Density Growth in Anisotropic Cosmologies of Bianchi Type I

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Density Growth in Anisotropic

Cosmologies of Bianchi Type I

Tomas Sverin

August 8, 2012

Master’s Thesis in Engineering Physics, 30 credits

Supervisor: Michael Bradley

Examiner: Mats Forsberg

UMEÅ UNIVERSITY

DEPARTMENT OF PHYSICS

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Abstract

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List of Figures

5.1. The background quantities energy density, expansion and shear. Ini-tial values at t0 = 1 are given by µ0 = 0.2, Θ0 = 3, Σ0 = 2

q

3

5 . . . 35

5.2. The growth of the density perturbation Dk and D⊥, where the initial

values at t0 = 1 are given by µ0 = 0.2, Θ0 = 3, Σ0 = ±2

q

3

5, and the

pressure is zero. . . 36 5.3. The growth of the density perturbation Dk, where the initial values

at t0 = 1 are given by µ0 = 0.2, Θ0 = 3, Σ0 = ±2

q

3

5, and for the

wave numbers k = kk/a10 = k⊥/a20 = 1, 5 and 20. The pressure is

given as p = 1₃µ. . . 38

5.4. The growth of the density perturbation Dkwith and without different

scale factors, where the initial values at t0 = 1 are given by µ0 = 0.2,

Θ0 = 3, Σ0 = ±2

q

3

5, and the pressure is given as p = 1

3µ. The

case with equal scale factors are given by kk/a10 = k⊥/a20 = 20,

and the case with different scale factors are given by kk/a10= 4 and

k⊥/a20= 20. . . 38

5.5. Additional terms in 4.20 when Hab 6= 0 for the wave numbers k =

kk/a10 = k⊥/a20 = 1, 5 and 20 . . . 39

B.1. The growth of the density perturbation Dk and D⊥, where the initial

values at t0 = 1 are given by µ0 = 0.2, Θ0 = 3, Σ0 = 0, and Λ = 0. . . 53

B.2. The growth of the density perturbation Dk and D⊥, where the initial

values at t0 = 1 are given by µ0 = 0.36, Θ0 = 2, Σ0 = 0. . . 54

B.3. The growth of the density perturbation Dk, where the initial values

q

3

5, and for the

wave numbers k = kk/a10= k⊥/a20 = 5 and 20. The pressure is given

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1. Introduction

The current best-fit model of the Universe is the ΛCDM model, which is spatially flat, isotropic and homogeneous on large scales. The constituents of this model include ordinary matter, radiation, a cosmological constant, Λ, and cold dark matter, see e.g. [1]. However, even though the Universe is assumed to be homogenous on large scales one can not exclude small anisotropies from observations, and therefore it can be of interest to study the effect of ansiotropies.

Perturbations of anisotropic cosmological models have been considered by many au-thors, e.g. [2]-[5]. Typically, the density perturbations grow faster in anisotropic models than in the isotropic models [6]. Most of these authors used methods depend-ing on the choice of gauge, like the perturbation theory of Lifshitz and Khalatnikov [7], or Bardeen’s gauge invariant theory [8]. The problem with using Bardeen-type variables is that they are defined to a particular coordinate system, and a conse-quence of this is that the their geometrical and physical meaning is unclear [6] [9]. In the case of the work by Lifshitz and Khalatnikov, the theory is well-known to be plagued by difficulties in interpretation, in particular concerning the choice of gauge, because physical results can only be worked out once the correspondence between the real inhomogeneous Universe and the background space-time has been made [6]. One can use covariantly defined objects, like the spatial curvature and kinematic quantities, rather than the metric as defining variables. This method makes it pos-sible to define perturbation variables that are zero on the background, and these variables are gauge invariant [9].

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1.1. Problem Statement and Objectives

Density growth in cosmological dust models of Bianchi type I have been studied by Osano, see [10], and an expanded study with this method would be to consider the effect of the density perturbations when the pressure and a cosmological constant are included in the cosmological model. In a similar way to the work done by Osano, the covariant 1+3 split of space-time will be used to study density growth in Bianchi type I models with a cosmological constant and pressure.

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2. Cosmological Models

A cosmological model represents the Universe at large scales, and it can be assumed that the space-time geometry is described by Einstein’s general theory of relativ-ity. Given the matter content, its evolution is then determined through Einstein’s equations.

First of all, the space-time geometry is represented on some averaging scale and de-termined by the metric gab(xµ). Furthermore, the matter is presented on the same

averaging scale, and its physical behavior must represent physically plausible mat-ter. The interaction between geometry and matter is described through Einstein’s relativistic gravitational field equations given by

Gab ≡ Rab−

1

2Rgab = Tab− Λgab, (2.1)

where Gab is the Einstein tensor, Rab the Ricci curvature scalar, R the scalar

curva-ture, gab the metric tensor, and Λ the cosmological constant. In this description of

Einstein’s field equations the geometrised units are characterised by c = 1 = 8πG/c4, and this means that all geometrical variables occurring have physical dimensions that are integer powers of the dimension [length].

The conservation of local energy-momentum is guaranteed because of the twice-contracted Bianchi identities

∇bGab = 0 ⇒ ∇bTab = 0, (2.2)

and it requires the cosmological constant Λ to satisfy ∇aΛ = 0, i.e., it is constant

in time and space.

Together, these determine the combined dynamical evolution of the model and the matter in it. To simplify things one often assume symmetries or special properties of some kind or another. In this study the matter description will be a perfect fluid with specified equation of state and the background geometry of the Universe will be assumed to be homogeneous.

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quantities, these then remain valid whatever coordinate system is chosen. The 1 + 3 covariant and tetrad approach are described in section 2.1 and section 2.2, respectively, and section 2.3 describes general and specific features about the Bianchi cosmological models.

2.1. 1 + 3 Covariant Description

The 1 + 3 covariant split of space-time is suitable, when it is possible to define a pre-ferred timelike direction, like the 4-velocity of matter of a prepre-ferred set of observers. The kinematic quantities of the 4-velocity, together with the energy-momentum ten-sors of matter sources, and the electric and magnetic parts of the Weyl curvature tensor are used rather than the metric. The fundamental equations are the Ricci and Bianchi identities, applied to the 4-velocity vector, with the Einstein’s field equa-tions included via algebraic relaequa-tions between the Ricci and the energy-momentum tensor. For more details see, e.g. [11]. This approach is also usable for perturbative calculations, as will be seen later.

2.1.1. Variables

2.1.1.1. Average 4-velocity of matter

In a cosmological space-time it is usually assumed that there is a well-defined pre-ferred motion of matter and hence a unique 4-velocity. This corresponds to prepre-ferred wordlines, and in each case the 4-velocitiy is

ua= dx

a

dt , uau

a

= −1, (2.3)

where t is proper time measured along the fundamental worldline.

2.1.1.2. Projection and Permutation Tensors

For a given ua_{, one defines projection tensors with properties}

U_ba = −uaub ⇒ UcaUbc= Uba, Uaa= 1, Uabub = ua, (2.4) and hab = gab+ uaub ⇒ hach c b = h a b, h a a= 3, habub = 0, (2.5)

where the first projects parallel to the 4-velocity vector ua, and the second projects onto the local 3-dimensional rest-space orthogonal to ua_{. Projections with h}b

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2.1 1 + 3 Covariant Description

vectors are denoted by angle brackets vhai ≡ ha

bvb, and the projected symmetric

tracefree (PSTF) of a tensor is given by

Thabi ≡ h(a_c hb)_d − 1 3h ab_h cd Tcd. (2.6)

There is also a volume element for the rest-spaces,

ηabc= udηdabc ⇒ ηabc = η[abc], ηabcuc= 0, (2.7)

where ηabcd is the 4-dimensional volume element (ηabcd = η[abcd], η0123 =

q

| det gab|).

We can define the covariant time derivative along the fundamental wordlines, where for any tensor Tab,

˙

Tab ≡ uc_∇

cTab, (2.8)

and the orthogonal projected covariant derivative ˜∇a, where for any tensor Tba,

˜ ∇cTba ≡ h a dh e bh f c∇fTed, (2.9)

and similarly for higher index tensors. ˜∇a is a proper 3-dimensional derivative if

and only if ua has zero vorticity.

2.1.1.3. Kinematical quantities

The covariant derivative of the 4-velocity, ua, can be split into its irreducible parts given by their symmetry properties

∇aub = −ua˙ub+ ˜∇aub = −ua˙ub +

1

3Θhab+ σab+ ωab, (2.10)

where the acceleration is defined as ˙ua ≡ ub∇bua, and the expansion of the fluid

is Θ = ˜∇aua . Furthermore σab ≡ ˜∇haubi is the tracefree symmetric shear tensor

(σabub = 0, σaa= 0, σab = σ(ab)) describing the rate of distortion of the matter flow,

and ωab ≡ ˜∇[aub] is the skew-symmetric vorticity tensor (ωab = ω[ab], ωabua = 0),

describing the rotation of the matter relative to a non-rotating frame.

2.1.1.4. Matter Tensor

The matter energy-momentum tensor Tab can be decomposed relative to ua in the

form

Tab= µuaub+ phab+ qaub+ qbua+ πab, (2.11)

where µ = Tabuaub

is the relativistic energy density relative to ua, qa= −Tbcubhca

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to ua, p = 1₃Tabhab

is the isotropic pressure, and πab = Tcdhchahdbi is the tracefree

anisotropic pressure.

The energy momentum tensor of a perfect fluid is obtained with the restriction

qa = πab = 0, (2.12)

which reduces Equation 2.11 to

Tab = µuaub+ phab. (2.13)

2.1.1.5. Weyl Curvature Tensor

The Riemann tensor Rabcd can be decomposed as

Rabcd = Cabcd+

1

2(gacSbd+ gbdSac− gadSbc− gbcSad)+ 1

12R (gacgbd− gadgbc) , (2.14) where Sab is the traceless part of the Ricci tensor

Sab ≡ Rab−

1

4Rgab, (2.15)

with the property Sa a = 0.

The Weyl curvature tensor Cabcd is the traceless part of the Riemann tensor, i.e.

Ca

bad = 0. The Weyl tensor is split relative to ua into electric (Eab) and magnetic

(Hab) parts according to Eab ≡ Cacbducud ⇒ Eaa = 0, Eab = E(ab), Eabub = 0, (2.16) Hab ≡ 1 2ηadeC de bcu c _{⇒ H}a a = 0, Hab = H(ab), Habub = 0. (2.17)

These tensors represent the free gravitational field, enabling gravitational action at a distance, and influence the motion of matter and radiation through the geodesic deviation equations for timelike and null vectors, respectively. In the geodesic equa-tion one can see that both the Weyl curvature tensor and the Ricci curvature tensor contribute to the geodesic deviation.

2.1.2. 1 + 3 Covariant Propagation and Constraint Equations

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2.1.2.1. Ricci Identities

The first set arise from the Ricci identities for the vector field ua_{, i.e.,}

(∇a∇b − ∇b∇a) uc= Rabcdud. (2.18)

On substituting from Equation 2.8, using Equation 2.1, and separating out the or-thogonally projected part into trace, symmetric tracefree, and skew-symmetric parts, and the parallel parts similarly, we obtain three propagation equations and three constraint equations. The propagation equations are

1. The Raychaudhuri equation ˙ Θ − ˜∇a˙ua = − 1 3Θ 2_{+ ( ˙u} a˙ua) − 2σ2+ 2ω2− 1 2(µ + 3p) + Λ, (2.19)

which is the basic equation of gravitational attraction. 2. The vorticity propagation equation

˙ ωhai− 1 2η abc_∇_˜ b˙uc = − 2 3Θω a + σa_bωb, (2.20)

giving a vorticity conservation law for a perfect fluid with acceleration potential Φ.

3. The shear propagation equation ˙σhabi− ˜∇ha˙ubi = −2 3Θσ ab_{+ ˙u}ha ˙ubi− σ_chaσbic− ωhaωbi− Eab−1 2π ab_{, (2.21)}

where the anisotropic pressure source term πab vanishes for a perfect fluid.

This shows how the tidal gravitational field, the electric Weyl curvature Eab,

directly induces shear (which then feeds into the Raychaudhuri and vorticity propagation equations, thereby changing the nature of the fluid flow).

The constraint equations are 1. The (0α)-equation 0 = (C1)a= ˜∇bσab− 2 3∇˜ a_{Θ + η}abch ˜ ∇bωc+ 2 ˙ubωc i + qa, (2.22)

showing how the momentum flux relates to the spatial inhomogeneity of the expansion. Note that the momentum flux is zero for a perfect fluid.

2. The vorticity divergence identity

0 = (C2) = ˜∇aωa− ( ˙uaωa) . (2.23)

3. The Hab-equation

0 = (C3)ab = Hab+ 2 ˙uhaωbi+ ˜∇haωbi − (curlσ)ab, (2.24)

characterising the magnetic Weyl curvature as being constructed from the ’distortion’ of the vorticity and the ’curl’ of the shear, curl σab _{= η}cdha_∇˜

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2.1.2.2. Twice-contracted Bianchi Identities

The second set of equations arise from the twice-contracted Bianchi identities. Pro-jecting parallel and orthogonal to ua_{, we obtain the propagation equations}

˙ µ + ˜∇aqa= −Θ (µ + p) − 2 ( ˙uaqa) − σabπab , (2.25) and ˙ qhai+ ˜∇a_{p + ˜}_∇ bπab = − 4 3Θq a_{− σ}a bq b_{− (µ + p) ˙u}a_{− ˙u} bπab− ηabcωbqc, (2.26)

which constitute the energy conservation equation and the momentum conserva-tion equaconserva-tion, respectively. For a perfect fluid, characterised by Equaconserva-tion 2.13, the propagation equations reduce to

˙

µ = −Θ (µ + p) , (2.27)

and the constraint equation

0 = ˜∇ap + (µ + p) ˙ua. (2.28)

2.1.2.3. Other Bianchi Identities

The third set of equations arise from the Bianchi identities

∇eRabcd+ ∇dRabec+ ∇cRabde = ∇[aRbc]de= 0. (2.29)

On using the splitting of Rabcd into Rab and Cabcd, the above 1 + 3 splitting of those

quantities, and Einstein’s field equations, the once-contracted Bianchi identities give two further propagation equations and two further constraint equations.

The propagation equations are the ˙E-equation

˙ Ehabi+ 1 2˙π habi_{− curlH}ab₋ 1 2curlπ ab _{= −}1 2(µ + p) σ ab_{− Θ}_Eab₊1 6π ab + 3σ_cha Ebic−1 6π bic_{− ˙u}ha qbi + ηcdha 2 ˙ucH bi d + ωc E_dbi+1 2π bi d , (2.30) and the ˙H-equation

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and the ’curls’ are defined as curlVab = ηcdha∇˜cV

bi

d , (2.32)

where Vab _{is an arbitrary tensor.}

Equation 2.30 and Equation 2.31 show how gravitational radiation arises, and to-gether they can be used to find a wave operator acting on Eab _{and H}ab_{, respectively.}

The constraint equations are the (divE)-equation 0 = (C4) a = ˜∇b Eab+ 1 2π ab₋1 3 ˜ ∇aµ + 1 3Θq a₋1 2σ a bq b _{− 3ω} bHab − ηabc σbdHcd− 3 2ωbqc , (2.33)

where one source term is the spatial gradient of the energy density, and the (divH)-equation 0 = (C5)a= ˜∇bHab+ (µ + p) ωa+ 3ωb Eab− 1 6π ab + ηabc ₁ 2 ˜ ∇bqc+ σbd E_cd+1 2π d c , (2.34)

where the fluid vorticity and shear act as source terms.

2.1.3. Irrotational Flow

For a barotropic perfect fluid we have

0 = qa = πab, p = p(µ) ⇒ ηabc∇˜a˙uc = 0. (2.35)

If we assume irrotational flow then we have the following implications [11]:

1. The fluid flow is hypersurface-orthogonal, and there exists a cosmic time func-tion t such that ua = −g

xb∇at, if additionally the acceleration vanishes,

we can set g to unity.

2. The metric of the orthogonal 3-spaces is hab.

3. From the Gauss embedding equation and the Ricci identities for ua_{, the Ricci}

tensor of the 3-spaces is given by

3_R

ab= − ˙σhabi− Θσab+ ˜∇ha˙ubi+ ˙uha˙ubi+ πab+

1 3hab 2µ − 2 3Θ 2_{+ 2σ}2_{+ 2Λ}_, (2.36) where σ2 _≡ 1 2σ ab_σ

ab, and Λ is the cosmological constant. The 3-space Ricci

scalar is given by

3_{R = 2µ −} 2

3Θ

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which is a generalised Friedmann equation, showing how the matter tensor determines the 3-space average curvature. These equations fully determine the curvature tensor3_R

abcd of the orthogonal 3-spaces.

In the case of irrotational flow Equation 2.19 and Equation 2.21 reduce to the fol-lowing propagation equations,

˙ Θ − ˜∇a˙ua = − 1 3Θ 2 + ˙ua˙ua− 2σ2− 1 2(µ + 3p) + Λ, (2.38) ˙σhabi− ˜∇ha˙ubi = −2 3Θσ ab_{+ ˙u}ha ˙ubi− σha_c σbic− Eab_, _(2.39)

Equation 2.20 reduces to ηabc_∇_˜

b˙uc= 0, and that is identically satisfied if p = p (µ).

The constraint equations, Equation 2.22 and Equation 2.24, reduce to 0 = (C1)a = ˜∇bσab− 2 3 ˜ ∇a_Θ, _(2.40) and 0 = (C3) ab = Hab− ηcdha_∇_˜ cσ bi d, (2.41)

respectively. The twice-contracted Bianchi identities can now be written in the form ˙ µ + ˜∇aqa = −Θ (µ + p) − 2 ( ˙uaqa) − σabπab , (2.42) ˜ ∇a_{p = − (µ + p) ˙u}a_, _(2.43)

and the other Bianchi identities yield ˙

Ehabi− curlHab _{= −}1

2(µ + p) σ

ab_{− ΘE}ab_{+ 3σ}ha

c Ebic+ 2ηcdha˙ucH bi

d , (2.44)

˙

Hhabi+ curlEab = −ΘHab+ 3σ_chaHbic− 2ηcdha_˙u cE

bi

d . (2.45)

In summary, we have a first-order system of equations that consists of six propaga-tion equapropaga-tions and six constraint equapropaga-tions. The system of equapropaga-tions is determinate once the fluid equations are given, and together they then form a complete set of equations [11].

2.2. Tetrad Description

The tetrad formalism is an approach that replaces a choice of local coordinates by the less restrictive choice of a local basis for the tangent bundle i.e. a locally defined set of four independent vectors called the tetrad.

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2.2 Tetrad Description

general relativity lies in the ability to choose the tetrad basis to reflect important physical aspects of the space-time.

It is suitable to use a constant metric ηij in a frame, for example Xi = Xiµ∂x∂µ, where

i = 0, 1, 2, 3. The dual basis can be written in the form ωi = ωi_µdxµ, and we have the relation ωi

µX µ

j = δji. Furthermore, the line element takes the form

ds2 = ηijωi⊗ ωj = ηijωiωj = gµνdxµdxν, (2.46)

where the tensor product ⊗ is defined below.

The formalism will here be briefly discussed, for more details see e.g. [12].

2.2.1. Tensor and Wedge Product

The tensor product between tensors ω and σ is written as ω ⊗ σ, and it acts on two vectors,

ω ⊗ σ (u, v) ≡ ω(u)σ(v), (2.47)

to produce a number.

The exterior product, also known as wedge product, of two vectors ω and σ is denoted as ω ∧ σ. The wedge product is anti-commutative, meaning that ω ∧ σ = − (σ ∧ ω) for all vectors ω and σ, and therefore it can be written in the form

ω ∧ σ = 1

2(ω ⊗ σ − σ ⊗ ω) . (2.48)

2.2.2. Exterior Differentiation: Structure Coefficients

The familiar quantity called differential of a function is known for every physicist. This concept is refined in modern differential geometry by the use of an operator

d, called the curl, gradient or exterior derivative operator, operating on forms. The

exterior derivative d involves four rules:

1. f scalar: df = f,µdxµ = Xi(f ) ωi ≡ f|iωi;

2. d on r-forms gives a r + 1 form; 3. d (dω) = 0;

4. d (ω ∧ σ) = dω ∧ σ + (−1)rω ∧ dσ (Note that functions are zero-forms).

Here r is the order of ω.

Let {ωi_{} be a basis of one-forms dual to a basis {X}

i} which has non-zero

commu-tators. The curl of any ωi _{is a two-form dω}i _{and hence a linear combination of the}

basis of two-vectors {ωi_{∧ ω}j_}:

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It can be shown that Di_jk are related to the structure coefficients C_jki of

[Xi, Xj] = Cjki Xi (2.50)

in the following way,

D_jki = −1 2C

i

jk. (2.51)

2.2.3. First and Second Cartan Structure Equations

Covariant differentiation is defined as

Ai;j ≡ Aµ;νXiµX ν

j, (2.52)

and can be decomposed in the following way

Ai;j = Ai|j− γijkAk, (2.53)

where γk

ij are the Ricci rotation coefficients. These can be written in the form

γ_ijk = ω_µkX_i;νµ X_jν = −ω_µ;νk X_iµX_jν, (2.54)

and Ai|j = Xj(Aµ).

As mention earlier, the metric is constant, and therefore the Ricci rotation coeffi-cients have the property

γijk = −γjik, (2.55)

where

γijk ≡ ηilγjkl . (2.56)

Hence, we can define the connection forms,

ωi_j ≡ γi jkω

k_, _(2.57)

ωij ≡ −ωji_. _(2.58)

We shall not consider covariant derivatives with non-zero torsion, because of their limited usefulness in general relativity, and therefore we have,

[U, V ] = LUV, (2.59)

where LU is the Lie derivative. For example, the Lie derivative of a tensor field T

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2.3 Bianchi Models

It can be shown that in a basis {Xi},

C_jki = γ_kji − γi

jk, (2.61)

where the structure coefficients are defined by Equation 2.51 [12]. This relation implies dωi = −1 2C i jkω j_{∧ ω}k_{= −γ}i jkω k_{∧ ω}j_, _(2.62)

and the definition of the connection forms (Equation 2.57) makes it possible to write it in the form,

dωi = ωk∧ ωi

k, (2.63)

which is, for zero torsion, the first Cartan structure equation.

The second Cartan structure equation defines the curvature forms of differential geometry and are equivalent to the Riemannian curvature, and for the interest of derivation of the equation one can see for example [12]. Only the equation will be presented here, and it is

dω_ji = −ω_ki ∧ ωk j + 1 2R i jklω k_{∧ ω}l_, _(2.64) where Ri

jkl is the Riemann curvature tensor for a metric in any basis.

In conclusion, we can use the first and second Cartan structure equations together with the connections forms to find the components of the Riemann tensor, and therefore also be able to find solutions to Einstein’s field equations by identifying the Ricci tensor with the energy-momentum tensor.

2.3. Bianchi Models

The Bianchi models are spatially homogeneous and anisotropic, and a general def-inition of homogeneity is to require that all comoving observers see essentially the same version of cosmic history. For a more detailed description, see e.g. [12]. The possible symmetries can be classified into classes usually called the Bianchi types, although there is one peculiar class of homogeneous solutions of the Einstein’s field equations, called Kantowski-Sachs solutions, that does not fit into this scheme. The Bianchi classification is based on the construction of spacelike hypersurfaces upon which it is possible to define at least three independent vector fields, ξi, that

satisfying the constraint

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This is Killing’s equation and the vectors that satisfy it are called Killing vectors. They are generators of the isometry group and satisfy a Lie-algebra by

[ξi, ξj] = Cijkξk, (2.66)

where the commutator is defined as

[ξi, ξj] ≡ ξiξj− ξjξi, (2.67)

and the Ck

ij are the structure constants of the isometry group. The structure

con-stants are antisymmetric in the sense that

C_ijk = −C_jik. (2.68)

Depending on their forms, the algebras are divided into nine different classes, the so called Bianchi classification [12]. The components of the metric, gij, describing a

Bianchi space are invariant under the isometry generated by infinitesimal translation of the Killing vector fields. In other words, the time-dependence of the metric is the same at all points, i.e., 3-space is homogeneous. The Einstein’s field equations relate the energy-momentum tensor Tij to the derivatives of gij, so if the metric is

invariant under a given set of operations, then so are the physical properties encoded by Tij.

The Friedmann models form special cases of Bianchi types. For example, the flat Friedmann model is a special case of the Bianchi type I model.

In general, exact solutions to Einstein’s field equations can be hard to find, but several special cases of Bianchi type solutions are known. One very well-known example of a class of exact vacuum solutions, which are useful illustrations of the sort of behavior one can obtain, are the Kasner solutions, belonging to the Bianchi type I class.

2.3.1. Bianchi Type I

The spatially homogeneous and anisotropic Bianchi type I space-times are described by the line element

ds2 = −dt2+ A2(t) dx2+ B2(t) dy2+ C2(t) dz2. (2.69) The commutators of the three Killing vectors satisfy [ξi, ξj] = 0, and the spatial

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2.3 Bianchi Models

depend only on the time coordinate. The fluid, which is orthogonal to the spa-tial surfaces, is geodesic and irrotational. These model are therefore covariantly characterised by,

˙u = ωa = 0, ∇˜aµ = ˜∇ap = ˜∇aΘ = 0, 3Rab = 0, (2.70)

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3. Background Solutions of Bianchi

Type I

Here the field equations for cosmological models of Bianchi type I are given and analysed. Some exact solutions of LRS (Locally Rotationally Symmetry) type are then presented. The tetrad approach is mainly used, but since the 1 + 3 covariant formalism will be used for the perturbative calculations in the next chapter, the evolution equations in the 1 + 3 formalism are also given in section 3.4.

3.1. Evolution Equations in Tetrad Approach

Bianchi type I models have a metric of the form,

ds2 = −dt2+ A2(t) dx2+ B2(t) dy2+ C2(t) dz2, ua= δa₀, (3.1) where A, B and C are expansion scale factors. We start to choose a basis

ω0 = dt, ω1 = A (t) dx, ω2 = B (t) dy, ω3 = C (t) dz, (3.2)

and the line element can now be written as

ds2 = −ω02+ω12+ω22+ω32. (3.3)

Starting from Cartan’s first structure equation (Equation 2.63) and using the con-nection form (Equation 2.57) and Equation 3.2 the non-zero Ricci rotation coeffi-cients are found as

γ₀₁1 = γ₁₁0 = ˙ A A, γ 2 02 = γ 0 22= ˙ B B, γ 3 03 = γ 0 33= ˙ C C, (3.4)

where the dot indicates derivation with respect to time, i.e. ˙A ≡ dA_dt.

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tensor are obtained R0₁₀₁ = −R0₁₁₀ = R1₀₀₁= −R1₀₁₀ = A¨ A, R0₂₀₂ = −R0₂₂₀ = R2₀₀₂= −R2₀₂₀ = ¨ B B, R0₃₀₃ = −R0₃₃₀ = R3₀₀₃= −R3₀₃₀ = ¨ C C, R1₂₁₂ = −R1₂₂₁ = −R₁₁₂2 = R2₁₂₁ = ˙ A ˙B AB, R1₃₁₃ = −R1₃₃₁ = −R₁₁₃3 = R3₁₃₁ = A ˙˙C AC, R2₃₂₃ = −R2₃₃₂ = −R₂₂₃3 = R3₂₃₂ = ˙ B ˙C BC. (3.5)

By using Equation 3.5 in the relation between the Riemann tensor and the Ricci curvature tensor, Rij = Rkikj, we find the diagonal components for the Ricci tensor

R00= − ¨ A A − ¨ B B − ¨ C C, R11= ¨ A A + ˙ A ˙B AB + ˙ A ˙C AC, R22= ¨ B B + ˙ A ˙B AB + ˙ B ˙C BC, R33= ¨ C C + ˙ A ˙C AC + ˙ B ˙C BC. (3.6)

Finally, we can find the expression for the trace of the Ricci curvature

R = ηabRab = 2 ¨ A A + ¨ B B + ¨ C C + ˙ A ˙B AB + ˙ A ˙C AC + ˙ B ˙C BC ! . (3.7)

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3.2 Evolution Equations and State Space

Now we have obtained a set of differential equations and by solving this we can find solutions in the metric form given by Equation 3.1.

3.2. Evolution Equations and State Space

This subsection is based on the work done by Goliath and Ellis, and can be found in [13]. By performing an expansion normalization we can start to define dimensionless variables according to Ω ≡ µ 3H2, ΩΛ ≡ Λ 3H2, Σαβ ≡ σαβ H . (3.12)

The density parameter Ω is related to ΩΛ and Σαβ by

Ω = 1 − Σ2− ΩΛ, (3.13)

where Σ2 _≡ ΣαβΣαβ

6 .

The shear is diagonal, and consequently there are only two independent components, and we can take them to be

Σ+ ≡

1

2(Σ22+ Σ33) , Σ− ≡ 1

2√3(Σ22− Σ33) , (3.14)

and with these definitions it follows that Σ2 _{= Σ}2

++ Σ2−. The reduced dynamical

systems become

Σ0_± = − (2 − q) Σ±, (3.15)

Ω0_Λ = 2 (1 + q) ΩΛ, (3.16)

where the deceleration parameter, q, is given by

q = 3 2(2 − γ) Σ 2₊1 2(3γ − 2) − 3 2γΩΛ. (3.17)

This three-dimensional dynamical system is compact, and has the following invariant submanifolds:

Σ+ = 0 : σ22 = −σ33

Σ− = 0 : the LRS submanifold, where Σ22 = Σ33

Ω = 0 : the vacuum boundary ΩΛ = 0 : the Λ = 0 submanifold

Of these, the last two constitute the boundary of the state space. The equilibrium points are given in Table 3.1.

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Σ2 _Ω

Λ Stability

F Friedmann 0 0 saddle

K Kasner circle 1 0 source

dS de Sitter 0 1 sink

Table 3.1.: Equilibrium points

3.3. LRS Bianchi Type I

We have σ22 = σ33 when Σ− = 0, which means the model will be locally rotational

symmetric (LRS) and the metric given by Equation 3.1 can now be written in the form

ds2 = −dt2+ A2(t) dx2+ B2(t)dy2+ dz2. (3.18)

Hence, the system of four equations in section 3.1 reduces to a system om three equations, 2 ˙ A ˙B AB + ˙ B B !2 = µ + Λ, (3.19) ¨ A A + ¨ B B + ˙ A ˙B AB = −p + Λ, (3.20) 2B¨ B + ˙ B B !2 = −p + Λ. (3.21)

This system of equations has been discovered by other authors, for example it can be found in [15] and [16], which findings are based on [17].

Elimination of the term (−p + Λ) from Equation 3.20 and Equation 3.21 gives ¨ B B + ˙ B B !2 −A¨ A − ˙ A ˙B AB = 0, (3.22)

which on integration yields

B2A − AB ˙˙ B = k, (3.23)

where k is a constant of integration.

By considering Equation 3.23 as a linear differential equation of A(t), where B(t) is an arbitrary function, we obtain

A = k1B + kB

ˆ

dt

B3_(t), (3.24)

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3.3 LRS Bianchi Type I

In similar way, if we consider Equation 3.23 as a linear differential equation of B(t), where A(t) is an arbitrary function, we obtain

B2 = k2A2− 2kA2

ˆ

dt

A3_(t), (3.25)

where k2 is a constant of integration. Therefore, for any B(t) from Equation 3.24,

one can obtain A (t), and vice versa.

With this knowledge it is possible to find exact solutions to the system of equations, depending on the conditions of the pressure and the cosmological constant. In this study we have found exact solutions for the dust case with and without cosmological constant.

3.3.1. Dust solution without cosmological constant

In the case of dust (p = 0) and no cosmological constant (Λ = 0), Equation 3.19, Equation 3.20 and Equation 3.21 reduce to

2 ˙ A ˙B AB + ˙ B B !2 = µ, (3.26) ¨ A A + ¨ B B + ˙ A ˙B AB = 0, (3.27) 2 ¨ B B + ˙ B B !2 = 0, (3.28)

which have the solutions

A(t) = a1t2/3− a2t−1/3, (3.29)

B(t) = b1t2/3, (3.30)

where a1, a2 and b1 are constants. One can see that for low t the solution approaches

the LRS Kasner solution, which has the metric

ds2 = −dt2+ t2adx2+ t2bdy2+ dz2, (3.31)

where a + 2b = 1 and a2 + 2b2 = 1[16]. Furthermore, the solution approaches the Friedmann solution

ds2 = −dt2+ t4/3dx2+ dy2+ dz2 (3.32)

for large t with equal scale factors.

By using Equation 3.29 and Equation 3.30 in Equation 3.26 the energy density can then be written in the form

µ(t) = 4

3

a1

t (a1t − a2)

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3.3.2. Dust solution with cosmological constant

In the case of dust (p = 0) and the presence of a cosmological constant (Λ 6= 0), Equation 3.19, Equation 3.20 and Equation 3.21 reduce to

2A ˙˙B AB + ˙ B B !2 = µ + Λ, (3.34) ¨ A A + ¨ B B + ˙ A ˙B AB = Λ, (3.35) 2B¨ B + ˙ B B !2 = Λ, (3.36)

which have the solutions

A(t) = a1exp   s Λ 3t  − a₂exp  −2 s Λ 3t  , (3.37) B(t) = b1exp   s Λ 3t  . (3.38)

where a1, a2and b1are constants. One can see that for low t the solution approaches

the LRS Kasner solution given by Equation 3.31, and for large t it approaches the de Sitter solution ds2 = −dt2+ exp  2 s Λ 3t   dx2+ dy2+ dz2. (3.39)

By using Equation 3.37 and Equation 3.38 in Equation 3.34 the energy density can be written in the form

µ(t) = 2Λa2exp −2qΛ 3t a1exp _q Λ 3t − a2exp −2qΛ 3t (3.40)

3.4. Evolution Equations in 1 + 3 Formalism

Since the 1 + 3 covariant formalism will be used for the perturbative calculations in the next chapter, the background equations are here also given in this formalism in terms of kinematic and curvature quantities. These are density µ, expansion Θ, shear σab, pressure p, and the electric part of the Weyl tensor Eab. A linear equation

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3.4 Evolution Equations in 1 + 3 Formalism

and the 3-Ricci scalar give restrictions on the above quantities. Together with the evolution and constraint equations in subsection 2.1.2 one finds

Eab = 1 3Θσab− σchaσ c bi, (3.41) µ = 1 3Θ 2_{− σ}2 _{− Λ.} _(3.42)

Due to the LRS symmetry the shear can be written as

σab = Σ ₃ 2nanb− 1 2hab , (3.43)

and in similar way the electric part of the Weyl curvature tensor is given by

Eab = ε ₃ 2nanb− 1 2hab , (3.44)

where na is the anisotropy vector and hab is the projection operator. From the

equations in subsection 2.1.2, Equation 3.43, and Equation 3.44, the set of evolution equations is reduced to ˙ µ = −γΘµ, (3.45) ˙ Θ = −1 3Θ 2₋3 2Σ 2₋3 2γ − 1 µ + Λ, (3.46) ˙ Σ = −ΘΣ, (3.47) ˙ ε = −Θε + 3 2Σε − 1 2γµΣ (3.48)

where two equations are dispensable due to the constraints given by Equation 3.41 and Equation 3.42. That the system is consistent can be seen by differentiating Equation 3.41 and Equation 3.42, and substituting Equation 3.45, Equation 3.46, Equation 3.47 and Equation 3.48.

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4. Density Perturbations

In this chapter we consider density perturbations of Bianchi type I models. These backgrounds have two extra terms when compared with the FLRW background, which are the shear tensor and the electric part of the Weyl tensor.

4.1. Propagation Equations

The zeroth-order variables are µ, p, Θ, σab and Eab, and their covariant time

deriva-tives. These variables are the background variables in this model, but as was seen in section 3.4, Eab and µ can be expressed in terms of the other variables and when

an equation of state is given also p is determined. We begin by applying the lin-earisation procedure discussed in section 2.1, to the full set of covariant equations. Taking vorticity to be zero, the linearisation process leads to the following propaga-tion equapropaga-tions, ˙ µ = −Θ (µ + p) , (4.1) ˙ Θ = ˜∇a˙ua− 1 3Θ 2_{− 2σ}2₋1 2(µ + 3p) + Λ, (4.2) ˙σhabi = ˜∇ha˙ubi− 2 3Θσab− σchaσ c bi − Eab, (4.3) ˙

Ehabi = −ΘEab+ 3σchaEbic −

1

2(µ + p) σab+ curlHab, (4.4)

˙

Hhabi = −ΘHab+ 3σchaHbic− curlEab. (4.5)

We will assume a perfect-fluid matter source p = p (µ), i.e. a barytropic equation of state, which implies that we have

p0 ≡ dp

dµ, (4.6)

and therefore Equation 2.28 can be written in the form ˙ua = −

p0 µ + p

˜

∇aµ (4.7)

Moreover, the equation of state can be chosen as

p = (γ − 1) µ, (4.8)

where γ is a constant. Causality then requires γ to be in the interval 0 ≤ γ ≤ 2. For example, γ = 1 implies that we have dust, and the matter source described as radiation is considered when γ = 4

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4.2. Evolution of Inhomogeneity

We follow the standard approach, where inhomogeneity variables are given by the comoving spatial gradients. In the case of Bianchi type I models, we have the spatial gradients of the energy density, the expansion and the shear scalar. Note that these inhomogeneity variables are constrained by the spatial gradient of the Ricci scalar from the 3-surface (Equation 2.37)

a ˜∇(3) a R = − 4 3aΘ ˜∇aΘ + 2a ˜∇aσ 2_{+ 2a ˜}_∇ aµ, (4.9)

where σ2 = 1₂σabσab, and a = a(t) is the scale factor.

The evolution of Equation 4.9 governs the growth of inhomogeneity. We can now define variables which represent each of the terms of the right hand side of this equa-tion. Hence, we define particular spatial gradients orthogonal to ua_{, characterising}

the inhomogeneity of space-time,

Da ≡

a ˜∇aµ

µ , (4.10)

Za ≡ a ˜∇aΘ, (4.11)

Ta ≡ a ˜∇aσ2, (4.12)

which are the comoving density gradient, the comoving expansion gradient and the shear scalar gradient, respectively. The average length scale a is determined by

˙a

a =

1

3Θ, (4.13)

so the volume of a fluid element varies as a3_.

As will be seen below, one more inhomogeneity variable will be needed. By using the traceless part of the 3-Ricci tensor

(3)_S bc = (3)Rbc− 1 3 (3)_Rh bc= − ˙σhbci− Θσbc+ ˜∇ha˙ubi, (4.14)

we define the auxiliary variable

Sa ≡ a ˜∇a

σbcSbc

, (4.15)

which is also a measure of inhomogeneity. Note that even though Sab is zero on the

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4.2 Evolution of Inhomogeneity

4.2.1. Evolution of the perturbation variables

The propagation equations for the gradients are obtained by taking the gradients ˜∇a

of the propagation equations in section 4.1 and then using the commutator between ’time’ and ’spatial’ derivatives acting on a scalar f , that to first order reduces to

˜ ∇af · = ˜∇af + ˙u˙ af −˙ 1 3Θ ˜∇af − σ b a∇˜bf, (4.16) as was shown in [18].

Taking the time derivative of Equation 4.10 by using Equation 4.1, Equation 4.6, Equation 4.8 and Equation 4.16, we obtain the following evolution equation

˙

Da+ σabDb+ γZa− (γ − 1) ΘDa= 0. (4.17)

The propagation of the comoving spatial derivative expansions in Equation 4.11 re-quires Equation 4.2, Equation 4.6, Equation 4.8 and Equation 4.16. The net result is the evolution equation

˙ Za+ 2 3ΘZa+ σ b aZb+ 2Ta+ 1 2µDa+ γ − 1 γ ! h ˜ ∇b∇˜b− 3σ2 i Da = 0, (4.18) where ˜∇a_∇_˜

a is the Laplace-Beltrami operator that generalises the Laplacian to a

curved space.

The propagation of the shear in Equation 4.12 requires Equation 4.2, Equation 4.6, Equation 4.8 and Equation 4.16, and reads

˙ Ta+2ΘTa+σabTb+2σ2Za−2 γ − 1 γ ! µΘσ2Da+Sa− γ − 1 γ ! σc_b∇˜a∇˜bDc= 0. (4.19)

This equation involves the new variable Sa. The evolution of this variable requires

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For details, see section A.1.

The system of equations reduces to the findings in [10] in the dust case, for which the equations reduce to a system of ordinary differential equations in time if Hab = 0.

In [10] it was also shown that this constraint is propagated for dust. Hab = 0 implies

that there is no information exchange via gravitational waves, such space-times are referred to as silent universes [14].

4.3. Isotropic Case

From the first-order equations in subsection 4.2.1 a single 4:th order for Da can be

derived in the generic case. For a dust background with no shear the equations reduce to ˙ Da+ Za = 0, (4.21) ˙ Za+ 2 3ΘZa+ 1 2µDa = 0, (4.22)

from which in this case, a second-order equation for Da can be derived,

¨ Da+ 2 3Θ ˙Da− 1 2µDa= 0, (4.23)

4.3.1. de Sitter Model

de Sitter model is a steady state solution in a constant curvature space-time. It is empty, because µ + p = 0, i.e., it does not contain ordinary matter, but rather a cosmological constant. It might seem unphysical to consider density perturbations in a vacuum solution, but the solutions might approximate perturbations around some background metrics with p µ Λ.

We find that Equation 4.23 reduces to ¨

Da+

2

3Θ ˙Da= 0, (4.24)

where the expansion is given by

Θ (t) =√3Λ. (4.25)

The solution for the growth of the energy density inhomogeneity is then

Da = C1exp  −2 s Λ 3t  + C₂, (4.26)

where C1 and C2 are constants, and we obtain that Da approaches a constant value

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4.3 Isotropic Case

4.3.2. Dust Universe without cosmological constant

For a flat FLRW dust universe, without Λ and with scale factor a = a0t2/3, the

energy density and expansion are given by µ(t) = _3t42 and Θ(t) =

2

t, respectively.

Equation 4.23 is the usual equation for growth of energy density inhomogeneity in dust universes without cosmological constant. The solutions to Equation 4.23 is then

Da= f1a(xα) t

2

3 + f_2a(xα) t−1, f˙_ia= 0,

where t is proper time along the flow lines [11], and we obtain that for late times the energy density perturbations grow unboundedly.

4.3.3. Dust Universe with cosmological constant

In the case of a dust universe with cosmological constant, Λ 6= 0, the solutions to the background propagation equations are

µ(t) = − 4ΛC exp √ 3Λt exp√3Λt+ C2 , (4.27) Θ (t) = √ 3Λ   exp√3Λt− C exp√3Λt+ C  , (4.28)

where C is a constant of integration.

We obtain that µ(t) → 0 and Θ(t) →√3Λ when t → +∞.

If C = −1 we have µ(t) > 0 and the Big Bang occurs at t = 0. By performing a Taylor expansion for low t we find that

µ(t) ≈ _√4Λ 3Λt2 = 4 3t2, (4.29) Θ(t) ≈√3Λ 1 + √ 3Λt + 1 1 +√3Λt − 1 ! ≈ 2 t, (4.30)

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5. Numerical Solutions

5.1. Equations

The propagation equations of the backgrounds are given in section 3.4, and because of the assumption that the background is spatially flat the energy density can be expressed in terms of the expansion, the shear scalar and the cosmological constant by using Equation 2.37 and Equation A.26 as

µ = 1

3Θ

2 ₋3

4Σ

2_{− Λ.} _(5.1)

The evolution equations of inhomogeneity have been worked out in subsection 4.2.1, where we choose a frame such that the background shear tensor becomes diagonal, and restricting to LRS this reduced to one independent component

σ_ab = diag Σ, −1 2Σ, − 1 2Σ . (5.2)

By using plane-wave harmonics as described in [6], first-order perturbations of the energy density can be written as

˜ ∇b_∇_˜ bDa = − k₁2 a2 1 + 2k 2 2 a2 2 ! Da, (5.3)

where k1 and k2 are the comoving wave numbers and a1 and a2 are the scale factors

in the preferred directions. Detailed derivations of the harmonic decomposition of plane waves can be find in section A.2. Furthermore, note that to first-order we also have σ_bc∇˜a∇˜bDc = −Σ k2 1 a2 1 − k 2 2 a2 2 ! Da. (5.4)

The evolution equations of the scale factors can be expressed in terms of the expan-sion and the shear scalar

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The system of equations in the first preferred direction is ˙ D1 = −ΣD1− γZ1+ (γ − 1) ΘD1, ˙ Z1 = − 2 3ΘZ1− ΣZ1− 2T1− 1 2µD1+ 9 4 γ − 1 γ ! Σ2D1+ γ − 1 γ ! k2 1 a2 1 + 2k 2 2 a2 2 ! D1, ˙ T1 = −2ΘT1− ΣT1− 3 2Σ 2_Z 1− S1+ 3 2 γ − 1 γ ! ΘΣ2D1− γ − 1 γ ! k2 1 a2 1 − k 2 2 a2 2 ! ΣD1, ˙ S1 = − 5 3ΘS1− 1 2Σ 2_T 1+ 1 3ΘΣ 2_Z 1− 1 2µΣ 2_D 1− 1 2ΣS1 −1 2 γ − 1 γ ! k2 1 a2 1 + 2k 2 2 a2 2 ! Σ2D1+ 1 3 γ − 1 γ ! k2 1 a2 1 − k 2 2 a2 2 ! µΘΣD1 −χ " −3 2µΣ 2_D 1+ ΘΣ2Z1− 3 2Σ 2_T 1 − 3 2ΣS1+ k2₁ a2 1 − k 2 2 a2 2 ! ΣZ1− k₁2 a2 1 + 2k 2 2 a2 2 ! T1 # . (5.7)

The system of equations in the second preferred direction is ˙ D2 = 1 2ΣD2− γZ2+ (γ − 1) ΘD2, ˙ Z2 = − 2 3ΘZ2+ 1 2ΣZ2− 2T2− 1 2µD2+ 9 4 γ − 1 γ ! Σ2D2+ γ − 1 γ ! k2 1 a2 1 + 2k 2 2 a2 2 ! D2, ˙ T2 = −2ΘT2+ 1 2ΣT2− 3 2Σ 2_Z 2− S2+ 3 2 γ − 1 γ ! ΘΣ2D2− γ − 1 γ ! k2 1 a2 1 − k 2 2 a2 2 ! ΣD2, ˙ S2 = − 5 3ΘS2− 1 2Σ 2 T2+ 1 3ΘΣ 2 Z2− 1 2µΣ 2 D2+ ΣS2 −1 2 γ − 1 γ ! k₁2 a2 1 + 2k 2 2 a2 2 ! Σ2D2+ 1 3 γ − 1 γ ! k₁2 a2 1 − k 2 2 a2 2 ! µΘΣD2 −χ " −3 2µΣ 2 D2+ ΘΣ2Z2− 3 2Σ 2 T2 − 3 2ΣS2+ k2 1 a2 1 − k 2 2 a2 2 ! ΣZ2− k2 1 a2 1 + 2k 2 2 a2 2 ! T2 # . (5.8)

Note that there is a coupling between the two directions, because of the relation

X2 = _a 1 a2 _k 2 k1 ! X1, (5.9)

where X is D, Z, T or S, and the derivation of this coupling relation can be found in section A.2.

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5.2 Background Propagation Equations

preferred direction will now be named as the perpendicular directions, with the labels D⊥, Z⊥, etc.

The time evolution of the density gradients are solved numerically in some cases. Since the system contains four different modes for each background the initial condi-tions can be chosen in many ways. In this work just an initial density perturbacondi-tions is assumed.

Throughout this section we choose the cosmological constant Λ = 1. Other initial values are given in the figure or text.

5.2. Background Propagation Equations

Figure 5.1 shows how the background propagation equations behave in backgrounds consisting of dust and radiation. The behavior are very similar, and one see that for large times the background solutions approach the point dS, which has been described in Table 3.1 as a sink point. A sink point is an equilibrium point that is a late-time attractor, i.e., when t → +∞ [14]. The expansion rate approaches a constant value, Θ = √3Λ, and both the density and the shear approach zero. The propagation equations are not shown in the case of negative shear, but from section 3.4 one can see that the behavior of the background propagation equations is the same with the only difference that the sign of the shear is negative. Hence, we obtain that the shear approaches zero regardless the sign of it, and therefore for large times the background solutions isotropise.

(a) Dust background, where the pressure is zero (b) Radiation background, where the pressure is

given by p = 1 3µ

Figure 5.1.: The background quantities energy density, expansion and shear. Initial values at t0 = 1 are given by µ0 = 0.2, Θ0 = 3, Σ0 = 2

q

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5.3. Dust

The time evolution of the density gradients are solved for numerically in the dust case, where γ = 1, and no consideration to wave number is required as can be seen from Equation 4.17, Equation 4.18, Equation 4.19 and Equation 4.20 with Hab = 0.

As mention earlier, we assume just an initial density perturbation, D0 = 0.001, and

hence Z0 = T0 = S0 = 0.

Figure 5.2 shows how the density gradient behaves in the dust solution. In part (a) we see that with positive shear Σ0 the density gradient in the anisotropic direction

decreases towards a constant value, and for the perpendicular direction the behavior is inversed and with different magnitude. If the shear Σ0 is negative the density

gradient in the anisotropy direction grows towards a constant value, and the behavior of the perpendicular direction is inversed but with different magnitude, which can be seen in part (b).

(a) Density gradients with positive shear (b) Density gradients with negative shear

Figure 5.2.: The growth of the density perturbation Dk and D⊥, where the initial

values at t0 = 1 are given by µ0 = 0.2, Θ0 = 3, Σ0 = ±2

q

3

5, and the pressure is

zero.

Note that all perturbations asymptotically approach constant values. This is consis-tent with that the background asymptotically approaches de Sitter, which we also have described in section 5.2.

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5.4 Radiation

times this universe approaches de Sitter model, and the numerical results coincide with the analytical analysis in subsection 4.3.1.

5.4. Radiation

The time evolution of the density gradients are solved numerically in the radiation case, where γ = 4

3, and wave numbers are required because spatial derivatives are

introduced in the evolution equations of inhomogeneities. In a similar way as in the dust case an energy density perturbation is assumed, whereas the other per-turbations are initially zero. The wave numbers in the preferred directions, kk and

k⊥, are initially assumed to be equal, and we can therefore define k ≡

kk

a10 =

k⊥

a20. The initially values of the scale factors are given by a10 = a1(t0) and a20 = a2(t0).

Furthermore, from Equation 5.9 we have that

D⊥= _a 10 a20 _k ⊥ kk ! Dk, (5.10)

and therefore we only show Dk.

In Figure 5.3 the growth of density perturbations in the anisotropic direction are shown for different values of the comoving wave number k. The initial values of the density perturbations at t0 = 1 are given by D0 = 0.001.

In part (a) we can see that for k = 1 the density gradient in the anisotropic direction reaches a minimum before it starts growing unboundedly, and in the perpendicular direction the density gradient increases without reaching any extremum point. For higher values of the wave number k the density gradients in the anisotropy direction show an oscillatory behavior with decreasing amplitude, and it damps off but does not look to fall off to zero. For corresponding k-values in the perpendicular direction the amplitudes intially increasing and thereafter start to damp off in the same way as described in the anisotropy direction. In the case of negative shear and k = 1 the density gradient in the anisotropic direction grows unboundedly without reaching any extremum point, which can be seen in part (b). Furthermore, for high k-values the density gradients in the anisotropic direction have oscillations with increasing amplitude before they are damped off. The energy density for k = 1 in the perpen-dicular direction behaves in the same way as described in the anisotropy direction. For higher k-values in the perpendicular direction the oscillations initially have an approximately constant amplitude that with time slowly start decreasing.

The same scenario that have been described for both positive and negative shear occur even when the initially values of the scale factors are different, this can also be seen as the wave numbers are different, kk/a10 6= k⊥/a20, and because of

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(a) Density gradients with positive shear Σ0 (b) Density gradients with negative shear Σ0

Figure 5.3.: The growth of the density perturbation Dk, where the initial values

q

3

5, and for the wave numbers

k = kk/a10= k⊥/a20= 1, 5 and 20. The pressure is given as p = 1₃µ.

(a) Density gradients with positive shear Σ0 (b) Density gradients with negative shear Σ0

Figure 5.4.: The growth of the density perturbation Dk with and without different

scale factors, where the initial values at t0 = 1 are given by µ0 = 0.2, Θ0 = 3,

Σ0 = ±2

q

3

5, and the pressure is given as p = 1

3µ. The case with equal scale factors

are given by kk/a10 = k⊥/a20 = 20, and the case with different scale factors are

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5.4 Radiation

consideration to different initial values of scale factors, where we have the ratio

a10/a20 = 5. The comoving wave number for each direction are the same, and

therefore D⊥ = 5Dk. In the case of positive shear the amplitude of the density

gra-dient decreasing more slowly with different scale factors in comparison with equal scale factors, and we note also that the wavelength of the oscillation increases with different initial values of the scale factors, which can be seen in part (a). In the perpendicular direction the amplitude of the density gradients have an approxi-mately constant value that with time slowly starts decreasing, but similar to the anisotropic direction the wavelength of the oscillation increases with time. In part (b) the same pattern for negative shear Σ0 as seen for positive shear is obtained,

but the increase of wavelength with time occur more rapidly. In general oscillation in different directions are found to get out of phase with time.

If the magnetic part of the Weyl tensor is nonzero it introduces additional terms in the equation for the auxiliary variable Sa. The behavior of these terms in the

anisotropic direction can be seen in Figure 5.5. For high k-values those terms show an oscillatory behavior that damps off and approaches constant values, and we notice that the additional terms are nonzero regardless the wave number, an therefore we have that Hab 6= 0. Hence, from Equation 2.41 we obtain that curlσab 6= 0.

(a) Additional terms with positive shear Σ0 (b) Additional terms with negative shear Σ0

Figure 5.5.: Additional terms in Equation 4.20 when Hab 6= 0 for the wave numbers

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6. Conclusions

A tetrad approach has been used to find exact solutions of LRS Bianchi type I, and those are dust universes with and without cosmological constant. Analysis of these solutions shows that they are valid and approaches expected results in certain boundaries. At late times the model approaches isotropy and describes an empty Universe.

A closed system for scalar perturbation on Bianchi type I cosmologies has been found in terms of gauge invariant variables. The extension to include the pressure in the cosmological model introduce terms with spatial derivatives in the evolution equations for the perturbative quantities in comparison to the findings in [10]. Due to the complexity of the governing equations, the choice of background, initial conditions and wave numbers, many different behaviors of the growth of the density perturbations can be obtained. In general the growth of density gradients is different in the direction of the anisotropy and the perpendicular direction, which should influence the formation of structures.

For long wavelength the presence of pressure makes the density gradients to grow un-boundedly in the preferred directions. For short wavelength the presence of pressure creates an oscillatory behavior for the energy density perturbations in the anisotropic and perpendicular directions, whereas the sign of the shear determine if the ampli-tude of the density gradient in the preferred directions initially increases or decreases. The oscillation damps off, but the amplitude of the density perturbations do not appear to approach zero. For late times the density gradients start to grow un-boundedly, which can be caused by numerical instabilities or real physical behavior, and this issue is discussed more in section B.2.

Density gradients with initially different scale factors show oscillatory behavior where the phase shift increase with time in comparison with initially equal scale factors.

Based on the numerical solutions it is possible to conclude that the magnetic part of the Weyl tensor is non-zero in the generalisation of the dust model to a cosmological model with pressure.

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Acknowledgments

I want to take this opportunity to thank my supervisor Michael Bradley at Umeå University, for his support and assistance in this thesis project. I can without a doubt say that without his help this thesis project would not have been finished in time.

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A. Detailed calculations

A.1. Evolution of the perturbation variables

In this section we present a more detailed calculation of the perturbative variables defined in subsection 4.2.1.

A.1.1. Density gradient D

a

Taking the time derivative of the comoving density gradient in Equation 4.1 and using Equation 4.16, we find that

˜ ∇aµ · − ˜∇aµ = ˙u˙ aµ −˙ 1 3Θ ˜∇aµ − σ c a∇˜cµ, (A.1)

and by using Equation 4.1, Equation 4.10 and Equation 4.13 we obtain

˜ ∇aµ · = −1 a(µ + p) ΘDa− 1 3aµΘDa+ 1 aµ ˙Da. (A.2)

Furthermore, use Equation 4.1 and Equation 4.11 to find ˜ ∇aµ = −˙ 1 a(1 + p 0 ) µΘDa− 1 a(µ + p) Za, (A.3)

and Equation 4.7 and Equation 4.10 to obtain ˙uaµ =˙

1

aµΘp

0

Da. (A.4)

Use Equation A.2, Equation A.3 and Equation A.4 in Equation A.1, multiply with

a and then use Equation 4.10 to obtain,

˙ Da− Θp µ Da+ 1 + p µ ! Za+ σabDb = 0. (A.5)

With the equation of state Equation A.5 becomes ˙

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A.1.2. Expansion gradient Z

a

Take the time derivative of the comoving spatial derivative expansion in Equation 4.11 by using Equation 4.16, to find that

˜ ∇aΘ · − ˜∇aΘ = ˙u˙ aΘ −˙ 1 3Θ ˜∇aΘ − σ c a∇˜cΘ. (A.7)

With help of Equation 4.2, Equation 4.7, Equation 4.10 and Equation 4.11, the terms in Equation A.7 can be written as

˜ ∇aΘ · = 1 3aΘZa+ 1 a ˙ Za, (A.8) ˜ ∇aΘ = −˙ µp0 a (µ + p) ˜ ∇a∇˜bDb− 2 aTa− 1 2aµDa− 3 2aµp 0 Da, (A.9) ˙uaΘ =˙ µΘ2p0 3a (µ + p)Da+ 2µp0σ2 a (µ + p)Da+ µ2p0 2a (µ + p)Da+ 3µpp0 2a (µ + p)Da− p0µ a (µ + p)ΛDa, (A.10) respectively. Substitute Equation A.8, Equation A.9 and Equation A.10 into Equation A.7 multiply with a and then use Equation 4.8 and Equation 4.11 to obtain,

˙ Za+ 2 3ΘZa+σ b aZb+ 1 2µDa+2Ta+ γ − 1 γ ! µ − 1 3Θ 2_{− 2σ}2_{+ Λ}_D a+ γ − 1 γ ! ˜ ∇a∇˜bDb = 0. (A.11) In this case ωab = 0, and on using the Ricci identities for the ˜∇-derivatives and the

zeroth-order relation (3)Rab = 1₃

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Rhab for the 3-dimensional Ricci tensor, we find

that ˜ ∇a∇˜bDb = ˜∇b∇˜bDa− 1 3     2µ − 2 3Θ 2_{+ 2σ}2_{+ 2Λ} | {z } =0     Da, (A.12)

Density Growth in Anisotropic Cosmologies of Bianchi Type I