1 Introduction↩︎

Two main challenges of the modern cosmology are the so called dark matter puzzle and dark energy problem. The former originates from the fact that the total luminous mass of galaxies and clusters of galaxies at large scales, does not provide sufficient gravitation to explain the observed dynamics of these systems. To overcome the problem, that is, to explain the rotation curves of spiral galaxies or the dynamics of the clusters of galaxies, one needs to consider an extra component of mass, which is uniformly distrusted around the galaxies and provides the necessary gravitation. However, one may argue that the problem is due to the flaw in the Newton’s law of gravitation at large scales. In this direction, alternative theories of gravitation have been speculated and debated. Among them, the so called MOND theory [1] is widely accepted, although its theoretical origin is still doubtful. Many attempts have been done to address the theoretical origin of the MOND theory. For example, in [2], [3], the authors argued that the origin of the MOND theory can be understood from Debye entropic gravity scenario. Other studies to disclose the origin of the dark matter puzzle have been carried out in [4]–[6]. In particular, very recently, it was suggested that primordial regular black holes produced during inflation can be regarded as the source of the dark matter [7], [8].

The latter comes from the fact that the observations of type Ia supernova explosions in high redshift galaxies confirm that our universe is currently undergoing a phase of accelerated expansion. This was an unexpected discovery and shook the foundations of the modern cosmology. In the context of standard cosmology, one needs to consider an extra unknown component of energy, which is usually called in the literatures as dark energy. What we know from dark energy is that it is smoothly filled all spaces and has anti-gravity nature which push our universe to accelerate. However, there is another way to justify the acceleration of the cosmic expansion, namely taking into account alternative theories of gravity such as \(f(R)\) gravity, Gauss-Bonnet gravity, or brane cosmology, etc.

On the other side, in recent years, there has been more theoretical progress on understanding the nature of gravity. It was argued that when the spacetime, as a large scales system, is considered as a thermodynamical system, then there is a profound connection between the laws of gravity describing the spacetime geometry and the laws of thermodynamics. The correspondence between the gravitational field equations and thermodynamics has been disclosed in three levels. At the first level, it was shown by Jacobson [9] that the hyperbolic second order partial differential Einstein equation for the spacetime metric has a predisposition to thermodynamic behavior. He disclosed that the gravitational Einstein equation can be derived from the relation between the horizon area and entropy, together with the Clausius relation \(\delta Q=T\delta S\) [9]. The correspondence between the first law of thermodynamics and gravitational field equations has been extended to \(f(R)\) gravity [10], Gauss-Bonnet gravity, the scalar-tensor gravity and more general Lovelock gravity [11]–[14]. In the context of Friedmann-Robertson-Walker (FRW) cosmology it has been confirmed that the first Friedmann equation on the apparent horizon can be translated to the first law of thermodynamics, \(dE=TdS+WdV\), and vice versa [15]–[22]. The correspondence between the first law of thermodynamics on the boundary and the gravitational field equations in the bulk sheds also the light on holography. These results further support the idea that gravitation on a macroscopic scale is a manifestation of thermodynamics. At the deeper level, it was shown that gravity is not a fundamental force and can be treated as an emergence phenomenon. Starting from the first principles, namely the holographic principle and the equipartition law of energy on the horizon degrees of freedom, Verlinde [23] argued that the change in the information of the system leads to an entropic force which can be translated to the law of gravity. The entropic nature of gravity has been widely explored (see e.g. [24]–[26] and references therein). In both approaches mentioned above one considers the spacetime as a pre-exist geometry. Is it possible to regard the spacetime itself as an emergent structure? In the deepest level, Padmanabhan proposed that the spatial expansion of the universe can be understood as the consequence of the emergence of space [27]. According to Padmanabhan’s proposal the cosmic space emerges as the cosmic time progress. In this approach the most fundamental notion namely the degrees of freedom of the matter fields in the bulk and the degrees of freedom on the boundary play crucial role. Indeed, by counting the difference between degrees of freedom on the boundary and in the bulk and equating it with the volume change, one is able to construct the dynamical equations describing the evolution of the universe [28]–[30].

Independent of the approach for dealing with the thermodynamics-gravity conjecture, the entropy expression associated with the boundary of system, plays a crucial role in extracting the gravitational field equations from thermodynamic arguments. In the cosmological background, any modification to the entropy associated with the apparent horizon of FRW universe, implies a modification to the Friedmann equations which leads to a modified cosmology [31]–[41]. Some of these modified cosmological models, inspired by thermodynamics-gravity correspondence, can explain the late-time acceleration of the cosmic expansion without invoking additional component of energy [34], [35].

In the present work, we are going to construct a general form of the entropy associated with the boundary, which may simultaneously address both flat rotation curves of spiral galaxies and the late time accelerated expansion. For this purpose, we start from a general expression for the MOND theory. Using an inverse approach, as well as the entropic force scenario for the MOND theory, we reconstruct a general form of the entropy. Applying the obtained entropy expression to the cosmological setup, we are able to extract the modified Friedmann equations by using thermodynamics-gravity conjecture. We shall also check the validity of the generalized second law of thermodynamics for the universe enveloped by the apparent horizon.

This paper is structured as follows. In section 2, we use the entropic nature of gravity and start from the MOND theory to construct the general form of the entropy associated with the horizon. In section 3, we start from the first law of thermodynamics on the apparent horizon and apply the modified entropy expression to establish corrections to the Friedmann equations. Given the general form of the entropy inspired by MOND theory, in section 4, we apply the entropic force scenario to construct the modified Friedmann equations. In section 5, we use the idea of emergence gravity and reconstruct the modified Friedmann equations. In section 6, we confirm that our cosmological model guaranties the generalized second law of thermodynamics for the universe enveloped by the apparent horizon. The last section is devoted to the closing remarks.

2 Corrections to entropy inspired by MOND theory↩︎

The Modified Newtonian dynamics (MOND) suggested by Milgrom to explain the flat rotation curves of the spiral galaxies [1]. According to the MOND theory, the Newton’s second law get modified for the large scales as \[\label{F0} F=m \mu \left(\frac{a}{a_0}\right)a,\tag{1}\] where \(a\) stands for the usual kinematical acceleration, which is taken as \(a=v^{2}/R\), and \(a_{0}=1.2\pm 0.27 \times 10^{-10}\) \(m/s^{2}\) is a constant [42]. Here \(\mu(x)\) is a real function satisfies the following boundary conditions, \[\label{mu} \mu(x)\approx\left\{ \begin{array}{ll} 1 \quad \quad {\rm for} \quad\quad {x}\gg 1, & \\ x \quad \quad {\rm for} \quad\quad {x}\ll 1. \end{array} \right.\tag{2}\] At large distance, at the galaxy out skirt, the kinematical acceleration ‘\(a\)’ is extremely small, smaller than \(10^{-10}\) \(m/s^{2}\) , i.e., \(a\ll a_{0}\), hence the function \(\mu (\frac{a}{a_{0}})=\frac{a}{a_{0}}\). Mathematically, for a galaxy with mass \(M\) and a star (particle) with mass \(m\), the Newton’s law of gravity get modified as \[\label{v} F=m \frac{a^2}{a_{0}}=\frac{GMm}{R^2}, \Rightarrow v= (GM a_{0})^{1/4}\approx cte.\tag{3}\] This implies that the velocity of star, on circular orbit from the galaxy-center is constant and does not depend on the distance; the rotational-curve is flat, as it observed.

Our aim here is to ansatz a general form for \(\mu(x)\) which satisfies conditions (2 ). Among several function which may satisfy condition (2 ), and motivated with the previous studies in this direction, we propose the following function, \[\begin{align} \label{mu1} \mu(x)=x(1+x^{\alpha})^{-1/{\alpha}}, \end{align}\tag{4}\] where \(\alpha>0\). Note that the interpolation function has usually been given the following functional form [42], [43] \[\begin{align} \label{mualph2} \mu(x)=\frac{x}{\sqrt{1+x^2}}. \end{align}\tag{5}\] However, an alternative simple interpolating function, \[\begin{align} \label{mualph1} \mu(x)=\frac{x}{1+x}, \end{align}\tag{6}\] was also propped, which provides a less sudden transition from the Newtonian to the MOND regime than does the standard function [44], [45]. It is clear that the general form we proposed in (4 ) reduces to functions (5 ) and (6 ) in the limiting case where \(\alpha=1,2\). One can easily check that (4 ) satisfies both conditions (2 ). On the other hand, since \(\alpha>0\) is a free parameter, one can expand expression (4 ), and reproduce all terms in the series expansion. If one choose other types of function \(\mu(x)\), their expansions may be similar to the proposed function (4 ), by suitably choosing the free parameter \(\alpha\). This discussion may justify our ansatz for function (2 ).

We consider a system that its boundary is not infinitely extended and forms a closed surface with spherical geometry. We can take the boundary as a storage device for information, i.e. a holographic screen. We also assume at the center of the holographic screen there is a mass \(M\) and at distance \(R\), mass \(m\) is located near the screen. Using the entropic force scenario [23], and taking into account a general form for the entropy associated with the holographic screen, we can write down the Newton’s law of gravitation as (\(k_B=\hbar=c=1\)) [25] \[\begin{align} \label{F2} F&=&\frac{GMm}{R^2}\times 4G \frac{ dS_h}{dA}\mid _{A=4\pi R^2}\nonumber\\&&=4 G m a \times \frac{ dS_h}{dA}\mid _{A=4\pi R^2}, \end{align}\tag{7}\] where \(a=GM/R^2\) is the acceleration of a particle with mass \(m\) which rotates at the distance \(R\) around the central mass \(M\). Equating expression (7 ) with Eqs. (1 ), after using (4 ), we find \[\label{mu2} \mu(x)=4G \frac{ dS_h}{dA}.\tag{8}\] The key point here is to recognize in Eq. (4 ), \(x=a/a_{0}\), with \(a_{0}=\gamma \pi M\), where \(\gamma\) is a parameter which can be constrained by observation. This implies that we can write \(x=4G/(\gamma A)\), where \(A=4\pi R^2\). Clearly, \(x\) can be calculated for each galaxy or cluster of galaxies. Since \(M\) is, at least, of order of a galaxy mass and \(a_{0}\simeq 10^{-10} m/s^2\ll 1\), hence \(\gamma\ll 1\). In terms of the horizon area, \(A\), the functional form of \(\mu\) can be written as \[\label{mu3} \mu(A)=\Big{\{}1+\left(\frac{\gamma A}{4G}\right)^{\alpha}\Big{\}}^{-1/{\alpha}}.\tag{9}\] Integrating Eq. (8 ), we find the entropy associated with the horizon as \[\label{Sh2} S_h=\frac{1}{4G}\int{\mu(A)dA}=\frac{1}{4G}\int{\Big{\{}1+\left(\frac{\gamma A}{4G}\right)^{\alpha}\Big{\}}^{-1/{\alpha}}dA}.\tag{10}\] The integral can be done in terms of hypergeometric function, \(_2F_1(a,b,c,z)\), and can be written in a compact form. The result is \[\label{Sh3} S_h=\frac{A}{4G} \times \text{{\;}} _{2}F_{1}\Big{\{} \frac{1}{\alpha},\frac{1}{\alpha},\frac{\alpha+1}{\alpha}, -\left( \frac{\gamma A}{4G}\right)^{\alpha}\Big{\}}.\tag{11}\] For \(\alpha=1\), the above expression for entropy reduces to \[\label{ShRenyi} S_h=\frac{1}{\gamma}\ln \left(1+\frac{\gamma A}{4G}\right),\tag{12}\] which is the well-known Renyi entropy [46]. On the other hand, for \(\alpha=2\), expression (11 ) restores \[\label{ShKan} S_h=\frac{1}{\gamma} \ln \left(\frac{\gamma A}{4G}+\sqrt{1+\left(\frac{\gamma A}{4G}\right)^2}\right),\tag{13}\] which is a deformed (dual) version of the Kaniadakis entropy [47]. Using the fact that \(_2F_1(a,b,c,z)\) has a convergent series expansion for \(|z| <1\), we can expand expression (11 ), up to linear term in \(\eta\), as

\[\label{Sh4} S_h=\frac{A}{4G} \Big{\{}1-\eta \left(\frac{ A}{4G}\right)^{\alpha}+...\Big{\}},\tag{14}\] where \[\eta=\frac{\gamma^{\alpha}}{\alpha(\alpha+1)}\ll 1.\] Finally, let us emphasize that although in Eq. (2 ), parameter \(\alpha>0\) can be any positive number, but for simplicity, in the remaining part of this work we assume \(\alpha\) is a positive integer number, although one can relax this assumption.

3 Modified Friedmann equations from the first law of thermodynamics↩︎

We consider a spatially homogeneous and isotropic universe with line elements \[ds^2={h}_{\mu \nu}dx^{\mu} dx^{\nu}+R^2(d\theta^2+\sin^2\theta d\phi^2),\] where \(R=a(t)r\), \(x^0=t, x^1=r\), the two dimensional metric is given by \(h_{\mu \nu}\)=diag \((-1, a^2/(1-kr^2))\), and \(k = -1,0, 1\), stands for open, flat, and closed universes, respectively. The dynamical apparent horizon, a marginally trapped surface with vanishing expansion, is determined by the relation \(h^{\mu \nu}\partial_{\mu}R\partial_{\nu}R=0\), which implies that the vector \(\nabla R\) is null on the apparent horizon surface. For FRW geometry, the explicit evolution of the apparent horizon radius reads [48]–[50] \[\label{radius} R=\frac{1}{\sqrt{H^2+k/a^2}},\tag{15}\] where \(H=\dot{a}/a\) is the Hubble parameter. The apparent horizon is a suitable boundary from thermodynamic arguments [18]. The surface gravity in a dynamical background should, in principle, include contributions from the time derivative of the apparent horizon radius. In this context, the Hayward-Kodama surface gravity, defined by [48]–[50] \[\label{kappa} \kappa = \frac{1}{2\sqrt{-h}} \partial_{\mu} \left( \sqrt{-h} h^{\mu \nu} \partial_{\nu} R \right),\tag{16}\] which is indeed the most general and invariant definition for surface gravity associated with a dynamical apparent horizon. Note that surface gravity includes terms involving both the Hubble parameter and its time derivative. The temperature associated with the dynamical apparent horizon is then defined as [48]–[50] \[\label{T} T_h=\frac{\kappa}{2\pi}=-\frac{1}{2 \pi R}\left(1-\frac{\dot{R}}{2HR}\right).\tag{17}\] To avoid negative temperature one can also define \(T=|\kappa|/2\pi\). Besides, when \(\dot{R}\ll 2HR\), which physically means that the radius of the apparent horizon is almost fixed, one may define \(T=1/(2\pi R )\) [17]. We further assume the energy content of the universe is in the form of perfect fluid with energy-momentum tensor \(T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu},\) where \(\rho\) and \(p\) are the energy density and pressure, respectively. As far as we know, there is no energy exchange between our universe and out of its boundary. As a result, we can assume the total energy-momentum inside the universe is conserved, which implies \(\nabla_{\mu}T^{\mu\nu}=0\). This leads to \[\label{Cont} \dot{\rho}+3H(\rho+p)=0.\tag{18}\] In addition, due to the volume change of the universe, a work density term is also appeared as [49] \[\label{Work} W=-\frac{1}{2} T^{\mu\nu}h_{\mu\nu}=\frac{1}{2}(\rho-p).\tag{19}\] Finally, we write down the first law of thermodynamics on the apparent horizon as \[\label{FL} dE = T_h dS_h + WdV.\tag{20}\] Next, we take the total energy inside the apparent horizon as \(E=\rho V\), with \(V=\frac{4\pi}{3}R^{3}\) is the volume enveloped by a 3-dimensional sphere with the area of apparent horizon \(A=4\pi R^{2}\). We now calculate the differential form of the total energy as, \[\label{dE1} dE=4\pi R^{2}\rho dR+\frac{4\pi}{3}R^{3}\dot{\rho} dt.\tag{21}\] Using the continuity equation (18 ), we can rewrite the above equation as \[\label{dE2} dE=4\pi R^{2}\rho dR-4\pi H R^{3}(\rho+p) dt.\tag{22}\] The key point here is to take the entropy associated with the apparent horizon of FRW universe in the form of Eq. (11 ) with \(A=4\pi R^{2}\) is the area of the apparent horizon and \(R\) is the horizon radius. After differentiating, we find \[\begin{align} \label{dSh} dS_h&=& \frac{dA}{4G}\Big{\{}1+\left(\frac{\gamma A}{4G}\right)^{\alpha}\Big{\}}^{-1/{\alpha}}, \end{align}\tag{23}\] Substituting relations (19 ), (22 ) and (23 ) in the first law of thermodynamics (20 ) and using definition (17 ) for the temperature as well as the continuity equation (18 ), after a little algebra, we find the differential form of the Friedmann equation as \[\label{Fried1} -2 \left(1+\beta R^{2\alpha}\right)^{-\frac{1}{\alpha}} \frac{dR}{R^{3}}= \frac{8\pi G}{3}d\rho,\tag{24}\] where we have defined \[\beta\equiv\left(\frac{\gamma \pi}{G}\right)^{\alpha}.\] Next, we integrate Eq. (24 ). The result is \[\label{Frie2} \frac{1}{R^{2}} \times \text{{\;}} _{2}F_{1}\Big{\{} \frac{1}{\alpha},-\frac{1}{\alpha},\frac{\alpha-1}{\alpha}, -\beta R^{2\alpha}\Big{\}}=\frac{8\pi G}{3}\rho,\tag{25}\] where we have absorbed the integration constant, which can be the energy density of the cosmological constant, in the total energy density, namely, \(\rho=\rho_m+\rho_{\Lambda}\). Substituting \(R\) from Eq.(15 ), we immediately arrive at \[\begin{align} \label{Fried3} &&\left(H^2+\frac{k}{a^2}\right)\times \text{{\;}} _{2}F_{1}\Bigg{\{} \frac{1}{\alpha},-\frac{1}{\alpha},\frac{\alpha-1}{\alpha}, -\beta \left(H^2+\frac{k}{a^2}\right)^{-\alpha}\Bigg{\}}=\frac{8\pi G}{3}\rho. \end{align}\tag{26}\] In this way, we derive the general form of the modified Friedmann equation inspired by the MOND theory. The second modified Friedmann equation can be easily derived by combining Eq. (26 ) with continuity equation (18 ). Using the fact that \(_2F_1(a,b,c,z)\) has a series expansion, we can write the Friedmann Eq. (26 ) in a compact series form (see appendix for details), \[\begin{align} \label{Fried3series} \sum_{n=0}^{\infty}\frac{C_n}{n!}\left(H^2+\frac{k}{a^2}\right)^{1-\alpha n} =\frac{8\pi G}{3}\rho, \end{align}\tag{27}\] where \[\begin{align} \label{Cn} C_n=(-\beta)^n \frac{\left(\frac{1}{\alpha}\right)_n\left(\frac{-1}{\alpha}\right)_n}{\left(\frac{\alpha-1}{\alpha}\right)_n}. \end{align}\tag{28}\] Since \(\beta<1\), we can expand the hypergeometric function up to the linear term in \(\beta\). This is equivalent to consider the first and the second term in series (27 ). We find \[\label{Fried4} \left(H^2+\frac{k}{a^2}\right)-\frac{\beta}{\alpha (1-\alpha)}\left(H^2+\frac{k}{a^2}\right)^{1-\alpha} = \frac{8\pi G}{3} \rho.\tag{29}\] Clearly, the above expression is ill-defined for \(\alpha=1\). In this case, one should start from expression (12 ) for the entropy to derive the modified Friedmann equation. It is easy to show that for \(\alpha=1\), the modified Friedmann equations gets the following form \[\label{Friedalpha1} \left(H^2+\frac{k}{a^2}\right)-\frac{\gamma \pi}{G} \ln \left(H^2+\frac{k}{a^2}\right)=\frac{8\pi G}{3} \rho.\tag{30}\] This is the modified Friedmann equation corresponds to the Reyni entropy (12 ). On the other hand for \(\alpha=2\), expression (29 ) restores \[\label{Friedalpha2} \left(H^2+\frac{k}{a^2}\right)+\frac{1}{2}\left( \frac{\gamma \pi}{G}\right)^2 \left(H^2+\frac{k}{a^2}\right)^{-1}=\frac{8\pi G}{3} \rho.\tag{31}\] This is the modified first Friedmann equation inspired by the deformed (dual) Kaniadakis entropy (13 ).

4 Entropic corrections to Newton’s law and Friedmann equations↩︎

In this section, we are going to apply the idea of entropic gravity and derive the correction terms to Newton’s law of gravity as well as corrections to Friedmann equations inspired by entropy expression (11 ). The idea that gravity is not a fundamental force and can be understood as an entropic force caused by changes in the information, when a material body moves away from the holographic screen, was suggested by Verlinde [23]. According to Velinde’s proposal when a test particle moves apart from the holographic screen, the magnitude of the entropic force on this body has the form \[\label{F} F\triangle x=T \triangle S,\tag{32}\] where \(\triangle x\) is the displacement of the particle from the holographic screen, while \(T\) and \(\triangle S\) are the temperature and the entropy change on the screen, respectively. Consider two masses, a test mass \(m\) and a spherically symmetric mass distribution \(M\) which is surrounded by surface \(\mathcal{S}\). The surface \(\mathcal{S}\) is located between \(m\) and \(M\) and \(m\) is very close to the surface as compared to its reduced Compton wavelength \(\lambda_m=\frac{\hbar}{m}\) (\(c=1\)). When a test mass \(m\) is a distance \(\triangle x = \epsilon \lambda_m\) away from the surface \(\mathcal{S}\), the change in the entropy (11 ) is given by \[\label{S3} \triangle S_h=\frac{\triangle A}{4G}\Big{\{}1+\left(\frac{\gamma A}{4G}\right)^{\alpha}\Big{\}}^{-1/{\alpha}},\tag{33}\] where \(A=4\pi R^2\) is the area of the surface \(\mathcal{S}\). The energy inside the surface is identified as \(E=M\). On the surface \(\mathcal{S}\), there live a set of “bytes" of information that scale proportional to the area of the surface so that that \(A=QN\), where \(N\) represents the number of bytes and \(Q\) is a fundamental constant. Note that \(N\) is the number of bytes and thus \(\triangle N=1\), hence we have \(\triangle A=Q\). According to the equipartition law of energy, the temperature \(T\) in terms of the total energy on the surface \(\mathcal{S}\) reads \[\label{E} T=\frac{2M}{Nk_B}.\tag{34}\] Substituting Eqs. (33 ) and (34 ) in Eq. (32 ), we arrive at \[\label{F3} F=-\frac{GMm}{R^2}\left(\frac{Q^2}{8\pi k_B \hbar \epsilon G^2}\right)\Big{\{}1+\left(\frac{\gamma A}{4G}\right)^{\alpha}\Big{\}}^{-1/{\alpha}}_{A=4\pi R^2},\tag{35}\] where we have assumed the force between \(m\) and \(M\) is attractive. In order to arrive at the modified Newton’s law of gravity, we define \(Q^2=8\pi k_B \hbar \epsilon G^2\). Taking this into account, we get \[\label{F4} F=-\frac{GMm}{R^2}\Big{\{}1+\left(\frac{\gamma A}{4G}\right)^{\alpha}\Big{\}}^{-1/{\alpha}}_{A=4\pi R^2}.\tag{36}\] Next, we can derive the dynamical equation for the Newtonian cosmology. We assume surface \(\mathcal{S}\) is the boundary of a spherical region with volume \(V\) and radius \(R= a(t)r\) where \(r\) is the radial co-moving coordinate. If we combine the second law of Newton for the test particle \(m\) near the surface, with the gravitational force (36 ), we reach \[\label{F6} F=m\ddot{R}=m\ddot{a}r=-\frac{GMm}{R^2}\left[1+\beta R^{2\alpha}\right]^{-1/{\alpha}}.\tag{37}\] Eq. (37 ) is nothing, but the modified Newton’s law of gravitation derived by taking the entropy associated with the holographic screen in the form of (11 ).

The energy density of the matter inside the volume \(V=\frac{4}{3} \pi a^3 r^3\), is \(\rho=M/V\). Thus, Eq. (37 ) can be rewritten as \[\label{F7} \frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\rho\left[1+\beta R^{2\alpha}\right]^{-1/{\alpha}}.\tag{38}\] This is the modified dynamical equation for Newtonian cosmology. On the other side, the active gravitational mass is defined as [51] \[\begin{align} \label{ActM} \mathcal{M} &=&2 \int_V{dV\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)u^{\mu}u^{\nu}}=(\rho+3p)\frac{4\pi}{3}a^3 r^3. \end{align}\tag{39}\] Replacing \(M\) with \(\mathcal{M}\) (\(\rho\rightarrow \rho+3p\))in Eq. (38 ), we find \[\label{F8} \frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho+3p)\left[1+\beta R^{2\alpha}\right]^{-1/{\alpha}}.\tag{40}\] Multiplying both sides of Eq. (40 ) with \(2\dot{a}a\) and using the continuity equation (18 ), after integrating we find \[\label{Frie1} \dot{a}^2+k=\frac{8\pi G}{3}\int{ d (\rho a^2) \Big{\{}1+\beta (ra)^{2 \alpha}\Big{\}}^{-1/{\alpha}}},\tag{41}\] where \(k\) is an integration constant which can be interpreted as the curvature constant. To calculate the integral, we first use the continuity equation (18 ), to find \[\label{rho} \rho=\rho_0 a^{-3(1+w)},\tag{42}\] where \(w=p/\rho\) is the equation of state parameter and \(\rho_0\) is the present value of the energy density. Substituting relation (42 ) in Eq. (41 ), after some calculations, we find the generalized form of the modified Friedmann equation as \[\begin{align} \label{edoxuyqc} \left(H^2+\frac{k}{a^2}\right) \times{_{2}F_{1}\left(\frac{1}{\alpha},-\frac{1+3w}{2 \alpha},\frac{2\alpha-1-3w}{2\alpha}, -\beta \left(H^2+\frac{k}{a^2}\right)^{-\alpha}\right)}^{-1}=\frac{8\pi G}{3} \rho. \end{align}\tag{43}\] Since \(\beta\ll1\), we can expand the hypergeometric function to find the modified Friedmann equation up to the first correction term. The result is \[\label{Frie3} \left(H^2+\frac{k}{a^2}\right)-\lambda\left(H^2+\frac{k}{a^2}\right)^{1-\alpha} = \frac{8\pi G}{3} \rho,\tag{44}\] where \[\label{beta} \lambda=\frac{\beta(1+3w)}{\alpha (2\alpha-1-3w)}.\tag{45}\] Again, we see that for \(\alpha=1\), the second term in Eq. (44 ) becomes a constant. The result obtained in Eq. (44 ) from entropic force scenario, is consistent with the one obtained from the first law of thermodynamics in Eq. (29 ), which further supports the validity of the entopic nature of gravity.

5 Modified Friedmann equations from emergence of cosmic space↩︎

In this section, we use the emergence scenario of gravity proposed by Padmanabhan [27] to find the corrections to the Friedmann equation based on the modified entropy expression given in Eq. (11 ). As we have seen in the previous sections, the correction terms are very small, thus in this section, for simplicity, we consider the entropy up to the first correction term given in Eq. (14 ).

According to Padmanabhan, for a pure de Sitter universe with Hubble constant \(H\), the holographic principle can be expressed in terms of \(N_{\rm sur}=N_{\rm bulk}\), where \(N_{\rm sur}\), and \(N_{\rm bulk}\), respectively, stand for degrees of freedom on the boundary and in the bulk. For our real universe, which is asymptotically de Sitter, as shown by a lot of astronomical observations, Padmanabhan proposed that in an infinitesimal interval \(dt\) of cosmic time, the increase \(dV\) of the cosmic volume is given by [27] \[\frac{dV}{dt}\propto \left(N_{\mathrm{sur}}-N_{\mathrm{bulk}}\right). \label{dV1}\tag{46}\] For a flat universe, Padmanabhan assumed the temperature and volume as \(T=H/2\pi\) and \(V=4\pi/3H^3\). The reason for this assumption comes from the fact that in this case one may consider our universe as an asymptotically de Sitter space. Padmanabhan proposed the difference between degrees of freedom on the horizon and in the bulk, leads to the expansion of our universe. Mathematically, he assumed [27] \[\label{dV} \frac{dV}{dt}=G(N_{\mathrm{sur}}-N_{\mathrm{bulk}}).\tag{47}\] Soon after Padmanabhan, his idea was extended to a nonflat universe by derivation of the Friedmann equations in Einstein, Gauss-Bonnet and more general Lovelock gravity with any spatial curvature [30]. It was argued that in this case one should replace the Hubble radius (\(H^{-1}\)) with the apparent horizon radius \(R=1/{\sqrt{H^2+k/a^2}}\), which is a generalization of Hubble radius for \(k\neq0\). The generalization of Eq. (47 ), for a nonflat universe was proposed as [30] \[\label{krvelmid} \frac{dV}{dt}=GRH \left(N_{\mathrm{sur}}-N_{\mathrm{bulk}}\right),\tag{48}\] For a flat universe, \(RH=1\), and Eq. (46 ) restores Eq. (47 ). The temperature associated with the apparent horizon is assumed to be \[\label{T2} T=\frac{1}{2\pi R}.\tag{49}\] The reason for taking this expression for temperature instead of relation (17 ) comes from the fact that here we would like to consider an equilibrium system, thus within an infinitesimal internal of time \(dt\) we propose \(\dot{R}\ll 2HR\), which physically means that the apparent horizon radius is fixed during an infinitesimal internal of time \(dt\), similar to de-Sitter Universe. Note that the proposal of Padmanabhan indeed relates the volume change \(dV\) in an infinitesimal interval \(dt\) of cosmic time to the degrees of freedom. Thus it is reasonable to neglect the dynamical terms in the Hayward surface gravity and approximate it as \(\kappa \simeq {1}/{R}.\) This approximation leads to the familiar expression for the horizon temperature [28]. Besides, since our universe is assumed to be asymptotically de Sitter, thus one should consider the temperature as (49 ). Only with this assumption, one can deduce the correct form of the Friedmann equations through Padmanabhan’s scenario [28]. Moreover, in the specific framework of Padmanabhana’s emergent gravity paradigm, relation (47 ) assumes that the system is near thermal equilibrium at each infinitesimal time step. In such a setting, taking the horizon radius as effectively constant during this short interval is physically meaningful and consistent with the idea of horizon thermodynamics in slowly varying spacetimes. Note that in section 3, one can also consider the temperature associated with the apparent horizon in the form of (49 ), however in this case one should apply the first law as \(-dE=TdS\) where \(-dE\) is the energy flux crossing the horizon and the volume term should be absent in the first law of thermodynamics [18].

Using the entropy expression (14 ), the number of degrees of freedom on the surface is given by \[\begin{align} \label{Nsur2} N_{\mathrm{sur}}&=&4 S_h=\frac{A}{G}\left[1-\eta \left(\frac{A}{4G}\right)^{\alpha}\right]=\frac{4\pi R^2}{G}+ 4\pi\zeta R^{2\alpha+2}, \end{align}\tag{50}\] where \(\zeta=-\eta \pi^{\alpha}/G^{\alpha+1}\) and we have taken \(A=4 \pi R^2\) as the area of the boundary. The total energy inside the apparent horizon is in the form of the Komar energy and is given by \[E_{\mathrm{Komar}}=|(\rho +3p)|V, \label{Komar}\tag{51}\] where \(V=4 \pi R^3/3\) is the volume of a sphere enveloped by the apparent horizon. The number of degrees of freedom of the matter field in the bulk can be obtained using the equipartition law of energy (\(k_B=1\)), \[N_{\mathrm{bulk}}=\frac{2|E_{\mathrm{Komar}}|}{T}. \label{Nbulk}\tag{52}\] Combining this relation with Eq. (51 ) and assuming, in an expanding universe, \(\rho+3p<0\), we find \[N_{\rm bulk}=-\frac{16 \pi^2}{3} R^4 (\rho+3p). \label{muliyoqp}\tag{53}\] Substituting relations (50 ) and (52 ) in assumption (46 ), after simplifying, we arrive at \[\begin{align} -2\frac{\dot{R}R^{-3}}{H}-2R^{-2}-2\zeta G R^{2\alpha-2}=\frac{8\pi G }{3}(\rho+3p). \label{Frgb11} \end{align}\tag{54}\] Next, we multiply both side of Eq. (54 ) by factor \(-\dot{a}a\), after using the continuity equation (18 ), we reach \[\label{Frgb2} \frac{d}{dt}\left( a^2 R^{-2}\right)+2\zeta G \dot{a}a R^{2\alpha-2} =\frac{8 \pi G }{3} \frac{d}{dt}(\rho a^2).\tag{55}\]
Taking into account the fact that \(R=a(t)r\) in the second term, we can integrate the above equation. The result is \[\label{Frgb3} R^{-2}+\frac{\zeta G}{\alpha} R^{2\alpha-2} =\frac{8 \pi G }{3} \rho.\tag{56}\] Using the fact that \(R=1/\sqrt{H^2+k/a^2}\), we finally get \[\label{Frgb4} \left(H^2+\frac{k}{a^2}\right)+ \chi \left(H^2+\frac{k}{a^2}\right)^{1-\alpha}=\frac{8\pi G}{3}\rho,\tag{57}\] where \[\chi=\frac{\zeta G}{\alpha}=-\frac{\beta}{\alpha^2(\alpha+1)}.\]

In this way, we obtain the modified Friedmann equation inspired by the MOND theory through the method of the emergence gravity. One can easily check that the result obtained here is consistent with those obtained in the previous sections from two other approaches, up to the leading order correction terms. Our studies therefore further support the viability of the Padmanabhan’s perspective of emergence gravity.

6 Generalized second law of thermodynamics↩︎

For a given modified entropy expression associated with the boundary of the system, one of the main question, which should be addressed is whether or not the entropy associated with the horizon can satisfy the generalized second law of thermodynamics. For an accelerated expanding universe, the generalized second law of thermodynamics have been investigated in the literatures [16], [52], [53].

Using Eqs. (18 ) and (24 ), we can find \[\label{dotR} \dot{R}= 4\pi G R^3 H (\rho+p) \left(1+\beta R^{2\alpha}\right)^{\frac{1}{\alpha}}.\tag{58}\] It is easy to show that \[\begin{align} \label{TSh1} T_{h} \dot{S_{h}}&=&4\pi H R^3 (\rho+p)\left(1-\frac{\dot{R}}{2H R}\right). \end{align}\tag{59}\] Since our universe is currently experiencing a phase of accelerated expansion, thus we may have \(\rho+p<0\), which implies the second law of thermodynamics may break down, \(\dot{S_{h}}<0\). Therefore we consider the generalized second law of thermodynamics. From the Gibbs equation we have [54] \[\label{Gib2} T_m dS_{m}=d(\rho V)+pdV=V d\rho+(\rho+p)dV,\tag{60}\] where \(T_{m}\) and \(S_m\) stand for the temperature and entropy of the matter fields in the bulk, respectively. We further assume there is no energy follow between the bulk and the boundary of the universe. This means that we can take \(T_m\approx T_h\) [54]. Thus from the Gibbs equation (60 ), one finds \[\label{TSm2} T_{h} \dot{S}_{m} =4\pi {R^2}\dot{R}(\rho+p)-4\pi {R^3} H(\rho+p).\tag{61}\] Combining Eqs. (58 ), (59 ) and (61 ), one can arrive \[\begin{align} \label{GSL3} T_{h}( \dot{S_{h}}+\dot{S_{m}})=8 \pi^2 G H R^{5}(\rho+p)^2 \left(1+\beta R^{2\alpha}\right)^{1/{\alpha}}. \end{align}\tag{62}\] This confirms that we have always \(\dot{S_{h}}+\dot{S_{m}}\geq 0\), which means that the time evolution of the total entropy, including the modified entropy associated with the apparent horizon plus the matter entropy inside the universe is a non decreasing function of time. This implies that the generalized second law of thermodynamics holds when the entropy associated with the apparent horizon is given by Eq. (11 ).

7 Closing remarks↩︎

Nowadays, it is a general belief that there is a profound connection between the laws of gravity and the laws of thermodynamics. It has been shown that the gravitational field equations can be derived from thermodynamic arguments in three levels. In the first level it was shown that the field equations of gravity can be derived from the first law of thermodynamics. In a deeper level, it was confirmed that gravity is an entropic force, which can be understood from statistical mechanics using two fundamental principles, namely the equipartition law of energy and the holographic principle. In the deepest level, it was argued that gravity (geometry) is not a pre-exist quantity and the cosmic space emerges as the cosmic time progress. This idea leads to extraction the Friedmann equations describing the evolution of the FRW universe by counting the degrees of freedom on the boundary and in the bulk.

In this paper, we have reconsidered thermodynamics-gravity correspondence to establish a general form of the entropy associated with the boundary. To this aim, we started from a general form of the MOND theory and then using an inverse approach for the entropic force scenario, we reconstructed the general form of the entropy associated with the boundary. We supposed the entropy associated with the apparent horizon of FRW universe has the same expression. This allows us to construct, using three methods, the modified Friedmann equations by starting from the modified entropy expression and applying the thermodynamics-gravity conjecture. We confirmed the consistency of the obtained results from three approaches, which further supports the idea of thermodynamics-gravity correspondence.

Based on the modified Friedmann equations derived here, one can establish a modified cosmological model. Thus, many issues remain to be addressed. First of all, the cosmological consequences of the obtained modified Friedmann equations should be studied. In particular, it is interesting to check whether or not the cosmological model based on these Friedmann equations can lead to an accelerated expansion without invoking dark energy. If this is the case, then we can assert that our model can explain both challenges of the modern cosmology without invoking unusual additional component of matter/energy. It is also worthy to explore cosmological parameters, growth of perturbations, inflationary models, and the early nucleosynthesis in the context of the modified Friedmann equations. In addition, one can start from the general form of the entropy given in this work, and reproduce the corrections to the Einstein field equations. These issues are under investigation and the results will be appeared in the future.

APPENDIX↩︎

The hypergeometric function \(_2F_1(a,b,c,z)\) is a mathematic function which is represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. The hypergeometric function \(_2F_1(a,b,c,z)\) has a series expansion as \[\begin{align} \label{Ap1} &&_2F_1(a,b,c,z)=\sum_{n=0}^{\infty}\frac{(a)_n (b)_n}{(c)_n}\frac{z^n}{n!}=1+\frac{ab}{c} \frac{z}{1!}+\frac{a(a+1) b (b+1)}{c(c+1)}\frac{z^2}{2!}\nonumber\\&&+\frac{a(a+1)(a+2) b (b+1)(b+2)}{c(c+1)(c+2)}\frac{z^3}{3!}+..., \end{align}\tag{63}\] where \[\label{Ap2} (q)_n=\left\{ \begin{array}{ll} 1 \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad\!\!{\rm for}\quad n=0, & \\ q (q+1)...(q+n-1)\quad {\rm for}\quad n>0. \end{array} \right.\tag{64}\] The series terminates if either \(a\) or \(b\) is a nonpositive integer, in which case the function reduces to a polynomial: \[\begin{align} \label{Ap3} _2F_1(-m,b,c,z)=\sum_{n=0}^{\infty}(-1)^n \frac{m!}{n!(m-n)!} \frac{(b)_n }{(c)_n}{z^n}. \end{align}\tag{65}\] The differentiation formula for the hypergeometric function is \[\begin{align} \label{Ap4} \frac{d^n}{dz^n}\text{{\;}} _2F_1(a,b,c,z)= \frac{(a)_n (b)_n }{(c)_n}\text{{\;}} _2F_1(a+n,b+n,c+n,z). \end{align}\tag{66}\] Many of the common mathematical functions can be expressed in terms of the hypergeometric function, or as limiting cases of it. Some typical examples are

\[\begin{align} \label{Ap5} _2F_1(1,1,2,-z)&=&\frac{\ln(1+z)}{z},\nonumber\\ _2F_1\left(\frac{1}{2},\frac{1}{2},\frac{3}{2},z^2\right)&=&\frac{\arcsin(z)}{z},\nonumber\\ _2F_1(a,b,b,z)&=& (1-z)^{-a}, \quad \quad (b \;\rm arbitrary). \end{align}\tag{67}\]

References↩︎