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 SE-901 87 UMEÅ SWEDEN

Abstract

This work generalises earlier works on the growth of the density perturbations in Bianchi type I cosmologies filled with dust to include also the effects of pressure and a positive cosmological constant. For the analysis the 1 + 3 covariant split of space- time formalism is used. As the perturbative variables we use scalar quantities that are zero on the background, and hence are gauge-invariant. These variables form a coupled closed system of first-order evolution equations, that is analysed numerically and analytically. For short wavelengths an oscillatory behavior is obtained, whereas for long wavelengths the energy density perturbations grow unboundedly.

Contents

1. Introduction 3 1.1. Problem Statement and Objectives ...... 4

2. Cosmological Models 5 2.1. 1 + 3 Covariant Description ...... 6 2.1.1. Variables ...... 6 2.1.2. 1 + 3 Covariant Propagation and Constraint Equations . . . . 8 2.1.3. Irrotational Flow ...... 11 2.2. Tetrad Description ...... 12 2.2.1. Tensor and Wedge Product ...... 13 2.2.2. Exterior Diﬀerentiation: Structure Coeﬃcients ...... 13 2.2.3. First and Second Cartan Structure Equations ...... 14 2.3. Bianchi Models ...... 15 2.3.1. Bianchi Type I ...... 16

3. Background Solutions of Bianchi Type I 19 3.1. Evolution Equations in Tetrad Approach ...... 19 3.2. Evolution Equations and State Space ...... 21 3.3. LRS Bianchi Type I ...... 22 3.3.1. Dust solution without cosmological constant ...... 23 3.3.2. Dust solution with cosmological constant ...... 24 3.4. Evolution Equations in 1 + 3 Formalism ...... 24

4. Density Perturbations 27 4.1. Propagation Equations ...... 27 4.2. Evolution of Inhomogeneity ...... 28 4.2.1. Evolution of the perturbation variables ...... 29 4.3. Isotropic Case ...... 30 4.3.1. de Sitter Model ...... 30 4.3.2. Dust Universe without cosmological constant ...... 31 4.3.3. Dust Universe with cosmological constant ...... 31

5. Numerical Solutions 33 5.1. Equations ...... 33 5.2. Background Propagation Equations ...... 35 5.3. Dust ...... 36

i Chapter 0 Contents

5.4. Radiation ...... 37

6. Conclusions 41

Acknowledgments 43

A. Detailed calculations 45 A.1. Evolution of the perturbation variables ...... 45 A.1.1. Density gradient Da ...... 45 A.1.2. Expansion gradient Za ...... 46 A.1.3. Shear gradient Ta ...... 47 A.1.4. Auxiliary variable Sa ...... 47 A.2. Harmonic decomposition ...... 50

B. Numerics 53 B.1. Test of Code ...... 53 B.2. Numerical Instabilities ...... 54

Bibliography 57

ii List of Figures

5.1. The background quantities energy density, expansion and shear. Ini- q 3 tial values at t0 = 1 are given by µ0 = 0.2, Θ0 = 3, Σ0 = 2 5 . . . . . 35 5.2. The growth of the density perturbation Dk and D⊥, where the initial q 3 values at t0 = 1 are given by µ0 = 0.2, Θ0 = 3, Σ0 = ±2 5 , and the pressure is zero...... 36 5.3. The growth of the density perturbation Dk, where the initial values q 3 at t0 = 1 are given by µ0 = 0.2, Θ0 = 3, Σ0 = ±2 5 , and for the wave numbers k = kk/a10 = k⊥/a20 = 1, 5 and 20. The pressure is 1 given as p = 3 µ...... 38 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, q 3 1 Θ0 = 3, Σ0 = ±2 5 , and the pressure is given as p = 3 µ. 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 at t0 = 1 are given by µ0 = 0.2, Θ0 = 3, Σ0 = ±2 5 , and for the wave numbers k = kk/a10 = k⊥/a20 = 5 and 20. The pressure is given 1 as p = 3 µ...... 54

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 authors, 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 depending 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 possible to define perturbation variables that are zero on the background, and these variables are gauge invariant [9].

This thesis consist of six chapters. The current chapter specifies the goals of the thesis project. Chapter 2 describes a 1+3 covariant approach and a tetrad approach, where the first approach is based on the works by George F R Ellis, Marco Bruni and Henk van Elst, and for second approach the standard work by Michael P. Ryan and Lawrence C. Shepley is used as reference. In chapter 3 the field equations for cosmological models of Bianchi type I are given and analysed, and some exact solutions of LRS type are then presented. Chapter 4 shows the theoretical results, and the numerical results are presented in chapter 5. In chapter 6 we draw conclusions from both the theoretical and numerical results.

3 Chapter 1 Introduction

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. A goal of this thesis project is to derive some cosmological solutions of Bianchi type I using the tetrad formalism, where a combination of analytical and numerical methods will be used in the absence of exact solutions. These will then be used as backgrounds for the study of density perturbations in anisotropic universes. The earlier work by Osano [10] on the growth of density perturbations in Bianchi type I cosmologies filled with dust are then extended to include the effects of pressure and a cosmological constant.

4 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 relativity. 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- µ termined by the metric gab (x ). Furthermore, the matter is presented on the same averaging scale, and its physical behavior must represent physically plausible matter. The interaction between geometry and matter is described through Einstein’s relativistic gravitational ﬁeld equations given by 1 G ≡ R − Rg = T − Λg , (2.1) ab ab 2 ab ab ab where Gab is the Einstein tensor, Rab the Ricci curvature scalar, R the scalar curvature, 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

ab ab ∇bG = 0 ⇒ ∇bT = 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. Space-time can be described via 1+3 covariantly defined variables and metric, where the latter can be based on a particular set of local coordinates or a particular tetrad. Here we will concentrate on the 1 + 3 covariant approach, but dealing also with the tetrad approach which serves as a completion to the 1 + 3 covariant approach. The basic point here is that because we have complete coordinate freedom in general relativity, it is preferable to describe physics and geometry by tensor relations and

5 Chapter 2 Cosmological Models 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 preferred timelike direction, like the 4-velocity of matter of a preferred set of observers. The kinematic quantities of the 4-velocity, together with the energy-momentum tensors 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 equations included via algebraic relations 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 preferred motion of matter and hence a unique 4-velocity. This corresponds to preferred wordlines, and in each case the 4-velocitiy is

dxa ua = , u ua = −1, (2.3) dt a 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

a a a c a a b a Ub = −u ub ⇒ Uc Ub = Ub ,Ua = 1,Uabu = u , (2.4) and

a c a a b hab = gab + uaub ⇒ hc hb = hb , ha = 3, habu = 0, (2.5) where the ﬁrst projects parallel to the 4-velocity vector ua, and the second projects a b onto the local 3-dimensional rest-space orthogonal to u . Projections with ha of

6 2.1 1 + 3 Covariant Description

hai a b vectors are denoted by angle brackets v ≡ hb v , and the projected symmetric tracefree (PSTF) of a tensor is given by 1 T habi ≡ h(ahb) − habh T cd. (2.6) c d 3 cd There is also a volume element for the rest-spaces,

d c ηabc = u ηdabc ⇒ ηabc = η[abc], ηabcu = 0, (2.7) q where ηabcd is the 4-dimensional volume element (ηabcd = η[abcd], η0123 = | det gab|). We can deﬁne the covariant time derivative along the fundamental wordlines, where for any tensor T ab,

˙ ab c ab T ≡ u ∇cT , (2.8)

˜ a and the orthogonal projected covariant derivative ∇a, where for any tensor Tb , ˜ a a e f d ∇cTb ≡ hdhbhc ∇f Te , (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 1 ∇ u = −u u˙ + ∇˜ u = −u u˙ + Θh + σ + ω , (2.10) a b a b a b a b 3 ab ab ab

b where the acceleration is defined as u˙ a ≡ u ∇bua, and the expansion of the fluid ˜ a ˜ is Θ = ∇au . Furthermore σab ≡ ∇haubi is the tracefree symmetric shear tensor b a (σabu = 0, σa = 0, σab = σ(ab)) describing the rate of distortion of the matter flow, ˜ a and ωab ≡ ∇[aub] is the skew-symmetric vorticity tensor (ωab = ω[ab], ωabu = 0), describing the rotation of the matter relative to a non-rotating frame.

2.1.1.4. Matter Tensor

a The matter energy-momentum tensor Tab can be decomposed relative to u in the form

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

a b a a b ca where µ = Tabu u is the relativistic energy density relative to u , q = −Tbcu h is the relativistic momentum density that corresponds to the energy ﬂux relative

7 Chapter 2 Cosmological Models

a 1 ab c d to u , p = 3 Tabh is the isotropic pressure, and πab = Tcdhhahbi is the tracefree anisotropic pressure. The energy momentum tensor of a perfect ﬂuid is obtained with the restriction

a q = 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

1 1 R = C + (g S + g S − g S − g S )+ R (g g − g g ) , (2.14) abcd abcd 2 ac bd bd ac ad bc bc ad 12 ac bd ad bc where Sab is the traceless part of the Ricci tensor

1 S ≡ R − Rg , (2.15) ab ab 4 ab

a with the property Sa = 0.

The Weyl curvature tensor Cabcd is the traceless part of the Riemann tensor, i.e. a a Cbad = 0. The Weyl tensor is split relative to u into electric (Eab) and magnetic (Hab) parts according to

c d a b Eab ≡ Cacbdu u ⇒ Ea = 0,Eab = E(ab),Eabu = 0, (2.16) 1 H ≡ η Cdeuc ⇒ Ha = 0,H = H ,H ub = 0. (2.17) ab 2 ade bc a ab (ab) ab

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 equation 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

There are three sets of equations to be considered from Einstein’s ﬁeld equations and their associated integrability conditions.

8 2.1 1 + 3 Covariant Description

2.1.2.1. Ricci Identities

The ﬁrst set arise from the Ricci identities for the vector ﬁeld ua, i.e.,

d (∇a∇b − ∇b∇a) uc = Rabcdu . (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 1 1 Θ˙ − ∇˜ u˙ a = − Θ2 + (u ˙ u˙ a) − 2σ2 + 2ω2 − (µ + 3p) + Λ, (2.19) a 3 a 2 which is the basic equation of gravitational attraction. 2. The vorticity propagation equation 1 2 ω˙ hai − ηabc∇˜ u˙ = − Θωa + σaωb, (2.20) 2 b c 3 b giving a vorticity conservation law for a perfect ﬂuid with acceleration potential Φ. 3. The shear propagation equation 2 1 σ˙ habi −∇˜ hau˙ bi = − Θσab +u ˙ hau˙ bi −σhaσbic −ωhaωbi − Eab − πab , (2.21) 3 c 2

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 2 h i 0 = (C )a = ∇˜ σab − ∇˜ aΘ + ηabc ∇˜ ω + 2u ˙ ω + qa, (2.22) 1 b 3 b c b c 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 ˜ a a 0 = (C2) = ∇aω − (u ˙ aω ) . (2.23)

3. The Hab-equation ab ab ha bi ˜ ha bi ab 0 = (C3) = H + 2u ˙ ω + ∇ ω − (curlσ) , (2.24) characterising the magnetic Weyl curvature as being constructed from the ab cdha ˜ bi ’distortion’ of the vorticity and the ’curl’ of the shear, curl σ = η ∇cσd .

9 Chapter 2 Cosmological Models

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

˜ a a ab µ˙ + ∇aq = −Θ(µ + p) − 2 (u ˙ aq ) − σabπ , (2.25) and 4 q˙hai + ∇˜ ap + ∇˜ πab = − Θqa − σaqb − (µ + p)u ˙ a − u˙ πab − ηabcω q , (2.26) b 3 b b b c which constitute the energy conservation equation and the momentum conservation equation, respectively. For a perfect ﬂuid, characterised by Equation 2.13, the propagation equations reduce to

µ˙ = −Θ(µ + p) , (2.27) and the constraint equation ˜ 0 = ∇ap + (µ + p)u ˙ a. (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 1 1 1 1 E˙ habi + π˙ habi − curlHab − curlπab = − (µ + p) σab − Θ Eab + πab 2 2 2 6 1 + 3σha Ebic − πbic − u˙ haqbi c 6 1 + ηcdha 2u ˙ Hbi + ω Ebi + πbi , c d c d 2 d (2.30) and the H˙ -equation 1 3 H˙ habi + curlEab − curlπab = − ΘHab + 3σhaHbic + ωhaqbi 2 c 2 1 − ηcdha 2u ˙ Ebi − σbiq − ω Hbi , (2.31) c d 2 c d c d

10 2.1 1 + 3 Covariant Description and the ’curls’ are defined as ab cdha ˜ bi curlV = η ∇cVd , (2.32) where V ab is an arbitrary tensor. Equation 2.30 and Equation 2.31 show how gravitational radiation arises, and together they can be used to find a wave operator acting on Eab and Hab, respectively. The constraint equations are the (divE)-equation 1 1 1 1 0 = (C )a =∇˜ Eab + πab − ∇˜ aµ + Θqa − σaqb − 3ω Hab 4 b 2 3 3 2 b b 3 − ηabc σ Hd − ω q , (2.33) bd c 2 b c where one source term is the spatial gradient of the energy density, and the (divH)- equation 1 0 = (C )a =∇˜ Hab + (µ + p) ωa + 3ω Eab − πab 5 b b 6 1 1 + ηabc ∇˜ q + σ Ed + πd , (2.34) 2 b c bd c 2 c where the fluid vorticity and shear act as source terms.

2.1.3. Irrotational Flow

For a barotropic perfect fluid we have a abc ˜ 0 = q = πab, p = p(µ) ⇒ η ∇au˙ c = 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- b tion t such that ua = −g x ∇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 1 2 3R = −σ˙ − Θσ + ∇˜ u˙ +u ˙ u˙ + π + h 2µ − Θ2 + 2σ2 + 2Λ , ab habi ab ha bi ha bi ab 3 ab 3 (2.36)

2 1 ab where σ ≡ 2 σ σab, and Λ is the cosmological constant. The 3-space Ricci scalar is given by 2 3R = 2µ − Θ2 + 2σ2 + 2Λ, (2.37) 3

11 Chapter 2 Cosmological Models

which is a generalised Friedmann equation, showing how the matter tensor determines the 3-space average curvature. These equations fully determine 3 the curvature tensor Rabcd of the orthogonal 3-spaces. In the case of irrotational ﬂow Equation 2.19 and Equation 2.21 reduce to the following propagation equations,

1 1 Θ˙ − ∇˜ u˙ a = − Θ2 +u ˙ u˙ a − 2σ2 − (µ + 3p) + Λ, (2.38) a 3 a 2 2 σ˙ habi − ∇˜ hau˙ bi = − Θσab +u ˙ hau˙ bi − σhaσbic − Eab, (2.39) 3 c

abc ˜ Equation 2.20 reduces to η ∇bu˙ c = 0, and that is identically satisﬁed if p = p (µ). The constraint equations, Equation 2.22 and Equation 2.24, reduce to 2 0 = (C )a = ∇˜ σab − ∇˜ aΘ, (2.40) 1 b 3 and

ab ab cdha ˜ bi 0 = (C3) = H − η ∇cσd , (2.41) respectively. The twice-contracted Bianchi identities can now be written in the form

˜ a a ab µ˙ + ∇aq = −Θ(µ + p) − 2 (u ˙ aq ) − σabπ , (2.42) ∇˜ ap = − (µ + p)u ˙ a, (2.43) and the other Bianchi identities yield 1 E˙ habi − curlHab = − (µ + p) σab − ΘEab + 3σhaEbic + 2ηcdhau˙ Hbi, (2.44) 2 c c d ˙ habi ab ab ha bic cdha bi H + curlE = −ΘH + 3σc H − 2η u˙ cEd . (2.45) In summary, we have a ﬁrst-order system of equations that consists of six propagation equations and six constraint equations. The system of equations 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. In the tetrad formalism all tensors are represented in terms of a chosen basis, and the advantage of the tetrad formalism over the standard coordinated-based approach to

12 2.2 Tetrad Description general relativity lies in the ability to choose the tetrad basis to reﬂect important physical aspects of the space-time. µ ∂ It is suitable to use a constant metric ηij in a frame, for example Xi = Xi ∂xµ , where i i µ i = 0, 1, 2, 3. The dual basis can be written in the form ω = ωµdx , and we have i µ i the relation ωµXj = δj. Furthermore, the line element takes the form

2 i j i j µ ν ds = ηijω ⊗ ω = ηijω ω = gµνdx dx , (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 1 ω ∧ σ = (ω ⊗ σ − σ ⊗ ω) . (2.48) 2

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: µ i i 1. f scalar: df = f,µdx = Xi (f) ω ≡ f|iω ; 2. d on r-forms gives a r + 1 form; 3. d (dω) = 0; 4. d (ω ∧ σ) = dω ∧ σ + (−1)r ω ∧ dσ (Note that functions are zero-forms). Here r is the order of ω. i Let {ω } be a basis of one-forms dual to a basis {Xi} which has non-zero commutators. The curl of any ωi is a two-form dωi and hence a linear combination of the basis of two-vectors {ωi ∧ ωj}:

i i j k dω = Djkω ∧ ω . (2.49)

13 Chapter 2 Cosmological Models

i i It can be shown that Djk are related to the structure coeﬃcients Cjk of

i [Xi,Xj] = CjkXi (2.50) in the following way, 1 Di = − Ci . (2.51) jk 2 jk

2.2.3. First and Second Cartan Structure Equations

Covariant diﬀerentiation is deﬁned as

µ ν Ai;j ≡ Aµ;νXi Xj , (2.52) and can be decomposed in the following way

k Ai;j = Ai|j − γijAk, (2.53)

k where γij are the Ricci rotation coeﬃcients. These can be written in the form

k k µ ν k µ ν γij = ωµXi;νXj = −ωµ;νXi Xj , (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

l γijk ≡ ηilγjk. (2.56)

Hence, we can deﬁne the connection forms,

i i k ωj ≡ γjkω , (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 ] = LU V, (2.59) where LU is the Lie derivative. For example, the Lie derivative of a tensor ﬁeld T αβ with respect to the vector ﬁeld X, where T has the components Tγδ , is given by

αβ αβ σ αβ σ ασ β αβ σ αβ σ (LX T )γδ = Tγδ,σa − Tγδ a,σ − Tγδ a,σ + Tσδ a,γ + Tγσ a,δ (2.60)

14 2.3 Bianchi Models

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

i i i Cjk = γkj − γjk, (2.61) where the structure coefficients are defined by Equation 2.51 [12]. This relation implies 1 dωi = − Ci ωj ∧ ωk = −γi ωk ∧ ωj, (2.62) 2 jk jk and the definition of the connection forms (Equation 2.57) makes it possible to write it in the form,

i k i dω = ω ∧ ωk, (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 1 dωi = −ωi ∧ ωk + Ri ωk ∧ ωl, (2.64) j k j 2 jkl

i where Rjkl 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 definition 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

ξi;j + ξj;i = 0. (2.65)

15 Chapter 2 Cosmological Models

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

k [ξi, ξj] = Cijξk, (2.66) where the commutator is deﬁned as

[ξi, ξj] ≡ ξiξj − ξjξi, (2.67)

k and the Cij are the structure constants of the isometry group. The structure constants are antisymmetric in the sense that

k k Cij = −Cji. (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 directions are mutually orthogonal and have different scale factors. The effective scale factor for the model is given by the average expansion scale factor a(t) = (ABC)1/3. One can see that Friedmann–Lemaître–Robertson–Walker (FLRW) metric is a sub- set of this model, which follows when the three scale factors are equal. The spatial sections of constant time in Bianchi type I models are flat and all the invariants

16 2.3 Bianchi Models depend only on the time coordinate. The ﬂuid, which is orthogonal to the spatial surfaces, is geodesic and irrotational. These model are therefore covariantly characterised by,

a ˜ ˜ ˜ 3 u˙ = ω = 0, ∇aµ = ∇ap = ∇aΘ = 0, Rab = 0, (2.70) where the variables have the deﬁnitions given in subsection 2.1.1.

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,

2 2 2 2 2 2 2 2 a a ds = −dt + A (t) dx + B (t) dy + C (t) dz , u = δ0 , (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

2 2 2 2 ds2 = − ω0 + ω1 + ω2 + ω3 . (3.3)

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

A˙ B˙ C˙ γ1 = γ0 = , γ2 = γ0 = , γ3 = γ0 = , (3.4) 01 11 A 02 22 B 03 33 C

˙ dA where the dot indicates derivation with respect to time, i.e. A ≡ dt . From Cartan’s second structure equation (Equation 2.64), where the connection forms are given by Equation 2.57, the following non-zero components of the Riemann

19 Chapter 3 Background Solutions of Bianchi Type I tensor are obtained A¨ R0 = −R0 = R1 = −R1 = , 101 110 001 010 A B¨ R0 = −R0 = R2 = −R2 = , 202 220 002 020 B C¨ R0 = −R0 = R3 = −R3 = , 303 330 003 030 C A˙B˙ R1 = −R1 = −R2 = R2 = , 212 221 112 121 AB A˙C˙ R1 = −R1 = −R3 = R3 = , 313 331 113 131 AC B˙ C˙ R2 = −R2 = −R3 = R3 = . (3.5) 323 332 223 232 BC By using Equation 3.5 in the relation between the Riemann tensor and the Ricci k curvature tensor, Rij = Rikj, we ﬁnd the diagonal components for the Ricci tensor A¨ B¨ C¨ R = − − − , 00 A B C A¨ A˙B˙ A˙C˙ R = + + , 11 A AB AC B¨ A˙B˙ B˙ C˙ R = + + , 22 B AB BC C¨ A˙C˙ B˙ C˙ R = + + . (3.6) 33 C AC BC Finally, we can ﬁnd the expression for the trace of the Ricci curvature ab R = η Rab A¨ B¨ C¨ A˙B˙ A˙C˙ B˙ C˙ ! = 2 + + + + + . (3.7) A B C AB AC BC

From the findings above it is possible to find a closed system of coupled second- order equations. We start from Equation 2.1, where the energy-momentum for a perfect fluid is given by Equation 2.13, and use Equation 3.1, Equation 3.6, and Equation 3.7 to obtain A˙B˙ A˙C˙ B˙ C˙ + + = µ + Λ, (3.8) AB AC BC A¨ B¨ A˙B˙ + + = −p + Λ, (3.9) A B AB A¨ C¨ A˙C˙ + + = −p + Λ, (3.10) A C AC B¨ C¨ B˙ C˙ + + = −p + Λ. (3.11) B C BC

20 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 µ Λ σ Ω ≡ , Ω ≡ , Σ ≡ αβ . (3.12) 3H2 Λ 3H2 αβ H The density parameter Ω is related to ΩΛ and Σαβ by 2 Ω = 1 − Σ − ΩΛ, (3.13) αβ 2 Σαβ Σ where Σ ≡ 6 . The shear is diagonal, and consequently there are only two independent components, and we can take them to be 1 1 Σ+ ≡ (Σ22 + Σ33) , Σ− ≡ √ (Σ22 − Σ33) , (3.14) 2 2 3 2 2 2 and with these deﬁnitions it follows that Σ = Σ+ + Σ−. The reduced dynamical systems become 0 Σ± = − (2 − q)Σ±, (3.15) 0 ΩΛ = 2 (1 + q)ΩΛ, (3.16) where the deceleration parameter, q, is given by 3 1 3 q = (2 − γ)Σ2 + (3γ − 2) − γΩ . (3.17) 2 2 2 Λ 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. In physical terms, this means that a generic perfect ﬂuid Bianchi type I is asymptotic to a Kasner model at the big bang, and is asymptotic at late time to the de Sitter model. Since the shear scalar Σ is zero for de Sitter model and non-zero at the Kasner model, all models isotropise into the future but generically have an anisotropic big bang [14]. In the case of no cosmological constant the ﬂat Friedmann model will act like a sink point, and therefore all models isotropise into the future [13].

21 Chapter 3 Background Solutions of Bianchi Type I

2 Σ ΩΛ 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˙ B˙ ! 2 + = µ + Λ, (3.19) AB B A¨ B¨ A˙B˙ + + = −p + Λ, (3.20) A B AB 2 B¨ B˙ ! 2 + = −p + Λ. (3.21) B B

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

2 B¨ B˙ ! A¨ A˙B˙ + − − = 0, (3.22) B B A AB which on integration yields

B2A˙ − ABB˙ = 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 dt A = k B + kB , (3.24) 1 ˆ B3(t) where k1 is a constant of integration.

22 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 dt B2 = k A2 − 2kA2 , (3.25) 2 ˆ A3 (t) 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˙ B˙ ! 2 + = µ, (3.26) AB B A¨ B¨ A˙B˙ + + = 0, (3.27) A B AB 2 B¨ B˙ ! 2 + = 0, (3.28) B B which have the solutions

2/3 −1/3 A(t) = a1t − a2t , (3.29) 2/3 B(t) = b1t , (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 + t2b dy2 + dz2 , (3.31) where a + 2b = 1 and a2 + 2b2 = 1[16]. Furthermore, the solution approaches the Friedmann solution ds2 = −dt2 + t4/3 dx2 + 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 4 a µ(t) = 1 . (3.33) 3 t (a1t − a2)

23 Chapter 3 Background Solutions of Bianchi Type I

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

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

s   s  Λ Λ A(t) = a1 exp  t − a2 exp −2 t , (3.37) 3 3 s  Λ B(t) = b1 exp  t . (3.38) 3

where a1, a2 and b1 are 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  s  2 2 Λ 2 2 2 ds = −dt + exp 2 t dx + dy + dz . (3.39) 3

By using Equation 3.37 and Equation 3.38 in Equation 3.34 the energy density can be written in the form q Λ 2Λa2 exp −2 3 t µ(t) = (3.40) q Λ q Λ a1 exp 3 t − a2 exp −2 3 t

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 of state, p = (γ − 1) µ, will be assumed. Bianchi type I geometries are spatially ﬂat and hence the equations Equation 2.36 and Equation 2.37 for the 3-Ricci tensor

24 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 1 E = Θσ − σ σc , (3.41) ab 3 ab cha bi 1 µ = Θ2 − σ2 − Λ. (3.42) 3

Due to the LRS symmetry the shear can be written as 3 1 σ = Σ n n − h , (3.43) ab 2 a b 2 ab and in similar way the electric part of the Weyl curvature tensor is given by 3 1 E = ε n n − h , (3.44) ab 2 a b 2 ab

a where n 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) 1 3 3 Θ˙ = − Θ2 − Σ2 − γ − 1 µ + Λ, (3.46) 3 2 2 Σ˙ = −ΘΣ, (3.47) 3 1 ε˙ = −Θε + Σε − γµΣ (3.48) 2 2 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. By choosing suitable initial values, numerical solutions that are physical adequate are possible to determine.

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 derivatives. 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 linearisation 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 propagation equations, µ˙ = −Θ(µ + p) , (4.1) 1 1 Θ˙ = ∇˜ u˙ a − Θ2 − 2σ2 − (µ + 3p) + Λ, (4.2) a 3 2 2 σ˙ = ∇˜ u˙ − Θσ − σ σc − E , (4.3) habi ha bi 3 ab cha bi ab 1 E˙ = −ΘE + 3σ Ec − (µ + p) σ + curlH , (4.4) habi ab cha bi 2 ab ab ˙ c Hhabi = −ΘHab + 3σhaHbic − 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 dp p0 ≡ , (4.6) dµ and therefore Equation 2.28 can be written in the form p0 u˙ = − ∇˜ µ (4.7) a µ + p a 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 4 radiation is considered when γ = 3 [13].

27 Chapter 4 Density Perturbations

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)

4 a∇˜ (3)R = − aΘ∇˜ Θ + 2a∇˜ σ2 + 2a∇˜ µ, (4.9) a 3 a a a

2 1 ab where σ = 2 σabσ , 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 equation. Hence, we deﬁne particular spatial gradients orthogonal to ua, characterising the inhomogeneity of space-time,

a∇˜ µ D ≡ a , (4.10) a µ ˜ Za ≡ a∇aΘ, (4.11) ˜ 2 Ta ≡ a∇aσ , (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˙ 1 = Θ, (4.13) a 3 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

1 (3)S = (3)R − (3)Rh = −σ˙ − Θσ + ∇˜ u˙ , (4.14) bc bc 3 bc hbci bc ha bi we deﬁne the auxiliary variable

˜ bc Sa ≡ a∇a σ Sbc , (4.15) which is also a measure of inhomogeneity. Note that even though Sab is zero on the background, it may be non-zero in the perturbated space-time. We now examine how these variables evolve.

28 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

· 1 ∇˜ f = ∇˜ f˙ +u ˙ f˙ − Θ∇˜ f − σb∇˜ f, (4.16) a a a 3 a a b 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

˙ b Da + σaDb + γZa − (γ − 1) ΘDa = 0. (4.17)

The propagation of the comoving spatial derivative expansions in Equation 4.11 requires Equation 4.2, Equation 4.6, Equation 4.8 and Equation 4.16. The net result is the evolution equation ! 2 1 γ − 1 h i Z˙ + ΘZ + σbZ + 2T + µD + ∇˜ b∇˜ − 3σ2 D = 0, (4.18) a 3 a a b a 2 a γ b a

˜ a ˜ where ∇ ∇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

γ − 1! γ − 1! T˙ +2ΘT +σbT +2σ2Z −2 µΘσ2D +S − σc∇˜ ∇˜ bD = 0. (4.19) a a a b a γ a a γ b a c

This equation involves the new variable Sa. The evolution of this variable requires Equation 4.15 and Equation 4.16, leading to

˙ 5 2 2 4 2 2 2 b 1 Sa + ΘSa + σ Ta − Θσ Za + µσ Da + σ Sb − √ σSa 3 3 9 3 a 3 2 γ − 1! 1 γ − 1! − σ2∇˜ b∇˜ D + µΘσc∇˜ ∇˜ bD 3 γ b a 3 γ b a c 4 √ +χ −2µσ2D + Θσ2Z − 2σT − 3σS − σb∇˜ ∇˜ bZ + ∇˜ ∇˜ bT = 0, a 3 a a a c a c b a (4.20) where  0, if H = 0 χ = ab 1, if Hab 6= 0

29 Chapter 4 Density Perturbations

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) 2 1 Z˙ + ΘZ + µD = 0, (4.22) a 3 a 2 a from which in this case, a second-order equation for Da can be derived, 2 1 D¨ + ΘD˙ − µD = 0, (4.23) a 3 a 2 a

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 2 D¨ + ΘD˙ = 0, (4.24) a 3 a where the expansion is given by √ Θ(t) = 3Λ. (4.25)

The solution for the growth of the energy density inhomogeneity is then  s  Λ Da = C1 exp −2 t + C2, (4.26) 3 where C1 and C2 are constants, and we obtain that Da approaches a constant value with time.

30 4.3 Isotropic Case

4.3.2. Dust Universe without cosmological constant

2/3 For a ﬂat FLRW dust universe, without Λ and with scale factor a = a0t , the 4 2 energy density and expansion are given by µ(t) = 3t2 and Θ(t) = 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

α 2 α −1 ˙ Da = f1a (x ) t 3 + f2a (x ) t , fia = 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 √ 4ΛC exp 3Λt µ(t) = − √ 2 , (4.27) exp 3Λt + C √ √ exp 3Λt − C  Θ(t) = 3Λ  √  , (4.28) exp 3Λt + C 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 4Λ 4 µ(t) ≈ = , (4.29) √ 2 2 3Λt 3t √ √ 1 + 3Λt + 1! 2 Θ(t) ≈ 3Λ √ ≈ , (4.30) 1 + 3Λt − 1 t which are expected in the case of ﬂat FLRW without Λ.

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 1 3 µ = Θ2 − Σ2 − Λ. (5.1) 3 4 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 1 1 σb = diag Σ, − Σ, − Σ . (5.2) a 2 2 By using plane-wave harmonics as described in [6], ﬁrst-order perturbations of the energy density can be written as

2 2 ! ˜ b ˜ k1 k2 ∇ ∇bDa = − 2 + 2 2 Da, (5.3) a1 a2 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

2 2 ! c ˜ ˜ b k1 k2 σb∇a∇ Dc = −Σ 2 − 2 Da. (5.4) a1 a2

The evolution equations of the scale factors can be expressed in terms of the expansion and the shear scalar

1 a˙ = Θ + Σ a , (5.5) 1 3 1 1 1 a˙ = Θ − Σ a . (5.6) 2 3 2 2

33 Chapter 5 Numerical Solutions

The system of equations in the ﬁrst preferred direction is

˙ D1 = −ΣD1 − γZ1 + (γ − 1) ΘD1, ! ! 2 2 ! ˙ 2 1 9 γ − 1 2 γ − 1 k1 k2 Z1 = − ΘZ1 − ΣZ1 − 2T1 − µD1 + Σ D1 + 2 + 2 2 D1, 3 2 4 γ γ a1 a2 ! ! 2 2 ! ˙ 3 2 3 γ − 1 2 γ − 1 k1 k2 T1 = −2ΘT1 − ΣT1 − Σ Z1 − S1 + ΘΣ D1 − 2 − 2 ΣD1, 2 2 γ γ a1 a2 5 1 1 1 1 S˙ = − ΘS − Σ2T + ΘΣ2Z − µΣ2D − ΣS 1 3 1 2 1 3 1 2 1 2 1 ! 2 2 ! ! 2 2 ! 1 γ − 1 k1 k2 2 1 γ − 1 k1 k2 − 2 + 2 2 Σ D1 + 2 − 2 µΘΣD1 2 γ a1 a2 3 γ a1 a2 " 2 2 ! 2 2 ! # 3 2 2 3 2 3 k1 k2 k1 k2 −χ − µΣ D1 + ΘΣ Z1 − Σ T1 − ΣS1 + 2 − 2 ΣZ1 − 2 + 2 2 T1 . 2 2 2 a1 a2 a1 a2 (5.7)

The system of equations in the second preferred direction is

1 D˙ = ΣD − γZ + (γ − 1) ΘD , 2 2 2 2 2 ! ! 2 2 ! ˙ 2 1 1 9 γ − 1 2 γ − 1 k1 k2 Z2 = − ΘZ2 + ΣZ2 − 2T2 − µD2 + Σ D2 + 2 + 2 2 D2, 3 2 2 4 γ γ a1 a2 ! ! 2 2 ! ˙ 1 3 2 3 γ − 1 2 γ − 1 k1 k2 T2 = −2ΘT2 + ΣT2 − Σ Z2 − S2 + ΘΣ D2 − 2 − 2 ΣD2, 2 2 2 γ γ a1 a2 5 1 1 1 S˙ = − ΘS − Σ2T + ΘΣ2Z − µΣ2D + ΣS 2 3 2 2 2 3 2 2 2 2 ! 2 2 ! ! 2 2 ! 1 γ − 1 k1 k2 2 1 γ − 1 k1 k2 − 2 + 2 2 Σ D2 + 2 − 2 µΘΣD2 2 γ a1 a2 3 γ a1 a2 " 2 2 ! 2 2 ! # 3 2 2 3 2 3 k1 k2 k1 k2 −χ − µΣ D2 + ΘΣ Z2 − Σ T2 − ΣS2 + 2 − 2 ΣZ2 − 2 + 2 2 T2 . 2 2 2 a1 a2 a1 a2 (5.8)

Note that there is a coupling between the two directions, because of the relation ! a1 k2 X2 = X1, (5.9) a2 k1 where X is D, Z, T or S, and the derivation of this coupling relation can be found in section A.2. From now on the notation of the ﬁrst preferred direction will be mention as the direction of anisotropy, with the labels Dk, Zk, etc. In similar way, the second

34 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 conditions can be chosen in many ways. In this work just an initial density perturbations 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 1 given by p = 3 µ Figure 5.1.: The background quantities energy density, expansion and shear. Initial q 3 values at t0 = 1 are given by µ0 = 0.2, Θ0 = 3, Σ0 = 2 5

35 Chapter 5 Numerical Solutions

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 q 3 values at t0 = 1 are given by µ0 = 0.2, Θ0 = 3, Σ0 = ±2 5 , and the pressure is zero.

Note that all perturbations asymptotically approach constant values. This is consistent with that the background asymptotically approaches de Sitter, which we also have described in section 5.2. In the case of no shear we obtain from Equation 5.5 and Equation 5.6 that the evolution of the scale factors are the same, this means that the dust background is Friedmann, F. In Figure B.1 we have a dust background without cosmological constant, and the density gradients in the anisotropic and perpendiculars are equal and grow unboundedly, which agree with the analytical analysis in subsection 4.3.2. In Figure B.2 we have a dust background with a cosmological constant, and we see that the density gradient grow towards the same higher constant value. For large

36 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 4 case, where γ = 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 perturbations are initially zero. The wave numbers in the preferred directions, kk and kk k⊥ k⊥, are initially assumed to be equal, and we can therefore define k ≡ = . a10 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 ! a10 k⊥ D⊥ = Dk, (5.10) a20 kk 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 perpendicular 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 Equation 5.10 the initial value of the density gradient in the anisotropic direction is affecting the initial value of the density gradient in the perpendicular direction. Figure 5.4 shows how the difference in the behavior of the density gradients with

37 Chapter 5 Numerical Solutions

(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 at t0 = 1 are given by µ0 = 0.2, Θ0 = 3, Σ0 = ±2 5 , and for the wave numbers 1 k = kk/a10 = k⊥/a20 = 1, 5 and 20. The pressure is given as p = 3 µ.

(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, q 3 1 Σ0 = ±2 5 , and the pressure is given as p = 3 µ. 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 5.4 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 gradient 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 approximately 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 k = kk/a10 = k⊥/a20 = 1, 5 and 20

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 unboundedly 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 amplitude 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 unboundedly, 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. The present analysis can be extended to study tensor perturbations, including gravitational waves, and the coupling between scalar and tensor perturbations.

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. I want to thank friends and family for supporting me and encouraging me.

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 Da

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

· 1 ∇˜ µ − ∇˜ µ˙ =u ˙ µ˙ − Θ∇˜ µ − σc∇˜ µ, (A.1) a a a 3 a a c and by using Equation 4.1, Equation 4.10 and Equation 4.13 we obtain

· 1 1 1 ∇˜ µ = − (µ + p)ΘD − µΘD + µD˙ . (A.2) a a a 3a a a a Furthermore, use Equation 4.1 and Equation 4.11 to ﬁnd 1 1 ∇˜ µ˙ = − (1 + p0) µΘD − (µ + p) Z , (A.3) a a a a a and Equation 4.7 and Equation 4.10 to obtain 1 u˙ µ˙ = µΘp0D . (A.4) a a a 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,

Θp p ! D˙ − D + 1 + Z + σbD = 0. (A.5) a µ a µ a a b

With the equation of state Equation A.5 becomes

˙ b Da − (γ − 1) ΘDa + γZa + σaDb = 0. (A.6)

45 Chapter A Detailed calculations

A.1.2. Expansion gradient Za

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

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

· 1 1 ∇˜ Θ = ΘZ + Z˙ , (A.8) a 3a a a a

µp0 2 1 3 ∇˜ Θ˙ = − ∇˜ ∇˜ bD − T − µD − µp0D , (A.9) a a (µ + p) a b a a 2a a 2a a

µΘ2p0 2µp0σ2 µ2p0 3µpp0 p0µ u˙ Θ˙ = D + D + D + D − ΛD , a 3a (µ + p) a a (µ + p) a 2a (µ + p) a 2a (µ + p) a a (µ + p) a (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,

2 1 γ − 1! 1 γ − 1! Z˙ + ΘZ +σbZ + µD +2T + µ − Θ2 − 2σ2 + Λ D + ∇˜ ∇˜ bD = 0. a 3 a a b 2 a a γ 3 a γ a b (A.11)

˜ In this case ωab = 0, and on using the Ricci identities for the ∇-derivatives and the (3) 1 (3) zeroth-order relation Rab = 3 Rhab for the 3-dimensional Ricci tensor, we ﬁnd that   ˜ ˜ b ˜ ˜ b 1  2 2 2  ∇a∇ Db = ∇b∇ Da − 2µ − Θ + 2σ + 2Λ Da, (A.12) 3  3  | {z } =0

Hence, the following evolution equation

! 2 1 γ − 1 h i Z˙ + ΘZ + σbZ + µD + 2T + ∇˜ b∇˜ − 3σ2 D = 0, (A.13) a 3 a a b 2 a a γ b a is obtained for Za.

46 A.1 Evolution of the perturbation variables

A.1.3. Shear gradient Ta

Taking the time derivative of Equation 4.12 and the use of Equation 4.16 leads to

a˙ · · 1 T˙ = a∇˜ σ2 + a∇˜ σ2 + au˙ σ2 − aΘ∇˜ σ2 − aσc∇˜ σ2, (A.14) a a a a a 3 a a c and with help of Equation 4.12 again and Equation 4.13 we obtain ˙ ˜ 2· 2· b Ta = a∇a σ + au˙ a σ − σaTb. (A.15) The ﬁrst term of the right hand side of Equation A.15 can be expressed in the form

˜ 2· ˜ bc bc ˜ a∇a σ = aσ˙ bc∇aσ + aσ ∇aσ˙ bc, (A.16) and from Equation 4.14 and Equation 4.15 we ﬁnd that ˜ bc bc ˜ ˜ 2 2 ˜ ˜ ˜ b c aσ˙ bc∇aσ + aσ ∇aσ˙ bc = −Sa − 2aΘ∇aσ − 2aσ ∇aΘ + aσbc∇a ∇ u˙ . (A.17) By using Equation 4.11, Equation 4.12 and Equation A.17 in Equation A.16, we ﬁnd that

! · γ − 1 a∇˜ σ2 = −S − 2ΘT − 2σ2Z + σ ∇˜ ∇˜ bDc . (A.18) a a a a γ bc a The second term of the right hand side of Equation A.15 can now be found with help of Equation 4.7, Equation 4.8 and Equation 4.10, which to ﬁrst-order leads to ! · γ − 1 au˙ σ2 = 2 µΘσ2D (A.19) a γ a Hence, from Equation A.15, Equation A.18 and Equation A.19 we ﬁnd the evolution equation γ − 1! γ − 1! T˙ +2ΘT +σbT +2σ2Z −2 µΘσ2D +S − σc∇˜ ∇˜ bD = 0. (A.20) a a a b a γ a a γ b a c

A.1.4. Auxiliary variable Sa

Taking the time derivative of Equation 4.15 leads to ˙ ˜ bc · bc · b Sa = a∇a σ Sbc +u ˙ a σ Sbc − σaSb. (A.21)

Use Equation 4.1, Equation 4.4 and multiply both sides with σbc to ﬁnd 2 2 4 2 2 2 1 σbcS˙ = − σ2∇˜ u˙ a + Θ2σ2 + σ2σ2 − µσ2 − Λσ2 − ΘσbcS − Θσbc∇˜ u˙ bc 3 a 9 3 3 3 3 bc 3 b c bc d bc d bc d bc +Θσ σdhbσci + 3σ σdhbEci + 2σ σ˙ dhbσci + σ (curlH)bc . (A.22)

47 Chapter A Detailed calculations

Then use Equation 2.6 to evaluate the PSTF tensors in Equation A.22, and the fact that

ab a habσ = σa = 0, (A.23) to obtain 2 2 4 2 2 2 σbcS˙ = − σ2∇˜ u˙ a + Θ2σ2 + σ2σ2 − µσ2 − Λσ2 − ΘσbcS bc 3 a 9 3 3 3 3 bc 1 − Θσbc∇˜ u˙ + Θσbcσ σd + 3σbcσ Ed + 2σbcσ˙ σd + σbc (curlH) 3 b c db c db c db c bc (A.24)

By using Equation 4.3 and Equation 4.14 it can be found that

2 2 16 2 2 2 σbcS˙ = − σ2∇˜ u˙ a + Θ2σ2 + σ2σ2 − µσ2 − Λσ2 − ΘσbcS bc 3 a 9 3 3 3 3 bc 1 − Θσbc∇˜ u˙ + σbcσdS − 3σbcσ σ σde + σbc (curlH) (A.25) 3 b c c db db ec bc To ﬁrst-order contractions of two, three and four PSTF tensor are given by [9] 3 3 9 x yab = XY, x ybcza = XY Z, x ybcz uda = XY ZU, (A.26) ab 2 ab c 4 ab cd 8 2 1 ab 3 2 respectively, so that, e.g., σ ≡ 2 σ σab = 4 Σ , and hence Equation A.25 yields 2 2 2 2 2 σbcS˙ = − σ2∇˜ u˙ a + Θ2σ2 − σ2σ2 − µσ2 − Λσ2 bc 3 a 9 3 3 3 2 bc 1 bc 1 bc ˜ bc − Θσ Sbc + √ σσ Sbc − Θσ ∇bu˙ c + σ (curlH) , (A.27) 3 3 3 bc √ 3 where σ = 2 Σ. Use Equation A.27 in Equation A.21 to ﬁnd

· 5 5 2 a∇˜ σbcS = − Θa∇˜ σbcS − aσbcS ∇˜ Θ − 2Sbca∇˜ S − a∇˜ σ2∇˜ u˙ b a bc 3 a bc 3 bc a a bc 3 a b 2 2 4 2 + Θ2a∇˜ σ2 + aσ2∇˜ Θ2 − σ2a∇˜ σ2 − µa∇˜ σ2 9 a 9 a 3 a 3 a 2 2 ˜ 2 ˜ 2 1 ˜ bc 1 bc ˜ − aσ ∇aµ − aΛ∇aσ + √ σa∇a σ Sbc + √ aσ Sbc∇aσ 3 3 3 3 1 h i − a∇˜ Θσbc∇˜ u˙ + a∇˜ σbc (curlH) . (A.28) 3 a b c a bc By using Equation 4.7 and Equation 4.8, Equation 4.10 and Equation 4.12 we ﬁnd that ! ! 2 2 γ − 1 1 2 γ − 1 − a∇˜ σ2∇˜ u˙ b = ∇˜ b∇˜ µ T + µσ2∇˜ ∇˜ bD , (A.29) 3 a b 3 γ µ b a 3 γ a b

48 A.1 Evolution of the perturbation variables and to ﬁrst-order we obtain " ! # p0 a∇˜ ∇˜ u˙ Sbc = a∇˜ ∇˜ − ∇˜ µ Sbc = 0. (A.30) a b c a b µ + p c

From Equation A.12, Equation A.29 and Equation A.30 we ﬁnd that Equation A.28 can be expressed on the form

· 5 2 4 2 2 4 2 a∇˜ σbcS = − ΘS − γµσ2D + Θσ2Z + Θ2 − µ − σ2 − Λ T a bc 3 a 3 a 9 a 9 3 3 3 a 2 γ − 1! 1 2 + σ2 ∇˜ ∇˜ bD − 2µ − Θ2 + 2σ2 + 2Λ D 3 γ b a 3 3 a ! 2 γ − 1 1 ˜ b ˜ 1 5 bc bc ˜ + ∇ ∇bµTa + √ σSa − aσ SbcZa − 2S a∇aSbc 3 γ µ 3 3 1 bc ˜ 1 ˜ bc ˜ ˜ h bc i +√ aσ Sbc∇aσ − a∇a Θσ ∇bu˙ c + a∇a σ (curlH) 3 3 bc (A.31)

Hence Da, Ta, Za and Sab are of ﬁrst-order, and vanish on the background, and therefore, we have   ! 2 γ − 1 1 ˜ b ˜ 1  2 2 2  0 = ∇ ∇bµ Ta = 2µ − Θ + 2σ + 2Λ Da 3 γ µ 3  3  | {z } =0 5 bc bc ˜ 1 bc ˜ = aσ SbcZa = 2S a∇aSbc = √ aσ Sbc∇aσ. (A.32) 3 3

· bc Evolution of the term au˙ a σ Sbc in Equation A.21 with help of Equation 4.7, Equation 4.3 and Equation 4.14, leads to

· p0 5 2 2 au˙ σbcS = − a ∇˜ µ[− ΘσbcS − SbcS − σ2∇˜ u˙ b + Θ2σ2 a bc µ + p a 3 bc bc 3 b 9 2 2 2 2 2 2 2 1 bc ˜ bc − σ σ − µσ − Λσ + √ σσ Sbc + ∇bu˙ cS ], (A.33) 3 3 3 3 and by using Equation 4.8 and Equation 4.10, the contribution to first-order is ! · 1 γ − 1 2 au˙ σbcS = 2µ − Θ2 + 2σ2 + 2Λ σ2D = 0, (A.34) a bc 3 γ 3 a because the background is spatially flat. The two last terms in Equation A.31 can to first-order be found as ! 1 1 γ − 1 − a∇˜ Θσbc∇˜ u˙ = − µΘσc∇˜ ∇˜ bD , (A.35) 3 a b c 3 γ b a c

49 Chapter A Detailed calculations and

h i 4 √ a∇˜ σbc (curlH) = 2µσ2D − Θσ2Z +2σT + 3σS +σb∇˜ ∇˜ bZ −∇˜ ∇˜ bT , a bc a 3 a a a c a c b a (A.36) respectively, where the derivation of Equation A.36 can be found in [9]. Finally, using Equation A.31, Equation A.32, Equation A.34, Equation A.35, and Equation A.36 in Equation A.21 the last evolution equation is given by

˙ 5 2 2 4 2 2 2 b 1 Sa + ΘSa + σ Ta − Θσ Za + µσ Da + σ Sb − √ σSa 3 3 9 3 a 3 2 γ − 1! 1 γ − 1! − σ2∇˜ b∇˜ D + µΘσc∇˜ ∇˜ bD 3 γ b a 3 γ b a c 4 √ + −2µσ2D + Θσ2Z − 2σT − 3σS − σb∇˜ ∇˜ bZ + ∇˜ ∇˜ bT = 0. a 3 a a a c a c b a (A.37)

A.2. Harmonic decomposition

To decompose ﬁrst-order perturbations of the Bianchi type I we use plane-wave harmonics

(k) α Q = exp (ikαx ) , (A.38) where α = 1, 2, 3 and kα is the comoving wave number. These are eigenfunctions of the Laplace-Beltrami operator (3)∇2 deﬁned on the three-hypersurface of constant time. We make the following deﬁnition

(k) ˜ (k) Qα ≡ a∇αQ , (A.39)

(k) α ˙ (k) with the properties Qα u = 0 and Qhαi = 0. It follows that

2 (3) 2 (k) kα (k) ∇ Qα = − 2 Qα , (A.40) aα since (3)R = 0. In the case of LRS Bianchi type I we have two preferred directions, and therefore Equation A.40 reduces to

" 2 # (3) 2 (k) k1 1 2 2 (k) ∇ Qα = − 2 + 2 k2 + k3 Qα , (A.41) a1 a2

50 A.2 Harmonic decomposition

and with the assumption that k2 = k3 we ﬁnd that

2 2 ! (3) 2 (k) k1 k2 (k) ∇ Qα = − 2 + 2 2 Qα . (A.42) a1 a2 For the derivation of Equation 5.9 we start to note that to ﬁrst-order when

(3) Rab = ωa = 0, (A.43) we have that ˜ ˜ ˜ ˜ ∇a∇bXc = ∇b∇aXc, (A.44) ˜ If Xc = a∇aX, then we have

˜ ˜ b ˜ ˜ ˜ b ∇a∇bX = ∇a∇b a∇ X , (A.45) which to ﬁrst-order can be written as

˜ ˜ b ˜ ˜ b ∇a∇bX = ∇b∇ Xa. (A.46)

By using the harmonic decomposition of plane waves we ﬁnd that ! ˜ ˜ b kα k1 1 k2 2 k3 3 ∇a∇bX = − X + X + X , (A.47) aα a1 a2 a2 2 2 2 ! ˜ ˜ b k1 k2 k3 ∇b∇ Xa = − 2 + 2 + 2 Xa. (A.48) a1 a2 a3

Now let α = 1 with k2 = k3 and X2 = X3 and use Equation A.47 and Equation A.48 in Equation A.46 to obtain ! a1 k2 X2 = X1. (A.49) a2 k1

B. Numerics

B.1. Test of Code

In this section we make a short discussion about the test of the code, and how it coincide with the analytical analysis and expected results for isotropically dust universes. In Figure B.1 the growth of density perturbation in a dust universe without cosmological constant and no shear is shown. The density gradient in the anisotropic and perpendicular directions grows equal and unboundedly, where the ﬁrst part is due to no shear and the second part coincide with the analytical analysis in subsection 4.3.2. The behavior of the density gradients corresponds to the expected analytical results, which validates the code.

(a) Density gradient in anisotropic direction (b) Density gradient in perpendicular direction

Figure 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.

In Figure B.2 we can see that density gradient are the same in the anisotropic and the perpendicular direction for a dust universe with a cosmological constant but with no shear, and they both grow towards a higher constant value. As discussed above, we expect the growth of density gradients to the same because of no shear. From Equation 4.26 we expect that the energy density perturbations approaching constant values, which our results also show.

53 Chapter B Numerics

(a) Density gradient in anisotropic direction (b) Density gradient in perpendicular direction

Figure 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. B.2. Numerical Instabilities

In Figure B.3 we can see that the density gradients start to grow unboundedly for high k-values and large times. One possible explanation can be numerical instabilities, which means that for large times the error increases and create the eﬀect of an unboundedly growth of density gradients.

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

Figure B.3.: The growth of the density perturbation Dk, where the initial values q 3 at t0 = 1 are given by µ0 = 0.2, Θ0 = 3, Σ0 = ±2 5 , and for the wave numbers 1 k = kk/a10 = k⊥/a20 = 5 and 20. The pressure is given as p = 3 µ.

Another possible explanation can be that this is an actual physical behavior, because

54 B.2 Numerical Instabilities we know that the physical k-values decrease with time, and therefore we also have the possibility that the wavelengths increases with time so they ﬁnally get larger than the size of the horizon, and thereafter start to grow unboundedly. To solve this issue one can try to decrease the step-length in the code, and see if the same behavior occurs. Unfortunately there was no time to perform these tests in this thesis project.

Bibliography

[1] D.N. Spergel et. al., Atrophys. J., 170, 377 (2007). [2] T.E. Perko, A. Matzner and L.C. Shepley, Phys. Rev. D, 6, 5 (1972). [3] B.L. Hu and T. Regge, Phys. Rev. Lett., 29, 1616 (1972). [4] R.B. Abbott, B. Bednarz and S.D. Ellis, Phys. Rev. D, 33, 2147 (1986). [5] A.G. Doroschkevish, Ya. B. Zeldovich and I.D. Novikov, Zh. Eksp. Teor. Fiz. 60, 3 (1971). Sov. Phys. JETP, 33, 5 (1971). [6] P. K. S. Dunsby, Phys. Rev. D, 48, 3562 (1993) [7] E.M. Lifshitz and I.M. Khalatnikov, Adv. Phys., 12, 185 (1963) [8] J.M. Bardeen, Phys. Rev. D, 22, 1882 (1980) [9] M. Bradley, P. K. S. Dunsby, M. Forsberg, and Z. Keresztes, Class. Quantum Grav, 29, 095023 (2012) [10] B. O. Osano, Thesis, Univ. of Cape Town (2008). [11] G. F. R. Ellis, H. van Elst. (2008). "Cosmological models". In Marc Lachièze- Rey. Theoretical and Observational Cosmology: Proceedings of the NATO Ad- vanced Study Institute on Theoretical and Observational Cosmology. NATO Science Series: Series C. 541. Cargèse Lectures 1998. Kluwer Academic. pp. 1–116. [12] M.P. Ryan Jr. and L.C. Shepley. (1975). Homogeneous Relativistic Cosmologies, Princeton University Press, Princton. [13] M. Goliath, and G. F. R. Ellis, Phys. Rev. D, 60(2) (1998) [14] G. F. R. Ellis and J. Wainwright. (1997). Dynamical Systems in Cosmology. Cambridge University Press, Cambridge. [15] S. Kumar and C. P. Singh, Astrophysics and Space Science, 312, 57-62 (2007) [16] S. Ram. International Journal of Theoretical Physics, Vol. 28, No. 8 (1989) [17] J. Hajj-Boutros. Journal of Mathematical Physics, 26, 2297 (1985). [18] R. Maartens, Phys. Rev. D, 55, 463 (1997).