#  Apparent Dark Matter Inspired by Einstein Equation of State

Kimet Jusufi<EMAIL_ADDRESS>Physics Department, State University
of Tetovo, Ilinden Street nn, 1200, Tetovo, North Macedonia  Ahmad Sheykhi
<EMAIL_ADDRESS>Physics Department and Biruni Observatory, College of
Science, Shiraz University, Shiraz 71454, Iran

###### Abstract

The purpose of this article is twofold. First, by means of Padmanabhan’s
proposal on the emergence nature of gravity, we recover the $\Lambda$CDM model
and the effect of the dark matter in the context of cosmology. Toward this
goal, we use the key idea of Padmanabhan that states cosmic space emerges as
the cosmic time progress and link the emergence of space to the difference
between the number of degrees of freedom on the boundary and in the bulk.
Interestingly enough, we show that the effect of the cold dark matter in the
cosmological setup can be understood by assuming an interaction between the
numbers of degrees of freedom in the bulk. In the second part, we follow the
Jacobson’s argument and obtain the modified Einstein field equations with
additional dark matter component emerging due to the interaction term between
dark energy and baryonic matter related by
$\Omega_{DM,0}=\sqrt{2\alpha\Omega_{M,0}\Omega_{DE,0}}$, where $\alpha$ is a
coupling constant. Finally, a correspondence with Yukawa cosmology is pointed
out, and the role of massive gravitons as a possibility in explaining the
nature of the dark sector as well as the theoretical origin of the Modified
Newtonian Dynamics (MOND) are addressed. We speculate that the interaction
coupling $\alpha$ fundamentally measures the entanglement between the
gravitons and matter fields and there exists a fundamental limitation in
measuring the gravitons wavelength.

##  I Introduction

After discovery of the black holes thermodynamics in 1970’s, physicists have
been speculating that there should be a correspondence between the laws of
gravity and the laws of thermodynamics. In 1995, Jacobson Jac  disclosed that
the Einstein’s field equations of gravity can be derived by applying the first
law of thermodynamics, $\delta Q=T\delta S$, on the boundary of spacetime,
where $\delta Q$ is the energy flux across the boundary, $T$ and $S$ are,
temperature and entropy associated with the boundary, respectively. This
derivation is of great importance and reveals that the first law of
thermodynamics, when is applied to a spacetime as a thermodynamic system at
large scales, can be translated to the field equations of gravity. Following
Jacobson, a lot of works have been done to address the correspondence between
the first law of thermodynamics and the field equations of gravity in
different setups Pad1 ; Pad2 ; Pad3 ; CaiKim ; Cai2 ; wang1 ; SheyWC ; SheyWC2
; SheyCQ ; SheyLog ; SheyPL ; DiGennaro:2022grw .

According to the $\Lambda$CDM cosmological model (which stands as a prominent
and widely accepted framework in describing the evolution of the Universe),
our Universe comprises three essential constituents: cold dark matter, dark
energy, and baryonic matter Springel:2006 . Cold dark matter, a substance that
eludes electromagnetic interaction and manifests solely through its
gravitational influence, offers a potential solution to many problems,
including the observed flat rotation curves of spiral galaxies Navarro:1997 ;
Navarro:2010 ; Moore:1999 ; Gilmore:2007 ; Salucci:2007 ; Naray:2008 . In
recent years, dark matter has been widely believed to be made of particles
beyond the standard model, which describes baryonic matter. However, today,
such an elusive dark matter particle remains undetected. On the other hand,
the cosmological constant gives the dark energy, which emerges as a
manifestation of energy permeating every corner of the Universe, frequently
associated with vacuum energy Weinberg:1989 .

As an alternative to dark matter, in addressing the flat rotation curves, the
Modified Newtonian Dynamics (MOND) theory was formulated by Milgrom Milgrom1 ;
Milgrom2 ; Milgrom3  (see also Ferreira:2009 ; Milgrom:2003 ; Tiret:2007 ;
Kroupa:2010 ; Cardone:2011 ; Richtler:2011 ; Bekenstein:2004 ). It was argued
that the theoretical origin of the MOND theory can be understood from Debye
entropic gravity perspective Li ; SheMOND . The idea that gravity is not a
fundamental interaction and can be regarded as an entropic force caused by the
changes in the information associated with the positions of material bodies,
was proposed by Verlinde Ver . Verlinde’s derivation of Newton’s law of
gravitation at the very least offers a strong analogy with a well understood
statistical mechanism. Therefore, this derivation opens a new window to
understand gravity from the first principles. The study on the entropic force
has raised a lot of attention recently (see Cai4 ; sheyECFE ; Ling ; Modesto ;
Yi ; Sheykhi2  and references therein). In the framework of mimetic gravity,
the MOND-like acceleration was recovered vagnozzi . According to the entropic
force scenario, dark matter represents an apparent manifestation, a
consequence of baryonic matter Verlinde:2016toy . Using the entropic force,
modified Friedmann equations have been obtained recently Jusufi:2022mir ;
Millano:2023ahb ; Jusufi:2023ayv .

Verlinde’s proposal on the entropic nature of gravity opened a new window on
the origin of gravity, however, it assumes the background geometry as a
preexist structure. An important question is how one can consider the
spacetime as an emergent structure? Padmanabhan Padmanabhan:2012ik  was the
first who answered this question and argued that the spatial expansion of our
universe can be regarded as the consequence of emergence of space and the
cosmic space is emergent as the cosmic time progresses. By calculating the
difference between the surface degrees of freedom and the bulk degrees of
freedom in a region of space and equating the result with the volume change of
the space, he derived the Friedmann equations describing the evolution of the
universe Padmanabhan:2012ik . Following Padmanabhan, a lot of works have been
carried out to investigate the emergence nature of gravity and various
aspects/extension of this theory have been addressed in the literatures Cai1 ;
Yang ; FQ ; Shey1 ; Shey2 ; FF ; Eune ; Shey3 .

In this paper, we would like to employ a novel idea and address the dark
matter problem in the context of Padmanabhan’s proposal of emergence gravity.
This leads to modification of Einstein equations of general relativity, too.
This paper is structured as follows. In the next section we recover the
$\Lambda$CDM model by assuming an interaction between baryonic matter and dark
energy in the bulk. In section III, Following Jacobson Jac , we recover the
modified Einstein’s field equation with an additional energy-momentum term
which can be regarded as the apparent dark matter. In section IV, we establish
a correspondence with Yukawa cosmology and explain the role of the massive
gravitons as a possible candidate to explain the nature of the dark sector and
the emergence of MOND. We finish with final remarks in section V.

##  II Recovering $\Lambda$CDM in Padmanabhan’s Emergent Universe

To explain why our Universe is expanding, Padmanabhan used the holographic
principle and the equipartition law of energy Padmanabhan:2012ik . According
to Padmanabhan the basic law governing the emergence of space must relate the
emergence of space to the difference between degrees of freedom on the surface
and in the bulk, namely $(N_{\rm sur}-N_{\rm bulk})$. He proposed relation
Padmanabhan:2012ik

$\frac{dV}{dt}=L_{P}^{2}(N_{\rm sur}-N_{\rm bulk}),$ (1)

where $V$ and $t$ are, respectively, the volume and the cosmic time in Planck
units. More generally, it is expected $(\Delta V/\Delta t)$ to be some
function of $(N_{\rm sur}-N_{\rm bulk})$ which vanishes when the latter does
(note also that $L_{P}^{2}=\hbar G/c^{3}$). In the present paper, we would
like to modify the above relation as

$\frac{dV}{dt}=L_{P}^{2}\left[N_{\rm sur}-\left(N_{\rm bulk}+N^{\rm int}_{\rm
bulk}\right)\right],$ (2)

where we assume there is an additional term for the degrees of freedom in the
bulk denoted by $N^{\rm int}_{\rm bulk}$, which originates from the
interaction between matter/energy components in the bulk. The origin of this
interaction term will be discussed later on. Such a universe obeys the
holographic principle where $N_{\rm sur}$, the number of degrees of freedom on
the spherical surface of Hubble radius $H^{-1}$, given by Padmanabhan:2012ik

$N_{\rm sur}=\frac{4\pi}{L_{P}^{2}H^{2}}.$ (3)

Here $N_{\rm bulk}$ is the effective number of degrees of freedom which obeys
the equipartition law of energy contained inside the universe
Padmanabhan:2012ik

$N_{\rm bulk}=\frac{|E|}{(1/2)k_{B}T}.$ (4)

We shall assume a multi-fluid matter inside the universe with proper Komar
energy density

$\displaystyle E$ $\displaystyle=$ $\displaystyle\sum_{i}(\rho_{i}+3p_{i})V,$
(5)

and hence we can write the equipartition law of energy as

$N_{\rm bulk}=-\sum_{i}\epsilon_{i}\frac{2(\rho_{i}+3p_{i})V}{k_{B}T},$ (6)

along with

$\frac{dV}{dt}=L_{P}^{2}\left[N_{\rm sur}-\left(\epsilon_{i}N_{\rm
bulk}+\epsilon^{\rm int}N^{\rm int}_{\rm bulk}\right)\right],$ (7)

where, in our definition, $\epsilon_{i}=-1$ stands for baryonic matter and
radiation and $\epsilon_{i}=1$ for dark energy [in order to ensure that
$N_{\rm bulk}>0$; for matter and radiation $\rho^{M}_{i}+3p^{M}_{i}>0$ and
$\rho^{R}_{i}+3p^{R}_{i}>0$, and $\rho_{DE}+3p_{DE}<0$ for dark energy,
respectively]. Furthermore, we take $\epsilon^{\rm int}=-1$, meaning that
$N^{\rm int}_{\rm bulk}>0$, which should behave as baryonic matter for reasons
that we will explain below.

The volume of the universe and the temperature associated with the horizon are
given by

$\displaystyle V=\frac{4\pi}{3H^{3}},\,\,\,\,\,T=\frac{H}{2\pi}.$ (8)

The novel idea is to consider an interaction term between baryonic matter and
dark energy in the bulk. As a result, we propose there is an additional
degrees of freedom which comes from the interaction term and assume, for pure
dimensional reasons, it can be written as

$\displaystyle N^{\rm int}_{\rm bulk}=-\epsilon^{\rm int}\sqrt{\alpha\ N_{\rm
bulk}^{\rm M}N_{\rm bulk}^{\rm DE}},$ (9)

where $\alpha$ is a coupling constant, $N_{\rm bulk}^{\rm M}$ and $N_{\rm
bulk}^{\rm DE}$ are the degrees of freedom of baryonic matter and dark energy
in the bulk, respectively. In the next section, we will give further physical
arguments about $\alpha$. Since $p_{M}=0$ for baryonic matter, and
$p_{DE}=-\rho_{DE}$ for dark energy fluid, then, for the interaction term, we
get

$\displaystyle N^{\rm int}_{\rm
bulk}=\sqrt{2\alpha\rho_{M}\rho_{DE}}\frac{2\,V}{k_{B}T}.$ (10)

Let us now obtain the standard Friedmann equation. Using Eqs. (7), (3), (6),
(8), (10) and $p_{i}=\omega_{i}\rho_{i}$, we get

$\displaystyle\frac{dV}{dt}$ $\displaystyle=$ $\displaystyle 4\pi
L_{P}^{2}\Big{[}\frac{1}{L_{P}^{2}H^{2}}+\sum_{i}\rho_{i}(1+3\omega_{i})\frac{4\pi}{3H^{4}}$
(11) $\displaystyle+$
$\displaystyle\sqrt{2\alpha\rho_{M}\rho_{DE}}\frac{4\pi}{3H^{4}}\Big{]},$

where we have set $k_{B}=1$. From the last equation, we obtain

$\frac{\ddot{a}}{a}=-\frac{4\pi
L_{P}^{2}}{3}\left[\sum_{i}\rho_{i}(1+3\omega_{i})+\sqrt{2\alpha\rho_{M}\rho_{DE}}\right].$
(12)

In the second term, we have an evolving matter given by
$\rho_{M}=\rho_{M,0}(1+z)^{3}$ and dark energy $\rho_{DE}=\rho_{DE,0}$, where
it is natural to identify this term as the dark matter for reasons we shall
argue below. Let us define the following equation as an evolving dark matter

$\frac{\rho_{DM}}{(1+z)^{3/2}}\equiv\sqrt{2\alpha\rho_{M}\rho_{DE}},$ (13)

with using $\rho_{M}=\rho_{M,0}(1+z)^{3}$ and $\rho_{DE}=\rho_{DE,0}$, leads
to

$\rho_{DM}=\rho_{DM,0}\,a^{-3},$ (14)

where $1+z=a^{-1}$ and $\rho_{DM,0}\equiv\sqrt{2\alpha\rho_{M,0}\rho_{DE,0}}$.
Note that assuming dark matter is pressureless, then Eq. (14) is also solution
of the continuity equation $\dot{\rho}_{DM}+3H\rho_{DM}=0$, as expected and
justifies ansatz (13).

If we multiply both sides of Eq. (12) by $2a\dot{a}$, and using the continuity
equation

$\dot{\rho}_{i}+3H(1+\omega_{i})\rho_{i}=0,$ (15)

where $H=\dot{a}/a$ is the Hubble parameter and
$\rho_{i}=\rho_{i,0}a^{-3(1+\omega_{i})}$, assuming several matter fluids with
a constant equation of state parameters $\omega_{i}$. We then arrive at

$d(\dot{a}^{2})=\frac{8\pi
L_{P}^{2}}{3}d\left(\sum_{i}\rho_{i,0}a^{-1-3\omega_{i}}+\rho_{DM,0}\,a^{-1-3\omega_{DM}}\right),$
(16)

where $\Omega_{DM}=0$. In the l.h.s, we get a constant of integration $k$ as
follows

$\dot{a}^{2}+k=\frac{8\pi L_{P}^{2}}{3}\int
d\left(\sum_{i}\rho_{i,0}a^{-1-3\omega_{i}}+\rho_{DM,0}a^{-1}\right).$ (17)

By solving the integral in the r.h.s, we get

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{8\pi
L_{P}^{2}}{3}\Big{(}\rho_{R,0}\,a^{-4}+\rho_{M,0}a^{-3}+\rho_{DM,0}a^{-3}$
(18) $\displaystyle+$ $\displaystyle\rho_{\rm
cur}a^{2}+\rho_{DE,0}\,a^{0}\Big{)},$

where we have also included the spatial curvature density

$\displaystyle\rho_{\rm cur}\equiv\frac{3k}{8\pi L_{P}^{2}a^{2}}.$ (19)

If we use the critical density

$\displaystyle\rho_{\rm crit}=\frac{3H_{0}^{2}}{8\pi L_{P}^{2}},$ (20)

we get the equation in terms of the density parameters or in terms of redshift
$z$ as follows

$E^{2}(z)=\Omega_{R,0}(1+z)^{4}+\Omega_{M,0}^{\rm tot}(1+z)^{3}+\Omega_{\rm
curv}(1+z)^{2}+\Omega_{DE,0},$ (21)

where $E(z)=H/H_{0}$, and

$\Omega_{DM,0}=\sqrt{2\,\alpha\,\Omega_{M,0}\Omega_{DE,0}},$ (22)

along with

$\displaystyle\Omega_{M,0}^{\rm
tot}=\Omega_{M,0}+\sqrt{2\,\alpha\,\Omega_{M,0}\Omega_{DE,0}}=\Omega_{M,0}+\Omega_{DM,0}.$
(23)

That suggests that dark matter can evolve due to the expansion of the
Universe, which is related to the fact that the baryonic matter evolves, and
the coupling parameter is a function of $z$, as we pointed out. As a special
case, if we consider a flat universe with $k=0$ for the late-time Universe,
which is consistent with the result obtained in Jusufi:2023xoa . We shall
elaborate more about this connection in this work. However, it is important to
see from these equations that dark matter is only an apparent effect due to
the interaction between the dark energy and matter. That is the key
distinction as compared to the $\Lambda$CDM model, whose respective Hubble
parameter as a function of the redshift for a flat space ($\Omega_{\rm
curv}=0$) is given by (written in our notation)

$H(z)=H_{0}\sqrt{\Omega_{R,0}(1+z)^{4}+\Omega_{M,0}^{\rm
tot}(1+z)^{3}+\Omega_{DE,0}},$ (24)

the last expression is obtained from Eq. (21) and it perfectly matches the
$\Lambda$CDM model.

##  III Einstein field equations with apparent dark matter

In this section, we will closely follow the seminal paper by Jacobson Jac .
Specifically, Jacobson derived the Einstein’s field equations of gravity
through the application of the first law of thermodynamics and one can view
the Einstein’s field equation as an equation of state. In the current study,
we aim to extend this framework, incorporating the influences of baryonic
matter, dark energy, and an additional term stemming from the interaction
between baryonic matter and dark energy. First, we need to consider the heat
flow across the horizon attributed to the energy carried by the matter field,
characterized by an energy-momentum tensor $T^{M}_{ab}$ representing baryonic
matter

$\delta Q^{M}=-\kappa\int_{H}\xi T^{M}_{ab}k^{a}\,k^{b}d\xi dS.$ (25)

In this equation with $S$ we represent the area of the horizon $H$. The vector
$k^{a}$ corresponds to the tangent vector of the horizon generators, further
$\kappa$ is known as the surface gravity and, finally, $\xi$ is an appropriate
affine parameter. We also introduce a dark energy term contribution

$\delta Q^{DE}=-\kappa\int_{H}\xi T^{DE}_{ab}k^{a}k^{b}d\xi dS.$ (26)

Building upon the concept introduced in the preceding section, which serves as
the fundamental idea in this paper, we incorporate an interaction term between
baryonic matter and dark energy. Consequently, we anticipate the emergence of
a contribution term

$\delta Q^{int.}=-\kappa\int_{H}\xi T^{int}_{ab}\,k^{a}\,k^{b}d\xi dS,$ (27)

where we have introduced an interaction term

$\displaystyle
T^{int}_{ab}=u_{a}u_{b}\sqrt{2\alpha\rho_{M,0}\rho_{DE,0}}\,\,a^{-3},$ (28)

where $\alpha$ is an interaction parameter between dark energy and baryonic
matter, $a$ is the scale factor, here 4-velocity of the prefect fluid with
$g^{ab}u_{a}u_{b}=-1$. For reasons that will be clarified later on we shall
identify the interaction term as the dark matter term, i.e. $T_{ab}^{DM}\equiv
T^{int}_{ab}$. We get

$\displaystyle
T_{ab}^{DM}=u_{a}u_{b}\sqrt{2\alpha\rho_{M,0}\rho_{DE,0}}\,\,a^{-3}\equiv\rho_{DM}u_{a}u_{b},$
(29)

According to the Raychaudhuri equation, for the change of the horizon area we
have

$\delta S=\int_{H}\theta d\xi dS=-\int_{H}\xi R_{ab}k^{a}\,k^{b}d\xi dS,$ (30)

in which $\theta$ is known as the expansion/contraction scalar. Finally, for
the first law of thermodynamics we obtain

$\displaystyle\delta(Q^{M}+Q^{DE}+Q^{int})=Td\mathbb{S},$ (31)

where $\mathbb{S}$ is the entropy and we note that $T$ is the temperature and
it reads $T=\kappa/2\pi$. Using the Bekenstein-Hawking relation between the
entropy and the horizon area $\mathbb{S}=S/4$, and with the help of the above
relations we get

$\int_{H}\xi\left(T^{M}_{ab}+T^{DE}_{ab}+T^{int}_{ab}\right)k^{a}k^{b}d\xi
dS=\frac{1}{8\pi G}\int_{H}\xi R_{ab}k^{a}k^{b}d\xi dS.$ (32)

From the last equation it is evident that one has

$\frac{1}{8\pi
G}\left(R_{ab}\,k^{a}\,k^{b}\right)=\left(T^{M}_{ab}+T^{DE}_{ab}+T^{int}_{ab}\right)k^{a}k^{b},$
(33)

which is valid for all null vector $k^{a}$. The last equation can be further
rewritten as

$\displaystyle 8\pi GT^{tot}_{ab}=R_{ab}+\zeta g_{ab},$ (34)

with $\zeta$ being some function. In addition we have defined the total
energy-momentum tensor to be

$\displaystyle T_{ab}^{tot}=T^{M}_{ab}+T^{DE}_{ab}+T_{ab}^{DM}.$ (35)

If we apply the covariant derivative to and we further utilize the contracted
Bianchi identity, we obtain:

$\displaystyle\nabla_{b}(R_{ab}+\zeta
g_{ab})=0\,\,\Rightarrow-\nabla_{b}\left(\frac{R}{2}\right)=\partial_{b}\zeta.$
(36)

We have also assumed $\nabla_{b}T^{tot}_{ab}=0$. From the last equation it is
easy to see that we get for the function $\xi$

$\displaystyle\xi=-\left(\frac{R}{2}\right)+C.$ (37)

In our case we can fix the constant $C$ to zero. But we further define

$\displaystyle T^{DE}_{ab}=-\frac{\Lambda}{8\pi G}g_{ab}.$ (38)

In this way we obtain the Einstein field equations as a thermodynamics
equation of state with dark matter

$\displaystyle R_{ab}-\frac{1}{2}Rg_{ab}+\Lambda g_{ab}=8\pi
G\left(T^{M}_{ab}+T_{ab}^{DM}\right).$ (39)

One should keep in mind that the dark matter is only an apparent effect and
obtained via the interaction of dark energy and matter. With this equation in
hand, we can apply to find the cosmological model of the universe. Let us now
use the Einstein equation with dark matter and extend our discussion of the
cosmological setup. Assuming the background spacetime to be spatially
homogeneous and isotropic, which is given by the Friedmann-Robertson-Walker
(FRW) metric

$ds^{2}=-dt^{2}+a^{2}\left[\frac{dr^{2}}{1-kr^{2}}+r^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2})\right],$ (40)

where we can further use $R=a(t)r$, $x^{0}=t,x^{1}=r$, the two dimensional
metric $h_{\mu\nu}$. 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 FRW universe

$R=ar={1}/{\sqrt{H^{2}+{k}/{a^{2}}}}.$ (41)

For the matter source in the FRW universe, we shall assume a perfect fluid
described by the stress-energy tensor

$T_{\mu\nu}=(\rho_{i}+p_{i})u_{\mu}u_{\nu}+p_{i}g_{\mu\nu}.$ (42)

this leads to the continuity equation have assumed several matter fluids with
a constant equation of state parameters $\omega_{i}$ and continuity equations
we have the expression for densities $\rho_{i}=\rho_{i0}a^{-3(1+\omega_{i})}$
(this includes baryonic matter $\omega_{i}=0$, radiation $\omega_{i}=1/3$,
dark matter $\omega_{i}=0$ and dark energy $\omega_{i}=-1$). with
$H=\dot{a}/a$ being the Hubble parameter. If we rewrite the Einstein equations
as

$\displaystyle R_{ab}=8\pi
G\left(T^{tot}_{ab}-\frac{1}{2}g_{ab}T^{tot}\right).$ (43)

From the Einstein field equations, we can obtain the Friedmann’s equation,
namely

$\frac{\ddot{a}}{a}=-\left(\frac{4\pi
G}{3}\right)\sum_{i}\left(\rho_{i}+3p_{i}\right).$ (44)

where we have the expression for densities
$\rho_{i}=\rho_{i0}a^{-3(1+\omega_{i})}$. Eq. (44) becomes

$\displaystyle\frac{\ddot{a}}{a}=$ $\displaystyle-\left(\frac{4\pi
G}{3}\right)\sum_{i}\left(1+3\omega_{i}\right)\rho_{i0}a^{-3(1+\omega_{i})}.$
(45)

Next, by multiplying $2\dot{a}a$ on both sides of Eq. (45), and by integrating
we obtain

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}+\frac{k}{a^{2}}=\frac{8\pi
G}{3}\sum_{i}\rho_{i0}a^{-3(1+\omega_{i})}$ (46)

or

$\displaystyle H^{2}+\frac{k}{a^{2}}=\frac{8\pi
G}{3}\sum_{i}\rho_{i0}a^{-3(1+\omega_{i})}$ (47)

We further assume, the energy-momentum tensor of each component is in the form
of the perfect fluid,

$\displaystyle T_{ab}^{i}=(\rho_{i}+p_{i})u_{a}u_{b}+p_{i}g_{ab}$ (48)

where $i=DM,DE,M$ and $u^{a}$ is the 4-velocity of the prefect fluid with
$g^{ab}u_{a}u_{b}=-1$. For the pressureless dark matter with $p_{DM}=0$, we
have

$\displaystyle{{T^{a}}_{b}}^{DM}=\rho_{DM}u^{a}u_{b}.$ (49)

Comparing with the equation

$\displaystyle{{T^{a}}_{b}}^{DM}=u^{a}u_{b}\sqrt{2\alpha\rho_{M,0}\rho_{DE,0}}\,\,a^{-3}.$
(50)

Equating Eqs. (49) and (50) we finally get

$\displaystyle\rho_{DM}=\sqrt{2\alpha\rho_{M,0}\rho_{DE,0}}(1+z)^{3}.$ (51)

Using the continuity equation for the matter, we have
$\rho_{M}=\rho_{M,0}(1+z)^{3}$, while for the cosmological constant (DE) the
energy density is constant, namely $\rho_{DE}=\rho^{0}_{DE}$, in the general
case we define

$\rho_{DM}=\sqrt{2\alpha\rho^{0}_{M}\rho^{0}_{DE}}(1+z)^{3},$ (52)

or

$\rho_{DM}=\rho^{0}_{DM}\,(1+z)^{3},$ (53)

that it we obtain the standard Friedmann equation

$\displaystyle\left(\frac{\dot{a}}{a}\right)^{2}$ $\displaystyle=$
$\displaystyle\frac{8\pi
G}{3}\Big{(}\rho_{R,0}\,a^{-4}+\rho_{M,0}a^{-3}+\rho_{DM,0}a^{-3}$ (54)
$\displaystyle+$ $\displaystyle\rho_{\rm
curv}a^{2}+\rho_{DE,0}\,a^{0}\Big{)},$

In addition, we have the spatial curvature density

$\displaystyle\rho_{\rm cur}=\frac{3k}{8\pi Ga^{2}}.$ (55)

If we use the critical density

$\displaystyle\rho_{\rm crit}=\frac{3H_{0}^{2}}{8\pi G},$ (56)

we get the equation in terms of the density parameters or in terms of redshift
$z$ as follows

$E^{2}(z)=\Omega_{R,0}(1+z)^{4}+\Omega_{M,0}^{\rm tot}(1+z)^{3}+\Omega_{\rm
curv}(1+z)^{2}+\Omega_{DE,0}.$ (57)

where $\Omega_{M,0}^{\rm
tot}=\Omega_{M,0}+\sqrt{2\,\alpha\,\Omega_{M,0}\Omega_{DE,0}}$. This result
coincides with Eq. (21) in the last section.

##  IV Correspondence with Yukawa cosmology and recovering MOND

###  IV.1 Correspondence with Yukawa cosmology

In this section, we would like to elaborate more on the nature of dark energy.
Einstein famously introduced the cosmological constant into the field equation
$G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi GT_{\mu\nu}/c^{4}$ to explain the
expansion of the Universe. Dark energy is commonly linked to represent the
energy density of the vacuum Carroll:2000fy ; hence, it is just a constant.
Specifically, the energy density can be calculated by integrating the vacuum
fluctuation energies using $\rho^{\rm vac}_{DE}\simeq\Lambda_{\rm cut}^{4}$
where we integrate up to a certain ultraviolet momentum cutoff $k_{\rm max}$,
we get $\rho^{\rm vac}_{DE}\sim\hbar k^{4}_{\rm max}$ Carroll:2000fy . That is
the origin of the famous discrepancy, namely, when one compares the
theoretical value with the observed value, that gives a huge discrepancy of
the order $\sim 10^{120}$. Speculation regarding a potential symmetry leads to
a vanishing vacuum energy. The problem surrounding the cosmological constant
remains unsolved and continues to challenge researchers. Certainly,
alternative concepts have been proposed to elucidate the cosmological
constant. In this context, we will present an argument for the involvement of
massive gravitons in clarifying the nature of the cosmological constant.
Moreover, we will draw attention to a correlation with Yukawa cosmology, the
subject of recent investigation in Jusufi:2023xoa ; Gonzalez:2023rsd . At
large distances, one can obtain a Yukawa-like potential

$\displaystyle\Phi=-\frac{GMm}{r}\left(1+\alpha\exp{(-r/\lambda)}\right),$
(58)

with the wavelength of massive graviton reads Visser:1997hd

$\lambda=\frac{\hbar}{m_{g}c}.$ (59)

It is interesting to note that the Yukawa potential has been obtained in
modified theories of gravity such as the $f(R)$ gravity Capozziello:2009vr ;
Benisty:2023qcv ). It was shown that the dark matter can be viewed as an
apparent effect, namely, dark matter is obtained via the long-range force
modification in terms of the following equation Gonzalez:2023rsd ;
Jusufi:2023xoa . In particular, one can obtain effectively the $\Lambda$CDM
using Gonzalez:2023rsd ; Jusufi:2023xoa

$\displaystyle\Omega_{DM,0}^{\Lambda\text{CDM}}=\sqrt{2\Omega^{\Lambda\text{CDM}}_{M,0}\Omega^{\Lambda\text{CDM}}_{DE,0}},$
(60)

where

$\Omega^{\Lambda\text{CDM}}_{DE,0}\sim\frac{c^{2}\alpha}{\lambda^{2}H_{0}^{2}}.$
(61)

We can therefore make the correspondence if we define the dark energy density
in the Padmanbhan universe as

$\Omega^{\rm Padmanbhan}_{DE,0}\sim\frac{c^{2}}{\lambda^{2}H_{0}^{2}}.$ (62)

In other words, there is an equivalence with the Yukawa cosmology
Jusufi:2023xoa  with two parameters: $\lambda$ that specifies the dark energy
density and it is related to the gravitons mass and $\alpha$ that specifies
the interaction between matter and dark energy. We can further write the
energy density of the cosmological constant contribution in terms of the
graviton mass. Since we found a correspondence with Yukawa cosmology, one can
see that the parameter $\alpha$ appears in the modified Newton’s potential
and, therefore, Newton’s law in large distances. The quest for a viable theory
grounded in a healthy and well-defined action that can integrate the concept
of massive gravitons remains an ongoing endeavour that has yet to reach a
conclusive resolution. the parameter $\alpha$ plays a crucial role in altering
Newton’s law. Yet, an intriguing question emerges regarding the fundamental
origin of this parameter. We shall present compelling arguments linking the
modified entropy to adjusted gravity, with $\alpha$ emerging as a consequence
of entropy modification. In some deep sense, $\alpha$ could potentially be
influenced by the entanglement entropy arising from the intricate interplay
between baryonic matter and the fluctuations in the gravitational field
(gravitons). One can already see a hint for this if we use
$\Omega_{\Lambda,0}^{\Lambda CDM}=\rho_{DE,0}/\rho_{\rm crit}$, where
$\rho_{\rm crit}=3H_{0}^{2}/(8\pi G)$ and $\rho_{DE,0}=\Lambda c^{2}/(8\pi G)$
one can obtain the cosmological constant in terms of $\alpha$, as follows

$\displaystyle\Lambda\sim\frac{3\alpha}{\lambda^{2}}.$ (63)

In this way, we get for the energy density of dark energy

$\displaystyle\rho_{DE}\sim\frac{3\,\alpha\,c^{2}}{8\pi G\lambda^{2}},$ (64)

which has the form of a holographic dark energy expression for the
cosmological constant. This signals that $\alpha$ fundamentally measures the
entanglement between the gravitons and matter fields.

###  IV.2 Recovering MOND

In this section, we would like to point out that the parameter $\alpha$ that
modifies the Newton’s law in the large distances, can explain theoretical
origin of the MOND theory. To see this, let us rewrite Eq. (12) as follows

$\frac{\ddot{R}}{R}=-\frac{4\pi
G}{3}\left[\sum_{i}\rho_{i}(1+3\omega_{i})+\sqrt{2\alpha\rho_{M}\rho_{DE}}\right],$
(65)

where we have restored the Newton’s constant $G$. This equation can also be
written as

$m\ddot{R}=-\frac{G\,m}{R^{2}}\left[\frac{4\pi
R^{3}}{3}\sum_{i}\rho_{i}(1+3\omega_{i})+\frac{4\pi
R^{3}}{3}\sqrt{2\alpha\rho_{M}\rho_{DE}}\right].$ (66)

Using the fact that the Komar mass [similar to Komar energy with
$p_{i}=\omega_{i}\rho_{i}$] reads

$\displaystyle M=\frac{4\pi R^{3}}{3}\sum_{i}\rho_{i}(1+3\omega_{i}),$ (67)

and if we consider only matter $[\omega_{i}=0$] and dark energy
$\omega_{i}=-1$] (we shall neglect radiation contribution), we get

$m\ddot{R}=-\frac{G\,m}{R^{2}}\left[\frac{4\pi
R^{3}}{3}(\rho_{M}-2\rho_{DE})+\frac{4\pi
R^{3}}{3}\sqrt{2\alpha\rho_{M}\rho_{DE}}\right].$ (68)

For Newton’s law in cosmological scales, we therefore get

$\vec{F}=m\,\vec{a}=-\left[\frac{G\,\,m\,(M-M_{DE})}{R^{2}}+\frac{G\,\,m}{R^{2}}V\,\sqrt{2\alpha\rho_{M}\rho_{DE}}\right]\vec{r_{0}},$
(69)

where $\vec{r_{0}}$ is some unit vector, and $M=4\pi R^{3}\rho_{M}/3$
(although in general $M$ depends on $R$) along with $M_{DE}=4\pi
R^{3}(2\rho_{DE})/3$. On cosmological scales, the dark energy mass $M_{DE}$
dominates over baryonic mass $M$, implying a negative sign which means an
expanding universe. Conversely, on galactic scales, the baryonic mass
dominates and the total force is toward the galactic center. Nevertheless, the
presence of the second term which depends on $\alpha$, yields an additional
contribution from the long-range gravity modification; such a term can be
attributed to the dark matter having a mass

$\displaystyle M_{DM}=\frac{4\pi R^{3}}{3}\sqrt{2\alpha\rho_{M}\rho_{DE}},$
(70)

we can restore Newton’s law of gravity

$\displaystyle\vec{F}=-\frac{G\,m\,(M-M_{DE}+M_{DM})}{R^{2}}\vec{r_{0}}.$ (71)

provided there is an extra mass term due to the dark matter. However, there is
another way of rewriting Eq. (66) as follows

$\displaystyle
a=\frac{G\,\,(M-M_{DE})}{R^{2}}+\sqrt{\alpha\left(\frac{G\,\,M}{R^{2}}\right)\left(\frac{G\,\,M_{DE}}{R^{2}}\right)}$
(72)

Let us define the acceleration due to the matter and the dark energy enclosed
in that region of volume $V$ as

$\displaystyle
a_{M}=\frac{G\,\,M}{R^{2}}\,\,\,\,\,\text{and}\,\,\,\,\,a_{DE}=\frac{G\,\,M_{DE}}{R^{2}},$
(73)

we obtain the total acceleration

$\displaystyle a=a_{M}-a_{DE}+\sqrt{\alpha a_{M}a_{DE}}.$ (74)

The last equation is the acceleration in MOND theory.

  * •

Cosmological scales

In cosmological scales, as we pointed out the dark energy dominates over
baryonic matter [in general it also should depend in the scale distance
$M=M(R)$], and for cosmological scales $R\sim 10^{26}\rm{m}$, $a_{M}\to 0$, or
$a_{M}<<a_{DE}$ and, from the lat equation we get $a=-a_{DE}$. It is clear
that the negative sign explains the expanding universe due to dark energy. One
can further check that

$\displaystyle a_{DE}=\frac{G\,\,M_{DE}}{R^{2}}\sim\frac{4\pi
G\rho_{DE}R}{3},$ (75)

where for the dark energy density we have

$\displaystyle\rho^{\rm cosmology}_{DE}\sim\frac{3\,\alpha\,c^{2}}{8\pi
G\lambda^{2}_{\rm cosmology}},$ (76)

hence we get

$\displaystyle a_{DE}\sim\frac{\alpha c^{2}R}{2\lambda^{2}_{\rm
cosmology}}\sim 10^{-10}\rm{m/s^{2}}.$ (77)

The last result gives the acceleration of the universe. Here we note that the
value of $\lambda_{\rm cosmology}$ is of the order of the observable radius of
the universe, i.e., $\lambda_{\rm cosmology}\sim 10^{26}$ m (see from
observation constrains reported in Jusufi:2023xoa ; Gonzalez:2023rsd ). In
addition, we have used $\alpha\sim 0.41$ along with $R\sim 10^{26}\rm{m}$, and
consequently the energy density of dark energy can be approximated to
$\rho_{DE}\sim 7\times 10^{-27}$ kg/m3.

  * •

Galactic scales

Let us now elaborate the case of galaxies where $R$ is of pc or kpc order.
However, near the galactic center in general the mass term is a function of
distance, i.e., $M(R)$. In other words, in the galactic center, we know that
baryoinic matter dominates compared to the dark energy ($a_{M}>>a_{DE})$,
hence we get

$\displaystyle a=a_{M}+\sqrt{\alpha\,a_{M}\,a_{DE}}.$ (78)

Further let us define $a_{0}$, by writing

$a_{0}=\alpha\,a_{DE},$ (79)

to get a MOND-like form for the acceleration

$\displaystyle a=a_{M}+\sqrt{a_{M}a_{0}}.$ (80)

The interesting part here is when one studies the large distances, say when
$R$ is of pc or kpc order. Hence let us take for the outer part of the galaxy
$R\sim 10^{17}$ m, we can try to compute $a_{0}$ which gives

$a_{0}=\alpha\,a_{DE}\sim\frac{4\pi\alpha^{2}G\rho_{DE}R}{3}\sim
10^{-20}\rm{m/s^{2}}!$ (81)

We have expected for $a_{0}$, as MOND predicts, $a_{0}\sim
10^{-10}\rm{m/s^{2}}$. To address this inconsistency, let’s initially
highlight that the energy density of dark energy on galactic scales is given
by

$\displaystyle\rho^{\rm galaxy}_{DE}\sim\frac{3\alpha\,c^{2}}{8\pi
G\lambda^{2}_{\rm galaxy}},$ (82)

and rewrite $a_{0}$ in terms of the energy density as follows

$\displaystyle a_{0}\sim\frac{\alpha^{2}\,c^{2}}{2\lambda^{2}_{\rm galaxy}}.$
(83)

As we shall elaborate the energy density of dark energy depends on the scale
of measurements (observations) due its dependence on $\lambda$. In some sense
there exists a fundamental limitations on measuring the dark energy density
due to the uncertainty principle when we measure $\lambda$. To see this more
clearly, let us go back in Eq. (59) and after we set $\lambda=\Delta x$, and
$\Delta p=m_{g}c$, for graviton we get the momentum-position uncertainty
relation

$\displaystyle\Delta x\Delta p\sim\hbar.$ (84)

Now lets see the implications of this relation. When $\lambda^{\rm
cosmology}\sim 10^{26}\,\rm{m}\,.$, i.e. in cosmological scales, we have more
uncertainty in position $\lambda^{\rm cosmology}=\Delta x^{\rm cosmology}\sim
10^{26}$ m, but more precision on the momentum $\Delta p^{\rm cosmology}$,
namely

$\displaystyle\Delta p^{\rm cosmology}\sim\frac{\hbar}{\Delta x^{\rm
cosmology}}\sim 10^{-60}\,\rm{N}\cdot\rm{s}\,.$ (85)

This simply means that when we apply the uncertainty relation for the graviton
entirety of the universe, our knowledge of position becomes more uncertain,
but we gain greater precision in measuring momentum. Conversely, when focusing
on galactic scales, the uncertainty in position decreases, approximately
$\lambda^{\rm galaxy}=\Delta x^{\rm galaxy}\sim 10^{19}$ m. However, this
reduction in position uncertainty is accompanied by an increase in momentum
uncertainty, denoted as $\Delta p^{\rm galaxy}$, specifically we can see this
from

$\displaystyle\Delta p^{\rm galaxy}\sim\frac{\hbar}{\Delta x^{\rm galaxy}}\sim
10^{-53}\,\rm{N}\cdot\rm{s}\,.$ (86)

In order to obtain a consistent result, we need $R\gtrsim 10^{15}$ m, namely

$a_{0}\sim\frac{\alpha^{2}\,c^{2}\,R}{2\lambda^{2}_{\rm galaxy}}\sim
10^{-10}\rm{m/s^{2}}.$ (87)

As we increase the scale of observations $R$, $\lambda$ will increase and
there is no sense to speak about $R>\lambda$. Our findings align with recent
observations related to the applicability of Newton’s law over substantial
distances, specifically employing wide binary systems as exemplified by
Hernandez and Chae Hernandez ; Chae:2023prf . Consequently, it is evident that
there are inherent constraints in accurately measuring the graviton wavelength
within the realm of cosmology. As we elaborated, the inconsistency that
emerges when comparing analyses conducted on galactic and cosmological scales,
can be explained through the prism of uncertainty relations.

##  V Final remarks and conclusions

The central and novel result we found in this paper is that by adopting the
emergence perspective of gravity, one can interpret the dark matter as an
effective/apparent phenomenon. In this viewpoint the dark matter appears as a
result of interaction between dark energy and baryonic matter which modifies
the numbers of degrees of freedom in bulk. This approach leads naturally to
derive the $\Lambda$CDM model of cosmology. The second important result is
that, we applied the Jacobson’s approach by taking into account an interaction
term between baryonic matter and dark energy which plays the role of dark
matter. We thus constructed the modified Einstein field equations as an
equation of state. The Einstein field equations obtained in this method,
contains energy momentum tensor of apparent dark matter as well as dark
energy. Both methods are consistent and we confirm how dark matter naturally
emerges by including an interaction term between dark energy and baryonic
matter. Finally, we obtained the $\Lambda$CDM model in cosmology and
established a correspondence with Yukawa cosmology. We then explored the
relation and the role of massive graviton in the dark sector. For the energy
density of dark energy we obtained a holographic dark energy expression which
implies that the interaction term $\alpha$ fundamentally measures the
entanglement between the gravitons and matter fields. Finally, a MOND-like
expression was obtained and the role of uncertainty relation for the
gravitons’ wavelength was discussed in cosmological/galactic observations.

###### Acknowledgements.

The work of A.Sheykhi is supported by Iran National Science Foundation (INSF)
under grant No. 4022705.

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