Linear Cosmological Perturbations and Cosmic Microwave Background Anisotropies. Carlo Baccigalupi

Transcription

1 Linear Cosmological Perturbations and Cosmic Microwave Background Anisotropies Carlo Baccigalupi January 21, 2015

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3 Contents 1 introduction things to know for attending the course plan of the lectures notation and micro-elements of general relativity Homogeneous and isotropic cosmology comoving coordinates and conformal time stress energy tensor expansion and conservation relativistic and non-relativistic matter, dark energy brief cosmological thermal history excercises question about the dynamics of cosmological abundances question about cosmic acceleration Classification of linear cosmological perturbations background and perturbation dynamics decomposition of cosmological perturbations in general relativity Fourier expansion of cosmological perturbations perturbations in the metric tensor scalar-type components vector-type components tensor-type components perturbations in the stress energy tensor scalar-type perturbations vector-type perturbations tensor-type perturbations gauge transformations metric tensor transformation stress energy tensor transformation gauge dependence, independence, invariance and spuriosity excercises question about the cosmological perturbations algebra and metric tensor fluctuations question about the cosmological perturbations algebra and stress energy tensor fluctuations question about gauge invariant formalism

4 4 CONTENTS 4 Linear cosmological perturbation dynamics analysis of the perturbed einstein and conservation equations effective horizon sub-horizon and super-horizon scales equations for scalar-type perturbations equations for vector-type perturbations equations for tensor-type perturbations super-horizon evolution sub-horizon evolution the Harrison Zel dovich spectrum the power spectrum of matter perturbations today excercises question about effective horizon and cosmological scales question about density perturbations in a cosmological constant dominated cosmology question about the super-horizon scalar-type peculiar velocity question about the Harrison Zel dovich power spectrum for gravitational waves question about structure formation in a cosmology which looks open Physics of the cosmic microwave background CMB phenomenology background and perturbations of a cosmological black body distribution polarization harmonic decomposition, angular power spectra, and Gaussian statistics lab frame ˆk frame Boltzmann equation gravitational scattering Thomson scattering harmonic expansion excercises question about the dipole Thomson scattering Cosmic microwave background anisotropies Solving the Boltzmann equation Tight coupling The CMB angular power spectrum total intensity polarization Relation with cosmological observables Status of CMB observation and future expectations excercises question about cosmological distances for decoupling question about angles and scales for the acoustic peaks question about the CMB angular power spectrum

5 CONTENTS question about the role of baryons in acoustic oscillations question about projection effects for CMB anisotropies question about an integral solution to CMB anisotropies question about Silk damping

6 6 CONTENTS

7 Chapter 1 introduction 7

8 8 CHAPTER 1. INTRODUCTION These are the lecture notes for the course of linear cosmological perturbations and cosmic microwave background anisotropies for PhD students at SISSA. Over the years, several students contributed to improve the quality of these notes, with suggestions and corrections. For this, i am thankful to Viviana Acquaviva, Enrico Barausse, Guido D Amico, Alessandro Lovato, Federico Stivoli. The aim of these lectures is to describe the physics of the relic radiation in cosmology, the Cosmic Microwave Background (CMB), in the framework of the theory of relativistic linear cosmological perturbations. At the end of the course, a student should be able to understand most of the modern scientific papers on this matter, ranging from the theoretical details of the CMB anisotropies, down to the physical content and implications of the current experimental data. In this chapter, we give an overview of the course, specifying the most important literature to look at, as well as reviewing the plan of the lectures. In the end, we also fix the notation we adopt, and introduce very basic elements of general relativity. 1.1 things to know for attending the course From the point of view of the description of the current cosmological picture, these lectures aim at being self consistent, altough who s attending might benefit from having familiarity with the physics of the Friedmann Robertson Walker (FRW) background cosmology, as well as any knowledge of cosmological perturbations. Moreover, the physics of the CMB is well known, as at the epoch in which it decouples from the other cosmological components, the system was close to thermal equilibrium characterized by a temperature of about 3000 K, for instance the same as the external region of a star, and interactions are well understood once the physical constituents are specified. On the other hand, some knowledge of general relativity is necessary in order to approach these lectures properly. Indeed, the whole cosmology is deeply rooted in general relativity, and with the level of precision of cosmological measurements nowadays, getting to the percent or better, it is no longer possible to take Newtonian shortcuts. This is relevant in particular for the whole scheme of linear cosmological perturbations, which is a most important part of the present work. The main text where studying while attending the course should be represented by these notes. Moreover, the students are welcome to consult the books and papers which represent the basis of this text. They are: The textbook by Scott Dodelson, Modern Cosmology, Elsevier Science, 2003, review paper by Hideo Kodama and Misao Sasaki, Progresses of Theoretical Physics Supplement 78, 1, 1984, papers by Wayne Hu and Martin White 1997, Phys. Rev. D 56, 596, Baccigalupi 1999, Phys. Rev. D 59, textbook by Andrew R. Liddle and David H. Lyth, Cosmological Inflation and Large Scale Structure, Cambridge Press 2000, may be also relevant.

9 1.2. PLAN OF THE LECTURES plan of the lectures The first part of the course is dedicated to a summary of the FRW cosmological background, setting the notation and estabilishing also a continuity with possible previous courses that the students may have attended. The lectures then cover the following topics: linear cosmological perturbation theory, CMB radiation in cosmology, Boltzmann equation for black body radiation in cosmology, CMB decoupling and anisotropies, harmonic expansion, phenomenology of CMB anisotropies, status of CMB observations and the knowledge we have from those up to now. The course consists of 14 lectures for the PhD students of the astrophysics and astroparticle courses. The program is the following: 1 th lecture: introduction and background cosmology (1-2.5), 2 nd lecture: classification and Fourier expansion of linear cosmological perturbations in general relativity (3.1,3.3), 3 rd lecture: perturbations to the metric and stress energy tensors, gauges ( ), 4 th lecture: gauge transformations and effects ( ), 5 th lecture: perturbed Einstein and conservation equations ( ), 6 th lecture: super-horizon evolution (4.1.6), 7 th lecture: sub-horizon evolution (4.1.7), 8 th lecture: Harrizon Zel dovich spectrum ( ), 9 th lecture: CMB physics ( ), 10 th lecture: Boltzmann equation, gravitational and Thomson scattering ( ), 11 th lecture: phenomenology of CMB anisotropies (6-6.2) 12 th lecture: the obserbed CMB angular power spectrum ( ), 13 th lecture: relation with cosmological observables (6.4), 14 th lecture: status of CMB observations and future expectations (6.5). The schedule above may undergo variations which will be notified in advance.

10 10 CHAPTER 1. INTRODUCTION 1.3 notation and micro-elements of general relativity Spacetime is described by three spatial coordinates plus time. Greek indeces run from 0 to 3, while latin indeces are used for spatial dimensions, from 1 to 3. We use x to indicate a generic spacetime point, x and ˆx for its spatial component and versor, respectively. The fundamental constants we use are the Boltzmann and Gravitational ones, indicated with k B and G, respectively. We work with unitary velocity of light, c = 1, and Planck constant, = 1. In general relativity, the infinitesimal distance from two spacetime points is defined as ds 2 = g µν dx µ dx ν, (1.1) where g µν (x) is the metric tensor. By an appropriate change of reference frame, at any given position x it is always possible to reduce the metric tensor to the Minkowski one, meaning that the system changes to the one which is in free fall locally in x. The signature of the metric tensor we adopt is the following: (, +, +, +). (1.2) Tensors with indeces down are covariant, while the ones with indeces up are controvariant. The inverse of the metric tensor is in the controvariant form, and is defined by g µρ g ρν = δ ν µ, (1.3) where δ ν µ is the Kronecker delta. We shall use the Kronecker delta in arbitrary index configuration: δ ν µ = δ µν = δ µν = 1 if µ = ν, 0 otherwise. (1.4) The Christoffel symbols are defined as usual as Γ αβγ = 1 2 ( gβγ x α Γ α βγ = 1 2 gαα ( gα β x γ + g αγ x β + g α γ x β g αβ x γ ) g βγ x α, ), (1.5) where the repeated indeces are summed everywhere, unless specified otherwise. The Riemann, Ricci and Einstein tensors are given by and R α βµν = Γα βν x µ Γα βµ x ν + Γα λµγ λ βν Γ α λνγ λ βµ, (1.6) R µν = R α αµν, (1.7) G µν = R µν 1 2 g µνr, (1.8) where R = R µ µ is the Ricci scalar. To simplify the notation, let us introduce the following conventions for derivation in general relativity: ;µ µ means covariant derivative with respect to x µ,

11 1.3. NOTATION AND MICRO-ELEMENTS OF GENERAL RELATIVITY11 a γij a means covariant derivative with respect to the unperturbed spatial metric in cosmology, γ ij, defined in (2.4).,µ means ordinary derivative with respect to x µ. A uni-dimensional tensor, or vector, v µ can be obtained via covariant derivation of a scalar quantity s as v µ = s ;µ = s,µ, (1.9) where the last equality holds for scalars only. covariant derivation of a vector as A tensor can be obtained via t µν = v µ;ν = v µ,ν v α Γ α µν, t ν µ = v ν ;µ = v ν,µ + v α Γ ν αµ, (1.10) Further covariant derivatives raise the rank of tensors: u µνρ = t µν;ρ = t µν,ρ t αν Γ α µρ t µα Γ α ρν u ν µρ = t ν µ;ρ = t ν µ,ρ t ν αγ α µρ + t α µγ ν ρα. (1.11) In this way the rank of tensors may be increased at will, keeping the sign conventions associated with covariant and controvariant indeces.

12 12 CHAPTER 1. INTRODUCTION

13 Chapter 2 Homogeneous and isotropic cosmology 13

14 14 CHAPTER 2. HOMOGENEOUS AND ISOTROPIC COSMOLOGY The FRW metric is built upon the hypothesis that there exists a global spatial hypersurface orthogonal to the time coordinate at all times, and that space is homogeneous and isotropic as seen from any location. Homogeneity means that at a given time, the physical properties, e.g. particle number density, velocity, spacetime curvature etc., are the same everywhere in space; isotropy means that in addition physical quantities do not depend on the observation direction. These assumptions simplify dramatically the structure of the metric tensor g µν. For orthoginal coordinates, no off-diagonal term is allowed. Moreover, the system remains with two degrees of freedom only; the first one is a global scale factor, fixing at each time the value of physical lengths; the second one is related to the spacetime curvature, as an homogeneous metric can be globally more or less curved. The form of the fundamental length element may be expressed as ( ) ds 2 = dt 2 + a(t) Kr 2 dr2 + r 2 dθ 2 + r 2 sin 2 θdφ 2, (2.1) where a(t) and K represent the global scale factor and the curvature, respectively; r, θ and φ are the usual spherical coordinates for radius, polar angle and angle on the plane orthogonal to the polar direction, respectively. The physical meaning of the scale factor can be read straightforwardly from the metric, and the only point to discuss concerns its dimensions. One may assign physical dimensions to the scale factor a or to the radial coordinate r; in this lectures, we choose the second option. Concerning the curvature, some more discussion is needed. The first point is about dimensions again; if r is dimensionless, K is also dimensionless. If r is a length, then K is the inverse of the square of a length. Moreover, if K = 0, then the spatial part of the length (2.1) is Minkowskian, and in this case the FRW metric is defined to be flat. If K > 0, there is a coordinate point in which the metric is singular, given by r H = ±1/ K; this means that an infinite physical distance corresponds to those coordinates, regardless of the value of the scale factor a, and the FRW metric is defined to be closed; note that this is in agreement with the assumption of homogeneity, as this property is the same as seen from all spacetime locations. If K < 0, the opposite happens, as there is no divergence, and the distance between two space points vanishes at infinity; in this case the FRW metric is defined to be open. Finally, note that one may always change the overall normalization of a or r in (2.1), and therefore, as a pure convention, we can restrict our attention to three relevant cases for K: for example, if K is chosen to be dimensionless, which is not the case here, K may be chosen to be 1, 0, 1, representing an open, flat and closed FRW cosmology, respectively. 2.1 comoving coordinates and conformal time Of course one might apply any change of coordinate to the FRW metric, giving it different looks. On the other hand the form (2.1) is the one that is most common for cosmological purposes. The reason is that the expansion, represented by the scale factor a, has been factored out of the spatial dependence. This leads us to the concept of comoving coordinates, i.e. at rest with respect to the cosmological expansion, or in other words defined by the spacetime points for which r = constant, θ = constant, φ = constant, (2.2)

15 2.1. COMOVING COORDINATES AND CONFORMAL TIME 15 Figure 2.1: A representation of cosmological expansion in comoving coordinates; distances between objects increase with time, but their coordinates, represented by labels here, stay constant. where r, θ and φ are coordinates in the frame where the metric assumes the form (2.1). In order to visualize, one may think that galaxies are the tracers of the cosmic expansion, or in other words, their motion is approximately described by (2.2). In the original Hubble picture of the cosmic expansion, this corresponds to assign the whole motion of galaxies to the cosmic expansion, giving them a fixed coordinate. A picture of comoving coordinates is given in figure 2.1. Although time does not enter in the discussion about comoving coordinates above, there is a very common time variable which may replace the ordinary time in (2.1). By performing the coordinate change dτ = dt a(t), (2.3) and adopting Cartesian coordinates instead of the spherical ones in (2.1), the FRW metric may be easily written as g µν a 2 0 (1 Kr 2 ) (1 Kr 2 ) 1 0 a2 γ µν (2.4) (1 Kr 2 ) 1 so that the cosmic expansion can be completely factored out, defining the comoving metric γ µν ; τ is the conformal time, and represents our time variable in the following, unless specified otherwise. We will indicate the conformal time derivative with, while those with respect to the ordinary time are indicated with the subscript t. It is also useful to define two different quantities describing the velocity of the expansion, i.e. H = a t a, H = ȧ a, (2.5) named ordinary and conformal Hubble expansion rates, respectively; as it is easy to see, the two are related by H = ah.

16 16 CHAPTER 2. HOMOGENEOUS AND ISOTROPIC COSMOLOGY 2.2 stress energy tensor The stress energy tensor specifies the content of spacetime, in terms of physical entities, i.e. particles and their properties. Here we describe only a perfect relativistic fluid, homogeneous and isotropic. As for the metric, these assumptions reduce substantially the complexity of the general expression for a stress energy tensor. The relevant quantities are just the energy density, ρ, and the pressure p, which need to depend on time only in the FRW background. The stress energy tensor in the mixed form with covariant and controvariant indeces is in direct relation with ρ and p: the (0, 0) component represent the energy density, while p is isotropically assigned to all directions as T ν µ ρ p p p, (2.6) where the minus to the energy density is due to the choice of our signature (1.2). The stress energy tensor may also be written as T µν = (ρ + p)u µ u ν + pg µν, (2.7) where u µ represents the quadri-velocity of a fluid element with respect to a given observer, with an affine parameter which for convenience may be taken as the conformal time itself: u µ = dxµ dτ. (2.8) In analogy with the normalization of the quadri-impulse of a particle with mass m, p µ p µ = m 2, the quadri-velocities are the momentum of a unitary mass particle, normalized as u µ u µ = 1. In comoving coordinates, where the u a = 0, this condition implies ( ) 1 u µ, 0, 0, 0. (2.9) a 2.3 expansion and conservation The Einstein and conservation equations G µν = 8πGT µν, T ;ν µν = 0 (2.10) reduce to two differential equations only, where the independent variable is the time τ; they determine the dynamics of the expansion, plus the conservation of energy, respectively. The first one is the Friedmann equation H 2 = 8πG 3 a2 ρ K, (2.11) which is equivalent to the Ray Choudhury equation describing the acceleration of the expansion: 3Ḣ = 4πGa2 (ρ + 3p). (2.12) The conservation equation becomes ρ + 3H(ρ + p) = 0. (2.13)

17 2.4. RELATIVISTIC AND NON-RELATIVISTIC MATTER, DARK ENERGY17 As it is evident, it is impossible to solve this system if some relation between pressure and energy density is given, p(ρ). For interesting cases, as those we shall see in the next Section, pressure is proportional to the energy density: p = wρ, (2.14) where w is defined to be the equation of state of the fluid. 2.4 relativistic and non-relativistic matter, dark energy So far we did not consider the case in which the stress energy tensor is made by more than one component, although in realistic conditions, several of those are present at the same time. In this case, the stress energy tensor we treated so far corresponds to the total one, i.e. the sum over those corresponding to each of the components, labeled by s, as follows: T µν = s st µν. (2.15) The single stress energy tensors may not be conserved as a result of mutual interactions between the different species. Therefore, the conservation equation for each component may be written as st ;ν µν = s Q µ, (2.16) where s Q µ represents the non-conservation. Since the total stress energy tensor must be conserved, the interactions between the different components must satisfy the condition sq µ = 0. (2.17) s The simplest example of cosmological component is represented by the nonrelativistic (nr) matter. The usual example is that of particles characterized by a temperature giving rise to a thermal agitation which is negligible with respect to the mass m, so that their momentum satisfies the approximation p µ p µ = m 2 g 00 (p 0 ) 2. (2.18) Whatever the interaction is, in this regime collisions are negligible. No collisions means no pressure, and therefore for this species the equation of state is simply zero. A most important cosmological component of this kind is commonly known as Cold Dark Matter (CDM). As it is easy to verify from the conservation equation, the time dependence of this component, assuming that it is decoupled from the others, may be expressed as a function of the scale factor as ρ nr a 3. (2.19) The next example is opposite in many respects. Relativistic (r) particles at thermal equilibrium are characterized by an energy which is dominated by thermal agitation rather than mass. The condition of thermal equilibrium implies that p r = 1 3 ρ r. (2.20)

18 18 CHAPTER 2. HOMOGENEOUS AND ISOTROPIC COSMOLOGY As it is easy to verify from the conservation equation, assuming again that the radiation is not coupled with other species, this implies ρ r a 4, (2.21) which has a direct intuitive meaning. Indeed, taking photons as an example, each of those carries an energy ω where ω is the photon frequency, redshifting with time as the inverse of the scale factor as a result of the stretching of all lengths. This is responsible for the extra 1/a power in (2.21) with respect to (2.19), which contains only the contribution from the dilution as a result of the expansion of the volume. A third case, most interesting and dense of theoretical implications, is the case in which the energy density is conserved, i.e. p de = ρ de, (2.22) where the subscript means dark energy, which is the class of cosmological species which produces an equation of state close or equal to 1. A constant vacuum energy density appeared for the first time in the form of a cosmological constant Λ, introduced by Einstein himself in the Einstein equations as a pure geometrical term: G µν + Λg µν = 8πGT µν. (2.23) Indeed, bringing it to the right hand side, and passing to the mixed form for indeces, one gets ( G ν µ = 8πG Tµ ν Λ ) 8πG δν µ, (2.24) which shows how a Λ term into the Einstein equation may be interpreted as a component of the stress energy tensor. Looking at (2.7) it is straightforward to verify that such component is characterized by p Λ = ρ Λ = Λ/8πG. 2.5 brief cosmological thermal history Starting from the Friedmann equation (2.11), it is straightforward to define the cosmological critical density which allows to write the same equation as ρ c = 3H2 8πG, (2.25) 1 = s Ω s, (2.26) where the cosmological abundance for a given species s is given by Ω s = ρ s ρ c, (2.27) and for consistency the curvature contribution is be also defined as Ω K = K H 2. (2.28)

19 2.5. BRIEF COSMOLOGICAL THERMAL HISTORY 19 Observations in cosmology measure the different abundances. According to the most recent ones (Komatsu et al., 2006), one has Ω de 73%, Ω K 0, Ω nr 27%, Ω r (2.29) Since the different components have a different time behavior, as described in the previous Section, the numbers above change with time. In particular, at early times, or sufficiently small values of the scale factor, Ω r was larger than the others, approaching 1 when going indefinitely back in time. The latter condition defines the radiation dominated era (RDE); that is followed by one in which the non-relativistic component dominates, which is called matter dominated era (MDE). At recent times, the expansion is getting dominated by the effect of the dark energy. We can give at this point a very brief chronology of the most relevant physical processes occurred during these epochs. Given the available observations, it is convenient but also reasonable to assume that in the beginning the universe was mainly characterized by relativistic components only, in mutual thermal equilibrium. This allows to assign a temperature T to the system, corresponding to the temperature of photons, equivalent to an energy scale E = k B T. The different cosmological species are maintained at equilibrium by their physical interactions. The cosmological expansion clearly works against the persistence of thermal equilibrium, as it dilutes the density of interacting objects. One by one, each component decouples from the rest of the system as expansion proceeds. The last one to decouple is represented by photons, and forms the CMB. Let us review the most important steps in the thermal cosmic history, going backward in time, up in temperature and energy: E 10 3 ev, the present epoch of star and galaxy formation and evolution (MDE), E 10 1 ev, photons decouple from baryons and leptons, origin of the CMB (MDE), E 10 0 ev, matter radiation equality, E 10 1 MeV, nucleosynthesis through deuterium formation from neutron proton scattering (RDE), E 10 0 MeV, neutrino decoupling from neutrino antineutrino annihilation in electron positron pairs, electron positron annihilation in two photons (RDE), E 10 2 GeV, symmetry breaking between electromagnetic and weak interaction (RDE),...? At higher energies, or earlier cosmological epochs, the electromagnetic and weak interactions should be described by a single one, named electroweak. Similar models exist for the unification of the strong interaction with the others, which should occur at earlier cosmological epochs. It is conceivable that also the gravitational interaction unifies with the other forces; although there is not a consensus model for that, one may guess that such epoch corresponds to

20 20 CHAPTER 2. HOMOGENEOUS AND ISOTROPIC COSMOLOGY the energy scale which may be formed combining the fundamental constant including the gravitational one: c 5 E P lanck = G 1019 GeV. (2.30) In the above expression, we have momentarily re-introduced the use of the speed of light and Planck constants. If this is indeed the scale at which gravity unifies with other forces, then a large gap seperates this scale from the highest one which may be probed directly in laboratories on the Earth, which is of the order of the TeV. Cosmology may provide access to relics of early stages in the cosmological history, where typical energy scales were much higher than that. As we will see later in these lectures, the CMB is a typical example, as it carries traces of the initial cosmological perturbations. 2.6 excercises question about the dynamics of cosmological abundances The cosmological critical energy density is ρ c = 3H2 8πG. (2.31) Today the ratio between non-relativistic matter energy density and the critical energy density, Ω nr 0 = ρ nr 0 /ρ c 0, is about 27%. The same ratio for the dark energy density is Ω de 0 = ρ de 0 /ρ c 0 = 73%. By describing the dark energy as a Cosmological Constant, p de = ρ de and the non-relativistic matter energy density as a pressureless component, p nr = 0, find the scale factor ratio of a/a 0 for which ρ de /ρ nr = 10 3, where a 0 is the value of the scale factor today question about cosmic acceleration Assuming that the universe today is described by a flat FRW with dark energy and matter characterized by Ω de = 73%, Ω nr = 27%, (2.32) compute the value a a or z a = 1/a a 1 at which the acceleration starts, i.e. when ä > 0; assume that the dark energy is described as a cosmological constant. In the hypothesis that the universe is completely matter dominated before that epoch 1, compute the cosmological time t a from a = 0 to a a, knowing that the Hubble expansion rate today is H 0 = 70 km/s/mpc. Compare the result with the age of the universe t 0 obtained integrating from a = 0 and a = 1, still in the hypothesis of matter domination. 1 Taking into account the radiation dominated era does not lead to substantial changes.

21 Chapter 3 Classification of linear cosmological perturbations 21

22 22CHAPTER 3. CLASSIFICATION OF LINEAR COSMOLOGICAL PERTURBATIONS The whole task we have is to define and understand small perturbations around the FRW metric, in a general relativistic framework. The perturbed FRW metric tensor may be defined as ḡ µν = g µν + δg µν = a 2 (γ µν + h µν ), (3.1) where the bar means background plus perturbations, i.e. the complete system; h µν (x) represents entirely the perturbation to the metric, and breaks the FRW assumptions completely. We limit our analysis to the linear regime, which may be simply expressed as h µν γ µν, (3.2) in any spacetime point x. Note that in a flat FRW background this corresponds to h µν 1. At the same time, the stress energy tensor is perturbed as T ν µ = T ν µ + δt ν µ, (3.3) and in this case linearity may be expressed in terms of the non-zero quantities in the background tensor, T µν : δt ν µ ρ, p. (3.4) One may question what is the motivation for dealing with linear perturbations only in cosmology. Indeed, structures we see around us are markedly non-linear in the density distribution, like galaxies or clusters of galaxies. The underlying assumption is that most of the cosmological evolution occurs in linear regime, while non-linearity is confined to very small scales, reaching the typical size of a galaxy or a cluster of galaxies only in cosmologically recent epochs. A great support for this assumption comes from the CMB itself. As it is well known the CMB temperature anisotropies are at the 10 5 level relatively to the average temperature over the whole sky, down to an angular resolution of a few arcminutes. This remarkable level of isotropy suggests that the density fluctuations, and cosmological perturbations in general, were in a linear regime at the epoch in which the CMB radiation decoupled from the rest of the system. Then, the linear approximation should be valid to describe the bulk of physical cosmology on a large interval of time and physical scales, before and after the CMB origin, breaking down only recently and on scales smaller than those of galaxy clusters. 3.1 background and perturbation dynamics Linearity allows to neglect the terms of second or higher orders, i.e. involving products of fluctuations. A direct consequence of this concerns the relativistic algebra of cosmological perturbations. Of course, in a perturbed FRW environment, tensor indeces are raised and lowered using the complete metric, ḡ µν. On the other hand, linearity allows to neglect the second or higher order terms. This implies non-trivial relations between fluctuations with different configurations of indeces. By using the usual general relativistic relation ḡ µν = ḡ µµ ḡ νν ḡ µ ν one gets δg µν = g µµ g νν δg µ ν, (3.5)

23 3.2. DECOMPOSITION OF COSMOLOGICAL PERTURBATIONS IN GENERAL RELATIVITY23 Figure 3.1: A representation of scalar-type (left) and vector-type (right) vector fields. where we used the fact that the background metric is diagonal. Another, most important consequence of the linear scheme is that the background dynamics is not influenced by perturbations:the prescription is no longer valid whenever h µν approaches unity or δtµ ν gets close to the background energy density or pressure. Therefore, any equation in cosmology splits in two, concerning the background and perturbation dynamics, respectively: { Ḡ µν = 8πG T Gµν = 8πGT µν µν, (3.6) δg µν = 8πGδT µν { T T µν ;ν ;ν = 0 µν = 0 δ(t µν) ;ν = 0. (3.7) Note that, on the contrary, perturbations are affected by the background dynamics. The scale factor evolution, as well as the dynamics of energy density and pressure, do appear explicitely in the perturbation equations, as we shall see in the following. 3.2 decomposition of cosmological perturbations in general relativity We now carry out a general discussion concerning the different components of the cosmological perturbations, h µν and δt µν. This is done on the basis of their physical properties in a general relativistic framework, yielding a simple but general classification, which is one of the basis of modern cosmology. The fluctuations depend on the spacetime point x (τ, x). The decomposition is made considering the spatial part only, while the perturbed Einstein and conservation equations (3.6, 3.7) give the time dependence of each species. Let us consider in general the space dependence of functions in general relativity, but restricting ourselves to the spatial metric at a fixed time, γ ij. Under spatial rotations, a function of the space may behave as a scalar, vector or tensor: s( x), v i ( x), t ij ( x). (3.8) While the physical nature of scalars is unique, the classification gets non-trivial already for vectors. Indeed, each vector function may be divided in full generality in a part which may be obtained via derivation of a scalar, and the rest. The

24 24CHAPTER 3. CLASSIFICATION OF LINEAR COSMOLOGICAL PERTURBATIONS first component is called scalar-type component of the vector, while the second one is called vector-type component of the vector. In a similar way, tensors may be divided in three components. The first one, called scalar-type component, comes from double derivation of a scalar function. The second one, called vector-type component, comes from derivation of a vector-type component of a vector. The remaining part is called tensor-type component. Technically, by derivation here we mean the covariant one with respect to the γ ij metric, as we are still in a general relativistic framework, even if the FRW assumptions are allowing us to restrict ourselves to space. Let us see all that in formulas now. For vectors, one has v i = s i + v i, (3.9) indicating scalar-type and vector-type components, respectively; mathematically, the relation above is also known as the Helmholtz equation. Still maintining full generality, s may be chosen is such in order to be responsible for the divergence of the whole vector function: γ ij s s i i = vi i = v. (3.10) This implies that the vector-type component is divergenceless: v i i = 0. (3.11) As we will see in a moment, this has a direct and intuitive physical meaning. For tensors, the decomposition may be written as t ij = s ij + v i j + t ij. (3.12) In analogy with what has been done for vectors, in general the scalar-type component is responsible for the trace of the tensor, obeying the equation γ ij s = t i i t, (3.13) while the vector-type is divergenless and obeys the same condition as in (3.11). This implies that the tensor-type component is traceless: t i i = 0. (3.14) We may further restrict the properties of t ij. The divergence of t ij is given by t ij j = s ij j + vi j j + tij j. (3.15) Within the constraint on s given by (3.13), we may require in full generality that the vector-type component satisfies which implies v i j j = tij j s ij j, (3.16) t j ij = 0. (3.17) The relations (3.17,3.14), together with the one above, are known as the transverse traceless conditions, respectively. The physical interpretation of this decomposition is rather simple. Consider for

25 3.3. FOURIER EXPANSION OF COSMOLOGICAL PERTURBATIONS 25 Figure 3.2: A representation of the relation between angles and scales in the CMB anisotropies. example the vector field of velocities caused by gravitational infall toward the center of some overdensity. That is an example of scalar-type component of a vector field, as the divergence of this field is non-null. Physically, the nature of such motion is clearly of scalar origin, because it is completely caused by the overdensity in the center. Consider now a vortex. In this case, no static density perturbation can cause such a motion, which is divergenless by construction. This is an example of a vector field which is made completely by its vector-type component. A graphical picture of the two types of vector fields is in figure 3.1. To finish with the examples, the tensor-type component of tensors describe to the tensor fluctuation modes in general relativity, i.e. the gravitational waves, as we see below. As a final mathematical remark, we notice that the classification above corresponds to the decomposition of the SO(3) group into its irreducible representations, up to spin 2 modes. 3.3 Fourier expansion of cosmological perturbations The perturbed Einstein and conservation equations will have in general the form A δ 1 + B δ 2 + C δ = 0, (3.18) where A, B, C,... are function of time only, while the δ i represent perturbations and their time derivatives; no products between perturbations are allowed in a linear theory. As a consequence, working in the Fourier space becomes feasible, as the absence of products between perturbed quantities makes each Fourier mode evolving independently on the others. Working in the Fourier space is particularly useful in the present context, since in cosmology observables are often probed by studying their angular distribution, which correspond to cosmological scales, or Fourier wavenumbers, at a given redshift. The CMB is a direct example of this. Indeed, as we shall see, the last scattering is a rather sudden

26 26CHAPTER 3. CLASSIFICATION OF LINEAR COSMOLOGICAL PERTURBATIONS event in cosmology, and the thickness of the last scattering surface would correspond today to a distance of about 10 Mpc, compared with almost Mpc from us. The angular distribution of CMB anisotropies directly corresponds to its Fourier expansion, expecially on small angular scales. This is graphically represented in figure 3.2. In this Section, therefore, we lay down the formalism necessary to study cosmological perturbations in the Fourier space, and that will be our framework in the following. We expand perturbations into a complete set of functions which are the eigenfunctions of the Laplace Beltrami operator γij f = γ ij f ij. For scalar functions, those eigenfunctions Y k ( x) are the solutions of the differential equation ( γij + k 2 )Y = 0. (3.19) where k = k and we drop the explicit arguments of Y k ( x) in the following. Note that in flat cosmology γij is just the Laplacian operator, and the solutions of the equation above are just plane waves: Y e i k x in flat FRW. (3.20) The scalar-type components of vectors is expanded in Fourier modes exploiting the simple derivation of the functions above, expressed as Y i = 1 k Y i, (3.21) where the factor and sign in front are purely conventional. Similarily, the scalartype component of tensors may be expanded in the following Fourier modes: Y ij = 1 k 2 Y ij γ ijy. (3.22) From (3.19) note that the expression above is traceless; indeed it will be convenient to express the trace of tensors separately. The vector-type component of vectors may be expanded into the vector solutions of the Laplace Beltrami operator, i.e. obeying with the divergenceless constraint ( γij + k 2 )Y (±1) i = 0, (3.23) γ ij Y (±1) i j = 0, (3.24) and the ±1 symbol indicates that there exist two independent solutions satisfying the divergenceless constraint. The vector-type component of tensors may be expanded into Y (±1) ij = 1 2k ( ) Y (±1) i j + Y (±1) j i, (3.25) where the sum is made in order to describe symmetric quantities like elements in the metric tensor are. Finally, the tensor-type component of tensors may be expanded into the tensor solutions of the Laplace Beltrami operator ( γij + k 2 )Y (±2) ij = 0, (3.26)

27 3.4. PERTURBATIONS IN THE METRIC TENSOR 27 with the transverse traceless constraint Y (±2) j ij = 0 = Y (±2)i i, (3.27) and as above the ±2 symbol indicates that there exist two independent solutions satisfying the transverse traceless constraint. From now on, we adopt the Fourier space as our default framework, working out the analysis for one generic mode; as already mentioned, unless otherwise specified, we drop the label k. 3.4 perturbations in the metric tensor We have now all the necessary tools to define and classify the different species of cosmological perturbations in general relativity. The classification is entirely done in terms of the properties under spatial rotations of the perturbation components, defined in section 3.2, and we consider those of the metric tensor first, δg µν ; with the addition of the background component, they represent the total metric tensor (3.1), ḡ µν = g µν + δg µν. We choose to work in Cartesian coordinates, where from (2.4) the background non-zero terms are g 00 = a 2, g 11 = a 2 /(1 Kr 2 ) = g 22 = g 33. We consider δg µν and define scalar and scalartype components, vector and vector-type components, tensor and tensor-type components, respectively scalar-type components The δg 00 quantity is clearly a scalar, since it does not contain any spatial index. In the Fourier space, it may be defined as δg 00 = 2a 2 AY, (3.28) where A and Y represent the Fourier amplitude and Laplace Beltrami operator eigenfunction at wavenumber k, respectively. As we already mentioned, unless specified otherwise we do not indicate the arguments τ in a, τ and k in A, k and x in Y, as in the following those dependences remain the same for all background terms, perturbations and Fourier eigenfunctions, respectively. Obviously, at any time the real space expression of the perturbations in the Fourier space may be got as follows: d 3 k δg 00 (τ, x) = (2π) 3 A(τ, k)y ( k, x). (3.29) The quantity δg 0i is a vector, and since the background term for that is zero, it coincides with the component of the total metric tensor, ḡ 0i. Just like any vector in our scheme, it contains scalar-type and vector-type components. Its scalar-type component may be defined as δg 0i = a 2 BY i. (3.30) The quantity δg ij contains scalar-type components on the diagonal, also yielding a perturbation in its trace, plus vector-type and tensor-type traceless contributions. The scalar-type component may be written as δg ij = 2a 2 (H L Y γ ij + H T Y ij ), (3.31)

28 28CHAPTER 3. CLASSIFICATION OF LINEAR COSMOLOGICAL PERTURBATIONS where H L and H T represent the amplitude at the Fourier mode k of the diagonal (L stays for longitudinal) and non-diagonal (T stays for transverse) components of the perturbations to the metric tensor. Indeed, in the mixed form the second term in the right hand side appears simply as 2H L Y δ j i, and Y ij is traceless as it is evident from (3.22). The transverse component, which as we see now has vector and tensor-type components as well, is also known as shear in the metric vector-type components In analogy with (3.30), we may define the vector-type contribution to the metric tensor fluctuation as δg (±1) 0i = a 2 B (±1) Y (±1) i. (3.32) The vector-type contribution to δg ij does not possess the trace part, reducing to the shear component only, which we may write as δg (±1) ij = 2a 2 H (±1) T Y (±1) ij, (3.33) representing the vector-type component of the metric shear tensor-type components The tensor-type fluctuations in the metric has only a shear component, and may be written as δg (±2) ij = 2a 2 H (±2) T Y (±2) ij. (3.34) This term represents the Fourier amplitude of cosmological gravitational waves, corresponding to the transverse traceless perturbation component. 3.5 perturbations in the stress energy tensor It is convenient to work with the indeces in the mixed form, as they have a direct physical meaning, e.g. represent energy density, pressure, momentum density etc.. As for the metric tensor, the total stress energy tensor may be written as in (3.3), where the background non-zero contributions are T0 0 = ρ, T1 1 = T2 2 = T3 3 = p. If different components are present in the stress energy tensor, their perturbations are still characterized by the quantities defined below, each one labeled with a species index s scalar-type perturbations 0 The perturbation component of T 0 concerns the energy density distribution. We may write it as δt0 0 = ρδy, (3.35) introducing the Fourier amplitude of the spatial fluctuations in the density contrast, δ(τ, k) δρ(τ, k)/ρ(τ). Similarily, the perturbed momentum concides with the corresponding component of the total stress energy tensor T 0 i and is expressed as δt0 i = (ρ + p)vy i, (3.36)

29 3.5. PERTURBATIONS IN THE STRESS ENERGY TENSOR 29 Figure 3.3: A representation of rotation and boost gauge transformations, in the left and right panel, respectively; in the two frames, both linearly close to the FRW one, the cosmological perturbations represented as the ball appears to have different coordinate location and peculiar velocity, respectively. where v represents the peculiar velocity, i.e. the velocity component with which the perturbed fluid drifts away from the Hubble flow, and ρ+p gives the effective momentum mass. Finally, the scalar-type stress energy perturbations are made by an isotropic or trace part, plus the traceless component, commonly known as shear, given by δtij = p(πl Y δij + πt Yij ), (3.37) where the first component, πl (as for the metric L means longitudinal) may be seen as the Fourier amplitude of the perturbation to the pressure contrast, πl (τ, ~k) δp(τ, ~k)/p(τ ), in analogy with what has been done for the energy density. The other component, πt (where again T means transverse), represents the stress energy shear component, as we see now has vector and tensor-type components as well vector-type perturbations In analogy with the definitions (3.36) and (3.37), but dealing with the vectortype components, we may write i(±1) δt0 = (ρ + p)v (±1) Y i(±1), j(±1) δti (±1) = pπt j(±1) Yi, (3.38) (3.39) which represent the vector-type contributions to the perturbed stress energy (±1) tensor; v (±1) may be seen as the Fourier amplitude of vorticity, while πt is the shear of vector-type origin tensor-type perturbations Finally, tensor-type perturbations affect the shear component only, and we may write those as j(±2) (±2) j(±2) δti = pπt Yi, (3.40) (±2) which represent the stress energy counterpart of the gravitational waves HT defined in (3.34).

30 30CHAPTER 3. CLASSIFICATION OF LINEAR COSMOLOGICAL PERTURBATIONS 3.6 gauge transformations As we have seen so far, the linearization of general relativity yields a quite simple decomposition of the different contributions on the basis of their geometrical properties. Linearity implies that in the Einstein equations no terms involving the product of two or more perturbations are considered, and that the equations themselves split in two sub-sets, one evolving the cosmological background, and one dealing with the perturbation evolution. In other words, linearization creates a new set of dynamical variables, the small perturbations around the FRW background, which are obviously not built-in general relativity. Of course, linear perturbations are different in different frames, and in particular on those differing by small amounts from the Hubble frame where the background metric is given by (2.4), yielding effects comparable to the perturbations themselves. Those frames are called gauges. The coordinate transformations between different gauges, involving coordinate shifts of the same order of the perturbations themselves, determine how the perturbed quantities in different gauges are related. We indicate gauge transformations as x µ = x µ + δx µ (x), (3.41) where δx µ, function of the spacetime point x, represents the coordinate shift between the new frame labeled with f, and the original one, f. Cosmological perturbations appear different in different gauges, as shown in figure 3.3 for a small rotation and boost, making a perturbation represented as a ball having different coordinate locations and velocity, respectively. A typical example of the second case is represented by the cosmological dipole, i.e. the dipole term in the angular expansion of the CMB temperature which is attributed to a Doppler shift due to the local motion, represented by a peculiar velocity v of our group of galaxies; the perturbation to the CMB temperature, and consequently our velocity, is measured to be rather small in units of the light velocity, so that the arguments of linear cosmological perturbation theory may be applied: ( ) δt = v (3.42) T dipole Indeed, in a frame moving with velocity v with respect to our local group of galaxies, no dipole would be seen. That is an example of gauge transformation, as the velocity shift is small as in (3.42). This example tells us two important aspects. First, cosmological perturbations, represented here by the CMB temperature, do depend on the chosen gauge. Second, perturbations may be zero in some gauges, and non-zero in others. We now treat more formally these issues. The coordinate shifts δx µ are functions of the spacetime point x and obey the same classification as we did in section 3.2: δx 0 is a scalar function, (3.43) δx i is a vector function, (3.44) composed by scalar-type and a vector-type components. We will express the gauge transformation laws in the Fourier space, therefore it is convenient to

31 3.6. GAUGE TRANSFORMATIONS 31 Figure 3.4: A representation of the Doppler shift on CMB photons, represented as γs, caused by our own peculiar motion with respect to the one where the CMB dipole is absent. define the Fourier amplitudes of δx µ : δx 0 d 3 k (τ, x) = (2π) 3 T (τ, k)y ( k, x) δx i d 3 k (τ, x) = (2π) 3 [L(τ, k)y i ( k, x) + L (±1) (τ, k)y i(±1) ( k, x)]. (3.45) Again we do not indicate explicitely the arguments of the functions T, L, L (±1) in the following. To find how perturbed quantities transform under gauge transformations it is enough to use the transformation laws of tensors in general relativity, keeping the linear order both in the perturbations and in the coordinate shifts as well, as we show now metric tensor transformation The general transformation of g µν between f and f is x ν ḡ µν (x) = xµ x µ x ḡ ν µ ν ( x), (3.46) where as usual the bar means background plus perturbations. It is easy to express the partial derivatives above as a function of the shifts δx µ : x µ x µ = δµ µ + δx µ,µ, x ν x ν = δν ν + δx ν,ν. (3.47) By keeping only the first order terms in the shifts, from (3.46) one obtains ḡ µν (x) = ḡ µν ( x) + ḡ µ ν( x)δx µ,µ + ḡ µν ( x)δx ν,ν. (3.48) To solve the relation above and find the transformation laws between perturbations, we need to express all quantities above in the same coordinate point.