Barrow Entropy Corrections to Friedmann Equations
aa r X i v : . [ g r- q c ] F e b Barrow Entropy Corrections to Friedmann Equations
Ahmad Sheykhi ∗ Department of Physics, College of Sciences, Shiraz University, Shiraz 71454, IranBiruni Observatory, College of Sciences, Shiraz University, Shiraz 71454, Iran
Inspired by the Covid-19 virus structure, Barrow argued that quantum-gravitational effects mayintroduce intricate, fractal features on the black hole horizon [Phys. Lett. B (2020) 135643].In this viewpoint, black hole entropy no longer obeys the area law and instead it can be given by S ∼ A δ/ , where the exponent δ ranges 0 ≤ δ ≤
1, and indicates the amount of the quantum-gravitational deformation effects. Based on this, and using the deep connection between gravity andthermodynamics, we disclose the effects of the Barrow entropy on the cosmological equations. Forthis purpose, we start from the first law of thermodynamics, dE = T dS + W dV , on the apparenthorizon of the Friedmann-Robertson-Walker (FRW) Universe, and derive the corresponding modifiedFriedmann equations by assuming the entropy associated with the apparent horizon has the form ofBarrow entropy. We also examine the validity of the generalized second law of thermodynamics forthe Universe enclosed by the apparent horizon. Finally, we employ the emergence scenario of gravityand extract the modified Friedmann equation in the presence of Barrow entropy which coincide withone obtained from the first law of thermodynamics. When δ = 0, the results of standard cosmologyare deduced. I. INTRODUCTION
The fast spreading of Covid-19 virus around the worldin 2020 and its continuation until now in 2021 providestrong motivations for many scientists to consider thestructure of this virus from different perspectives. In-spired by the fractal illustrations of this virus, recentlyBarrow proposed a new structure for the horizon geom-etry of black holes [1]. Assuming a infinite diminishinghierarchy of touching spheres around the event horizon,he suggested that black hole horizon might have intricategeometry down to arbitrary small scales. This fractalstructure for the horizon geometry, leads to finite volumeand infinite (or finite) area. Based on this modificationto the area of the horizon, the entropy of the black holesno longer obeys the area law and will be increased due tothe possible quantum-gravitational effects of spacetimefoam. The modified entropy of the black hole is given by[1] S h = (cid:18) AA (cid:19) δ/ , (1)where A is the black hole horizon area and A is thePlanck area. The exponent δ ranges as 0 ≤ δ ≤ δ = 0, the area law is restoredand A → G , while δ = 1 represents the most intricateand fractal structure of the horizon. Although the cor-rected entropy expression (1) resembles Tsallis entropyin non-extensive statistical thermodynamics [2–4], how-ever, the origin and motivation of the correction, as wellas the physical principles are completely different. Someefforts have been done to disclose the influences of Barrow ∗ [email protected] entropy in the cosmological setups. A holographic darkenergy model based on entropy (1) was formulated in [5].It was argued that this scenario can describe the historyof the Universe, with the sequence of matter and darkenergy eras. Observational constraints on Barrow holo-graphic dark energy were performed in [6]. A modifiedcosmological scenarios based on Barrow entropy was pre-sented in [7] which modifies the cosmological field equa-tions in such a way that contain new extra terms actingas the role of an effective dark-energy sector. The gener-alized second law of thermodynamics, when the entropyof the Universe is in the form of Barrow entropy, was in-vestigated in [8]. Other cosmological consequences of theBarrow entropy can be followed in [9–16].The profound connection between gravitational fieldequations and laws of thermodynamics has now been wellestablished (see e.g. [17–29] and references therein). Ithas been confirmed that gravity has a thermodynamicalpredisposition and the Einstein field equation of generalrelativity is just an equation of state for the spacetime.Considering the spacetime as a thermodynamic system,the laws of thermodynamics on the large scales, can betranslated as the laws of gravity. According to “gravity-thermodynamics” conjecture, one can rewrite the Fried-mann equations in the form of the first law of themody-namics on the apparent horizon and vice versa [30–33].In line with studies to understand the nature of grav-ity, Padmanabhan [34] argued that the spacial expansionof our Universe can be understood as the consequenceof emergence of space. Equating the difference betweenthe number of degrees of freedom in the bulk and onthe boundary with the volume change, he extracted theFriedmann equation describing the evolution of the Uni-verse [34]. The idea of emergence spacetime was alsoextended to Gauss-Bonnet, Lovelock and braneworld sce-narios [35–39].In the present work, we are going to construct the cos-mological field equations of FRW universe with any spe-cial curvature, when the entropy associated with the ap-parent horizon is in the form of (1). Our work differs from[7] in that the author of [7] derived the modified Fried-mann equations by applying the first law of thermody-namics, T dS = − dE , to the apparent horizon of a FRWuniverse with the assumption that the entropy is given by(1). Note that − dE in [7] is just the energy flux crossingthe apparent horizon, and the apparent horizon radiusis kept fixed during an infinitesimal internal of time dt .However, in the present work, we assume the first lawof thermodynamics on the apparent horizon in the form, dE = T dS + W dV , where dE is now the change in theenergy inside the apparent horizon due to the volumechange dV of the expanding Universe. This is consistentwith the fact that in thermodynamics the work is donewhen the volume of the system is changed. Besides, in[7], the author focuses on a flat FRW universe and mod-ifies the total energy density in the Friedmann equationsby considering the contribution of the Barrow entropy inthe field equations, as a dark-energy component. Here,we consider the FRW Universe with any special curva-ture and modify the geometry (gravity) part (left handside) of the cosmological field equations based on Barrowentropy. The approach we present here is more reason-able, since the entropy expression basically depends onthe geometry (gravity). For example, the entropy ex-pressions differ in Einstein, Gauss-Bonnet or f ( R ) grav-ities. Any modifications to the entropy should changethe gravity (geometry) sector of the field equations andvice versa. We shall also employ the emergence idea of[34] to derive the modified cosmological equations basedon Barrow entropy. Again, we assume the energy den-sity (and hence the number of degrees of freedom in thebulk) is not affected by the Barrow entropy, while thehorizon area (and hence the number of degrees of free-dom on the boundary) get modified due to the changein the entropy expression. Throughout this paper we set κ B = 1 = c = ~ , for simplicity.This paper is outlined as follows. In the next section,we derive the modified Friedmann equations, based onBarrow entropy, by applying the first law of thermody-namics of the apparent horizon. In section III, we exam-ine the validity of the generalized second law of thermo-dynamics for the universe filled with Barrow entropy. Insection IV, we derive the modified Friedmann equationsby applying the emergence scenario for the cosmic space.We finish with conclusions in the last section. II. MODIFIED FRIEDMAN EQUATIONSBASED ON BARROW ENTROPY
Our starting point is a spatially homogeneous andisotropic universe with metric ds = h µν dx µ dx ν + ˜ r ( dθ + sin θdφ ) , (2)where ˜ r = a ( t ) r , x = t, x = r , and h µν =diag( − , a / (1 − kr )) represents the two dimensional met- ric. The open, flat, and closed universes corresponds to k = 0 , , −
1, respectively. The boundary of the Universeis assumed to be the apparent horizon with radius [37]˜ r A = 1 p H + k/a . (3)From the thermodynamical viewpoint the apparent hori-zon is a suitable horizon consistent with first and secondlaw of thermodynamics. Also, using the definition of thesurface gravity, κ , on the apparent horizon [31], we canassociate with the apparent horizon a temperature whichis defied as [31, 37] T h = κ π = − π ˜ r A (cid:18) − ˙˜ r A H ˜ r A (cid:19) . (4)For ˙˜ r A ≤ H ˜ r A , the temperature becomes T ≤
0. Thenegative temperature is not physically acceptable andhence we define T = | κ | / π . Also, within an infinites-imal internal of time dt one may assume ˙˜ r A ≪ H ˜ r A ,which physically means that the apparent horizon radiusis fixed. Thus there is no volume change in it and onemay define T = 1 / (2 π ˜ r A ) [30]. The profound connectionbetween temperature on the apparent horizon and theHawking radiation has been disclosed in [40], which fur-ther confirms the existence of the temperature associatedwith the apparent horizon.The matter and energy content of the Universe is as-sumed to be in the form of perfect fluid with energy-momentum tensor T µν = ( ρ + p ) u µ u ν + pg µν , (5)where ρ and p are the energy density and pressure, re-spectively. Independent of the dynamical field equations,we propose the total energy content of the Universe satis-fies the conservation equation, namely, ∇ µ T µν = 0. Thisimplies that ˙ ρ + 3 H ( ρ + p ) = 0 , (6)where H = ˙ a/a is the Hubble parameter. Since our Uni-verse is expanding, thus we have volume change. Thework density associated with this volume change is de-fined as [41] W = − T µν h µν . (7)For FRW background with stress-energy tensor (5), thework density is calculated, W = 12 ( ρ − p ) . (8)We further assume the first law of thermodynamics onthe apparent horizon is satisfied and has the form dE = T h dS h + W dV, (9)where E = ρV is the total energy of the Universe en-closed by the apparent horizon, and T h and S h are, re-spectively, the temperature and entropy associated withthe apparent horizon. The last term in the first law isthe work term due to change in the apparent horizonradius. Comparing with the usual first law of thermo-dynamics, dE = T dS − pdV , we see that the work term − pdV is replaced by W dV , unless for a pure de Sitterspace where ρ = − p , where the work term W dV reducesto the standard − pdV .Taking differential form of the total matter and energyinside a 3-sphere of radius ˜ r A , we find dE = 4 π ˜ r A ρd ˜ r A + 4 π r A ˙ ρdt. (10)where we have assumed V = π ˜ r A is the volume en-veloped by a 3-dimensional sphere with the area of appar-ent horizon A = 4 π ˜ r A . Combining with the conservationequation (6), we arrive at dE = 4 π ˜ r A ρd ˜ r A − πH ˜ r A ( ρ + p ) dt. (11)The key assumption here is to take the entropy associatedwith the apparent horizon in the form of Barrow entropy(1). The only change needed is to replace the black holehorizon radius with the apparent horizon radius, r + → ˜ r A . If we take the differential form of the Barrow entropy(1), we get dS h = d (cid:18) AA (cid:19) δ/ = (cid:18) πA (cid:19) δ/ d (cid:0) ˜ r δA (cid:1) = (2 + δ ) (cid:18) πA (cid:19) δ/ ˜ r δA ˙˜ r A dt (12)Inserting relation (8), (11) and (12) in the first law ofthermodynamics (9) and using definition (4) for the tem-perature, after some calculations, we find the differentialform of the Friedmann equation as2 + δ πA (cid:18) πA (cid:19) δ/ d ˜ r A ˜ r − δA = H ( ρ + p ) dt. (13)Combining with the continuity equation (6), arrive at − δ πA (cid:18) πA (cid:19) δ/ d ˜ r A ˜ r − δA = 13 dρ. (14)Integration yields − δ πA (cid:18) πA (cid:19) δ/ Z d ˜ r A ˜ r − δA = ρ , (15)which results2 + δ − δ (cid:18) πA (cid:19) δ/ πA r − δA = ρ , (16) where we have set the integration constant equal to zero.The integration constant may be also regarded as the en-ergy density of the cosmological constant and hence it canbe absorbed in the total energy density ρ . Substituting˜ r A from Eq.(3) we immediately arrive at2 + δ − δ (cid:18) πA (cid:19) δ/ πA (cid:18) H + ka (cid:19) − δ/ = ρ (cid:18) H + ka (cid:19) − δ/ = 8 πG eff ρ, (18)where we have defined the effective Newtonian gravita-tional constant as G eff ≡ A (cid:18) − δ δ (cid:19) (cid:18) A π (cid:19) δ/ . (19)Equation (18) is the modified Friedmann equation basedon the Barrow entropy. Thus, starting from the first lawof thermodynamics at the apparent horizon of a FRWuniverse, and assuming that the apparent horizon areahas a fractal features, due to the quantum-gravitationaleffects, we derive the corresponding modified Friedmannequation of FRW universe with any spatial curvature. Itis important to note that in contrast to the Friedmannequation derived in [7], here the energy density ρ is notinfluenced by the Barrow entropy, while the effect of themodified entropy contributes to the geometry sector ofthe field equations. In the limiting case where δ = 0, thearea law of entropy is recovered and we have A → G .In this case, G eff → G , and Eq. (18) reduces to thestandard Friedmann equation in Einstein gravity.We can also derive the second Friedmann equation bycombining the first Friedmann equation (18) with thecontinuity equation (6). If we take the derivative of thefirst Friedmann equation (18), we arrive at2 H (cid:18) − δ (cid:19) (cid:18) ˙ H − ka (cid:19) (cid:18) H + ka (cid:19) − δ/ = 8 πG eff ρ. (20)Using the continuity equation (6), we arrive at (cid:18) − δ (cid:19) (cid:18) ˙ H − ka (cid:19) (cid:18) H + ka (cid:19) − δ/ = − πG eff ( ρ + p ) . (21)Now using the fact that ˙ H = ¨ a/a − H , and replacing ρ from the first Friedmann equation (18), we can rewritethe above equation as (cid:18) − δ (cid:19) (cid:18) ¨ aa − H − ka (cid:19) (cid:18) H + ka (cid:19) − δ/ = − πG eff p − (cid:18) H + ka (cid:19) − δ/ . (22)After some simplification and rearranging terms, we find(2 − δ ) ¨ aa (cid:18) H + ka (cid:19) − δ/ + (1 + δ ) (cid:18) H + ka (cid:19) − δ/ = − πG eff p. (23)This is the second modified Friedmann equation govern-ing the evolution of the Universe based on Barrow en-tropy. For δ = 0 ( G eff → G ), Eq. (23) reproduces thesecond Friedmann equation in standard cosmology2 ¨ aa + H + ka = − πGp. (24)If we combine the first and second modified Friedmannequations (18) and (23), we can obtain the equation forthe second time derivative of the scale factor. We find(2 − δ ) ¨ aa (cid:18) H + ka (cid:19) − δ/ = − πG eff δ ) ρ + 3 p ]= − πG eff ρ [(1 + δ ) + 3 w ] , (25)where w = p/ρ is the equation of state parameter. Takinginto account the fact that 0 ≤ δ ≤
1, the condition forthe cosmic accelerated expansion (¨ a > δ ) + 3 w < −→ w < − (1 + δ )3 . (26)When δ = 0, which corresponds to the simplest hori-zon structure with area law of entropy, we arrive at thewell-known inequality w < − / δ = 1, which implies the most intricate andfractal structure, we find w < − /
3. This implies that, inan accelerating universe, the fractal structure of the ap-parent horizon enforces the equation of state parameterto become more negative.In summary, in this section we derived the modifiedcosmological equations given by Eqs. (18) and (25) inBarrow cosmology. These equations describe the evolu-tion of the universe with any spacial curvature, when theentropy associated with the apparent horizon get modi-fied due to the quantum-gravitational effects. We leavethe cosmological consequences of the obtained Friedmannequations for future studies, and in the remanning partof this paper, we focus on the generalized second lawof thermodynamics as well as derivation of Friedmannequation (18) from emergence perspective.
III. GENERALIZED SECOND LAW OFTHERMODYNAMICS
Our aim here is to investigate another law of thermo-dynamics, when the horizon area of the Universe has afractal structure and the associated entropy is given byBarrow entropy (1). To do this, we consider the gen-eralized second law of thermodynamics for the Universe enclosed by the apparent horizon. Our approach herediffers from the one presented in [8]. Indeed the authorsof [8] modified the total energy density in the Friedmannequations based on Barrow entropy. The cosmologicalfield equations given in relations (3) and (4) of [8] arenothing but the standard Friedmann equations, in a flatuniverse, with additional energy component which acts asa dark energy sector [42]. However, as we mentioned inthe introduction, here the effects of the Barrow entropyenter the gravity (geometry) part of the cosmological fieldequations. Thus, we assume the energy component of theUniverse is not affected by the Barrow entropy. Besideswe consider the FRW universe with any special curvature,while the authors of [8] only considered a flat universe.In the context of the accelerating Universe, the general-ized second law of thermodynamics has been explored in[43–45].Combining Eq. (14) with Eq. (6) and using (19), weget 1˜ r − δA (2 − δ ) ˙˜ r A = 8 πG eff H ( ρ + p ) . (27)Solving for ˙˜ r A , we find˙˜ r A = 8 πG eff − δ H ˜ r − δA ( ρ + p ) . (28)Since δ ≤
1, thus the sign of ρ + p determines the sign of˙˜ r A . In case where the dominant energy condition holds, ρ + p ≥
0, we have ˙˜ r A ≥
0. Our next step is to calculate T h ˙ S h , T h ˙ S h = 12 π ˜ r A (cid:18) − ˙˜ r A H ˜ r A (cid:19) ddt (cid:18) AA (cid:19) δ/ = 2 + δ π (cid:18) − ˙˜ r A H ˜ r A (cid:19) (cid:18) πA (cid:19) δ/ ˜ r δA ˙˜ r A (29)Substituting ˙˜ r A from Eq. (28) and using definition (19),we reach T h ˙ S h = 4 πH ˜ r A ( ρ + p ) (cid:18) − ˙˜ r A H ˜ r A (cid:19) . (30)For an accelerating universe, the equation of state param-eter can cross the phantom line ( w = p/ρ < − ρ + p <
0. As a result, the second law of thermodynamics,˙ S h ≥
0, no longer valid. In this case, one can considerthe total entropy of the universe as, S = S h + S m , where S m is the entropy of the matter field inside the apparenthorizon. Therefore, one should study the time evolutionof the total entropy S . If the generalized second law ofthermodynamics holds, we should have ˙ S h + ˙ S m ≥
0, forthe total entropy.From the Gibbs equation we have [46] T m dS m = d ( ρV ) + pdV = V dρ + ( ρ + p ) dV, (31)where here T m stands for the temperature of the matterfields inside the apparent horizon. We further proposethe thermal system bounded by the apparent horizon re-mains in equilibrium with the matter inside the Universe.This is indeed the local equilibrium hypothesis, whichyields the temperature of the matter field inside the uni-verse must be uniform and the same as the temperatureof its boundary, T m = T h [46]. In the absence of localequilibrium hypothesis, there will be spontaneous heatflow between the horizon and the bulk fluid which is notphysically acceptable for our Universe. Thus, from theGibbs equation (31) we have T h ˙ S m = 4 π ˜ r A ˙˜ r A ( ρ + p ) − π ˜ r A H ( ρ + p ) . (32)Next, we examine the generalized second law of thermo-dynamics, namely, we study the time evolution of thetotal entropy S h + S m . Adding equations (30) and (32),we get T h ( ˙ S h + ˙ S m ) = 2 π ˜ r A ( ρ + p ) ˙˜ r A . (33)Substituting ˙˜ r A from Eq. (28) into (33) we reach T h ( ˙ S h + ˙ S m ) = 16 π − δ G eff H ˜ r − δA ( ρ + p ) ≥ , (34)which is clearly a non-negative function during the his-tory of the Universe. This confirms the validity of thegeneralized second law of thermodynamics for a universewith fractal boundary, namely when the associated en-tropy with the apparent horizon of the universe is in theform of Barrow entropy (1). IV. EMERGENCE OF MODIFIED FRIEDMANNEQUATION
In his proposal [34], Padmanabhan argued that gravityis an emergence phenomena and the cosmic space is emer-gent as the cosmic time progressed. He argued that thedifference between the number of degrees of freedom onthe holographic surface and the one in the emerged bulk,is proportional to the cosmic volume change. In this re-gards, he extracted successfully the Friedmann equationgoverning the evolution of the Universe with zero spacialcurvature [34]. In this perspective the spatial expansionof our Universe can be regarded as the consequence ofemergence of space and the cosmic space is emergent,following the progressing in the cosmic time. Accordingto Padmanabhan’s proposal, in an infinitesimal interval dt of cosmic time, the increase dV of the cosmic volume,is given by [34] dVdt = G ( N sur − N bulk ) , (35)where G is the Newtonian gravitational constant. Here N sur and N bulk stand for the number of degrees of free-dom on the boundary and in the bulk, respectively. Fol-lowing Padmanabhan, the studies were generalized to Gauss-Bonnet and Lovelock gravity [35]. While the au-thors of [35] were able to derive the Friedmann equa-tions with any spacial curvature in Einstein gravity, theyfailed to extract the Firedmann equations of a nonflatFRW universe in higher order gravity theories [35]. In[37], we modified Padmanabhan’s proposal in such a waythat it could produce the Friedmann equations in higherorder gravity theories, such as Gauss-Bonnet and Love-lock gravities, with any spacial curvature. The modifiedversion of relation (35) is given by [37] dVdt = G ˜ r A H − ( N sur − N bulk ) . (36)Comparing with the original proposal of Padmanabhanin Eq. (35), we see that in a nonflat universe, the volumeincrease is still proportional to the difference between thenumber of degrees of freedom on the apparent horizonand in the bulk, but the function of proportionality is notjust a constant, and is equal to the ratio of the apparenthorizon and Hubble radius. For flat universe, ˜ r A = H − ,and one recovers the proposal (35).Our aim here is to derive the modified Friedmann equa-tion based on Barrow entropy from emergence proposalof cosmic space. Inspired by Barrow entropy expression(1), let us define the effective area of the apparent hori-zon, which is our holographic screen, as e A = A δ/ = (cid:0) π ˜ r A (cid:1) δ/ . (37)Next, we calculate the increasing in the effective volumeas d e Vdt = ˜ r A d e Adt = 2 + δ π ˜ r A ) δ/ ˙˜ r A = 12 δ + 2 δ − π ) δ/ ˜ r A ddt (cid:0) ˜ r δ − A (cid:1) . (38)Our first key assumption here is to specify the correctexpression for the number of degrees of freedom on theapparent horizon, N sur . Motivated by (38) and following[37], we choose N sur = 4 π ˜ r δA G eff , (39)where we have used (19). We also assume the tempera-ture associated with the apparent horizon is the Hawkingtemperature, which is given by [35] T = 12 π ˜ r A , (40)and the energy contained inside the sphere with volume V = 4 π ˜ r A / E Komar = | ( ρ + 3 p ) | V. (41)Employing the equipartition law of energy, we can definethe bulk degrees of freedom as N bulk = 2 | E Komar | T . (42)In order to have N bulk >
0, we take ρ + 3 p < N bulk = − π r A ( ρ + 3 p ) , (43)The second key assumption here is to take the correctform of expression (36). To write the correct proposal, wemake replacement G → Γ − and V → e V in the proposal(36) and rewrite it asΓ d e Vdt = ˜ r A H − ( N sur − N bulk ) . (44)where Γ = 4 /A δ/ . Substituting relations (38), (39)and (43) in Eq. (44), after simplifying, we arrive at4 A δ (cid:18) πA (cid:19) δ/ ˜ r δA ˙˜ r A = ˜ r A H − " ˜ r δA G eff + 4 π r A ( ρ + 3 p ) . (45)Using definition (19), the above equation can by furthersimplified as(2 − δ )˜ r δ − A ˙˜ r A H − r δ − A = 8 πG eff ρ + 3 p ) . (46)If we multiply the both side of Eq. (46) by factor 2 ˙ aa ,after using the continuity equation (6), we arrive at ddt (cid:0) a ˜ r δ − A (cid:1) = 8 πG eff ddt ( ρa ) . (47)Integrating, yields (cid:18) H + ka (cid:19) − δ/ = 8 πG eff ρ, (48)where in the last step we have used relation (3), and setthe integration constant equal to zero. This is indeed themodified Friedmann equation derived from emergence ofcosmic space when the entropy associated with the ap-parent horizon is in the form of Barrow entropy (1).The result obtained here from the emergence approachcoincides with the obtained modified Friedmann equa-tion from the first law of thermodynamics in section II.Our study indicates that the approach presented here isenough powerful and further supports the viability of thePadmanabhan’s perspective of emergence gravity and itsmodification given by Eq. (44). V. CONCLUSIONS
Recently, and motivated by the Covid-19 virus struc-ture, Barrow proposed a new expression for the black hole entropy [1]. He demonstrated that taking into accountthe quantum-gravitational effects, may lead to intricate,fractal features of the black hole horizon. This complexstructure implies a finite volume for the black hole butwith infinite/finite area for the horizon. In this view-point, the deformed entropy associated with the blackhole horizon no longer obeys the area law and increasescompared to the area law due to fractal structure of thehorizon. The amount of increase in entropy depends onthe amount of quantum-gravitational deformation of thehorizon which is characterized by an exponent δ .Based on Barrow’s proposal for black hole entropy andassuming the entropy associated with the apparent hori-zon of the Universe has the same expression as black holeentropy, we investigated the corrections to the Firedmannequations of FRW universe, with any spacial curvature.These corrections come due to the quantum-gravitationalfractal intricate structure of the apparent horizon. Todo this, and motivated by the “gravity-thermodynamics”conjecture, we proposed the first law of thermodynamics, dE = T h dS h + W dV , holds on the apparent horizon ofFRW universe and the entropy associated with the ap-parent horizon is given by Barrow entropy (1). Startingfrom the first law of thermodynamics and taking the en-tropy in the form of Barrow entropy (1), we extractedmodified Friedmann equations describing the evolutionof the Universe. Then, we checked the validity of thegeneralized second law of thermodynamics by consider-ing the time evolution of the matter entropy togetherwith the Barrow entropy associated with the apparenthorizon. We also employed the idea of emergence grav-ity suggested by Padmanabhan [34] and calculated thenumber of degrees of freedom in the bulk and on theboundary of universe. Subtracting N sur and N bulk andusing the modification of Padmanabhan’s proposal givenin Eq. (44), we were able to extract Friedmann equationswhich is modified due to the presence of Barrow entropy.This result coincides with the one obtained from the firstlaw of thermodynamics. Our study confirms the viabilityof the emergence gravity proposed in [34, 37].Many interesting topics remain for future consider-ations. The cosmological implications of the modifiedFriedmann equations and the evolution of the Universecan be addressed. The influences of the modified Fried-mann equations on the gravitational collapse, structureformation and galaxies evolution can be investigated.The effects of the fractal parameter δ on the thermalhistory of the Universe, as well as anisotropy of CMB arealso of great interest which deserve exploration. Thesestudies belong beyond the scope of the present work andwe leave them for the future projects. Acknowledgments
I thank Shiraz University Research Council. [1] John D.Barrow,
The area of a rough black hole , Phys.Lett. B , 135643 (2020), [arXiv:2004.09444].[2] C. Tsallis, L. J. L. Cirto,
Black hole thermodynamical en-tropy , Eur. Phys. J. C , 2487 (2013), [arXiv:1202.2154].[3] M. Tavayef a, A. Sheykhi a,b, Kazuharu Bamba c, H.Moradpour, Tsallis holographic dark energy , Phys. Lett.B , 195 (2018), [arXiv:1804.02983].[4] M. A Zadeh, A Sheykhi, H Moradpour, K Bamba,
Noteon Tsallis holographic dark energy , Eur. Phys. J. C ,940 (2011), [arXiv:1806.07285].[5] E. N. Saridakis, Barrow holographic dark energy , Phys.Rev. D , 123525 (2020), [arXiv:2005.04115].[6] F. K. Anagnostopoulos, S. Basilakos, E. N. Saridakis,
Observational constraints on Barrow holographic dark en-ergy , Eur. Phys. J. C , 826 (2020), [arXiv:2005.10302].[7] E. N. Saridakis, Modified cosmology through spacetimethermodynamics and Barrow horizon entropy,
JCAP ,031 (2020), [arXiv:2006.01105].[8] E. N. Saridakis and S. Basilakos, The generalizedsecond law of thermodynamics with Barrow entropy ,[arXiv:2005.08258].[9] E. M. C. Abreu, et. al.,
Barrow’s black hole entropy andthe equipartition theorem , Eur. Phys. Lett. , 40005(2020), [arXiv:2005.11609].[10] A. A. Mamon, et. al.,
Dynamics of an Interacting BarrowHolographic Dark Energy Model and its ThermodynamicImplications , The European Physical Journal Plus ,134 (2021), [arXiv:2007.16020].[11] E. M. C. Abreu, J. A. Neto,
Barrow black hole corrected-entropy model and Tsallis nonextensivity , Phys. Lett. B , 135805 (2020), [arXiv:2009.10133].[12] J. D. Barrow, S.s Basilakos, E. N. Saridakis,
BigBang Nucleosynthesis constraints on Barrow entropy ,[arXiv:2010.00986].[13] S. Srivastava, U. K. Sharma,
Barrow holographic darkenergy with Hubble horizon as IR cutoff,
Int. J. Geom.Methods Mod. Phys. Vol. , No. 1, 2150014 (2021),[arXiv:2010.09439].[14] B. Das, B. Pandey, A study of holographic dark energymodels using configuration entropy , [arXiv:2011.07337].[15] U. K. Sharma, G. Varshney, V. C. Dubey,
Barrow age-graphic dark energy, doi 10.1142/S0218271821500218,[arXiv:2012.14291].[16] A. Pradhan, A. Dixit, V. K. Bhardwaj,
Barrow HDEmodel for Statefinder diagnostic in FLRW Universe ,[arXiv:2101.00176].[17] T. Jacobson,
Thermodynamics of Spacetime: The Ein-stein Equation of State , Phys. Rev. Lett. , 1260 (1995),[arXiv:gr-qc/9504004].[18] T. Padmanabhan, Gravity and the Thermodynam-ics of Horizons , Phys. Rept. , 49 (2005),[arXiv:gr-qc/0311036].[19] T. Padmanabhan, Thermodynamical Aspects of Grav-ity: New insights, Rept. Prog. Phys. , 046901 (2010),[arXiv:0911.5004].[20] A. Paranjape, S. Sarkar, T. Padmanabhan, Ther-modynamic route to Field equations in Lanczos-Lovelock Gravity , Phys. Rev. D , 104015 (2006),[arXiv:hep-th/0607240].[21] A. V. Frolov and L. Kofman, Inflation and deSitter Thermodynamics , JCAP , 009 (2003), [arXiv:hep-th/0212327].[22] B. Wang, E. Abdalla and R. K. Su,
Relating Fried-mann equation to Cardy formula in universes with cos-mological constant , Phys. Lett. B , 394 (2001),[arXiv:hep-th/0101073].[23] E. Verlinde,
On the Origin of Gravity and the Laws ofNewton , JHEP , 029 (2011), [arXiv:1001.0785].[24] R.G. Cai, L. M. Cao and N. Ohta,
Friedmann equationsfrom entropic force,
Phys. Rev. D , 061501 (2010),[arXiv:1001.3470].[25] R. G. Cai, L. M. Cao and Y. P. Hu, Corrected Entropy-Area Relation and Modified Friedmann Equations , JHEP , 090 (2008), [arXiv:0807.1232].[26] A. Sheykhi,
Entropic corrections to Friedmann equations ,Phys. Rev. D , 104011 (2010), [arXiv:1004.0627].[27] A. Sheykhi, Thermodynamics of apparent horizon andmodified Friedmann equations , Eur. Phys. J. C , 265(2010), [arXiv:1012.0383].[28] A. Sheykhi and S. H. Hendi, Power-law entropic cor-rections to Newton law and Friedmann equations , Phys.Rev. D , 044023 (2011), [arXiv:1011.0676].[29] S. Nojiri, S. D. Odintsov and E. N. Saridakis, Modifiedcosmology from extended entropy with varying exponent
Eur. Phys. J. C , no.3, 242 (2019), [arXiv:1903.03098[gr-qc]].[30] R. G. Cai and S. P. Kim, First Law of Ther-modynamics and Friedmann Equations of Friedmann-Robertson-Walker Universe , JHEP , 050 (2005),[arXiv:hep-th/0501055].[31] M. Akbar and R. G. Cai,
Thermodynamic behaviorof the Friedmann equation at the apparent horizon ofthe FRW universe , Phys. Rev. D , 084003 (2007),[arXiv:hep-th/0609128].[32] A. Sheykhi, B. Wang and R. G. Cai, Thermody-namical Properties of Apparent Horizon in WarpedDGP Braneworld , Nucl. Phys. B , 1 (2007),[arXiv:hep-th/0701198].[33] A. Sheykhi, B. Wang and R. G. Cai,
Deep connec-tion between thermodynamics and gravity in Gauss-Bonnet braneworlds , Phys. Rev. D , 023515 (2007),[arXiv:hep-th/0701261].[34] T. Padmanabhan, Emergence and Expansion of CosmicSpace as due to the Quest for Holographic Equipartition ,[arXiv:1206.4916].[35] R. G. Cai,
Emergence of Space and Spacetime Dynamicsof Friedmann-Robertson-Walker Universe , JHEP , 016(2012), [arXiv:1207.0622].[36] K. Yang, Y. X. Liu and Y. Q. Wang, Emergence of Cos-mic Space and the Generalized Holographic Equipartition ,Phys. Rev. D , 104013 (2012), [arXiv:1207.3515].[37] A. Sheykhi, Friedmann equations from emergence ofcosmic space , Phys. Rev. D , 061501(R) (2013),[arXiv:1304.3054].[38] A. Sheykhi, M. H. Dehghani, S. E. Hosseini, Emergence of spacetime dynamics in entropy cor-rected and braneworld models , JCAP , 038 (2013),[arXiv:1309.5774].[39] A. Sheykhi A, M. H. Dehghani, S. E. Hosseini, Fried-mann equations in braneworld scenarios from emer-gence of cosmic space , Phys. Lett. B , 23 (2013),[arXiv:1308.2668]. [40] R.G. Cai, L.M. Cao, Y.P. Hu,
Hawking Radiation of Ap-parent Horizon in a FRW Universe , Class. Quantum.Grav. Unified first law of black-hole dynamicsand relativistic thermodynamics , Class. Quant. Grav. ,3147 (1998), [arXiv:gr-qc/9710089].[42] A. Sheykhi, Thermodynamics of interacting holographicdark energy with the apparent horizon as an IRcutoff,
Class. Quantum Grav. (2010) 025007,[arXiv:0910.0510].[43] B. Wang, Y. Gong, E. Abdalla, Thermodynamics of anaccelerated expanding universe , Phys. Rev. D , 083520 (2006), [arXiv:gr-qc/0511051].[44] J. Zhou, B. Wang, Y. Gong, E. Abdalla, The generalizedsecond law of thermodynamics in the accelerating uni-verse , Phys. Lett. B , 86 (2007), [arXiv:0705.1264].[45] A. Sheykhi, B. Wang,
Generalized second law of ther-modynamics in GB braneworld,
Phys. Lett. B , 434(2009), [arXiv:0811.4478].[46] G. Izquierdo and D. Pavon,
Dark energy and the gen-eralized second law , Phys. Lett. B633