a4 a2(t) For f (t) > 0, the solution for δ(t) is periodic and the perturbation where ρ1 = ρ(t1), a1 = a(t1) and we have again considered the flat universe (k = 0), although the k =±1cases can also be easily propagate in the medium as sound waves. For f (t) < 0, however, solved. The above equation can be rewritten as lead to imaginary frequency, indicating a growth of δ(t) with time. In what follows we are interested in the second case where f (t) < 4−2β 2 2 da vs κ −2β 0 and further assume 2 4π Gρ. In this case Eq. (72)can be = B1a , (64) a dt written 2a˙ where δ¨ + δ˙ − 4π Gρδ = 0. (74) a 8π L2 ρ a4 p 1 1 In the framework of standard model of cosmology where a t ∝ B1 ≡ . (65) ( ) 3 t2/3, Eq. (74)can be further rewritten as Eq. (64) admits a solution of the form, 4 2 δ¨ + δ˙ − δ = 0, (75) 1−β/2 3t 3t2 = 1/4 2t a(t) B1 . (66) 2/3 2 − β which has a solution in the form δ(t) = δ0t , and thus the perturbation grows with time. On the other hand, when the Friedmann When β = 1, we arrive at equations are governed by the Tsallis cosmology, the scale factor √ (4−2β)/3 1/4 in the matter dominated era is, a(t) ∝ t . On substituting in a(t) = 2B t1/2, (67) 1 Eq. (74), we arrive at which is the corresponding solution in standard cosmology. The − 4 3 4 − 2β 4 2β second time derivative of the scale factor is obtained as δ¨ + (2 − β)δ˙ − δ = 0. (76) 3t 2 3t −β/2 β 1/4 2 − + a¨(t) =− B t (1 β/2) < 0. (68) The solution of the above equation is given by 2 1 2 − β δ(t) = δ × This shows that in the radiation dominated era, the universe has 0⎛ ⎞ been in an decelerated phase (a¨(t) < 0). Now, we calculate the en- − − 2 − −2β ⎝ 5 4β 2(β 2) 4 2β β−1⎠ ergy density, the Hubble and the deceleration parameters in the Bessel J , −6 t . (77) 6β − 6 9β − 9 3 radiation dominated era. We find 1 To have an insight on the form of δ(t), let us consider a special ρ(t) ∝ , (69) t4−2β case where β = 5/4at the early stage of the universe. In this case a˙ 2 − β the expansion of solution (77)is given by H(t) = = , (70) a 2t 1/2 δ(t) = δ0 1 − α1t + α2t + ... (78) aa¨ β q(t) =− = . (71) 2 a˙ 2 − β where αi are constants. Thus, the growth of energy differs in Tsallis cosmology compared to the standard cosmology. It is important Therefore, for β<2, we have always q > 0(a¨(t) < 0), which con- to note that this study is very brief and we leave the details of firms that we have a decelerated universe in the radiation dom- investigations on the density perturbation in Tsallis cosmology for inated era. This implies that in the early stage of the universe future studies. where the relativistic particles have been dominated, our Universe has been in a decelerated phase and at the late time it undergoes 6. Conclusion and discussion an accelerated expansion. In summary, Tsallis cosmology can explain the evolution of the universe from the early deceleration to It was already pointed out by Gibbs in 1902 that in systems the late time acceleration, without invoking dark energy, which is whose partition function diverges, like gravitational system, the consistent with recent observations. Boltzmann–Gibbs (BG) theory cannot be applied. As a result, the thermodynamical entropy of such nonstandard systems is not de- 5.3. Density perturbation scribed by an additive entropy but with appropriately generalized nonadditive entropies. Based on this, and using the statistical ar- In order to study the density perturbation, we follow the guments, Tsallis and Cirto [1]argued that the microscopic mathe- method which was first introduced by James Jeans [59]. Assuming matical expression of the thermodynamical entropy of a black hole a flat FRW universe in the matter-dominated era, the fundamental does not obey the area law and can be modified as in Eq. (1). differential equation governing the fractional density perturbation Besides, it is well-known that the entropy of the whole universe, δ = ρ1/ρ in an expanding universe becomes [59] considered as a system with apparent horizon radius has a similar expression to the entropy associated with black hole horizon. The ˙ 2 2 ¨ + 2a ˙ + vs κ − = only change is needed replacing the black hole horizon radius r+ δ δ 4π Gρ δ 0, (72) ˜ a a2 by the apparent horizon radius r A of the Universe. In this paper, we have assumed the entropy associated with 2 = where vs p1/ρ1 is the speed of sound, ρ1 and p1 stand for the the apparent horizon of FRW universe is in the form of the Tsallis perturbed part of the energy density and pressure, respectively, entropy and examined its consistency with the laws of thermody- and κ is the co-moving wavevector. The solution of the above namics. For this purpose, we first assumed the first law of thermo- equation depends on the sign of the function dynamics, dE = ThdSh + WdV, holds on the apparent horizon and A. Sheykhi / Physics Letters B 785 (2018) 118–126 125 the Tsallis entropy has the form (1). Then, we showed that the first Acknowledgements law of thermodynamics on the apparent horizon can be rewritten in the form of the modified Friedmann equations of a FRW I am grateful to the referee for very constructive comments universe with any spatial curvature. We have also examined the which helped me improve the paper significantly. I also thank generalized second law of thermodynamics, by studying the time Shiraz University Research Council. 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