# Unification of Dark Matter and Dark Energy
in a Modified Entropic Force Model

Zhe Chang1,2111E-mail<EMAIL_ADDRESS>Ming-Hua Li1,2222E-mail:
<EMAIL_ADDRESS>Xin Li2,3333E-mail<EMAIL_ADDRESS>1Institute of High Energy
Physics, Chinese Academy of Sciences, 100049 Beijing, China
2Theoretical Physics Center for Science Facilities, Chinese Academy of
Sciences
3Institute of Theoretical Physics, Chinese Academy of Sciences, 100190
Beijing, China

###### Abstract

In Verlinde’s entropic force scenario of gravity, Newton’s laws and Einstein
equations can be obtained from the first pinciples and general assumptions.
However, the equipartition law of energy is invalid at very low temperatures.
We show clearly that the threshold of the equipartition law of energy is
related with horizon of the universe. Thus, a one-dimension Debye (ODD) model
in the direction of radius of the modified entropic force (MEF) maybe suitable
in description of the accelerated expanding universe. We present a Friedmann
cosmic dynamical model in the ODD-MEF framework. We examine carefully
constraints on the ODD-MEF model from the Union2 compilation of the Supernova
Cosmology Project (SCP) collaboration, the data from the observation of the
large-scale structure (LSS) and the cosmic microwave background (CMB), i.e.
SNe Ia+LSS+CMB. The combined numerical analysis gives the best-fit value of
the model parameters $\zeta\simeq 10^{-9}$ and $\Omega_{m0}=0.224$, with
$\chi_{min}^{2}=591.156$. The corresponding age of the universe agrees with
the result of D. Spergel et al.Spergel2003  at $95\%$ confidence level. The
numerical result also yields an accelerated expanding universe without
invoking any kind of dark energy. Taking $\zeta(\equiv 2\pi\omega_{D}/H_{0})$
as a running parameter associated with the structure scale $r$, we obtain a
possible unified scenario of the asymptotic flatness of the radial velocity
dispersion of spiral galaxies, the accelerated expanding universe and the
Pioneer 10/11 anomaly in the entropic force framework of Verlinde.

###### pacs:

95.36.+x,95.35.+d,98.80.-k

##  I 1\. Introduction

The law of gravity, which was first discovered by Isaac Newton in the 18th
century and later reformulated by Albert Einstein in the early 20th century,
still remains unclear in the microscopic level nowadays. Enlightened by
Hawking et al.’s workHawking ; Bekenstein ; Hawking2  about the black hole
entropy in 1970’s, JacobsonJacobson  got the gravitational field equations as
Einstein’s, starting from the first law of thermodynamics. Further research of
PadmanabhanPadmanabhan1 ; Padmanabhan2  also gives gravity a thermodynamical
interpretation. These results prompt people to take a statistical physics
point of view on gravity. Recently, VerlindeVerlinde  reinterpreted gravity as
an entropic force caused by a change of amount of information associated with
the positions of bodies of matter. Newton’s second law could be obtained with
the introduction of Unruh temperatureUnruh . He also got the Newton’s law of
gravity and the Komar’s definition of massKomar  in a static curved space in
relativistic case from the holographic principleSusskind  and black hole
thermodynamics.

Lots of work have been done to reveal the implications of the entropic force
interpretation of gravity. To name a few: derivation of Friedmann equation in
the entropic force frameworkShu ; Cai1 ; Cai3 ; Cai4 , the corresponding
Newton gravity formulation in loop quantum gravitySmolin , the construction of
holographic actions from black hole entropyMakela ; Caravelli , the entropic
force scenario of holographic dark energyLiM , the generalization of the
Newton’s potential in the relativistic caseTian , and the modified entropic
force(MEF) due to the modification of the equipartition law of energy at very
low temperatureGao , etc.

In Verlinde’s paperVerlinde , the equipartition law of energy of free
particles plays a key role in the derivation of Newton’s laws. GaoGao  pointed
out that the equipartition law of energy does not hold at very low
temperatures. He made a modification simply using the three-dimension Debye
model in solid state physics. His work provides an explanation for the
accelerated expanding universe without invoking dark energy. However, we note
that only the vibration in the direction of radius is related with an
observable quantity and the “low temperature” should be fixed by a threshold.
So that, a one-dimension Debye(ODD) function is suitable to be used to revise
the free particle’s equipartition law of energy in the study of cosmologyXin
Li . In fact, such a modification leads us to the famous modification of
Newtonian dynamics(MOND). MOND was constructed to account for the asymptotic
flatness of the rotational velocity curves of spiral galaxiesTrimble ; Rubin ;
TF .

In this paper, we show clearly that the threshold of equipartition law of
energy is related with horizon of the universe. A Friedmann cosmic dynamical
model is set up in the framework of the ODD-MEF. We examine carefully
constraints on the ODD-MEF model from the Union2 compilation of the Supernova
Cosmology Project(SCP) collaboration, the data from the observation of the
large-scale structure(LSS) and the cosmic microwave background(CMB), i.e. SNe
Ia+LSS+CMB. Results yield an accelerated expanding universe without invoking
any kind of dark energy. Furthermore, by taking $\zeta(\equiv
2\pi\omega_{D}/H_{0})$ as a running parameter associated with the structure
scale $r$, we obtain a possible unified scenario of the asymptotic flatness of
the radial velocity dispersion of spiral galaxies, the accelerated expanding
universe and the Pioneer 10/11 anomaly in the entropic force framework of
Verlinde.

The paper is organized as follows. In section 2, a brief review on Verlinde’s
work and the connection between the ODD-MEF model and MOND is presented. We
derive the corresponding Friedmann equation and consider the cosmological
constraints on it in setion 3. In section 4, an alternative approach to the
Pioneer 10/11 anomaly is presented. In section 5, by taking $\zeta$ as a
running parameter associated with the structure scale $r$, we suggest a
unified scenario of the asymptotic flatness of the rotation curves of spiral
galaxies, the accelerated expanding universe and the Pioneer 10/11 anomaly in
the framework of the ODD-MEF. Conclusions and further discussions can be found
in section 6.

##  II 2\. From Modified Entropic Force To Mond

Think of a closed holographic screen and a free particle of mass $m$ near it
on the side that spacetime has already emerged. The particle moves towards the
screen, traveling a distance $\triangle x$ before merging into it. According
to BekensteinBekenstein , the change of entropy of the screen associated with
the amount of information stored on it is

$\triangle S=2\pi k_{B}\frac{mc}{\hbar}\triangle x\ ,$ (1)

where $k_{B}$ is the Boltzmann constant and $c$ denotes the speed of light.
The particle will experience an effective macroscopic force due to the
statistical tendency to increase its entropy. This is described by

$F\triangle x=T\triangle S\ ,$ (2)

where $T$ is the temperature of the screen. By introducing Unruh’s relation
between acceleration and temperatureUnruh

$k_{B}T=\frac{1}{2\pi}\frac{\hbar a}{c}\ ,$ (3)

one recovers the second law of Newton

$F=ma\ .$ (4)

Suppose the holographic screen has a radius $R$, the area of which is $A=4\pi
R^{2}$. In theory of emergent space, each fundamental bit occupies one unit
cell, the area of which on the screen is defined as
$L_{p}\equiv\sqrt{G\hbar}$. $L_{p}$ is the Planck length. $G$ is mere a new
constant here, which is later identified with Newton’s constant. Let’s denote
the number of bits by $N$, which is given as

$N\equiv\frac{Ac^{3}}{L_{p}^{2}}=\frac{Ac^{3}}{G\hbar}=\frac{4\pi
R^{2}c^{3}}{G\hbar}\ .$ (5)

Each bit represents a microscopic degree of freedom. The total energy of the
screen is given by the equipartition law

$E=\frac{1}{2}Nk_{B}T\ .$ (6)

Assuming that the energy of the screen is proportional to the mass $M$ that
would emerge in the part of space enclosed by the screen itself as

$E=Mc^{2}\ $ (7)

and using the equation (4), (5), (6) and (7), one obtains the familiar
lawVerlinde :

$F=G\frac{Mm}{R^{2}}\ .$ (8)

Thermodynamics and statistical physics tells us that the equipartition law of
energy is valid only when the temperature is not very low. On the other hand,
the Debye model in solid state physics is found to be successful in describing
experimental results for most of the solid objects at very low
temperaturesDebye . It should be noticed that the threshold of low energy for
each direction of vibration is different, because it is related with the
structure scale in the radial direction. In the case of astronomy and
cosmology, the only temperature threshold is related with horizon of the
universe. Thus, we focus on the ODD-MEF model throughout this paper. The
equipartition law of energy can be rewritten as

$E=\frac{1}{2}Nk_{B}T\mathfrak{D}(x)\ ,$ (9)

where the one-dimension Debye function is defined as

$\mathfrak{D}(x)\equiv\frac{1}{x}\int^{x}_{0}\frac{y}{e^{y}-1}dy\ .$ (10)

$x$ is related to the Debye frequency $\omega_{D}$ and defined as

$x\equiv\frac{\hbar\omega_{D}}{k_{B}T}=\frac{2\pi c\omega_{D}}{a}\ .$ (11)

Combining the equations (4), (5), (7) and (9), we obtain the modified Newton’s
law of gravitation

$\frac{GM}{R^{2}}=a\mathfrak{D}\left(\frac{2\pi c\omega_{D}}{a}\right)\ .$
(12)

There are two limit cases for the equation (12). One is the high temperature
limit, with $x\ll 1$. We have

$\mathfrak{D}(x)\approx\frac{1}{x}\int^{x}_{0}dy=1.$ (13)

Therefore, the modified equipartition law of energy returns to the equation
(6), with which Newton’s law of gravity the equation (8) is recovered.

The other is the low temperature limit, with $x\gg 1$. The Debye function
$\mathfrak{D}(x)$ reduces to

$\mathfrak{D}(x)\approx\frac{1}{x}\int^{\infty}_{0}\frac{y}{e^{y}-1}dy=\frac{\pi^{2}}{6x}.$
(14)

Then the equation (12) reads

$\frac{GM}{R^{2}}=\frac{\pi}{12c\omega_{D}}a^{2}=\frac{a^{2}}{a_{0}}\ ,$ (15)

where the constant $a_{0}$ is defined as

$a_{0}\equiv\frac{12c\omega_{D}}{\pi}.$ (16)

Thus, $x$ can be rewritten as

$x=\frac{2\pi c\omega_{D}}{a}\ =\frac{\pi^{2}a_{0}}{6a}\ $ (17)

and the equation (10) turns out to be

$\displaystyle\mathfrak{D}(x)=\frac{6}{\pi^{2}}\frac{a}{a_{0}}\int_{0}^{\frac{\pi^{2}a_{0}}{6a}}\frac{y}{e^{y}-1}dy\
.$ (18)

Defining the function

$\mu(t)=\frac{6}{\pi^{2}}t\int_{0}^{\frac{\pi^{2}}{6t}}\frac{y}{e^{y}-1}dy\ ,$
(19)

where $t=a/a_{0}$, one can immediately obtain the gravity law of MONDXin Li :

$\frac{GM}{R^{2}}=a\mathfrak{D}\left(\frac{2\pi
c\omega_{D}}{a}\right)=a\mu(x)\ .$ (20)

Note that the definition of $\mu(x)$ in the equation (19) has the
propertyMilgrom

$\lim_{x\gg 1}\mu(x)=1~{}~{}~{}{\rm and}~{}~{}~{}\lim_{x\ll 1}\mu(x)=x\ $ (21)

as demanded by MOND.

The equation (20) implies the modified Poisson equation for the gravitational
potential $\phi$

$\nabla\cdot(\mu(|\nabla\phi|/a_{0})\nabla\phi)=4\pi G\rho\ ,$ (22)

where $\rho$ is the energy density of matter sources.

In MOND, in order to explain the observed rotational velocity curves of spiral
galaxies, $a_{0}$ is suggested to be of the order of

$a_{0}\sim~{}10^{-8}cm/s^{2}\ .$ (23)

MilgromMilgrom ; Milgrom1  found that

$2\pi a_{0}\approx cH_{0}\ .$ (24)

Combining the equation (16) and (24), we get that

$H_{0}\approx 24\omega_{D}\ .$ (25)

Thus, the corresponding wave length of $\omega_{D}$ is

$\lambda<c/\omega_{D}=c/24H_{0}\ .$ (26)

It really should be less than the cosmological horizon $L_{h}\sim c/H_{0}$.
Thus, Milgrom’s relation between the threshold of the acceleration $a_{0}$ and
the Hubble constant $H_{0}$ has a deep origin.

##  III 3\. The Cosmological Constraints

###  III.1 3.1 The Modified Friedmann Model

For convenience, we adopt the natural unit $c=k_{B}=1$ in this section.

The $(3+1)$-demension Friedmann-Robertson-Walker(FRW) metric can be written as

$ds^{2}=h_{ab}dx^{a}dx^{b}+\tilde{r}^{2}d\Omega_{2}^{2}\ ,$ (27)

where $\tilde{r}=a(t)r$ and the two-dimension metric $h_{ab}={\rm
diag}(-1,a^{2}/(1-kr^{2}))$, with $k=0,-1,1$ refers to a flat, open and closed
universe respectively. Given
$h^{ab}\partial_{a}\tilde{r}\partial_{b}\tilde{r}=0$, one gets the radius of
the apparent horizon(AH)Hawking3 :

$\tilde{r}_{A}=\frac{1}{\sqrt{H^{2}+k/a^{2}}}\ .$ (28)

$H\equiv\dot{a}/a$ is the Hubble parameter and the dot “ $\cdot$ ” denotes the
derivative with respect to time $t$. The area of the AH is
$A=4\pi{\tilde{r}_{A}}^{2}$. The number of bits of information stored on the
AH is

$N=4\pi\tilde{r}_{A}^{2}/L_{p}^{2}\ .$ (29)

Suppose from time $t$ to $t+dt$, the radius of the AH changes from
$\tilde{r}_{A}$ to $\tilde{r}_{A}+d\tilde{r}_{A}$. The full derivative of the
identity (9) gives the change of the total energy on the screen as

$dE=\frac{1}{2}N\mathfrak{D}(x)dT+\frac{1}{2}T\mathfrak{D}(x)dN+\frac{1}{2}NTd\mathfrak{D}(x)\
.$ (30)

Let’s first do formal reductions of the above identity. The temperature of the
AH, the so-called Hawking temperatureHawking2 ; Cai2 , is

$T_{A}=\hbar/(2\pi\tilde{r}_{A})\ .$ (31)

The change of $T_{A}$ is of the form

$dT_{A}=-\frac{\hbar}{2\pi\tilde{r}_{A}^{2}}d\tilde{r}_{A}\ .$ (32)

From the equation (29), we get

$dN=\frac{8\pi\tilde{r}_{A}}{L_{p}^{2}}d\tilde{r}_{A}\ .$ (33)

The full derivative of the one-dimension Debye function takes the form

$d\mathfrak{D}(x)=\left[-\frac{1}{x}\mathfrak{D}(x)+\frac{1}{e^{x}-1}\right]dx\
.$ (34)

The identity (11) gives

$dx=-\frac{x}{T}dT\ .$ (35)

Thus, the equation (34) can be rewritten as

$d\mathfrak{D}(x)=\left[\mathfrak{D}(x)-\frac{x}{e^{x}-1}\right]\frac{dT_{A}}{T_{A}}\
.$ (36)

Making use of the equations (29), (31), (32), (33) and (36 ), one can rewrite
the equation (30) as

$\displaystyle dE=$
$\displaystyle\left[\frac{1}{2}\cdot\frac{4\pi\tilde{r}_{A}^{2}}{L_{p}^{2}}\cdot\left(-\frac{\hbar}{2\pi\tilde{r}_{A}^{2}}d\tilde{r}_{A}\right)+\frac{1}{2}\cdot\left(\frac{\hbar}{2\pi\tilde{r}_{A}}\right)\cdot\left(\frac{8\pi\tilde{r}_{A}}{L_{p}^{2}}d\tilde{r}_{A})\right)\right]\cdot\mathfrak{D}(x)$
$\displaystyle+\frac{1}{2}NT_{A}\cdot\left[\mathfrak{D}(x)-\frac{x}{e^{x}-1}\right]\frac{dT_{A}}{T_{A}}$
$\displaystyle=$
$\displaystyle\left(\frac{x}{e^{x}-1}\right)\frac{d\tilde{r}_{A}}{G}\ .$ (37)

From the definition of the AH (28) and remembering $H\equiv\dot{a}/a$, we have

$d\tilde{r}_{A}=-H\tilde{r}_{A}^{3}\left(\dot{H}-\frac{k}{a^{2}}\right)dt\ .$
(38)

Thus, we get

$dE=-H\tilde{r}_{A}^{3}\left(\dot{H}-\frac{k}{a^{2}}\right)\left(\frac{x}{e^{x}-1}\right)\frac{dt}{G}\
.$ (39)

Next, we consider the energy-momentum tensor of a perfect isotropic fluid
$T_{\mu\nu}=(\rho+p)U_{\mu}U_{\nu}+pg_{\mu\nu}$, where $\rho$ and $p$ are
respectively the energy density and the pressure. This is just the case of the
matter in the universe. The energy flow of the matter isShu

$dE=4\pi\tilde{r}_{A}^{2}T_{\mu\nu}k^{\mu}k^{\nu}dt=4\pi\tilde{r}_{A}^{3}(\rho+p)Hdt,$
(40)

where the Killing vector of the horizon $k^{\mu}=(1,-Hr,0,0)$.

Identifying the equation (39) and (40), one obtains

$\left(\frac{x}{e^{x}-1}\right)\left(\dot{H}-\frac{k}{a^{2}}\right)=-4\pi
G(\rho+p).$ (41)

Combining the equation (41) with the energy conservation equation

$\dot{\rho}+3H(\rho+p)=0\ ,$ (42)

we get

$\left(\frac{x}{e^{x}-1}\right)\left(\dot{H}-\frac{k}{a^{2}}\right)=-\frac{4\pi
G}{3H}\dot{\rho}\ .$ (43)

Notice that we consider a spatially flat, matter dominated universe throughout
this paper, namely $k=0$ and
$\rho=\rho_{m}=\rho_{m0}\,(1+z)^{3}=\rho_{m0}\,a^{-3}$. The equation (III.1)
multiplying by $2Hdt/H_{0}^{2}$ makes

$\displaystyle\left(\frac{x}{e^{x}-1}\right)\frac{2H\dot{H}}{H_{0}^{2}}dt$
$\displaystyle=$ $\displaystyle-\frac{8\pi G}{3H_{0}^{2}}\dot{\rho}dt\ ,$
$\displaystyle\left(\frac{x}{e^{x}-1}\right)\frac{d(H^{2})}{H_{0}^{2}}$
$\displaystyle=$ $\displaystyle-\frac{8\pi G}{3H_{0}^{2}}d\rho\ .$ (44)

With the notations

$\Omega_{m0}\equiv\frac{8\pi
G\rho_{m0}}{3H_{0}^{2}}\,,~{}~{}~{}~{}~{}~{}~{}E\equiv\frac{H}{H_{0}}=\frac{\zeta}{x}\,,~{}~{}~{}~{}~{}~{}~{}\zeta\equiv\frac{H_{D}}{H_{0}}\equiv\frac{2\pi\omega_{D}}{H_{0}}\,,$
(45)

one obtains

$\left(\frac{x}{e^{x}-1}\right)dE^{2}=\Omega_{m0}~{}da^{-3}\ .$ (46)

$E$ is also called the reduced Hubble parameter.

Instead of numerically solving the differential equation (46) to get the
modified Friedmann equation of the MEF model, we need to find an approximated
expression of it. Note that $E(z)$ grows rapidly as the redshift $z$
increases. Suppose that $x=\zeta/E(z)$ is a small quantity today, i.e. $x\ll
1$, so that we can expand $x/(e^{x}-1)$ in powers of $x$ near $x=0$:

$\frac{x}{e^{x}-1}=1-\frac{x}{2}+{\cal
O}(x^{2})~{},~{}~{}~{}~{}~{}~{}~{}~{}x\ll 1\ .$ (47)

Then we get the approximated expression of the equation (46)

$\displaystyle\Omega_{m0}~{}da^{-3}$ $\displaystyle\thickapprox$
$\displaystyle\left(2E-\zeta\right)dE\ .$ (48)

Integrating the equation (48), we have

$E^{2}-\frac{3}{2}\,\zeta E=\Omega_{m0}\,a^{-3}+{\rm const.}\ ,$ (49)

where “ const. ” is an integral constant and should be determined by
$E(a=1)=1$. Finally, one obtains a quadratic equation for $E$,

$E^{2}-\zeta E=\Omega_{m0}a^{-3}+(1-\zeta-\Omega_{m0})\ .$ (50)

Note that $E$ only allows of a positive value by its definition (45). Solving
the equation (50), one gets the Friedmann equation in the ODD-MEF model,

$E(a)=\frac{1}{2}\zeta+\frac{1}{2}\left[(\zeta-2)^{2}+4\Omega_{m0}(a^{-3}-1)\right]^{1/2}\
,$ (51)

or equivalently,

$E(z)=\frac{1}{2}\zeta+\frac{1}{2}\left[(\zeta-2)^{2}+4\Omega_{m0}((1+z)^{3}-1)\right]^{1/2}\
.$ (52)

As long as $\zeta\rightarrow 0$, the ODD-MEF model reduces to the
$\Lambda$-CDM model, in which

$E(a)=\left[\Omega_{m0}a^{-3}+\left(1-\Omega_{m0}\right)\right]^{1/2}\ .$ (53)

In astronomy, distance modulus is defined asCarroll

$\mu_{th}(z)\equiv 5\log_{10}[d_{L}(z)({\rm Mpc})]+25\ ,$ (54)

where $d_{L}$ is the luminosity distance:

$\displaystyle d_{L}(z)$ $\displaystyle=$
$\displaystyle\frac{(1+z)}{H_{0}}\int_{(1+z)^{-1}}^{1}\frac{da}{a^{2}E(a)}$
(55) $\displaystyle=$
$\displaystyle\frac{(1+z)}{H_{0}}\int_{(1+z)^{-1}}^{1}\frac{2da}{a^{2}\cdot\left\\{\zeta+\left[(\zeta-2)^{2}+4\Omega_{m0}(a^{-3}-1)\right]^{1/2}\right\\}}\
.$

Finally, we give the specific form of the deceleration parameter $q$ as a
function of $E(z)$. Remembering $H\equiv\dot{a}/a$, $a=1/(1+z)$ and

$\displaystyle\frac{d}{dt}=\frac{da}{dt}\cdot\frac{dz}{da}\cdot\frac{d}{dz}=-\frac{H}{a}\frac{d}{dz}\
,$ (56)

one can easily get the following identities

$q\equiv-\frac{\ddot{a}}{aH^{2}}=-1+(1+z)\,E^{-1}\frac{dE}{dz}\ .$ (57)

###  III.2 3.2 The Numerical Study

In this subsection, we consider the cosmological constraints on the ODD-MEF
model from the observational data. First, we limit ourselves to the 557 SNe Ia
from the Union2 compilation of the Supernova Cosmology Project (SCP)
collaboration . Next, we include the data from the observation of the large-
scale structure (LSS) and the cosmic microwave background (CMB) as useful
complements for the SNe Ia data to put a joint constraint on the parameters of
the model.

The Union2 compilation consists of 557 SNe Ia, which is the largest published
and spectroscopically confirmed sample of SNe Ia so farSCP . The $\chi^{2}$
statistic of the sample is given by

$\chi_{SN}^{2}(\zeta,\Omega_{m0})=\sum\limits_{i=1}^{557}\frac{\left[\mu_{obs}(z_{i})-\mu_{th}(z_{i};\zeta,\Omega_{m0})\right]^{2}}{\sigma^{2}(z_{i})}\
,$ (58)

where $\mu_{obs}(z_{i})$ and $\sigma(z_{i})$ are respectively the observed
value and the $1\sigma$ uncertainty of the distance modulus of 557 Union2 SNe
Ia. Given the redshift $z_{i}$ and the parameter values of $\zeta$ and
$\Omega_{m0}$, one can get the theoretical, model-related value of the
distance modulus $\mu_{th}$ according to the equation (54). By minimizing the
$\chi_{SN}^{2}$, one obtains the best-fit values of the parameters:
$\zeta\simeq 10^{-7}$ and $\Omega_{m0}=0.192$, with a minimum value of
$\chi^{2}$ as $\chi_{min}^{2}=574.583$.

In addition, we consider the cosmic age in the ODD-MEF model. The age of the
universe is given by

$t_{0}=H_{0}^{-1}\int_{0}^{\infty}\frac{dz}{(1+z)E(z)}\ ,$ (59)

where $H_{0}^{-1}$ represents the Hubble time, with the value
$H_{0}^{-1}=9.778h^{-1}$Gyr. We take $h$ to be $0.72$. According to D. Spergel
et al.Spergel2003 , the age of the universe is $t_{0}=13.7\pm 0.2$Gyr. This
can also be written in the dimensionless age parameter $H_{0}t_{0}\simeq
0.99$, with $1\sigma$ confidence level range $0.96\lesssim H_{0}t_{0}\lesssim
1.05$Zhang . We show these in FIG.1, together with the $70\%$, $95\%$ and
$99\%$ confidence level contours in the $\Omega_{m0}$-$\zeta$ parameter plane.

As we see, the SNe Ia data alone are not enough to put a very strong
constraint on the parameters. The parameter $\zeta$ can still take a value
close to $1$. This fact invalidates the approximation (47). To recognize this,
one should notice that $E(z)\equiv H/H_{0}=\zeta/x=1$ today. A large $x$
implies a large $\zeta$. Moreover, one can see that the best-fit result of the
SNe Ia data is almost ruled out by the age of the universe at $1\sigma$
confidence level ($0.96\lesssim H_{0}t_{0}\lesssim 1.05$). So in what follows,
we use the data from the observation of the large-scale structure (LSS) and
the cosmic microwave background (CMB) to enforce a more rigorous constraint on
the model parameters.

Here, we only take into account the most conservative and robust informations
from the LSS and the CMB observationsZhang . For the LSS, we use the distance
parameter ${\cal A}$. It can be obtained from a spectroscopic sample of 46,748
luminous red galaxies of the Sloan Digital Sky Survey (SDSS). It is defined
asSDSS ; Eisenstein

${\cal
A}\equiv\Omega_{m0}^{1/2}\,E(z_{b})^{-1/3}\left[\frac{1}{z_{b}}\int_{0}^{z_{b}}\frac{dz}{E(z)}\right]^{2/3}\
,$ (60)

where $z_{b}=0.35$. $E(z)$ is given by equation (52). The value of ${\cal A}$
has been determined to be ${\cal A}_{0}=0.469\pm\sigma_{\cal A}$, where the
$1\sigma$ uncertainty $\sigma_{\cal A}=0.017$Eisenstein .

For the CMB, we use the measurement of shift parameter $\cal R$ alone. It is
defined asWang ; Tegmark97

${\cal R}\equiv\Omega_{m0}^{1/2}\int_{0}^{z_{\ast}}{dz\over E(z)}~{},$ (61)

where the redshift of recombination is found to be ${z_{\ast}}=1091.3$
according to the WMAP 7-year (WMAP7) dataWMAP7 . The value of $\cal R$ has
been updated to ${\cal R}_{0}=1.725\pm\sigma_{\cal R}$, where the $1\sigma$
uncertainty $\sigma_{\cal R}=0.018$WMAP7 .

To include both the LSS and the CMB observation as well as the 557 Union2 SNe
Ia data to perform a combined numerical analysis of the parameters, we use a
$\chi^{2}$ statistic as

$\chi^{2}=\chi_{SN}^{2}+\chi_{LSS}^{2}+\chi_{CMB}^{2}\ ,$ (62)

where $\chi_{SN}^{2}$ is given by the identity (58). $\chi_{CMB}^{2}$ and
$\chi_{LSS}^{2}$ are the contributions from the CMB and the LSS data, which
are defined as

$\chi^{2}_{LSS}=\frac{({\cal A}-{\cal A}_{0})^{2}}{\sigma_{\cal
A}^{2}}~{}~{}~{}and~{}~{}~{}\chi^{2}_{CMB}=\frac{({\cal R}-{\cal
R}_{0})^{2}}{\sigma_{\cal R}^{2}}\ ,$ (63)

where ${\cal A}$ and $\cal R$ are given by the identity (60) and (61)
respectively. By minimizing such a $\chi^{2}$, one gets the best-fit parameter
values $\zeta\simeq 10^{-9}$ and $\Omega_{m0}=0.224$, with
$\chi_{min}^{2}=591.156$. These results are shown in FIG.2. For comparison, we
also present the result for the SNe Ia+LSS fit in FIG.3, of which we use a
$\chi^{2}$ statistic

$\chi^{2}=\chi_{SN}^{2}+\chi_{LSS}^{2}\ .$ (64)

The best-fit parameter values are $\zeta\simeq 10^{-8}$ and
$\Omega_{m0}=0.213$, with $\chi_{min}^{2}=584.831$.

One can see that after using the LSS and the CMB data, we successfully put a
much more rigorous constraint on $\zeta$. It is within the range
$0\leq\zeta\leq 0.09$ at $95\%$ confidence level. This result confirms that
our approximation (47) is reasonable. And it is more than one order smaller
than Wei’s predictionWei . In addition, the best-fit result of the combined
analysis is in accordance with the age of the universe given by D. Spergel et
al.Spergel2003  at $1\sigma$ confidence level.

The corresponding distance modulus $\mu_{th}(z)$ is shown in FIG.4. And the
corresponding reduced Hubble parameter $E(z)$ and the deceleration parameter
$q(z)$ are plotted in FIG.5 and FIG.6. Notice that $q=0$ at the redshift
$z_{t}=0.906$ for the SNe Ia+LSS+CMB fit. $z_{t}$ is called the transition
redshift. This fact implies that the universe goes from a decelerated
expanding period to an accelerated expanding period at a time point in the
early epoch, in the absence of any dark matter or dark energy.

##  IV 4\. The Pioneer 10/11 Anomaly

Last, but not the least, let us consider the Pioneer 10/11 anomaly in the ODD-
MEF model. The Pioneer 10 and 11 spacecraft, which were launched in 1972 and
1973, are the two most accurately navigated vehicles in the solar system. The
radio-metric data received by them at the heliocentric distances ranging from
$20$-$70$ AU has consistently indicated the existence of a small, anomalous
frequency drift changing with a rate of $6\times 10^{-9}$ Hz/sAnderson , which
was later interpreted as a constant, sunward accelerationTuryshev1

$a_{P}=(8.74\pm 1.33)\times 10^{-10}~{}m/s^{2}\ .$ (65)

This apparent violation of the Newton’s law of gravity has been known as the
Pioneer anomaly, the nature of which still remains unclear to date.

Several mechanisms have been developed to explain for the anomaly. To name a
fewTuryshev2 , it could be attributed to the possible systematic errors due to
the gas leaks from the propulsion system, the gravitational effect of the
Kuiper Belt Objects or dust, or the influence of the expansion of the
universe, etc. In our previous workXin Li2 , we have also provided a possible
explanation of the Pioneer 10/11 anomaly in the framework of Finsler geometry.

Here, we present another possible interpretation of the Pioneer anomaly in the
ODD-MEF model. Starting from the identity (12) and using the approximation
(47), we have

$\displaystyle\frac{GM}{R^{2}}=a\mathfrak{D}(x)$ $\displaystyle=$
$\displaystyle a\cdot\frac{1}{x}\int^{x}_{0}\frac{y}{e^{y}-1}dy$ (66)
$\displaystyle=$ $\displaystyle a-\frac{\pi cw_{D}}{2}\ .$

We may rewrite it as

$a=\frac{GM}{R^{2}}+\frac{\pi cw_{D}}{2}\ .$ (67)

In the above identity, $GM/R^{2}$ is the conventional Newtonian prediction of
the gravitational acceleration at a distance $R$ from the source with mass
$M$. We take the left hand side of the equation (67), i.e. $a$, to be the
acceleration that the spacecraft actually experiences. Then the second term at
the right hand side should be interpreted as the abnormal acceleration
$a_{P}$, namely

$a=\frac{GM}{R^{2}}+a_{P}\ ,$ (68)

where

$a_{P}\equiv\frac{\pi cw_{D}}{2}\ .$ (69)

Suppose that the parameter takes a value as $\zeta\approx 5$ over the
heliocentric distance ranging from $20$ to $70$ AU. This is the structure
scale where the abnormal acceleration $a_{P}$ has been detected. Using the
definition (45) of $\zeta$, we get the magnitude of $\omega_{D}$

$w_{D}=\frac{H_{0}}{2\pi}\zeta\approx 2.0\times 10^{-18}~{}{\rm Hz}\ .$ (70)

With the identity (69), one gets the abnormal acceleration predicted by the
ODD-MEF model

$a_{P}\equiv\frac{\pi cw_{D}}{2}~{}\approx~{}8.74\times 10^{-10}~{}m/s^{2}\ .$
(71)

It agrees with the detected result (65). The plus sign in the equation (68)
illustrates that the direction of $a_{P}$ is in the same direction as the
conventional Newtonian prediction $GM/R^{2}$, which is pointing towards the
sun. This is in accordance with the detection.

##  V 5\. A Unified Scenario of Anomalies in Gravity

In section 4, we found that $\zeta\approx 5$ provides a possible explanation
for the Pioneer 10/11 anomaly in the solar system. On the other hand, to
account for the observations of the SNe Ia, the LSS and the CMB, $\zeta$ is
found to be $\zeta\sim 10^{-9}$. This was demonstrated in section 3.

In the following, we will find out the magnitude of $\zeta$ corresponding to
MOND. Making use of

$w_{D}=\frac{H_{0}}{2\pi}\zeta$ (72)

and the identity (16), we get

$\displaystyle a_{0}$ $\displaystyle=$
$\displaystyle\frac{6cH_{0}}{\pi^{2}}\zeta\ .$ (73)

The above identity multiplying $2\pi$ makes

$2\pi a_{0}=\frac{12\zeta}{\pi}cH_{0}\ .$ (74)

Note that in the equation (24), $2\pi a_{0}\approx cH_{0}$. So we have

$\frac{12\zeta}{\pi}\approx 1\ ,$ (75)

which implies

$\zeta\approx 0.25\ .$ (76)

In summary, we obtain three different magnitudes of $\zeta$ with respect to
three different structure scales:

$\zeta\sim\begin{cases}10^{-9}\ ,&10^{8}\sim
10^{10}~{}$pc~{}~{}~{}~{}~{}$(\text{SNe Ia+LSS+CMB})\\\ 0.25\ ,&10^{3}\sim
10^{4}~{}$pc~{}~{}~{}~{}~{}~{}$(\text{MOND})\\\ 5\ ,&10^{-4}\sim
10^{-3}~{}$pc~{}~{}~{}$(\text{Pioneer 10/11 anomaly})\end{cases}\ ~{}~{}~{},$
(77)

where $1~{}$pc$=2.06\times 10^{5}$ AU$=3.09\times 10^{16}$ m$=3.26$ l.y.. One
can see that $\zeta$ is found to be of a declining value while the relevant
structure scale increases.

##  VI 6\. Conclusions and Remarks

VerlindeVerlinde  interpreted gravity as an entropic force due to the tendency
of the system to increase its entropy, which is associated with the change of
positions of material bodies. With the holographic principle, the total
entropy of the system can be calculated via the area $A$ and the temperature
$T$ of its boundary. With Unruh’s relation of the acceleration and the
temperature experienced by the accelerated observer, one can easily recover
the laws of Newton. However, at very low temperatures, the equipartition law
of energy is no longer valid. We noted that the threshold of low temperature
is related to the specific dimension of the direction. In the case of
astronomy and cosmology, the only threshold comes from the limited radius of
the solar system , the galaxies and the universe. We made a modification by
revising the equipartition law of energy with the one-dimension Debye function
(10). It was referred as the ODD-MEF model.

With the definition (16), we deduced the famous formula of MOND. To consider
the cosmological constraints on the model, we derived the Friedmann equation
(52). We made use of the Union2 compilation of the Supernova Cosmology
Project(SCP) collaboration, the data from the observation of the large-scale
structure(LSS) and the cosmic microwave background(CMB), i.e. SNe Ia+LSS+CMB,
to put a joint constraint on the parameters $\zeta$ and $\Omega_{m0}$. The
combined numerical analysis gives the best-fit values of the model parameters
$\zeta\simeq 10^{-9}$ and $\Omega_{m0}=0.224$, with $\chi_{min}^{2}=591.156$.
In FIG.2, one can see that the best-fit result agrees with the age prediction
of the universe by D. Spergel et al.Spergel2003  and the WMAP collaboration at
$1\sigma$ confidence level. And the parameter $\zeta$ is successfully limited
to a narrow range $0\leq\zeta\leq 0.09$ at $95\%$ confidence level. It is more
than one order smaller than Wei’s prediction $0\leq\zeta\leq 0.2$Wei .

In the ODD-MEF model, we introduced a dimensionless, structure scale-related
parameter $\zeta$. Phenomenological study indicated that different values of
$\zeta$ corresponds to dynamics at different distance scales, as in the
identity (77). Therefore, we are allowed to suggest a unified scenario of the
asymptotic flatness of the rotation curves of spiral galaxies, an accelerated
expanding universe, and the Pioneer 10/11 anomaly in the solar system in a
single framework. Thus, we got a possible unification of dark matter and dark
energy in the one-dimensional Debye model of the modified entropic force.

Since $\zeta$ is a running parameter associated with the structure scale $r$,
a specific form of the function $\zeta(r)$ should be given. As one foresees,
this function should have the monotonically decreasing property as shown in
(77). Such a running $\zeta$ also implies that, other than the ODD-MEF model,
there exists a more fundamental theory, which taking the model as its
effective scenario. To find the specific $\zeta(r)$ and such a fundamental
theory is one of the main subjects of our future investigation.

###### Acknowledgements.

##  VII Acknowledgments

This work was supported by the National Natural Science Fund of China under
Grant No. 10575106 and No. 10875129.

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Figure 1: The best-fit $70\%$, $95\%$ and $99\%$ confidence regions in the
$\Omega_{m0}$-$\zeta$ plane with the isochrones of constant $H_{0}t_{0}$, for
the Union2 compilation of the Supernova Cosmology Project(SCP) collaboration.
The dark gray dashed line denotes constraints from the age of the universe at
the $1\sigma$ confidence level $0.96\lesssim H_{0}t_{0}\lesssim 1.05$, with a
central value $H_{0}t_{0}\simeq 0.99$. It can be seen that the best-fit result
of the SNe Ia data is almost excluded by the age of the universe at the
$1\sigma$ confidence level. The isochrones are labeled for the case of
$H_{0}=72~{}{\rm km~{}s^{-1}~{}Mpc^{-1}}$. The dark gray thick solid lines
indicate the age of the universe $t_{0}$ in the unit of Hubble time
$H_{0}^{-1}$. The dot with the coordinate $(0.192,10^{-7})$ represents the
best-fit value, with $\chi_{min}^{2}=574.583$. The best-fit result is almost
excluded by E. Komatsu et al.WMAP7  in $1\sigma$ confidence level.

Figure 2: The best-fit $70\%$, $95\%$ and $99\%$ confidence regions in the
$\Omega_{m0}$-$\zeta$ plane with the isochrones of constant $H_{0}t_{0}$, for
the Union2 compilation of the Supernova Cosmology Project(SCP) collaboration
plus the data from the observation of the large-scale structure(LSS) and the
cosmic microwave background(CMB), i.e. SNe Ia+LSS+CMB. The dark gray dashed
lines denote constraints from the age of the universe at $1\sigma$ confidence
level $0.96\lesssim H_{0}t_{0}\lesssim 1.05$, with a central value
$H_{0}t_{0}\simeq 0.99$. The isochrones are labeled for the case of
$H_{0}=72~{}{\rm km~{}s^{-1}~{}Mpc^{-1}}$. The dark gray thick solid lines
indicate the age of the universe $t_{0}$ in the unit of Hubble time
$H_{0}^{-1}$. The dot with the coordinate $(0.224,10^{-9})$ represents the
best-fit value, with $\chi_{min}^{2}=591.156$. The best-fit result is in
accordance with D. Spergelet al.Spergel2003  at $1\sigma$ confidence level. A
small $\zeta$, $0\leq\zeta\leq 0.09$ within $95\%$ confidence level, is
obtained.

Figure 3: The best-fit $70\%$, $95\%$ and $99\%$ confidence regions in the
$\Omega_{m0}$-$\zeta$ plane with the isochrones of constant $H_{0}t_{0}$, for
the Union2 compilation of the Supernova Cosmology Project(SCP) collaboration
and the data from the observation of the large-scale structure(LSS), i.e. SNe
Ia+LSS. The dark gray dashed lines denote constraints from the age of the
universe at $1\sigma$ confidence level $0.96\lesssim H_{0}t_{0}\lesssim 1.05$,
with a central value $H_{0}t_{0}\simeq 0.99$. The isochrones are labeled for
the case of $H_{0}=72~{}{\rm km~{}s^{-1}~{}Mpc^{-1}}$. The dark gray thick
solid lines indicate the age of the universe $t_{0}$ in the unit of Hubble
time $H_{0}^{-1}$. The dot with the coordinate $(0.213,10^{-8})$ represents
the best-fit value, with $\chi_{min}^{2}=584.831$. The $95\%$ confidence level
of the parameter $\zeta$, i.e. $0\leq\zeta\leq 0.17$, is much smaller than the
SNe Ia-only case.

Figure 4: The distance modulus $\mu$ versus redshift $z$ for the Union2
compilation of the Supernova Cosmology Project(SCP) collaboration plus the
data from the observation of the large-scale structure(LSS) and the cosmic
microwave background(CMB), i.e. SNe Ia+LSS+CMB. The observation consists of
557 SNIa data points with $1\sigma$ error bars. The theoretically computed
result $\mu_{th}$ with the best-fit parameter values $\zeta=10^{-9}$ and
$\Omega_{m0}=0.224$ are plotted in black solid line for the one-dimension
Debye model of modified entropic force.

Figure 5: The reduced Hubble parameter $E$ versus redshift $z$ for three fits,
namely SNe Ia-only, SNe Ia+LSS, and SNe Ia+LSS+CMB, denoted by the dark gray
dotted line, dashed line, and the solid line, respectively. It can be seen
that $E(z)$ grows rapidly as the redshift $z$ increases.

Figure 6: The deceleration parameter $q$ versus redshift $z$ for three fits,
namely SNe Ia-only, SNe Ia+LSS, and SNe Ia+LSS+CMB. Note that $q=0$ at the
redshift $z_{t}=1.034$ for SNe Ia only, $z_{t}=0.948$ for SNe Ia+LSS, and
$z_{t}=0.906$ for SNe Ia+LSS+CMB. $z_{t}$ is called the transition redshift,
which implies that the universe goes from a decelerated expanding period to an
accelerated expanding period at a time point in the early epoch.