# Modified Friedmann Equations From Debye Entropic Gravity

A. Sheykhi1,2 111<EMAIL_ADDRESS>and Z. Teimoori1 1Department of
Physics, Shahid Bahonar University, P.O. Box 76175, Kerman, Iran
2 Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O.
Box 55134-441, Maragha, Iran

###### Abstract

A remarkable new idea on the origin of gravity was recently proposed by
Verlinde who claimed that the laws of gravitation are no longer fundamental,
but rather emerge naturally as an entropic force. In Verlinde derivation, the
equipartition law of energy on the holographic screen plays a crucial role.
However, the equipartition law of energy fails at the very low temperature.
Therefore, the formalism of the entropic force should be modified while the
temperature of the holographic screen is very low. Considering the Debye
entropic gravity and following the strategy of Verlinde, we derive the
modified Newton’s law of gravitation and the corresponding Friedmann equations
which are valid in all range of temperature. In the limit of strong
gravitational field, i.e. high temperature compared to Debye temperature,
$T\gg T_{D}$, one recovers the standard Newton’s law and Friedmann equations.
We also generalize our study to the entropy corrected area law and derive the
dynamical cosmological equations for all range of temperature. Some limits of
the obtained results are also studied.

##  I Introduction

Although gravity is the most universal force of nature, however, the origin of
it in quantum level is still unclear. This is due to the fact that it is
remarkably hard to combine gravity with quantum mechanics compared with all
the other forces, and hence the final theory of the quantum gravity has not
been established yet. The universality of gravity suggests that its emergence
should be understood from general principles that are independent of the
specific details of the underlying microscopic theory.

According to Einstein’s theory of general relativity, the concept of gravity
has strongly connected to the spacetime geometry. Indeed, Einstein field
equations tell us that the presence of energy or stress causes the deformation
of the spacetime geometry. In 1970’s Bekenstein and Hawking HB  discovered
black holes thermodynamics. With combination of quantum mechanics and general
relativity they predicted that a black hole behaves like a black body,
emitting thermal radiations, with a temperature proportional to its surface
gravity at the black hole horizon and with an entropy proportional to its
horizon area. The Hawking temperature and the horizon entropy together with
the black hole mass obey the first law of thermodynamics, $dM=TdS$. Since the
discovery of black hole thermodynamics in 1970’s, physicists have been
speculating that there should be a direct relation between thermodynamics and
Einstein equations. This is expected because the geometrical quantities like
horizon area and surface gravity are proportional to entropy and horizon
temperature, respectively. After Bekenstein and Hawking a lot of works have
been done to disclose the connection between themodynamics and gravity B ; D .
In 1995 Jacobson Jac  put forward a great step and showed that the Einstein
field equation is just an equation of state for the spacetime and in
particular it can be derived from the proportionality of entropy and the
horizon area together with the fundamental relation $\delta Q=TdS$. Following
Jacobson, however, an overwhelming flood of papers has appeared which attempt
to show that there is indeed a deeper connection between gravitational field
equations and horizon thermodynamics. It has been shown that, not only in
Einstein gravity but also in a wide variety of theories, the gravitational
field equations for the spacetime metric has a predisposition to thermodynamic
behavior. This result, first pointed out in Pad , has now been demonstrated in
various theories including f(R) gravity Elin  and cosmological setups Cai2 ;
Cai3 ; CaiKim ; Wang ; Cai33 ; Shey0 ; Shey1 ; Shey2 . For a recent review on
the thermodynamical aspects of gravity and complete list of references see
Pad0 .

Recently, Verlinde Ver  has invented a conceptual theory that gravity is no
longer fundamental, but is emergent. According to Verlinde, one can start from
the first principles, and gravity appears as an entropic force naturally and
unavoidably in a theory in which space is emergent through a holographic
scenario. Similar discoveries are also made by Padmanabhan Padm  who observed
that the equipartition law for horizon degrees of freedom combined with the
Smarr formula leads to the Newton’s law of gravity. In addition, Verlinde’s
arguments reveal a fact that the key to understanding gravity is information
(or entropy). In Verlinde derivation the holographic principle and the
equipartition law of energy play a crucial role. The holographic principle was
originally proposed by ’t Hooft Hooft  and then developed in cosmology by
Susskind and others Sus . According to this principle the combination of
quantum mechanics and gravity requires the three dimensional world to be an
image of data that can be stored on a two dimensional projection much like a
holographic image. The studies on the entropic gravity scenario have arisen a
lot attention recently Cai4 ; Smolin ; Li ; Tian ; Vancea ; Modesto ; Sheykhi1
; BLi ; Sheykhi2 ; Gu ; other ; mann .

It is well-known that the equipartition law of energy for a system of
particles only valid for the situation where the kinetic energy of the
particles is much larger than the effective interacting potential between
particles. This means that the equipartition law of energy break down at very
low temperatures. It is found that Debye model, which modified the
equipartition law of energy, is in good agreement with experimental results
for most solid objects. According to Verlinde, we know that the gravity can be
explained as an entropic force, it means that the gravity may have a
statistical thermodynamics explanation. Therefore, the formalism of the
entropic force should be modified while the temperature of the holographic
screen is very low. This means that Newtonian gravity takes a different form
in the background of an extreme weak gravitational field. In the present work,
inspired by the Debye’s model in statistical thermodynamics, we generalize the
formalism of the entropic gravity to the very law temperatures.

The outline of the present paper is as follows. The modified Newton’s law of
gravitation and the corresponding Friedmann equations which are valid in all
range of temperature are extracted in the next section. Sec. III is devoted to
the derivation of the Entropy corrected Friedmann equations in Debye entropic
force scenario. The paper ends with a conclusion, which appears in Sec. IV.

##  II Debye Entropic Gravity and Friedmann Equation

Consider a closed holographic screen and a free particle of mass $m$ near it
on the side that spacetime has already emerged. In Verlide’s picture when the
particle has an entropic reason to be on one side of the screen and the screen
carries a temperature, it will experience an effective macroscopic force due
to the statistical tendency to increase its entropy. This is described by

$F=T\frac{\triangle S}{\triangle x}.$ (1)

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. According to the Unruh formula, the temperature
in Eq. (1) associated with the holographic screen is

$k_{B}T=\frac{\hbar g}{2\pi c}.$ (2)

where $g$ represents the proper gravitational acceleration on the screen which
is produced by the matter distribution inside the screen. Suppose we have a
mass distribution $M$ which induces a holographic screen $\mathcal{S}$ at some
distance $R$ that has encoded on it gravitational information. Suppose we have
also a test mass $m$ which is assumed to be very close to the holographic
screen as compared to its reduced Compton wavelength
$\lambda_{m}=\frac{\hbar}{mc}$. Assuming the holographic screen forms a closed
surface. The key statement is that we need to have a temperature in order to
have a force. One can think about the boundary as a storage device for
information. Assuming that the holographic principle holds, the maximal
storage space, or total number of bits, is proportional to the area $A$. Let
us denote the number of used bits by $N$. It is natural to assume that this
number will be proportional to the area $A=4\pi R^{2}$. Thus we write

$A=NQ,$ (3)

where $Q$ is a fundamental constant which should be specified later. Note that
$N$ is the number of bits and thus for one unit change we have $\triangle
N=1$, hence from (3) we find $\triangle A=Q$. Motivated by Bekenstein’s area
law of black hole entropy, we assume the entropy associated with the
holographic screen obey the area law, namely

$S=\frac{A}{4\ell_{p}^{2}},$ (4)

where $\ell_{p}^{2}=G\hbar/c^{3}$ is the Planck length. Following Ver  we also
assume the energy on the holographic screen is proportional to the mass
distribution $M$ that would emerge in the part of space enclosed by the screen

$E=Mc^{2}.$ (5)

According to statistical thermodynamics the equipartition law of energy for
free particle only valid for the situation in which the kinetic energy of
particle is much larger than the effective interacting potential between them.
Therefore, the equipartition law of energy fails in the very low temperatures.
It is found that Debye model, which modified the equipartition law of energy,
is in good agreement with experimental results for most solid objects.
Following Verlinde’s scenario, the laws of gravitation are no longer
fundamental, but rather emerge naturally as an entropic force. It means that
the gravity may have a statistical thermodynamics origin. Thus, any
modification of statistical mechanics should modify the laws of gravity
accordingly. Motivated by this point, we modify the equipartition law of
energy as

$E=\frac{1}{2}Nk_{B}T\mathcal{D}(x),$ (6)

where the Debye function is defined by

$\mathcal{D}(x)\equiv\frac{3}{x^{3}}\int_{0}^{x}\frac{y^{3}}{e^{y}-1}dy.$ (7)

Here $x$ is related to the temperature $T$ as follows

$x\equiv\frac{T_{D}}{T}=\frac{\hbar\omega_{D}}{Tk_{B}},$ (8)

where $T_{D}$ is the Debye critical temperature and $\omega_{D}$ is the Debye
frequency. Combining Eqs. (3), (5) and (6), we obtain the temperature of the
holographic screen as

$T=\frac{2Mc^{2}Q}{4\pi R^{2}k_{B}\mathcal{D}(x)}.$ (9)

Substituting Eqs. (9) and (4) in (1), and using relation $\triangle A=Q$, we
get

$F=-\frac{Mm}{R^{2}}\frac{1}{\mathcal{D}(x)}\left(\frac{Q^{2}c^{3}}{8\pi\ell_{p}^{2}k_{B}\hbar}\right)$
(10)

where we have taken $\triangle x=-\frac{\hbar}{mc}$ for one fundamental unit
change in the entropy and the entropy gradient points radially from the
outside of the surface to inside. In order to derive the Newton’s law of
gravitation we must define $Q^{2}=8\pi k_{B}\ell_{p}^{4}$. Finally we reach

$F=-G\frac{Mm}{R^{2}}\frac{1}{\mathcal{D}(x)}.$ (11)

The corresponding gravitational acceleration is obtained as

$g=\frac{GM}{R^{2}}\frac{1}{\mathcal{D}(x)}.$ (12)

Using relation (2) we can define the Debye acceleration $g_{D}$ which is
related to the Debye temperature as

$T_{D}=\frac{\hbar g_{D}}{2\pi k_{B}c},\ \ \
x=\frac{T_{D}}{T}=\frac{g_{D}}{g}$ (13)

Eq. (11) is the Newton’s law of gravitation resulting from Debye entropic
force. Let us study two different limits of Eq. (11). In the strong
gravitational field limit, i.e. at high temperature, $T\gg T_{D}$ ($x\ll 1$),
the Debye function reduces to

$\mathcal{D}(x)\approx 1.$ (14)

As a result in this limit, the usual Newtonian gravity is restored. In the
weak gravitational field limit, i.e. at very law temperature, $T\ll T_{D}$
($x\gg 1$) we have

$\mathcal{D}(x)=\frac{\pi^{4}}{5x^{3}}=\frac{\pi^{4}}{5}\left(\frac{g}{g_{D}}\right)^{3}$
(15)

In this limit, the Newton’s law is modified as

$F=-\frac{5GMm}{R^{2}}\frac{g_{D}^{3}}{\pi^{4}g^{3}},$ (16)

while the gravitational acceleration becomes

$g=\left(\frac{5GMg_{D}^{3}}{\pi^{4}}\right)^{\frac{1}{4}}\frac{1}{\sqrt{R}}\
\ \Rightarrow\ g\propto\frac{1}{\sqrt{R}}$ (17)

Therefore in this limit the gravitational field differs from Newtonian
gravity. Let us then consider the cosmological implications of the presented
model. We assume the background spacetime is spatially homogeneous and
isotropic which is described by the line element

$ds^{2}={h}_{\mu\nu}dx^{\mu}dx^{\nu}+R^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}),$ (18)

where $R=a(t)r$, $x^{0}=t,x^{1}=r$, the two dimensional metric
$h_{\mu\nu}$=diag $(-1,a^{2}/(1-kr^{2}))$. Here $k$ denotes the curvature of
space with $k=0,1,-1$ corresponding to flat, closed, and open universes,
respectively. The dynamical apparent horizon, a marginally trapped surface
with vanishing expansion, is determined by the relation
$h^{\mu\nu}\partial_{\mu}R\partial_{\nu}R=0$. A simple calculation gives the
apparent horizon radius for the Friedmann-Robertson-Walker (FRW) universe

$R=ar=\frac{1}{\sqrt{H^{2}+k/a^{2}}}.$ (19)

The matter source in the FRW universe is assumed as a perfect fluid with
stress-energy tensor

$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu}.$ (20)

Now we are in a position to derive the dynamical equation for Newtonian
cosmology. Consider a compact spatial region $V$ with a compact boundary
$\mathcal{S}$, which is a sphere with physical radius $R=a(t)r$. Note that
here $r$ is a dimensionless quantity which remains constant for any
cosmological object partaking in free cosmic expansion. Combining the second
law of Newton for the test particle $m$ near the surface, with gravitational
force (11) we get

$m\ddot{R}=m\ddot{a}r=-G\frac{Mm}{R^{2}}\frac{1}{\mathcal{D}(x)}.$ (21)

We also assume $\rho=M/V$ is the energy density of the matter inside the the
volume $V=\frac{4}{3}\pi a^{3}r^{3}$. Thus, Eq. (21) can be rewritten as

$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\rho\frac{1}{\mathcal{D}(x)}$ (22)

This is the dynamical equation for Newtonian cosmology which is valid in all
range of the temperature. For strong gravitational field
($\mathcal{D}(x)\simeq 1$) we reach the well-known formula

$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}\rho.$ (23)

Next we want to derive the Friedmann equations of FRW universe. For this
purpose we need to employ the concept of the active gravitational mass
$\mathcal{M}$ Pad3 , since this quintity produces the acceleration in general
relativity. From Eq. (22) with replacing $M$ with $\mathcal{M}$ we have

$\mathcal{M}=-\frac{\ddot{a}a^{2}r^{3}}{G}\mathcal{D}(x)$ (24)

On the other side, the active gravitational mass is defined as Cai4

$\mathcal{M}=2\int_{V}{dV\left(T_{\mu\nu}-\frac{1}{2}Tg_{\mu\nu}\right)u^{\mu}u^{\nu}}.$
(25)

A simple calculation gives

$\mathcal{M}=(\rho+3p)\frac{4\pi}{3}a^{3}r^{3}.$ (26)

Equating Eqs. (24) and (26) we obtain

$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho+3p)\frac{1}{\mathcal{D}(x)}.$ (27)

This is the modified acceleration equation for the dynamical evolution of the
FRW universe. Multiplying $\dot{a}a$ on both sides of Eq. (27), and using the
continuity equation

$\dot{\rho}+3H(\rho+p)=0,$ (28)

after integrating we find

$H^{2}+\frac{k}{a^{2}}=\frac{8\pi G}{3a^{2}}\int\frac{d(\rho
a^{2})}{\mathcal{D}(x)}.$ (29)

This is the first Friedmann equation resulting from Debye entropic force. Eqs
(29) and (28) together with the equation of state $p=w\rho$ govern the
evolution of the universe. It is important to note that since $x=x(T)$ and the
temperature is also a function of scale factor namely, $T=T(a)$, thus in
general we cannot integrate Eq. (29) and derive the simplified result. When
$x\ll 1\Rightarrow\mathcal{D}(x)\approx 1$, the well-known Friedmann equation
in standard cosmology is recovered. For $x\gg 1$, using Eq. (15) we find

$H^{2}+\frac{k}{a^{2}}=\frac{8\pi
G}{3}\rho\frac{5}{\pi^{4}}\left(\frac{g_{D}}{g}\right)^{3}.$ (30)

In order to derive the second Friedmann equation (29), we have to combine the
first Friedmann equation with continuity equation (28). Let us put $k=0$ for
simplicity, which has been confirmed by recent observations. Differentiating
Eq. (29) we find

$\displaystyle 2HdH=\frac{8\pi G}{3}\left[-\frac{2}{a^{2}}\frac{da}{a}\int
d(\rho
a^{2})\frac{1}{\mathcal{D}(x)}+d\rho\frac{1}{\mathcal{D}(x)}+2\rho\frac{da}{a}\frac{1}{\mathcal{D}(x)}\right]$
(31)

Multiplying $\frac{3}{2}$ on both sides of Eq. (31) and dividing by $dt$, we
find

$\displaystyle 3H\dot{H}=-\frac{8\pi G}{a^{2}}\frac{\dot{a}}{a}\int d(\rho
a^{2})\frac{1}{\mathcal{D}(x)}+4\pi G\dot{\rho}\frac{1}{\mathcal{D}(x)}+8\pi
G\rho\frac{\dot{a}}{a}\frac{1}{\mathcal{D}(x)}.$ (32)

Using the continuity equation (28), the above equation can be written as

$\displaystyle-\left[\dot{H}\mathcal{D}(x)+\frac{8\pi
G}{3a^{2}}\mathcal{D}(x)\int d(\rho a^{2})\frac{1}{\mathcal{D}(x)}-\frac{8\pi
G}{3}\rho\right]=4\pi G(\rho+p).$ (33)

For $x\ll 1$ we have $D(x)\approx 1$, and the well-known second Friedmann
equation of FRW universe in flat spacetime is recovered, namely

$\displaystyle-\dot{H}=4\pi G(\rho+p).$ (34)

It is worth noting that the Friedmann equation in Debye entropic force
scenario was first studied in Gao . Let us stress the difference between our
derivation in this section and that of Gao . The author of Gao  has derived
Friedmann equations, following the method of FW , by applying the
equipartition law of energy, $E=NT/2$, to the apparent horizon of a FRW
universe with the assumption that the apparent horizon has the temperature
$T=\hbar/(2\pi R)$, where $R$ is the apparent horizon radius. Thus the total
energy change of the system is obtained as FW

$\displaystyle dE=\frac{1}{2}NdT+\frac{1}{2}TdN=\frac{dR}{G},$ (35)

during the infinitesimal time interval $dt$, where the apparent horizon radius
evolves from $R$ to $R+dR$. Indeed, the above equation is just the first law
of thermodynamics in the form $dE=TdS$ on the apparent horizon, where
$T=\hbar/(2\pi R)$ and $S=A/(4\hbar G)$ is the entropy of the system which
assumed to obey the area-law and $A=4\pi R^{2}$ is the apparent horizon area.
While in the present work we have not employed the first law of thermodynamics
for deriving the modified Friedmann equations. Therefore, our result is
independent of the definition of the temperature in a dynamical spacetime.

##  III Entropic Corrected Friedmann Equation in Debye entropic gravity

In this section we would like to consider the effects of the quantum
correction terms to the entropy expression, on the laws of gravity in Debye
model of entropic gravity. The result we will obtain are valid in all range of
temperature. The correction terms to the entropy expression originate from the
loop quantum gravity (LQG). The quantum corrections provided to the entropy-
area relationship leads to the curvature correction in the Einstein-Hilbert
action and vice versa Zhu ; Suj . In the presence of quantum corrections the
entropy takes the following form Zhang

$S=\frac{A}{4\ell_{p}^{2}}-\beta\ln{\frac{A}{4\ell_{p}^{2}}}+\gamma\frac{\ell_{p}^{2}}{A}+\mathrm{const},$
(36)

where $\beta$ and $\gamma$ are dimensionless constants of order unity. These
corrections arise in the black hole entropy in LQG due to thermal equilibrium
fluctuations and quantum fluctuations Rovelli . We will show that these
corrections modify the Newton’s law of gravitation as well as the Friedmann
equations. First of all we rewrite Eq. (36) in the following form

$S=\frac{A}{4\ell_{p}^{2}}+{s}(A),$ (37)

where $s(A)$ represents the correction terms in the entropy expression. In
this case the entropy change is obtained as

$\triangle S=\frac{\partial S}{\partial A}\triangle
A=\left(\frac{1}{4\ell_{p}^{2}}+\frac{\partial{s}(A)}{\partial
A}\right)\triangle A.$ (38)

Substituting Eqs. (9) and (38) in Eq. (1) and using relations $\triangle
x=-\frac{\hbar}{mc}$ and $\triangle A=Q$, one can easily find

$F=-\frac{Mm}{R^{2}\mathcal{D}(x)}\left(\frac{Q^{2}c^{3}}{2\pi
k_{B}\hbar}\right)\left[\frac{1}{4\ell_{p}^{2}}+\frac{\partial{s}}{\partial
A}\right]_{A=4\pi R^{2}}.$ (39)

If we define $Q^{2}\equiv 8\pi k_{B}\ell_{p}^{4}$, as before, we immediately
derive the modified Newton’s law of gravity in Debye entropic gravity

$F=-G\frac{Mm}{R^{2}}\frac{1}{\mathcal{D}(x)}\left[1-\frac{\beta}{\pi}\frac{\ell_{p}^{2}}{R^{2}}-\frac{\gamma}{4\pi^{2}}\frac{\ell_{p}^{4}}{R^{4}}\right].$
(40)

In the absence of correction terms ($\beta=\gamma=0$), the above equation
reduces to the result of the previous section. Let us study two different
limit of the above equation. In the strong gravitational limit, i.e. at high
temperature, $T_{D}\ll T$ ($\mathcal{D}(x)\approx 1)$ we have

$F=-G\frac{Mm}{R^{2}}\left[1-\frac{\beta}{\pi}\frac{\ell_{p}^{2}}{R^{2}}-\frac{\gamma}{4\pi^{2}}\frac{\ell_{p}^{4}}{R^{4}}\right],$
(41)

which is exactly the result obtained in Sheykhi1 . When $\beta=\gamma=0$, one
recovers the well-known Newton’s law. on the other hand, at very law
temperature $T_{D}\gg T$ we have $\mathcal{D}(x)=\frac{\pi^{4}}{5x^{3}}$ and
Eq. (40) reduces to

$F=-G\frac{Mm}{R^{2}}\frac{5}{\pi^{4}}\frac{g_{D}^{3}}{g^{3}}\left[1-\frac{\beta}{\pi}\frac{\ell_{p}^{2}}{R^{2}}-\frac{\gamma}{4\pi^{2}}\frac{\ell_{p}^{4}}{R^{4}}\right].$
(42)

To derive Friedmann equation we follow the method of the previous section.
Combining the second law of Newton for the test particle $m$ near the screen
with gravitational force (42) we obtain

$F=m\ddot{R}=m\ddot{a}r=-\frac{MmG}{a^{2}r^{2}}\frac{1}{\mathcal{D}(x)}\left[1-\frac{\beta}{\pi}\frac{\ell_{p}^{2}}{R^{2}}-\frac{\gamma}{4\pi^{2}}\frac{\ell_{p}^{4}}{R^{4}}\right]$
(43)

which from it we can derive the acceleration equation

$\frac{\ddot{a}}{a}=-\frac{4\pi
G}{3}\rho\frac{1}{\mathcal{D}(x)}\left[1-\frac{\beta}{\pi}\frac{\ell_{p}^{2}}{R^{2}}-\frac{\gamma}{4\pi^{2}}\frac{\ell_{p}^{4}}{R^{4}}\right].$
(44)

With the entropic corrections terms, the active gravitational mass
$\mathcal{M}$ will be modified accordingly. The active gravitational mass
$\mathcal{M}$ in this case is obtained as

$\mathcal{M}=-\frac{\ddot{a}a^{2}}{G}r^{3}\mathcal{D}(x)\left[1-\frac{\beta}{\pi}\frac{\ell_{p}^{2}}{R^{2}}-\frac{\gamma}{4\pi^{2}}\frac{\ell_{p}^{4}}{R^{4}}\right]^{-1}.$
(45)

Equating the above equation with Eq. (26) yields

$\frac{\ddot{a}}{a}=-\frac{4\pi
G}{3}(\rho+3p)\frac{1}{\mathcal{D}(x)}\left[1-\frac{\beta}{\pi}\frac{\ell_{p}^{2}}{R^{2}}-\frac{\gamma}{4\pi^{2}}\frac{\ell_{p}^{4}}{R^{4}}\right].$
(46)

Next we multiply the both sides of the above equation by $a\dot{a}$, after
using the continuity equation (28) and integrating we find

$\displaystyle H^{2}+\frac{k}{a^{2}}$ $\displaystyle=$
$\displaystyle\frac{8\pi G}{3a^{2}}\left[\int\frac{d(\rho
a^{2})}{\mathcal{D}(x)}-\frac{\beta}{\pi}\frac{\ell_{p}^{2}}{r^{2}}\int\frac{d(\rho
a^{2})}{a^{2}\mathcal{D}(x)}-\frac{\gamma}{4\pi^{2}}\frac{\ell_{p}^{4}}{r^{4}}\int\frac{d(\rho
a^{2})}{a^{4}\mathcal{D}(x)}\right].$ (47)

Where $k$ is an integration constant. Unfortunately, the above equation cannot
be integrated in general for an arbitrary $\mathcal{D}(x)$. In the limiting
case $D(x)\approx 1$, the integrations can be done following the method
developed in Sheykhi1 . We find ( see Sheykhi1  for details)

$\displaystyle\left(H^{2}+\frac{k}{a^{2}}\right)+\frac{\beta\ell_{p}^{2}(1+3\omega)}{3\pi(1+\omega)}\left(H^{2}+\frac{k}{a^{2}}\right)^{2}+\frac{\gamma\ell_{p}^{4}(1+3\omega)}{4\pi^{2}(5+3\omega)}\left(H^{2}+\frac{k}{a^{2}}\right)^{3}=\frac{8\pi
G}{3}\rho.$ (48)

Again we see that in the absence of correction terms $(\beta=0,\gamma=0)$ the
well-known Friedmann equation is recovered. For $x\gg 1$
($\mathcal{D}(x)=\frac{\pi^{4}}{5x^{3}})$ Eq. (47) can be written

$\displaystyle H^{2}+\frac{k}{a^{2}}$ $\displaystyle=$
$\displaystyle\frac{8\pi G}{3}\left[\frac{1}{a^{2}}\int d(\rho
a^{2})\frac{5x^{3}}{\pi^{4}}\right.$ (49)
$\displaystyle\left.-\frac{\beta}{\pi}\frac{\ell_{p}^{2}}{a^{2}r^{2}}\int\frac{d(\rho
a^{2})}{a^{2}}\frac{5x^{3}}{\pi^{4}}-\frac{\gamma}{4\pi^{2}}\frac{\ell_{p}^{4}}{a^{2}r^{4}}\int\frac{d(\rho
a^{2})}{a^{4}}\frac{5x^{3}}{\pi^{4}}\right].$

##  IV conclusion

Verlinde proposal on the entropic origin of the gravity is based strongly on
the assumption that the equipartition law of energy holds on the holographic
screen induced by the mass distribution of the system. However, from the
theory of statistical mechanics we know that the equipartition law of energy
does not hold in the limit of very low temperature. By low temperature, we
mean that the temperature of the system is much smaller than Debye
temperature, i.e. $T\ll T_{D}$. It was demonstrated that the Debye model is
very successful in interpreting the physics at the very low temperature. Since
the discovery of black holes thermodynamics, physicist have been thought that
the gravitational systems such as black hole and our universe can also be
regarded as a thermodynamical system. Hence, it is expected that the
equipartition law of energy for the gravitational system should be modified in
the limit of very low temperature (or very weak gravitational field).

In this paper inspired by the Verlinde proposal and following the Debye model
of equipartition law of energy in statistical thermodynamics, we modified the
entropic gravity. First, we studied the Debye entropic gravity and derived the
modified Newton’s law of gravitation and the corresponding Friedmann equations
which are valid in all range of temperature. We found that the modified
entropic force returns to the Newton’s law of gravitation while the
temperature of the holographic screen is much higher than the Debye
temperature. Then we extended our study to the case where there are correction
terms such as logarithmic correction in the entropy expression. In this case
we again reproduced the gravitational equations for all range of temperature.
Our study shows a deep connection between Debye entropic gravity and modified
Friedmann equation. The microscopic statistical thermodynamical model of
spacetime may shed light on the origin of the Debye entropic gravity and the
microscopic origin for the Newton’s law of gravity and also Friedmann
equations in cosmology.

###### Acknowledgements.

This work has been supported financially by Research Institute for Astronomy
and Astrophysics of Maragha (RIAAM), Iran.

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