key: cord-0266891-plw1nn5j
authors: Sheykhi, Ahmad
title: Barrow Entropy Corrections to Friedmann Equations
date: 2021-02-11
journal: nan
DOI: 10.1103/physrevd.103.123503
sha: 21ac1f2e4890c530311f855daf5f411f6b98a653
doc_id: 266891
cord_uid: plw1nn5j

Inspired by the Covid-$19$ virus structure, Barrow argued that quantum-gravitational effects may introduce intricate, fractal features on the black hole horizon [Phys. Lett. B {bf808} (2020) 135643]. In this viewpoint, black hole entropy no longer obeys the area law and instead it can be given by $Ssim A^{1+delta/2}$, where the exponent $delta$ ranges $0leqdeltaleq1$, and indicates the amount of the quantum-gravitational deformation effects. Based on this, and using the deep connection between gravity and thermodynamics, we disclose the effects of the Barrow entropy on the cosmological equations. For this purpose, we start from the first law of thermodynamics, $dE=TdS+WdV$, on the apparent horizon of the Friedmann-Robertson-Walker (FRW) Universe, and derive the corresponding modified Friedmann equations by assuming the entropy associated with the apparent horizon has the form of Barrow entropy. We also examine the validity of the generalized second law of thermodynamics for the Universe enclosed by the apparent horizon. Finally, we employ the emergence scenario of gravity and extract the modified Friedmann equation in the presence of Barrow entropy which coincide with one obtained from the first law of thermodynamics. When $delta=0$, the results of standard cosmology are deduced.

The fast spreading of Covid-19 virus around the world in 2020 and its continuation until now in 2021 provide strong motivations for many scientists to consider the structure of this virus from different perspectives. Inspired by the fractal illustrations of this virus, recently Barrow proposed a new structure for the horizon geometry of black holes [1] . Assuming a infinite diminishing hierarchy of touching spheres around the event horizon, he suggested that black hole horizon might have intricate geometry down to arbitrary small scales. This fractal structure for the horizon geometry, leads to finite volume and infinite (or finite) area. Based on this modification to the area of the horizon, the entropy of the black holes no longer obeys the area law and will be increased due to the possible quantum-gravitational effects of spacetime foam. The modified entropy of the black hole is given by [1] 

where A is the black hole horizon area and A 0 is the Planck area. The exponent δ ranges as 0 ≤ δ ≤ 1 and stands for the amount of the quantum-gravitational deformation effects. When δ = 0, the area law is restored and A 0 → 4G, while δ = 1 represents the most intricate and fractal structure of the horizon. Although the corrected entropy expression (1) resembles Tsallis entropy in non-extensive statistical thermodynamics [2] [3] [4] , however, the origin and motivation of the correction, as well as the physical principles are completely different. Some efforts have been done to disclose the influences of Barrow * asheykhi@shirazu.ac.ir entropy in the cosmological setups. A holographic dark energy model based on entropy (1) was formulated in [5] . It was argued that this scenario can describe the history of the Universe, with the sequence of matter and dark energy eras. Observational constraints on Barrow holographic dark energy were performed in [6] . A modified cosmological scenarios based on Barrow entropy was presented in [7] which modifies the cosmological field equations in such a way that contain new extra terms acting as the role of an effective dark-energy sector. The generalized second law of thermodynamics, when the entropy of the Universe is in the form of Barrow entropy, was investigated in [8] . Other cosmological consequences of the Barrow entropy can be followed in [9] [10] [11] [12] [13] [14] [15] [16] . The profound connection between gravitational field equations and laws of thermodynamics has now been well established (see e.g. [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] and references therein). It has been confirmed that gravity has a thermodynamical predisposition and the Einstein field equation of general relativity 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 be translated as the laws of gravity. According to "gravitythermodynamics" conjecture, one can rewrite the Friedmann equations in the form of the first law of themodynamics on the apparent horizon and vice versa [30] [31] [32] [33] . In line with studies to understand the nature of gravity, Padmanabhan [34] argued that the spacial expansion of our Universe can be understood as the consequence of emergence of space. Equating the difference between the number of degrees of freedom in the bulk and on the boundary with the volume change, he extracted the Friedmann equation describing the evolution of the Universe [34] . The idea of emergence spacetime was also extended to Gauss-Bonnet, Lovelock and braneworld scenarios [35] [36] [37] [38] [39] .

In the present work, we are going to construct the cosmological field equations of FRW universe with any spe-cial curvature, when the entropy associated with the apparent horizon is in the form of (1). Our work differs from [7] in that the author of [7] derived the modified Friedmann equations by applying the first law of thermodynamics, T dS = −dE, to the apparent horizon of a FRW universe with the assumption that the entropy is given by (1) . Note that −dE in [7] is just the energy flux crossing the apparent horizon, and the apparent horizon radius is kept fixed during an infinitesimal internal of time dt. However, in the present work, we assume the first law of thermodynamics on the apparent horizon in the form, dE = T dS + W dV , where dE is now the change in the energy inside the apparent horizon due to the volume change dV of the expanding Universe. This is consistent with the fact that in thermodynamics the work is done when the volume of the system is changed. Besides, in [7] , the author focuses on a flat FRW universe and modifies the total energy density in the Friedmann equations by considering the contribution of the Barrow entropy in the field equations, as a dark-energy component. Here, we consider the FRW Universe with any special curvature and modify the geometry (gravity) part (left hand side) of the cosmological field equations based on Barrow entropy. The approach we present here is more reasonable, since the entropy expression basically depends on the geometry (gravity). For example, the entropy expressions differ in Einstein, Gauss-Bonnet or f (R) gravities. Any modifications to the entropy should change the gravity (geometry) sector of the field equations and vice versa. We shall also employ the emergence idea of [34] to derive the modified cosmological equations based on Barrow entropy. Again, we assume the energy density (and hence the number of degrees of freedom in the bulk) is not affected by the Barrow entropy, while the horizon area (and hence the number of degrees of freedom on the boundary) get modified due to the change in 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 on Barrow entropy, by applying the first law of thermodynamics of the apparent horizon. In section III, we examine the validity of the generalized second law of thermodynamics for the universe filled with Barrow entropy. In section IV, we derive the modified Friedmann equations by applying the emergence scenario for the cosmic space. We finish with conclusions in the last section.

Our starting point is a spatially homogeneous and isotropic universe with metric

wherer = a(t)r, x 0 = t, x 1 = r, and h µν =diag (−1, a 2 /(1 − kr 2 )) represents the two dimensional met-ric. The open, flat, and closed universes corresponds to k = 0, 1, −1, respectively. The boundary of the Universe is assumed to be the apparent horizon with radius [37] 

From the thermodynamical viewpoint the apparent horizon is a suitable horizon consistent with first and second law of thermodynamics. Also, using the definition of the surface gravity, κ, on the apparent horizon [31] , we can associate with the apparent horizon a temperature which is defied as [31, 37] 

Forṙ A ≤ 2Hr A , the temperature becomes T ≤ 0. The negative temperature is not physically acceptable and hence we define T = |κ|/2π. Also, within an infinitesimal internal of time dt one may assumeṙ A ≪ 2Hr A , which physically means that the apparent horizon radius is fixed. Thus there is no volume change in it and one may define T = 1/(2πr A ) [30] . The profound connection between temperature on the apparent horizon and the Hawking radiation has been disclosed in [40] , which further confirms the existence of the temperature associated with the apparent horizon. The matter and energy content of the Universe is assumed to be in the form of perfect fluid with energymomentum tensor

where ρ and p are the energy density and pressure, respectively. Independent of the dynamical field equations, we propose the total energy content of the Universe satisfies the conservation equation, namely, ∇ µ T µν = 0. This implies thatρ

where H =ȧ/a is the Hubble parameter. Since our Universe is expanding, thus we have volume change. The work density associated with this volume change is defined as [41] 

For FRW background with stress-energy tensor (5), the work density is calculated,

We further assume the first law of thermodynamics on the apparent horizon is satisfied and has the form

where E = ρV is the total energy of the Universe enclosed by the apparent horizon, and T h and S h are, respectively, the temperature and entropy associated with the apparent horizon. The last term in the first law is the work term due to change in the apparent horizon radius. Comparing with the usual first law of thermodynamics, dE = T dS − pdV , we see that the work term −pdV is replaced by W dV , unless for a pure de Sitter space where ρ = −p, where the work term W dV reduces to the standard −pdV .

Taking differential form of the total matter and energy inside a 3-sphere of radiusr A , we find

where we have assumed V = 4π 3r 3

A is the volume enveloped by a 3-dimensional sphere with the area of apparent horizon A = 4πr 2 A . Combining with the conservation equation (6), we arrive at

The key assumption here is to take the entropy associated with the apparent horizon in the form of Barrow entropy (1). The only change needed is to replace the black hole horizon radius with the apparent horizon radius, r + → r A . If we take the differential form of the Barrow entropy (1), we get

Inserting relation (8), (11) and (12) in the first law of thermodynamics (9) and using definition (4) for the temperature, after some calculations, we find the differential form of the Friedmann equation as

Combining with the continuity equation (6), arrive at

Integration yields

which results

where we have set the integration constant equal to zero. The integration constant may be also regarded as the energy density of the cosmological constant and hence it can be absorbed in the total energy density ρ. Substituting r A from Eq.(3) we immediately arrive at

The above equation can be further rewritten as

where we have defined the effective Newtonian gravitational constant as

Equation (18) is the modified Friedmann equation based on the Barrow entropy. Thus, starting from the first law of thermodynamics at the apparent horizon of a FRW universe, and assuming that the apparent horizon area has a fractal features, due to the quantum-gravitational effects, we derive the corresponding modified Friedmann equation of FRW universe with any spatial curvature. It is important to note that in contrast to the Friedmann equation derived in [7] , here the energy density ρ is not influenced by the Barrow entropy, while the effect of the modified entropy contributes to the geometry sector of the field equations. In the limiting case where δ = 0, the area law of entropy is recovered and we have A 0 → 4G. In this case, G eff → G, and Eq. (18) reduces to the standard Friedmann equation in Einstein gravity. We can also derive the second Friedmann equation by combining the first Friedmann equation (18) with the continuity equation (6) . If we take the derivative of the first Friedmann equation (18), we arrive at

(20) Using the continuity equation (6), we arrive at

Now using the fact thatḢ =ä/a − H 2 , and replacing ρ from the first Friedmann equation (18), we can rewrite the above equation as

After some simplification and rearranging terms, we find

This is the second modified Friedmann equation governing the evolution of the Universe based on Barrow entropy. For δ = 0 (G eff → G), Eq. (23) reproduces the second Friedmann equation in standard cosmology

If we combine the first and second modified Friedmann equations (18) and (23), we can obtain the equation for the second time derivative of the scale factor. We find

where w = p/ρ is the equation of state parameter. Taking into account the fact that 0 ≤ δ ≤ 1, the condition for the cosmic accelerated expansion (ä > 0), implies

When δ = 0, which corresponds to the simplest horizon structure with area law of entropy, we arrive at the well-known inequality w < −1/3 in Friedmann cosmology, while for δ = 1, which implies the most intricate and fractal structure, we find w < −2/3. This implies that, in an accelerating universe, the fractal structure of the apparent horizon enforces the equation of state parameter to become more negative. In summary, in this section we derived the modified cosmological equations given by Eqs. (18) and (25) in Barrow cosmology. These equations describe the evolution of the universe with any spacial curvature, when the entropy associated with the apparent horizon get modified due to the quantum-gravitational effects. We leave the cosmological consequences of the obtained Friedmann equations for future studies, and in the remanning part of this paper, we focus on the generalized second law of thermodynamics as well as derivation of Friedmann equation (18) from emergence perspective.

Our aim here is to investigate another law of thermodynamics, when the horizon area of the Universe has a fractal structure and the associated entropy is given by Barrow entropy (1). To do this, we consider the generalized second law of thermodynamics for the Universe enclosed by the apparent horizon. Our approach here differs from the one presented in [8] . Indeed the authors of [8] modified the total energy density in the Friedmann equations based on Barrow entropy. The cosmological field equations given in relations (3) and (4) of [8] are nothing but the standard Friedmann equations, in a flat universe, with additional energy component which acts as a dark energy sector [42] . However, as we mentioned in the introduction, here the effects of the Barrow entropy enter the gravity (geometry) part of the cosmological field equations. Thus, we assume the energy component of the Universe is not affected by the Barrow entropy. Besides we 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 generalized second law of thermodynamics has been explored in [43] [44] [45] .

Combining Eq. (14) with Eq. (6) and using (19), we get

Solving forṙ A , we finḋ

Since δ ≤ 1, thus the sign of ρ + p determines the sign oḟ r A . In case where the dominant energy condition holds, ρ + p ≥ 0, we haveṙ A ≥ 0. Our next step is to calculate T hṠh ,

Substitutingṙ A from Eq. (28) and using definition (19), we reach

For an accelerating universe, the equation of state parameter can cross the phantom line (w = p/ρ < −1), which means that the dominant energy condition may violate, ρ+p < 0. As a result, the second law of thermodynamics, S h ≥ 0, no longer valid. In this case, one can consider the total entropy of the universe as, S = S h + S m , where S m is the entropy of the matter field inside the apparent horizon. Therefore, one should study the time evolution of the total entropy S. If the generalized second law of thermodynamics holds, we should haveṠ h +Ṡ m ≥ 0, for the total entropy. From the Gibbs equation we have [46] T m dS m = d(ρV ) + pdV = V dρ + (ρ + p)dV,

where here T m stands for the temperature of the matter fields inside the apparent horizon. We further propose the thermal system bounded by the apparent horizon remains in equilibrium with the matter inside the Universe. This is indeed the local equilibrium hypothesis, which yields the temperature of the matter field inside the universe must be uniform and the same as the temperature of its boundary, T m = T h [46] . In the absence of local equilibrium hypothesis, there will be spontaneous heat flow between the horizon and the bulk fluid which is not physically acceptable for our Universe. Thus, from the Gibbs equation (31) we have

Next, we examine the generalized second law of thermodynamics, namely, we study the time evolution of the total entropy S h + S m . Adding equations (30) and (32), we get

Substitutingṙ A from Eq. (28) into (33) we reach

which is clearly a non-negative function during the history of the Universe. This confirms the validity of the generalized second law of thermodynamics for a universe with fractal boundary, namely when the associated entropy with the apparent horizon of the universe is in the form of Barrow entropy (1).

In his proposal [34] , Padmanabhan argued that gravity is an emergence phenomena and the cosmic space is emergent as the cosmic time progressed. He argued that the difference between the number of degrees of freedom on the holographic surface and the one in the emerged bulk, is proportional to the cosmic volume change. In this regards, he extracted successfully the Friedmann equation governing the evolution of the Universe with zero spacial curvature [34] . In this perspective the spatial expansion of our Universe can be regarded as the consequence of emergence of space and the cosmic space is emergent, following the progressing in the cosmic time. According to Padmanabhan's proposal, in an infinitesimal interval dt of cosmic time, the increase dV of the cosmic volume, is given by [34] 

where G is the Newtonian gravitational constant. Here N sur and N bulk stand for the number of degrees of freedom on the boundary and in the bulk, respectively. Following Padmanabhan, the studies were generalized to Gauss-Bonnet and Lovelock gravity [35] . While the authors of [35] were able to derive the Friedmann equations with any spacial curvature in Einstein gravity, they failed to extract the Firedmann equations of a nonflat FRW universe in higher order gravity theories [35] . In [37] , we modified Padmanabhan's proposal in such a way that it could produce the Friedmann equations in higher order gravity theories, such as Gauss-Bonnet and Lovelock gravities, with any spacial curvature. The modified version of relation (35) is given by [37] dV dt = Gr

Comparing with the original proposal of Padmanabhan in Eq. (35) , we see that in a nonflat universe, the volume increase is still proportional to the difference between the number of degrees of freedom on the apparent horizon and in the bulk, but the function of proportionality is not just a constant, and is equal to the ratio of the apparent horizon and Hubble radius. For flat universe,r A = H −1 , and one recovers the proposal (35) . Our aim here is to derive the modified Friedmann equation based on Barrow entropy from emergence proposal of cosmic space. Inspired by Barrow entropy expression (1), let us define the effective area of the apparent horizon, which is our holographic screen, as

Next, we calculate the increasing in the effective volume as

Our first key assumption here is to specify the correct expression for the number of degrees of freedom on the apparent horizon, N sur . Motivated by (38) and following [37] , we choose

where we have used (19) . We also assume the temperature associated with the apparent horizon is the Hawking temperature, which is given by [35] 

and the energy contained inside the sphere with volume

Employing the equipartition law of energy, we can define the bulk degrees of freedom as

In order to have N bulk > 0, we take ρ + 3p < 0 [34] . Thus the number of degrees of freedom in the bulk is obtained as

The second key assumption here is to take the correct form of expression (36) . To write the correct proposal, we make replacement G → Γ −1 and V → V in the proposal (36) and rewrite it as

where Γ = 4/A 1+δ/2 0 . Substituting relations (38) , (39) and (43) in Eq. (44) , after simplifying, we arrive at

Using definition (19) , the above equation can by further simplified as

If we multiply the both side of Eq. (46) by factor 2ȧa, after using the continuity equation (6), we arrive at

Integrating, yields

where in the last step we have used relation (3), and set the integration constant equal to zero. This is indeed the modified Friedmann equation derived from emergence of cosmic space when the entropy associated with the apparent horizon is in the form of Barrow entropy (1). The result obtained here from the emergence approach coincides with the obtained modified Friedmann equation from the first law of thermodynamics in section II. Our study indicates that the approach presented here is enough powerful and further supports the viability of the Padmanabhan's perspective of emergence gravity and its modification given by Eq. (44) .

Recently, and motivated by the Covid-19 virus structure, Barrow proposed a new expression for the black hole entropy [1] . He demonstrated that taking into account the quantum-gravitational effects, may lead to intricate, fractal features of the black hole horizon. This complex structure implies a finite volume for the black hole but with infinite/finite area for the horizon. In this viewpoint, the deformed entropy associated with the black hole horizon no longer obeys the area law and increases compared to the area law due to fractal structure of the horizon. The amount of increase in entropy depends on the amount of quantum-gravitational deformation of the horizon which is characterized by an exponent δ.

Based on Barrow's proposal for black hole entropy and assuming the entropy associated with the apparent horizon of the Universe has the same expression as black hole entropy, we investigated the corrections to the Firedmann equations of FRW universe, with any spacial curvature. These corrections come due to the quantum-gravitational fractal intricate structure of the apparent horizon. To do 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 of FRW universe and the entropy associated with the apparent horizon is given by Barrow entropy (1). Starting from the first law of thermodynamics and taking the entropy in the form of Barrow entropy (1), we extracted modified Friedmann equations describing the evolution of the Universe. Then, we checked the validity of the generalized second law of thermodynamics by considering the time evolution of the matter entropy together with the Barrow entropy associated with the apparent horizon. We also employed the idea of emergence gravity suggested by Padmanabhan [34] and calculated the number of degrees of freedom in the bulk and on the boundary of universe. Subtracting N sur and N bulk and using the modification of Padmanabhan's proposal given in Eq. (44), we were able to extract Friedmann equations which is modified due to the presence of Barrow entropy. This result coincides with the one obtained from the first law of thermodynamics. Our study confirms the viability of the emergence gravity proposed in [34, 37] .

Many interesting topics remain for future considerations. The cosmological implications of the modified Friedmann equations and the evolution of the Universe can be addressed. The influences of the modified Friedmann equations on the gravitational collapse, structure formation and galaxies evolution can be investigated. The effects of the fractal parameter δ on the thermal history of the Universe, as well as anisotropy of CMB are also of great interest which deserve exploration. These studies belong beyond the scope of the present work and we leave them for the future projects.

The area of a rough black hole

Black hole thermodynamical entropy

Tsallis holographic dark energy

Note on Tsallis holographic dark energy

Barrow holographic dark energy

Observational constraints on Barrow holographic dark energy

Modified cosmology through spacetime thermodynamics and Barrow horizon entropy

The generalized second law of thermodynamics with Barrow entropy

Barrow's black hole entropy and the equipartition theorem

Dynamics of an Interacting Barrow Holographic Dark Energy Model and its Thermodynamic Implications

Barrow black hole correctedentropy model and Tsallis nonextensivity

Big Bang Nucleosynthesis constraints on Barrow entropy

Barrow holographic dark energy with Hubble horizon as IR cutoff

A study of holographic dark energy models using configuration entropy

Barrow agegraphic dark energy

Barrow HDE model for Statefinder diagnostic in FLRW Universe

Thermodynamics of Spacetime: The Einstein Equation of State

Gravity and the Thermodynamics of Horizons

Thermodynamic route to Field equations in Lanczos-Lovelock Gravity

Inflation and de Sitter Thermodynamics

Relating Friedmann equation to Cardy formula in universes with cosmological constant

On the Origin of Gravity and the Laws of Newton

Friedmann equations from entropic force

Corrected Entropy-Area Relation and Modified Friedmann Equations

Entropic corrections to Friedmann equations

Thermodynamics of apparent horizon and modified Friedmann equations

Power-law entropic corrections to Newton law and Friedmann equations

Modified cosmology from extended entropy with varying exponent

First Law of Thermodynamics and Friedmann Equations of Friedmann-Robertson-Walker Universe

Thermodynamic behavior of the Friedmann equation at the apparent horizon of the FRW universe

Thermodynamical Properties of Apparent Horizon in Warped DGP Braneworld

Deep connection between thermodynamics and gravity in Gauss-Bonnet braneworlds

Emergence and Expansion of Cosmic Space as due to the Quest for Holographic Equipartition

Emergence of Space and Spacetime Dynamics of Friedmann-Robertson-Walker Universe

Emergence of Cosmic Space and the Generalized Holographic Equipartition

Friedmann equations from emergence of cosmic space

Emergence of spacetime dynamics in entropy corrected and braneworld models

Friedmann equations in braneworld scenarios from emergence of cosmic space

Hawking Radiation of Apparent Horizon in a FRW Universe

Unified first law of black-hole dynamics and relativistic thermodynamics

Thermodynamics of interacting holographic dark energy with the apparent horizon as an IR cutoff

Thermodynamics of an accelerated expanding universe

The generalized second law of thermodynamics in the accelerating universe

Generalized second law of thermodynamics in GB braneworld

Dark energy and the generalized second law

I thank Shiraz University Research Council.