j. c. fabris et al- density perturbations in a universe dominated by the chaplygin gas

8/3/2019 J. C. Fabris et al- Density Perturbations in a Universe Dominated by the Chaplygin Gas

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DENSITY PERTURBATIONS IN A UNIVERSEDOMINATED BY THE CHAPLYGIN GAS

J. c. Fabris", S.V.B. Ooncalves? and P.E. de Souza"Departamento de Ffsica, Universidade Federal do Espirito Santo, CEP29060-900,

Vitoria, Espirito Santo, BrazilAbstract

We study the fate of density perturbations in a Universe dominate by the Chap-lygin gas, which exhibit negative pressure. In opposition a other models of perfectfluid with negative pressure, there is no instability in the small wavelength limit,due to the fact that the sound velocity for the Chaplygin gas is positive. We showthat it is possible to obtain the value for the density contrast observed in large scalestructure of the Universe by fixing a free parameter in the equation of state of thisgas. The negative character of pressure must be significant only very recently.PACS number(s): 98.80.Bp, 98.65.Dx

1 IntroductionThe nature of "missing mass" that apparently dominate the Universe today is one ofthe most intriguing problem in modern physics. This problem appears at various lev-els, beginning with the rotation curve of spiral galaxies [1], and extending to the totalmass of the Universe as it is deduced through the first acoustic peak of the spectrum ofthe anisotropy of cosmic background radiation [2]. A fraction of this missing mass hasbeen called dark energy since it does not cluster, remaining a smooth component of theUniverse. This problem has acquired a most dramatic face through the results comingfrom the observation of supernova type la, which indicate that the Universe today is inan accelerated expansion [3,4]. Hence, the inflationary paradigma, first conceived for theprimordial Universe, may be applied to our actual Universe.

Observations today favor a cosmological scenario where Ob I"V 0.02, Om I"V 0.3 andOA I"V 0.7, with these terms representing the baryonic, cold dark matter and cosmologicalconstant fraction of the total matter content of the Universe. Hence, the Universe todaymust be nearly flat OK I"V o . The acceleration of the Universe, as infered from thesupernova results, indicate an effective equation of state p =op, with a I"V -0.67. Allthese values are not precisely determined by observations. But, the comparison of thedifferent observational results indicates values which are near to those displayed above.

It is possible that a cosmological constant may be the dominant component of matterin the Universe. In this way, the cosmological constant problem was revived. However,

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other types of matter with negative pressure have been considered as candidate of theso-called dark energy of the Universe: fluids of topological defects (domain walls, strings)[5], quintessence (a self-interacting scalar field minimally coupled to gravity) [6], etc. Allthese possibilities have their advantages as well as their disavantages.

The cosmological constant, for example, may have its origin in the energy of the vac-uum quantum state. But, in order to reconcile the observed value and the theoreticalvalue predicted by quantum field theory, a fine tunning of 10120 orders of magnitude mustbe made [7]. Topological defects faces different difficulties: it leads to a non gaussian spec-trum of perturbations, in apparent disagreement with observations [8]. Moreover, as thesupernova observations indicate, the most probable candidate from different topologicaldefects would be a gas of domain walls. But, it is very difficult to implement a cosmologi-cal model where the domain walls can effectivelly become the dominant component of thematter content of the Universe without become relativistic, acquiring a positive equationof state [5].

Quintessence seems to be, until now, the most robust candidate for dark energy. But,in order to implement a consistent quintessence model, different possible potential termsmay be used, with in general a very poor theoretical justification. It must be stressed,however, that supergravity theories predict self-interacting scalar field terms which maylead to a coherent cosmological scenario [9]. However, the existence of some many differentquintessence scenario pleads for a more fundamental approach.

Recently, a different matter component with negative pressure has been presented aspossible candidate for this dark energy: the Chaplygin gas [10].

The Chaplygin gas has an equation of state of the typeAp=--p (1 )

where A is a constant. This exotic fluid has an interesting origin. It has been firstconsidered in problems connected with fluid dynamics. But, later connections with thefundamental physics have appeared. Taking the Nambu-Goto action of string theory, theChaplygin fluid appears after considering d-branes in a d + 2 dimensional space-time, inthe light cone gauge [11].

An interesting point concerning the Chaplygin gas is the fact that, even if it comesfrom a brane model in string theory, it obeys the equations of a newtonian fluid. Theequation of state (1) implies more symmetries than those typical of the galilean group,which is a consequence of its relativistic origin. In particular, a hamiltonian descriptionof an irrotational fluid with the equation of state (1) exhibits symmetries with the samedimension as the Poincare group.

In this work, we will investigate a cosmological model where the Universe is dominatedby the Chaplygin gas, which in our four dimensional world can be a phenomenologicalrepresentation of a gas of membranes. We will mainly interested in the evolution of densityperturbations, trying to verify if the dominance of the Chaplygin gas is compatible withthe formation of structure. In order to do so, we will exploit the newtonian descriptionof the Chaplygin gas. But, it will verified that the results are consistent with what wecan expect from a full relativistic formulation of the problem. These results indicate that

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structure can be formed in a phase where the Chaplygin gas dominates. Moreover, in spiteof the negative pressure typical of this gas, the model is stable even in its hydrodynamicalformulation.

In fact, it will be shown, using the newtonian approach, that the Chaplygin gas ex-hibit growing modes in the begining of the material phase. Later, it behaves most asa cosmological constant. Hence, initially the Chaplygin gas clusters, becoming later asmooth component of the Universe. In this sense, the accelerating Universe predicted bythe Chaplygin gas model has many different features compared with a cosmological con-stant or quintessence model. In order to fit observations, cold dark matter, which remainsclustered during all the evolution of the material phase of the Universe, is yet necessary.But the fact that the dark energy component has not remained smooth during all thehistory of the Universe, may lead to different previsions for the different cosmologicalparameter. The present study is a first step in view of the construction of a realistic cos-mological model based on the Chaplygin gas, and a two-fluid model, taking into accountthe contribution of cold dark matter, may be near to what we expect to be a realisticscenario.

This paper is organized as follows. In next section we describe the Chaplygin gas andsome of its main features. In section 3 we sketch a relativistic treatment of a cosmologicalmodel with the Chaplygin gas. In order to obtain analytical expression, we devellop anewtonian analysis at the background and perturbative levels in section 4. In section 5we present our conclusions.

2 The Chaplygin gasThe Chaplygin gas is generically characterised by the equation of state (1). This equationof state was first conceived in connection with the thermodynamics of adiabatic process.More rececently, however, it has acquired an interesting connection with d-branes in ad + 2 dimensional space-time.

The d-branes may be expressed through the aid of the Nambu-Goto action,_f Old ( )d ex- ox,SNG - d d ...d -1 det 8a 8f3 (2 )where X!' are the d+2 target space-time variable, while i parametrizes the d-dimensionalbrane. The action (2) can be rewritten under different forms using convenient parametriza-tions. If the light-cone parametrization is choosed, with

(3 )

and X+ =vIAo, the Nambu action reduces to [12](4 )

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A very curious aspect of this action is that the variables () and p obey the newtonianhydrodynamic equations for an irrotational fluid with the pressure given by (1):

(5 )Since the Chaplygin gas satisfy the newtonian equations of motion, it is invariant

under the galilean transformations. However, due to the specific form of the equationof state, it admits also some other symmetries. In particular, it is invariant under timerescaling

(6 )as well as the space-time mixing

..........1 2 ()t -+ t + w . r + " 2 wr -+ f+ z e(7 )(8 )

where wand ware the generators of the new symmetries. Summing up all the symmetries,it is easy to verify that the total dimensionality of the symmetry group of the Chaplygingas is the same as of the Poincare group. It means that the relativistic character of theChaplygin gas is somehow hidden behind its newtonian expressions.

It is interesting to note that, if another parametrization were choosed, the cartesianparametrization, the action (2) would take the form a Born-Infeld-like action. Moreover,the Chaplygin gas admits a supersymmetric extension. A very comphreensive review ofall the features of Chaplygin gas is [13].

3 A relativistic analysisIn order to investigate the cosmological consequences of the Chaplygin gas, let us firstconsider its coupling to the Einstein's equations:

1R /w - " 2 gttvR 87rGT tt v(p + p )u ttU V _ e"A

(9 )

p p

(10)(11)

Considering the flat Friedmann-Robertson- Walker metric,ds2 =dt2 - a2(t) (dx2 + dy2 + dz2)

the equations of motion read(12)

87rGp (13)

p Ap

(14)(15)

o

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In [10], some consequences for the evolution of the Universe of such exotic fluid havebeen investigate. It was found that, in the relativistic context, the density is connectedto the scale factor as P = J A + ~ . (16)Hence, initially the Universe behaves as it was dominated by a dust fluid, and thenthe density becomes asymptotically constant, revealing that the cosmological constantbecomes the dominant component of the Universe. There is an intermediate phase, whichcan be described by a cosmological constant mixed with a stiff matter fluid. In thissense, this exotic fluid may be a candidate for describing the Universe as the supernovaobservational results indicate it must be.

Ifwe want to analyze the evolution of density perturbations for the cosmological modeldescribed above, we must perturbed the Einstein's equation. Hence, we introduce in theEinstein's equations the quantities

p=p+Jp (17)where the first term in the right-hand-side of each expression means the backgroundsolution described before, and the second one is a small perturbation around it. Inorder to compute the evolution of the perturbations, we may employ the so-called gaugeinvariant formalism [14] or fix a coordinate condition, taking care of verifying in the end ifthe final solutions are not artifact of coordinates. We choose to work in the synchronouscoordinate condition httO =0; the residual coordinate freedom is easily identified in thiscase, and we can fix the real physical solutions. The perturbed equations are

.. ah+2-h a. 1 .Jp + (p + p )O + 3 (p + p )Jp - 2 (P + p )h

(p + p)iJ + ( p + jJ)O + 5~(p + p )Oa

87rG(Jp + 3Jp ) (18)(19)(20)

where n is the wavelength of the perturbations, which appears in the last expression dueto the fact that we performed a plane wave expansion to describe the spatial behaviourof the perturbations, and 0 =JUi,i. From the pressure expression, we have

(21)The background expressions are complicated enough to have explicit expression for

the perturbed quantities. Closed expression can be found only in the asymptotic region.Initially, the Universe behaves as it was dominated by a pressurelless fluid. The solutionsfor the perturbed equations in this case are well known [15], reading, for the densitycontrast J,

(22)5


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In the other asymptotic, the Universe behaves as it was dominated by a cosmologicalconstant. It is well-known that density perturbation in a Universe dominated by a cos-mological term is zero. There is an intermediate region, where the Chaplygin gas exhibita behaviour typical of a mix of stiff matter fluid and a cosmological term. However, thiscase exhibits background solutions which do not allow simple expressions for the per-turbed quantities, even if we consider the perturbations on the cosmological term as zero,as we must expect.

4 A newtonian approachAs we have seen, the relativistic study of this problem leads to some technical difficulties.Considering that we can approximate the evolution of the Universe filled by such fluidby the three phases described before, the behaviour of density perturbation in the initialand final phases are already known ( c 5 dust ex t2/3 and c 5 ee =0 respectively). Hence in orderto obtain non-trivial effects we must consider the intermediate phase. However, as statedbefore, such phase admits analytical solutions for the background, but it seems that noanalytical solutions can be obtained for the perturbed equations.

Since our intention is to have explicit expressions for density perturbations, in orderat least to evaluate what happens for the evolution of structure in a Universe dominate bythe Chaplygin gas, we will exploit the fact that the action (4) leads to the hydrodynamicnewtonian equations. Of course, the true nature of the problem is relativistic, sincewe regard the Chaplygin gas as a phenomenological representation of a gas of d-branes.However, we are interested in the material phase of the evolution of the Universe, wheregalaxies begin to be formed, for which the newtonian approximation is quite realistic.Moreover, in the limit of small wavelength of the perturbations, the relativistic problemreduces to the newtonian problem. Since, in general, perfect fluid models with negativepressure leads to instabilities at this regime, the newtonian approximation to be employedhere can test very well the consistency of the Chaplygin gas model.

Coupling to (4) a term representing the gravitational potential we have the followingequations [16]:ap ( _,) 0 (23)t + V. pv

11+ 11.V11 Vp (24)--VpV2 47fGp (25)

If these equations describe an expanding homogeneous and isotropic Universe their solu-tion is

P op=-a3_ , _ ,av=r-a (26)

The "scale factor" a(t) obeys a Friedmann-like equation, even in the newtonian approx-imation [16], and its solution, for the marginal "flat" case, is a(t) ex t2/3. Notice thatthese solutions are independent of the pressure. Indeed, since in a homogeneous Universe

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p =p(t), there is no pressure gradient, and the presence of pressure does not influencethe evolution of the Universe.

We turn now to the perturbative level. Introducing in equations (23,24,25) the quan-tities P =P o + S p , v =V o + O v , p =P o + O p and e p =e p o + O e p , where each quantity has beenexpressed as a sum of the background solution plus a small perturbation, we can combinethe resulting perturbed equation in order to obtain a unique differential equation for thedensity contrast 0 = 8p [16]:p

(27)In this expression v ; = ? p is the the sound velocity and n is the wavenumber of theperturbation, which appears due to the fact that we expanded the spatial dependenceof the perturbed quantities in plane waves. Since all the problem is being treated in anewtonian framework, no gauge problem, connected with the coordinate transformationfreedom typical of the corresponding relativistic problem, appears in this case, and allsolutions of equation (27) have a physical meaning [15]. An important property of theChaplygin gas is that, in spite of exhibiting negative pressure, its sound velocity is positive.This does not happen in general: fluids with negative pressure quite frequently exhibitinstabilities, at least in their hydrodynamical representation, due to an imaginary soundvelocity [17, 18]. The Chaplygin gas is an important exception to this general feature ofperfect fluids with negative pressure.

Inserting the background solutions and the expression for the sound velocity in (27),we determine the behaviour of density perturbations. The solution reads

(28)where v =5/14 and ~2 =~~: g G . In principle C1 and C2 are constants that depend on n.Notice that the expression for density contrast for perfect fluids with negative pressureexhibit, in general, modified Bessel functions, which reflects their instability in the smallwavelength limit [17, 18].

The solution (28) has two asymptotic regime. For t -+ 0, there is a growing modegiven by 0+ ex t2/3. This coincides with the evolution of density perturbations in a dustfluid in General Relativity. On the other hand, for t -+ 00, the density contrast oscillateswith decreasing amplitude, O ex t-4/3 cos (~2ne /3 + ~),where ~ is a phase, approachinga zero value asymptotically. Hence, for large values of time, the solution approachesthe result for density perturbation in General Relativity with a cosmological constant,O ee =o . The transition from a "dust" phase to a "cosmological constant" phase is smoothand the moment it happens is essentially dictated by the value of the constant A in (4).This behaviour inverts what happens in the traditional case of density perturbations innewtonian gravity with positive pressure[16]. In this case, taking the equation of statep ex p ' Y , I> 4/3, the density contrast initially oscillates, and after it grows with 0+ ex t2/3.

These features have some interesting consequences. In fact, let us suppose that atthe moment of decoupling of radiation and matter, ti I"V lOllS, the amplitude of density

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perturbations in a Universe dominated by the Chaplygin gas, we performed a newtoniananalysis since the Chaplygin gas obeys formally the equations of newtonian hydrodynamic.However, the newtonian treatement is also justified by the fact that it approximates thegeneral features of the full relativistic problem in the phase where galaxies form, which isthe phase we are most interested on.

One of the main conclusions of this work is that a Universe dominated by the Chaplygingas admits an initial phase of growing perturbations, with the the same rate as in dustcase of the cosmological standard model, from which it follows decreasing oscillations,which asymptotically go to zero. Hence, even if the newtonian analysis is limited forthe study of the Universe in large scales, in this case the newtonian solution interpolatesconveniently an initial dust phase ( O d ex t2/3) with a cosmological constant phase ( O e e =0).Even if the newtonian background model is very different from the relativistic one, sinceit gives a dust-like solution for any value of time, the perturbative newtonian analysis isconsistent with the different phases predicted by expressions (1,16), which comes fromrela tivistic considerations.

There is an initial phase where the Chaplygin gas agglomerates. Later, it tends tobecome a smooth component of the total matter existing in the Universe. This fact hastwo main consequences: first, since clusters of galaxies which are near us need clustereddark matter, the model presented here must be complemented by some kind of cold darkmatter component; second, the evaluation of cosmological parameters must be revised,since this dark energy has an initial behaviour more near cold dark matter, approachinglater a genuine cosmological constant behaviour. This demands, at least, a two-fluidmodel in order to try to construct a realistic scenario, and perharps a full relativistictreatment of the evolution of density perturbations.

Another important feature to be emphasized is that the Chaplygin gas, in spite ofpresenting negative pressure, is stable at small scale, in opposition to what happens ingeneral with perfect fluids with negative pressure. This is due to the fact that the soundvelocity in the Chaplygin gas is positive, while in many other perfect fluid models withnegative pressure the sound velocity becomes imaginary.Acknowledgements: The remarks made by the anonimous referees were very impor-

tant in order to improve the present version of this paper. We thank CNPq (Brazil) forpartial financial support.

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j. c. fabris et al- density perturbations in a universe dominated by the chaplygin gas

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