holographic dark energy and cosmic coincidence

5
Physics Letters B 628 (2005) 206–210 www.elsevier.com/locate/physletb Holographic dark energy and cosmic coincidence Diego Pavón a , Winfried Zimdahl b a Departamento de Física, Universidad Autónoma de Barcelona, 08193 Bellaterra (Barcelona), Spain b Institut für Theoretische Physik, Universität zu Köln, D-50937 Köln, Germany Received 5 May 2005; accepted 8 August 2005 Available online 30 September 2005 Editor: L. Alvarez-Gaumé Abstract In this Letter we demonstrate that any interaction of pressureless dark matter with holographic dark energy, whose infrared cutoff is set by the Hubble scale, implies a constant ratio of the energy densities of both components thus solving the coin- cidence problem. The equation of state parameter is obtained as a function of the interaction strength. For a variable degree of saturation of the holographic bound the energy density ratio becomes time dependent which is compatible with a transition from decelerated to accelerated expansion. 2005 Elsevier B.V. All rights reserved. Nowadays there is a wide consensus among cos- mologists that the Universe has entered a phase of ac- celerated expansion [1]. The debate is now centered on when the acceleration did actually begin, whether it is to last forever or it is just a transient episode and, above all, which is the agent behind it. Whatever the agent, usually called dark energy, it must possess a negative pressure high enough to violate the strong energy con- dition. A number of dark energy candidates have been put forward, ranging from an incredibly tiny cosmo- logical constant to a variety of exotic fields (scalar, tachyon, k-essence, etc.) with suitably chosen poten- E-mail addresses: [email protected] (D. Pavón), [email protected] (W. Zimdahl). tials [2]. Most of the candidates, however, suffer from the coincidence problem, namely: Why are the matter and dark energy densities of precisely the same order today? [3]. Recently, a new dark energy candidate, based not in any specific field but on the holographic princi- ple, was proposed [4–9]. The latter, first formulated by ’t Hooft [10] and Susskind [11], has attracted much attention as a possible short cut to quantum gravity and found interesting applications in cosmology—see, e.g., [12]—and black hole growth [13]. According to this principle, the number of degrees of freedom of physical systems scales with their bounding area rather than with their volume. In this context Cohen et al. reasoned that the dark energy should obey the afore- said principle and be constrained by the infrared (IR) 0370-2693/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2005.08.134

Upload: diego-pavon

Post on 26-Jun-2016

219 views

Category:

Documents


2 download

TRANSCRIPT

Page 1: Holographic dark energy and cosmic coincidence

b

infraredthe coin-le degree

transition

Physics Letters B 628 (2005) 206–210

www.elsevier.com/locate/physlet

Holographic dark energy and cosmic coincidence

Diego Pavóna, Winfried Zimdahlb

a Departamento de Física, Universidad Autónoma de Barcelona, 08193 Bellaterra (Barcelona), Spainb Institut für Theoretische Physik, Universität zu Köln, D-50937 Köln, Germany

Received 5 May 2005; accepted 8 August 2005

Available online 30 September 2005

Editor: L. Alvarez-Gaumé

Abstract

In this Letter we demonstrate that any interaction of pressureless dark matter with holographic dark energy, whosecutoff is set by the Hubble scale, implies a constant ratio of the energy densities of both components thus solvingcidence problem. The equation of state parameter is obtained as a function of the interaction strength. For a variabof saturation of the holographic bound the energy density ratio becomes time dependent which is compatible with afrom decelerated to accelerated expansion. 2005 Elsevier B.V. All rights reserved.

cos-f ac-n

it isovent,tiveon-

eeno-ar,ten-

m

noti-

hvityee,

oftheral.ore-IR)

Nowadays there is a wide consensus amongmologists that the Universe has entered a phase ocelerated expansion[1]. The debate is now centered owhen the acceleration did actually begin, whetherto last forever or it is just a transient episode and, aball, which is the agent behind it. Whatever the ageusually called dark energy, it must possess a negapressure high enough to violate the strong energy cdition. A number of dark energy candidates have bput forward, ranging from an incredibly tiny cosmlogical constant to a variety of exotic fields (scaltachyon, k-essence, etc.) with suitably chosen po

E-mail addresses: [email protected](D. Pavón),[email protected](W. Zimdahl).

0370-2693/$ – see front matter 2005 Elsevier B.V. All rights reserveddoi:10.1016/j.physletb.2005.08.134

tials [2]. Most of the candidates, however, suffer frothe coincidence problem, namely:Why are the matterand dark energy densities of precisely the same ordertoday? [3].

Recently, a new dark energy candidate, basedin any specific field but on the holographic princple, was proposed[4–9]. The latter, first formulatedby ’t Hooft [10] and Susskind[11], has attracted mucattention as a possible short cut to quantum graand found interesting applications in cosmology—se.g.,[12]—and black hole growth[13]. According tothis principle, the number of degrees of freedomphysical systems scales with their bounding area rathan with their volume. In this context Cohen etreasoned that the dark energy should obey the afsaid principle and be constrained by the infrared (

.

Page 2: Holographic dark energy and cosmic coincidence

D. Pavón, Z. Zimdahl / Physics Letters B 628 (2005) 206–210 207

r-the

orR

n oftherktedthettlyuni-

nen-in

ow-to

into

ionhatin-see

ards.ghic

ion

t

aleatethetheuslyxpa

is

of

in-anthergyin-hcat

ng.n--

eenfer-f the

ei-

ossde-

cutoff [14]. In line with this suggestion, Li has agued that the dark energy density should satisfyboundρX � 3M2

pc2/L2, wherec2 is a constant and

M2p = (8πG)−1 [7]. He discusses three choices f

the length scaleL which is supposed to provide an Icutoff. The first choice is to identifyL with the Hub-ble radius,H−1. Applying arguments from Hsu[6],Li demonstrates that this leads to a wrong equatiostate, namely that for dust. The second option isparticle horizon radius. However, this does not woeither since it is impossible to obtain an acceleraexpansion on this basis. Only the third choice,identification ofL with the radius of the future evenhorizon gives the desired result, namely a sufficiennegative equation of state to obtain an acceleratedverse.

Here, we point out that Li’s conclusions rely othe assumption of an independent evolution of theergy densities of dark energy and matter which,particular, implies a scalingρM ∝ a−3 of the matterenergy densityρM with the scale factora(t). Any in-teraction between both components will change, hever, this dependence. The target of this Letter isdemonstrate that as soon as an interaction is takenaccount, the first choice, the identification ofL withH−1, can simultaneously drive accelerated expansand solve the coincidence problem. We believe tmodels of late acceleration that do not solve the cocidence problem cannot be deemed satisfactory (however,[15]).

Let us reconsider the argument Li used to discthe identification of the IR cutoff with Hubble’s radiuSetting L = H−1 in the above bound and workinwith the equality (i.e., assuming that the holograpbound is saturated) it becomesρX = 3c2M2

P H 2. Com-bining the last expression with Friedmann’s equatfor a spatially flat universe, 3M2

P H 2 = ρX + ρM , re-

sults in ρM = 3(1 − c2)M2P H 2. Now, the argumen

runs as follows: the energy densityρM varies asH 2,which coincides with the dependence ofρX on H .The energy density of cold matter is known to scasρM ∝ a−3. This corresponds to an equation of stpM � ρM , i.e., dust. Consequently, this should beequation of state for the dark energy as well. Thus,dark energy behaves as pressureless matter. Obviopressureless matter cannot generate accelerated esion, which seems to rule out the choiceL = H−1.

,

,n-

This is exactly Li’s conclusion. What underlies threasoning is the assumption thatρM andρX evolve in-dependently. However if one realizes that the ratiothe energy densities

(1)r ≡ ρM

ρX

= 1− c2

c2,

should approach a constant, finite valuer = r0 forthe coincidence problem to be solved, a differentterpretation is possible, which no longer relies onindependent evolution of the components. Givenunknown nature of both dark matter and dark enethere is nothing in principle against their mutualteraction (however, in order not to conflict with “fiftforce” experiments[16] we do not consider baryonimatter) to the point that assuming no interactionall is not less arbitrary than assuming a coupliIn fact, this possibility is receiving growing attetion in the literature[17–19] and appears to be compatible not only with SNIa and CMB data[20] buteven favored over non-interacting cosmologies[21].On the other hand, the coupling should not be sas an entirely phenomenological approach as difent Lagrangians have been proposed in support ocoupling—see[22] and references therein.

As a consequence of their mutual interaction nther component conserves separately,

(2)ρM + 3HρM = Q, ρX + 3H(1+ w)ρX = −Q,

though the total energy density,ρ = ρM + ρX, does.HereQ denotes the interaction term, andw the equa-tion of state parameter of the dark energy. Without lof generality we shall describe the interaction as acay process withQ = ΓρX whereΓ is an arbitrary(generally variable) decay rate. Then we may write

(3)ρM + 3HρM = ΓρX

and

(4)ρX + 3H(1+ w)ρX = −ΓρX.

Consequently, the evolution ofr is governed by

(5)r = 3Hr

[w + 1+ r

r

Γ

3H

].

In the non-interacting case (Γ = 0) and for a con-stant equation of state parameterw this ratio scalesasr ∝ a3w. If we now assumeρX = 3c2M2 H 2, this

P
Page 3: Holographic dark energy and cosmic coincidence

208 D. Pavón, Z. Zimdahl / Physics Letters B 628 (2005) 206–210

n

e of

oiceost.

niednUs-

-

atiohedi-

m-eter

sion

ic

bleim-nt

. In

tedotelve

tion.nce-

d-ois

gainori-toby

eter

na-

s

t be-

en-

se

definition implies

(6)ρX = −9c2M2P H 3

[1+ w

1+ r

],

where we have employed Einstein’s equationH =−3

2H 2[1 + w1+r

]. Inserting(6) in the left-hand sideof the balance equation(4) yields a relation betweethe equation of state parameterw and the interactionrateΓ , namely,

(7)w = −(

1+ 1

r

3H.

The interaction parameterΓ3Htogether with the ratio

r determine the equation of state. In the absencinteraction, i.e., forΓ = 0, we havew = 0, i.e., Li’sresult is recovered as a special case. For the chρX = 3c2M2

P H 2 an interaction is the only way thave an equation of state different from that for duAny decay of the dark energy component (Γ > 0)into pressureless matter is necessarily accompaby an equation of statew < 0. The existence of ainteraction has another interesting consequence.ing the expression(7) for Γ in (5) provides us withr = 0, i.e.,r = r0 = const. Therefore, if the dark energy is given byρX = 3c2M2

P H 2 and if an interactionwith a pressureless component is admitted, the rr = ρM/ρX is necessarily constant, irrespective of tspecific structure of the interaction. Under this contion we have [cf.(1)]

(8)c2 = 1

1+ r0.

At variance with[7,9], the fact thatc2 is lower thanunity does not prompt any conflict with thermodynaics. For the case of a constant interaction paramΓ

3H≡ ν = const, it follows that

(9)

ρ,ρM,ρX ∝ a−3m

(m = 1+ w

1+ r0= 1− ν

r

),

while the scale factor obeysa ∝ tn with n = 2/(3m).Consequently, the condition for accelerated expanis w/(1+ r0) < −1/3, i.e.,ν > r0/3.

Accordingly, the expression for the holographdark energy with the identificationL = H−1 fits wellinto the interacting dark energy concept. The Hubradius is not only the most obvious but also the splest choice. It is not only compatible with a consta

ratio between the energy densities but requires ita sense, the holographic dark energy withL = H−1

together with the observational fact of an acceleraexpansion almost calls for an interacting model. Nthat the interaction is essential to simultaneously sothe coincidence problem and have late acceleraThere is no non-interacting limit, since in the abseof interaction, i.e.,Q = Γ = 0, there is no acceleration.

Obviously, a change ofr0 demands a corresponing change ofc2. Within the framework discussed sfar, a dynamical evolution of the energy density ratioimpossible. As a way out it has been suggested ato replace the Hubble scale by the future event hzon [23]. Here we shall follow a different strategyadmit a dynamical energy density ratio. Motivatedthe relation(8) in the stationary caser = r0 = const,we retain the expressionρX = 3c2M2

P H 2 for the darkenergy but allow the so far constant parameterc2 tovary, i.e., c2 = c2(t). Since the precise value ofc2

is unknown, some time dependence of this paramcannot be excluded. Then this definition ofρX implies

(10)ρX = −9c2M2P H 3

[1+ w

1+ r

]+ (c2)·

c2ρX,

which generalizes Eq.(6). Using now the expressio(10) for ρX on the left-hand side of the balance eqution (4), leads to

(11)(c2)·

c2= −3H

r

1+ r

[w + 1+ r

r

Γ

3H

].

A vanishing left-hand side, i.e.,c2 = const, consis-tently reproduces(7). Comparing the right-hand sideof Eqs. (11) and (5)yields (c2)·/c2 = −r/(1 + r),whose solution is

(12)c2(1+ r) = 1.

The constant has been chosen to have the correchavior (8) for the limit r = r0 = const. We concludethat if the dark energy is given byρX = 3c2M2

P H 2 andc2 is allowed to be time dependent, this time depdence must necessarily preserve the quantityc2(1+r).The time dependence ofc2 thus fixes the dynamicof r (and vice versa). Sincer is expected to decreasin the course of cosmic expansion,r < 0, this is ac-companied by an increase inc2, i.e.,(c2)· > 0.

Page 4: Holographic dark energy and cosmic coincidence

D. Pavón, Z. Zimdahl / Physics Letters B 628 (2005) 206–210 209

h

tionved todel

tro-

n-

ionint

otThe

sat-ovet at

ant

dt,ted

ystu-

rkra-he

gyofra-

s.),cetier

ingries,

on-ris,

alNJ,

00)

509

72

03)

82

18

04)

06

Solving(11) for the equation of state parameterw

we find

(13)w = −(

1+ 1

r

)[Γ

3H+ (c2)·

3Hc2

].

For (c2)· = 0 one recovers expression(7). It is ob-vious, that both a decreasingr and an increasingc2

in (13) tend to makew more negative compared witw = −(1+ 1

r) Γ

3Hfrom (7). A variation of thec2 para-

meter can be responsible for a change in the equaof state parameterw. Such a change to (more) negativalues is required for the transition from decelerateaccelerated expansion. For a specific dynamic moassumptions about the interaction have to be induced. This may be done, e.g., along the lines of[18,19]. However, as is well known, the holographic eergy must fulfill the dominant energy condition[24]whereby it is not compatible with a phantom equatof state (w < −1). This automatically sets a constraonΓ andc2.

It is noteworthy that in allowingc2 to vary, contraryto what one may think, the infrared cutoff does nnecessarily change. This may be seen as follows.holographic bound can be written asρX � 3c2M2

p/L2

with L = H−1. Now, Li and Huang[7–9]—as well asourselves—assume that the holographic bound isurated (i.e., the equality sign is assumed in the abexpression). Since the saturation of the bound is noall compelling, and the “constant”c2(t) increases withexpansion (asr decreases) up to reaching the constvalue (1 + r0)

−1, the expressionρX = 3c2(t)M2pH 2,

in reality, does not imply a modification of the infrarecutoff, which is stillL = H−1. What happens is thaasc2(t) grows, the bound gets progressively saturaup to full saturation when, asymptotically,c2 becomesa constant. In other words, the infrared cutoff alwaremainsL = H−1, what changes is the degree of saration of the holographic bound.

In this Letter we have shown thatany interac-tion of a dark energy component with densityρX =3c2M2

P H 2 (andc2 = const) with a pressureless damatter component necessarily implies a constanttio of the energy densities of both components. Tequation of state parameterw is determined by theinteraction strength. A time evolution of the enerdensity ratio is uniquely related to a time variationthec2 parameter. Under this condition a decreasingtio ρM/ρX sendsw to lower values.

References

[1] A.G. Riess, et al., Astron. J. 116 (1998) 1009;S. Perlmutter, et al., Astrophys. J. 517 (1999) 565;S. Perlmutter, et al., in: V. Lebrun, S. Basa, A. Mazure (EdProceedings of the IVth Marseille Cosmology ConferenWhere Cosmology and Particle Physics Meet, 2003, FronGroup, Paris, 2004;A.G. Riess, et al., Astrophys. J. 607 (2004) 665.

[2] S. Carroll, in: W.L. Freedman (Ed.), Measuring and Modelthe Universe, in: Carnegie Observatory, Astrophysics Sevol. 2, Cambridge Univ. Press, Cambridge, 2004;T. Padmanbhan, Phys. Rep. 380 (2003) 235;J.A.S. Lima, Braz. J. Phys. 34 (2004) 194;V. Sahni, astro-ph/0403324;V. Sahni, in: P. Brax, et al. (Eds.), Proceedings of the IAP Cference on the Nature of Dark Energy, Frontier Group, Pa2002.

[3] P.J. Steinhardt, in: V.L. Fitch, D.R. Marlow (Eds.), CriticProblems in Physics, Princeton Univ. Press, Princeton,1997.

[4] P. Horava, D. Minic, Phys. Rev. Lett. 85 (2000) 1610;P. Horava, D. Minic, Phys. Rev. Lett. 509 (2001) 138.

[5] K. Enqvist, M.S. Sloth, Phys. Rev. Lett. 93 (2004) 221302.[6] S.D.H. Hsu, Phys. Lett. B 594 (2004) 13.[7] M. Li, Phys. Lett. B 603 (2004) 1.[8] Q.-G. Huang, Y. Gong, JCAP 08 (2004) 006.[9] Q.-G. Huang, M. Li, JCAP 08 (2004) 013.

[10] G. ’t Hooft, gr-qc/9310026.[11] L. Susskind, J. Math. Phys. (N.Y.) 36 (1995) 6377.[12] B. Wang, E. Abdalla, T. Osada, Phys. Rev. Lett. 85 (20

5507;M. Cataldo, N. Cruz, S. del Campo, S. Lepe, Phys. Lett. B(2001) 138;R. Horvat, Phys. Rev. D 70 (2004) 087301;G. Geshnizjani, D.J.H. Chung, N. Afshordi, Phys. Rev. D(2005) 023517.

[13] P.S. Custodio, J.E. Horvath, Gen. Relativ. Gravit. 35 (201337.

[14] A.G. Cohen, D.B. Kaplan, A.E. Nelson, Phys. Rev. Lett.(1999) 4971.

[15] B. McInnes, astro-ph/0210321;R.J. Scherrer, Phys. Rev. D 71 (2005) 063519.

[16] B. Ratra, P.J.E. Peebles, Phys. Rev. D 37 (1988) 3406;K. Hagiwara, et al., Phys. Rev. D 66 (2002) 010001.

[17] L. Amendola, Phys. Rev. D 62 (2000) 043511;G. Mangano, G. Miele, V. Pettorino, Mod. Phys. Lett. A(2003) 831;S. del Campo, R. Herrera, D. Pavón, Phys. Rev. D 70 (20043540;G. Farrar, P.J.E. Peebles, Astrophys. J. 604 (2004) 1;Z.-K. Guo, R.-G. Cai, Y.-Z. Zhang, JCAP 05 (2005) 002;Z.-K. Guo, Y.-Z. Zhang, Phys. Rev. D 71 (2005) 023501;R.-G. Cai, A. Wang, JCAP 0503 (2005) 002;B. Gumjudpai, T. Naskar, M. Sami, S. Tsujikawa, JCAP(2005) 007;R. Curbelo, T. Gonzáles, I. Quirós, astro-ph/0502141.

Page 5: Holographic dark energy and cosmic coincidence

210 D. Pavón, Z. Zimdahl / Physics Letters B 628 (2005) 206–210

21

ev.

5)

1.

0)

[18] W. Zimdahl, D. Pavón, L.P. Chimento, Phys. Lett. B 5(2001) 133.

[19] L.P. Chimento, A.S. Jakubi, D. Pavón, W. Zimdahl, Phys. RD 67 (2003) 083513.

[20] G. Olivares, F. Atrio, D. Pavón, Phys. Rev. D 71 (200063523.

[21] M. Szydłowski, astro-ph/0502034.[22] S. Tsujikawa, M. Sami, Phys. Lett. B 603 (2004) 113.[23] B. Wang, Y. Gong, E. Abdalla, Phys. Lett. B 624 (2005) 14[24] D. Bak, S.-J. Rey, Class. Quantum Grav. 17 (2000) L83;

E.E. Flanagan, D. Marolf, R.M. Wald, Phys. Rev. D 62 (200084035.