Dark Energy as a Holographic Ricci Component of the Universe

Moulay-Hicham Belkacemia, Mariam Bouhmadi-Lopezb,c,∗, Ahmed Errahmania, Taoufiq Oualia

aLaboratory of Physics of Matter and Radiation, Mohammed I University, BP 717, Oujda, MoroccobDepartment of Theoretical Physics, University of the Basque Country UPV/EHU, P.O. Box 644, 48080 Bilbao, Spain

cIKERBASQUE, Basque Foundation for Science, 48011, Bilbao, Spain

Abstract

The holographic Ricci dark energy model is a very interesting proposal to describe the present acceleration of theuniverse. However, it turns out that a Friedmann-Lemaıtre-Robertson-Walker filled with this kind of fluid might facea big rip singularity in its future. Here we propose a way to smooth such a future doomsday on this kind of model.

Keywords: Late-Time Acceleration of the Universe, Dark Energy, Holographic Ricci Dark Energy and ModifiedTheories of Gravity

1. Introduction

From several astrophysical observations (supernovaetype Ia [1], cosmic microwave background [2], largescale structure [3], etc.), it is now widely accepted thatthe universe is undergoing a state of accelerating expan-sion. On the other hand, dark energy (characterised withnegative pressure) is the simplest and may be the mostphysical cause for the current acceleration of the uni-verse [4].

Now what is dark energy? We are far from giving ananswer to this question. Nevertheless, we would like tohighlight that there are several promising candidates astheoretical directions to the dark energy problem fromthe point of view of fundamental physics. An importantcandidate, inspired on applying the holographic princi-ple to the universe as a whole, was advanced and namedthe holographic dark energy scenario [5, 6] whose en-ergy density is inversely proportional to the square ofan appropriate length, L, that characterises the size ofthe system, in this case the universe, and representing

∗Corresponding authorEmail addresses: hicham.belkacemi@gmail.com

(Moulay-Hicham Belkacemi), mariam.bouhmadi@ehu.es (MariamBouhmadi-Lopez), ahmederrahmani1@yahoo.fr (AhmedErrahmani), ouali_ta@yahoo.fr (Taoufiq Ouali)

the infra-red (IR) cutoff of it. One of the natural choicesof this length, L, is the inverse of the Hubble rate. How-ever, this choice does not induce acceleration in a homo-geneous and isotropic universe [6] (see Ref. [7, 8] for anexample where a modification of the model presentedin [6] can explain the current acceleration of the uni-verse). Another choice for the length L was suggestedby Gao et al. [9] (see also Ref. [10]), in which the IRcutoff of the holographic Ricci dark energy (HRDE) wastaken to be the Ricci scalar curvature, i.e. L2 ∝ 1/R.When the size of the universe is characterised in such away, we end up with the holographic Ricci dark energy(HRDE) model [9], whose energy density reads:

ρH = 3βM2P

(12

dH2

dx+ 2H2

), (1)

where MP is the Planck mass, x = − ln(z + 1) = ln(a),z is the redshift and β is a dimensionless parameter thatmeasures the strength of the holographic component.

As we will review on the next section, a spatially flatFriedmann-Lemaıtre-Robertson-Walker (FLRW) uni-verse filled with this kind of matter accelerates andtherefore the HRDE can play the role of dark energy onthe Universe. It turns our that the asymptotic behaviourof the Universe depends crucially on the value acquiredby β: (i) if 1/2 < β the universe is asymptotically de Sit-

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ter, otherwise (ii) the universe faces a big rip singularity[11, 12] in its future evolution.

Our main purpose in this paper is to see if we can ap-pease the big rip appearing in some cases on the HRDEby invoking some infra-red and ultra-violet curvaturecorrections. This two corrections can be quite impor-tant to remove the big rip singularity which takes placeon the future and at high energy. The curvature cor-rections will be modeled within a 5-dimensional brane-world model with an induced gravity (IG) term on thebrane and a Gauss-Bonnet term in the bulk [13].

2. The Holographic Ricci Dark Energy Model

We start revising the HRDE model [9]. More pre-cisely, we consider a flat FLRW universe in the pres-ence of non-relativistic matter and a HRDE component[9]. The Friedmann equation for this model reads

a2

a2 =1

3M2P

(ρm + ρH), (2)

where ρm and ρH denote the energy density of matterand the HRDE component, respectively. The pressure-less matter, ρm is self-conserved, that is

ρm = 3M2PH2

0Ωm(1 + z)3, (3)

where Ωm is the dimensionless energy density parame-ter defined as

Ωm =ρ0

m3M2

PH20

, (4)

Furthermore, the HRDE density is proportional to theinverse of the Ricci scalar curvature radius R:

R = 6(H + 2H2). (5)

So that, the HRDE density is defined as [9]

ρH = 3βM2P

(12

dH2

dx+ 2H2

), (6)

where β is a dimensionless parameter that measures thestrength of the holographic component. By rewritingEq. (3) in terms of x = − ln(z + 1) and substituting ittogether with Eq. (6) in Eq. (2) the Friedmann equationcan be rewritten as [9]

E2 = Ωme−3x + β(

12

dE2

dx+ 2E2

). (7)

Therefore, the dimensionless energy density parameterof the HRDE component can be written as

ΩH = β(

12

dE2

dx+ 2E2

). (8)

By evaluating the Friedmann equation (7) at thepresent time, we obtain a constraint on the dimension-less parameters of the model:

1 = Ωm + ΩH0. (9)

After solving the Friedmann equation (7), we get

E2(z) =2

2 − βΩm(1 + z)3 + Ωβ(1 + z)4− 2β , (10)

where β � 12 , 2, andΩβ is an integration constant. Then,

by evaluating the solution (10) at the present time, weobtain

1 =2Ωm2 − β + Ωβ , (11)

which is a complementary constraint to that given inEq. (9).

Substituting E(z) given in Eq. (10) in Eq. (6), we ob-tain the HRDE density:

ρH = 3M2PH2

0[β

2 − βΩm(1 + z)3 + Ωβ(1 + z)4− 2β

]. (12)

Notice that, in the HRDE model, it is assumed thatthe energy density of the different components fillingthe universe is conserved and in particular the one cor-responding to the HRDE. So that, by substituting the en-ergy density (12) in the conservation law ρH+3H(ρH+pH) = 0, we obtain the HRDE pressure, pH:

pH = −3M2PH2

0

(2

3β− 1

3

)Ωβ(1 + z)4− 2

β . (13)

Finally, the total energy density, ρtot = ρm + ρH,reads

ρtot = 3M2PH2

0[2Ωm2 − β (1 + z)3 + Ωβ(1 + z)4− 2

β

]. (14)

Before continuing, we notice that the term (1+ z)4− 2β on

the previous equation induces acceleration if and only if0 < β < 1. We will impose this condition; 0 < β < 1,to ensure late-time acceleration. In the far future, asz tends to −1, the universe would be dominated by theholographic dark energy. If the range of the holographicparameter satisfies 0 < β < 1

2 , then the energy density(14) and the Hubble rate (10) diverge as well as H andpH; therefore, the universe hits a big rip singularity [11,12]. Notice that for 0 < β < 1

2 , the future singularityis avoided only for vanishing Ωβ; i.e. on absence of

M.-H. Belkacemi et al. / Nuclear Physics B (Proc. Suppl.) 246–247 (2014) 187–190188

a HRDE component. However, Ωβ has a crucial rolein the acceleration of an “holographic” universe, hencethe big rip singularity is unavoidable within the HRDEscenario unless 1 > β > 1

2 . Indeed, on that case; 1 >β > 1

2 , the universe would be asymptotically de Sitter.We will see on the next section, if this kind of fu-

ture singularity can be avoided by invoking some cur-vature corrections on the action which induce infra-redand ultra-violet modifications of general relativity.

3. The HRDE Model with Curvature Corrections

We consider a DGP brane-world model, where thebulk action contains a GB curvature term. The bulk cor-responds to two symmetric pieces of a 5-dimensional(5d) Minkowski space-time. The brane is spatially flatand its action contains an IG term. We assume thatthe brane is filled with matter and a HRDE component.Then, the modified Friedmann equation reads [13]:

H2 =1

3M2Pρ +ε

rc

(1 +

8α3

H2)

H, (15)

where H is the brane Hubble parameter, ρ = ρm +ρH isthe total cosmic fluid energy density of the brane whichcan be described through a cold dark matter component(CDM) with energy density ρm and a holographic Riccidark energy component with energy density ρH. Theparameters rc and α correspond to the cross over scaleand the GB parameter, respectively, both of them be-ing positive. The parameter ε in Eq. (15) can take twovalues: ε = 1, corresponding to the self-acceleratingbranch in the absence of any kind of dark energy [13];and ε = −1, corresponding to the normal branch whichrequires a dark energy component to accelerate at late-time (see for example [7, 15, 14]). For simplicity, wewill keep the terminology: (i) self-accelerating branchwhen ε = 1 and (ii) normal branch when ε = −1.

The modified Friedmann equation (15) can be rewrit-ten as

dEdx=

−Ωme−3x+(2β−1)E2

+2ε√Ωrc (1+ΩαE2

)EβE , (16)

where E(z) = H/H0 and

Ωm =ρm0

3M2PH2

0

, Omegarc =1

4r2cH2

0

, and

Ωα =83αH2

0 (17)

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0 0.05 0.1 0.15 0.2 0.25

β−

β+

βlimΩα

Figure 1: Plot of the parameters β± and βlim, defined in Eqs. (20) and(19), respectively, versus the parameter Ωα. We have used the valuesq0 ∼ −0.7, and Ωm ∼ 0.27. The parameter βlim defines the borderline between the normal branch (βlim < β) and the self-acceleratingbranch (β < βlim).

are the usual convenient dimensionless parameters andthe subscripts 0 denotes the present value (we will fol-low the same notation as in [14, 7, 16]). By evaluatingthe modified Friedmann equation at present and impos-ing that the brane is currently accelerating, we obtaina constraint on the parameter β which depends on thechosen brane{

β < βlim for ε = +1,β > βlim for ε = −1, (18)

where

βlim =1 −Ωm1 − q0

. (19)

An estimation of βlim can be obtained as follows: thebrane would behave roughly (to be consistent with thepresent observations) as the ΛCDM leading to βlim ∼0.43.

Even though the modified Friedmann equation (16)cannot be solved analytically, we can obtain the futureasymptotic behaviour of the brane which reads: (i) Ifβ < βlim or β− ≤ β, the brane is asymptotically de Sit-ter. (ii) If βlim < β < β−, the brane faces a big freezesingularity in its future [17], where (see also Fig. 1)

β± =1 + Ωα ± 2

√Ωα(1 −Ωm)

2[1 + Ωα ±

√Ωα(1 − q0)

] . (20)

We have completed and confirmed those results bysolving numerically the cosmological evolution of thebrane. We refer the reader to [16] for more details. Ouranalysis shows that even though the infra-red and ultra-violet effect can appease the big rip appearing on the

M.-H. Belkacemi et al. / Nuclear Physics B (Proc. Suppl.) 246–247 (2014) 187–190 189

HRDE model, it cannot remove them completely. Wewould like as well to point out that when the GB term isswitched off a little rip event [18] can show up which ismuch milder than a big rip or a big freeze. The little riphas been previously found on brane-world models [19].

4. Conclusions

We present a dark energy model based on a HRD en-ergy brane-world model of the Dvali-Gabadadze-Porratiscenario with a GB term in the bulk. The reason forinvoking curvature corrections, for example through abrane-world scenario, is to try to smooth the doom-days present on a standard 4-dimensional HRD energymodel. It turns out that the model presented here canonly partially remove those doomsdays.

Acknowledgement

M.B.L. acknowledges the kind invitation to partici-pate on the 9th International CosPA Asia Pacific Sym-posium held on Taipei (Taiwan) in 2012. She also ac-knowledges the support of the Basque Foundation forScience IKERBASQUE. A.E. and T.O. are supportedby CNRST, through the fellowship URAC 07/214410.This work was supported by the Portuguese AgencyFCT through PTDC/FIS/111032/2009.

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