dark energy perturbations and cosmic coincidence

Dark energy perturbations and cosmic coincidence

Javier Grande,* Ana Pelinson,† and Joan Sola‡

High Energy Physics Group, Departament d’ECM, and Institut de Ciencies del Cosmos, Universitat de Barcelona,Avinguda Diagonal 647, E-08028 Barcelona, Catalonia, Spain(Received 6 December 2008; published 17 February 2009)

While there is plentiful evidence in all fronts of experimental cosmology for the existence of a

nonvanishing dark energy (DE) density D in the Universe, we are still far away from having a

fundamental understanding of its ultimate nature and of its current value, not even of the puzzling fact

that D is so close to the matter energy density M at the present time (i.e. the so-called ‘‘cosmic

coincidence’’ problem). The resolution of some of these cosmic conundrums suggests that the DE must

have some (mild) dynamical behavior at the present time. In this paper, we examine some general

properties of the simultaneous set of matter and DE perturbations ðM; DÞ for a multicomponent DE

fluid. Next we put these properties to the test within the context of a nontrivial model of dynamical DE

(the XCDM model) which has been previously studied in the literature. By requiring that the coupled

system of perturbation equations for M and D has a smooth solution throughout the entire

cosmological evolution, that the matter power spectrum is consistent with the data on structure formation,

and that the ‘‘coincidence ratio’’ r ¼ D=M stays bounded and not unnaturally high, we are able to

determine a well-defined region of the parameter space where the model can solve the cosmic coincidence

problem in full compatibility with all known cosmological data.

DOI: 10.1103/PhysRevD.79.043006 PACS numbers: 95.36.+x, 04.62.+v, 11.10.Hi

I. INTRODUCTION

Undoubtedly the most prominent accomplishment ofmodern cosmology to date has been to provide strongindirect support for the existence of both dark matter(DM) and dark energy (DE) from independent data setsderived from the observation of distant supernovae [1], theanisotropies of the cosmic microwave background (CMB)[2], the lensing effects on the propagation of light throughweak gravitational fields [3], and the inventory of cosmicmatter from the large-scale structures (LSS) of theUniverse [4,5]. But, in spite of these outstanding achieve-ments, modern cosmology still fails to understand theultimate physical nature of the components that build upthe mysterious dark side of the Universe, most conspicu-ously the DE component of which the first significantexperimental evidence was reported 10 years ago fromsupernovae observations. The current estimates of the DEenergy density yield

expD ’ ð2:4 103 eVÞ4 and it is

believed that it constitutes roughly 70% of the total energydensity budget for an essentially flat Universe. The bigquestion now is: what is it from the point of view offundamental physics? One possibility is that it is theground state energy density associated to the quantum fieldtheory (QFT) vacuum and, in this case, it is traditional toassociate D to ¼ =8G, where is the cosmologi-cal constant (CC) term in Einstein’s equations. The prob-lem, however, is that the typical value of the(renormalized) vacuum energy in all known realistic

QFT’s is much bigger than the experimental value. Forexample, the energy density associated to the Higgs poten-tial of the standard model (SM) of electroweak interactionsis more than 50 orders of magnitude larger than the mea-sured value of D.Another generic proposal (with many ramifications) is

the possibility that the DE stands for the current value ofthe energy density of some slowly evolving, homogeneous,and isotropic scalar field (or a collection of them). Scalarfields appeared first as dynamical adjustment mechanismsfor the CC [6,7] and later gave rise to the notion ofquintessence [8]. While this idea has its own merits (espe-cially concerning the dynamical character that confers tothe DE) it has also its own drawbacks. The most obviousone (often completely ignored) is that the vacuum energyof the SM is still there and, therefore, the quintessence fieldjust adds up more trouble to the whole fine-tuning CCproblem [9,10]!Next-to-leading is the ‘‘cosmological coincidence prob-

lem,’’ or the problem of understanding why the presentlymeasured value of the DE is so close to the matter density.One expects that this problem can be alleviated by assum-ing that D is actually a dynamical quantity. While quin-tessence is the traditionally explored option, in this paperwe entertain the possibility that such dynamics could be theresult of the so-called cosmological ‘‘constants’’ (like; G; . . . ) being actually variable. It has been proven in[11] that this possibility can perfectly mimic quintessence.It means that we stay with the parameter and make it‘‘running,’’ for example, through quantum effects [12–14].1 However, in [17] it was shown that, in order to have

*[email protected]†[email protected]‡[email protected] 1For a general discussion, see [15,16].

PHYSICAL REVIEW D 79, 043006 (2009)

1550-7998=2009=79(4)=043006(27) 043006-1 2009 The American Physical Society

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an impact on the coincidence problem, the total DE in thiscontext should be conceived as a composite fluid made outof a running and another entity X, with some effectiveequation of state (EOS) parameter !X, such that the totalDE density and pressure read D ¼ þ X and pD ¼ þ!XX, respectively. We call this system theXCDM model [17]. Let us emphasize that X (called‘‘the cosmon’’) is not necessarily a fundamental entity; inparticular, it need not be an elementary scalar field. Asremarked in [17], X could represent the effective behaviorof higher order terms in the effective action (includingnonlocal ones). This is conceivable, since the Bianchiidentity enforces a relation between all dynamical compo-nents that enter the effective structure of the energy-momentum tensor in Einstein’s equations, in particular,between the evolving and other terms that could emergeafter we embed general relativity in a more general frame-work [18,19]. Therefore, at this level, we do not impose amicroscopic description for X and in this way the treatmentbecomes more general.2 The only condition defining X isthe DE conservation law, namely, we assume that D ¼ þ X is the covariantly self-conserved total DE density.

In this paper, we analyze the combined dynamics of DEand matter density perturbations for such conserved DEdensity D. The present study goes beyond the approxi-mate treatment presented in [23], where we neglected theDE perturbations and estimated the matter perturbations ofthe XCDM model using an effective (variable) EOS we

for the composite fluid ð; XÞ. The main result was that asizeable portion of parameter space was still compatiblewith a possible solution of the cosmic coincidence prob-lem. The ‘‘effective approach’’ that we employed in [23]was based on three essential ingredients: (i) the use of theeffective EOS representation of cosmologies with variablecosmological parameters [11]; (ii) the calculation of thegrowth of matter density fluctuations using the effectiveEOS of the DE [24]; and (iii) the application of the, so-called, ‘‘F test’’ to compare the model with the LSS data,i.e. the condition that the linear bias parameter, b2ðzÞ ¼PGG=PMM, does not deviate from the CDM value bymore than 10% at z ¼ 0, where PMM / ðM=MÞ2 isthe matter power spectrum and PGG is the galaxy fluctua-tion power spectrum [4,5]—see [23,25] for details. Thisthree-step methodology turned out to be an efficientstreamlined strategy to further constrain the region of theoriginal parameter space [17]. However, a full-fledgedanalysis of the system of cosmological perturbations inwhich the DE and matter fluctuations are coupled in adynamical way remained to be performed. This kind ofanalysis is presented here.

The structure of the paper is as follows. In the nextsection, we outline the meaning of the cosmic coincidenceproblem within the general setting of the cosmologicalconstant problem. In Sec. III, the basic equations forcosmological perturbations of a multicomponent fluid inthe linear regime are introduced. In Sec. IV, we describethe general framework for addressing cosmological pertur-bations of a composite DE fluid with an effective EOS. InSec. V, we describe some generic features of the cosmo-logical perturbations for the dark energy component. Theparticular setup of the XCDM model is focused inSec. VI. In Secs. VII and VIII, we put the XCDM modelto the stringent test of cosmological perturbations andshow that the corresponding region of parameter spacebecomes further reduced. Most important, in this regionthe model is compatible with all known observational dataand, therefore, the XCDM proposal can be finally pre-sented as a robust candidate model for solving the cosmiccoincidence problem. In Sec. IX, we offer a deeper insightinto the correlation of matter and DE perturbations. In thelast section, we present the final discussion and deliver ourconclusions.

II. THE COINCIDENCE PROBLEM AS A PART OFTHE BIG CC PROBLEM

The cosmic coincidence problem is a riddle, wrapped inthe polyhedric mystery of the cosmological constant prob-lem [9,10], which has many faces. Indeed, we shouldclearly distinguish between the two main aspects whichare hidden in the CC problem. In the first place, we havethe ‘‘old CC problem’’ (the ugliest face of the CC conun-drum), i.e. the formidable task of trying to explain therelatively small (for particle physics standards) measuredvalue of or, more generally, of the DE density [1],roughly exp

D 1047 GeV4, after the many phase transi-

tions that our Universe has undergone since the very earlytimes, in particular, the electroweak Higgs phase transitionassociated to the standard model of particle physics, whosenatural value is in the ballpark of EW G2

F 109 GeV4

(GF being Fermi’s constant). The discrepancy EW=expD ,

which amounts to some 56 orders of magnitude, is thebiggest enigma of fundamental physics ever.3 Apart fromthe induced CC contribution from phase transitions, wehave the pure vacuum-to-vacuum quantum effects. Sincethe (renormalized) zero point energy of a free particle ofmass m contributes m4 to the vacuum energy density[12,13], it turns out that even a free electron contributes anamount more than 30 orders of magnitude larger than theaforementioned experimental value of D. Only light neu-trinos m 103 eV, or scalar particles of similar mass,could contribute just the right amount, namely, if these

2See e.g. [20–22] for recent constraints on CDM, XCDM,and quintessencelike models. The margin for the energy den-sities and EOS parameter !X is still quite high. In the XCDMcase, the fact that is running provides an even wider range ofphenomenological possibilities.

3See e.g. [13,15,16] for a summarized presentation with spe-cific insights closely related to the present approach.

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particles would be the sole active degrees of freedom in ourpresent cold Universe (see [12]).

On the other hand, the cosmic coincidence problem [26]is that second (‘‘minor’’) aspect of the CC problem ad-dressing the specific question: ‘‘why just now?’’ i.e. whydo we find ourselves in an epoch t ¼ t0 where the DEdensity is similar to the matter density, Dðt0Þ ’ Mðt0Þ? Inview of the rapidly decreasing value of MðaÞ 1=a3,where a ¼ aðtÞ is the scale factor, it is quite puzzling toobserve that its current value is precisely of the same orderof magnitude as the vacuum energy or, in general, the darkenergy density D. It is convenient to define the ‘‘cosmiccoincidence ratio’’

rðaÞ ¼ DðaÞMðaÞ ¼

DðaÞMðaÞ ; (1)

where ðDðaÞ;MðaÞÞ are the corresponding densitiesnormalized with respect to the current critical density 0

c 3H2

0=8G. For0M ’ 0:3 and0

D ’ 0:7, we have r0 ’ 2:3,which is of Oð1Þ. However, in the standard cosmologicalCDM model, where D is constant and Mða ! 1Þ !0, the ratio r grows unboundedly with the expansion of theUniverse. So the fact that r0 ¼ Oð1Þ is regarded as a puzzlebecause it suggests that t ¼ t0 is a very special epoch of ourUniverse. One could also consider the inverse ratio r1 ¼MðaÞ=DðaÞ, which goes to zero with the expansion. Thecoincidence problem can be equivalently formulated eitherby asking why r is not very large now or why r1 is notvery small. Solving the coincidence problem would be tofind either (1) a concrete explanation for r and r1 being oforder one at present within the standard cosmologicalmodel, or (2) a modified cosmological model (compatiblewith all known cosmological data) insuring that theseratios do not undergo a substantial change, say by morethan 1 order of magnitude or so, for a very long period ofthe cosmic history that includes our time.

In a very simplified way, let us summarize some of thepossible avenues that have been entertained to cope withthe coincidence puzzle:

(i) Quintessence and the like [8,27–30]. One postulatesthe existence of a set of cosmological scalar fieldsi

essentially unrelated to the rest of the particle phys-ics world. The DE produced by these fields has aneffective EOS parameter!D * 1which causes D

to decrease always with the expansion (i.e.dD=da < 0), but at a pace slower (on average)than that of the background matter. Thus, it finallycatches up with it and D emerges to the surface, i.e.the condition D > M eventually holds (presum-ably near our time). In this framework, there is thepossibility of self-adjusting and tracker solutions[8,27], where the DE keeps track of the matterbehavior and ultimately dominates the Universe. Itrequires to take some special forms of the potential,and in some cases the Lagrangian involves nonca-

nonical kinetic terms. For example, in the simplecase of a single scalar field and the exponentialpotential VðÞ expð=MPÞ one finds that thecoincidence ratio becomes fixed at the value

r ¼ 3ð1þ!mÞ2 3ð1þ!mÞ

; (2)

where!m is the EOS of the background matter (i.e. 0or 1=3, depending on whether cold or relativisticmatter dominates, respectively). So, at the presenttime, r ¼ 3=ð2 3Þ, and by the appropriate choiceof one can match the current experimental value.But, of course, the choice of the potential was ratherpeculiar and the field itself is completely ad hoc.

Moreover, it has a mass m ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiV00ðÞp H

1033 eV (as it follows from a self-consistent solu-tion of Einstein’s equations); such mass scale is 30orders of magnitude below the mass scale associated

to the DE, which is 1=4D 103 eV. In this sense, it

looks a bit unnatural to aim at solving the CC prob-lem by introducing a field whose extremely tinymass creates another cosmological puzzle.On the other hand, within the context of interactivequintessence models [28] (whose main leitmotif isprecisely trying to cure the coincidence problem),the coupling ofi and the matter components makesan allowance for energy exchange between the twokinds of fields, and as a result the ratio (1) can beconstant or slowly variable, whereas in other imple-mentations one can achieve an oscillatory trackingbehavior of r, although the construction is essentiallyad hoc [29]. Another generalization leads tok-essence models [30] (characterized by noncanon-ical kinetic terms), where fine-tuning problems in thetracking can be disposed of but the dominant back-ground component can be tracked only up to matter-radiation equality and is lost immediately afterwards(as the DE is immediately prompted into a CC-likebehavior). In one way or another, however, all vari-ants of quintessence suffer from several drawbacks,and, in particular, from the following generic one:they assume (somehow implicitly) that the remain-ing fields of the particle physics spectrum (i.e. thosewhich were already there from the very beginning)have nothing to do with the CC problem. As a resultof such a bold assumption, the (likely real) vacuumproblem of the conventional fields in QFT is merelytraded for the (likely fictitious) vacuum problem ofquintessence, which is no less acute because no realexplanation is provided for the smallness of thecurrent D value versus m4 (where m is any typicalmass scale in particle physics). Hence we are back tothe same kind of CC problem we started with.

(ii) Phantom energy [31]. It is motivated by the fact that,observationally speaking, the effective EOS of the

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DE cannot be excluded to satisfy!D & 1 near ourpresent time. As indicated above, many quintes-sencelike models in reality are hybrid constructionscontaining a mixture of fields with a phantom com-ponent. The reason is that one wants to give allow-ance for a ‘‘CC-crossing’’ !D ¼ 1 near our time.While phantom energy shares with quintessence theuse of scalar fields i, here the DE produced bythese fields is always increasing with the expansion,dD=da > 0, even after the relation D > M isfulfilled. The consequence of this ever growingbehavior of the dark component is that one endsup with a superaccelerated expansion of theUniverse that triggers an eventual disruption of allforms of matter (the so-called ‘‘big rip’’). Whencomputing the fraction of the lifetime of theUniverse where the ratio (1) stays within givenbounds before the ‘‘doomsday,’’ one finds that itcan be sizeable.

(iii) Nonlocal theories. There is some renewed interestin these kinds of theories in which the emphasis isplaced on the existence of possible nonlocal struc-tures in the effective action [32]. It has recentlybeen emphasized in [16] that the dynamical evolu-tion of the vacuum energy should come from aresummation of terms in the effective action lead-ing to nonlocal contributions of the formRF ðG0RÞ, for some unknown function F of di-mension 2, where G0 is the massless Green’s func-tion (G0 1=h). The canonical possibility wouldbe F ¼ M2G0R, where M is a parameter withdimension of mass. This situation leads to an ef-fective evolution of the CC of the form M2H2 during the matter-dominated epoch, whereasin the radiation era the effective CC would approxi-mately be zero (because R T

’ 0 for relativisticmatter, see (12) and (13) below). As a result, thecoincidence puzzle could somehow be understoodfrom the fact that the CC may start to be prepon-derant at some point once the onset of the matter-dominated epoch is left behind.

(iv) Of course many other ideas have been explored. Forinstance, one may introduce special fluids with verypeculiar EOS, such as the Chaplygin gas [33],which behaves as pressureless matter at early times(!D ’ 0) and as vacuum energy at present (!D ’1). Although there is some connection with bra-neworld cosmology, this proposal suffers from thesame problem as quintessence in that it supersedesthe vacuum state of traditional fields by the newvacuum of that peculiar fluid. Finally, let us men-tion the anthropic models, which fall in a quitedifferent category, in the sense that one does notlook for a solution of the coincidence problemexclusively from first principles of QFT or string

theory, but rather through the interplay of the ‘‘hu-man factor.’’ Basically, one ties the value of theratio (1) to the time when the conditions arise forthe development of intelligent life in the Universe,in particular, of cosmologists making observationsof the cosmos. This variant has also a long story, butwe shall refrain from entering the details, see e.g.[9,34].

III. COSMOLOGICAL PERTURBATIONS FOR AMULTICOMPONENT FLUID IN THE LINEAR

REGIME

In the remainder of the paper we concentrate on studyingsome general properties of the cosmological perturbations,both of matter and DE, and the implications they may haveon the coincidence problem within models characterizedby having running vacuum energy and other DE compo-nents. According to cosmological perturbation theory, allenergy density components, including the dark energy,should fluctuate and contribute to the growth of thelarge-scale cosmological structures. In this section, wediscuss the general framework of linear density perturba-tions in models composed of a multicomponent DE fluidbesides the canonical matter.In the following we use the standard metric perturbation

approach [35] and consider simultaneous density and pres-sure perturbations

N ! N þ N; pN ! pN þ pN; (3)

for all the components (N ¼ 1; 2; . . . ) of the fluid, includ-ing matter and all possible contributions from the multi-component DE part. At the same time, we have metric andvelocity perturbations for each component:

gB ! g ¼ gB þ g; UN ! U

N þ U

N; (4)

where gB is the background metric. The four-vector ve-

locity UN in the comoving coordinates has the following

components and perturbations ðU0N;U

iNÞ ¼ ð1; 0Þ !

UN ¼ ð0; vi

NÞ, where viN is the three-velocity of the Nth

component of the fluid in the chosen coordinate system. Asa background space-time, we assume the homogeneousand isotropic Friedmann-Lemaıtre-Robertson-Walker(FLRW) metric with flat space section, hence gB ¼diagf1;a2ðtÞijg, where a is the scale factor.

In order to derive the set of perturbed equations, let usfirst introduce Einstein’s equations:

R 12Rg ¼ 8GT; (5)

where G is the Newton constant and T is the total

energy-momentum tensor of matter and dark energy.Both the background and perturbed metric are assumedto satisfy these equations. The total energy-momentumtensor of the system is assumed to be the sum of the perfectfluid form for each component:


043006-4

T ¼ XN

TN ¼ X

N

½pNg þ ðN þ pNÞUNU

N : (6)

The components of T are the following:

T00 ¼ X

N

N T; (7)

Tij ¼ X

N

pNij pT

ij; (8)

where T and pT are the total energy density and pressure,respectively.

Perturbations on the metric and on the energy-momentum tensor are uniquely defined for a given per-turbed space-time, provided we make a gauge choice. Thelatter means that we choose a specific coordinate system; inthis way, four out of the 10 components g h of the

metric perturbation can be fixed at will. Here we haveadopted the synchronous gauge, widely used in the litera-ture, in which the four preassigned values of the metricperturbations are h0i ¼ 0 and h00 ¼ 0. Setting h ¼a2, in this gauge the perturbed, spatially flat,

FLRW metric takes on the form

ds2 ¼ gdxdx ¼ dt2 a2ðtÞðij þ ijÞdxidxj

¼ a2ðÞ½d2 ðij þ ijÞdxidxj; (9)

where in the last equality we have expressed the result alsoin terms of the conformal time , defined through d ¼dt=a. We may compare (9) with the most general pertur-bation of the spatially flat FLRW metric,

ds2 ¼ a2ðÞ½ð1þ 2c Þd2 !iddxi

ðð1 2Þij þ ijÞdxidxj; (10)

consisting of the 10 degrees of freedom associated to thetwo scalar functions c , , the three components of thevector function !iði ¼ 1; 2; 3Þ, and the five components ofthe traceless ij. Clearly, the synchronous gauge (9) is

obtained by setting c ¼ 0, !i ¼ 0 and absorbing thefunction into the trace of ij. In this way, ij in (9)

contributes six degrees of freedom. As we will see in amoment, in practice only the nonvanishing trace of themetric disturbance will be necessary to perform the analy-sis of cosmic perturbations in this gauge. To within firstorder of perturbation theory, such a trace is given by

h gh ¼ gijhij ¼ hiia2

¼ ii; (11)

where gij is the inverse of gij ¼ a2ðtÞðij þ ijÞ, and it isunderstood that the repeated Latin indices are summedover 1, 2, 3.

For the physical interpretation, notice that the synchro-nous gauge is associated to a coordinate system in whichthe cosmic time coordinate is comoving with the fluidparticles (g00 ¼ 1, i.e. c ¼ 0), which is the reason for its

name and also explains why this gauge does not have anobvious Newtonian limit. In fact, this gauge choice isgenerally appropriate for the study of fluctuations whosewavelength is larger than the Hubble radius ( dH H1). Actually, any mode satisfies this condition at suffi-ciently early epochs, and in this regime the effects of thespace-time curvature are unavoidable.Next we wish to compute the perturbations in the syn-

chronous gauge. To start with, it is convenient to rewriteEinstein’s equations (5) as follows:

R ¼ 8GS; S T 12gT; (12)

where T ¼ T is the trace of (6), hence

T ¼ XN

ðN 3pNÞ: (13)

For the calculation of the perturbations, we can use any ofthe components of Einstein’s equations. However, since weare going to use the conservation law rT

¼ 0 to derive

additional fluctuation equations, it is convenient to perturbthe (00)-component of (12) because the other componentsare well known not to be independent of the conservationlaw. Thus, using (6), (12), and (13) we obtain

S00 ¼ T00 1

2T ¼ 1

2

XN

ðN þ 3pNÞ ) S00

¼ 1

2

XN

ðN þ 3pNÞ: (14)

On the other hand, a straightforward calculation shows thatthe perturbation of the (00)-component of the Ricci tensorcan be written as follows:

R00 ¼ 1

2

@2h

@t2H

@h

@t¼ 1

2hþHh; (15)

where we have used (11) and defined the ‘‘hat variable’’

h @

@t

hiia2

¼ h

: (16)

The overhead circle ( ) indicates partial differentiation

with respect to the cosmic time (i.e. f @f=@t, for any f),

in order to distinguish it from other differentiations to be

used later on. Therefore, H ¼ a=a is the ordinary expan-

sion rate in the cosmic time t.Since the fluctuations S00 and R00 from (14) and (15)

are constrained by (12), we find

hþ 2Hh ¼ 8G

XN

ðN þ 3pNÞ: (17)

If we substitute (16) in the previous expression, a second-order differential equation in the original variable (11)ensues. In terms of the conformal time , it can be writtenas follows:


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€hþH _h ¼ 8Ga2XN

ðN þ 3pNÞ; (18)

where the dot (_) indicates differentiation with respect to

(i.e. _f df=d) and H _a=a ¼ Ha is the expansionrate in the conformal time.

The Friedmann equation can be written in terms of thenormalized densities as

H2 ¼ 8G

3T ¼ H2

0

XN

N; (19)

where H0 is the present value of the Hubble parameter andN N=

0c are the normalized densities with respect to

the current critical density 0c 3H2

0=8G.

The subsequent step is to perform perturbations on theconservation law rT

¼ 0, as this will provide the

additional independent equations. Using (6), the previouslaw reads explicitly as follows:X

N

frðN þ pNÞUNU

N þ ðN þ pNÞ

½UNrU

N þU

NrU

Ng ¼ g

XN

rpN: (20)

For any four-velocity vector, we have UNU

N ¼ 1 and,

therefore, we have the orthogonality relation UNrU

N ¼

0. In this way, by contracting Eq. (20) with UN we find the

simpler resultXN

½UNrN þ ðN þ pNÞrU

N ¼ 0: (21)

Let us emphasize that the sumP

N in this equation need notrun necessarily over all the terms of the cosmic fluid. Itmay hold for particular subsets of fluid components that areoverall self-conserved. In particular, it could even hold foreach component, if they would be individually conserved.In our case, it applies to the specific matter component andalso, collectively, to the multicomponent DE part. It isstraightforward to check that, in the FLRWmetric, we have

rUN ¼ 3H ðN ¼ 1; 2; . . .Þ: (22)

Using this relation, it is immediate to see that, in thecomoving frame, Eq. (21) boils down to

XN

½N þ 3HðN þ pNÞ ¼ 0: (23)

Moreover, perturbing (22) in the synchronous gauge, we

find (using 0 ¼ h=2 for the perturbed Christoffel

symbol involved in the covariant derivative) the usefulresult

H ¼ 1

3ðrU

N Þ ¼

1

3

N h

2

; (24)

where we have introduced the notation N rðUN Þ ¼

riðUiNÞ (with U0

N ¼ 0) for the covariant derivative of

the perturbed three-velocity UiN ¼ vi

N . Equipped withthese formulas, the perturbed equation (23) immediatelyleads to

XN

N þðN þpNÞ

N h

2

þ 3HðN þpNÞ

¼ 0:

(25)

The previous result could have equivalently been obtainedby setting ¼ 0 in (20) and perturbing the correspondingequation. An independent relation can be obtained bysetting ¼ i in (20) within the comoving frame and carry-ing out the perturbation. Using the relevant Christoffelsymbol i

0j ¼ Hij and Eq. (22) we obtain, after some

calculations,

XN

fðN þ pNÞU i

N þ ½N þ pN þ 5HðN þ pNÞUi

Ng

¼ XN

ripN: (26)

The final step is obtained by computing the divergenceri on both sides of this equation. To within first orderof perturbation theory, we obtain riripN ¼gikrirkpN ¼ ð1=a2Þr2pN for the action of ri onthe right-hand side (r.h.s.) of (26), whereas on its left-hand side (l.h.s.) we can use the variable N previously

defined. Moreover, we have riU i

N ¼ N owing to the

commutativity of coordinate differentiation and perturba-tion operations. In this way, the final outcome reads

XN

fðN þ pNÞN þ ½N þ p

N þ 5HðN þ pNÞNg

¼ k2

a2XN

pN; (27)

where it is furthermore understood that we have used theFourier decomposition for all the perturbation variables:

fðt;xÞ ¼Z d3k

ð2Þ3 fðt; kÞeikx; (28)

where f ¼ ðh; N; pN; NÞ. In Fourier space, the per-turbation variables are denoted with the same notation, butthey are the Fourier transforms of the original ones, so theirarguments are t and k because the space variable x has beentraded for the wave number k jkj. The latter will bemeasured in units of h Mpc1, where h ’ 0:7 is the re-duced Hubble constant—not to be confused with the traceof the synchronous perturbed metric, Eq. (11). In theseunits, the linear regime corresponds to length scales ‘k1 with wave numbers k < 0:2h Mpc1, i.e. ‘ >5h1 Mpc. Notice that, if desired, one can easily rewritethe above perturbation equations (25) and (27) in confor-

mal time simply by using f ¼ _f=a for any f.


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We have obtained three basic sets of perturbation equa-tions (17), (25), and (27) for the four kinds of perturbation

variables ðhðt; kÞ; Nðt; kÞ; pNðt; kÞ; Nðt; kÞÞ. It is thusclear that the evolution of the cosmic perturbations canbe completely specified only after we assume some rela-tion of the pressure perturbation pN and the densityperturbation N for each fluid. If the perturbations areadiabatic, then that relation is simply

pN ¼ c2a;NN; (29)

where c2a;N is the adiabatic speed of sound for each fluid,

defined as

c2a;N _pN

_N

; (30)

where the dot differentiation here is with respect to what-ever definition of time. Notice that if the various compo-nents would have an EOS of the form pN ¼ wNN , withconstant EOS parameter wN , then c2a;N ¼ wN . However,

even in this case the mixture has a variable effective EOSparameter, as we will see in the next section.

On the other hand, if the perturbations are nonadiabatic,there is an entropy contribution to the pressure perturbation[36]:

pNN ¼ pN c2a;NN; (31)

where N ðpNÞnonadiabatic=pN is the intrinsic entropyperturbation of the Nth component, representing the dis-placement between hypersurfaces of uniform pressure anduniform energy density [37]. For covariantly conservedcomponents, a gauge-invariant relationship between pN

and N for a general nonadiabatic stress is given by [37–40]:

pN ¼ c2s;NN þ 3Ha2ðN þ pNÞðc2s;N c2a;NÞNk2

;

(32)

where c2s;N can be regarded as a rest frame speed of sound.

We will refer to c2s;N as the effective speed of sound, in the

sense that we treat the cosmic fluid effectively as hydro-dynamical matter. Since (32) is gauge invariant, the per-turbed quantities in this expression can be computed, inparticular, within the synchronous gauge. In this way, wecan consistently substitute (32) in the Eqs. (17), (25), and(27) to eliminate the perturbation pN . This allows us,finally, to solve for the three basic sets of perturbations

ðhðt; kÞ; Nðt; kÞ; Nðt; kÞÞ: (33)

IV. PERTURBATIONS FOR A COMPOSITE DEFLUID WITH AVARIABLE EFFECTIVE

EQUATION OF STATE

In this section, we apply the linear matter and darkenergy density perturbations to a general class of models

in which the DE fluid is a composite and covariantly self-conserved medium and matter is also canonically con-served. From Eq. (23), in the matter-dominated epoch(pM ¼ 0), the matter component M satisfies

0MðaÞ þ

3

aMðaÞ ¼ 0: (34)

Here we found it convenient to trade the differentiationwith respect to the cosmic time ( ) for the differentiationwith respect to the scale factor. The latter is denoted by a

prime (i.e. f0 df=da for any f; hence f ¼ aHf0). The

scale factor is related to the cosmological redshift z byaðzÞ 1=ð1þ zÞ, where we define að0Þ a0 ¼ 1 at thepresent time. It follows that the normalized matter densityevolves as

MðaÞ ¼ 0Ma

3; (35)

where 0M is the corresponding current value. Since the

total DE is also globally conserved, from Eq. (23) we alsoobtain

0D þ 3

að1þ weÞD ¼ 0; (36)

where we is the effective EOS parameter and D ¼ 1 þ2 þ . . . is the total density of the multicomponent DEfluid. For a composite DE model, in which the DE is amixture of fluids with individual EOS pi ¼ !ii (i ¼1; 2; . . . ; n), the effective EOS parameter is defined as

we ¼ pD

D

¼ !11 þ!22 þ . . .

1 þ 2 þ . . .: (37)

The Hubble expansion in terms of the normalized den-sities, in the matter-dominated period, follows from (19)

H2 ¼ 8G

3T ¼ H2

0ð0Ma

3 þDðaÞÞ; (38)

where D is the normalized total DE density DðaÞ DðaÞ=0

c.The nonadiabatic perturbed pressure (32) for the total

DE component can be written in terms of the effective EOSas4

pD ¼ c2sD þ 3Ha2ð1þ weÞDðc2s c2aÞDk2

; (39)

where we have omitted for simplicity the subindex ‘D’from the adiabatic and effective speeds of sound of the DE(i.e. c2a c2a;D, c

2s c2s;D; this convention will be used

throughout the text). The units are taken to be the lightspeed c ¼ 1 and @ ¼ 1 such that the Planck scale is defined

by MP ¼ G1=2 ¼ 1:22 1019 GeV. In these units, usu-

4In the particular case where the DE is a purely running, onehas to consider the perturbations , but since ! ¼ 1 itturns out that D (here ) plays no role and hence it is enoughto consider the relation p ¼ , see Ref. [41] for details.


043006-7

ally 0 c2s 1 for a general DE model. In this range, onecan show that for constant EOS parameter there is smallsuppression on the DE fluctuation D as c2s increases[37]. Near-zero (but not vanishing) sound speed today ispossible in models like k-essence, for example, in whichthe EOS parameter is positive until the matter-dominatedtriggers a change to negative pressure; in this kind ofmodel, it is even possible to have c2s > 1, a regime forwhich the growth of the DE density perturbations is sup-pressed [42].

In a nonperfect fluid, spatial inhomogeneities in T

imply shear viscosity in the fluid. In this case, a possiblecontribution to shear through a ‘‘viscosity parameter’’ c2visshould also be taken into account [43]. In principle, c2s is anarbitrary parameter. Nevertheless, the limit whereðc2s ; c2visÞ ! ð1; 0Þ corresponds exactly to a scalar field

component with canonical kinetic term [44].Using the total DE conservation law (36), we can write

the total DE adiabatic sound speed (30) as

c2a p0D

0D

¼ we a

3

w0e

ð1þ weÞ : (40)

The perturbed equations (25) and (27) for the (con-served) matter component (for which pM ¼ pM ¼ 0)can be written as differential equations in the scale factor:

0M ¼ 1

aH

M h

2

; (41)

0M ¼ 2

aM; (42)

where M M=M is the relative matter fluctuation(density contrast). According to Eq. (42), the matter veloc-ity gradient is decaying (M / a2). Assuming the con-ventional initial condition 0M Mða ¼ 1Þ ¼ 0, we haveMðaÞ ¼ 0ð8 aÞ. So, the perturbed matter set of coupledequations (41) and (42) yields the simple relation

h ¼ 2aH0M: (43)

Let us also define the relative fluctuation of the DEcomponent, D D=D. Using the nonadiabatic per-turbed pressure (39) and the DE conservation law (36), wecan write the perturbed equations (25) and (27) for the self-conserved DE fluid in the following way:

0D ¼ ð1þ weÞ

aH

1þ 9a2H2ðc2s c2aÞ

k2

D h

2

3

aðc2s weÞD; (44)

0D ¼ 1

að2 3c2sÞD þ k2

a3H

c2sD

ð1þ weÞ ; (45)

where in the last equation we have used (40) to eliminatec2a. Moreover, from (45) one can see that a negligible DE

sound speed (c2s 0) causes the velocity gradient to decay(D / a2), as in the case of matter [Eq. (42)]. If weassume the conventional initial conditions 0M ¼ 0D ¼ 0,we have M ¼ D ¼ 0ð8 aÞ. In this case, the total DEfluid is comoving with the matter as long as the Universeand perturbations evolve, which is a very particular case.Actually, for this case, the k (scale) dependence disappearsfrom the equations. On the other hand, from Eq. (45) onecan see that, if we neglect the DE perturbations, D 0,for a constant c2s we obtain again M ¼ D ¼ 0 and thescale independence.However, D modifies the evolution of the metric

fluctuations according to the perturbed Einstein equation(17); and, in turn, this causes the corresponding evolutionof the matter perturbations through Eq. (43). We can writedown the appropriate form of the perturbation equation asfollows. First, we define the ‘‘instantaneous’’ normalized

densities at a cosmic time t, ~N ¼ N=c, where c ¼3H2=8G is the critical density at the same instant of time.(Notice that the new densities equal the previously defined

ones, i.e. ~N ¼ N , only at t ¼ t0.) Making use of thedefinition of the relative DM fluctuation M and of thenonadiabatic perturbed pressure (39), we may cast Eq. (17)in the following way:

h0 þ 2

ah 3H

a~MM

¼ 3H

a~D

ð1þ 3c2sÞD þ 9a2Hðc2s c2aÞ D

k2

: (46)

Next we use Eq. (43) to eliminate h from (46). With thehelp of

H0ðaÞ ¼ 4G

aHðaÞ ½MðaÞ þ ð1þ weðaÞÞDðaÞ (47)

¼ 3HðaÞ2a

1þ weðaÞrðaÞ

1þ rðaÞ

(48)

and

~MðaÞ ¼ 1

1þ rðaÞ ;~DðaÞ ¼ rðaÞ

1þ rðaÞ ; (49)

where rðaÞ is the cosmic coincidence ratio (1) between theDE and matter densities, we may finally rewrite (46) asfollows:

00MðaÞ þ

3

2

1 weðaÞrðaÞ

1þ rðaÞ0MðaÞa

3

2

1

1þ rðaÞMðaÞa2

¼ 3

2HðaÞ rðaÞ

1þ rðaÞð1þ 3c2sÞDðaÞ

a2þ 9ðc2s c2aÞDðaÞ

k2

:

(50)

If we would neglect the DE fluctuations ðD ¼ 0; D ¼ 0Þ,the r.h.s. of the previous equation would vanish. Underthese conditions, one would be left with a decoupled,


043006-8

second-order, homogeneous differential equation that de-termines the matter perturbations M. As could be ex-pected, the resulting equation coincides with Eq. (2.16)of Ref. [23], where the approximation of neglecting the DEperturbations was made right from the start as an inter-mediate procedure to investigate the amount of lineargrowth of the matter perturbations and put constraints onthe parameter space of theXCDMmodel. This procedurewas called ‘‘effective’’ in that reference, since all theinformation about the DE is exclusively encoded in thenontrivial EOS function we ¼ weðaÞ. Let us write thehomogeneous equation as follows:

00MðaÞ þ

3

2ð1 we

~DÞ0MðaÞa

3

2ð1 ~DÞMðaÞ

a2¼ 0;

(51)

and let us assume a time interval not very large such that~D and we remain approximately constant. Looking for a

power-law solution of (51) in the limit ~D 1, we find,for the growing mode,

M a13ð1weÞ ~D=5 a

1 3ð1 weÞ

5~D lna

: (52)

Since we < 0 for any conceivable form of DE, this equa-tion shows very clearly that we should expect a growthsuppression of matter perturbations (i.e. M an, with

n < 1) whenever a (positive) DE density ~D is presentwithin the horizon. Physically speaking, we associate thiseffect to the existence of negative pressure that producescosmological repulsion of matter.

However, being the DE density nonconstant in general(D 0), the DE perturbations themselves (and not onlythe value of the background DE density) should act as asource for the matter fluctuations. This effect is preciselyencoded in the inhomogeneous part of Eq. (50), i.e. in itsr.h.s which is, in general, nonzero for D, D 0. In orderto better appreciate this effect, let us consider anothersimplified situation where the analytical treatment is stillpossible, namely, let us assume an adiabatic regime (c2s ¼c2a) with roughly constant EOS (we ’ const.) at very largescales (for which k in Eq. (45) is very small, and hence theD component becomes negligible). Under these condi-tions, Eq. (44) greatly simplifies as follows:

0D ¼ ð1þ weÞ

aH

h

2¼ ð1þ weÞ0

M; (53)

where in the second step we have used (43). The rates ofchange of the perturbations for matter and DE, therefore,become proportional in this simplified setup. To be moreprecise, we see from (53) that for we * 1 (quintessence-like behavior of the composite DE fluid) the matter fluc-tuations that are growing with the expansion (0

M > 0)trigger fluctuations in the DE also growing with the ex-pansion (0

D > 0), whereas for we & 1 (phantomlikebehavior of the DE) we meet exactly the opposite situation,

i.e., increasing fluctuations in the matter density (0M > 0)

lead to decreasing fluctuations in the DE (0D < 0). Note

that upon trivial integration of (53) one finds D ¼ ð1þweÞM þ C, whereC is a constant determined by the initialconditions. For C ¼ 0 one obtains a result that fits with thewell-known adiabatic initial condition relating the densitycontrasts of generic matter and DE components [35],

M

1þ wM

¼ D

1þ wD

; (54)

where, for nonrelativistic matter, we havewM ¼ 0, and!D

is, in this case, the effective EOS we of the composite DEfluid. Since a positive DE density always leads to cosmo-logical repulsion, it follows from (53) that one shouldexpect some inhibition (resp. enhancement) of the mattergrowth for the quintessencelike (resp. phantomlike) case.Although the previous example illustrates the impact of

the DE fluctuations on the matter growth for a simplesituation, a more complete treatment is required in thegeneral case. In practice, this means that we have tonumerically solve the system (43)–(46) or, if desired, re-place the last equation with the second-order inhomoge-neous equation (50) whose r.h.s. depends on the densitycontrast and the velocity gradient for the DE, D, D 0.Notice that the presence of overdensity DE perturbations(D > 0) does not necessarily imply the inhibition of thecorresponding matter perturbations since the coefficient1þ 3c2s in front of D on the r.h.s. of Eq. (50) is positivefor nonadiabatic DE perturbations. Only for c2s ¼ c2a wemeet the aforementioned possibility because c2a ’ we isusually negative, unless we is rapidly decreasing with theexpansion, see Eq. (40). In this sense, the discussion above,based on Eq. (53), is only valid at very large scales,specifically for k-modes whose length scale ‘ k1 isoutside the sound horizon (cf. Sec. V).5 However, atsmaller scales, especially at scales inside the sound hori-zon, and for a general nonadiabatic regime, we need tosolve the aforesaid complete system of equations for thebasic set of perturbation variables for the metric, matter,and DE:

ðhða; kÞ; Mða; kÞ; Dða; kÞ; Dða; kÞÞ: (55)

In this way, we have extended the effective treatment of theDE perturbations presented in Ref. [23], and we are nowready to better assess the scope of its applicability. InSec. VII, we will apply this general formalism to theXCDM model.

5In Sec. IX, we will see that this particular situation doesaccommodate very well the realistic dynamics of the cosmicperturbations for matter and DE during the early stages of theevolution.


043006-9

V. SOME GENERIC FEATURES OF THE DEPERTURBATIONS

In the present section, we summarize some character-istic features of the DE perturbations. Many propertieswhich are, in principle, common to any model with aself-conserved DE, will be later exemplified in Sec. IXwithin the nontrivial context of the so-called XCDMmodel [17,23].

A. Divergence at the CC boundary

In general, the EOS of the DE will be a dynamicalquantity, we ¼ weðaÞ. In many models, the EOS maychange from quintessencelike ( 1<we <1=3) tophantom (we <1) behavior or vice versa, acquiringtherefore the value we ¼ 1 (also referred to as the ‘‘CCboundary’’) at some instant of time. This is problematicsince, as we shall see next, the equations for the perturba-tions diverge at that point.

The divergence at the CC boundary is common to anyDE model and has been thoroughly studied in the literature(see e.g. [45,46]). The problem can be readily seen bydirect inspection of Eqs. (44) and (45). Note that, eventhough c2a diverges at the crossing [cf. (40)], the combina-tion ð1þ weÞc2a remains finite and, therefore, Eq. (44) iswell behaved. Thus, the problem lies exclusively in the(1þ we) factor in the denominator of (45). One mightthink that the divergence can be absorbed through a re-definition of the variables, but this is not the case.

Getting around this difficulty is not always possible. It iswell known that there is no way for a single scalar fieldmodel to cross the CC boundary [45]. The simplest way toavoid this problem is to assume two fields ðQ;PÞ, e.g. onethat works as quintessence wQ >1 and dominates the

DE density until the CC-crossing point, and beyond it theother field retakes the evolution with a phantom behaviorwP <1, or the other way around; see [46,47] for a de-tailed discussion and specific parametrizations of we. Aswe will see, in the XCDM model the additional restric-tions needed to avoid this divergence will further constrainthe physical region of the parameter space.

B. Unbounded growth for adiabatic DE perturbations

Another well-known problem is the unbounded growthof the DE perturbations for a negative squared speed ofsound c2s . As already mentioned in the previous section, inthe adiabatic case we have c2a ’ we, which is in generalnegative as long as the EOS parameter is not varying toofast. As a result, the adiabatic DE perturbations may lead toexplosive growth unless extra degrees of freedom areassumed (see e.g. [44] for discussion).

In order to better see the origin of the problem, let usrewrite Eqs. (44) and (45) in terms of conformal time ,

which is easily done by making use of _f ¼ a2Hf0 (forany f):

_D ¼ að1þ weÞ1þ 9H 2ðc2s c2aÞ

k2

D h

2

3H ðc2s weÞD; (56)

_ D ¼ H ð2 3c2sÞD þ k2

a

c2sD

ð1þ weÞ : (57)

As in Sec. III, we have definedH ¼ _a=a Ha. If we usethe two equations above and Eq. (46) to get a second-orderdifferential equation for D, we arrive at

€D ¼ k2c2sD þOðh; M; D; DÞ; (58)

where the second term on the r.h.s. represents other termslinear in these variables. Assuming that the various pertur-bations are initially more or less of the same order, we seethat the first term on the r.h.s. of (58) will dominateprovided

kZ

csd

1: (59)

Notice that, for constant sound velocity, this conditionsimply tells us that the wavelength of the modes satisfies‘ k1 s, where s ¼ cs is the ‘‘sound horizon.’’Equation (59) is a generalization of this condition forarbitrary sound speed, in which case the sound horizon isgiven by

s ¼Z

0csd ¼

Z a

0

csd~a

~a2Hð~aÞ (60)

and constitutes a characteristic scale for the DE perturba-tions. As we will see next, the DE is expected to be smoothfor scales well below it [43,48].For scales well inside the sound horizon, (58) becomes

the equation of a simple harmonic oscillator, whose solu-tion is (in what follows, we assume constant c2s for sim-plicity):

D ¼ C1eicsk þ C2e

icsk; (61)

where C1 and C2 are constants. We see that, for c2s < 0 (i.e.imaginary cs) and neglecting the decaying mode, the DEperturbations grow exponentially. Obviously, this situationis unacceptable for structure formation.6 On the other hand,if c2s > 0 the DE density contrast oscillates. Since M

grows typically as the scale factor a, this ensures that theratio D=M D=a ! 0 with the expansion. In other

6Let us note that this kind of problem need not occur when theDE is a pure running , for then the perturbation is nolonger an independent dynamical variable and is, in fact, deter-mined by an algebraic function of the other perturbation varia-bles, see [41] for details.


043006-10

words, this tells us that the DE is going to be a smoothcomponent (as it is usually assumed) as long as we are wellinside the sound horizon. This feature is treated in moredepth in the following section.

C. Smoothness of DE below the sound horizon

As a matter of fact, Eq. (58) is an oversimplification. Inaddition to having a term proportional to D, we also haveone depending on its first derivative. So we can write thatequation more precisely as follows:

€D ¼ D1D þD2_D þOðh; M; DÞ; (62)

which gives us not just a simple, but a damped harmonicoscillator. The coefficients D1 and D2 are, in general,functions of the conformal time . So the DE densitycontrast does not only oscillate, but its amplitude decreaseswith time. Indeed, it was shown in [43] that the quantity

ðrestÞD ¼ D þ 3Hað1þ weÞD

k2; (63)

which corresponds to the density contrast in the DE restframe, oscillates with an amplitude A that decreases ac-cording to

A / c1=2s að1þ3weÞ=2: (64)

The damped oscillations of the DE density contrast areclearly seen in the XCDM model, as we will show inSec. IX.

Finally, we may ask ourselves whether the scales rele-vant to the LSS surveys [4] are well inside the soundhorizon or not. Note that, in a matter-dominated universewith negligible CC term and constant cs, we have H2 ¼H2

00Ma

3 and the sound horizon (60) takes the simple

form

s ¼ 2cs

H0ð0MÞ1=2

ffiffiffia

p: (65)

Thus, in general, we expect that the size of the soundhorizon at present (a0 ¼ 1) should be roughly of the orderof the Hubble length H1

0 . On the other hand, the obser-

vational data concerning the linear regime of the matterpower spectrum lie in the range 0:01h Mpc1 < k <0:2h Mpc1 [4], which corresponds to length scales ‘k1 comprised in the interval ð600H0Þ1 < ‘< ð30H0Þ1,hence well below the sound horizon (at least for c2s not tooclose to 0). Therefore, according to the previous discus-sions, we expect the DE density to be smooth at thosescales, and indeed it will be so for the XCDM model.Nevertheless, as we will see in Sec. IX, the larger the scale‘ or the smaller the speed of sound cs, the more importantthe DE perturbations are, because then (59) is not such agood approximation.

VI. THE XCDM MODEL AS A CANDIDATE TOSOLVE THE COSMIC COINCIDENCE PROBLEM

TheXCDMmodel [17] provides an interesting way ofexplaining the so-called cosmological coincidence prob-lem (cf. Sec. II). The idea is related to the possibility ofhaving a dynamical component X, called the ‘‘cosmon,’’7

which interacts with a running cosmological constant. Ifthe matter components are canonically conserved, thecomposite DE ‘‘fluid’’ made out of X and the running will be a self-conserved medium too. The dynamics of theXCDM universe is such that its composite DE mayenforce the existence of a stopping point after manyHubble times of cosmological expansion. As a result, thismodified FLRW-like universe can remain for a long whilein a situation where the coincidence ratio (1) does notchange substantially from the time when the DE becamesignificant until the remote time in the future when thestopping point is attained. Subsequently, the Universe re-verses its motion till the big crunch.The total DE density and pressure for the XCDM

universe are obtained from the sum of the respective CCand X components:

D ¼ þ X; pD ¼ p þ pX: (66)

The evolving CC density ðtÞ ¼ ðtÞ=8G of themodel is motivated by the quantum field theoryformulation in curved space-time by which the CC is asolution of a renormalization group equation. Following[13,15,16,19,49], the CC density emerges in general as aquadratic function of the expansion rate:

ðHÞ ¼ 0 þ 3

8M2

PðH2 H20Þ; (67)

where 0 ¼ ðH ¼ H0Þ is the present value. The dimen-

sionless parameter is given by

12

M2

M2P

; (68)

where M is an effective mass parameter representing theaverage mass of the heavy particles of the grand unifiedtheory (GUT) near the Planck scale, after taking intoaccount their multiplicities. Depending on whether theyare bosons or fermions, ¼ þ1 or ¼ 1, respectively.For example, for M ¼ MP one has jj ¼ 0, where

0 1

12’ 2:6 102: (69)

On physical grounds, we expect that this value of jjshould be the upper bound for this parameter. In the next

7Originally, the cosmon appeared in Ref. [7] as a scalar fieldlinked to the mechanism of dynamical adjustment of the CC. Inthe present context, the entity X is also differentiated from theCC, but if taken together they form a composite and interactiveDE medium.


043006-11

section, we will see if we can pinpoint a region of parame-ter space compatible with this expectation.

The energy density associated to the cosmon componentX is obtained from the total DE conservation law (36) andthe composite form (66),

0XðaÞ þ 0

ðaÞ ¼ 3

að1þ wXÞXðaÞ; (70)

where wX is the effective EOS parameter of X,

pX wXX: (71)

In principle, wX could be a function of the scale factor.However, a simpler assumption that allows us to perform acompletely analytic treatment is to consider that the Xcomponent behaves as a barotropic fluid with a constantEOS parameter in one of the following two ranges: !X *1 (quintessencelike cosmon) or !X & 1 (phantomlikecosmon). On the other hand, the EOS parameter for therunning component still remains as the cosmologicalconstant one, w ¼ 1, i.e.

p : (72)

From these assumptions, it is easy to find the followingrelation between the effective EOS parameter of the totalDE fluid (37) and the EOS parameter of the cosmon wX:

ð1þ weðaÞÞDðaÞ ¼ ð1þ wXÞXðaÞ: (73)

The normalized density of the cosmon component,XðaÞ ¼ XðaÞ=0

c, can be obtained from the previousrelations after solving the differential equation (70). Inthis equation, we have 0

ðaÞ ¼ ð3=8ÞM2PdH

2=da from

(67), and the derivative dH2=da ¼ 2HðaÞH0ðaÞ can beexplicitly computed from (47) upon using (35) and (73).One finally obtains the differential equation

0XðaÞ þ

3

að1þ!X ÞXðaÞ ¼ 30

Ma4: (74)

With the boundary condition that the current value of X is0X, the solution of (74) can be written in the following

way8:

XðaÞ ¼ 0X½ð1þ bÞa3ð1þwXÞ ba3; (75)

where in the above equations we have used the notations

b 0M

ð!X Þ0X

; (76)

ð1þ wXÞ: (77)

As we will discuss in more detail below, the parameter must remain small (jj< 0:1) in order to be compatiblewith primordial nucleosynthesis. For ¼ 0 the CC density(67) becomes constant. In this case, the two parameters(76) and (77) vanish and Eq. (75) boils down to thesimplest possible form, which is characteristic of a self-conserved monocomponent system,

XðaÞ ¼ 0Xa

3ð1þwXÞ: (78)

It is only in this particular situation where the cosmon Xcould be a self-conserved scalar field with its own dynam-ics. But in general this is not so because in QFT in curvedspace-time we have good reasons to expect a running

[13,15,16], and hence 0. Therefore, if the total DE isto be conserved, the dynamics of X is not free anymore andbecomes determined as in (75).The normalized total DE density D ¼ D=

0c is given

by

DðaÞ ¼10

1 wX

0M

wX

a3ð1þwXÞ þ0

1

þ 0Ma

3

wX ; (79)

where the various current normalized densities satisfy therelation 0

M þ0D ¼ 0

M þ0 þ0

X ¼ 1, which may

be called the ‘‘XCDM cosmic sum rule.’’ Using (73),the effective EOS of the DE in the XCDM model cannow be obtained explicitly,

weðaÞ ¼ 1þ ð1þ!XÞXðaÞDðaÞ ; (80)

with XðaÞ and DðaÞ given by (75) and (79),respectively.The total DE density (79) varies in such a way that the

ratio (1) can remain under control, which is the clue forsolving the coincidence problem in a dynamical way [17].Indeed, the explicit computation of such a ratio yields

rðaÞ ¼

10

0Mð1 Þ

wX

wX

a3ðwXÞ þ ð0

Þa3ð1 Þ0

M

þ

wX ; (81)

and it can be bounded due to the existence of a maximum

(triggered by the a3ðwXÞ term in the previous for-mula). Moreover, rðaÞ given above stays relatively constant(typically not varying more than 1 order of magnitude) fora large fraction of the history of the Universe and for asignificant region of the parameter space [17]. In contrast,in the standard concordanceCDMmodel, the CC densityremains constant, ¼ 0

, and the coincidence ratio

grows unstoppably with the cubic power of the scale factor,rðaÞ ¼ 0

a3=0

M. In this scenario, it is difficult to explain

why the constant ¼ 0 is of the same order of magni-

8The fact that the evolution of the cosmon X is completelydetermined by the dynamics of the running (67), togetherwith the hypothesis of total DE conservation (70), implies that Xcannot be generally assimilated to a scalar field, which has itsown dynamics. In fact, as we have already mentioned, X is to beviewed in general as an effective entity within the context of theeffective action of QFT in curved space-time.


043006-12

tude as the matter density right now: 0M. Let us point out

that the standard model ratio is just that particular case of(81) for which ¼ 0 (no running CC) and 0

X ¼ 0 (nocosmon).

Before closing this section, we would like to make aremark and some discussion concerning the quadratic evo-lution law (67) for the cosmological term. This equationwas originally motivated within the framework of therenormalization group (RG) of QFT in curved space-time[12–14,18,49] (see also [15] for a short review). We pointout a criticism against this approach that recently appearedin the literature [50]. While the nature of this criticism wasamply rejected in [16] (see below for a summary), it is fairto say that the question of whether a rigorous RG approachin cosmology is feasible is still an open question andremains a part of the CC problem itself. Although it isnot our main aim here to focus on this fundamental issue,let us briefly sketch the situation along the lines ofRef. [16], to which we also refer the reader for a summaryof the rich literature proposing different RG formulationsof cosmology both in QFT in curved space-time and inquantum gravity.

The RG method in cosmology treats the vacuum energydensity as a running parameter and aims at finding afundamental differential relation (renormalization groupequation) of the form

d

d ln¼ ðP;Þ; (82)

which is supposed to describe the leading quantum con-tributions to it, where is a function of the parameters Pof the effective action (EA) and is a dimensional scale.The appearance of this arbitrary mass scale is characteristicof the renormalization procedure in QFT owing to theintrinsic breaking of scale invariance by quantum effects.The quantity in (82) is a (-dependent) renormalizedpart of the complete QFT structure of the vacuum energy.Depending on the renormalization scheme, the scale canhave a more or less transparent physical meaning, but thephysics should be completely independent of it. Such(overall) independence of the observable quantities isactually the main message of the RG; but, remarkablyenough, the dependence of the individual parts is alsothe clue of the RG technique to uncover the leading quan-tum effects.

Essential for the RG method in cosmology is to under-stand that, in order for the vacuum energy to acquiredynamical properties, we need a nontrivial external metricbackground. The dynamical properties of this curved back-ground (e.g. the expanding FLRW space-time, character-ized by the expansion rate H) are expected to induce afunctional dependence ¼ ðHÞ. The latter shouldfollow from parametrizing the quantum effects with thehelp of the scale and then using some appropriatecorrespondence of with a physical quantity, typically

with H in the cosmological context, although there areother possibilities [12–14]. In this way, one expects toestimate the subset of quantum effects reflecting the dy-namical properties of the nontrivial background. Although cancels in the full EA, the RG method enables one toseparate the relevant class of quantum effects responsiblefor the running. The procedure is similar to the RG in ascattering process in QCD; the parametrization of thequantum effects in terms of is the crucial strategy tofinally link them with the energy of the process through thecorrespondence ! q (where q is a typical momentum ofthe scattering process) at high energy. One can also pro-ceed in the same way in QED and electroweak theory(although here one can adopt more physical subtractionschemes, if desired). The RG technique can actually beextended to the whole particle physics domain. In cosmol-ogy, however, the situation is more complicated, partlybecause (as remarked above) the physical scale behindthe quantum effects is not obvious. Still, one expects thatit should be related with the expanding metric backgroundand hence the expansion rate H can be regarded as areasonable possibility. On this basis, the heuristic argu-ments exhibited in [13,14,16,18,19,49] combined with thegeneral covariance of the EA suggest that the solution ofthe RG equation (82) should lead to the kind of quadraticlaw (67) that we have used.According to [16], the point of view of Ref. [50] is

incorrect on two main accounts: first, because they try todisprove the running through the overall cancellation of thearbitrary scale in the EA; and second, because theyneglect the essential role played by the nontrivial metricbackground. As emphasized in [16], the cancellation of in the EA cannot be argued as a valid criticism because thisfact is a built-in feature of the RG and it was neverquestioned. If this would be a real criticism, it wouldalso apply to QED, QCD, or any other renormalizableQFT, and nevertheless this is no obstacle for using theRG method in these theories as an extremely useful strat-egy to extract the dependence of the quantum effects on thephysical energy scale of the processes, in particular, the so-called running coupling constants gs ¼ gsðqÞ and e ¼ eðqÞof the strong and electromagnetic interactions. Moreover,in the absence of a nontrivial metric background, there isno physical running of the vacuum energy, even thoughthere is still dependence of the various parts of the EAand, in particular, of the CC, see e.g. [51]. Therefore, at theend of the day such criticism seems to go against theessence of the RG method and its recognized ability toencapsulate the leading quantum effects on the physicalobservables.In cosmology, the principles of the RG should be the

same, but there are two main stumbling blocks that preventfrom straightforwardly extending the method in practice[16], to wit: (i) the aforesaid lack of an obvious/uniquecorrespondence of with a cosmological scale defining


043006-13

the physical running, and also (no less important) (ii) thehuge technical problems related with the application of theRG within a physical (momentum-dependent) renormal-ization scheme in a curved background. These difficultiesare unavoidable here because we are dealing with QFT inthe infrared regime and moreover the metric expansionscannot be performed on a flat background; indeed, therecannot be a flat background in the presence of a cosmo-logical term. While these two problems remain unresolvedin a completely satisfactory manner, it is legitimate to usethe phenomenological approach and the educated guess(e.g. the general covariance of the EA) to hint at therunning law. This is the guiding principle followed in theaforementioned references and that led to Eq. (67).

Finally, let us emphasize that irrespective of whethersuch a law can be substantiated within the strict frameworkof the RG, the present study remains perfectly usefulsimply treating (67) as an acceptable type of a phenome-nological variation law and keeping also in mind thatadding the cosmon may contribute to the resolution ofthe pressing cosmic coincidence problem.

VII. DARK ENERGY PERTURBATIONS IN THEXCDM MODEL

In this section, we further elaborate on the conditions tobound the ratio r ¼ rðaÞ and discuss the constraints on theparameter space, in particular, the impact of the DE per-turbations on these constraints. In Sec. V, we discussedanalytically some generic features about DE perturbations.In principle, those results should apply to any model inwhich the DE is self-conserved. The XCDM model,given its peculiarities (a composite DE which results in acomplicated evolution of the effective EOS), constitutes anontrivial example of that kind of model. In this sense, it isinteresting to use the XCDM to put our general predic-tions to the test. At the same time, this will allow us toimpose new constraints on the parameter space of themodel, improving its predictivity.

The parameter space of theXCDMmodel was alreadytightly constrained in [23]. In that work, the matter densityfluctuations were analyzed under the assumption that theDE perturbations could be neglected. As a first approxi-mation this is reasonable since, as we have discussed inSec. V, the DE is expected to be smooth at scales wellbelow the sound horizon. Thus, we will take the results of[23] as our starting point and will check numerically thegoodness of that approximation. Finally, we will furtherconstrain the parameter space using the full approachpresented in the present work. Let us summarize the con-straints that were imposed in [23]:

(1) Nucleosynthesis bounds: As already commented,the ratio (81) between DE and matter densitiesshould remain relatively small at the nucleosynthe-sis time, in order not to spoil the big bang modelpredictions on light-element abundances. Requiring

that ratio to be less than 10%, it roughly translatesinto the condition jj< 0:1, where the parameter was defined in Eq. (77).

(2) Solution of the coincidence problem: In Ref. [17],where the XCDM model was originally intro-duced as a possible solution to the coincidenceproblem, it was shown that there is a large subvo-lume of the total XCDM parameter space forwhich the ratio rðaÞ remains bounded and near thecurrent value r0 (say, jrðaÞj & 10r0, where r0 7=3) during a large fraction of the history of theUniverse. Thus, the fact that the matter and DEdensities are comparable right now may no longerbe seen as a coincidence. Such a solution of thecoincidence problem is related to the existence of afuture stopping (and subsequent reversal) of theUniverse expansion within the relevant region ofthe parameter space.

(3) Current value of the EOS parameter: Recent studies(see e.g. [2]) suggest that the value of the DEeffective EOS should not be very far from 1 atpresent. Although these results usually rely on theassumption of a constant EOS parameter (and thusare not directly applicable to the XCDM model),we adopted a conservative point of view and stuck tothem by enforcing the condition j1þ weða ¼ 1Þj 0:3 on the EOS function (80).

(4) Consistency with LSS data: As said before, in [23]we studied the growth of matter density fluctuationsunder the assumption that the DE was smooth on thescales relevant to the linear part of the matter powerspectrum. From the fact that the standard CDMmodel provides a good fit to the observational data,we took it as a reference and imposed that theamount of growth (specifically, the matter powerspectrum), of our model did not deviate by morethan 10% from the CDM value (‘‘F-test’’ condi-tion). This condition can also be justified from theobserved galaxy fluctuation power spectrum, see[23] for more details.

The upshot of that analysis was that there is still a bigsubvolume of the three-dimensional XCDM parameterspace ð;!X;

0Þ satisfying simultaneously the above

conditions.9 The projections of that volume onto the threeperpendicular planes ð;0

Þ, ð;!XÞ, and ð0; !XÞ are

displayed in Figs. 1 and 2 (shaded regions). These regionswhere already determined in Ref. [23]. In the next section,we will discuss how the final set of allowed points becomesfurther reduced when we take into account the analysis ofthe DE perturbations.

9In Ref. [23], we took a prior for the normalized matter densityat present, specifically0

M ¼ 0:3. This means that0D ¼ 0:7 for

a spatially flat universe. For better comparison with those results,we keep this prior also in the present work.


043006-14

A. Divergent behavior at the CC boundary

As discussed in Sec. V, if the effective EOS of the modelcrosses the CC boundary (we ¼ 1) at some point in thepast, the perturbation equations will present a real diver-gence. Obviously, this circumstance makes the numericalanalysis unfeasible at the points of parameter space af-fected by the singularity.

In the absence of an apparent mechanism to get aroundthis singularity, we are forced to restrict our parameterspace to the subregion where the solution of the perturba-tion equations (43)–(46) is regular, namely, by removingthose points of the parameter space that present such acrossing in the past, because these points cannot belong toa well-defined history of the Universe. In the absence of amore detailed definition of the cosmon entity X, this newconstraint is unavoidable. This should not be considered asa drawback of the model, for even in the case when oneuses a collection of elementary scalar fields to represent theDE, one generally meets the same kind of divergent be-havior as soon as the CC boundary is crossed, unless somespecial conditions are arranged. In other words, even if thecomponents of the DE are as simple as, say, elementaryscalar fields with smooth behavior and well-defined dy-namical properties (including an appropriately chosen po-tential), there is no a priori guarantee that the CC boundarycan be crossed safely [45]. It is possible to concoct ingen-ious recipes, see e.g. [46], such that the perturbation equa-tions become regular at the CC boundary, but the procedureis artificial in that one must introduce new fields (one

quintessencelike and another phantomlike) satisfying spe-cial properties such that their respective EOS behaviorsmatch up continuously at the CC crossing. Apart from thefact that fields with negative kinetic terms are not verywelcome in QFT, one cannot just replace the original fieldswith the new ones without at the same time changing theoriginal DE model. As we will see below, in the XCDMcase the absence of CC crossing projects out a region of theparameter space which is significantly more reduced, andtherefore the predictive power of the model becomes sub-stantially enhanced.In Sec. 6 of the first reference in [17], it was shown that

the necessary and sufficient condition for having a CC-boundary crossing in the past within theXCDMmodel isthat the parameter b given in (76) is positive. As can bereadily seen, this will happen whenever and 0

X havethe same sign (where we use the fact that !X < 0 and

-0.4 -0.2 0 0.2 0.4ν

-2

-1

0

1

2Ω

Λ0

FIG. 1. Projection of the 3D physical volume of the XCDMmodel onto the 0

plane. Shaded area: points that satisfy all

the constraints in [23], see also Sec. VII of the present work.Striped area: points that are not affected by the divergence at theCC boundary discussed in Sec. VII A. The final allowed region isthe one both shaded and striped. As a result of considering theDE perturbations, the possible values of the parameters becomestrongly restricted, which implies a substantial improvement inthe predictive power of the model.

-0.4 -0.2 0 0.2 0.4ν

-2

-1.5

-1

-0.5

ωX

(a)

-2 -1 0 1 2ΩΛ

0

-2

-1.5

-1

-0.5

ωX

(b)

FIG. 2. Projection of the 3D physical volume of the XCDMmodel onto the !X and 0

!X planes. Shaded: points

that satisfy all the constraints in [23]. Striped: points that, inaddition, are not affected by the divergence at the CC boundarydiscussed in Sec. VII A. The final allowed region is the one bothshaded and striped.


043006-15

jj j!Xj in the relevant region of parameter space).From the cosmic sum rule of the XCDM model, wehave 0

X ¼ 10M 0

; thus, using our prior 0M ¼

0:3, the set of allowed points (for which 0X has different

sign from ) are those comprised in the striped areas inFigs. 1 and 2. This leaves us with a very small allowedregion in each plane, which is just the correspondingintersection of the shaded area and the striped one.

At the end of the day, it turns out that most of the pointsin the shaded area in Figs. 1 and 2 (viz. those allowed bythe conditions stated in the previous section and the analy-sis of [23]) are ruled out by the new constraint emergingfrom the DE perturbations analysis, and hence we end upwith a rather definite prediction for the values of theXCDM parameters. In particular, we find from thesefigures that only small positive values of are allowed,at most of order 102. Let us emphasize that this is invery good agreement with the theoretical expectationsmentioned in Sec. VI. Recall that, from the point of viewof the physical interpretation of in Eq. (68), we expected in the ballpark of 0 102 at most—see Eq. (69)—since the masses of the particles contributing to the runningof the CC should naturally lie below the Planck scale.10 Letus mention that the interesting bounds on obtained inRef. [52] on the basis of the so-called generalized secondlaw of gravitational thermodynamics would suggest thatonly the effective mass near the Planck mass is allowed.However, let us point out that such a study has beenperformed without including the nontrivial effect fromthe cosmon.

A very important consequence of the dark energy per-turbative constraint is that the effective EOS of the DE canbe quintessencelike only, i.e. 1<we <1=3. To provethis statement, let us start from Eq. (73). For the currentvalues of the parameters, this equation can be rewritten as

ð1þ w0eÞ0

D ¼ ð1þ!XÞ0X; (83)

where w0e weða ¼ 1Þ is the value of the effective

EOS parameter at the present time. Looking at Figs. 2(a)and 2(b), we realize the following two relevant features:first, the cosmon component is necessarily phantomlike(!X <1) in the allowed region by the DE perturbations;and, second, its energy density at present is negative;namely, 0

X ¼ 0:70 < 0 because from Fig. 2(b) we

have 0 > 0:7. Therefore, since the r.h.s. of (83) is con-

strained to be positive and0D ¼ 0:7> 0, we are enforced

to have w0e >1. However, the fulfillment of this condi-

tion at present implies its accomplishment in the past, i.e.

weðaÞ>1ð8 a < 1Þ, otherwise there would have been acrossing of the CC boundary at some earlier time, which isexcluded by the analysis of the DE perturbations. Theupshot is that the EOS of the DE in the XCDM modelcan only appear effectively as quintessence (q.e.d.). Inreality, it only mimics quintessence, of course, as its ulti-mate nature is not; such DE medium is a mixture made outof running vacuum energy and a compensating entity thatinsures full energy conservation of the compound system.Dark energy components X with negative energy density

are peculiar in cosmology since, in contrast to standard DEcomponents, they satisfy the strong energy condition (likeordinary matter), and as a result the gravitational behaviorof X is attractive rather than repulsive. Because of thisdouble resemblance with matter and phantom DE(although with the distinctive feature X < 0), such com-ponents can be called ‘‘phantom matter’’ [17]. Being X ingeneral an effective entity, such ‘‘phantom matter’’ behav-ior is actually nonfundamental.

B. Adiabatic speed of sound in the XCDM

In the Eqs. (43)–(46), we assumed the most general casein which the perturbations could be nonadiabatic.Moreover, we have shown that the adiabatic case usuallyleads to an unphysical exponential growth of the perturba-tions as a result of c2a in (40) being negative. Next we willcheck that, indeed, the most common situation in theXCDM model is to have c2a < 0. Notwithstanding, adia-batic perturbations are not completely forbidden in thepresent framework, as there is a small region of the pa-rameter space for which c2a could be positive.From Eqs. (40) and (73), and making use of the DE

conservation law (36), we find that the adiabatic speed ofsound for the XCDM model can be cast as follows:

c2aðaÞ ¼ 1 a

3

0XðaÞ

XðaÞ : (84)

With the help of (75), we can rewrite the last expression as

c2aðaÞ ¼ 0X

XðaÞ ð1þ bÞð!X Þa3ð1þ!XÞ: (85)

We want to find out the condition for this expression to bepositive. As ð!X Þ< 0 (remember that !X < 0 andjj< 0:1, due to nucleosynthesis constraints), that condi-tion simply reads

0X

XðaÞð1þ bÞ< 0: (86)

The cosmon energy density XðaÞ cannot vanish because,in such a case, the perturbation equations would diverge.Indeed, XðaÞ ¼ 0 corresponds to a CC-boundary cross-ing at some value of the scale factor in the past, cf. Eq. (80).Thus, being XðaÞ a continuous function, it must have thesame sign in the past as that of its present value, i.e.

10Let us clarify that the tighter bounds on determined inRef. [41] are possible only because, in the latter work, the DE isnot conserved and there is no cosmon. As we have shown in [23],a running cosmological constant model without a self-conservedDE cannot solve the coincidence problem in a natural waybecause the required values of are too large and, hence,incompatible with the physical interpretation of this parameter.


043006-16

0X=XðaÞ> 0ð8 a < 1ÞÞ. In short, the final condition

that ensures that c2a > 0 is

ð1þ bÞ< 0: (87)

If this condition would be satisfied, then c2a > 0would holdfor the entire past history of the Universe and, under thesecircumstances, the adiabatic equations may be used. Itturns out that the relation (87) can be satisfied in theXCDM model, although only in a narrow range of theparameter space. In fact, from the definition of the parame-ter b in (76), the expectation that j!Xj ¼ Oð1Þ, and ne-glecting , we see that (87) is approximately equivalent to

0X *

0M

!X

¼ OðÞ: (88)

Given the fact that was found to be positive and small(cf. Figs. 1 and 2) and, at the same time, 0

X < 0 (seeprevious section), the above condition does not leave muchfreedom within the allowed parameter space, roughly & 0

X < 0. This narrow strip is, however, not neces-sarily negligible; e.g. if we take of order of 0 102

[cf. Eq. (69)], this possibility is still permitted in theparameter space, see Figs. 1 and 2. In such a case, thepresent cosmon density could still be of the order or larger(in absolute value) than, say, the current neutrino contri-bution to the energy density of the Universe (0

103).No matter how tiny (in absolute value) a negative cosmoncontribution to the energy density is, it suffices to take careof the cosmic coincidence problem along the lines that wehave explained. Therefore, the adiabatic contribution isperfectly tenable, but the numerical analysis of the subse-quent sections remains essentially the same (as we havechecked) independently of whether the sound speed of theDE medium is adiabatic or not. For this reason, in whatfollows we will assume the more general situation of non-adiabatic perturbations, with the understanding that adia-batic ones can do a similar job in the corresponding regionof the parameter space.

VIII. THE MATTER POWER SPECTRUM

In this section, we compare the matter power spectrumpredicted by the XCDM model with the observed galaxypower spectrum measured by the 2dFGRS survey [4]. TheXCDM matter power spectrum is found by evolving theperturbation equations (43)–(46) from a ¼ ai to thepresent (a0 ¼ 1), where ai 1 is the scale factor atsome early time, but well after recombination. In theseequations, we must of course use the expansion rate (38)with the full DE density (79).

In order to set the initial conditions at a ¼ ai, we use theprediction from the standard CDM model. Indeed, thestandardCDMmodel provides a good analytical fit to the2dFGRS observed galaxy power spectrum. Taking this fitas our starting point, we compute analytically the values ofthe CDM perturbations at an arbitrary scale factor. Since

the DE does not play an important role until very recently,we may assume that the initial matter and metric perturba-tions at a ¼ ai for the XCDM model are the same as forthe CDM model.

A. Initial matter and metric perturbations

As previously commented, the perturbed equations areevolved from some ai 1 to a0 ¼ 1. The value of ai isunimportant, provided that it lies well after recombination(to insure that all the processes encoded in the transferfunction have already taken place). For definiteness, wetake ai ¼ 1=500, i.e. cosmological redshift z 500. Nextwe specify the initial conditions at a ¼ ai.For the matter and metric perturbations, the initial con-

ditions in the standard cosmological model can be com-puted analytically [35,36]. The matter perturbed equationsin the standard CDM model are (43) and (50) takingDðaÞ ¼ DðaÞ ¼ 0. In these conditions, the r.h.s. of thedifferential equation (50) vanishes and we are left with ahomogeneous equation. In the standard model, this equa-tion is just Eq. (51) with we ¼ 1, i.e.

00MðaÞ þ

3

2ð2 ~MðaÞÞ

0MðaÞa

3

2~MðaÞMðaÞ

a2¼ 0:

(89)

Since matter is conserved, we have ~MðaÞ ¼ðH0=HðaÞÞ20

M=a3. Moreover, from Eq. (48) (again with

we ¼ 1) one can also see that ~MðaÞ ¼ð2a=3ÞH0ðaÞ=HðaÞ. Hence, the homogeneous equa-tion (89) can be conveniently cast as follows:

00MðaÞ þ

3

aþH0ðaÞ

HðaÞ0MðaÞ

3

20

M

H20

H2ðaÞMðaÞa5

¼ 0:

(90)

This equation is now in a standard form and can be solvedanalytically [53]. Let us first introduce the variableDðaÞ ¼MðaÞ=ref (the so-called growth factor), where ref is thematter density contrast at some initial scale factor. In theinitial matter era (a ¼ ai 1) wherein the cosmologicalterm is negligible, we have H2ðaÞ=H2

0 ¼ 0Ma

3 in very

good approximation; then, by imposing the boundary con-ditions DðaÞ / a and D0ðaÞ ¼ 1, the growing solution of(90) is simply DðaÞ ¼ a, as can be easily checked. In thegeneral case, one can find the solution for the growingmode which reduces to the previous one deep into thematter-dominated era. The final result reads

DðaÞ ¼ MðaÞref

¼ 50M

2

HðaÞH0

Z a

0

d~a

ð~aHð~aÞ=H0Þ3: (91)

In particular, for HðaÞ=H0 ¼ ð0MÞ1=2a3=2 it just boils

down to the solution DðaÞ ¼ a corresponding to the earlymatter-dominated epoch, as expected.


043006-17

Furthermore, Eq. (43) leads to

hðaÞ ¼ 2aHðaÞD0ðaÞ

DðaÞ MðaÞ; (92)

and for DðaÞ ¼ a it renders the initial condition

hðaiÞ ¼ 2HðaiÞMðaiÞ (93)

for the metric fluctuation. Later on, when the DE (i.e. 0 >

0 in the CDM) starts to play a role, the matter (andmetric) fluctuations become suppressed. The suppressionis given by the value of the growth factorDðaÞ, which is nolonger proportional to the scale factor.11 From (91) it isclear that MðaÞ=DðaÞ is a constant, which can be writtenas MðaiÞ=ai at early times (when DðaiÞ ¼ ai) and asMða0Þ=Dða0Þ at the present time. Therefore,

MðaiÞ ¼ aiD0

Mða0Þ; (94)

hðaiÞ ¼ 2HðaiÞ aiD0

Mða0Þ; (95)

where D0 Dða0Þ and the subindex in HðaiÞ has beenadded to emphasize that the initial value of the Hubbleparameter is to be computed within the CDM model.

Note that, instead of setting the value of hðaiÞ, we couldhave chosen to put initial conditions on the derivative of thedensity contrast, 0

M. In that case, as it is evident from (94),we would have that 0

MðaiÞ ¼ Mða0Þ=D0. Then the initialvalue of the metric fluctuation is constrained by Eq. (43) to

be hðaiÞ ¼ 2HXðaiÞaiMða0Þ=D0, where now HXðaiÞ isthe XCDM value of the Hubble function. Note that this

value of hðaiÞ is not exactly the same as that in (95), sinceHðaiÞ and HXðaiÞ are not identical. However, being thatthe difference is rather small (as we have checked numeri-cally), the behavior of the perturbations does not dependsignificantly on that choice.

The Eqs. (94) and (95) give us the initial conditions atai ¼ 1=500 for the matter and metric perturbations interms of the density contrast today Mða0Þ. We associatethe latter with the 2dFGRS observed galaxy power spec-trum fitted in the CDM model, as detailed below.

The matter power spectrum of the CDM model can beapproximated as [41]:

PðkÞ jMðkÞj2 ¼ AkT2ðkÞ g2ð0

TÞg2ð0

MÞ; (96)

where 0T ¼ 0

M þ0. It assumes a scale-invariant

(Harrison-Zeldovich) primordial spectrum, as genericallypredicted by inflation. This primordial spectrum is modi-fied when taking into account the physical properties ofdifferent constituents of the Universe, in particular, theinteractions between them. All these effects are encodedinto the scale-dependent transfer function TðkÞ, whichdescribes the evolution of the perturbations through theepochs of horizon crossing and radiation/matter transition.The growth at late times which, in the CDM model, isindependent of the wave number, is described by thegrowth function gðÞ. Finally, A is a normalization factor.The transfer function can be accurately computed by

solving the coupled system formed by the Einstein and theBoltzmann equations. Although a variety of numerical fitshave been proposed in the literature, here we use the so-called BBKS transfer function [54]:

TðkÞ ¼ lnð1þ 2:34qÞ2:34q

½1þ 3:89qþ ð16:1qÞ2 þ ð5:46qÞ3

þ ð6:71qÞ41=4; (97)

where

q ¼ qðkÞ k

ðhÞ Mpc1(98)

and is the Sugiyama’s shape parameter [35,55]

0Mhe

0B

ffiffiffiffiffiffiffiffiffih=0:5

pð0

B=0MÞ: (99)

On the other hand, for the growth function we assume thefollowing approximation [56]:

gðÞ ¼ 5

2

4=7 0

þ1þ

2

1þ0

70

1;

(100)

which reflects the suppression in the growth of perturba-tions caused by a positive cosmological constant. Thenormalization coefficient A is related to the CMB anisot-ropies through [41]

A ¼ ð2lHÞ4 62

5

Q2rms-PST20

; (101)

where Qrms-PS is the quadrupole amplitude of the CMBanisotropy (see below for more detailed explanations),lH H1

0 ’ 3000h1 Mpc is the Hubble radius, and T0 ’2:725 K, the present CMB temperature. Therefore, thevalue of the normalization factor A could in principle beinferred from measurements of the CMB. However, wehave obtained it by fitting the power spectrum (96) to the2dFGRS observed galaxy power spectrum [4], as discussedbelow.We assume h ¼ 0:7 for the reduced Hubble parameter

and a spatially flat universe with 0M ¼ 0:3 (hence 0

¼0:7 for the flatCDMmodel) in order to be consistent withour assumption in previous analyses [17,23]. The fit to the

11Notice that if ~ is small and essentially constant, then thegrowth factor DðaÞ takes the approximate form DðaÞ an, withn ¼ 1 6 ~=5< 1, as it follows from (52) for we ¼ 1, orfrom (89). This demonstrates, if only roughly, the suppressionbehavior in an explicit analytic way. In the general CDM case,however, the solution for the growing mode is given by (91), inwhich H is the full expansion rate of the standard model.


043006-18

2dFGRS observed galaxy power spectrum P2dFðkÞ is ob-tained assuming the matter budget composed of a baryonicpart 0

B ¼ 0:04 and a dark matter contribution 0DM ¼

0:26. In order to calculate the best fit, we use the formula(96) to minimize the -square distribution

2 1

ndof

Xk

½P2dFðkÞ PðkÞ22ðkÞ (102)

in terms of the normalization A. There are 39 values of k inthe 2dFGRS data, so the number of degrees of freedom isndof ¼ 38. We find, as best fit, the value

A ¼ 8:99 105h4 Mpc4 (103)

with 2 ¼ 0:43. From (101) we see that this value of Aimplies Qrms-PS ’ 20:85 K. Let us now clarify thatQrms-PS is not the observed quadrupole CMB anisotropy(usually denoted Qrms), but rather the value derived from afit to the entire CMB power spectrum (PS). For a power-law spectrum with n ¼ 1 (i.e. for a scale-invariant PS), theCOBE team obtained Qrms-PS ¼ 18 1:6 K [57], andmoreover they found that the observed Qrms is smallerthan the fitted Qrms-PS. Whether this is a chance result ofcosmic variance or reflects the physical cosmology is notknown [57]. Our fitted value forQrms-PS falls within the 2range of the corresponding COBE value (although whenthe quadrupole itself is not used in the fit, the COBEuncertainties become larger [57]). As several authorshave noted [55,58], such a normalization may be inade-quate for models with a cosmological constant, given thefact that the CMB spectra of -dominated models is quitedifferent from a simple power law, especially at largescales (low multipoles). For instance, in [58] an alternativenormalization for theCDMmodel is proposed, which forh ¼ 0:8 and 0

¼ 0:7 yields Qrms-PS ¼ 22:04 K, withan error of the order of 11%, which is in agreement withour result. In general one can find a number of differentvalues forQrms-PS in the literature depending on the kind ofanalysis performed or the data set used, and this is why wepreferred to compute the normalization directly from a fitto the matter power spectrum. Finally, let us emphasize thatour value for Qrms-PS lies within the 95% confidence inter-val for the observed quadrupole anisotropy (Qrms) by bothCOBE [57] and WMAP [59].

Therefore, we will assume (94) and (95) as the initialconditions for the matter and metric perturbations, identi-fying Mða0Þ with MðkÞ from the formula (96) using thefitted coefficient (103).

B. The XCDM matter power spectrum

The procedure discussed above helps us to set initialconditions for the matter and metric perturbations in theXCDM model. However, since the CDM model doesnot include DE perturbations, we should set independentinitial conditions on DðaiÞ and DðaiÞ. As already dis-

cussed, the scales relevant to the matter power spectrumremain always well below the sound horizon (65) and weexpect negligible DE perturbations at any time. Thus, themost natural choice for the initial values of the DE pertur-bations is

DðaiÞ ¼ 0; DðaiÞ ¼ 0: (104)

Indeed, this is not the only reasonable choice. For in-stance, we could have also assumed the adiabatic initialcondition (54) for the DE density contrast, i.e. DðaiÞ ¼ð1þ weðaiÞÞMðaiÞ, with MðaiÞ given by Eq. (94). Again,it has been checked that the evolution of the perturbationsdoes not depend significantly on the particular initial con-dition used.Assuming the initial conditions (94), (95), and (104) for

the matter, metric, and DE perturbations at ai, respectively,we can solve the perturbed equations (43)–(46).Equivalently, we can solve (44) and (45) to obtainðD; DÞ and then (50) to get the matter density fluctuationstoday Mðk; a ¼ 1Þ for any dark energy model. In particu-lar, we can (as a consistency check) solve the perturbationequations for the CDM model [in that case DðaÞ ¼DðaÞ ¼ 0, so the only equations needed are (43) and(46)]. In doing so, we recover exactly the spectrum P

defined in (96).Now we proceed to compute the spectrum of the

XCDM, PXðkÞ. In order to better compare the shapeof the different spectra and the goodness of their fit to the2dFGRS observed galaxy power spectrum, we will nor-malize them at the smallest length scale considered ‘k1, i.e. at k ¼ 0:2, taking the CDM spectrum (96) asreference. To this purpose, we introduce a normalizationfactor AX in the matter power spectrum:

PXðkÞ AXjMðk; a ¼ 1Þj2: (105)

Notice that the normalization factor AX gives us thedifference in the matter power spectrum of the model withrespect to that of the CDM at the specific scale k ¼ 0:2.Let us clarify that the reason for choosing this scale for thenormalization is that, as discussed in Sec. V, the smaller thelength scale (i.e. the higher the value of k) the less impor-tant the DE perturbations are. Therefore, at k ¼ 0:2, thematter power spectrum of the model should not dependsignificantly on whether we consider the effect of the DEperturbations; in particular, it should be independent of thespeed of sound cs. For larger scales, however, the DEperturbations can be more significant and, as we shall seebelow, they may introduce some differences in the shape ofthe power spectrum, which are nevertheless small in thelinear regime.The XCDM power spectrum was calculated for two

fiducial values of the DE speed of sound, c2s ¼ 1 and c2s ¼0:1 and several combinations of the parameters , !X, and0

. For values of the parameters allowed in Figs. 1 and 2


043006-19

(shaded and striped region) we find that AX 1 (within10% of accuracy).

In Fig. 3(a), we put together the 2dFGRS observedgalaxy power spectrum, the CDM spectrum, and thenormalized and unnormalized XCDM one, for the setof parameters0

¼ 0:8, ¼ 0 2:6 102, andwX ¼1:6, which are allowed in Figs. 1 and 2. For these values,we have obtained a normalization factor AX ffi 1:1 and anaccurate agreement between the XCDM power spectrumand PðkÞ. This was expected since we are assumingallowed values of the parameters, i.e., values already con-sistent with LSS data according to the ‘‘effective’’ ap-proach used in [23] (cf. the discussion in Sec. VII).Therefore, the predicted power spectrum from theXCDM ought to be very close to the CDM one, which

is in fact what we have substantiated now by explicitnumerical check.However, for values of the parameters out of the allowed

region in Fig. 1 the predicted matter power spectrum candiffer significantly from the CDM one, PðkÞ. This oc-curs mainly for points that do not satisfy the F-test condi-tion [23], even if the other observable constraints [namelythe ones related to nucleosynthesis and the present value ofthe EOS (cf. Sec. VII)] are fulfilled. Let us remind that theF test consists in requiring that the matter power spectrumof the model under consideration (in this case, theXCDM model) differs from that of the CDM in lessthan 10%, under the assumption that DE perturbations canbe neglected. Given the fact (explicitly analyzed here) thatthe DE perturbations should not play a very important role,it is reasonable to expect that the F test should be approxi-mately valid even when we do not neglect the DE pertur-bations. Thus, the XCDM model should exhibit a largedeviation in the amount of growth with respect to theCDM precisely for those points failing the F test.Points of this sort are those located in the striped region,but outside the shaded one in Fig. 1. For these points, weshould expect an anomalously large normalization factorAX (namely, the factor that controls the matching of thetwo overall shapes) and, at the same time, we may alsoobserve an evident scale dependence in the power spec-trum, i.e. some significant difference in the predicted shapeas compared to the CDM one. Such potentially relevantscale dependence (or k dependence) is introduced by theDE perturbations themselves through the last term on ther.h.s of (39) and is eventually fed into Eqs. (44)–(46).As a concrete example, let us consider Fig. 3(b) where

we compare the 2dFGRS observed galaxy power spectrumand PðkÞ with the XCDM matter power spectrumPXðkÞ for the following set of parameters: 0

¼þ0:35, ¼ 0:2, and wX ¼ 0:6. These values fulfillthe nucleosynthesis bound (constraint No. 1 in Sec. VII),specifically, we have jj ¼ 0:08 for these parameters(meaning that DE density at the nucleosynthesis timerepresents roughly only 8% of the total energy density);and satisfy also the current EOS constraint (No. 3 inSec. VII): w0

e ¼ 0:8. However, this choice of parameterslargely fails to satisfy the constraint No. 4, i.e. the F test:indeed, we find F ¼ 2:06, which implies that the discrep-ancy in the amount of growth with respect to the CDMwhen we neglect DE perturbations is more than 200%. Asexpected, for such set of parameters we encounter a largenormalization factor for the two fiducial DE sound speedsc2s ¼ 1 and c2s ¼ 0:1 that we are using in our analysis(on average AX ’ 2:7). This is reflected in the evidentgap existing between the upper and lower set of curves inFig. 3(b). The lower set reflects the real growth jMðkÞj2 ofmatter perturbations before applying the normalizationfactor. Such a normalization consists in the following: forthe smallest scale available in the data, theXCDM curves

FIG. 3. The CDM power spectrum PðkÞ (dot-dashed line)versus the normalized and non-normalized spectrum predictedby the XCDM model, for DE sound speeds c2s ¼ 0:1 (dashedline) and c2s ¼ 1 (solid/gray line): (a) for a set of parametersallowed by the analysis of Ref. [23] (cf. shaded and stripedregion in our Figs. 1 and 2), 0

¼ 0:8, ¼ 0 2:6 102,

and wX ¼ 1:6. The corresponding curves PðkÞ and PXðkÞcoincide in this case; (b) for a set of parameters not allowed bythe F test [23] (points in the striped, but nonshaded, region in ourFig. 1), 0

¼ þ0:35, ¼ 0:2, and wX ¼ 0:6. In this case,

PXðkÞ presents a slight deviation as compared to PðkÞ at largescales (i.e. at small k). The lower set of curves in (b) displays thereal (non-normalized) growth, see the text.


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have been shifted upwards until they match up with thestandard CDM prediction. Apart from the overall gapbetween the two sets of curves, we also find a significantshape deviation with respect to the standard CDM modelat large scales, as can be seen in Fig. 3(b). The smaller thespeed of sound, the greater the deviation, as discussed inmore depth in the next section.

IX. MATTER AND DARK ENERGY DENSITYFLUCTUATIONS

As we have discussed in Sec. V, the DE fluctuations D

should oscillate and become eventually negligible as com-pared to the matter fluctuations M, especially at smallscales (inside the sound horizon). However, as also notedabove, for values that significantly violate the F test [23],the power spectrum and its shape can be noticeably differ-ent from that of the CDM model (cf. Fig. 3(b)). Thissuggests that, under such circumstances, the DE densityperturbations are not completely negligible owing to thefact that the term which depends on k in the perturbationequations is also proportional to D—see Eq. (45). Inaddition, there appears a suppression of the growth ofmatter fluctuations in comparison with the growth pre-dicted by the CDM model. This inhibition of mattergrowth is characteristic of cosmologies where the DEbehaves quintessencelike, i.e. when the DE density de-creases with the expansion, whereas phantomlike DE (in-creasing with the expansion) would cause the oppositeeffect (an enhancement of the power). A similar situationwas also observed in Ref. [41] for models with pure run-ning, where in the case > 0 (in which decreases withthe expansion) there is an inhibition of growth while for < 0 (when increases with the expansion) there is anenhancement—see also [60–62] for other studies.

Let us clarify that these differences in the amount ofgrowth are present even if we neglect the DE perturbations,see [23]. In fact, the effect of the latter is very small,specially for allowed values of the parameters, and be-comes noticeable only at large scales. At these scales, wefind that the DE perturbations tend to compensate thesuppression produced at the background level. This slightenhancement is greater the smaller the DE sound speed is.Such a feature can be appreciated in Fig. 4, where wecompare the growth of the matter fluctuations at a largescale ‘ k1 (with k ¼ 0:01) predicted by both theCDM and XCDM models for the two fiducial soundspeeds of the DE considered before and for the same valuesof the parameters as in Fig. 3.

The growth of matter density fluctuations for theXCDM model is in agreement with the predicted oneby the CDM model (the dot-dashed and black line) inFig. 4(a), for the set of parameters in the allowed region,whereas in Fig. 4(b) we see the previously commentedsuppression for the set of parameters not satisfying the Ftest. The former case can be compared with Fig. 5(a) in

which we have assumed the same set of allowed parame-ters; as expected, we find completely negligible DE fluc-tuations today and in the recent past, in agreement with theF-test assumption [23] which means completely negligibleDE fluctuations at large scales and a maximum 10% ofdeviation from the CDM growth of matter density fluc-tuations. On the other hand, values of the parameters notsatisfying the F test present not only suppression onthe growth of matter density fluctuations, as shown inFig. 4(b), but also larger DE fluctuations today and in therecent past, as shown in Fig. 5(b).Furthermore, as discussed in Sec. V, the growth of DE

fluctuations is expected to oscillate at small scales andrapidly decay, the fact that legitimates our assumptionsfor the initial conditions of the DE perturbations. Weshow these oscillations for an allowed set of parametersin Fig. 6. Similar behavior is obtained for values of theparameters not allowed by the F test and for both DE soundspeeds c2s ¼ 1 and c2s ¼ 0:1. The amplitude of the DEgrowth starts negligible ( 103) and rapidly decays tozero, as shown in Fig. 6.

FIG. 4. The XCDM matter density fluctuations at a fixedlarge scale k ¼ 0:01 (in units of h Mpc1) as a function of thescale factor a in comparison with those predicted by the CDMmodel (dot-dashed line). For the former we have assumed thesame values of the parameters and meaning of the lines as inFig. 3: (a) for the set of parameters in the allowed region; (b) forthe set of parameters not allowed by the F test in [23].


043006-21

In Fig. 7 we plot the present value of the DE perturba-tions as a function of the wave number for the two sets ofallowed (Fig. 7(a)) and nonallowed (Fig. 7(b)) parametersused in the previous plots. We see that the DE perturbations

are negligible at small scales (large k), whereas they be-come larger at larger scales. This is because by increasingthe scale we are getting closer to the sound horizon, asdiscussed in Sec. V. We also see that the DE perturbationsare larger for the parameters not allowed by the F test,which explains why the shape of the matter powerspectrum differs from that of the CDM in this case(cf. Fig. 3(b)]. However, when comparing with Fig. 4, wesee that even at the largest explored scale (k ¼ 0:01) theratio D=M remains rather small, staying at the level of103. Finally, let us comment that the DE density contrastcan become negative with the evolution, as in this casehappens for c2s ¼ 1.As discussed in Sec. V, the decay of the DE perturba-

tions takes place once the term proportional to k2 in (45)becomes dominant. This same term is also responsible forthe exponential growth of the (DE) perturbations in theadiabatic case or, more generally, whenever c2s is negative.In order to better appreciate its influence, it is useful tocompare the evolution of the DE perturbations in theadiabatic case (for the most common situation where c2a <0) and the nonadiabatic one (c2s > 0) with the scenario inwhich c2s ¼ 0, since in the latter the term proportional to k2

disappears from the equations. This is precisely what hasbeen done in Fig. 8(a) for the allowed set of parameters

FIG. 5. The XCDM growth of DE fluctuations at the largescale k ¼ 0:01 (in units of h Mpc1) as a function of the scalefactor a. We have assumed the same values of the parameters asin Fig. 3 for DE sound speed c2s ¼ 0:1 (dashed line) and c2s ¼ 1(solid/gray line): (a) for the set of parameters in the allowedregion; (b) for the set of parameters not allowed by the F test in[23].

FIG. 6. The XCDM growth of DE fluctuations for a smallscale k ¼ 0:2 (in units of h Mpc1) and the same set of allowedparameters assumed before, and for DE sound speed c2s ¼ 0:1.

FIG. 7. The scale dependence of DE fluctuations today (a ¼ 1)for the same values of parameters and meaning of the lines as inFig. 5. They rapidly decay at small scales (large k).


043006-22

used throughout this section. In that figure, the evolution ofthe DE density contrast at a sufficiently large scale ‘ k1

for which the DE perturbations can be sizeable is shown(namely at k ¼ 0:01). We illustrate the effect for threedifferent regimes of the speed of sound: c2s ¼ c2a (withc2a < 0), c2s ¼ 0, and c2s ¼ 0:1. The gray lines representthe evolution of D when the last term in Eq. (44) isneglected, showing indeed that the qualitative behaviorof the perturbations in the adiabatic and c2s ¼ 0:1 casesdoes not stem from that term.

For c2s ¼ 0:1, the scale considered is initially (i.e. atai ¼ 1=500) larger than the sound horizon, and thus theterm proportional to k2 is negligible at the beginning of theevolution. The same is true for the adiabatic case because,in the asymptotic past, the effective EOS of the XCDMmodel resembles that of matter/radiation (weðaÞ ! wm fora ! 0) [17] and, thus, we have c2s ¼ c2a ’ we ’ 0 in thematter-dominated epoch. (We recall that in all our discus-sion we remain in the matter epoch, equality being at a104). Therefore, the term proportional to k2 is initiallyunimportant in all the three cases and this makes theperturbations to evolve in a nearly identical fashion at thesefirst stages, as can be clearly seen from Fig. 8(a).As the evolution continues, the curve corresponding to

c2s ¼ 0:1 begin to depart from the others. This occursmainly from the instant when the sound horizon is crossed,i.e. when the wavelength of the k-mode gets comparable tothe sound horizon; such an instant can be defined throughthe condition ks ¼ , similarly as in [43]. Then, the termproportional to k2 begins to dominate, which in turn makesthe DE velocity gradient D to rapidly increase and the DEperturbations to decay. Later on, the term proportional to k2

becomes important also in the adiabatic case. Because ofthe different sign (c2a < 0), the effect that it triggers is nowopposite to the one observed in the c2s ¼ 0:1 case; thus,instead of getting stabilized, the DE perturbations initiatean exponential growth.The previous features can be further assessed in a quan-

titative way by comparing the numerical importance (inabsolute value) of the two terms inside the curly brackets in(44). In Fig. 8(b), we plot the ratio between these two termsfor both the adiabatic case (with c2a < 0) and the non-adiabatic situation (c2s ¼ 0:1) (note that the term propor-tional to H2=k2 may be neglected for the sub-Hubbleperturbations we are dealing with). Comparison withFig. 8(a) reveals that it is precisely when the term propor-tional to D stops being negligible that the evolution of theperturbations begins to depart from the c2s ¼ 0 case. Theabsolute value of the adiabatic speed of sound is alsoshown, in order to illustrate the ultimate reason for thestartup of the exponential growth in the adiabatic mode: itis only when c2a begins to depart significantly from 0 thatthe term proportional to k2 becomes important, which inturn triggers a rocket increase of the velocity gradient D.As we have discussed in connection to Fig. 8(a), the

initial evolution of the perturbations in theXCDMmodelis nearly the same for any of the three values of the speed ofsound. In fact, in the adiabatic case, and given the behaviorof the effective EOS in the asymptotic past, the conditionsthat lead to the simplified setup (53) hold. Therefore, that isthe equation initially controlling the evolution of D. Forc2s ¼ 0 we arrive at exactly the same equation, whereas forpositive c2s the resulting equation only differs by the lastterm in (44), which, as it has been previously discussed,happens to be negligible at least during the first stages of

10-2

10-1 2·10

-1

1a

10-3

10-2

10-1

1

101

102

103

δ D(a

)k=

0.01

(a)

cs2 = ca

2

cs2 = 0

cs2 = 0.1

k·λs = π

10-2

10-1 2·10

-1

1a

10-3

10-2

10-1

1

101

2θD

(a)

/ h(a

) k=

0.01

cs2 = ca

2

cs2 = 0.1

k·λs = π(b)

^

ca2(a)

FIG. 8. (a) Evolution of the DE perturbations for the same setof parameters as in Fig. 3(a), at the large scale k ¼ 0:01 and forthree different speeds of sound: c2s ¼ c2a < 0 (solid line), c2s ¼ 0(dashed line), and c2s ¼ 0:1 (dot-dashed line). For the latter, theDE perturbations begin to decay after the sound horizon crossing(characterized by the condition ks ¼ ), whereas in the adia-batic case D starts to grow exponentially at a ’ 0:2. The lastterm in Eq. (44) may be neglected (gray lines) without alteringthe qualitative behavior, which is triggered by the D-term and,ultimately, by the one proportional to k2 in (45); (b) Comparisonof the two terms inside the curly brackets in (44) for the c2s ¼ 0:1and adiabatic cases. When the D-term becomes important, thestabilization (resp. unbounded growth) of the nonadiabatic (resp.adiabatic) perturbations becomes manifest.


043006-23

the evolution. Notice that for we ’ const:, Eq. (53) inte-grates to DðaÞ ¼ ð1þ weÞM þ C, where C is a constantdetermined by the initial conditions. In Sec. IV, we pointedout that C ¼ 0 corresponds to the adiabatic initial condi-tion (54). However, for the alternate initial condition (104),and taking into account that weðaiÞ ’ wM ¼ 0 for theXCDM model in the early matter-dominated epoch[17], we have C ¼ MðaiÞ and thus DðaÞ ¼ MðaÞ MðaiÞ. From here we find that the ratio between DE andmatter perturbations in the early times of the evolutionreads:

DðaÞMðaÞ

¼ 1 MðaiÞMðaÞ : (106)

This simple predicted behavior is confirmed from thenumerical analysis in Fig. 9, where again the allowed set ofparameters has been used. We see that the ratio D=M

starts being 0, and subsequently as the matter perturbationsgrow the last term in (106) diminishes, until the asymptoticvalue D=M ¼ 1 is reached. This value is maintaineduntil the conditions leading to (53) cease to be valid. Inthe adiabatic and c2s > 0 cases, this happens when the termproportional to k2 on the r.h.s. of Eq. (45) can no longer beneglected. On the other hand, when c2s ¼ 0, the D=M ’1 regime is abandoned at the point when the effective EOSstarts acquiring sizable negative values. Moreover, beingthe term proportional to k2 absent, it is now the last term onthe r.h.s. of Eq. (44)—which was irrelevant for the othertwo cases—the one that tends to stabilize the DE pertur-bations (see also Fig. 8(a)).

From the detailed analysis that we have presented here,we conclude that the approximation of neglecting the DEperturbations can be justified [23]. But this does not meanthat the computation of these perturbations is useless.Indeed, the issue at stake here is not so much the quanti-tative impact of the DE fluctuations upon the matter powerspectrum—which is actually negligible, as we have seen—but rather the fact that the DE perturbations may be con-sistently defined in a certain subregion of the parameterspace only. Of course this subregion cannot be detectedwithin the context of the effective approach [23].Therefore, in general, the computation of the DE perturba-tions may have a final quantitative bearing on this kind ofanalysis since it may further restrict the physical region ofthe parameter space in a very significant way. In short, eventhough the simultaneous account of the DE perturbationshas a small numerical effect on the matter power spectrumwithin the domain where the full system of cosmologicalperturbations is well defined, it may nevertheless prove tobe a highly efficient method for excluding large regions ofparameter space where that system is ill defined. The up-shot is that the combined analysis of the DE and matterperturbations may significantly enhance the predictivity ofthe model, as we have indeed illustrated in detail for thenontrivial case of the XCDM model of the cosmicevolution.

X. DISCUSSION AND CONCLUSIONS

In this paper, we have addressed the impact of thecosmological perturbations on the coincidence problem.In contrast to the previous study [23], where this problemwas examined in a simplified ‘‘effective approach’’ inwhich the DE perturbations were neglected, in the presentwork we have taken them into account in a full-fledgedmanner. We find that the results of the previous analysiswere reasonable because the DE perturbations generallytend to smooth at scales below the sound horizon.However, the inclusion of the DE perturbations provedextremely useful to pin down the physical region of theparameter space and also to put the effective approachwithin a much larger perspective and to set its limitations.First of all, we have performed a thorough discussion on

the coupled set of matter and DE perturbations for ageneral multicomponent fluid. This has prepared theground to treat models in which the DE is a compositemedium with a variable EOS. We have concentrated onthose cases in which the DE, despite its composite nature,is described by a self-conserved density D. Notice that ifmatter is covariantly conserved, the covariant conservationof the DE is mandatory. In particular, this is the situationfor the standard CDM model, although in this case theself-conservation of the DE appears through a trivial cos-mological constant term, ¼ 0

, which remains imper-

turbable throughout the entire history of the Universe. Onemay nevertheless entertain generalized frameworks where

10-2

10-1 2·10

-1

1a

10-3

10-2

10-1

1

101

102

δ D(a

)/δ m

(a)

k=0.

01

cs2 = ca

2

cs2 = 0

cs2 = 0.1

k·λs = π

FIG. 9. Evolution of the ratio between DE and matter pertur-bations for the same parameter values as in Fig. 8. The specialregime of perturbations (53) is seen to be approximately realizedduring the first stages of the evolution, for any of the threeconsidered speeds of sound. Since we ! 0 for small a in theXCDM model, we expect D=M ’ 1 [cf. Eq. (106)] until theconditions leading to (53) no longer hold. One can clearlyconfirm this situation in the figure.


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the DE is not only self-conserved, but is nontrivial anddynamical. This is not a mere academic exercise; forinstance, in quantum field theory in curved space-timewe generally expect that the vacuum energy should be arunning quantity [13,15,16]. Therefore, in such cases, theCC density becomes an effective parameter that mayevolve typically with the expansion rate, ¼ ðHÞ,and constitutes a part of the full (dynamical) DE of thecomposite cosmological system with variable EOS. Inthese circumstances, if the gravitational coupling G isconstant, the running CC density ¼ ðHÞ cannot becovariantly conserved unless other terms in the effectiveaction of this system compensate for the CC variation. Wehave called the effective entity that produces such com-pensation ‘‘X’’ or cosmon and denoted with X its energydensity. Therefore, D ¼ þ X is the self-conservedtotal DE density in this context, which must be dealt withtogether with the ordinary density of matter M. A genericmodel of this kind is what we have called the XCDMmodel [17,23].

Furthermore, from general considerations based on thecovariance of the effective action of QFT in curved space-time [13,15,16], we expect that the running CC density ¼ ðHÞ should be an affine quadratic law of theexpansion rateH, see Eq. (67). Using this guiding principleand the ansatz of self-conservation of the DE, we find thatthe evolution of X, and hence of D, becomes completelydetermined, even though its ultimate nature remains un-known. In particular, X is not a scalar field in general.

The XCDM model was first studied in [17] as apromising solution to the cosmic coincidence problem, inthe sense that the coincidence ratio r ¼ D=M can stayrelatively constant, meaning that it does not vary in morethan 1 order of magnitude for many Hubble times. Themain aim of the present paper was to make a further step toconsolidate such a possible solution of the coincidenceproblem, specifically from the analysis of the coupledsystem of matter and DE perturbations. Let us remarkthat this has been a rather nontrivial test for the XCDMmodel. Indeed, after intersecting the region where the DEperturbations of this model can be consistently defined,with the region where the coincidence problem can besolved [17,23], we end up with a significantly more re-duced domain of parameter space where the model canexist in full compatibility with all known cosmologicaldata. The main conclusion of this study is that the predic-tivity of the model has substantially increased. Therefore,it can be better put to the test in the next generation ofprecision cosmological observations, which include thepromising DES, SNAP, and PLANCK projects [63].

Interestingly enough, we have found that the final regionof the parameter space is a naturalness region which ismore accessible to the aforementioned precision experi-

ments. For example, we have obtained the bound 0 < &0 102 for the parameter that determines the runningof the cosmological term. This bound is perfectly compat-ible with the physical interpretation of from its definition(68). Moreover, our analysis indicates that the cosmonentity X behaves as ‘‘phantom matter’’ [17], i.e. it satisfies!X <1 with negative energy density. This result is aclear symptom (actually an expected one) from its effectivenature. It is also a welcome feature; let us recall [17] thatphantom matter, in contrast to the ‘‘standard’’ phantomenergy, prevents the Universe from reaching the big ripsingularity. Finally, perhaps the most noticeable (and ex-perimentally accessible) feature that we have uncoveredfrom the analysis of the DE perturbations in the XCDMmodel, is that the overall EOS parameter we associated tothe total DE density D behaves effectively as quintessence(we * 1) in precisely the region of parameter spacewhere the cosmic coincidence problem can be solved. Inother words, quintessence is mimicked by the XCDMmodel in that relevant region, despite the fact that there isno fundamental quintessence field in the present frame-work. A detailed confrontation of the various predictions ofthe XCDM model [in particular, the kind of dependencewe ¼ weðzÞ] with the future accurate experimental data[63], may eventually reveal these features and even allowto distinguish this model from alternative DE proposalsbased on fundamental quintessence fields.To summarize: we have demonstrated that the set of

cosmological models characterized by a composite, andcovariantly conserved, DE density D in which the vacuumenergy is a dynamical component [specifically, onethat evolves quadratically with the expansion rate, seeEq. (67)], proves to be a distinguished class of modelsthat may provide a consistent explanation of why D isnear M, in full compatibility with the theory of cosmo-logical perturbations and the rest of the cosmological data.Remarkably, such a class of models is suggested by theabove-mentioned renormalization group approach to cos-mology. We conclude that the XCDM model can belooked upon as a rather predictive framework that mayoffer a robust, and theoretically motivated, dynamical so-lution to the cosmic coincidence problem.

ACKNOWLEDGMENTS

The authors are grateful to Julio Fabris for useful dis-cussions. We have been supported in part by MEC andFEDER under Project No. FPA2007-66665 and also byDURSI Generalitat de Catalunya under ProjectNo. 2005SGR00564. The work of J. G. is also financedby MEC under BES-2005-7803. We acknowledge the sup-port from the Spanish Consolider-Ingenio 2010 programCPAN CSD2007-00042.


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dark energy perturbations and cosmic coincidence

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