Emergent Dark Matter in Late Time Universe on Holographic Screen
Rong-Gen Cai Sichun Sun Yun-Long Zhang
hep-th astro-ph.CO gr-qc hep-ph




 Self-force on an arbitrarily coupled scalar charge in cylindrical  thin-shell spacetimes
    C. Tomasini[e-mail: ctomasini@df.uba.ar], E. Rub n de Celis[e-mail: erdec@df.uba.ar], C. Simeone[e-mail: csimeone@df.uba.ar]
 Departamento de F sica, Facultad de Ciencias Exactas y Naturales, Universidad de
 Buenos Aires and IFIBA, CONICET, Ciudad Universitaria, Buenos Aires 1428, Argentina.
    December 30, 2023
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§ INTRODUCTION



The origin of the dark matter and the dark energy is one of the most important issues in current high energy physics and cosmology. From observations, only 5% of the energy components of the current universe is visible to us. At different scales, ranging from the galactic scale to the cosmic microwave background (CMB) scale, there are many observed phenomena to test for various models of dark matter and dark energy <cit.>. The cold dark matter model which treats the dark matter as collisionless particles is successful at CMB and large scales, but at the galactic scale, some discrepancies were proposed <cit.>. Moreover, the particle dark matter remains elusive from the direct detection so far <cit.>. One of the alternatives to the cold dark matter is the modified Newtonian dynamics or modified gravity <cit.>, which focuses on the small scale crisis that cold dark matter cannot explain. Although those modified gravity theories seem to be less successful in producing the universe evolution scenario consistent  with CMB and large scale structure data, they can explain multiple features in galaxy rotational curves such as Tully-Fisher relation <cit.>, Renzo's Rule <cit.>,  etc.

Recently, E. Verlinde proposed an emergent modified gravity  scenario from volume contribution of entanglement entropy in the de Sitter spacetime <cit.>, which leads to the apparent dark matter.  It is also related to the idea that  Einstein gravity can be an  emergent phenomenon  as  the entropy force with area law <cit.>. Although the Verlinde's derivation in <cit.> received some doubts on the consistency in the literature <cit.>,  we find several Verlinde's key ideas rather inspiring. One is the possibility that our macroscopic notions of spacetime and gravity emerge from an underlying microscopic description, encouraged by the recent development of entanglement  entropy and quantum information. Another one is viewing the dark matter as merely a gravitational response of the baryonic matter on the spacetime, so as to derive the dark matter distribution around the galaxy, the Tully-Fisher relation.


In this paper, we propose a new viewpoint beyond Verlinde's emergent gravity, which can be considered as a  (3+1) dimensional holographic screen embedded into a higher dimensional flat spacetime. We identify the holographic stress-energy tensor as that of the total dark components including the dark energy and dark matter.
We first construct a toy model, which provides a constraint relation among  the densities of dark matter, dark energy, and baryonic matter, in the case of considering the Lambda cold dark matter (ΛCDM) parameterization.
Furthermore, we generalize our toy model to the holographic Friedmann-Robertson-Walker (FRW) universe in a flat bulk and propose a new parameterization from the holographic model. The effective dark matter and dark energy are emergent and are identified with the Brown-York stress-energy tensor  <cit.>. We also compare our approach to the well studied Dvali-Gabadadze-Porrati (DGP) braneworld model <cit.>.

To produce the galaxy rotational curves, we further sketch a holographic elastic model with a de Sitter boundary and fix an inconsistency in the Verlinde's paper proposed in <cit.>. We recover the Tully-Fisher relation from the first law of thermodynamics and elasticity of the “de Sitter medium”.
The elasticity can also be realized in blackfold approaches <cit.> or some holographic models <cit.>. Notice here that we adopt the novel idea of elasticity of dark matter in the Verlinde's paper. Because the elasticity seems to capture the nature that the apparent dark matter is only the response of the presence of the baryonic matter. In the end, we also comment on the relation of the current construction in this paper with different scenarios such as some braneworld models and holographic models of the universe in the literature. 


In section <ref>, we firstly introduce the toy de-Sitter model in a flat bulk, which leads to the relation between dark matter component and baryonic matter component of the current universe.
In section <ref>, we generalize our toy model to the holographic FRW universe in a flat bulk and compare it with the DGP braneworld scenario.
In section <ref>, we reproduce the Tully-Fisher relation, with the help of holographic elasticity model and Verlinde's assumptions. We briefly compare and discuss the connection between our toy model and other scenarios, such as other braneworld models, holographic gravity, emergent gravity and summarize our results in section <ref>.





§ EMBEDDING  DE-SITTER UNIVERSE IN A FLAT BULK
 






 
We consider a 3+1 dimensional time-like hypersurface with induced metric g_μν and extrinsic curvature _μν, which is embedded into a 4+1 dimensional flat bulk spacetime. 
After adding the  stress-energy tensor T_μν of the baryonic matter and radiation, which is localized on the hypersurface,
we assume that the induced Einstein field equations on the hypersurface are modified as

    R_μν-  1/2R    g_μν - 1/L_μν -  g_μν =  κ_4  T_μν.

The length scale L is related to the positive cosmological constant Λ = 3/L^2.
The Einstein constant κ_4=  8π G/c^4, G is the Newton gravitational constant and c is the speed of light. Equivalently, we can rewrite the above modified Einstein field equations in (<ref>) as

    R_μν-  1/2R  g_μν   =  κ_4 T_μν+ κ_4_μν ,
    _μν   ≡1/κ_4 L_μν -  g_μν.

They are expected to govern the late time evolution of our universe.  _μν will turn out to be the Brown-York stress-energy tensor  <cit.> induced from higher dimensional space time. We will see L is related to the higher dimensional coupling constant κ_5 through L=κ_5/κ_4. At the cosmological scale, we assume that T_μν only includes the stress-energy tensor of baryonic matter and radiation. While _μν in (<ref>) represents the total dark components in our universe, such as the dark energy and dark matter.

We are going to  consider the parameterization in ΛCDM model describing the evolution of the  late time universe. In detail, we take the FRW metric in 3+1 dimensions, which assumes that our universe is uniform and isotropic at large scale, with scale factor a(t), 

    s_4^2    =g_μνx^μ x^ν =  -c^2 t^2 +a(t)^2
        r^2/1- k r^2  +r^2 Ø_2
    .

In the spatial flat ΛCDM model with k=0, it contains a positive cosmological constant Λ, which contributes to the dark energy with density parameter Ω_Ł, cold dark matter density parameter  Ω_D, and baryon density parameter Ω_B.
The Friedmann equation is given by

    H(t)^2/H_0^2 =  Ω_Λ +  Ω_D/a(t)^3 + Ω_B/a(t)^3 + Ω_R/a(t)^4.

H(t) is the Hubble parameter and H_0≡ H(t_0) is the Hubble constant today at t=t_0.
 
    H(t)≡ȧ(t)/a(t),       H_0^2  = κ_4 c^4 /3ρ_c ⇒ ρ_c =3 /κ_4H_0^2/c^4.

ȧ(t) is the derivative with respect to the time t and ρ_c is the critical energy density of the universe.
If requiring a(t_0)=1, from (<ref>) we have  1=Ω_Ł  +Ω_B +Ω_D +Ø_R.
Since the radiation density parameter Ω_R∼ 10^-4 is very small,
in the late time universe we will simply take Ω_D+Ω_B+Ω_Ł≃ 1.
After neglecting the radiation component Ø_R in (<ref>), the Friedmann equation of the late time spatial flat ΛCDM  universe can be written as
 
    H(t)^2/H_0^2 =  Ω_Λ +  Ω_D/a(t)^3 + Ω_B/a(t)^3≡_Ł +_D+_B.

We have introduced the time dependent notations with tilde, which satisify
 
    Ω_Ł=_Ł |_t=t_0,  Ω_D=_D |_t=t_0,   Ω_B=_B |_t=t_0 .


Based on the modified Einstein field equations (<ref>) and the Hamiltonian constraint from the consistent embedding in higher dimensional flat bulk, we are going to show an interesting constraint relation between these parameters,

    :  Ω_D^2   =1/2Ω_Ł (Ω_D- Ω_B).

Let us compare with the constraint relation in the Verlinde's emergent gravity <cit.>,

    Verlinde:  Ø_D^2    = 4 3Ø_B.

In the current universe both of these two relations (<ref>) and (<ref>) are remarkably well obeyed. 
Taking the observation values within the ΛCDM model <cit.>, with a bit priori choice of the parameters as
 
    Ω_Ł≃0.685,  Ω_D≃0.265,    Ω_B≃0.050 ,


we can calculate the following differences,  

    δ_   ≡Ω_D^2 - 1/2Ω_Ł (Ω_D- Ω_B)≃-0.003    ,  
    δ_V   ≡Ω_D^2 - 4 3Ω_B ≃ 0.004   .

Thus, our relation (<ref>) holds as well as the Verlinde's (<ref>) with minor difference in approximation.
We will show  how to derive this constraint equation (<ref>) in the following sections.





 §.§ Hamiltonian Constraint From Hypersurface Embedding


Similar to the formula (<ref>),
let us write down the Einstein equation in d dimensional spacetime as

    R_μν-  1/2 R   g_μν=κ_d
        T_μν + _μν
    ,

with μ,ν=0,1,...,(d-1), and κ_d=  8π G_d/c^4.
T_μν is the stress-energy tensor of baryonic matter and radiation,
and _μν is the effective dark components of our universe, which can include both of the dark energy and dark matter.
The trace of above equations yields the Ricci scalar

    R     = -2κ_d/d-2
        T +
    .


Now we assume that the geometry with metric g_μν can be embedded into one higher dimensional spacetime, as a hypersurface with the normal vector ^A, and the indices A, B = 0,1,...,d.
We can define the induced metric on the hypersurface g^A_  B= ^A_  B-^A_B
as well as the extrinsic curvature  _μν≡ g_μ^ A g_ν^ B∇̃_(A_B),
 with μ, ν are the indices on the hypersurface, which depend on the coordinate choices.
 ∇̃ is the covariant derivative associated with the bulk metric _AB.
Even though there exists  matter in the late time universe, we require them to be localized on the hypersurface,
such that we still have _AB^A^B=𝒯_AB^(d+1)^A^B=0 .
Thus, considering the Gauss equations, the Hamiltonian constraint equation of the hypersurface leads to

    0= 2 _AB^A^B   ≡^2- _μν^μν  -R.

On the other hand, the momentum constraint equations  _AB^Ag^N_ν=𝒯_AB^(d+1)^Ag^N_ν=0 lead to

    0=_AB^Ag^N_ν   = ∇_μ^μ_ν - g^μ_ν=0.

∇ is the covariant derivative associated with the metric g_μν on the hypersurface.

Next we assume that the stress-energy tensor of the dark components in (<ref>) can be given by the Brown-York stress-energy tensor  associated with the hypersurface <cit.>,

    _μν    =1/κ_d+1_μν -  g_μν,

and κ_d+1 is the Einstein's constant in d+1 dimensions.
Replacing the extrinsic curvature by the Brown-York stress-energy tensor,
the Hamiltonian constraint relation (<ref>) becomes

    ^2/d-1-_μν^μν =  R   /( κ_d+1)^2  .

Then by plugging (<ref>) into (<ref>), we have

    ^2/d-1-_μν^μν =
    - κ_d/( κ_d+1)^22/d-2T+.

While the momentum constraint equations (<ref>) lead to ∇^μ_μν=0.




 


 §.§ Holographic de-Sitter Screen in a Flat Bulk
  

Firstly we set that the stress-energy tensor of the baryonic matter and radiation in the Einstein field equations (<ref>) vanish, T_μν=0.
As a warm up, let us consider the hypersurface as the d dimensional de Sitter spacetime,

    s_d^2    =g_μνx^μ x^ν= -c^2 t^2 +e^2(ct/L)
        r^2  +r^2 Ø_d-2
    ,

which can be embedded into the d+1 dimensional flat spacetime
  
    s^2_d+1   
    =η_AB x^A x^B=-  X_0^2+ X_i^2,

with i=1,2,...,d.
The vacuum Einstein field equations associated with the (d+1)-dimensional flat bulk metric (<ref>)
turn out to be _AB=0.
Let us study the embedding of de Sitter hypersurface (<ref>) in more details.
It is a hyperbolic spacetime with radius L,

    L^2=- T^2+ X_i^2,    ^A=1/L(X_0,X_i).

where ^A is the normal vector of the hypersurface pointing outwards.
The cosmological constant Λ_d =(d-1)(d-2)/2 L^2 will play the role of the dark energy.
Notice that to balance the Einstein field equations (<ref>) with the induced de Sitter metric g_μν in (<ref>), one requires either the cosmological constant or the apparent dark energy term.

Interestingly, for the pure de Sitter spacetime (<ref>),
after considering  normal vector (<ref>) which leads to the extrinsic curvature ^μν=1/L g^μν,
the Brown-York stress-energy tensor  (<ref>) turns out to be ^μν = ^μν_Ł = - 1/κ_d+1d-1/L g^μν.
Then we arrive at the stress-energy tensor of apparent dark energy,
 
    ^μν_Ł   = - Λ_d/κ_d g^μν,  when κ_d+1/κ_d  = 2 L/d-2 .

One can see that the Einstein field equations in (<ref>) with the de Sitter metric (<ref>) are naturally satisfied

    R_μν-  1/2 R   g_μν=κ_d_μν^Ł .



From (<ref>) we read out the dark energy density formula
 
    ρ̃_Ł  = u_μ u_ν/c^4^μν_Ł = Λ_d/κ_d c^2  .

After considering (<ref>) with T=0, we have the identity

    _Ł^2/d-1-^Ł_μν_Ł^μν  = - ρ̃_Ł c^2/d-1_Ł .

Thus, assuming ^μν  =^μν_Ł≡ - Λ/κ_d g_μν in the constraint equation (<ref>),
 the pure de Sitter spacetime satisfies the above identity automatically. Notice here that  the Brown-York stress-energy tensor  plays the role of dark energy and there is no baryonic matter or dark matter yet in the set-up.






 §.§ Emergent Dark Matter on Holographic Screen



Next, we consider to add a small amount of baryonic matter and radiation in with the uniform and isotropic distribution.
Considering (<ref>)(<ref>),
our assumption for the constraint relation (<ref>) becomes

    ^2/d-1-_μν^μν = - ρ̃_Ł c^2/d-1[ T+ ].

This is the main constraint relation in this section.
Since in Einstein field equations (<ref>),
T_μν is the stress-energy tensors of baryonic matter and radiation,

    T^μν   =T_B^μν +T_R^μν,     T_B^μν  =(ρ_B  )u^μ u^ν,   T_R^μν  =(ρ_R  )u^μ u^ν+ p_R h^μν,

where ρ_B is the mass density baryonic matter, h^μν=g^μν +u^μ u^ν and u^μ is the velocity in d dimensions. The dark energy and dark matter are all assumed to be related to the extrinsic curvature of the hypersurface embedded in the higher dimensional flat bulk.
We take the Brown-York stress-energy tensor _μν, which is playing the role of dark energy and dark matter, 

    ^μν   =  _Ł^μν+_D^μν,    _Ł^μν= -(ρ_Ł c^2)  g^μν,  _D^μν=(ρ_D ) u^μ u^ν+p_D h^μν.

Putting them back into the constraint equation (<ref>), we have
 
    (ρ_Λ+ρ_D) [ dρ_Λ -(d-2) ρ_D- 2(d-1)  p_D/c^2] 
    
    =    ρ̃_Ł{ dρ_Ł+ρ_B+[ ρ_D - (d-1)p_D/c^2]+[ ρ_R - (d-1)p_R/c^2] }.
 
If setting ρ̃_Ł=ρ_Ł  and with equation (<ref>), we arrive at

    ρ_D^2    = ρ_Ł/d-2
        ρ_D-ρ_B-ρ_R+(d-1)p_R/c^2
    
     -d-1/d-2p_D/c^22ρ_D+ ρ_Λ.


When d=4, the stress-energy tensor of radiation is traceless  -ρ_R c^2+  3 p_R =0.
Keeping the pressure p_D of the dark matter in the constraint relation (<ref>) leads to
 
    ρ_D^2    = ρ_Ł/2 (1+3 _D)[ ρ_D(1-3_D)-ρ_B ],
         _D≡p_D/ρ_D c^2  .

_D denotes the effective state equation of the emergent dark matter, which can be time dependent in general.

Dividing both sides of (<ref>) by the squire of the critical energy density ρ_c^2 in (<ref>), we obtain the generalized constraint relation
 
    _D^2 =_Ł/2 (1+3 _D) [  _D(1-3 _D) - _B].

The components have been identified as
 
    _Ł≡ρ_Ł/ρ_c,  _D≡ρ_D/ρ_c,  _B≡ρ_B/ρ_c,

 which can be  time dependent in general case.



We will take the assumption that the evolution of the late time universe is governed by the ΛCDM parameterization,
and the total dark components are identified as the Brown-York stress-energy tensor in (<ref>).



We also assume  the emergent  dark matter is pressureless at t=t_0 for now and discuss the otherwise later in this paper.
Through setting _D= 0 in (<ref>), and considering (<ref>), we can obtain our main toy constraint in (<ref>),
 
    Ω_D^2 =1/2Ω_Ł (Ω_D- Ω_B).

 

If we further use Ω_Ł +Ω_D +Ω_B≃ 1 in the late time universe, then

    Ω_D^2 =   1/3  (Ω_D- Ω_B+ Ω_B^2).

Considering Ω_D≃ 5 Ω_B from (<ref>), as well as Ω_B≃0.05 ≪ 1,
 we can also arrive at the Verlinde's  Ω_D^2 ≃4/3Ω_B in (<ref>).
On the other hand, since Ω_B+ Ω_D≲Ω_Ł,
despite being not so precise, the de Sitter background is still a good approximation. However, if we consider the dark matter in smaller scales around the galaxies and compare with galactic rotational curves, we need to consider the effects of back-reaction of baryonic matter. This is the same situation in the earlier universe when matter or radiation dominates  in 
the energy of the universe and can not be treated as perturbations on the background anymore. In such cases, this toy model turns out to be not enough, we will resort to the more complicated model in the next section.






§ HOLOGRAPHIC FRW UNIVERSE AND EMERGENT DARK MATTER




In this section, we consider a more consistent embedding of the FRW metric into one higher dimensional flat spacetime <cit.>. We assume that the stress-energy tensor of the total dark components, including dark matter and dark energy, is provided by the holographic stress-energy tensor on the FRW hypersurface.
In  section <ref>, we firstly review the consistent embedding of the FRW hypersurface in a flat bulk. In section <ref>, we review the usual parameterization in DGP braneworld model with the Z_2 symmetry along the hypersurface. In section <ref>, we further develop our viewpoint on the holographic FRW universe in a flat bulk, using a different boundary condition from <ref>. We also discuss its connection to the well studied DGP braneworld model. In particular, we show that under  special parameter choice, the constraint relation (<ref>) in our toy model can be recovered in late time universe.
 



 §.§ Embedding FRW Universe in a Flat Bulk



Consider a 4+1 dimensional flat bulk  with action _5 and metric _AB, along with the  3+1 dimensional time like hypersurface  with action _4 and induced metric g_μν.
The total action is given by _5+_4, where

    _5=1/2 κ_5∫_^5 x √(-g̃)   + 1/κ_5∫_^4x √(-g) , 
       _4= 1/2 κ_4∫_^4x √(-g) R + ∫_^4x √(-g)L_  .

 is the trace of extrinsic curvature of the hypersurface , and L_ is the Lagrange density of matter localized on the hypersurface.
If choose the Gaussian normal coordinates of the bulk metric _AB, we have

    s^2_5 =_ABx^Ax^B =  ^2 + _μνx^μx^ν .

We assume the hypersurface  located at =0, which is the shared boundary of the half bulk _+ for the region >0 and the half bulk _- for the region <0.


The bulk equations of motion are given by the variation of the total action _5+_4 with the bulk metric _AB,

    1/κ_5_AB-1/2g̃_AB +1/κ_4R_μν-1/2 R g_μν^μ_ A^ν_ Bδ()   =  T_μν^^μ_ A^ν_ Bδ() .

With the matching junction condition at the hypersurface =0,

    _μν^+ - _μν^-  + 1/κ_4 G_μν   =T_μν^,

where G_μν≡ R_μν-1/2 R g_μν.
The effective stress-energy tensor from extrinsic curvature is

    _μν^±≡1/κ_5^±_μν -^± g_μν.
 
We include the baryonic matter, radiation and other effective terms in the Lagrangian L_, which leads to the stress-energy tensor

    T_μν^   =-2/√(-g)δ/δ g^μν∫_ d^4x √(-g)L_.

The extrinsic curvature is ^±_μν≡_μ^A _ν^B ∇̃_(A^±_B)  |_, and ^± is chosen as the normal vector of  along the ± directions, respectively.

We consider that our universe is uniform and isotropic at large scale, and take the spatially flat FRW metric in d=4 dimensions, 
   
    s^2_4 =    -c^2  t^2 +a(t)^2
        r^2 + r^2 Ø_2
    .

The consistent embedding in higher dimensional flat  spacetime has been discussed in <cit.>,
where the bulk metric (<ref>) in Gaussian normal coordinates is

    s^2_5 =_ABx^Ax^B =  ^2  - (,t)^2   c^2  t^2 +(,t)^2
        r^2 + r^2 Ø_2
     .

The consistent embedding metric functions are solved as  <cit.>,

    (,t)^2    =a(t)^2 +  ^2  ȧ(t)^2/c^2± 2 √(a(t)^2ȧ(t)^2/c^2+) ,
    (,t)     =_t(,t)/ȧ(t)  .

Here  is the integration constant, with dimension of [L]^-2.

 
In the following section <ref>, we will choose the Z_2 symmetry along the brane similar as the usual DGP model, where the parameter  is neglected.
 In section <ref>, we will consider the FRW brane as a cutoff hypersurface in the flat bulk and present an alternative interpretation  as the holographic FRW(hFRW) universe, with a non-zero parameter .
 







 §.§ Braneworld Scenario: DGP Model with Z_2 Symmetry


 
In the usual DGP model <cit.>,  the Z_2 symmetry along the brane in the bulk has been imposed,
which leads to the boundary condition ^ +_μν=-^ -_μν, as well as the modified Einstein field equations on the brane

    1/κ_4 G_μν   =T_μν^ + _μν^ ,  _μν^≡^ -_μν- ^ +_μν= 2/κ_5^-_μν -^- g_μν.

In the self accelerating branch of the DGP model (sDGP),
T_μν^ includes the baryonic matter and dark matter, while _μν^ provides the effective dark energy.
In the normal branch of the DGP model (nDGP),
depending on the parameterization, a cosmological constant needs to be supplemented.
In the usual setup, =0 in (<ref>) was chosen, then the metric (<ref>) becomes more simple,

    s^2_5     = ^2 - 
        1±||/cä(t)/ȧ(t)
    ^2 c^2  t^2 +
        1± ||/cȧ(t)/a(t)
    ^2 a(t)^2r^2 +r^2 Ø_2.

The above two equations will lead to the modified Friedmann equation on the brane,

    H(t)^2    = κ_4 c^4 /3
        ρ_ (t) + ρ_(t)
    ,

as well as the acceleration equation

    Ḣ(t)+H(t)^2   =- κ_4 c^4 /6
        ρ_ (t) + ρ_(t)+ 3  p_ (t) + p_(t)/c^2
    .

The energy conservation equations for each component are

    ρ̇_ı(t)   =-3 H(t) 
        ρ_ı(t)+ p_ı(t)/c^2
    ,    ı=,   .

Plugging the metric (<ref>) into the stress-energy tensor _μν^ in (<ref>), we can read out the effective energy density and pressure

    ρ_(t)    =±2/κ_5 c^3  3H(t), 
    
    p_(t)    =∓2/κ_5 c [ 3 H(t)  + Ḣ(t)/H(t)].


The positive ρ_ and negative p_ correspond to the sDGP branch. The negative ρ_ and positive p_ correspond the nDGP branch, and the extra effective cosmological constant is required in equation (<ref>).
Since _μν^ in (<ref>) is proportional to the Brown-York stress-energy tensor,
it is natural to see that the Hamiltonian constraint equation on the brane in the bulk (<ref>) is satisfied and leads to

    _^2/3-^_μν_^μν =
    - κ_4/( κ_5/2)^2
        T_ + _
    .

Compared with the constraint relation in our toy model (<ref>), the coupling constant κ_5 is replaced by κ_5/2 in (<ref>). It is due to the double copies of the Brown-York stress-energy tensor in ^_μν  at (<ref>) in the DGP model.

Here we pay attention to the sDGP branch, where the modified Friedmann equation (<ref>) is summarized as

    H(t)^2/H_0^2 = Ω_M/a(t)^3  + √(Ω_)H(t)/H_0,
        Ω_=c^2/^2 H_0^2,    ≡(κ_5/2)/κ_4.

 If considering a(t_0)=1 and H(t_0)=H_0 in (<ref>),
we have 1=Ω_M+ √(Ω_).
Compared with (<ref>) in our toy model, =L/2 is due to the Z_2 symmetry of the sDGP brane, which includes the double copies of the Brown-York stress-energy tensor.

Equivalently,

    H(t)^2/H_0^2   =  Ω_/2+ Ω_M/a(t)^3
    +  
        Ω_^2 /4+ Ω_Ω_M/a(t)^3
    ^1/2,    Ø_M=1-√(Ø_) .

Notice that to make the presentation more clear, we did not include the contribution from spatial curvature Ω_K and radiation  Ω_R.
It is easy to see that if setting Ω_M = 0 in (<ref>), we will have the Friedmann equation of the de-Sitter Universe.
In the self accelerating branch of the DGP model, Ω_M=Ω_B + Ω_D, including both of the components of baryonic matter and dark matter, while Ω_ℓ is the component of the effective dark energy.
More detailed study of the phenomenology of the DGP model can be found in <cit.>.
 
In order to recover the constraint relation (<ref>) in the toy model, we need to give a different interpretation of these parameters in the sDGP model,








    _Ł   =    Ω_  ,    _D =  Ω_(t) -Ø_,    _B≡Ω_M/a(t)^3= H(t)^2/H_0^2 - Ω_(t)  , 
    _D   =-1-1/3 H(t)Ω̇_(t)/Ω_(t) -Ø_,        Ω_(t)≡ρ_/ρ_c = √(Ω_)H(t) /H_0   .

In particular, taking the derivative of (<ref>) and eliminating Ω_M with (<ref>)  again will lead to the identical relation of Ḣ(t), as well as _D from (<ref>),

    Ḣ(t)   =  - 3 H(t)^2H(t)/H_0-√(Ø_)/2 H(t)/H_0-√(Ø_), 
    _D   = - H(t)/H_0-√(Ø_)/2 H(t)/H_0-√(Ø_)  .

One can check that the general constraint relation (<ref>) is satisfied after putting back the expressions (<ref>) and (<ref>), at any cosmological time t.



From (<ref>) and (<ref>), it is clear to see that only in the very late time universe that H(t)→ H_0√(Ø_), we can have _D(t)→ 0, which is a bit different with the ŁCDM model.
If setting Ω_M=Ω_B, equating the right hand side of (<ref>) with that in the late time ŁCDM model (<ref>) at a(t_0)=1, we arrive at  Ω_D^2 =Ω_ŁΩ_B-Ω_D. It is quite different from our toy constraint relation (<ref>), because it requires Ø_B>Ø_D, which does not match with the observed parameters in (<ref>). Thus, if taking the sDGP model with Friedmann equation in (<ref>) and setting Ω_M=Ω_B, it can not recover our toy constraint relation (<ref>).
 
  

In the next section, we will give an alternative interpretation of the embedding scenario as the holographic FRW universe. In particular, we will turn on the parameter I, which is usually set to be zero in the the previous studies of the  DGP models <cit.>.
It will become clear that only if keeping this parameter I in the embedding function (<ref>), 
we can recover the constraint relation (<ref>) in our toy model.
 








 §.§ Holographic Scenario: Holographic FRW Universe



In the previous subsection, we have studied the dynamics of a FRW hypersurface, which is embedded into the higher dimensional flat spacetime. 
The physical picture is related to the traditional braneworld models <cit.>, or the blackfold approaches with higher dimensional embedding <cit.>.

In this subsection, we will give a new physical interpretation of the FRW hypersurface in a flat bulk with the embedding metric (<ref>). It can be reduced to our toy model with a de-Sitter hypersurface in the flat bulk.
From the viewpoint of the cutoff holography in the flat spacetime <cit.>,
we can drop the manifold _- in the flat bulk, such that the hypersurface  at =0 plays the role of   the holographic boundary of the manifold _+.  
Or from the viewpoint of membrane paradigm <cit.>, the manifold _+ can be replaced by the quasi-local Brown-York stress-energy tensor on the hypersurface .
It is also equivalent to set  _μν^+ =0 in the junction condition (<ref>), and the Einstein field equation becomes

    1/κ_4 G_μν   =T_μν^M + _μν^,

where the Brown-York stress-energy tensor  on  is

    _μν^   =-2/√(-g)δ(_5)/δ g^μν= _μν^- =1/κ_5^-_μν -^- g_μν.

Again the Hamiltonian constraint equation  (<ref>)  is automatically satisfied

    _^2/3-^_μν_^μν =
    - κ_4/( κ_5)^2[ T_ + _].


We will choose the negative branch in (<ref>), such that the energy density and pressure in ^_μν are given by

    (κ_5 c^2) ρ_   = - 3 ∂_(,t)/(,t)|_=0=  3[H(t)^2/c^2 + /a(t)^4]^1/2 , 
    
    (κ_5 ) p_   =
        2∂_(,t)/(,t)+∂_(,t)/(,t)
    |_=0
    =-  [ Ḣ(t)+3 H(t)^2 /c^2 + /a(t)^4]  /[H(t)^2/c^2 + /a(t)^4]^1/2 .

The modified Friedmann equation is,

    H(t)^2    = κ_4 c^4 /3
        ρ_ (t) + ρ_(t)
    ,     H_0^2  = κ_4 c^4 /3ρ_c  .

And the energy conservation equation remains

    ρ̇_ı(t)   =-3 H(t) 
        ρ_ı(t)+ p_ı(t)/c^2
    ,    ı=,  .

Again we use the same setting in (<ref>), considering that ρ_c  =  3/κ_4H_0^2/c^4,
we have
 
    Ω_Ł=ρ_Ł/ρ_c=c^2/L^2 H_0^2,    ρ_Ł  =  3/κ_41/c^2 L^2,    
    L=κ_5/κ_4 .









Putting (<ref>) into (<ref>), the modified Friedmann equation is summarized as

    H(t)^2/H_0^2 = Ω_M/a(t)^3 + Ω_Ł^1/2
        H(t)^2/H_0^2 + Ω_/a(t)^4
    ^1/2 ,    Ω_≡  c^2/H_0^2.
 
Or equivalently,

    H(t)^2/H_0^2   = Ω_Ł/2 + Ω_M/a(t)^3 + 
        Ω_Ł^2 /4+ Ω_ŁΩ_M/a(t)^3 + Ω_ŁΩ_/a(t)^4
    ^1/2.


We named this Scenario as the holographic FRW(hFRW) model. Instead of using the ΛCDM parameterization in (<ref>), we has a different set of parameters in the hFRW model. 
Notice here that by setting Ω_M=Ω_B+Ω_D and Ω_=0, we can recover the usual Friedmann equation  (<ref>) of the sDGP model.
While if setting Ω_M=Ω_B and turning  the parameter Ω_I, it can be shown that one  is able to recover our toy constraint relation (<ref>).

 
Firstly, we need to match these parameters in hFRW model with that in the constraint relation (<ref>),

    _Ł   = Ω_Ł,     _D =  Ω_(̋t) -Ø_Ł,     _B≡Ω_M/a(t)^3
    = H(t)^2/H_0^2 -Ω_(̋t), 
    
    _D    =-1-1/3 H(t)Ω̇_(̋t)/Ω_(̋t)-Ø_Ł,      Ω_(̋t)≡ρ_/ρ_c
    = Ω_Ł^1/2[ H(t)^2/H_0^2 + Ω_/a(t)^4]^1/2  .

In particular, taking the derivative of (<ref>) and eliminating Ω_M with (<ref>)  again will lead to the identical relation of Ḣ(t), as well as _D(t) from (<ref>),

    Ḣ(t)   =  - 3 H(t)^2
        √( H(t)^2/H_0^2+Ø_I/a(t)^4) -√(Ø_Ł)
    
      - 1/3Ø_I/a(t)^4/H(t)^2/H_0^2/2√( H(t)^2/H_0^2+Ø_I/a(t)^4)- √(Ø_Ł)  , 
    _D   = - 
        √( H(t)^2/H_0^2+Ø_I/a(t)^4) -√(Ø_Ł)
       - 1/3Ø_I/a(t)^4/
        √( H(t)^2/H_0^2+Ø_I/a(t)^4) -√(Ø_Ł)
    / 2√( H(t)^2/H_0^2+Ø_I/a(t)^4)- √(Ø_Ł)   .

One can check that the general constraint relation (<ref>) is satisfied automatically
after plugging in above quantities (<ref>) and (<ref>), at any cosmological time t. 









Now let us compare it with the late time evolution of ΛCDM model with Friedmann equation  (<ref>).

If only setting Ω_M=Ω_B, and equalizing the right hand side of (<ref>) and (<ref>) at a(t_0)=1, we arrive at

    Ω_D^2 = Ω_ŁΩ_ -  Ω_ŁΩ_D-Ω_B.


Thus, once taking

    Ω_=3/2Ω_D-Ω_B,

we can recover our toy constraint relation in (<ref>).
Plugging the  ΛCDM parameterization
(<ref>) into the effective energy density (<ref>) and pressure (<ref>) from the Brown-York stress-energy tensor, and considering (<ref>), we have

    ρ_≃̋ρ_c(Ω_Ł+Ω_D),     p_≃̋- (ρ_c c^2)Ω_Ł.

Thus, it is consistent with the ansatz in our toy model (<ref>) with ρ_D=ρ_cΩ_D and p_D=0, as well as ρ_Λ=ρ_cΩ_Ł=- p_Λ/c^2.

Finally, we summarize the normalized Hubble parameters H(z)/H_0 in terms of the redshift z in various models.
The redshift z is related to the scale factor via a(t)/a(t_0)=1/(1+z).
Considering (<ref>)(<ref>)(<ref>) and setting a(t_0)=1, we have


    ΛCDM:  H(z) /H_0   =  √(Ω_Λ +   Ω_M(1+z)^3), 
    
    sDGP:  H(z)/H_0    = √(Ω_/2+ Ω_M(1+z)^3
    +  Ω_ℓ/2[ 1 +  4 Ω_M/Ω_ℓ(1+z)^3]^1/2),
    
    hFRW:  H(z) /H_0     =√(Ω_Ł/2 + Ω_B(1+z)^3 
    +Ω_Λ/2[ 1+  
    4 Ω_B/Ω_Λ(1+z)^3 + 
    4 Ω_I/Ω_Λ(1+z)^4]^1/2).

Taking (<ref>) and (<ref>), the associated state equations _D(z) for various models are

    ΛCDM:  _D(z)   = 0   , 
    
    sDGP: _D(z)    =  - H(z)/H_0-√(Ø_)/2 H(z)/H_0-√(Ø_)  ,
    
    hFRW:  _D(z)   = - 
        √( H(z)^2/H_0^2+ Ω_I(1+z)^4) -√(Ω_Λ)
       - 1/3Ø_I(1+z)^4/
    
        √( H(z)^2/H_0^2+ Ω_I(1+z)^4) -√(Ω_Λ)
    / 2√( H(z)^2/H_0^2+ Ω_I(1+z)^4)- √(Ø_Ł)  .

Again in order to make the presentation simpler, we here neglected the contribution of radiation Ω_R and spatial curvature Ω_K, which can be easily included in the equations above. Here including Ω_ in (<ref>) turns the value of _D(z) from negative to positive in the late time universe, and thus effectively contributes to the emergent dark matter.
 




In Figure <ref>, we plot the reduced Hubble parameters H(z)/H_0 and the state equations _D(z) of the emergent dark matter in terms of the redshift z in various models. 
The Friedmann equation of spatial flat ŁCDM model is in (<ref>), with the  parameters in (<ref>). 
The Friedmann equation of sDGP model is in (<ref>), with the fitting parameter Ø_M=0.21  <cit.>.
The Friedmann equation of our hFRW model is in (<ref>), 
with a special choice of the parameters Ø_M=Ø_B, Ø_I=3/2(Ø_D-Ø_B), along with the values in (<ref>).
More detailed studies of this non-zero Ω_ and fitting parameters in the hFRW model will appear in our future work.












 



§ TOWARDS HOLOGRAPHIC DE SITTER BRANE WITH ELASTICITY





 
In the above section <ref>, inspired by the emergent gravity by Verlinde in <cit.>, we have proposed the emergent dark matter on the de-Sitter hypersurface in a flat bulk, which gives rise to the similar mechanism as in <cit.>.
In section <ref>, we have generalized the holographic  de-Sitter scenario to the time evolution case with a FRW hypersurface in a flat bulk.
However, the above setups still lack of the elasticity in the Verlinde's emergent gravity <cit.>.
In the holographic models,  the elastic property can usually be realized in the blackfold approaches <cit.>, or by including the effective mass terms in the bulk of holographic models <cit.>.
In this section, we will consider the embedding of a de Sitter hypersurface in the flat bulk with effective massive gravity terms, where the holographic elasticity is implemented.



On the other hand, in both section <ref> and section <ref>, we considered the uniform and isotropic metric at the cosmological scale. While in this section, aiming at a comparison with Verlinde's  derivation on the Tully-Fisher relation, we focus on the response at galactic scales of the de-Sitter brane. Instead of the uniform baryonic matter, we need to add the spherically symmetric baryonic matter.
We are also trying to reconcile the inconsistency in Verlinde's emergent gravity pointed out by <cit.>. We present a consistent derivation of Tully-Fisher relation in the frame of the elastic model and try to resolve some issues in the original Verlinde's story.
 




 §.§ Holographic Stress Tensor and Verlinde's  Apparent Dark Matter



 In order to embed the d-dimensional Verlinde's emergent gravity with elasticity into a higher dimensional bulk spacetime,
we sketch the more general total action as _d+1+_d, where

    _d+1=1/2 κ_d+1∫_^d+1 x √(-g̃)
        _d+1-2 Ł_d+1 +
    
    + 1/κ_d+1∫_^dx √(-g)𝒦, 
       _d= 1/2 κ_d∫_^dx √(-g)R_d -2Ł_d+  ∫_^dx √(-g)L_ .

In the bulk manifold  with metric g̃_AB, the Lagrangian density  represents the effective term which can provide the holographic elasticity. On the boundary  with induced metric g_μν, the trace of the extrinsic curvature is 𝒦.
Like in our toy model in section <ref>, we can set Λ_d+1=0 in the bulk, and study the holographic elastic response of the screen after putting in the baryonic matter.
One may also add Λ_d in the boundary action _d, which can contribute to the total cosmological constant on the boundary theory, or the tension of the boundary brane.


The viscous or elastic response of the boundary theory in the holographic description is encoded in the transverse-traceless tensor mode of the metric perturbations in the bulk. In the toy model in section <ref>, we embed the d-dimensional de Sitter spacetime as the hypersurface in the (d+1)-dimensional flat bulk in Einstein gravity. The usual holographic solid model is on the flat boundary in AdS_d+1 spacetime in massive gravity  <cit.>. However, one may embed the dS-sliced coordinates into the AdS_d+1 spacetime and obtain dS_d as the boundary hypersurface. It is foreseeable that the elastic solid model can be generalized into the case with the de Sitter boundary with the tension term <cit.>.
Although the detailed construction is still left to be done and we leave it to a future work, for now, we assume the above action can capture the feature of the elastic theory. In any case, the discussion for the rest of this section is actually independent of the holographic construction and it only uses the theory that describes the elastic solid with different modulus values in a de Sitter background.

For the d-dimensional de Sitter hypersurface embedded in the higher dimensional flat bulk spacetime, 
instead of the expanding coordinate in (<ref>),
we now consider the static coordinate patch described by the metric
 
    s_d^2    =g_μνx^μx^ν  =-f(r) t^2 +ĥ_ijx^ix^j ,  
    f(r) = 1-r^2/L^2,
    s_d-1^2    =ĥ_ijx^ix^j  = 1/f(r) r^2 +r^2 Ω_d-2^2,  
    i, j= 1,2,... d-1.

Here ĥ_ij is the induced metric on the spacial slice.
One can also define the projection tensor h^μν≡ g^μν +u^μ u^ν, along with the d velocity u^μ.
The Brown-York stress-energy tensor on the boundary of the bulk with the action (<ref>)  is given by

    ^μν   =  - 2/√(-g)δ(_d+1)/δg_μν≡ρ̃_D u^μ u^ν +^μν ,

where ρ̃_D is the effective holographic energy density induced from higher dimension, and we introduced the covariant stress tensor ^μν which satisfies ^μνu_μ=0.
The stress tensor _ij and strain tensor _ij are given by

    _ij≡h_i^μh_j^ν_μν,    _ij = h_i^μh_j^ν_μν,    _μν≡∇_(μ_̆ν),

and _μ is the shift vector associated with the deformation.


Now we take a detour to the Verlinde's derivation, where the displacement _i is caused by the presence of the baryonic matter, and the metric solution in (<ref>) becomes

    f(r) = 1-r^2/L^2+2Φ_B,    Φ_B≡ -8π G_d/(d-2)Ω_d-2M_B/r^d-3.

We consider the simple case that M_B =∫ ^r_B_0 ρ_D(r')A(r') dr' is the constant total mass of baryonic matter in the galaxy, with the characteristic scale r_B.
In the deep-MOND regime we are looking at in this section, r_B < r≪ L  <cit.>,
such that f(r) ≈ 1+ 2Φ_B will be taken in the following derivations <cit.>.

One issue in the Verlinde's derivation raised by <cit.>, is that the displacement _i=Φ_B/a_0n_i is identified, with a_0=c H_0.
When choosing d=4 in (<ref>), the baryonic matter induces a Newtonian potential Φ_B ∼ -GM_B/r, then the apparent dark matter surface density Σ_D ∼_ij scales as 1/r^2 at the large r.
However, to produce the Tully-Fisher relation or flat rotational curves in galaxies, Σ_D ∼_ij has to scale as 1/r at the large r. Thus, this is the inconsistency in the Verlinde's original story  <cit.>.
In the following, we try to circumvent this issue and see whether this assumption of the displacement can be abandoned. This displacement _i=Φ_B/a_0n_i is important  in <cit.>, where the ADM mass definition of the de Sitter spacetime is related to the strain tensor through
M=1/a_0∮ __∞σ_ijn_j d A_i.
The problem with this argument is for a pure de Sitter space with the positive cosmological constant in Einstein gravity, it can not have the elastic property on its own. The elaborated derivation <cit.> avoids the facts that one needs to go beyond the theory of Einstein gravity to have the correct elastic dark matter.
One way out is to embed the de Sitter hypersurface in higher dimensional bulk, such that the elasticity will emerge from the holographic brane,

and the effective Einstein field equations will be modified.
In the following, we propose a way to resolve this issue by employing a model of holography with the bulk action (<ref>) and reproduce the elastic dark matter response formula as the baryonic Tully-fisher relation in <cit.>.





 §.§ Emergent Tully-Fisher Relation from Holographic Elasticity






Considering the fact that δ_ij=-δĥ_ij /2 in <cit.>

and (<ref>),
the spacial components are
 
    _ij   =1/√(-g)δ(√(-g) ) /δ^ij
     =-2/√(-g)δ( _d+1 ) /δĥ^ij
    =   2 μ_ij +  λδ_ij  (^k_ k) ,

with shear elastic modulus μ and bulk modulus λ+2μ/(d-1). 
By tuning the effective Lagrangian  in the bulk (<ref>), we expect the designed values of μ coming out of the holographic engineering with only the traceless stress tensor <cit.>, such that the bulk modulus vanishes λ+2μ/(d-1)=0.
We can define the traceless part '_ij as below,

    _ij   =2 μ '_ij,    
     '_ij  = _ij- 1/d-1δ_ij (^k_ k) .

This describes only the shear deformation, without changing the volume of the body.  The  stress tensor _ij and strain tensor _ij then only have the traceless part '_ij. 
We can diagonalize the elastic strain tensor _ij and stress tensor _ij simultaneously, since they are symmetric and linearly related. Their eigenvalues are called the principal strain and stress values.We define (r) as the largest eigenvalue of the traceless part of the strain tensor,

    (r)    ≡'_ij  n^i n^j .

We adopt the volume formula of the entropy change Δ S(r) of the de Sitter spacetime by the total baryonic matter within a radius r in <cit.>,

    Δ S(r)= - 2 π  M_B    r/ħ,     T_dS= ħ a_0/2π,

where  T_dS is the  Gibbons-Hawking temperature of de Sitter spacetime.
Notice that this relation is somewhat ad-hoc here since it assumes that holographic elastic de Sitter brane has the same entropy formula as the pure de Sitter space. We can put forth a naive argument that nevertheless, it can still have Δ S(r) ∝  M_B r.
The change of the free energy then follows the volume law of the thermodynamics of the “de Sitter medium” with 
    Δ F (r)= - T_dSΔ S(r) =a_0   M_B  r .



Now slightly different from assuming the displacement  _i in Verlinde's <cit.>,
 we start from the variation of the holographic free energy density F in (<ref>).
Similarly, we will only consider the leading order contribution in terms of M_B with a fixed background metric in (<ref>),
where the effects of r^2/L^2 will also be neglected.
If we only consider the shear modes in (<ref>) and do not consider the variation of background metric, the free energy density formula becomes

    δ(√(-g))   =√(-g)_ijδ^ij =  μ δ ( √(-g)'_ij'^ij).

The change of total free energy F=∫_𝒱_d-1 within a radius r is approximately

    Δ F (r) = μ∫_0^r A()( '_ij'^ij)≳μd-1/d-2∫_0^r  A() ()^2.

Here  the area is A(r)=Ω_d-2r^d-2. We will take the last equal sign to be approximately true by defining (r) as the largest eigenvalue of the traceless part of _ij and assuming the other principal strains are equal and summed to -. This step is similar to the Verlinde's derivation.
After differentiating both equations (<ref>) and (<ref>) with respect to r, we arrive at

    μd-1/d-2  A(r) (r)^2 =a_0  M_B  .

One notices here that if we identify M_D(r) =∫ ^r_0 ρ_D(r')A(r') dr' as the total apparent dark matter mass enclosed inside radius r,
and assume the surface density of the apparent dark matter Σ_D and baryonic matter Σ_B as

    Σ_D(r) ≡M_D(r)/A(r)= μ/a_0(r),    Σ_B(r) ≡M_B/A(r),

then from (<ref>) we can arrive at the response relation as below,

    Σ_D(r)^2 = μ/a_0d-2/d-1Σ_B(r),    μ =  a_0^2/16 π G.

The value of shear elastic modulus μ has been chosen to be the same as in Verlinde's <cit.>,
although we take a different ansatz for the pre-factor in Σ_D(r).


When d=4, and considering g_D(r) = GM_D(r)/r^2, g_B(r) = GM_B /r^2, the equation (<ref>) leads to the same relation in Verlinde's <cit.>,

    g_D(r)^2= a_0/6 g_B(r).

 In the deep-MOND regime that g_D(r)≫ g_B(r), the above conclusion leads to the baryonic Tully-Fisher relation that,

    v_f^4 =  a_0/6GM_B ,      g_D(r) =v_f^2 /r ,

where v_f is the asymptotic velocity of the flattened galaxy rotation curve.
 
Notice here that we haven't explicitly constructed the theory with baryonic mass M_B on top of the elastic background. To do so requires more ingredients in the theory and it can become complicated, for such examples, see e.g. <cit.>.
We may compare the dark matter density here as in the toy model in section <ref>.
The strain tensor ϵ_μν in (<ref>) is related to the extrinsic curvature _μν.
The stress tensor _μν in (<ref>) contributes to the total Brown-York stress-energy tensor _μν.
For the toy model in section<ref>, we employ the hypersurface displacement in extrinsic curvature as the elastic displacement tensor, whereas in the holographic model in this section, the extra fields with  in the bulk is introduced. The boundary term in action (<ref>) is also related to the Brown-York stress-energy tensor in the modified Einstein equations (<ref>) in the toy model.
This leads to the questions in the previous section that whether we can realize this response in a model with braneworld, although the answer is not clear at this moment.









§ DISCUSSION AND CONCLUSION
  




In this final section, we will further discuss the difference and connections between our approach and some well-studied scenarios, especially the braneworld and the holographic universe.
Following the Verlinde's model for the apparent dark matter, there seem to be three crucial conditions for his construction so far. First, there is the background entropy, which distributes evenly in the volume. Second, the positive cosmological constant, provides the thermal bath with the Gibbons-Hawking temperature T_dS=ħ a_0/2π. The last, the apparent dark matter is only the response to the presence of the baryonic matter.
The braneworld scenario may offer something similar to above-mentioned conditions and becomes a natural playground for the Verlinde's emergent gravity. The elastic medium full of entropy can be explained from higher dimensions though additional brane dynamics, by treating our (3+1) dimensional spacetime as the boundary of the bulk spacetime. Cosmological constant and the standard model can be easily implemented with braneworld in the literature <cit.>.
Most interestingly, the branes with tensions and dynamics, may react to the matter fields we put in, with extra terms introduced. Especially the extrinsic curvature, a concept valid only from higher dimensional spacetime, may describe the “elastic response" nature of the apparent dark matter from the Verlinde's theory.


We comment on the possible constraints from gravitational wave observations.
Recently it is argued that two relativistic models of modified Newtonian dynamics seem inconsistent with observations <cit.>. Modified gravity models are constrained from two aspects.
One is the constraint of the energy loss rate from ultra high energy cosmic rays, which indicates that gravitational waves should propagate at the speed of light.
The other is the observed gravitational waveforms from LIGO, which are consistent with Einstein's gravity and suggest that the gravitational wave should satisfy linear equations of motion in the weak-field limit.
Although Verlinde suggested similar modifications of Newtonian dynamics as in MOND theory <cit.>, which emerges with a different underlining physical origin,
there are no covariant equations of motion for the gravitational waves. For our toy model in the previous sections,
the induced dark components can be viewed as dark energy and dark matter stress-energy tensor and they behave the same as the particle dark matter in the ΛCDM models at the leading order,
so it can pass the above-mentioned constraints <cit.>.

To be more specific, the extra apparent dark sector as the extra term in Einstein equation fills the space as dark medium and interacts with propagating photons in it. In our induced gravity, gravitational field equations are modified as

    R_μν-1/2g_μνR = κ_4 [ T_μν+_μν].

The Bianchi identity leads to
0≡∇^μ G_μν= κ_4 ∇^μT_μν+ κ_4 ∇^μ_μν.
If we did not put additional sources in the bulk, the Brown-York stress-energy tensor  (<ref>) itself is conserved ∇^μ_μν=0.
Thus it is similar to the effects of real dark matter, and it does not conflict with the observations from LIGO so far <cit.>.




Let us further compare our toy model to the other well studied braneworld models <cit.>
other than DGP.  Except for the constraint equations,
we also have the dynamical Einstein equations in higher dimensions.
For example, in BDL (Binetruy-Deffayet-Langlois) model <cit.>,
 the FRW metric is also embedded into one higher dimensional flat spacetime.
After including the baryonic matter T_μν on the brane, in principle we can also define the effective stress-energy tensor from the braneworld model

    ^_μν≡_μν + ( g_μσ- _μσ)^σ_ ν- 1/2^2-_ρσ^ρσg_μν+...  ,

which includes the description of  the hypersurface evolution, and _μν is associated with the bulk Weyl tensor. It might be interesting to derive the Tully-Fisher relation based on this formula.
One thing we notice is that in most of the braneworld models <cit.>, the gravity leaks to the extra dimension at large scale, so the gravitational force scales like 1/r^d-2, comparing to the Newtonian force 1/r^2. To have the observed flat galaxy rotational curves, the gravitational force has to scale like 1/r. Naively this does not work. 
While one may try to extend the DBI action of the branes with more fields to capture the elastic behaviors of the dark matter response theory, see for example <cit.>. One not only needs to add tension term to the D-brane action as the cosmological constant, the extra scalar/vector fields and its response with baryonic matter are also needed, see for example <cit.>.

The extra dimensions in the braneworld setups may also have some extra effects on the gravitational waves production and propagation. If we indeed take a higher dimensional point of view, we expect one extra breezing mode on top of two polarized propagating modes <cit.>. The extra breezing mode is constrained by the current experiments. There are also multiple massive Kaluza-Klein gravitational modes associated with the extra dimensions. Although those massive modes decay fast and may not reach the gravitational waves detector, they are constrained as well by the gravitational waves signal templates from the binary black hole signals.

Our model does not necessarily have observable effects from extra dimensions. It is still quite interesting to ask whether we can probe extra dimensions, although there is no evidence from experiments so far.  One recent study may come from gravitational waves physics in <cit.>, in addition to the long searching constraints from colliders and precision measurements of gravity.



In the traditional holographic theories for our 3+1 dimensional universe <cit.>, gravity in the bulk is encoded into the field theory on the boundary.
For the models which consider the universe as the boundary of  4+1  dimensional AdS <cit.>, there is an effective contribution from the holographic stress-energy tensor, which can be identified as the stress-energy tensor of dark energy and/or dark matter.
In our construction, the holographic screen is embedded into one higher dimensional flat spacetime, which is inspired by the holographic hydrodynamics in Rindler spacetime <cit.>.
It is named as Rindler fluid, which is described by the Brown-York stress-energy tensor on the accelerating cutoff hypersurface in a flat bulk. The dynamics is governed by the Hamiltonian and momentum constraint equation on the hypersurface. 
What is more, it will be interesting to see how the entanglement can happen in our toy model with the de Sitter boundary <cit.>. The entanglement between two cosmological horizons may have an impact on the gravity as suggested by Verlinde <cit.>.
It is shown in <cit.>, that the entangled pair in 3+1 dimensions can be described by the wormhole in 4+1 dimensional bulk spacetime. Thus, it is more clear to see the entanglement through the embeddings of wormholes into a higher dimensional bulk.



In summary, we construct a model for the dark components of our universe, where the dark sector originates from the induced stress-energy tensor of higher dimensional spacetime. In this holographic picture, there is only baryonic matter and radiation in the late time universe and dark matter is considered as the response of baryonic matter from the geometric effects.
In our approach, the toy model and the more developed hFRW model are partly borrowed from the Verlinde's emergent gravity in a subtle way. We choose to start from a holographic de Sitter screen in higher dimensional flat spacetime, with the known covariant relativistic formulas. The toy model produces one additional constraint of the late time universe components from ΛCDM parameterization. We then relate our toy model to the DGP braneworld with the new interpretation of the dark matter.
Moreover, we suggest a new holographic scenario with a set of parameters for the late time universe evolution. 
Although it has been pointed out that there are some inconsistencies in the Verlinde's emergent gravity <cit.>,
the idea that considers the dark matter as the response of baryonic matter is still quite interesting <cit.>. In section <ref> we fix an inconsistency of Verlinde's story and re-derived the Tully-Fisher relation.
In the future, it is interesting to relate our holographic model to the other well-motivated dark matter models (see for example <cit.>), 
as well as the emergent cosmology from quantum entanglement or thermodynamical laws (see for example  <cit.>). 




§ ACKNOWLEDGMENTS



We thank  Y.  F.  Cai, S.  P.  Kim, B.  H.  Lee, T.  Liu, C.  Park, M.  Sasaki, J.  Soda, H.  Tye, S.  J.  Wang, M. Yamazaki
for helpful conversations,  as well as the anonymous referees for very helpful comments and suggestions.
R.  G.  Cai was supported by the National Natural Science Foundation of China (No.11690022, No.11375247, No.11435006, No.11647601), Strategic Priority Research Program of CAS (No.XDB23030100), Key Research Program of Frontier Sciences of CAS.
S.  Sun was supported by MOST and NCTS in Taiwan.
Y.  L.  Zhang was supported by Young Scientist Training Program in APCTP, which is funded by the Ministry of Science, ICT & Future Planning(MSIP), Gyeongsangbuk-do and Pohang City.



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