Brane cosmology, Weyl fluid, and density perturbations

Supratik Pal*

Physics and Applied Mathematics Unit, Indian Statistical Institute, 203 B.T.Road, Kolkata 700 108, India(Received 14 May 2008; published 7 August 2008)

We develop a technique to study relativistic perturbations in the generalized brane cosmological

scenario, which is a generalization of the multifluid cosmological perturbations to brane cosmology. The

novelty of the technique lies in the inclusion of a radiative bulk which is responsible for bulk-brane energy

exchange, and in turn, modifies the standard perturbative analysis to a great extent. The analysis involves a

geometric fluid—called the Weyl fluid—whose nature and role have been studied extensively both for the

empty bulk and the radiative bulk scenario. Subsequently, we find that this Weyl fluid can be a possible

geometric candidate for dark matter in this generalized brane cosmological framework.

DOI: 10.1103/PhysRevD.78.043517 PACS numbers: 98.80.�k, 98.80.Cq

I. INTRODUCTION

During the last few years braneworld gravity hasemerged as a more general theory of gravity, mainly dueto the possibility of explaining the gravitational phe-nomena observed in the four-dimensional (4D)universefrom a broader perspective [1,2]. Subsequently, develop-ments of the cosmological sector [3] were brought forth,motivated by the challenges facing any theory of cosmol-ogy, be it general relativity (GR) or modified gravity, inexplaining predictions from the highly accurate observa-tional data [4–6]. In spite of great complications, thecosmological aspects of this scenario did show some prom-ising features. To mention a few, brane cosmology natu-rally gives rise to singularity-free bouncing and cyclicuniverses [7]. Also, in this theory, the universe does notneed any special initial condition for the inflation to start sothat the isotropy is built in the theory [8]. Even the possi-bility of inflation without any 4D inflaton field is in vogue[9]. Brane cosmology thus results in interesting physicswhich needs further investigations.

In this scenario, the bulk spacetime is either AdS5 [10]or a generalized version of it. The generalized globalstructure depends upon whether the bulk has only a cos-mological constant or there are any nonstandard modelfields minimally or nonminimally coupled to gravity orto brane matter. When the bulk is empty consisting only ofa cosmological constant, the bulk metric in which aFriedmann-Robertson-Walker (FRW) brane can be consis-tently embedded is given by a five-dimensionalSchwarzschild–anti-de Sitter (Sch-AdS5) or a Reissner-Nordstrom anti-de Sitter (RNAdS5) black hole[2,3,7,11,12]. A subsequent generalization of this scenariocan be obtained when the bulk is not necessarily empty butit consists of a radiative field, resulting in a Vaidya–anti-deSitter (VAdS5) black hole for the bulk metric [3,13–22]. A‘‘black hole in the bulk’’ scenario provides us with a novelway of visualizing cosmological phenomena on the 4D

universe. In this scenario, the brane is moving in thebulk, with its radial trajectory being identified with thescale factor of the 4D world, so that the expansion of theuniverse is a realization of the radial trajectory of the branein the bulk.The most notable contribution from bulk geometry on

the brane is, perhaps, an additional term in the Friedmannequations, which arise from the projection of the bulkWeyltensor onto the brane. The precise role of this term, com-patible to FRW background on the brane, is to supply ageometric perfect fluid whose nature is governed by thecontents of the bulk (in turn, bulk geometry) we choose.For an empty bulk, it is radiationlike and is called the darkradiation. There is extensive study in the literature eitherby setting it to zero for practical purposes or by attributinga very small value to it, constrained by nucleosynthesisdata (< 3% of total radiation energy density of the uni-verse) [2,23]. Examples include metric-based perturba-tions [24], density perturbations on large scalesneglecting dark radiation [25], or including its effects[26], curvature perturbations [27] and the Sachs-Wolfeeffect [28], vector perturbations [29], tensor perturbations[30], and cosmic microwave background (CMB) anisotro-pies [31]. In all the cases, the effect has been found tomodify the standard analysis very little, as expected fromits radiationlike behavior.On the contrary, when the bulk is not necessarily empty,

the nature of this entity is no longer radiationlike, rather itdepends upon the contents in the bulk, which is reflected bythe VAdS5 bulk geometry [3,13–22,32,33]. It is thus im-portant to determine the nature as well as the role of thisentity, called in general the Weyl fluid, in the cosmologicaldynamics and perturbations, and find if this scenario hassome advantages over others. A recent work [21] hasshown its significance as a possible dark matter candidateby Newtonian analysis of perturbations, followed by someconfrontation with observations [34]. However, as in GR,the Newtonian analysis of gravitational instability is lim-ited in the sense that it fails to account for the perturbationson scales larger than the Hubble radius. One needs relativ-*supratik_v@isical.ac.in

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istic analysis valid at super-Hubble scales as well. Further,in order to test the braneworld scenario observationally, weneed a complete description of the evolution of densityperturbations in the most general brane cosmological sce-nario provided by this VAdS5 bulk. With these motiva-tions, we develop here a technique for relativistic densityperturbations valid for this generalized brane cosmology,which will act as a natural extension of the covariantperturbations of the general relativistic framework [35] tobraneworld scenario. We further show in the subsequentdiscussions that the Weyl fluid can play a crucial role inlate-time cosmologies as a geometric candidate for darkmatter albeit its actual material existence.

II. BRANE DYNAMICS WITH WEYL FLUID

As mentioned, we shall concentrate on the most generalbulk scenario, for which the bulk geometry is given by aVaidya–anti-de Sitter metric

dS25 ¼ �fðr; vÞdv2 þ 2drdvþ r2d�23 (2.1)

where �3 is the 3-space having flat, spherical, or hyper-

boloidal symmetry, fðr; vÞ ¼ k� �5

6 r2 � mðvÞ

r2, andmðvÞ is

the resultant of the variable mass of the Vaidya black holeand radiation field. This type of bulk can exchange energywith the brane as a null flow along the radial direction[3,13–22]. Consequently, the brane matter conservationequation is modified to

_�þ 3_a

að�þ pÞ ¼ �2 (2.2)

where is the null flow characterizing the VAdS5 bulk bythe radiation field of a null dust Tbulk

AB ¼ qAqB, whichleads to the above equation by using

r�T�� ¼ �2TbulkAB n

AgB� (2.3)

(where nA are the normals to the surface), which gives

r�T�� ¼ �2 u� (2.4)

(u� are the unit velocity vectors), and readily leads to

Eq. (2.2). This modified conservation equation, with thehelp of the Bianchi identity on the brane r�G�� ¼ 0,

leads to another constraint equation

r�E�� ¼ 6�2

�r�S�� þ 2

3

��25

�_ þ 3

_a

a

�� 3�2

�u�

þ 2

3�25~r� (2.5)

where � is the brane tension and E�� and S�� are, respec-tively, the projected bulk Weyl tensor and the quadraticcontribution from the brane energy-momentum tensor tothe Einstein equation on the brane [21]. The above equa-tion governs the evolution of the Weyl fluid �� (so namedsince it is a fluidlike contribution from the bulkWeyl tensorto the brane). For FRW geometry on the brane, this is given

by

_� � þ 4_a

a�� ¼ 2 � 2

3

��5

�

�2�_ þ 3

_a

a

�(2.6)

so that this quantity evolves as [16,21]

�� ¼ Cð�Þa4

/ 1

að4��Þ(2.7)

which gives a general, physically relevant behavior for theWeyl fluid. Here, � is the proper time on the brane.Obviously, contrary to the Sch-AdS5 bulk, here Cð�Þ isevolving, and consequently, the Weyl fluid no longer be-haves like radiation. To a brane-based observer, the cos-mological dynamics is now governed by an effectiveperfect fluid, the components of which are given by [15,21]

�eff ¼ �þ �2

2�þ Cð�Þ

a4(2.8)

peff ¼ pþ �

2�ð�þ 2pÞ þ Cð�Þ

3a4: (2.9)

The anisotropic components of the Weyl fluid, viz., q�� and

���� vanish, in order that the VAdS5 bulk be compatible to

FRW geometry on the brane. The Friedmann equation andthe covariant Raychaudhuri equation, expressed in terms ofthese effective quantities, are respectively [22]

H2 ¼ �24

3�eff þ�

3� k

a2(2.10)

_H ¼ ��24

2ð�eff þ peffÞ þ k

a2� �2

5

3 : (2.11)

In the Newtonian analysis of perturbations on the brane,one considers small fluctuations of the effective density�effð ~x; �Þ ¼ ��effð�Þð1þ effð ~x; �ÞÞ and the effective gravi-tational potential �effð ~x; �Þ ¼ �eff

0 þeff on the hydro-

dynamic equations for this effective perfect fluid andobtains for a barotropic fluid a single second-order equa-tion in terms of Fourier mode

d2effk

d�2þ 2

_a

a

deffk

d��

�4�G ��eff �

�c2effs k

a

�2�effk ¼ 0

(2.12)

where c2effs is the square of the effective sound speed[2,21]. The above perturbation equation of the effectivefluid can account for the required amount of gravitationalinstability if the Weyl density redshifts more slowly thanbaryonic matter density, so that it can dominate over matterat late times, which is realized when 1<�< 4 inEq. (2.7). Now, for late-time behavior, we can drop thequadratic terms in Eqs. (2.8) and (2.9) so that the effective

density is given by �eff ¼ �ðbÞ þ �� which is now consti-

tuted of the usual matter (baryonic) density �ðbÞ and anadditional density contribution from the Weyl fluid. This

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Weyl density, being geometric, is essentially nonbaryonic.Consequently, we can decompose Eq. (2.12) to get theindividual evolution equations for the perturbation foreach of the fluids

d2ðbÞ

d�2þ 2

_a

a

dðbÞ

d�¼ 4�G ��ðbÞðbÞ þ 4�G ���� (2.13)

d2�

d�2þ 2

_a

a

d�

d�¼ 4�G ���� þ 4�G ��ðbÞðbÞ (2.14)

where ðbÞ and � are the fluctuations of baryonic matter

and Weyl fluid, respectively. With�ðbÞ � ��, the relevantgrowing mode solutions are given by [21]

�ðzÞ ¼ �ð0Þð1þ zÞ�1 (2.15)

ðbÞðzÞ ¼ �ðzÞ�1� 1þ z

1þ zN

�(2.16)

with the input that the late-time behavior of the expansionof the universe in Randall-Sundrum-II braneworlds (RS II)is the same as the standard cosmological solution for thescale factor [13,36] where the scale factor is related to theredshift function by a / ð1þ zÞ�1.

The solutions reveal that at a redshift close to zN, the

baryonic fluctuation ðbÞ almost vanishes but the Weylfluctuation � still remains finite. So, even if the baryonicfluctuation is very small at a redshift of zN � 1000, asconfirmed by CMB data [4], the fluctuations of the Weylfluid still had a finite amplitude during that time, whereasat a redshift much less than zN the baryonic matter fluctua-tions are of equal amplitude as the Weyl fluid fluctuations.This is precisely what is required to explain the formationof structures we see today. Thus, the Newtonian analysis ofperturbations on the brane is capable of explaining struc-ture formation (within its limit) by Weyl fluid, devoid ofany material existence of dark matter. Hence, the Weylfluid acts as a possible geometric candidate for dark matter.

III. RELATIVISTIC PERTURBATIONS WITHWEYL FLUID

The Newtonian analysis depicted so far turns out to bean useful tool to study perturbations on the brane after theuniverse enters the Hubble length. A more complete pic-ture can be obtained if one studies relativistic analysis ofperturbations, which include the evolution of the universeat the super-Hubble scale as well. In this section we shalldevelop a multifluid perturbative technique in order todiscuss relativistic perturbation relevant for brane cosmol-ogy. This will be carried through in the subsequent sectionsfor the purpose of analysis for different braneworld scenar-ios. Our basic motivation in the attempt to develop a multi-fluid perturbative technique is governed by the realizationobtained from Newtonian analysis that, contrary to theSch-AdS5 bulk scenario, the Weyl fluid may not be so

insignificant as to neglect its effects at late time, if wehave a general bulk geometry. Consequently, in a generalbrane cosmological scenario, along with baryonic matter,the universe consists of a considerable amount of Weylfluid as well.Before going into the details, let us jot down here the

major points in addressing relativistic perturbations on thebrane.(i) Here the cosmological dynamics is governed by a

two-fluid system. One of the components of the

system is a material fluid �ðbÞ—the baryonic mattercontent on the brane. The second component is ageometric fluid ��—the Weyl fluid. The total (effec-tive) density for the system on the brane is given by

�eff ¼ �ðbÞ þ ��.(ii) Though there are two components of the effective

fluid, the Weyl fluid being a geometric entity, thereis a single material fluid in the analysis. As a result,there will be no entropy perturbation as such.

(iii) For the same reason, there is no peculiar velocityfor the Weyl component, leading to v� ¼ 0.

(iv) The anisotropic components of the Weyl fluid beingabsent so as to fit it into an FRW background, wewill set q�� ¼ 0 ¼ ��

�� right from the beginning.

As a result, each component of the two-fluid systembehaves individually like a perfect fluid, resultingin a perfect fluid behavior for the effective fluid as awhole.

(v) These two fluids interact and exchange energy be-tween them, which is governed via the bulk-braneenergy exchange and the backreaction of the systemon the brane.

(vi) Since there is energy exchange between these twofluids, the conservation equation for each individualis now modified. These modified forms of the con-servation equations have been explained inEqs. (2.2) and (2.6).

Because of the interaction between the two fluid com-ponents, each of the two modified conservation equationscan be written in terms of the contribution from the inter-action as

_� ðiÞ þ�ð�ðiÞ þ pðiÞÞ ¼ IðiÞ (3.1)

where � ¼ 3 _aa is the volume expansion rate, a superscript

ðiÞ denotes the quantities for the i-th fluid, and IðiÞ is thecorresponding interaction term. It readily follows fromEqs. (2.2) and (2.6) that the interaction terms, when writtenexplicitly, are given by

IðbÞ ¼ �2 (3.2)

I� ¼ 2 � 2

3

��5

�

�2�_ þ 3

_a

a

�: (3.3)

For relativistic perturbations, we express the densities ofeach of the contributing fluid components in terms of

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dimensionless parameters as

��ðbÞ ¼ �2�ðbÞ

3H20

; ��� ¼ �2��

3H20

(3.4)

with the first one for baryonic matter and the second onefor Weyl fluid. Considering the nature of the Weyl fluid asdiscussed in the previous section, we find that the densityparameter for the Weyl fluid is given by

��� ¼ 2C0

a4��0 H20

(3.5)

where C0 is the onset value for the Weyl parameter Cð�Þ.For completion, we mention here that there can, in

principle, appear two more dimensionless parameters,one each for the cosmological constant and the branetension arising in the brane cosmological context. They are

�� ¼ �

3H20

; �� ¼ �2�20

6�H20

(3.6)

with the total density satisfying the critical value

�tot ¼Xi

�i ¼ ��ðbÞ þ��� þ�� þ�� ¼ 1: (3.7)

Here, and throughout the rest of this article, we haveconsidered a spatially flat universe with k ¼ 0. InEq. (3.6), the first one is relevant if one considers cosmo-logical constant in this brane universe while studying theexpansion history of the universe whereas the one due tothe brane tension is relevant in the high-energy early uni-verse (inflationary) phase but is negligible for low-energylate-time phenomena such as structure formation. Thus thebaryonic density and the Weyl density are the only tworelevant contributions in the scenario being discussed here.In what follows we shall restrict ourselves to the discussionof the Einstein–de Sitter brane universe for which�� ¼ 0leading to �tot � ��ðbÞ þ��� ¼ 1.

We now express the comoving fractional gradients of theeffective density and expansion relevant in the brane cos-mology as

�ðiÞ� ¼ a

�ðiÞD��ðiÞ (3.8)

Zeff� ¼ aD�� (3.9)

�eff� ¼ a

�effD��

eff : (3.10)

As already discussed, both baryonic matter and Weylfluid behave individually as perfect fluid components,which means the effective flux arising from the peculiarvelocities of each component vanish to zero order, con-firming that the perturbations considered here are gauge-invariant at the first order.

With the above notations, the linearized evolution equa-tion for the density perturbations in the braneworld isobtained by taking spatial gradient of the modified conser-vation equations. After linearization, it turns out to be

_�ðiÞ� ¼

�3HwðiÞ � IðiÞ

�ðiÞ

��ðiÞ� � ð1þ wðiÞÞZeff

�

� c2effs IðiÞ

�ðiÞð1þ weffÞ�eff� � 3aHIðiÞ�

�ðiÞ þ a

�ðiÞD�IðiÞ

(3.11)

where wðiÞ ¼ pðiÞ=�ðiÞ is the equation of state for i-th fluid

and c2ðiÞs ¼ _pðiÞ= _�ðiÞ is the sound speed squared for thatspecies, with the corresponding quantities for the effectivetotal fluid being, respectively,

weff ¼ 1

�eff

Xi

�ðiÞwðiÞ (3.12)

c2effs ¼ 1

�effð1þ weffÞXi

c2ðiÞs �ðiÞð1þ wðiÞÞ: (3.13)

In the relativistic perturbations, contrary to theNewtonian analysis, we further have an evolution equationfor the effective expansion gradient, which depends on theeffective fluid. This is obtained by taking spatial gradientof the modified Raychaudhuri Eq. (2.11) and is given in thebraneworld scenario by

_Z eff� þ 2HZeff

� ¼ ��2

2�eff�eff � c2effs

1þ weffD�D

��eff�

þ �25

1þ weffc2effs �eff

� � a�25D� :

(3.14)

It should be mentioned here that since the evolution ofthe expansion gradient is dependent on the curvature per-turbations, the latter should not remain strictly constant inthis multifluid perturbation scenario. However, thoughthere is a significant energy exchange between brane mat-ter and Weyl fluid at early times, we shall see from the nextsection that the energy exchange between the two fluids isalmost in equilibrium at late times, so that the local curva-ture perturbations can safely be considered to be constantfor all practical purpose. One should, however, consider thevariation of this term while analyzing the inflationaryphase, for instance. We follow this argument right fromhere in order to avoid mathematical complicacy.As in GR, we find that while discussing perturbations in

brane cosmology, it is advantageous to express the aboveequations in terms of covariant quantities. These densityperturbations are governed by the fluctuation of the follow-ing covariant projections:

�ðiÞ ¼ aD��ðiÞ� (3.15)

�eff ¼ aD��eff� (3.16)

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Zeff ¼ aD�Zeff� : (3.17)

Consequently, the covariant density perturbation equa-tion and expansion gradient on the brane, when expressedin terms of the above covariant quantities, are obtainedstraightaway from Eqs. (3.11) and (3.14). They are givenby

_� ðiÞ ¼�3HwðiÞ � IðiÞ

�ðiÞ

��ðiÞ � ð1þ wðiÞÞZeff

� c2effs IðiÞ

�ðiÞð1þ weffÞ�eff � 3a2HD�IðiÞ�

�ðiÞ þ a2

�ðiÞD2IðiÞ

(3.18)

_Z eff þ 2HZeff ¼ ��2

2�eff�eff � ac2effs

1þ weffD2�eff

þ �25

1þ weffc2effs �eff � a2�2

5D2 :

(3.19)

In deriving the above covariant perturbation equations, wehave considered those kinds of perturbations for which,like the unperturbed Weyl fluid, the anisotropic stressesand fluxes for the perturbed Weyl fluid are also vanishing.

These sets of equations provide the key informationabout the perturbation in brane cosmology. In the subse-quent section, we shall try to analyze these relativisticperturbation equations and obtain possible consequences.

IV. SOLUTIONS AND ANALYSIS

A. Empty bulk: Noninteracting fluids

Let us now discuss the special scenario when the bulk isempty for which the VAdS5 bulk reduces to Sch-AdS5. Inthis case, there is no question of energy exchange betweenthe brane and the bulk. Consequently, there is no interac-tion between brane matter and Weyl fluid as such, whichreveals from Eq. (3.1) the fact that the individual conser-vation equation for each of the components is preserved.Thus, the Weyl fluid evolves in this case as

�� / a�4 (4.1)

with the Weyl parameter � now being zero, so that forempty bulk, the Weyl fluid behaves like radiation, forwhich this is called dark radiation.

Since in this case, there is no interaction between the twocomponents of the effective fluid and, also, there is no nullflow from the bulk to the brane (or vice versa), we candrop the interaction terms and the terms involving in theanalysis. As a result, the covariant perturbation Eqs. (3.18)and (3.19) are vastly simplified. They are now given by

_� ðiÞ ¼ 3HwðiÞ�ðiÞ � ð1þ wðiÞÞZeff (4.2)

_Z eff þ 2HZeff ¼ ��2

2�eff�eff � ac2effs

1þ weffD2�eff : (4.3)

Taking the time derivative of the above two equationsand combining them, we obtain the evolution equations fordensity perturbations of the two fluids:

€� ðbÞ þ 2H _�ðbÞ ¼ �2

2�eff�eff (4.4)

€� � þ 2H _�� ¼ 4

3

�2

2�eff�eff þ ��

�2H2 � �2

2

�þH _��

(4.5)

where the first equation is for baryonic matter while thesecond one is for dark radiation.Recall that the amount of dark radiation is constrained

by the nucleosynthesis data to be at most 3% of the totalradiation density of the universe. So, it redshifts at a fasterrate than ordinary matter on the brane so that the matter onthe brane becomes dominant on the Weyl fluid at late time.Hence, it is expected that the dark radiation does not playany significant role in late-time cosmologies. It is obvious

from the fact that in this case,�ðbÞ � ��, which when putback into the above equations, leads to �ðbÞ � ��, so thatthe dark radiation fluctuation does not contribute substan-tially at late times. The Sch-AdS5 bulk scenario thus failsto explain structure formation with only baryonic matterand dark radiation. One needs cold dark matter in thetheory and the dark radiation can, at best, slightly modifythe standard perturbative analysis.

B. General bulk: Interacting Weyl fluid

The general scenario, however, is different from theempty bulk case since now the Weyl fluid exchangesenergy with brane matter through interactions and is thedominant contribution of the effective fluid in the pertur-bation equations. Here, the evolution of perturbations forthe individual fluids is governed by Eqs. (3.18) and (3.19),which now include the effects of the interaction terms aswell as the effect of null radiation through the term involv-ing . With these inclusions, the equations become a bittoo complicated and it is almost impossible to have ananalytical solution from these complicated equations.However, the equations turn out to be tractable if weincorporate certain simplifications following physical ar-guments, without losing any essential information as such.The simplifications we incorporate are as follows:(i) The null flow from the brane to the bulk is a

function of time only. This means that we are con-sidering only the time-evolution for the null radia-tion, at least on the brane, which is relevant for itslate-time behavior in perturbation analysis.

(ii) The energy exchange between the two fluids is inequilibrium, i.e., the energy received by the Weylfluid is the same as the energy released by brane

matter, so thatPiIðiÞ ¼ IðbÞ þ I� ¼ 0. Hence, no

extra energy is leaked to the bulk from the brane

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at late time (though at early time there may be someleakage of energy from the brane to the bulk). Thisbasically describes the late-time behavior, consis-tent with the fact that the standard evolution history(scale factor) is regained in this scenario at the‘‘matter-dominated’’ era [13,36].

Now, we have shown that in this generalized braneworldscenario, the Weyl fluid, in general, evolves as (refer toEquation (2.7))

�� ¼ C0a�ð4��Þ (4.6)

with the parameter� in the range 1<�< 4 so that it is thedominant contribution in the two-fluid system. The energyexchange between the components of the system being inequilibrium, we find from Eq. (2.7) that the Weyl fluid nowbehaves as

�� / a�3=2 (4.7)

with the parameter � ¼ 52 . This readily suggests that the

Weyl fluid actually redshifts more slowly than ordinarymatter and, hence, can dominate over matter at late times,reflecting one of the fundamental properties of dark matter.This also provides a more stringent bound for the value of� from theoretical ground alone (which was predictedfrom Newtonian analysis to fall within 1 to 4).

We now take the time derivative of the covariant pertur-bation Eqs. (3.18) and (3.19), and rearrange terms so as toobtain a single second-order differential equation for eachof the fluids. Thus, the equation describing evolution ofscalar perturbations of matter on the brane turns out to be

€�ðbÞ þ 2H _�ðbÞ ¼ �2

2�eff�eff � c2effs �2

5

1þ weff�eff

þ 4H

�ðbÞ

��ðbÞ þ c2effs �eff

1þ weff

�

þ�2

�ðbÞ

��ðbÞ þ c2effs �eff

1þ weff

���(4.8)

(note an overdot outside the square bracketed term of theabove equation) whereas the scalar perturbation equationfor the Weyl fluid on the brane is given by

€�� þ 2H _�� ¼ 4

3

�2

2�eff�eff

� c2effs �eff

1þ weff

�7H

�� þ 4�25

3þ 2 _

��

�

� c2effs_�eff

1þ weff

2

��

þ ���2H2 � �2

2� 7H

�� � 2 _

��

�

þ _���H � 2

�� ��25

3

�: (4.9)

Recall from the discussions following Eq. (4.7) that inthis scenario the Weyl fluid is the dominant component ofthe effective fluid. Consequently, the evolution equationfor the Weyl fluid at late times is radically simplified by

using �ðbÞ � �� since the Weyl fluid is now the dominantcontribution. With the energy exchange between the twofluids being in equilibrium, the expression for the null flowfurther simplifies the above equation so that it can now berecast in the following form:

€� � þ A

t_�� �

�B

tþ C

t2

��� ¼ 0 (4.10)

where the constants A, B, C are readily determined fromthe constraint equations. These constants are given by

A ¼ 2

3þ 5

2

� 0

��0

��2

3

a0H0

�2=3

(4.11)

B ¼ 2

3�2�0

�2

3

a0H0

�3=2 þ

�1þ �2

6��0

��2

3

a0H0

�2=3

(4.12)

C ¼ A

4� 19

18: (4.13)

The above Eq. (4.10) for �� turns out to be somewhattractable. One of its solutions is given by

�� � tð1=2Þ�ðA=2ÞBesselI½ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2Aþ A2 þ 4C

p; 2

ffiffiffiffiB

p ffiffit

p �:(4.14)

The above solution, consisting of a Bessel function, isfound to be a growing function. Therefore, the evolutionequation for the Weyl fluid, indeed, shows a growing modesolution, which is required to explain the growth of pertur-bations at late times. Thus, the relativistic perturbationtheory relevant in brane cosmology gives rise to a fluidwhich is very different from dark matter in origin andnature but has the potentiality to play the role of darkmatter in a cosmological context. It is worthwhile to notethat, to a brane-based observer, the nature of the Weyl fluidis determined from bulk geometry arising from the radia-tion flow in the bulk. That is why the Weyl fluid can betreated as a geometric candidate for dark matter.Figure 1 depicts a qualitative behavior of the growth of

Weyl fluid perturbations with time. The figure once againshows that the evolution of perturbations of Weyl fluid isvery different from cold dark matter (CDM), which makesthe theory distinct from standard analysis with CDM. We,however, note that since the dynamics here is completelydifferent from the standard one involving CDM, one can-not comment conclusively on the merits/demerits of Weylfluid over CDM right from here. One has to reformulateand estimate different cosmological parameters in thiscontext and confront them with observations for a moreconclusive comment. For example, the relation of thetransfer function with the potential will now be replaced

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by a novel relation with the effective potential discussed inthe brane cosmological context [21]. As a result, the varia-tion of the growth function with the scale factor may not bethe same as usually needed in the standard cosmologicalparadigm. It is to be seen if this analysis of perturbationwith the Weyl fluid fits in this new brane cosmologicalframework, which is not a trivial exercise, we suppose. Theinterested reader may further refer to [37] for an overallview on how different cosmological parameters are devel-oped in a specific theoretical framework.

However, even at this stage, our model does show someagreement with observational results. From the recentstudies on confronting braneworld models with observa-tions [34] by obtaining the luminosity distance for FRWbranes with the Weyl fluid, it is found that a certain amountof Weyl fluid with 2 � 3 is in nice agreement withsupernovae data. From the relativistic perturbations dis-cussed in this article we have found a specific value for �,namely � ¼ 5

2 , which falls within this region. Thus the

braneworld model of perturbations fits well in this obser-vational scenario. We hope an extensive study in thisdirection will lead to more interesting results to make amore conclusive remark.

V. SUMMARYAND OPEN ISSUES

In this article, we have developed a technique for rela-tivistic perturbations valid for a general brane cosmologi-cal scenario. The essential distinction of our analysis fromthe studies on brane cosmological perturbations available

in the literature is that here the geometrical effect of thebulk on the brane—the so-called Weyl fluid—plays a verycrucial role in determining the nature of the evolution ofdensity perturbations. This is materialized from the real-ization that in the general brane cosmological scenarioobtained from Vaidya–anti-de Sitter bulk, the Weyl fluidplays a significant role in controlling the dynamics on thebrane, contrary to the earlier results based on dark radia-tion. Our results are, in a sense, a generalization of themultifluid covariant perturbation formalism in a branecosmological framework. Further, we have solved the per-turbation equations and found that the perturbation of theWeyl fluid grows at late time, and thus, this component ofbraneworld gravity plays a significant role in late-timecosmology to act as a possible geometric candidate fordark matter. We have discussed some of the implications offluctuations involving it and have mentioned some observ-able sides of this model as well.An important issue is to fit this theoretical model with

current observational data. Recently there has been someprogress in this direction [34]. An extensive study onconfronting this braneworld model with observations in amore rigorous method can provide us with necessary in-formation on the merits and demerits of the formalism. Tothis end, a thorough study of different parameters related tocosmological perturbation is to be performed. As men-tioned in the previous section, the different cosmologicalparameters need to be reformulated in this framework. Thenext step is to estimate them and confront them withobservations. For example, it is to be seen if the powerspectrum, redefined in this paradigm with the Weyl fluidacting as a dark matter candidate, fits with the highlyaccurate observational data.Further, analysis of different types of metric-based per-

turbations, namely, scalar, vector, and tensor as well asrelated issues like CMB anisotropy, Sachs-Wolfe effect,etc., has to be studied in detail in this brane cosmologicalframework with a significant Weyl fluid. An extensivestudy in this direction is essential, which we hope toaddress in the near future. Also, to apply this formalismin the framework of braneworld models of dark energy [38]remains another interesting issue.

ACKNOWLEDGMENTS

The author thanks Biswajit Pandey, Ratna Koley andVarun Sahni for stimulating discussions.

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