Vorticity and Gravitational Wave Perturbations on Cosmological Backgrounds Using the 1+1+2 Covariant Split of Spacetime

MS

E

P

Vorticity and Gravitational Wave

Perturbations on Cosmological

Backgrounds Using the 1+1+2

Covariant Split of Spacetime

Author:

Robin Törnkvist

Supervisor: Prof. Michael Bradley

Master’s Thesis, 30HP Engineering Physics, 300 HP

Department of Physics Umeå University

Abstract

Acknowledgements

Contents

2.3 Perturbation Theory and the Gauge Problem . . . 8

3 1+1+2 Covariant Split of Spacetime 10 3.1 1+3 Covariant Formalism . . . 10

3.2 1+1+2 Covariant Formalism . . . 13

3.3 Infinitesimal Frame Transformation of the Dyad . . . 16

4 Perturbations of LRS Class II Cosmological Backgrounds 17 4.1 Cosmological Background . . . 17

4.2 Linearization . . . 19

4.3 Harmonic Expansion . . . 22

4.4 New Equations and Degrees of Freedom . . . 23

4.5 Final Evolution and Constraint Equations . . . 25

4.6 Vorticity . . . 27

5 Geometrical Optics Approximation 31 5.1 Evolution Equations . . . 31

5.2 Gravitational and Shear Waves . . . 32

5.2.1 Even Parity . . . 32

5.2.2 Odd Parity . . . 33

6 Concluding Remarks 34 A Vector and Tensor Spherical Harmonics Identities 36 B Harmonic Expansion of Linearized Equations 38 B.1 Even Parity . . . 38

B.2 Odd Parity . . . 40

C Dependent Harmonic Coefficients 43 C.1 Even Parity . . . 43

C.2 Odd Parity . . . 45

Chapter 1

Introduction

Cosmological models based on general relativity has for the last century been an in-contestable method of describing our universe at its largest scales, with an immense amount of verified measurable predictions. Cosmological redshift, the expansion of spacetime and the cosmic microwave background are among these verified re-sults, [1–3] and there are still predictions that we have yet to directly verify, e.g. the possible existence and nature of dark matter. These models also often include a nonvanishing cosmological constant, which could act as an explanation for what is called dark energy, an energy introduced to explain the acceleration of the expan-sion of the universe. One of the predictions resulting from general relativity in the early twentieth century was that of gravitational waves, [4, 5], and the first direct measurement of such waves was recently made, [6], verifying their existence. Such waves are also predicted to exist on a cosmological scale, and in recent years an in-terest has developed in studying what effect an anisotropic universe would have on these waves. This has been examined in the case of locally rotationally symmet-rical (LRS) class II cosmological backgrounds with the matter content assumed to be a perfect fluid, using perturbation theory and assuming a vanishing vorticity on the perturbed model, [7, 8]. The results shows decoupled, source free and damped cosmological gravitational waves with a certain dependency on the anisotropy.

In this thesis we present, for the first time, a general treatment of perturbed LRS class II cosmological backgrounds with nonvanishing vorticity on the perturbed model. We start by assuming a LRS class II cosmological background, which is characterized by that the vorticity, twist of the 2-sheets and magnetic part of the Weyl tensor all vanish, [9, 10]. We will also assume that the matter content is de-scribed by a perfect fluid and a nonvanishing cosmological constant. Motivated by the anisotropy of the background, which gives us a preferred spatial direction, we approach the perturbation using the 1+1+2 covariant formalism. This entails first splitting spacetime in the 1+3 covariant formalism, [11], where the velocity of the matter is the preferred direction. We then get 3-surfaces which we can further de-compose in a direction which on the background will be the direction of anisotropy, and on the perturbed model is left general until further calculations require a choice. This gives us all relevant quantities in the 1+1+2 formalism, [12].

We approach the perturbation of the model in terms of defining all our variables as gauge invariant quantities. This is done by finding the quantities which vanish on the background, which by the Stewart-Walker lemma, [13], are guaranteed to be gauge invariant. After linearizing the propagation and constraint equations given in the 1+1+2 formalism we then decompose the gauge invariant variables using a har-monic decomposition. This further reduces our system of scalar, tensor and vector equations into a system of only scalar time evolution and constrain equations. This system is solved and results in a set of eight variables with associated time evolution equations, from which all other variables are given.

degree of freedom in the even sector, and one degree of freedom in the odd sector. The time evolution equations for the two vorticity variables completely decouples from the other perturbed variables, though some of the other perturbed variables time evolution equations contain source terms due to vorticity. We find that the vor-ticity can be solved for exactly by assuming a linear equation of state. This gives a solution to the vorticity which only depends on time through the expansion co-efficients, and on integration constants depending on the comoving wavenumbers through the initial conditions.

We compare our result in the geometrical optics approximation, i.e. the high fre-quency limit, to previous research where the same model has been studied with the exception of vanishing vorticity on the perturbed model, see [7, 8]. The resulting evolution equations for the six remaining harmonic coefficients turn out to be iden-tical to those found when the vorticity vanishes on the perturbed model, except for a source term in the evolution of the density, which in turn acts as a source term for the shear waves. The last four variables represent source-free gravitational waves.

The outline of the thesis is as follows. In Chapter 2 we introduced some pre-liminary theory required to explain the main method and results of the thesis. We start by looking at the main axioms of general relativity, and proceed to look at some of their implications. We end the chapter by examining the approach we will use to solve Einstein’s field equations, namely perturbation theory, and we discuss the problem of gauge invariance which can follow from this approach.

In Chapter 3 we present the 1+1+2 covariant formalism, in which we separate the vector, covector and tangent spaces of spacetime into 1+1+2 directions, where one direction is given by the average velocity of matter ua, one direction is given by the direction of spatial anisotropy na_{, and the last two directions are left general and}

form a 2-sheet. The formalism is introduced by first looking at the 1+3 decomposi-tion, and then further decomposing the quantities given there in to 1+1+2 directions. We also look at the possibility of fixing the dyad given by our two directions ua_and

na, and what freedom we have in this fixing.

In Chapter 4 we proceed to look at the main work of the thesis. We start by defin-ing our cosmological background and gauge invariant variables, and then proceed to linearize the equations governing the dynamics of the perturbed model in terms of these variables. Next we decompose the variables in terms of harmonics, and solve the new system of equations in terms of the harmonic coefficients. We end the chapter by looking at the general evolution of the vorticity, at which rate it falls off for some specific cases, and how it relates to the conservation of angular momentum. In Chapter 5 we look at the time evolution equations of the remaining coefficients given in the previous chapter, except for the vorticity, in the geometrical optics ap-proximation. The resulting wave equations are analyzed, and compared to previous research.

Chapter 2

Preliminaries

In this thesis we will, as the title suggests, look at the evolution of vorticity and gravitational waves on cosmological backgrounds. Before we can delve in to the technical details and results though, we must first look at some background theory which will act as a basis for our forthcoming reasoning.

Since we are interested in describing the universe at large scales, we start this chapter focused on the theory of General Relativity, by summarizing some of the main definitions and axioms, and then present some of the important implications that they result in. We then end the chapter by looking at perturbation theory and the gauge problem, following [14, 15], by explaining the key points in the approach we will later use to end up with equations that describe actual physical quantities on our perturbed spacetime.

2.1

General Relativity

General relativity, a topic introduced as a complete theory in 1915 by Albert Einstein, [16], is one of the main areas of modern physics and currently our best way of, fun-damentally, explaining the universe at larger scales than those in the realm governed by quantum physics. The basis of general relativity is that gravity is described by the geometry, more specifically the curvature, of spacetime and not introduced as a force acting on matter in it. It also ties together matter and the geometry of spacetime in the sense that the matter together with the initial and boundary conditions defines geometry, and then geometry determines the motion of matter. General relativity is a vast topic filled with a great many interesting results and predictions, however, we will not be able to cover them all here. Instead, we will in this section outline the basic axioms of general relativity, and look specifically at how the equations gov-erning the geometry and motion of matter arises, in a simplified manner. For the reader interested in a more in depth explanation of the subject, e.g. [17] and [18] are recommended, whose methods we will follow in this section.

Before we proceed to look at the axioms of general relativity, we need to briefly discuss the topic of differential geometry, which introduces objects known as mani-folds, defined below.

Definition 2.1.1. Let M be a topological space. Then M is a manifold if the following axioms hold,

(i) M is provided with a family of pairs {(Ui, ϕi)}.

(ii) {Ui} is a family of open sets that cover M . The map ϕi is a homeomorphism

from Uito some open subset U of Rm. We call m the dimension of M .

(iii) Given Uiand Uj such that Ui∩ Uj 6= ∅, the transition map ψji = ϕj· ϕ−1i from

Given an m-dimensional manifold one can, for each point p ∈ M , define a tangent space TpM which is a m-dimensional vector space. One can subsequently define

a cotangent space for p, which is the dual space of TpM, and also tensors spaces

containing (p, q)-tensors whose basis elements are constructed based on the basis elements of the tangent and cotangent spaces. These spaces can then be bundled together to create the tangent bundle, cotangent bundle and tensors bundles, on which we can define tangent fields, cotangent fields and tensor fields, defined over the whole manifold. It is also possible to choose a set of basis vectors for the tangent space, known as a tetrad, and this implicitly creates a basis for the cotangent and tensor spaces.

In this thesis we frequently use vectors, covectors and (p, q)-tensors in our equa-tions, and when we introduce them it is understood that they are actually fields, or rather the coefficients of fields, for vector, covector and tensor fields. I.e. if we in-troduce a (1, 2)-tensor Tµνρ it is implied that Tµνρ = Tµνρ(p)for some p ∈ M . The

construction of the spaces listed above are rather intricate, and to fully understand the details and complexities of manifolds one must be well-versed with the topic of differential geometry, which unfortunately is a topic we will not cover in depth in this thesis. We will assume that the reader has a basic understanding of this topic, and any reader interested in a more in depth analysis of this fascinating area of math-ematics is referred to e.g. [19] for a thorough explanation, or [20] for an introduction with a focus on its connections to physics.

One of the fundamental axioms of general relativity is the following.

Axiom 2.1.2. The geometrical aspect of the universe we exist in can be modeled by a four-dimensional pseudo-Rimeannian manifold that is locally homeomorphic to M4, which we call spacetime.

A pseudo-Riemannian manifold is a manifold with a defined (0, 2)-tensor field gab,

known as the metric, which acts as an inner product on the tangent spaces. The word pseudo here indicates that the metric does not have to be positive definite, instead it obeys the weaker condition that it has to be a non-degenerate bilinear form. This axiom can be simplified into saying that the universe around us is not flat, i.e. it is not a Euclidian space, though it can locally be described as a flat Minkowski space. On a global scale however, there exists curvature in spacetime, described by the metric gab, and the dimensions of space and time cannot be seen as completely independent

of one another.

The next axiom of importance is stated as as follows.

Axiom 2.1.3. Any free massive particle will follow a timelike geodesic in spacetime, and any free massless particle will follow a null-geodesic in spacetime.

From this axiom we see that the physical laws are independent of coordinates, and hence are geometrical. With a particle following a timelike geodesic we mean that if the particles position is described by xµ(s), where s is the chosen parameter of the motion along the geodesic, it holds that

gµν

dxµ

ds dxν

ds < 0. (2.1)

Similarly, for a null-geodesic we have that

gµν

dxµ ds

dxν

ds = 0. (2.2)

dealing with the shortest distance between points on a curved spacetime, which is not a straight line, though locally it will be the straightest line possible. Among oth-ers, we get that light bends around gravitational fields, and also the redshift of light, though we will not look at those phenomena in detail here.

Before we proceed, it should be mentioned that one can reobtain Newton’s laws of gravity by what is known as the weak field approximation, i.e. one assume that

gµν = ηµν+ hµν (2.3)

where ηµν is the metric of flat Minkowski space, and hµν is a small correction to the

metric, but not so small that it can be neglected. This describes approximately flat space, i.e. when the curvature is small and the speed of an object is low, and by looking at the geodesic equations for a moving object one obtains the classical New-tonian laws, showing that they are indeed correct as an approximation in special scenarios.

We would be remiss if we in a summary of general relativity did not also mention the axiom known as the equivalence principle, an axiom of special relativity which Einstein extended to arrive at general relativity, in the first ever expose of the subject found in [21]. We will not present it as a formal axiom though, since it is embedded in our two previous axioms. The axiom can be summarized as follows. In a non-rotating laboratory that is freely falling in a gravitational field for a short time, the laws of physics will be the same as in an inertial frame where the gravitational field is absent. Here, an inertial frame is mentioned, which is a set of coordinates in which the metric takes the form

gµν = ηµν+ O(x2). (2.4)

Hence, the metric is approximately the metric of Minkowski space, and so the laws of physics are then in that scenario those of special relativity. This axiom was a guid-ing principle in the development of general relativity, and is in the version we have presented embedded in the axiom that states that spacetime is a pseudo-Riemannian manifold, since this implies that spacetime is locally described as Minkowski space. Lastly, we look at the axiom which govern the interaction between geometry and matter, which is given below.

Axiom 2.1.4. The interaction between the geometry of spacetime and its matter con-tent is determined by Einstein’s field equations. They are stated as

Gab= Rab−

1

2Rgab = Tab− Λgab. (2.5) Here, Gabis known as the Einstein tensor, Λ is the cosmological constant and Tab is

the energy momentum tensor containing all forms of energy and momentum in the system. The tensor Rabis known as the Ricci tensor and is defined as

Rab = Rcacb (2.6)

and the Ricci scalar R is the defined as

R = gabRab. (2.7)

The last two variables above are defined using the Riemann curvature tensor Ra bcd

which encodes the curvature of spacetime, defined as

where the Christoffel symbol Γa

bcis expressed using the metric as

Γa_bc= 1 2g

ad_(∂

cgdb+ ∂bgdc− ∂dgbc) . (2.9)

The Riemann curvature tensor has the useful properties that are known as the first Bianchi identity

R_a[bcd]= 0 (2.10) and the second Bianchi identity,

Rab[cd;e]= 0. (2.11)

Equation (2.5), combined with the twice-contracted Bianchi identities gives us that

∇aGab= 0 (2.12)

which in turn implies that

∇aTab = 0, (2.13)

and so the total energy momentum is locally conserved. Equation (2.5), (2.12) and (2.13) completely determine the geometry and hence the dynamical evolution of matter in spacetime. One way of finding a solution to Einstein’s field equations is to find the metric gab, though we will in this thesis use another approach. Instead

of finding the metric, we will look for solutions to the other variables in the equa-tions, which in turn describes the metric. The main variable of interest is then Rabcd,

which can be found by instead looking at R, Raband the Weyl curvature tensor Cabcd

defined as

Cabcd= Rabcd− Eabcd− Gabcd (2.14)

where Eabcd = 1 2(gacSbd+ gbdSac− gadSbc− gbcSad) , (2.15) Gabcd = 1 12R (gacgbd− gadgbc) (2.16) and Sab= Rab− 1 4Rgab. (2.17)

The Weyl curvature tensor is trace free, i.e. Ca_bad = 0, and therefore not seen in Einstein’s field equations. To get a complete description that is equivalent to finding the metric in this approach we would also need a complete tetrad description, which then for each basis vector eawill give us new relevant quantities through ∇aeb. In

2.2

Cosmological Models

With general relativity defined, to some extent, we can proceed to define a central concept that we will use in this thesis, namely a cosmological model. A cosmological model is supposed to represent the universe for some scale, and for large scales we will assume that the spacetime geometry is described by the theory of general relativity. This is as mentioned previously true for any scale much larger than the Planck length, but in the context of cosmological models we will use the word large to indicate scales which includes the whole universe. We then have a cosmological model if we specify the following, [15].

(i) The spacetime geometry for some averaging scale. This will be determined by the metric of spacetime, and this metric must be compatible with the current astronomical observational data that exists.

(ii) The matter present and its physical behaviour, on the same averaging scale as used to describe the geometry. This is done through the energy momentum tensor of each matter component and the associated equations governing their behaviour.

(iii) The interaction of geometry and matter. Since we assume that general relativity is the theory that governs this interaction, this will be given by Einstein’s field equations, found in equation (2.5).

(iv) The observational relations which the model will predict. This must include the predictions of both discrete sources and background radiation. These observa-tional relations must also imply a theory of structure growth for both large and small scales, and radiation absorption and emission.

On the largest scales, the universe seems to be approximately both homogeneous and isotropic, which makes these assumptions a good way to construct simple cos-mological models, such as the simplest known as Friedmann universe, which is based on the Robertson-Walker metric. We also know that the universe is expanding, and that the expansion is accelerating, hence a nonvanishing cosmological constant Λis usually assumed to explain this phenomenon. Simple approximations regard-ing the matter are also useful, and it is usually assumed that the matter behaves approximately like a perfect fluid, and so is characterized by a pressure p and a mass density ρ. One can further simplify for present day and assume the matter to behave like dust with p = 0, or in earlier times of the universe by assuming that the matter behaves like radiation with p = ρc2_{/3. The approach when these simpler}

models have been researched is then to move on to slightly more complex models, which one can do by e.g. changing to an anisotropic model, as we will later do in this thesis.

2.3

Perturbation Theory and the Gauge Problem

In order to solve Einstein’s field equations on a cosmological background, we will in this thesis use perturbation theory. The general idea is simple, we start out with an idealized cosmological background, e.g. the LRS class II background defined in the previous section, where many variables vanish. We define this idealized background as S. We know however that the universe is not this simple due to all the structure we can observe, hence we need to improve this model. This is done by a mapping Φ : S → S, where S is a more realistic version of spacetime. We then get for e.g. the metric that

δgab= gab− gab (2.18)

where g_abis the metric of the idealized spacetime, gab = Φ(gab)and δgabis the

per-turbation of the metric and should be sufficiently small. This approach presents us with a problem though, known as the gauge problem. Imagine for a moment that we start with a perturbed model S, obtained by some specific mapping Φ as above. Is it then possible to recreate that idealized model S, i.e. could we obtain gab from

gab? The answer, if the only restrictions to Φ are the ones listed above, is no. This

is due to the fact that no unique tensorial averaging process was defined that will give us gab from gab. This problem presents what is known as gauge freedom in

defining our perturbation. Since we do not have a unique inverse to our new per-turbed variables, they depend specifically on which mapping we choose, and so the equations for the perturbations will contain both physical modes, and gauge modes. The physical modes will correspond to real physical variation, but the gauge modes corresponds only to how we choose Φ. Hence, we have a gauge problem in that we have to distinguish between what in our equations on S that correspond to real physical phenomenas and what is simply a result of our choice of mapping.

There are different ways to approach this problem, one of which is to fix the gauge in some manner, and then keep track of what gauge freedom remains, see e.g. [14] for a more general discussion of this method, or [22] for a more specific example. There is also the approach of defining gauge invariant objects, which can be seen in the highly influential article [23] by Bardeen. There is however, another way of approaching the problem compared to the approach of Bardeen. This is also done by using gauge invariant quantities, but relies heavily on a lemma presented in [13], which is presented in a rewritten and slightly shorter version below.

Lemma 2.3.1(Stewart-Walker Lemma). If a quantity T c···d

a···b vanishes on the

back-ground space-time, that quantity is gauge-invariant to all orders.

This lemma is incredibly useful since it tells us that all quantities that vanish on our spacetime background S will be invariant under the choice of Φ, and so our mapping does not matter since it will always define the same perturbation. Hence, if we can describe the perturbed model S solely in terms of quantities that vanish on the background, then all quantities on S will be gauge invariant and represent actual physical quantities. This also simplifies the process of finding the solutions on the perturbed model, since for a quantity T c···d

a···b that vanish on the background

we get that

δT_a···bc···d= T_a···bc···d− T_a···bc···d= T_a···bc···d (2.19) and so our perturbations δT c···d

a···b directly obey Einstein’s field equations on the

Chapter 3

1+1+2 Covariant Split of Spacetime

In order to find the equations governing the geometry of spacetime and the evolu-tion of matter we will use a method known as the 1+1+2 covariant split of space-time. This method utilizes 1+1+2 covariantly defined variables in order to describe the metric, and will greatly simplify our equations. It is a favorable method when dealing with an anisotropic cosmological background with a preferred direction in space, which is what we will look at in the next chapter.

The basic idea of this split is to describe all our tensors, scalars and vectors in terms of 1+1 specific vector fields which belong to the tangent bundle of our space-time manifold. The remaining 2 variables will be left as free variables. After we have implemented this method we will have two preferred directions, and we also have the freedom to change the basis vectors of the tangent bundle, i.e. the tetrad description, to some degree. This will allow us to further simplify our equations by setting certain variables to zero.

In order to properly define the 1+1+2 covariant description though, we must first look at the 1+3 covariant description. We start this chapter by doing this, following the approach of [15]. We then move on to look at the 1+1+2 covariant description based on the work done in [12]. Lastly, we look at the remaining freedom in chang-ing dyad to obtain the possibility of further simplifychang-ing our equations, followchang-ing [8].

3.1

1+3 Covariant Formalism

We will assume that for our chosen cosmological background there exists a family of preferred worldlines on the spacetime manifold, which pointwise represent the average motion of matter. This family of worldlines will at each given point have a 4-velocity ua_{defined as}

ua= dx

a

dτ , (3.1)

where τ is the proper time along the worldlines and the identity uaua = −1holds.

We will assume that this velocity is unique for each point. Using ua, we can now define two unique tenors which act as projection tensors,

Ua_b = −uaub, hab = gab+ uaub. (3.2)

We see that these tensors obey the identities

Ua_cUc_b = Ua_b, Ua_a = 1, U_abub = ua (3.3)

and

This shows that both tensors are well defined projection tensors. The tensor Ua b

projects an element in the direction of the 4-velocity, and hab projects an element in

the direction orthogonal to ua_{, onto what we will refer to as the 3-surfaces.}

Using the vector and tensors above we can now define two new derivatives, firstly the covariant time derivative along ua, which for a tensor T_a···bc···dacts as

˙

T_a···bc···d= ue∇eTa···bc···d, (3.5)

where ∇ais the covariant derivative. Secondly, we have the fully orthogonally

pro-jected covariant derivative, which for a tensor T_a···bc···dacts as DeTa···bc···d= haf· · · h

g b h

c

p · · · hqdhre∇rT_{f ···g}p···q. (3.6)

We will also introduce a shorthand notation for the orthogonal projection of vec-tors and the orthogonally projected trace-free symmetric part of tensors, using the angle brackets as vhai= ha_bvb, Thabi=  h(a_chb)_d−1 3h ab_h cd  Tcd. (3.7)

Also, we can define a new volume element,

ηabc= udηabcd, ηabc= η[abc], ηabcuc= 0. (3.8)

We can now decompose any 4-vector into a scalar parallel to uaand a 3-vector part orthogonal to ua_{. Any second rank tensor can also be decomposed into a scalar,}

vector and projected, trace-free and symmetric 3-tensor parts. We use this to express the covariant derivative of uaas

∇aub= −uau˙b+

1

3Θhab+ σab+ ωab. (3.9) Our new quantities given in this decomposition are the acceleration

˙

ua= ub∇bua, u˙aua= 0, (3.10)

the rate of volume expansion

Θ = Daua, (3.11)

the projected, symmetric and trace-free rate of shear tensor

σab= Dhaubi, σab= σ(ab), σabub = 0, σaa = 0, (3.12)

and the projected, skew-symmetric and trace-free vorticity

ωab= D[aub], ωab= ω[ab], ωabub = 0. (3.13)

We can also define a new kinematical quantity based on our definition of the vortic-ity, which is called the vorticity vector,

ωa= 1 2η

abc_ω

bc, ωaua= 0, ωabωb = 0. (3.14)

In the same manner as with the covariant derivative of the 4-velocity, we can decompose the energy-momentum tensor Tabas

where we now have the relativistic energy density relative to ua_,

µ = Tabuaub, (3.16)

the relativistic momentum density

qa= −Tbcubhca, qaua= 0, (3.17)

the isotropic pressure

p = 1 3Tabh

ab_{, (3.18)}

and lastly the anisotropic pressure,

πab= Tcdhchahdbi, πab = π(ab), πabub = 0, πaa = 0. (3.19)

Next, we can decompose the quantities which describe the Riemann curvature tensor, given in Section 2.1. We start with the Maxwell field strength tensor Fab

which split into an electric part, Ea, and a magnetic part Ha, defined as

Ea= Fabub, Eaua= 0, (3.20) and Ha= 1 2ηabcF bc_{, H} aua= 0. (3.21)

To see the full decomposition of Fabwe refer the reader to [15].

Lastly, we decompose the Weyl conformal curvature tensor Cabcd into an

‘elec-tric’, Eab, and ‘magnetic’, Hab, part in the same manner as with Fab,

Eab = Cacbducud, Eab= E(ab), Eabub = 0, Eaa = 0, (3.22) and Hab= 1 2ηadeC de bcuc, Hab = H(ab), Habub = 0, Haa = 0. (3.23)

It is now possible to decompose the Riemann curvature tensor and all the prop-agation and constraint equations in terms of the new variables given above. As discussed in the previous chapter, the equations governing the interaction between matter and geometry, and the evolution of matter, are Einstein’s field equations with a nonvanishing Λ, and their associated integrability conditions. The resulting equa-tions will be the set of Ricci identities for ua,

2∇[a∇b]uc= Rab dc ud (3.24)

the once contracted Bianchi identity

∇_[aRbc]de= 0 (3.25)

and the twice contracted Bianchi identity

∇_a  Rab−1 2g ab_R  = 0 (3.26)

and also Maxwell’s field equations. The Ricci identities for ua_{divide into three}

˙ Θ − Dau˙a= − Θ2 3 + ˙u a_u˙ a− 2σ2+ 2ω2− 1 2(µ + 3p) + Λ. (3.27) The commutation relations for Da and the dot derivative will also provide us

with certain propagation equations and constraints. Since the decomposition of all the equations above does not introduce any new variables though, and we are mainly interested in those equations in the 1+1+2 formalism, we refer the interested reader to [15] for the full set of these equations in the 1+3 formalism.

3.2

1+1+2 Covariant Formalism

With the 1+3 covariant split defined we can now move on to separate spacetime further, into what is known as the 1+1+2 formalism. As noted in the beginning of this chapter, this is especially useful when dealing with an anisotropic cosmologi-cal background, since we then have a preferred spacelike direction. We will follow the work presented in [12] in defining this formalism, and in doing so start by in-troducing a new unit vector, na, which resides on the 3-surfaces, i.e. naua = 0and

nana= 1. This vector will act as our next direction in which we will split the tangent

spaces of the spacetime. As previously, we use this new vector to define a projection tensor Na

b as

Na_b = ha_b− nanb= gab+ uaub− nanb (3.28)

and this tensor obeys the identities

Nabna= 0, Nabua= 0, Naa = 2. (3.29)

Hence, Na_b projects vectors onto surfaces, or as we will sometimes call them, 2-sheets.

This new direction vector and projection tensor now allows us to decompose any 3-vector ψaresiding on a 3-surface, i.e. ψaua= 0, as

ψa= Ψna+ Ψa (3.30)

where

Ψ = ψana, Ψa= Nabψb = ψa. (3.31)

Above we have introduced the bar notation on the index to denote that the vector is projected with respect to naon that index. We also have that each tensor ψab which

is projected onto the 3-surface and is symmetric and traceless, i.e. ψab = ψhabior

ψab = ψ(ab), ψabub = 0, ψaa = 0, (3.32)

can be decomposed into

˜ Ψ = nanbψab, (3.34) ˜ Ψa= ˜Ψa= Nabncψbc, (3.35) ˜ Ψab = ψ{ab}=  N_(acN_b)d− 1 2NabN cd  ψcd. (3.36)

Here we have introduced, similar to the angle brackets, the curly brackets to denote the symmetric, trace-free and projected with respect to napart of a tensor.

As with the decomposition with respect to ua_{, we can now define two new}

derivatives with respect to na_{. First we have the derivatives along n}a _{in the}

3-surfaces, which for any tensor T_a···bc···dacts as

b

T_a···bc···d= neDeTa···bc···d. (3.37)

Secondly, we have the projected derivative on the 2-surfaces, which are projected on every free index as

δeTa···bc···d= NejNaf· · · N g b N

c

h · · · NidDjTf ···gh···i. (3.38)

In the 1+3 formalism we defined a new volume element for the 3-surfaces, ηabc,

and we can extend this method in the 1+1+2 formalism by defining the alternating Levi-Civita 2-tensor, which acts as a volume element on the 2-surfaces, as

εab = ηabcnc= udηdabcnc, εabnb = 0, ε(ab)= 0. (3.39)

The following identities also hold,

ηabc= naεbc− nbεac+ ncεab, εacεbc = Nab, (3.40)

εabεcd= NacNbd− NadNbc, εabεab= 2. (3.41)

We can now decompose the covariant derivative of na, in the direction orthogo-nal to ua_{, into}

Danb = naab+

1

2φNab+ ξεab+ ζab. (3.42) This introduces three new variables, which when traveling along naare the acceler-ation of na_,

aa= ncDcna, (3.43)

the 2-sheet expansion

φ = δana, (3.44)

the rotation of na_{, or the twisting of the 2-sheet}

ξ = 1 2ε

ab_δ

anb, (3.45)

and lastly the shear of na_,

We can also decompose the time derivative of ua_{, using equation (3.30) and (3.31),}

since ˙uaua= 0. We then get that

˙ ua= ub∇bua= Ana+ Aa where A = na_u˙ a, Aa= Nabu˙b. (3.47) Also, since ˙na_n

a= 0we can decompose the change of naalong uaas

˙na= ub∇_bna= Bua+ αa, (3.48) where

B = −ua˙na, αa= Nab˙nb. (3.49)

Now, we have that ua_n

a= 0which implies that ua˙na = −nau˙a. This in turn implies

that, from equation (3.47) and (3.49),

B = A. (3.50)

Lastly, we can decompose all relevant 3-vectors and 2-tensors that we defined in the 1+3 formalism, using equation (3.30) and (3.33). They become

ωa= Ωna+ Ωa, (3.51) σab= Σ  nanb− 1 2Nab  + 2Σ(anb)+ Σab, (3.52) Eab= E  nanb− 1 2Nab  + 2E_(an_b)+ Eab, (3.53) Hab= H  nanb− 1 2Nab  + 2H(anb)+ Hab, (3.54) qa= Qna+ Qa, (3.55) πab= Π  nanb− 1 2Nab  + 2Π(anb)+ Πab. (3.56)

We can now rewrite all equations governing the geometry of spacetime and evo-lution of matter, see the previous section, in terms of the variables defined above. We omit these equations in their most general form in this thesis, since we are mainly interested in the linearized version of them, which will be presented in Chapter 4. The interested reader can find the propagation and constraint equations to all orders in [12] though. Also, we get a new set of equations which are a result from the Ricci identities for na, i.e.

2∇[a∇b]nc= Rab dc nd. (3.57)

As in the 1+3 formalism, the commutation relations for δaand the hat derivative

3.3

Infinitesimal Frame Transformation of the Dyad

We have now expressed all our variables and equations in the 1+1+2 formalism, meaning that all variables are expressed using uaand na, and these two vectors acts as a dyad (ua, na). We will later fix this dyad, to zeroth order, so that ua is our preferred timelike vector. We will however, still have some freedom in choosing the direction of nasince we will later leave that general on the perturbed model. This is achieved by an infinitesimal frame transformation from our current dyad (ua, na)to a new dyad (ua_{, n}a_{). This can be defined as, to first order,}

ua= ua+ va+ νna, uava= nava= 0, (3.58)

and

na= na+ la+ mua, uala= nala= 0. (3.59)

Since the new dyad also obeys

uaua= −1, nana= 1, uana= 0 (3.60)

we see that ν = m. As mentioned earlier, we will always have a fixed ua_{, and so any}

transformation we do must not affect ua. Hence, va = ν = m = 0, which means that

our only freedom in changing the dyad lies in la. The variables which are affected

by a change of laare, φ = φ + δala, (3.61) ξ = ξ + εabδalb, (3.62) Σa= Σa− 2Σ 3 la, (3.63) aa= aa+ bla, (3.64) αa= αa+ ˙la, (3.65) ζ_ab= ζab+ δ{alb}, (3.66) E_a= Ea− 3E 2 la. (3.67)

Hence, we will later have the freedom to set one of these variables to zero by fixing na_.

Chapter 4

Perturbations of LRS Class II Cosmological

Backgrounds

With the required background theory covered we can now begin our main analysis of this thesis, namely of the evolution of perturbations of a perfect fluid on a LRS class II cosmological background, with the matter behaving like a perfect fluid with nonzero vorticity on the perturbed model. This scenario with the exception of the vorticity being zero, i.e. the perturbed fluid being irrotational, has been studied before, see [8] and [7]. We will compare our results to these previous findings, and look specifically at what effect a nonzero vorticity has on the final equations.

In this chapter we start by specifying a general cosmological background and look at which variables are of interest to us. We then linearize the equations given in [12] for a 1+1+2 covariant split of spacetime in terms of our relevant variables. We move on to perform a harmonic expansion on the linearized equations, resulting in new propagation equations and constraints for the scalar harmonic coefficients. Next, we reduce the system to eight free variables and end up with their evolution equations, effectively giving us the evolution of all our original variables. Lastly, we look at the evolution of the vorticity. We examine the general case, and also some specific cases to see at which rate it falls off, and its relation to the conservation of angular momentum.

4.1

Cosmological Background

The cosmological background of interest for us is the perfect fluid LRS class II space-time, which is completely characterized by the fact that ξ = Hab = ωab = 0, see e.g.

[9]. These backgrounds are spatially homogeneous, but anisotropic with a preferred direction given by na_{. We choose the square of the line element to be}

ds2= −dt2+ a2₁(t)dz2+ a2₂(t) dϑ2+ fK(ϑ)dϕ2
, (4.1)

where a1(t)and a2(t)are scale factors determining the rate of expansion, ϑ and ϕ are

regular spherical coordinates and fK(ϑ)is a function that determines the geometry

of the 2-sheets. The constant K takes the values +1, −1 and 0 giving f1(ϑ) = sin2ϑ

for a sphere, f−1(ϑ) = sinh2ϑfor a pseudo-sphere and f0(ϑ) = 1 for a plane. We

will not choose a value for K at the moment, instead we will keep the geometry of the 2-sheets general. We will evaluate the differences of geometry, which in our case turn out to be irrelevant, in the final equations of Section 4.4.

We can without loss of generality, see [7], choose the 4-velocity ua_{of a comoving}

observer, and the direction of anisotropy na_{on this background to be}

e0= u = ∂ ∂t, e1 = n = 1 a1(t) ∂ ∂z. (4.2)

all Cartan invariants, see [24]. An implication of this choice of frame is that the acceleration vanishes, i.e. A = 0.

The background is now determined by the only nonvanishing variables

{µ, p, E, Θ, Σ, φ, a₁, a2}, (4.3)

however, using e.g. equation (102) and (104) in [12] we get the constraint

φE = φΣ = 0, (4.4)

and so motivated by this we set φ = 0. The only models with φ 6= 0 in the LRS class II cosmological models are negatively curved Friedmann models with certain problems in the harmonic decomposition that we will later use, see [7]. Hence, we omit the case when φ = 0. From [25] we get that

Θ = ˙a1 a1 + 2˙a2 a2 , (4.5) Σ = 2 3  ˙a1 a1 − ˙a2 a2  (4.6) and so our background is now completely determined by the remaining nonvanish-ing variables

S(0) = {µ, p, E , Θ, Σ}. (4.7) The evolution equations given by equations (94), (95) and (96), and the constraint in equation (100), all found in [12], now reduces to

˙ Θ = −Θ 2 3 − 3 2Σ 2₋1 2(µ + 3p) + Λ, (4.8) ˙ Σ = 2 3(µ + Λ) + Σ2 2 − ΣΘ − 2 9Θ 2_{, (4.9)} ˙ µ = −Θ (µ + p) , (4.10) and 3E = −2 (µ + Λ) − 3Σ2+2 3Θ 2_{+ ΣΘ, (4.11)}

where Λ is the cosmological constant. If we also choose an equation of state, i.e. p = p(µ), the background is completely determined by Σ, Θ and µ.

Lastly, the two dimensional scalar curvature, R, of the 2-surfaces is defined as

R = NacNbdR_abcd (4.12)

where Rabcdis the two dimensional Riemann tensor defined by

1

2Rabcdψ

d_{= δ}

[aδb]ψc− Ωεabψ˙c+ ξεabψb_c (4.13) for some arbitrary 2-vector ψa_{. This reduces R in our case to}

where the last equality relates R to K, which value depends on the geometry of the 2-surfaces as explained earlier in this section.

4.2

Linearization

With the background determined we are now ready to look at the perturbed model, which we start doing by making some assumptions. First of all, we assume that the perturbed model is described by a perfect fluid with 4-velocity ua, which will also be our preferred timelike vector as on the background, i.e. e0 = u. This assumption

partially fixes the frame to zeroth order, though we still have the freedom to fix na, which we will do later in Section 4.4. Secondly, we choose to look at barotropic perfect fluids, which implies that p = p(µ) and πab _{= q}a _{= 0, and hence Π = Π}a ₌

Πab= Q = Qa= 0on the perturbed model.

The variables we have in the 1+1+2 formalism are all defined with respect to ua and na_{, and so they are not necessarily frame invariant. This presents us with a}

problem, since the frame choice will not completely fix the mapping between our background model and the perturbed model. However, as discussed in Section 2.3, the variables that vanish on the background will be gauge invariant, according to the Stewart-Walker lemma, [13]. Hence, for the nonvanishing variables on the back-ground, found in the set S(0) in equation (4.7), we instead look at their vanishing gradients on the perturbed model,

Xa= δaE, Va= δaΣ, Wa= δaΘ,

µa= δaµ, pa= δap.

(4.15) In order to find the complete propagation of the variables above we also need to find their hat-derivatives, though we will see in Section 4.3 that these are completely described in terms of the δa derivatives and Ω, hence we do not define any new

variables for them.

The newly defined variables above all vanish on the background, which gives us the following set of all first order nonzero variables which vanish on the background,

S(1) = {φ, ξ, A, Ω, H, Xa, Va, Wa, µa, pa, aa, αa, Aa, Ωa, Σa, Ea, Ha, ζab, Σab, Eab, Hab}.

(4.16) With the background and perturbed model well defined, and the gauge problem taken care of, we can now linearize the relevant equations in [12], taking into account the minor misprints as noted in [8]. In this linearization, the variables in S(0)_{, and}

their time derivatives, are considered to be of zeroth order, and the variables in S(1) are considered to be of first order.

We start by linearizing the commutation relations for scalars, found in [12], which we will need in order to rewrite the evolution and propagation equations in terms of our new variables defined by the gradients of the nonvanishing quantities. To first order for some scalar ψ they become

δaδbψ − δbδaψ = 2εabΩ ˙ψ. (4.20)

These commutation relations will also provide us with some useful identities when we later move on to a harmonic expansion of the variables in Section 4.3.

With the commutation relations at hand we proceed to linearize the main equa-tions in terms of the variables given in S(1). The nontrivial evolution equations on the perturbed model then become, to first order,

˙ φ =  Σ −2Θ 3   1 2φ − A  + δaαa, (4.21) ˙ ξ = 1 2  Σ −2Θ 3  ξ + 1 2εabδ a_αb₊1 2H, (4.22) ˙ ζ{ab}= 1 2  Σ −2Θ 3  ζab+ δ{aαb}− εc{aHb}c, (4.23) ˙ Ω =  Σ − 2Θ 3  Ω + 1 2εabδ a_Ab_{, (4.24)} ˙ Σ{ab}=  Σ −2Θ 3  Σab+ δ{aAb}− Eab, (4.25) ˙ H = 3 2  Σ −2Θ 3  H − ε_abδaEb_{− 3Eξ, (4.26)} ˙ µa= 1 2  Σ −2Θ 3  µa+ ˙µAa− (µ + p)Wa− Θ (µa+ pa) , (4.27) ˙ Xa=  Σ − 2Θ 3  Xa+ ˙EAa− 1 2(µ + p − 3E )Va −1 2(µa+ pa)Σ − E Wa+ εbcδaδ b_Hc_, (4.28) ˙ Va− 2 3 ˙ Wa= 3 2  Σ −2Θ 3   Va− 2 3Wa  +1 3(µa+ 3pa) − δaδbAb+ Σ −˙ 2 ˙Θ 3 ! Aa− Xa. (4.29)

The equations containing a mixture of evolution and propagation contributions are

˙ Ea+ 1 2εabHb b ₌ 3 4εabδ b_{H −} 1 2  µ + p − 3E 2  Σa+ 3E 4 εabΩ b + 3Σ 4 − Θ  E_a−3E 2 αa+ 1 2εbcδ b_Hc a, (4.34) ˙

E{ab}− εc{aHb_b}c= −ε_c{aδcH_b}− 1 2(µ + p)Σab−  3Σ 2 + Θ  Eab− 3E 2 Σab, (4.35) ˙ H_a−1 2εabEb b _{= −}3 4εabX b₋1 2εbcδ b_Ec a+  3Σ 4 − Θ  H_a+3E 4 εaba b₋3E 2 εabA b_, (4.36) ˙

H_{ab}+ εc{aEb_b}c= ε_c{aδcE_b}−  3Σ 2 + Θ  H_ab+ 3E 2 εc{aζ c b} . (4.37)

The equations containing only propagation contributions are

b φ = 2Θ 2 9 + ΘΣ 3 + δaa a₋2 3(µ + Λ) − E − Σ 2_, (4.38) b ξ =  Σ + Θ 3  Ω + 1 2εabδ a_ab_{, (4.39)} b ζ{ab}= δ{aab}+  Σ + Θ 3  Σab− Eab, (4.40) b Va− 2 3Wca= −δaδbΣ b_{− ε} bcδaδbΩc− 3Σ 2 δaφ + 2  ˙ Σ − 2 3 ˙ Θ  εabΩb, (4.41) b Σa− εabΩbb= 1 2Va+ 2 3Wa− εabδ b_{Ω − δ}b_Σ ab− 3Σ 2 aa, (4.42) b Ω = −δaΩa, (4.43) b

Σ{ab}= δ{aΣb}− εc{aδcΩb}− εc{aHb}c+

And lastly, the constraints are δaΩa+ εabδaΣb = H − 3Σξ, (4.50) 1 2δaφ − εabδ b_{ξ − δ}b_ζ ab=  Σ 2 − Θ 3   εabΩb− Σa  − Ea, (4.51) Va− 2 3Wa+ 2εabδ b_{Ω + 2δ}b_Σ ab = −2εabHb, (4.52) pa= −(µ + p)Aa. (4.53)

4.3

Harmonic Expansion

Motivated by the work done in [8] we will now expand our variables into harmonics. This method is incredibly useful since it will reduce our system of tensor equations into a system of scalar equations, where each scalar depends only on time, due to our choice of frame fixing. Firstly, the scalars can be expanded, for a general scalar Ψ, as

Ψ = X

kk,k⊥

ΨS_kkk⊥Pkk_Qk⊥_{. (4.54)}

Here, the scalar function ΨS

kkk⊥ depends only on time, see e.g [12, 25]. The next scalar function Pkk_{is the eigenfunction of the Laplacian b∆ = n}a_∇

anb∇b, and is also

constant on the 2-sheets, i.e.

b ∆Pkk _{= −}k 2 k a2 1 Pkk_{, δ} aPkk = ˙Pkk = 0. (4.55)

The dimensionless constants kk are comoving wavenumbers in the direction of na,

i.e. the direction of anisotropy. The harmonics were first introduced in [26] and [27], and are eigenfunctions of the two-dimensional Laplace-Beltrami operator δ2 _{= δ}

aδa, δ2Qk⊥ _{= −}k 2 ⊥ a2₂Q k⊥_, b Qk⊥_{= ˙Q}k⊥ _{= 0. (4.56)}

Here the dimensionless constants k⊥ are comoving wavenumbers in the direction

perpendicular to na_{, the 2-sheets. Depending on the geometry of the 2-sheets,}

de-fined by R, the wavenumbers can be either discrete or continuous, e.g. for R > 0 the topology of the 2-sheets are that of spheres and so the harmonics are spherical harmonics with k2

⊥ = l(l + 1)for discrete values of l. As noted earlier though, we

leave the geometry of the 2-sheets general and so R is left undefined.

In the same way as above, we can expand vectors and tensors in harmonics, as done in [26, 28, 29]. A vector Ψais expanded in an even and odd sum as

Ψa= X kk,k⊥ Pkk  ΨV_k k,k⊥Q k⊥ a + Ψ V kk,k⊥Q k⊥ a  (4.57)

Qk⊥

a = a2δaQk⊥, Q k⊥

a = a2εabδbQk⊥. (4.58)

For a tensor Ψabthe expansion takes the form

Ψab= X kk,k⊥ Pkk  ΨT_kk,k⊥Qk⊥ ab + Ψ T kk,k⊥Q k⊥ ab  (4.59) where ΨT

kk,k⊥ is a scalar depending only on time, and in a similar manner as before the even and odd tensor harmonics are defined as,

Qk⊥ ab = a 2 2δ{aδb}Qk⊥, Q k⊥ ab = a22εc{aδcδb}Qk⊥. (4.60)

There also exists some useful identities concerning the above vector and tensor har-monics, which are presented in Appendix A.

We will in the following part of this thesis suppress the subscript of kk and k⊥

on the scalars in the harmonic decompositions, i.e. ΨS_kkk⊥ = ΨS, ΨV_kkk⊥ = ΨV and ΨT

kkk⊥ = Ψ

T_{, since the superscripts S, V and T should be sufficient to distinguish}

these quantities from their non-harmonic counterpart.

4.4

New Equations and Degrees of Freedom

With the definitions given in the previous section we can now decompose the equa-tions given in Section 4.2. Each equation separates to an even and odd part, and the full set of equations is presented in Appendix B. There is now a total of 30 evolution equations, 28 constraints and 37 variables in the form of scalars from the harmonic decomposition of the variables in S(1)_{. The system also decouples into an even and}

odd sector, with the even variables

n

φS, AS, ΩV, HV, ΣV, EV, XV, VV, WV, µV, pV, aV, AV, αV, ΣT, ET, HT, ζTo, and the odd variables

n

ξS, ΩS, ΩV, HS, HV, ΣV, EV, XV, VV, WV, µV, pV, aV, AV, αV, ΣT, ET, HT, ζT o

. Before we proceed to solve the system we need to compare some results with previous studies of similar problems, in order to approximate how many of the vari-ables in our new system of equations that we may express in terms of other free variables. In [8] they approach the same problem as in this thesis, except for some differences in the geometry, and most relevant for us, with the vorticity vanishing on the perturbed model. Hence we now have three new variables compared to that sce-nario, ΩS_{, Ω}V _{and Ω}V_{. Also, if the vorticity vanishes the commutation relations will}

set five of the odd parity scalars to zero, and these variables are in our case nonzero. Hence, we should expect eight new variables compared to that scenario. However, we find in equation (B.44) that

ΩV = ia2kk a1k2_⊥

and so we can express ΩV _{in terms of Ω}S_{. Also, by using the commutation relation}

given in equation (4.20) we find that for a vector of the form Za = δaψ, for some

scalar ψ it holds that

εab[δa, δb]ψ = εab(δaZb− δbZa) = 4 ˙ψΩ. (4.62)

By expanding Za and Ω in terms of harmonic coefficients and using the identities

found in Section 4.2 and Appendix A we find the following identities, µV = 2a2µ˙ k2_⊥ Ω S_{, (4.63)} VV = 2a2Σ˙ k2 ⊥ ΩS, (4.64) WV = 2a2 ˙ Θ k2 ⊥ ΩS, (4.65) XV = 2a2E˙ k_⊥2 Ω S_{, (4.66)} pV = 2a2p˙ k2 ⊥ ΩS. (4.67)

Thus, the five variables that vanished when the vorticity was zero on the perturbed model can now be expressed in terms of ΩS and so they introduce no new degrees of freedom.

As we noted earlier in this chapter, the hat-derivatives of the nonvanishing quan-tities in S(0)are still needed to get a complete picture of the propagation of the vari-ables in S(0). In the case of vanishing vorticity on the perturbed model, these quan-tities are completely described in terms of the even part of the harmonic expansion of the delta-derivatives, due to the commutation relations. This is not the case now, however, since each hat derivate of a quantity in S(0) vanish on the background we can expand them in harmonics. For a scalar ψ where, as before, Za= δaψwe get that

b ψ = X kk,k⊥ ˜ ψS_k kk⊥P kk_Qk⊥_{. (4.68)}

Using the decomposition above and the even part of the commutation relation found in equation (4.19) we get that

˜ ψ = 2a2ψΩ˙ V +ia2kk a1 ZV (4.69)

and so we get that

˜ E = 2a₂EΩ˙ V +ia2kk a1 XV, (4.73) ˜ p = 2a2pΩ˙ V + ia2kk a1 pV. (4.74)

Hence, the hat-derivatives introduce no new degrees of freedom compared to the case of vanishing vorticity on the perturbed model. This means that we should only expect the vorticity to introduce at most two new degrees of freedom.

Two other notable differences from the previous study is that AV vanishes due to one of the constraints when the vorticity is zero, and that they set aa = 0which

implies that aV _{= a}V _{= 0. However, we see from equation (B.55) that A}V _{can in}

our case be expressed as a function of pV, which we showed above is a function of ΩS. Also, we have yet to fix the frame in terms of naon the perturbed model, which gives us the same possibility to remove two degrees of freedom, e.g. aa _{= 0which}

implies that aV = aV = 0, as seen in Section 3.3. Hence there are no new degrees of freedom introduced from AV or aafor a nonvanishing vorticity on the perturbed

model.

To summarize, we should in total get at most two new degrees of freedom, and using this fact we can move on to solve the system of equations.

4.5

Final Evolution and Constraint Equations

We now want to solve our new system of equations for the harmonic coefficients. First we note that we still have the freedom to partially fix the frame in choosing a direction for na. Based on the reasoning in Section 4.4 and the equations given

in Section 3.3 we choose na _{so that a}

a = 0, which will simplify our calculations

greatly. Also, we choose to look at adiabatic matter perturbations, i.e. p = p(µ), which implies that pV = c2_sµV, where c2s = ˙p/ ˙µis the square of the matter speed of

sound.

Now, in a rather heuristic approach based on the reasoning in the previous chap-ter, we assume that all the even variables can be expressed in terms of the five free variables

n

ΩV, µV, ΣT, ET, HTo and all the odd variables in terms of the three free variables

n

ΩS, ET, HTo.

We could of course choose eight of the other variables instead, but this choice is motivated by the results of [8], where the variables related to H and E represent gravitational waves. Our choice of eight free variables turns out to be correct, and we get the remaining even variables

n

φS, AS, HV, ΣV, EV, XV, VV, WV, pV, aV, AV, αV, ζT o

, and odd variables

n

ξS, ΩV, HS, HV, ΣV, EV, XV, VV, WV, µV, pV, aV, AV, αV, ΣT, ζT o

all of which can be expressed in terms of the eight free variables. These relations are presented in Appendix C, and the consistency of all the expressions have been checked against the propagation equations and constraints found in Appendix B, using MATLAB. The equations of interest are now the evolution equations, which for the even parity sector become

˙ ΩV = − 2Θ 3 + Σ 2 + ˙ p µ + p  ΩV, (4.75) ˙ ΣT =  Σ − 2Θ 3  ΣT − c 2 s a2(µ + p) µV − ET, (4.76) ˙ ET = − _ik k a1 + C17 a2C13  HT −1 2(µ + p) Σ T ₋ C14 a2C13 ΩV −  Θ + 3Σ 2 + C19 a2C13  ET − C18 a2C13 µV, (4.77) ˙ µV = Σ 2 − 4Θ 3 + C18 C13 (µ + p)  µV + (µ + p) C17 C13 +ikka2 a1  HT − (µ + p) 3a2Σ 2 − C19 C13  ET − (µ + p) _2ik k a1 −C14 C13  ΩV − (µ + p) Ra 2 2− k⊥2 a2 −C15 C13 +ikka2C16 a1  ΣT, (4.78) ˙ HT = ikk a1  2C19 3a2ΣC13 − 1  ET + 2 3a2Σ _ik kC14 a1C13 + µ + p  ΩV +  2 3Σ  _ik kC17 a1a2C13 −Ra 2 2− k⊥2 2a2₂ − 3E 2  − Θ − 3Σ 2  HT + 2ikkC18 3a1a2ΣC13 µV, (4.79)

where all Ciare defined in Appendix C. For the odd parity sector we get that

˙ ET = ikk a1 1 − 1 k_⊥2D4 − a 2 2(3E + µ + p) k2 ⊥D4 Ra22− k⊥2
! HT −3 2 2Θ 3 + Σ + Σ k_⊥2D4 + a 2 2Σ (3E + µ + p) k2 ⊥D4 Ra22− k⊥2
! ET −  _ik kD5 a1k⊥2D4 + a2 Ra2 2− k⊥2 (3E + µ + p) Ra 2 2− k2⊥ a2k⊥2 + D3+ ikka2D5 a1k⊥2D4  ΩS (4.82) where as previously, all Diare defined in Appendix C.

We can already note an interesting fact about the vorticity, namely that its evolu-tion equaevolu-tions, (4.75) and (4.80), completely decouple from the other variables and do not show any direct dependence of kk or k⊥. The other evolution equations we

end up with are quite cumbersome to solve though, and so in order to find solutions we will move on to examine them in the high frequency limit. However, the factor R is still undefined, and in the high frequency limit it becomes negligible, since it cor-responds to the curvature of the 2-sheets. Therefore, we want to examine the effects different values of R might have before moving on.

We recall from equation (4.14) that

R = 2K

a2₂, (4.83)

where K is equal to ±1 or 0. By using this relation in the evolution equations above and evaluating for the three different cases of K, using MATLAB, we find that there is no significant difference for the three cases. The coefficients connected to the free variables change slightly, but the evolution equations are still dependent on the same free variables. Since the three cases for different K only vary the evolution equations presented above slightly, and we are interested in the high frequency limit where they do not change the evolutions equations above at all, we omit them from this thesis. It should be noted that it is of course possible that these different cases have an influence for small values of the comoving wave numbers, which would corre-spond to large scale perturbations. Before we proceed to examine the high frequency limit we will first look at the evolution of the vorticity, since it is decoupled from the other variables and independent of the comoving wave numbers.

4.6

Vorticity

The evolution equations for the vorticity, ˙ ΩS=  Σ −2Θ 3 − ˙ p µ + p  ΩS, (4.84) ˙ ΩV = − Σ 2 + 2Θ 3 + ˙ p µ + p  ΩV (4.85)

are completely decoupled from the other variables and do not contain any direct dependence on kk or k⊥, which means that they remain the same in the high and

p = (γ − 1) µ (4.86) and using equation (4.5) and (4.6). This results in the solutions

ΩS = CΩaγ−11 a 2(γ−2)

2 (4.87)

and

ΩV = C_Ωaγ−2₁ a2γ−3₂ (4.88) for some constants related to the boundary conditions, CΩand C_Ω. Hence the

evolu-tion of the vorticity is completely decoupled from all other perturbed quantities and depend only on time through a1and a2, and the comoving wavenumbers kk and k⊥

through the initial conditions.

We can examine the two solutions above in the case of the matter behaving like dust, respectively radiation. For the case of dust we have that γ = 1 and so

ΩS= CΩ a2 2 , ΩV = CΩ a1a2 (4.89) and for radiation we have that γ = 4/3 which results in

ΩS = CΩa 1 3 1 a 4 3 2 , ΩV = CΩ a 2 3 1a 1 3 2 . (4.90)

A further interesting case to examine the solutions above in is the flat Friedmann case, when Σ = 0. We see from equation (4.84) and (4.85) that ˙ΩS = ΩV in this case, and from equation (4.6) we get that that a1 = a2 = a. Looking at the ratio against Θ

we see for dust, with a ∼ t23 and hence Θ ∼ 2t−1from equation (4.5), that ΩS Θ = ΩV Θ ∼ 1 t13 , (4.91)

and so the vorticity falls of faster than Θ, which is to be expected. For radiation we have that a ∼ t12 and so we get that

ΩS Θ = ΩV Θ ∼ t 1 2 (4.92)

which indicates that the vorticity perturbations does not fall of over time, but instead they grow. This is not a problem though, since first order perturbation theory is only valid for a limited time in the case of radiation.

We end this section by looking at what implications the results above has on the conservation of angular momentum L, which we know can be expressed as

L ∼ mR2Ω (4.93)

with m being the mass, R being the radius and Ω the vorticity of some comoving volume. From the first law of thermodynamics for a perfect fluid, a law which we can derive from Einstein’s field equations, we have that

with U = µV being the energy and V the volume. In the flat Friedmann case, Σ = 0, we can solve the vorticity from its evolution equations using the same method as above, and we get that

ΩS = ΩV = CΩa3γ−5 (4.95)

for some constant CΩ, where a = a1 = a2. Since there is no preferred spatial

di-rection for the expansion in this scenario we have that V = a3_{, which means that}

equation (4.94) can be written as

d µa3
= − (γ − 1) µd a3

(4.96) which reduces to

a3dµ + 3a2µda = −3 (γ − 1) a2da (4.97) and we rewrite this as

dµ µ = −

3γda

a . (4.98)

This can be solved as

µ = Cµ1a

−3γ

(4.99) for some constant Cµ1. Now, since we know that the mass obeys the relation m ∼ µV ∼ a−3γa3and the radius obeys R ∼ a we get from equation (4.93) that

L ∼ a−3γa3· a2· a3γ−5= a0 = 1. (4.100) This shows that the angular momentum is independent of the expansion coefficient, and hence preserved, as expected since the vorticity is unaffected by the other per-turbations to first order in perturbation theory.

For the more general case with Σ 6= 0 we have to look separately at the scalar and vector part of the vorticity, and since we now have a preferred spatial direction for the expansion we do not expect the angular momentum to be preserved in any di-rection on the 2-sheets. However, we still have a rotational symmetry on the 2-sheets and so in the directions orthogonal to the 2-sheets we expect the angular momentum to be preserved. Looking firstly in the direction of anisotropy, we have the vorticity given by equation (4.87) and the volume as V ∼ a1a22. Rewriting equation (4.94) as

above, using these relations, gives us dµ µ = −γ  da1 a1 +2da2 a2  (4.101) which means that

µ = Cµ2a

−γ 1 a

−2γ

2 (4.102)

for some constant Cµ2. Using the relation above and R ∼ a2in equation (4.93) shows that L ∼ a−γ₁ a−2γ₂ a1a22· a22· a γ−1 1 a 2(γ−2) 2 = a01a02= 1 (4.103)

For the direction orthogonal to the anisotropy we have the vorticity given by equation (4.88), and since V ∼ a1a22as previously we get the same expression for µ

given by equation (4.102). Since we have a rotational symmetry on the 2-sheets it is arbitrary which direction we choose to look at, and so we let the radius orthogonal to the vorticity be given by some vector R = a1zbz + a2xbxfor some real fixed values zand x, withbzin being in the direction of anisotropy andbxsome arbitrary direction in the 2-sheet. It follows that R2 _{∼ a}2

1+ a22and so equation (4.93) gives us that

Chapter 5

Geometrical Optics Approximation

We now turn our attention to what is known as the geometrical optics approxima-tion, that is, the high frequency limit. We define this limit with respect to our vari-ables as k2_k a2₁, k2_⊥ a2₂  Θ 2_{, Σ}2_{, E , µ, p.} (5.1) We will start by using this approximation to get a simplified version of the evolution equations found in Section 4.5. After that we rewrite the new equations to clearly show that they represent waves, and compare them to previous research.

5.1

Evolution Equations

We start by introducing two new variables which will be recurring in the final equa-tions, k2 = k 2 ⊥ a2 2 +k 2 k a2 1 (5.2) and ˜ k2 = k 2 ⊥ a2₂ + 2 k_k2 a2₁. (5.3)

We now look at the high frequency limit for the evolution equations given in Sec-tion 4.5, keeping the first and second order terms in each equaSec-tion separately. For the even parity sector we get that

˙ ΩV = − 2Θ 3 + Σ 2 + ˙ p µ + p  ΩV, (5.4) ˙ ΣT =  Σ − 2Θ 3  ΣT − c 2 s a2(µ + p) µV − ET, (5.5) ˙ ET ₌ a1˜k2 2ikk HT −1 2(µ + p) Σ T _{− Θ +} 3Σ 2 − 3k2_⊥Σ 2a2₂k˜2 ! ET_{, (5.6)} ˙ µV = a2k2(µ + p) ΣT − a2a1˜k2 2ikk (µ + p) HT −2ikk a1 (µ + p) ΩV, (5.7) ˙ HT = −ikk a1 1 + k 2 ⊥ a2 2k˜2 ! ET − 3k 2 kΣ a2 1k˜2 + Θ ! HT. (5.8)

˙ ΩS=  Σ −2Θ 3 − ˙ p µ + p  ΩS, (5.9) ˙ HT = −a1˜k 2 2ikk ET − 3k 2 kΣ a2₁k˜2 + Θ ! HT, (5.10) ˙ ET = ikk a1 1 + k 2 ⊥ a2 2˜k2 ! HT −3 2 2Θ 3 + Σ 1 − k_⊥2 a2 2k˜2 !! ET. (5.11) If we compare these equations to the ones given when the vorticity vanishes on the perturbed model, [8] or [7], we see that we get the same evolution equations, except for a source term depending on ΩV in the evolution equation for µV_{, and two}

new decoupled evolution equations for ΩS_{and Ω}V_.

5.2

Gravitational and Shear Waves

We saw in the previous section that the remaining evolution equations are almost the same as in the case of vanishing vorticity, and so motivated by [8] we rewrite our evolution equations in the form of second order differential equations, in order to examine which variables will correspond to damped waves. The general wave equation for a time dependent scalar X is given by

¨

X + 2ζω ˙X + ω2X = Z (5.12)

where ζ represents the damping parameter, ω represents the undamped angular fre-quency and Z = Z(t) is some factor which produces forced oscillations. For ζ < 1 we get the real angular frequency from ωp1 − ζ2_{, and so the propagation speed of}

the wave is given by ωp1 − ζ2_/k.

5.2.1 Even Parity

qE = Θ + 3Σ 2 + E Σ− d dtln  a1k˜2  , (5.16) q_H= Θ +3Σ 2 + E Σ− d dtln  k2 a1k˜2  , (5.17) qΣ = 2Σ + 5Θ 3 . (5.18)

We once again compare with the case when the vorticity vanishes on the back-ground, and find that the evolution equations are the same, except for a term con-taining ΩV in equation (5.15). Hence, HT and ET _{represents damped, decoupled}

gravitational waves which to leading order propagates at the speed of light. The remaining variable ΣT represents shear waves which propagate with the speed of sound, and at higher order is both damped and acted on by a force produced by the gravitational wave HT and the vorticity ΩV. The last variable µV _{represent density}

waves which propagate at the speed of sound, and to higher order are also damped and undergo forced oscillations similar to the shear. We have omitted much of the calculations and reasoning regarding the above results, since it is almost identical to previous research, see [7, 8].

5.2.2 Odd Parity

For the odd parity we can rewrite the evolution equations as ¨ ET + q_EE˙T + k2ET = 0, (5.19) ¨ HT _{+ q} HH˙T + k2HT = 0, (5.20) where q_E = 3 Σ + 2Θ 3 − k_⊥2Σ a2₂k˜2 ! − d dtln 1 a1 1 + k 2 ⊥ a2₂˜k2 !! , (5.21) qH= 3 Σ + 2Θ 3 − k_⊥2Σ a2₂k˜2 ! − d dtln  a1˜k2  . (5.22)

The solutions above are independent of the vorticity and so exactly the same as in the case when the vorticity vanishes on the perturbed model. Hence, both ET and HT _{represent decoupled gravitational waves which to leading order propagate at}

Chapter 6

Concluding Remarks

In this thesis we have extended previous research looking at perturbations of LRS class II cosmological backgrounds with vanishing vorticity on the perturbed model, [7, 8], to include a nonvanishing vorticity on the perturbed model. As in the previ-ous research, we have utilized the 1+1+2 covariant formalism, and the results are in terms of time evolution equations of harmonic coefficients. In the case of nonvan-ishing vorticity on the perturbed model we end up with eight harmonic coefficients from which all other quantities can be described, and they decouple into the even and odd sector represented by

n

ΩV, µV, ΣT, ET,HTo and

n

ΩS, ET, HTo,

respectively. This means that the vorticity has introduced two new degrees of free-dom, in ΩV and ΩS.

It turns out that the time evolution of the vorticity completely decouples from the other perturbed variables, and can be solved for exactly if we assume a linear equation of state, p = (γ − 1) µ. The resulting vorticity is then

ΩS = CΩaγ−11 a 2(γ−2)

2 (6.1)

and

ΩV = C_Ωaγ−2₁ a2γ−3₂ (6.2) where CΩ and C_Ω are constants related to the boundary conditions. The solutions

above also show that angular momentum is preserved in the direction of anisotropy, and it is also preserved for arbitrary directions in the 2-sheets for the flat Friedmann case, Σ = 0 or equivalently a1 = a2. It is not preserved for arbitrary directions on

the 2-sheets for a1 6= a2though, which is expected.

The remaining variables were analyzed in the high frequency limit, in which their evolution equations were the same as in the case with vanishing vorticity on the perturbed model, except for a source term containing the vorticity in the time evolution of the density, µV_{. The density in turn acts as a source term for the shear,}

ΣT, and so the shear wave get an extra source term from the vorticity. The other four variables, ET_{, ¯H}T_{, ¯E}T _{and H}T _{all represents source-free, damped gravitational}

waves as in the case with vanishing vorticity on the perturbed model.

which could allow for observations of gravitational waves such as those predicted here.