Anisotropy in Bianchi-type brane cosmologies

0.22 MB·

by  T. Harko

gr-qc/0401069
Anisotropy in Bianchi-type brane cosmologies
T. Harko and M. K. Mak
∗ †
Department of Physics, The University of Hong Kong, Pokfulam Road, Hong Kong
(Dated: January 16, 2004)
Thebehaviorneartheinitialstateoftheanisotropyparameterofthearbitrarytype,homogeneous
and anisotropic Bianchi models is considered in the framework of the brane world cosmological
models. The matter content on the brane is assumed to be an isotropic perfect cosmological fluid
obeying a barotropic equation of state. To obtain the value of the anisotropy parameter at an
arbitrary moment an evolution equation is derived, describing the dynamics of the anisotropy as a
function of the volume scale factor of the Universe. The general solution of this equation can be
obtainedinanexactanalyticalformfortheBianchiIandVtypesandinaclosedformforallother
homogeneous and anisotropic geometries. The study of the values of the anisotropy in the limit of
small timesshowsthat forall Bianchitypespace-timesfilled with anon-zeropressurecosmological
fluid,obeyingalinearbarotropicequationofstate,theinitialsingularstateonthebraneisisotropic.
4
This result is obtained by assuming that in thelimit of small times theasymptotic behaviorof the
0
scalefactorsisofKasner-type. Forbraneworldsfilledwithdust,theinitialvaluesoftheanisotropy
0
coincide in both braneworld and standard four-dimensional general relativistic cosmologies.
2

n PACSnumbers: 04.20.Jb,04.65.+e,98.80.-k
a
J

6
1
I. INTRODUCTION

1
v
The idea [1] that our four-dimensional Universe might be a three-brane, embedded in a higher dimensional space-
9
time,hasattractedmuchattention. Accordingtothebrane-worldscenario,thephysicalfieldsinourfour-dimensional
6
0 space-time,whichareassumedtoariseasfluctuationsofbranesinstringtheories,areconfinedtothethreebrane. Only
1 gravity can freely propagate in the bulk space-time, with the gravitational self-couplings not significantly modified.
0 This model originated from the study of a single 3-brane embedded in five dimensions, with the 5D metric given
4 by ds2 = e f(y)η dxµdxν +dy2, which, due to the appearance of the warp factor, could produce a large hierarchy
− µν
0
betweenthescaleofparticlephysicsandgravity. Evenifthefifthdimensionisuncompactified,standard4D gravityis
/
c reproducedonthebrane. Hencethismodelallowsthepresenceoflarge,oreveninfinitenon-compactextradimensions.
q Our brane is identified to a domain wall in a 5-dimensional anti-de Sitter space-time.
- The Randall-Sundrum model was inspired by superstring theory. The ten-dimensional E E heterotic string
r 8 8
×
g theory, which contains the standardmodel of elementary particle, could be a promising candidate for the description
v: of the real Universe. This theory is connected with an eleven-dimensional theory, M-theory, compactified on the
i orbifold R10 S1/Z2 [2]. In this model we have two separated ten-dimensional manifolds. For a review of dynamics
X ×
and geometry of brane Universes see [3].
r The static Randall-Sundrum solution has been extended to time-dependent solutions and their cosmologicalprop-
a
erties have been extensively studied [4]-[37]. In one of the first cosmological applications of this scenario, it was
pointed out that a model with a non-compact fifth dimension is potentially viable, while the scenario which might
solve the hierarchy problem predicts a contracting Universe, leading to a variety of cosmological problems [12]. By
adding cosmological constants to the brane and bulk, the problem of the correct behavior of the Hubble parameter
on the brane has been solved by Cline, Grojean and Servant [13]. As a result one also obtains normal expansion
during nucleosynthesis, but faster than normal expansion in the very early Universe. The creation of a spherically
symmetric brane-worldin AdS bulk has been considered,from a quantum cosmologicalpoint ofview, with the use of
the Wheeler-de Witt equation, by Anchordoqui, Nunez and Olsen [14].
The effective gravitational field equations on the brane world, in which all the matter forces except gravity are
confined on the 3-brane in a 5-dimensional space-time with Z -symmetry have been obtained, by using a geometric
2
approach, by Shiromizu, Maeda and Sasaki [15, 16]. The correct signature for gravity is provided by the brane with
positive tension. If the bulk space-time is exactly anti-de Sitter, generically the matter on the brane is requiredto be
spatiallyhomogeneous. The contractionofthe 5-dimensionalWeyltensorwiththe normaltothe braneE givesthe
IJ
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
2
leading order corrections to the conventional Einstein equations on the brane. The four-dimensional field equations
for the induced metric and scalar field on the world-volume of a 3-brane in the five-dimensional bulk, with Einstein
gravity plus a self-interacting scalar field, have been derived by Maeda and Wands [17].
Realistic brane-worldcosmologicalmodels require the considerationof more generalmatter sourcesto describe the
evolutionanddynamics ofthe veryearlyUniverse. The influence ofthe bulk viscosityofthe matter onthe brane has
been considered, for an isotropic flat Friedmann-Robertson-Walker (FRW) geometry, in [20] and, for a Bianchi type
I geometry, in [21]. The first order rotationalperturbations of isotropic FRW cosmologicalmodels have been studied
in [22] and [23].
ThegeneralsolutionofthefieldequationsforananisotropicbranewithBianchitypeIandVgeometry,withperfect
fluid and scalar fields as matter sources, has been obtained in [27]. Expanding Bianchi type I and V brane-worlds
always isotropize. Anisotropic Bianchi type I brane-worlds with a pure magnetic field and a perfect fluid have also
been analyzed [32]. Limits on the initial anisotropy induced by the 5-dimensional Kaluza-Klein graviton stresses by
usingtheCMBanisotropieshavebeenobtainedbyBarrowandMaartens[33]. Thedynamicsofaflat,isotropicbrane
Universe with two-component matter source: a perfect fluid with a linear barotropic equation of state and a scalar
fieldwithapower-lawpotentialhasbeeninvestigatedin[34]. Solutionsforwhichthescalarfieldenergydensityscales
asapower-lawofthescalefactor(socalledscalingsolutions)havebeenobtainedandtheirstabilityanalysisprovided.
A family of Bianchitype braneworldswith anisotropyhas been constructed in [35], by solving the five-dimensional
field equations in the bulk. The cosmological dynamics on the brane has been analyzed by also including the Weyl
term,andtherelationbetweentheanisotropyonthebraneandtheWeylcurvatureinthebulkhasbeendiscussed. In
thesemodels,itisnotpossibletoachievegeometricanisotropyforaperfectfluidorscalarfield–thejunctionconditions
require anisotropic stresses on the brane. But in an anti-de Sitter bulk, the solutions can isotropize and approach
a Friedmann type brane. Bianchi I type brane cosmologies with scalar matter self-interacting through combinations
of exponential potentials have been studied in [36]. Such models correspond in some cases to inflationary Universes.
In the brane scenario, as happens in standard four-dimensional general relativity, an increase in the number of fields
assists inflation. The asymptotic behavior of Bianchi I brane worlds was considered in [37]. As a consequence of the
nonlocal anisotropic stresses induced by the bulk, in the limit in which the mean radius goes to infinity the brane
does not isotropizeandthe nonlocalenergydoes notvanish. The inflationdue to the cosmologicalconstantmight be
prevented by the interaction with the bulk.
Thestudyofanisotropichomogeneousbraneworldcosmologicalmodels[24]-[30]hasshownanimportantdifference
between these models and standard four-dimensional general relativity, namely, that brane Universes are born in an
isotropicstate. ForBianchitype I andV geometriesthis type ofbehaviorhas beenfound bothby exactly solvingthe
gravitationalfieldequations[27],orfromthequalitativeanalysisofthemodel[26]. Ageneralanalysisoftheanisotropy
in spatially homogeneous brane world cosmological models has been performed by Coley [28], who has shown that
the initial singularity is isotropic,and hence the initial conditions problem is naturally solved. Consequently,close to
the initial singularity, these models do not exhibit Mixmaster or chaotic-like behavior [29]. Based on the results of
the study of homogeneousanisotropiccosmologicalmodels Coley [29] conjecturedthat the isotropic singularitycould
be ageneralfeature ofbranecosmologies. This conclusionhas been contestedas aresultofa perturbativeanalysisof
the dimensionless shear [38]. The existence of a decaying mode in scalar perturbations that grows unbounded in the
past seems to suggest that anisotropy also grows unbounded in the limit of small times [38]. This result is based on
including,throughperturbations,genericinhomogeneitiesinthe cosmologicalmodelandtheseareresponsibleforthe
unboundedgrowthoftheanisotropynearthesingularity. However,inaqualitativenumericalstudyoftheasymptotic
dynamicalevolutionofspatiallyinhomogeneousbrane-worldcosmologicalmodelsclosetotheinitialsingularity,Coley,
He and Lim [39] have shown that spatially inhomogeneous G brane cosmological models with one spatial degree of
2
freedom always have an initial singularity, which is characterizedby the fact that spatial derivatives are dynamically
negligible. From the numerical analysis they have also found that there is an initial isotropic singularity in all of
these spatially inhomogeneous brane cosmologies, including the physically important cases of radiation and a scalar
field source. The numerical studies indicate that the singularity is isotropic for all relevant initial conditions. A
similar result has been obtained by using the covariant and gauge-invariant approach for the analysis of the linear
perturbations of the isotropic model , which is a past attractor in the phase space of homogeneousBianchi models
b
F
onthebrane[40]. ThereforeonecanconcludethatbraneUniversesarebornwithnaturallybuilt-inisotropy,contrary
to standard four-dimensional general relativistic cosmology [41]. The observed large-scale homogeneity and isotropy
of the Universe can therefore be explained as a consequence of the initial conditions.
Onthe otherhandmostofthe studiesregardingthe behaviorofthe anisotropyinbraneworldcosmologicalmodels
have been done at a qualitative level, and have not provided explicit exact representations for the anisotropy. Also
many observationally important questions, like, for example, the maximum value of the anisotropy or the moment
in the evolution of the Universe when this maximum occurred have not been answered yet. It is also not clear for
what type of equation of state or range of parameters of the cosmological fluid on the brane the initial singularity is
isotropic, or what is the effect of an inflationary phase on the initial anisotropy.
3
It is the purpose of the present paper to consider the evolution and dynamics of the anisotropy of homogeneous
anisotropic(Bianchitype)braneworldcosmologicalmodelsinasystematicway. Asafirststepanevolutionequation
for the anisotropy parameter (describing differences in the time expansion of the Universe along the three principal
axis) is derived. From mathematical point of view it is a separable first order differential equation for Bianchi types
I and V, and a Bernoulli type equation for the other Bianchi types. By integrating the evolution equation, one can,
generally, obtain the anisotropy parameter as a function of the energy density and the volume scale factor of the
Universe. The study of the behavior of the anisotropy parameter near the singular state shows that generally the
initialvalueofthisparameterisdependentontheequationofstateofthecosmicmatter. Ahighdensitycosmological
fluid obeying a barotropic equation of state starts its evolution on the brane from an isotropic geometry, while the
expansion of a pressureless dust, in an anisotropic space-time, is similar to the standard four-dimensional general
relativistic one. But for inflationary models the behavior of the anisotropy parameter is similar in both brane world
cosmologicalmodels and standard four-dimensional general relativity.
Thepresentpaperisorganizedasfollows. Thebasicequationsofthebraneworldcosmologicalmodelsarepresented
in Section II. The evolution of the anisotropy of Bianchi type I and V models is considered in Section III. In Section
IV the anisotropy of arbitrary Bianchi type models is analyzed. Finally, in Section V, we discuss and conclude our
results.
II. GRAVITATIONAL FIELD EQUATIONS IN THE BRANE WORLD MODEL
On the 5-dimensional space-time (the bulk), with the negative vacuum energy Λ and brane energy-momentum as
5
source of the gravitationalfield, the Einstein field equations are given by
G =k2T , T = Λ g +δ(Y) λg +Tmatter , (1)
IJ 5 IJ IJ − 5 IJ − IJ IJ
In this space-time a brane is a fixed point ofthe Z symmetry. In the(cid:2)followingcapital(cid:3)Latinindices run inthe range
2
0,...,4, while Greek indices take the values 0,...,3.
Assuming a metric of the form ds2 = (n n +g )dxIdxJ, with n dxI = dχ the unit normal to the χ = const.
I J IJ I
hypersurfaces and g the induced metric on χ = const. hypersurfaces, the effective four-dimensional gravitational
IJ
equations on the brane (the Gauss equation), take the form [15, 16]:
G = Λg +k2T +k4S E , (2)
µν − µν 4 µν 5 µν − µν
where S is the local quadratic energy-momentum correction,
µν
1 1 1
S = TT T αT + g 3TαβT T2 , (3)
µν µν µ να µν αβ
12 − 4 24 −
(cid:0) (cid:1)
and E is the non-local effect from the free bulk gravitational field, the transmitted projection of the bulk Weyl
µν
tensor C , E =C nAnB, with the property E E δµδν as χ 0.
ThefoIuArJ-dBimenIJsionalIcAoJsBmologicalconstant,Λ,andthIeJco→upliµnνgcIonJstant,k ,→aregivenbyΛ=k2 Λ +k2λ2/6 /2
4 5 5 5
and k2 =k4λ/6, respectively, with λ the vacuum energy on the brane.
4 5 (cid:0) (cid:1)
The Einstein equation in the bulk and the Codazziequation, also imply the conservationof the energy momentum
tensor of the matter on the brane, D T ν = 0. Moreover, the contracted Bianchi identities on the brane imply that
ν µ
the projected Weyl tensor should obey the constraint D E ν =k4D S ν.
ν µ 5 ν µ
For any matter fields (scalar field, perfect or dissipative fluids, kinetic gases etc.) the general form of the brane
energy-momentum tensor can be covariantly given as [42]
T =ρu u +ph +π +2q u . (4)
µν µ ν µν µν (µ ν)
The decomposition is irreducible for any chosen 4-velocity uµ. Here ρ and p are the energy density and isotropic
pressure, and h = g +u u projects orthogonal to uµ. The energy flux obeys q = q , and the anisotropic
µν µν µ ν µ <µ>
stress obeys π =π , where angular brackets denote the projected, symmetric and trace-free part:
µν <µν>
1
V =h νV , W = h αh β hαβh W . (5)
<µ> µ ν <µν> (µ ν) − 3 µν αβ
(cid:20) (cid:21)
The symmetry properties of E imply that in general we can decompose it irreducibly with respect to a chosen
µν
4-velocity field uµ as [18]
1
E = k4 u u + h + +2 u , (6)
µν − U µ ν 3 µν Pµν Q(µ ν)
(cid:20) (cid:18) (cid:19) (cid:21)
4
where k = k /k , is a scalar, a spatial vector and a spatial, symmetric and trace-free tensor. For
5 4 µ µν
U Q P
homogeneous models = 0 and = 0 [25]. Hence the only non-zero contribution from the 5-dimensional
µ µν
Q P
Weyl tensor from the bulk is given by the scalar or ”dark radiation” term .
U
In the following we shall consider only homogeneous and anisotropic brane geometries. In particular, those which
havespace-likesurfacesofhomogeneity(i.e. aG actingsimplytransitivelyonaV ). Thecorrespondingcosmological
3 3
models fall into nine classes of equivalence, the so-called Bianchi models. It is useful to classify the nine types into
two disjoint groups,depending on the different properties of the isometry groups (the Lie groups),called class A and
class B.
At each point of the space-time on the brane we take a local orthonormaltetrad, where the metric can be written
as
3
ds2 = ω0 2+ ωi 2, (7)
−
i=1
(cid:0) (cid:1) X(cid:0) (cid:1)
where the differential one-forms ωµ will be taken as ω0 = dt and ωi = a (t)Ωi, where Ωi are time-independent
i
differential1-formsanda (t),i=1,2,3arethe cosmicscalefactors. The time independent1-formsobeythe relations
i
dΩi = 1ci Ωk Ωl, where the ci are the canonical structure constants and denotes the exterior product [43].
−2 kl ∧ kl ∧
Although the Ansatz for the metric givenby Eq. (7) is not the mostgeneralone that one can chose,it is sufficientto
display all the main features of the behavior of Bianchi geometries.
With the help of the scale factors one can define the following variables: V = 3 a (volume scale factor),
i=1 i
H = a˙ /a ,i = 1,2,3 (directional Hubble parameters), H = (1/3) 3 H (mean Hubble parameter) and ∆H =
i i i i=1 i Q i
H H, i = 1,2,3. By using the definitions of H and V we immediately obtain H = V˙/3V, where a dot denotes
i− P
the derivative with respect to the cosmologicaltime t.
According to the definition of the energy-momentum tensor on the brane, Eq. (4), in the general case of Bianchi
typegeometries,thesymmetryofthespace-timeallowsdifferentspatialcomponentsofT . Thereareseveralphysical
µν
processesthat could generateananisotropicenergy momentum tensor,with T1 =T2 =T3, like magnetic fields, heat
1 6 2 6 3
transfer and/or viscous dissipative processes in the cosmological fluid on the brane. The most important of these
processesarethebulkviscoustypedissipativeprocesses,whicharethemainsourcesofentropygenerationintheearly
Universe. However,the effect ofthe bulk viscosityofthe cosmologicalfluid onthe branecanbe consideredby adding
totheusualthermodynamicpressurepthebulkviscouspressureΠandformallysubstitutingthepressuretermsinthe
energy-momentum tensor by p =p+Π [21]. Therefore the consideration of the bulk viscosity of the cosmological
eff
fluiddoesnotleadtoananisotropicpressuredistribution. Theviscousdissipativeanisotropicstressπ ofthematter
µν
onthe branesatisfiestheevolutionequationτ hβhνπ˙ +π = 2ησ ηT (τ uν/2ηT) π [42],whereη isthe
2 α β µν αβ − αβ− 2 ;ν αβ
shearviscositycoefficient, τ2 =2ηβ2, with β2 the thermodynamiccoefficienhtfor the tensor dissipaitivecontributionto
the entropy density and σ is the shear tensor. Generally, it is assumed that the dissipative contribution from the
αβ
shear viscosity in the early Universe can be neglected, η 0 [42]. Consequently, in the followings we consider that
≈
the anisotropicstressesofthe matter onthe brane alsovanish, π 0. We suppose thatinEq. (4)the heattransfer
µν
≈
is zero, that is, we take q =0. All these approximationsare standardin the analysis of the physics of the very early
µ
Universe. Therefore in the following we assume that the pressure distribution of the cosmological fluid on the brane
is isotropic and the fluid pressure satisfies a barotropic equation of state of the form p=p(ρ).
For any homogeneous model the conservation equations of the energy density of the matter ρ on the brane and of
the dark radiation can be written as
U
ρ˙+3(ρ+p)H =0, (8)
˙ +4H =0, (9)
U U
leading to a general dependence of ρ and of V of the form
U
C
0 0
V = , = U , (10)
w U V4/3
where
dρ
w=exp (11)
ρ+p(ρ)
(cid:20)Z (cid:21)
and C 0 and 0 are constants of integration.
0 0
≥ U ≥
5
ThemodifiedEinsteingravitationalfieldequationsonthebranecanbewrittenintheformofthestandardEinstein
four-dimensional field equations,
G =k2T(eff), (12)
µν 4 µν
where T(eff) = Λg /k2 + T + k4S 1/k2E [18]. Then the effective total energy density, pressure,
µν − µν 4 µν 5 µν − 4 µν
anisotropic stress and energy flux for a perfect fluid are ρ(eff) = Λ/k2 + ρ(1+ρ/2λ) + 6/k4λ , p(eff) =
(cid:0) (cid:1) 4 4 U
p Λ/k2 + (ρ/2λ)(ρ+2p) + 2/k4λ , π(eff) = 6/k4λ and q(eff) = 6k4λ . Since for homogeneous
− 4 4 U µν 4 Pµν µ 4 Qµ (cid:0) (cid:1)
cosmological models = = 0, it follows that in the case of a perfect cosmological fluid there is a close anal-
µ µν
Q P (cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
ogy between the gravitational field equations on the brane and standard four-dimensional general relativity, with
the role of the standard energy density and pressure played by ρ(eff) and p(eff), respectively. The formal analogy
between standard four-dimensional general relativity and brane world cosmology allows the immediate extension of
the Collins-Hawking definition of isotropization of a cosmologicalmodel [44] to the case of brane Universes.
Hence, we say that a brane world cosmological model approaches isotropy if the following four conditions hold as
t : i) the Universe is expanding indefinitely and H > 0 ii) T(eff)00 > 0 and T(eff)0i/T(eff)00 0, i = 1,2,3.
→ ∞ →
T(eff)0i/T(eff)00 representsan averagevelocity of the matter onthe brane relative to the surfaces of homogeneity. If
this does not tend to zero, the Universe would not appear homogeneous or isotropic iii) the anisotropy in the locally
measuredHubble constantσ/H tends to zero,σ/H 0 andiv)the distortionpartofthe metric tends toa constant.
In condition iii) σ2 = (1/2)σ σµν represents the s→hear of the normals n . For a metric of the form ds2 = dt2
µν µ
−
exp(2α)[exp(2β)] ΩµΩν,whereΩµareone-formsthatarenotexactingeneral,αisatimedependentfunctionandβ
µν
isasymmetrictracelessmatrix,thesheartensorisdefinedasσ =[exp(β)]. [exp( β)] +[exp(β)]. [exp( β)]
µν λµ − λν λν − λµ
[44]. For Bianchi class A and B models the shear is given by σ2 =σ σµν/2=(1/2) 3 H2 3H2 .
µν i=1 i −
Spatially homogeneous models can be divided in three classes: those which have les(cid:16)s than the escape(cid:17)velocity (i.e.,
P
those whose rate of expansion is insufficient to prevent them from recollapsing), those which have just the escape
velocity and those which have more than the escape velocity [44]. Models of the third class do not tend, generally,to
isotropy. In fact the only types which can tend toward isotropy at arbitrarily large times are types I, V, VII and
0
VII . Fortype VII there isnononzeromeasuresetofthesemodelswhichtendstoisotropy[44]. The Bianchitypes
h h
that drive flat and open Universes away from isotropy in the Collins-Hawking sense are those of type VII.
As an indicator of the degree of anisotropy of a cosmological model one can take the mean anisotropy parameter,
defined according to [43]
1 3 ∆H 2
i
A= . (13)
3 H
i=1(cid:18) (cid:19)
X
For an isotropic cosmological model H = H = H = H and A 0. The anisotropy parameter is an important
1 2 3
≡
indicator of the behavior of anisotropic cosmological models, since in standard four-dimensional general relativity it
is finite even for singular states (for example, A = 2 for Kasner-type geometries [20]). The time evolution of A is a
good indicator of the dynamics of the anisotropy.
For a homogeneous brane world model filled with a perfect fluid satisfying a barotropic equation of state the
conditions i), ii) and iv) of the Collins-Hawking definition of isotropization are naturally satisfied. Hence, if we
consider only expanding brane worlds, H > 0 and condition i) holds. From the choice of the matter content, and
due to the symmetries of the energy-momentum tensor, we have T(eff)00 ρ(eff) > 0 and T(eff)0i 0. The choice
≡ ≡
of the geometry implies that condition iv) is also satisfied. Therefore, the value of the parameter σ/H is the main
indicator of the isotropic/anisotropic behavior of a brane world cosmological model. As one can see from Eq. (13)
the quantityσ/H isproportionaltothesquarerootoftheanisotropyparameter,σ/H √Aandso,accordingtothe
∼
Collins-Hawkingdefinition, if A 0, a brane worldcosmologicalmodel will isotropize in the large time limit t .
→ →∞
The formal mathematical similarity between standard four-dimensional general relativity and brane world theory
can be also used to extend the Hawking-Penrose singularity theorem [45] to brane cosmologies. From the definition
of the effective energy density ρ(eff) and of the effective pressure p(eff) it follows that for a linear barotropic fluid
with p = (γ 1)ρ, 1 γ 2, the effective energy-momentum tensor on the brane satisfies both the strong and
weak energy−condition≤s, wh≤ich can be expressed as T(eff)µν (1/2)gµνT(eff) u u 0 and T(eff)µνu u 0,
µ ν µ ν
− ≥ ≥
respectively, where u is an arbitrary timelike four-vector. The first of these conditions implies that the sum of the
µ
(cid:0) (cid:1)
local energy density and pressure is non-negative, ρ(eff)+p(eff) 0 and ρ(eff)+3p(eff) 0. The second condition
requiresthatthelocalenergydensitybenon-negativeineveryobse≥rver’srestframe,ρ(eff) ≥0andρ(eff)+p(eff) 0.
≥ ≥
ThereforetheBianchi-typebranespacetimesfilledwithaperfectlinearbarotropicfluidaresingular,sincetheysatisfy
the following conditions: a) R uµuν 0 for all timelike vectors uµ b) u R u uλuε =0 for a vectoruµ tangent
µν [µ ν]λε[σ ρ]
≥ 6
to some geodesic c) there are no closed time-like curves and d) either i) there is a closed trapped surface or ii) there
is a point p for which uµ <0 for all of the vectors uµ tangent to the past light cone of p [45].
;µ
6
III. EVOLUTION OF THE ANISOTROPY IN BIANCHI TYPE I AND V MODELS
For a better understanding of the dynamics of the anisotropy in the brane world we consider first in detail the
evolutionofthe meananisotropyparameterAinBianchitype I andV geometries. Fromaformalpointofviewthese
two geometries are described by the line element
ds2 =−dt2+a21(t)dx2+a22(t)e−2αxdy2+a23(t)e−2αxdz2. (14)
The metric for the Bianchi type I geometry formally corresponds to the case α = 0, while for the Bianchi type V
case we have α=1.
Tostudy the time behaviorofthe anisotropyparameterAnearthe initialsingularpointonthe braneweneedonly
the Einstein field equations involving the time derivatives of H ,i=1,2,3, and which are given by
i
1 d 2α2 k2 k4 1
(VH )=Λ+ + 4 (ρ p) 5ρp+ k4 U0 , i=1,2,3. (15)
V dt i V2/3 2 − − 12 3 V4/3
For α=0 we obtain the (ii),i=0 field equations for Bianchi type I geometry, while α=1 gives the Bianchi type
6
V equations on the brane world.
By summing Eqs. (15) we find
1 d 2α2 k2 k4 1
(VH)=H˙ +3H2 =Λ+ + 4 (ρ p) 5ρp+ k4 U0 . (16)
V dt V2/3 2 − − 12 3 V4/3
Subtraction of Eq. (16) from Eqs. (15) gives
K
i
∆H =H H = , i=1,2,3, (17)
i i
− V
with K , i=1,2,3 constants of integration satisfying the consistency condition
i
3
K =0. (18)
i
i=1
X
With the use of Eq. (17) the anisotropy parameter defined in Eq. (13) becomes
K2 9K2
A= = , (19)
V2H2 V˙2
where K2 = (1/3) 3 K2. Taking the time derivative of Eq. (19) we obtain the following evolution equation for
i=1 i
the anisotropy:
P
dA H˙ +3H2
= 2 A, (20)
dt − H
or, equivalently,
1 dA 2 2 2α2 k2 k4 1
= V H˙ +3H2 = V Λ+ + 4 (ρ p) 5ρp+ k4 U0 . (21)
A2dV −3K2 −3K2 V2/3 2 − − 12 3 V4/3
(cid:16) (cid:17) (cid:20) (cid:21)
Taking into account that dρ/dV = (ρ+p)/V, Eq. (21) can be written as
−
1 dA 2C2 1 k2 k4 1
A2 dρ = 3K02(ρ+p)w2 Λ+2α2C0−2/3w2/3+ 24 (ρ−p)− 125ρp+ 3k4U0C0−4/3w4/3 , (22)
(cid:20) (cid:21)
with the general solution given by
3K2
A(ρ)= , (23)
−2C02 Λ+2α2C0−2/3w2/3+ k242 (ρ−p)− k1254ρp+ 31k4U0C0−4/3w4/3 [(ρ+p)w2]−1dρ−C
R h i
where C is an arbitrary integration constant.
7
Eq. (23) gives the general representation of the anisotropy parameter as a function of the energy density of the
cosmological fluid for the Bianchi type I and V space-times in the brane world scenario. For a cosmological fluid for
which the thermodynamic pressurep obeys a linear barotropicequationof state ofthe formp=(γ 1)ρ, γ =const.,
1 γ 2, we have ρ=ρ /Vγ, with ρ 0 a constantofintegration. ρ canbe expressedinterms−ofC as ρ =Cγ.
≤ ≤ 0 0 ≥ 0 0 0 0
Hence the anisotropy equation Eq. (23) can be immediately integrated to give the general exact dependence of the
anisotropy parameter on the volume scale factor for Bianchi type I and V geometries:
3K2
A(V)= , (24)
ΛV2+3α2V4/3+k2ρ V2 γ + k54ρ2V2 2γ +k4 V2/3+C
4 0 − 12 0 − U0
where the arbitrary integration constant C =0 is related, via the field equations, to the constant K2 by the relation
6
C =2K2/3 [27]. The singular state at t=0 is characterized by the condition V(0)=0. The value of the anisotropy
parameter for t=0 depends on the equation of state of the cosmologicalfluid. Hence for 1<γ 2, from Eq. (24) it
≤
follows
lim A(V)=0,1<γ 2. (25)
V 0 ≤
→
Therefore the singular state of the high density Bianchi type I and V brane cosmological models is isotropic, with
A(0)=0. For the case of the pressureless dust filled anisotropic brane Universes, p=0 and γ =1. In this case
36K2
lim A(V)= ,γ =1. (26)
V 0 k4ρ2+12C
→ 5 0
The singular state of the dust filled brane Universe is anisotropic, with A(0) = 0. In the case of the standard
6
four-dimensional general relativity (SGR), the behavior of the anisotropy parameter is different from the case of
brane cosmological models. SGR is recovered if the limits k 0, k 0 and Λ are taken simultaneously.
5 5
→ → → −∞
Therefore the anisotropy parameter in standard four-dimensional general relativity ASGR is given by
3K2
ASGR(V)= . (27)
ΛV2+3α2V4/3+k2ρ V2 γ +C
4 0 −
Near the singular state, (cid:0) (cid:1)
3K2
lim ASGR(V)= >0, γ [1,2]. (28)
V 0 C ∀ ∈
→
In fact, one canshow that for barotropicmatter filled standardfour-dimensionalgeneralrelativistic Bianchi type I
and V models ASGR(0) 2 [27].
≤
Thereforeinthe case γ =1 weobtainthe followingrelationbetweenthe initialvalues ofthe anisotropyparameters
A and ASGR of the brane world models and of the SGR, respectively:
ASGR(0)
A(0)= . (29)
1+ k54ρ20
12C
In Bianchi type I and V geometries there is also a simple proportionality relation between the shear scalar σ2 and
the anisotropy parameter:
3
1 3
σ2 =σ σik/2= H2 3H2 = AH2. (30)
ik 2 i − ! 2
i=1
X
IV. BEHAVIOR OF ANISOTROPY IN ARBITRARY TYPE BIANCHI MODELS
For arbitrary type Bianchi geometries the (ii),i = 1,2,3 components of the gravitational field equations on the
brane can be represented in the following general form:
1 d k2 k4 1
(VH )=F (a ,a ,a )+Λ+ 4 (ρ p) 5ρp+ k4 U0 , i=1,2,3, (31)
V dt i i 1 2 3 2 − − 12 3 V4/3
8
Bianchi type c1 c2 c3
I 0 0 0
II 1 0 0
VI0 1 -1 0
VII0 1 1 0
VIII 1 1 -1
IX 1 1 1
TABLE I: Valuesof theconstants ci,i=1,2,3, for the Bianchi typeA models [44], [46].
where F (a ,a ,a ),i=1,2,3 are functions which depend on the Bianchi type. For the class A models [46]
i 1 2 3
c a2 c a2 2 c a2 2
F (a ,a ,a )= 2 2− 3 3 − 1 1 , (32)
1 1 2 3 2V2
(cid:0) (cid:1) (cid:0) (cid:1)
where the constants c ,i=1,2,3 define the Bianchi type, and are given in the table.
i
F (a ,a ,a ) and F (a ,a ,a ) can be obtained by a cyclic permutation of the elements in the numerator of F .
2 1 2 3 3 1 2 3 1
For Bianchi types V and VI
h
a2+q2 b2
F (a ,a ,a )= 2 0 0 +2 , (33)
1 1 2 3 − a2 a4a2
1 2 3
a2+a q b2
F (a ,a ,a )= 2 0 0 0 2 , (34)
2 1 2 3 − a2 − a4a2
1 2 3
a2 a q
F (a ,a ,a )= 2 0− 0 0, (35)
3 1 2 3 − a2
1
with a ,q and b constants.
0 0
Ifwetakeq =b=0weobtainBianchitypeV,forq ,b=0weobtainBianchitype VI (h=0),whileforq = 1
0 0 h 0
6 6 −
we have Bianchi type III [46].
For Bianchi types IV and VII (h=0) the functions F are slightly more complicated. Thus for Bianchi type IV
h i
6
2 a2
F (a ,a ,a )= + 3 , (36)
1 1 2 3 a2 2a2a2
1 1 2
2 a2 a2
F (a ,a ,a )= + 3 + 3 f˙2, (37)
2 1 2 3 a2 2a2a2 2a2
1 1 2 2
2 a2 a2
F (a ,a ,a )= 3 + 3 f˙2, (38)
3 1 2 3 a2 − 2a2a2 2a2
1 1 2 2
with f corresponding to the off-diagonal term [47]. For Bianchi type VII (h=0)
h
6
a2+a2 a˙ a˙ a˙
F (a ,a ,a )=2∆2a2a2 1 2 2 3 1 2 , (39)
1 1 2 3 1 2a2 a2 a − a − a
1− 2 (cid:18) 3 1 2(cid:19)
F (a ,a ,a )= F (a ,a ,a ), (40)
2 1 2 3 1 1 2 3
−
F (a ,a ,a ) 0, (41)
3 1 2 3
≡
9
where ∆=constant=0 [48].
6
Generally there is one more field equation, the (00) equation, but it will not be used in the proof of the results.
By adding Eqs. (31) on obtains
1 d k2 k4 1
(VH)=H˙ +3H2 =F (a ,a ,a )+Λ+ 4 (ρ p) 5ρp+ k4 U0 , (42)
V dt 1 2 3 2 − − 12 3 V4/3
where
3
1
F (a ,a ,a )= F (a ,a ,a ). (43)
1 2 3 i 1 2 3
3
i=1
X
Substraction of Eq. (42) from Eq. (31) and integration of the resulting equation gives
K 1 ∆F [a (V),a (V),a (V)]
i i 1 2 3
∆H =H H = + dV,i=1,2,3, (44)
i i
− V 3V H
Z
where
∆F (a ,a ,a )=F (a ,a ,a ) F (a ,a ,a ),i=1,2,3, (45)
i 1 2 3 i 1 2 3 1 2 3
−
with the property
3
∆F (a ,a ,a )=0, (46)
i 1 2 3
i=1
X
and K ,i=1,2,3 are constants of integration satisfying the condition 3 K =0.
i i=1 i
Therefore for an arbitrary Bianchi type geometry the anisotropy parameter can be represented in the following
P
exact form:
K2+G2+L K2+G2+L
A= =9 , (47)
V2H2 V˙2
where
3
1
K2 = K2, (48)
3 i
i=1
X
1 3 ∆F 2
G2 = idV , (49)
27 H
i=1(cid:20)Z (cid:21)
X
and
3
2 ∆F
i
L= K dV. (50)
i
9 H
i=1 Z
X
Taking the time derivativeof Eq. (47), and changingthe time variableto V, it followsthat in anarbitraryBianchi
type geometry the anisotropy parameter satisfies a Bernoulli type first order differential equation of the form
dA d
= ln K2+G2+L A
dV dV −
(cid:20) (cid:21)
2(cid:0)V (cid:1) k2 k4 1
F (a ,a ,a )+Λ+ 4 (ρ p) 5ρp+ k4 U0 A2, (51)
3(K2+G2+L) 1 2 3 2 − − 12 3 V4/3
(cid:20) (cid:21)
with the general solution given by
3 K2+G2+L
A(V)= , (52)
V F (a1,a2,a3)+Λ+ k242(cid:0)(ρ−p)− k1254ρ(cid:1)p+ 31k4U0V−4/3 dV +C
R h i
10
where C is an arbitrary constant of integration.
Forabranecosmologicalfluidobeyingalinearbarotropicequationofstatep=(γ 1)ρ,theanisotropyparameter
−
for arbitrary Bianchi type cosmological models is given by
3V2γ K2+G2+L
A(V)= , (53)
V2γ VF[a1(V),a2(V),a3(V)]dV +ΛV2((cid:0)1+γ)+k42ρ0V2(cid:1)+γ + k1254ρ20V2+k4U0V2(1/3+γ)+CV2γ
We shallconsiRder in the following the behaviorof the anisotropyparameterfor arbitrarytype Bianchispace-times,
filledwithalinearbarotropiccosmologicalfluid,obeyinganequationofstateoftheformp=(γ 1)ρ,1 γ 2. For
this form of cosmological matter the conditions ρ(eff) > 0, ρ(eff)+p(eff) > 0 and ρ(eff)+3p(−eff) > 0≤are≤satisfied
on the brane. Therefore the corresponding cosmological models are singular [45]. Since in most Bianchi types one
does not know any exact solution, in order to find the behavior of the anisotropy at early times, it is necessary to
use some asymptotic solutions obtained, near the singularity,by approximate methods, in the limit of small values of
the time parameter. In standard four-dimensional general relativity it is concluded that for all Bianchi types there
exists a Kasner-like ”vacuum phase” near the singularity, that is, in general, Einstein’s vacuum equations are the
first order approximation of the equations with a nonvanishing matter term for t 0. This idea has been proposed
→
a long time ago by Belinskii, Lifshitz and Khalatnikov [49]. The general argument comes from the consideration of
a Kasner type metric ds2 = dt2 tp1dx2 tp2dy2 tp3dz2, with p ,i = 1,2,3 constants satisfying the conditions
i
3 p = 3 p2 =1. For a line−ar barotr−opic equa−tion of state it follows from the Bianchi identity (8) that energy
i=1 i i=1 i
density of the matter behaves like ρ t γ(p1+p2+p3). Therefore for asymptotes fulfilling the Kasner constraints the
−
P P ∼
matter term may be neglected in comparison with terms coming from the geometric part of the field equations and
containing the second time derivatives. A Kasner type behavior near the singularity for a Bianchi type IX brane
world geometry has been discussed in [28]
From Eq. (53) one can derive the behavior of the anisotropyparameter near the singular state on the brane for all
Bianchitypes. WesupposethatthebraneUniversestartsitsevolutionfromasingularstate,witha (0)=0,i=1,2,3.
i
For t>0 and for an expanding geometry, the scale factors are monotonically increasing functions of time, which for
small t can be represented in the Kasner form
a Vmi,i=1,2,3, (54)
i
∼
with m constants satisfying the conditions 0 m <1,i=1,2,3 and 3 m =1.
i ≤ i i=1 i
Theexistenceofasucharepresentationforthescalefactorsnearthesingularpointonthebranehasbeenexplicitly
P
proven,inthecaseofBianchitypeIandVgeometries,in[27]. ThemeanHubbleparameterbehavesnearthesingular
state like H V 1.
−
∼
With these assumptions it is easy to find the behavior of the integrals involving the functions F and G . For
i i
Bianchi class A models, VF [a (V),a (V),a (V)]dV can be written, near the singular point, as a sum of terms
1 2 3
of the form V4mi,i=1,2,3 and V2mi+2mj,i=j:
R 6
3 3
VF[a (V),a (V),a (V)]dV = α V4mi + β V2mi+2mj, (55)
1 2 3 i ij
Z i=1 i=j=1
X 6X
where α ,i=1,2,3 and β ,i,j =1,2,3 are constants.
i ij
Thus, for example, for the integral involving the function F (a ,a ,a ) we obtain
1 1 2 3
c2 c c c2 c2
VF (a ,a ,a )dV = 2 V4m2 2 3 V2(m2+m3)+ 3 V4m3 1 V4m1. (56)
1 1 2 3
8m − 2(m +m ) 8m − 8m
Z 2 2 3 3 1
For the other Bianchi types, the integral is a sum of terms of the form V2−2mi,i=1,2,3 or V2+2m3−2m1−2m2 and
cyclic permutationof this term. Due to the relationm +m +m =1 the last expressioncan be transformedto the
1 2 3
form V4mi,i=1,2,3. In the limit V 0, all these terms tend to zero.
→
ThebehaviorofthefunctionG2 andLcanbeanalyzedinasimilarway,andonecaneasilyshowthatlim G2 =
V 0
lim L = 0. Therefore for all Bianchi type cosmological models the anisotropy parameter on the brane→has the
V 0
gener→al property lim A = 0. In the limit of large times, V and all Bianchi models (except type VII)
V 0
isotropize, with A 0→. → ∞
→
V. DISCUSSIONS AND FINAL REMARKS
Thetimeevolutionoftheanisotropyparameterforalinearbarotropicfluidinthebraneworldmodelisverydifferent
fromthe standardfour-dimensionalgeneralrelativisticcase. Inhomogeneousandanisotropicbraneworldmodels the