Corrections to Friedmann equations inspired by Kaniadakis entropy

Adopting the thermodynamics-gravity conjecture, and assuming the entropy associated with the apparent horizon of the Friedmann-Robertson-Walker (FRW) universe has the form of the generalized Kaniadakis entropy, we extract the modified Friedmann equations describing the evolution of the universe using the first law of thermodynamics on the apparent horizon. We then investigate the validity of the generalized second law of thermodynamics for the universe enclosed by the apparent horizon.


I. INTRODUCTION
Inspired by the laws of black holes mechanics, the profound connection between the laws of gravity and the first law of thermodynamics was established [1].This connection is usually called thermodynamics-gravity conjecture/correspondence and is now generally accepted from theoretical point of view.The deep connection between gravity and thermodynamics has been well established at three levels.At the first level, it was argued that the field equations of gravity can be written in the form of the first law of thermodynamics on the horizon and vice versa (see [1][2][3][4][5][6][7][8][9][10][11] and references therein).At the second level, it was argued that the field equations of gravity can be derived from statistical mechanics.In this approach, by starting from the first principles, namely the holographic principle and the equipartition law of energy on the horizon degrees of freedom, one can obtain the gravitational field equations [12].This approach is known as entropic force scenario, which states that gravity is not a fundamental force and can be emerged from the change in the information of the system.This scenario has attracted a lot of attentions (see e.g.[13][14][15][16][17]).At the third level, it was argued that the spatial expansion of the universe can be understood as the consequence of the emergence of space.Thus, there is no pre-exist geometry or spacetime, and the cosmic space emerges as the cosmic time progress [18].Emergence scenario of gravity has been generalized to Gauss-Bonnet, Lovelock and braneworld frameworks [19][20][21][22][23].
In this Letter we would like to study the effects of the generalized Kaniadakis entropy on the cosmological field equations.Modified Friedmann equations based on the generalized Kaniadakis entropy was already explored in [35].Starting from the relation −dE = T dS on the ap- * asheykhi@shirazu.ac.ir parent horizon of FRW universe, the influence of the Kaniadakis entropy was explored [35].Note that here −dE is the energy flux crossing the horizon within an infinitesimal period of time dt, while T and S are, respectively, the temperature and the entropy associated with the apparent horizon [35].It was shown that the modified Friedmann equations contain new extra terms that constitute an effective dark energy sector depending on the Kaniadakis parameter K.It was also argued that the dark energy equation of state parameter deviates from standard ΛCDM cosmology at small redshifts, and remains in the phantom regime during the history of the universe [35].Our work differs from [35] in that we modify the geometry part of the cosmological field equations, and we assume the energy/matter content of the universe is not affected by the generalized Kaniadakis entropy.We believe this is more reasonable, since entropy is a geometrical quantity and any modification to it should change the geometry part of the field equations.In addition, since our universe is expanding, thus we consider the work term (due to the volume change) in the first law of thermodynamics and write it as dE = T dS + W dV .Cosmological implications of the modified Friedmann equations based on generalized Kaniadakis entropy, and its influences on the early baryogenesis and primordial Lithium abundance have been investigated in [36].Other cosmological consequences of the Kaniadakis entropy have been carried out in [37][38][39][40][41].
The structure of this Letter is as follows.In section II, we review the origin of the Kaniadakis entropy and its application to black hole physics.In section III, we start from the first law of thermodynamics and derive corrections to the Friedmann equations through the generalized Kaniadakis entropy.In section IV, we confirm the validity of the generalized second law of thermodynamics in this scenario.We finish with conclusion in the last section.

II. KANIADAKIS HORIZON ENTROPY
In this section, we review the origin and formalism of the generalized Kaniadakis entropy.Kaniadakis entropy is one-parameter entropy which generalizes the classical Boltzmann-Gibbs-Shannon entropy.It originates from a coherent and self-consistent relativistic statistical theory.The advantages of Kaniadakis entropy is that it preserves the basic features of standard statistical theory, and in the limiting case restore it [42,43].The general expression of the Kaniadakis entropy is given by [42,43] with k B is the Boltzmann constant, and Here K is called the Kaniadakis parameter which is a dimensionless parameter ranges as −1 < K < 1, and measures the deviation from standard statistical mechanics.
In the limiting case where K → 0, the standard entropy is restored.
In such a generalized statistical theory the distribution function becomes [42,43] where Let us note that the chemical potential µ can be fixed by normalization [42,43].Alternatively, Kaniadakis entropy can be expressed as [44][45][46][47][48][49]] Here P i is the probability in which the system to be in a specific microstate and W represents the total number of the system configurations.Throughout this paper we set It is also interesting to apply the Kaniadakis entropy to the black hole thermodynamics.It is well known that the entropy of the black hole, in Einstein gravity, obeys the so called Bekenstein-Hawking entropy, which states that the entropy of the black hole horizon is proportional to the area of the horizon, S BH = A/(4G).Now we assume P i = 1/W , and using the fact that Boltzmann-Gibbs entropy is S ∝ ln(W ), and S = S BH , we get W = exp [S BH ] [37].
Substituting P i = e −SBH into Eq.( 7) we arrive at When K → 0 one recovers the standard Bekenstein-Hawking entropy, S K→0 = S BH .Considering the fact that deviation from the standard Bekenstein-Hawking is small, we expect that K ≪ 1.Therefore, we can expand expression (8) as The first term is the usual area law of black hole entropy, while the second term is the leading order Kaniadakis correction term.In the next section, we shall apply the above expression to extract the modified friedmann equations.

III. CORRECTIONS TO THE FRIEDMANN EQUATIONS
Consider a spatially homogeneous and isotropic spacetime which is described by the line elements where r = a(t)r, x 0 = t, x 1 = r, and h µν =diag (−1, a 2 /(1 − kr 2 )) represents the two dimensional subspace.The parameter k denotes the spatial curvature of the universe with k = −1, 0, 1, corresponds to open, flat, and closed universes, respectively.The radius of the apparent horizon, which is a suitable horizon from thermodynamic viewpoint, is given by [10] rA = 1 The associated temperature with the apparent horizon is given by [6, 21] We also assume the matter/energy content of the universe has the form of the perfect fluid with energymomentum tensor, T µν = (ρ + p)u µ u ν + pg µν , where ρ and p are, respectively, the energy density and pressure.The conservation equation holds for the total matter and energy of the universe, namely ∇ µ T µν = 0.In the background of FRW geometry this reads ρ + 3H(ρ + p) = 0, where H = ȧ/a is the Hubble parameter.The work density associated with the volume change of the universe is defined by [50] It is a matter of calculations to show that In order to extract the Friedmann equations from thermodynamics-gravity conjecture, we assume the first law of thermodynamics, holds on the apparent horizon.The total energy of the universe enclosed by the apparent horizon is E = ρV , while T h and S h are temperature and entropy associated with the apparent horizon, respectively.One can easily show that where A is the volume enveloped by a 3dimensional sphere with the area of apparent horizon A = 4πr 2 A .Using the conservation equation, we find We assume the entropy of the apparent horizon is in the form of the generalized Kaniadakis entropy (9).In order to apply entropy (9) to the universe, we need to replace the horizon radius of the black hole with the radius of the apparent horizon, namely, r + → rA .Therefore, we write the apparent horizon entropy as where S = A/(4G) = πr 2 A /G. Taking differential form of the Kaniadakis entropy (18), we get where Inserting relations ( 14), ( 17), (19) and (20) in the first law of thermodynamics (15) and using definition (12) for the temperature, after some calculations, we find the differential form of the Friedmann equation as Using the continuity equation, we reach where we have defined Integrating Eq. ( 22), we arrive at where Λ is an integration constant which can be interpreted as the cosmological constant.Substituting rA from Eq.( 11), we arrive at where ρ Λ = Λ/(8πG).This is the modified Friedmann equation inspired by the generalized Kaniadakis entropy.When α → 0, we find the Friedmann equations in standard cosmology.We can also derive the second modified Friedmann equation by combining the first modified Friedmann equation (25) with the continuity equation.If we take the time derivative of the first Friedmann equation ( 25), after using the continuity equation, we arrive at In this way, we derive the modified Friedmann equations given by Eqs. ( 25) and ( 26) when the entropy associated with the apparent horizon is in the form of the generalized Kaniadakis entropy.Let us note that the modified Friedmann equations through Kaniadakis entropy was explored in [35].Our approach in this work has several differences with the one discussed in [35].First, the authors of [35] have modified the total energy density in the Friedmann equations.The Friedmann equations derived in [35] have the form of the standard Friedmann equations, with additional dark energy component that reflects the effects of the corrected entropy.However, in our approach the modified Kaniadakis entropy affects the geometry (gravity) part of the cosmological field equations, and the energy content of the universe does not change.From physical point of view, our approach is reasonable, since basically the entropy depends on the geometry of spacetime (gravity part of the action).As a result, any modification to the entropy should affect directly the gravity side of the dynamical field equations.Second, the authors of [35] applied the first law of thermodynamics, −dE = T dS, on the apparent horizon and obtained the modified Friedmann equations through Kaniadakis entropy.Here −dE is the energy flux crossing the apparent horizon within an infinitesimal period of time dt.While in the present work, we take the first law of thermodynamics as dE = T dS + W dV , where dE is now the change in the total energy inside the apparent horizon.Third, the authors of [35] assume the apparent horizon radius rA is fixed and consider the associated temperature as T = 1/(2πr A ), while in this work, due to the cosmic expansion, we assume the apparent horizon radius changes with time.Therefore, we include term W dV in the first law of thermodynamics (15).

IV. GENERALIZED SECOND LAW OF THERMODYNAMICS
Next we explore the validity of the generalized second law of thermodynamics when the entropy associated with the horizon is Kaniadakis entropy (18).In the context of an accelerating universe, the generalized second law of thermodynamics were explored in [51][52][53].
Combining Eq. ( 22) with continuity equation yields Solving for ṙA , we find When the dominant energy condition holds, ρ + p ≥ 0, we have ṙA ≥ 0. We then calculate T h Ṡh , In an accelerated universe one may have w = p/ρ < −1, indicating the violation of the dominant energy condition, ρ + p < 0. In this case, the inequality Ṡh ≥ 0 no longer valid and one should consider the time evolution of the total entropy, namely the entropy associated with the horizon and the matter field entropy inside the universe, S = S h + S m .The Gibbs equation implies [54] T m dS m = d(ρV where the temperature and the entropy of the matter fields inside the universe are denoted by T m and S m , respectively.We propose the boundary of the universe is in thermal equilibrium with the matter field inside the universe.This implies the temperature of both part are equal T m = T h [54].If one relax the local equilibrium hypothesis, then one should observe an energy flow between the horizon and the bulk fluid, which is not physically acceptable.From the Gibbs equation ( 30) one may write Next, we consider the time evolution of the total entropy S h + S m .Combining Eqs. ( 29) and (31), we arrive at Substituting ṙA from Eq. ( 28) into (32) we reach In summary, when the horizon entropy has the form of the generalized Kaniadakis entropy 18, the generalized second law of thermodynamics still holds for a universe enclosed by the apparent horizon.

V. CONCLUSION
It is widely accepted that there is a correspondence between the laws of gravity and the laws of thermodynamics.This connection allows to extract the field equations of gravity by starting from the first law of thermodynamics on the boundary of the system and vice versa.In this approach the entropy associated with the boundary of the system plays a crucial role.Any modification to the entropy expression modifies the field equations of gravity.
In this work, by assuming the entropy associated with the apparent horizon of the FRW universe is in the form of the generalized Kaniadakis entropy, and starting from the first law of thermodynamics, dE = T h dS h +W dV , we extracted the modified Friedmann equations describing the evolution of the universe with any spatial curvature.Since entropy is a geometrical quantity, we expect any correction to the entropy expression modifies the gravity (geometry) part of the gravitational field equations.Therefore, we keep fixed the energy content of the universe, as it is more reasonable.In the obtained Friedmann equations, the cosmological constant appears as a constant of integration.We have also explored the time evolution of the total entropy, including the entropy of the apparent horizon together with the entropy of the matter field inside the horizon.We found out that the total entropy is always a non-decreasing function of time which confirms that the generalized second law of thermodynamics holds for the universe with corrected Kaniadakis entropy.
The obtained modified Friedmann equations ( 25) and (26) provide a background to investigate a new cosmology based on Kaniadakis entropy.Many issues can be explored in this direction.One can study the cosmological implications of the modified Friedmann equations and study the evolution of the universe from early deceleration to the late time acceleration.One may also investigate the inflationary scenarios, Big Bang nucleosynthesis, as well as the growth of perturbations in this setup.Dark energy scenarios, including the holographic and agegraphic dark energy models, can be verified based on Kaniadakis modified Friedmann equations.