Newtonian Versus Relativistic Cosmology

Transcription

1 Newtonian Versus Relativistic Cosmology Master Thesis by Samuel Flender 1st corrector: Prof. Dr. Domini Schwarz 2nd corrector: Prof. Dr. Dietrich Bödeer University of Bielefeld January 2012

2 Contents 1 Introduction 4 2 The homogeneous and isotropic universe Basic equations Content of the universe Ination Evolution of scales The perturbed universe: Newtonian treatment Denitions Perturbed elds Perturbed Newtonian equations Continuity equation Poisson equation Euler equation Summary Solutions First order solutions Initial conditions Transfer function Second order solutions Discussion Solutions during Λ-domination Newtonian cosmological simulations The perturbed universe: relativistic treatment The perturbed metric Gauge transformations Gauge transformations of scalars, vectors and tensors Gauge transformation of metric perturbations Gauge transformations of velocity and density perturbations Gauge-invariant quantities Gauges Relativistic equations in the conformal Newtonian gauge Solutions Solutions in the conformal Newtonian gauge Solutions in other gauges Gauge-invariant solutions Vector and tensor perturbations

3 5 Quantication of relativistic corrections Relativistic corrections to Newtonian simulations Relativistic corrections to the observed power spectrum Relativistic corrections in ΛCDM models Summary and outloo 57 Appendix 58 Appendix A: Helmholtz decomposition theorem Appendix B: Statistical properties of Gaussian perturbations Appendix C: Perturbations in the inaton eld Appendix D: Meszaros equation Appendix E: Identity of synchronous and comoving gauge References 66 Acnowledgments 68 3

4 Figure 1: The large scale structure of the universe. This map was created using data from the Sloan digital sy survey. Each dot represents a galaxy. [source: 1 Introduction Cosmology is the study of the origin and structure of the universe. We have good observational evidence that the universe was isotropic and homogeneous at early times, the best evidence being the small uctuations (about 10 5 ) of the cosmic microwave bacground (CMB) radiation [1]. However, the structure of the universe that we observe today is very inhomogeneous on scales below 100 Mpc 1 : there are galaxies (on scales of 1 Mpc) and clusters of galaxies (on scales of 10 Mpc). On scales of 100 Mpc there are superclusters of galaxies, which are separated by large voids [2], see gure 1. The theory of structure formation connects the early homogeneous universe with the inhomogeneous universe today: It is assumed that there exist initially small density perturbations, which grow due to gravitational attraction. The mathematical tool to describe this scenario accurately is relativistic cosmological perturbation theory, where one considers small perturbations on the homogeneous and isotropic Friedmann-Robertson-Waler (FRW) metric. However, in this theory there are complications arising due to the freedom of gauge, i.e. choosing the correspondence between perturbed and bacground quantities. Another approach is Newtonian cosmological perturbation theory, which results in perturbed Newtonian equations (continuity equation, Euler equation and Poisson equation) on a FRW bacground. 1 Distances in cosmology are usually measured in units of Millions of parsec (Megaparsec, Mpc), where one parsec (pc) is roughly 3.26 lightyears. 4

5 Figure 2: Result of the Millennium simulation. [source: One has to eep in mind that Newtonian theory is wrong in that it assumes instantaneous gravitational interaction and innite speed of light. However, it turns out that Newtonian and relativistic cosmological perturbation theory coincide for scales much smaller than the Hubble horizon, which is basically the size of the observable universe. Cosmological N-body simulations use Newtonian cosmology to move particles and simulate the density growth in the universe. These simulations can be used to test our understanding of structure formation. A famous simulation is the Millennium run (see gure 2), which simulated the behaviour of particles (each particle representing a dar matter clump with a mass of 10 9 suns) in a box of 1Gpc side length [3]. But the question is, how reliable are these simulations, since they use Newtonian rather than relativistic equations. In this wor, we want to study the relativistic corrections to the Newtonian equations. Cosmological perturbation theory is a major tool to do this. The outline is as follows: In part 2, we will summarize our current understanding of the universe and give the most important equations describing it. In part 3, we are going to consider Newtonian perturbation theory on an expanding FRW bacground. We will derive and solve the perturbed Newtonian equations up to second order. In part 4, we will introduce relativistic cosmological perturbation theory and derive and solve the relativistic evolution equations for the perturbation variables. In part 5, we will quantify the relativistic corrections to cosmological simulations and the observed matter power spectrum. 5

6 2 The homogeneous and isotropic universe 2.1 Basic equations This section resumes the basic equations in cosmology. For a reference, see e.g. [4, 5]. We set c = 1. The Friedmann equation relates the Hubble parameter to the content of the universe: H 2 = 8πG 3 ρ(t) a 2, (2.1) where a is the scale factor, H 1 a a t = ȧ a is the Hubble parameter, is the curvature of the universe and ρ(t) is its energy density. This can consist of matter density, radiation density, or something else. Dene the critical energy density and the relative energy density Then we can rewrite the Friedmann equation as ρ c 3H2 8πG (2.2) Ω(t) ρ(t) ρ c (t). (2.3) 1 Ω(t) = a 2 H 2. (2.4) From this equation it can be seen that the universe has critical energy density (Ω = 1) if and only if the curvature vanishes. Furthermore, an overcritical universe (Ω > 1) has positive curvature and an undercritical universe (Ω < 1) has negative curvature. The equation of state connects the energy density of a given source of energy and the pressure P that is created by it, where w is the equation of state parameter. P = wρ, (2.5) Another important equation is the continuity equation, which tells how much the energy density will change in time, Inserting the equation of state, we nd ρ + 3H(ρ + P ) = 0. (2.6) ρ + 3Hρ(1 + w) = 0, (2.7) which has the solution ρ = ρ 0 a 3(1+w), (2.8) where ρ 0 = ρ(t 0 ) is the energy density today (we scale a such that a(t 0 ) = 1). For a spatially at single uid universe, the Friedmann equation then becomes ȧ 2 = 8πG 3 ρ 0a (1+3w). (2.9) 6

7 Figure 3: Content of the universe today. [source: map.gsfc.nasa.gov/universe/uni_matter.html] Given an equation of state parameter w we can solve this dierential equation for the scale factor a(t). Finally, we introduce the Raychaudhuri equation, which can be derived from the Friedmann equation and the continuity equation. It gives the acceleration of the scale factor in terms of energy density and pressure, 2.2 Content of the universe ä a = 4πG (ρ + 3P ). (2.10) 3 Now we discuss the dierent components of our universe. The universe contains: Matter. This is to a small part (Ω b,0 0.05) baryonic matter, which is mostly hydrogen. Everything that we see, stars, galaxies and clusters of galaxies, but also interstellar dust is contained in this number. However, various observations suggest that there is another type of matter that we cannot see directly, but its gravitational eects [6]. We call this type of matter dar matter. Matter density ρ m, dened as mass per volume, scales as ρ m (t) a 3, because volume scales as a 3. Thus, the equation of state parameter is w m = 0. In other words, matter does not create any pressure. Today, baryonic and dar matter together mae up a relative energy density of Ω m,0 0.28, but the biggest part of it comes from dar matter, Ω dm, Radiation. Radiation density scales as ρ γ a 4, because the number density of photons scales as n γ a 3 and the energy of each photon scales as E γ 1 λ a 1. Hence, the equation of state parameter is w γ = 1 3. Measurements show that today Ω γ,0 10 4, which comes mainly from CMB (cosmic microwave bacground) photons. Curvature. It is not nown for sure if the universe is spatially at. If curvature exists, then it acts eectively as an energy density, as can be seen from the Friedmann equation, with ρ a 2, so that w = 1 3. However, measurements show that the eect of curvature cannot be large today, Ω,

8 ~a 4 m ~a 3 =const radiation domination matter domination dar energy domination today a Figure 4: Overview over the dierent epochs of our universe (in a logarithmic scaling). Because radiation, matter and dar energy density scale dierently with time, there exist eras where one of them is dominating. Λ. Dar energy (also called vacuum energy or cosmological constant Λ) is a component of the universe whose energy density does not change with time. Consequently, the equation of state parameter is w Λ = 1, whence it creates negative pressure. Measurements of Type IA supernovae show that the universe today is expanding with an accelerating scale factor. These measurements can be explained by the existence of a cosmological constant with a relative energy density of Ω Λ, Thus, the biggest contribution to the energy density of the universe today comes from dar energy [7]. Now that we now how the dierent components of the universe scale, we can give an approximation when which component was dominant. Current cosmological models say that after the big bang, there was a short time when the universe was dar energy dominated, so that it expanded exponentially. This period is called ination. It broe down when the slow-roll conditions were violated, to be explained below. After this, the universe got radiation-dominated (a t 1/2 ) until the time of radiation-matter equality at a eq = Ωγ,0 Ω m, Then it became matter-dominated (a t 2/3 ) until the time of ( ) 1/3 matter-λ equality at Ωm,0 a mλ = Ω Λ, Thus, today (a0 = 1) we live at a time just after the cosmological constant overtoo matter and started to accelerate the expansion of the universe, see gure 4 for an illustration. 2.3 Ination Ination is a phase of accelerated expansion of the universe which is assumed to have happened for a very short time after the Big Bang. The precise denition of ination is an epoch with accelerating scale factor, ä > 0. (2.11) 8

9 In order to have ination, one typically introduces a scalar eld with negative pressure, that drives the ination. We call this scalar eld the inaton ϕ. Let its Lagrangian be L = 1 2 ϕ,µϕ,µ V (ϕ) (2.12) with some potential V (ϕ). Then the inaton eld has the energy density ρ ϕ = 1 2 ϕ2 + V (ϕ) (2.13) and the pressure P ϕ = 1 2 ϕ2 V (ϕ). (2.14) Note that for slowly varying ϕ, we have P ϕ ρ ϕ, which gives w ϕ 1, whence the inaton eld has negative pressure. Now we substitute the above equations into the Friedmann and Raychaudhuri equations, which gives: where V,ϕ dv dϕ and M Pl is the reduced Planc mass, H 2 = ( ) 1 1 3MPl 2 2 ϕ2 + V (ϕ), (2.15) ϕ + 3Hϕ = V,ϕ, (2.16) M Pl 1 8πG. (2.17) One can show that these relations reduce to H 2 1 3M 2 Pl V (ϕ), (2.18) 3Hϕ V,ϕ, (2.19) if the slow-roll conditions, ɛ 1, (2.20) η 1, (2.21) are satised, where ɛ and η are the slow-roll parameters, dened as: ɛ M Pl 2 ( ) 2 V,ϕ, 2 V (2.22) η MPl 2 V,ϕϕ V. (2.23) 9

10 dominating component w a(t) H(t) H 1 = (ah) 1 Λ 1 a(t) e Λt H(t) = Λ H 1 = 1 Λ e Λt a 1 1 radiation 3 a(t) t 1/2 H(t) = 1 2t H 1 = 2t 1/2 a matter 0 a(t) t 2/3 H(t) = 2 3t H 1 = 3 2 t1/3 a 1/2 Table 1: Evolution of the scale factor, the Hubble parameter and the Hubble horizon during dar energy, radiation and matter domination. To see the connection between slow-roll conditions and ination, rewrite the condition for ination as ä a = Ḣ + H2 > 0, (2.24) or, equivalently, Substituting the slow-roll equations gives Ḣ < 1. (2.25) H2 Ḣ H 2 M Pl 2 2 ( V,ϕ V ) 2 = ɛ. (2.26) Hence, if the slow-roll conditions are satised (ɛ 1), then ination taes place (ä > 0). The duration of ination is controlled by the magnitude of η. The potential has to be at enough (which corresponds to a small η) so that ɛ stays small. As an example, consider the inaton potential The slow-roll parameters are V (ϕ) = 1 2 m2 ϕ 2. (2.27) ɛ = η = 2M 2 Pl ϕ 2. (2.28) Hence, ination occurs as long as ϕ 2 2MPl 2, and it breas down near the minimum of the potential, when ϕ 2 2MPl 2 [8]. 2.4 Evolution of scales Later we will discuss the evolution of density perturbations in Fourier space, that is, the perturbations on a given comoving scale 1. An important question is if the considered scale is larger or smaller than the Hubble horizon at the time. The (comoving) Hubble horizon is given by the inverse of the comoving Hubble parameter, H 1 = (ah) 1. Let us evaluate how this horizon develops in time during dierent periods of the universe. In order to do this, we use eq. (2.9) in order to nd the scale factor a(t). The results are listed in table 1. Note that during ination (Λ-domination) the comoving Hubble scale decreases, or in other words, the horizon shrins. During this period, all scales leave the horizon. Later, when ination ends, the scales start to re-enter the horizon during the radiation- and matterdominated era, small scales before large scales (see gure 5). We normalize the horizon so that today 10

11 comoving scale Inflation horizon comoving scale time Figure 5: A given comoving scale 1 leaves the horizon during ination and re-enters later, during radiation or matter domination. (not to scale!) Scale 1 [Mpc] [Mpc 1 ] a H z H 4286 (horizon today) (equality scale) Table 2: Dierent scales with their scale factor a H and redshift z H = 1 a H 1 of horizon entry. The scale factor at radiation-matter equality is given for a ΛCDM model and the Hubble horizon today is given for h = 0.7. (at a = 1) it is the radius of the whole visible universe, R H 3000h 1 Mpc 2. Thus, H 1 = R H a 1/2 for a a eq. (2.29) We then normalize the horizon before a eq to match the above relation: H 1 = R H a for a < a eq. (2.30) a 1/2 eq Table 2 shows some scales with their scale factor of horizon entry a H. 2 h 0.7 is the reduced Hubble constant, dened by H 0 = 100h m s 1 Mpc 1. 11

12 3 The perturbed universe: Newtonian treatment 3.1 Denitions In cosmology, there are two dierent types of coordinates used, proper coordinates r and comoving coordinates x. These are related to each other by the scale factor a, r = ax. (3.1) Consequently, we need to dene the two dierent Nabla operators, x and r r, which are related to each other by r = 1. (3.2) a In the same way, we can dene two dierent times, related to each other by the scale factor: dt = adτ. (3.3) We call t the proper time and τ the conformal time. In our notation, a dot denotes a partial derivative with respect to proper time t, a prime with respect to conformal time τ. parameter H ȧ a a and the conformal Hubble parameter H a. The absolute velocity u is dened as u dr dt = d(ax) = adτ We dene the Hubble da dτ a x + dx dτ = Hx + v = Hr + v = v H + v, (3.4) where v H Hr = Hx is the Hubble velocity or Hubble ow, while the comoving (peculiar) velocity v dx dτ measures the velocity relative to the Hubble ow. The convective derivative in the (t, r)-system is given by d dt = t + u r, (3.5) which follows simply from the chain rule. In the (τ, x)-system the convective derivative is given by d dτ = τ + v. (3.6) Because proper time and conformal time are related by dt = adτ, the convective derivative transforms as 3.2 Perturbed elds d dt = 1 d a dτ. (3.7) Now we are going to consider perturbations up to the second order. We start from the continuity equation, Poisson equation and Euler equation on an expanding, spatially at FRW bacground. We will consider a universe lled only with pressureless matter and no cosmological constant. This model 12

13 is called the Einstein-de Sitter model. The relevant elds are the matter density ρ, the gravitational potential φ and the absolute velocity u. Now we split the elds into a bacground part (denoted with a superscript (0)) and a perturbed part. The perturbed part itself can be further split into rst order perturbations, second order perturbations (superscripts (1), (2)) and so on: ρ = ρ (0) + δρ = ρ (0) + ρ (1) + ρ (2) +... = ρ (0) (1 + δ (1) + δ (2) +...), (3.8) φ = φ (0) + δφ = φ (0) + φ (1) + φ (2) +..., (3.9) u = v H + v = v H + v (1) + v (2) +..., (3.10) where we have dened the n-th order density contrast δ (n), δ (n) ρ(n). (3.11) ρ (0) Here, v H = Hr = Hx is considered as the bacground velocity, and the peculiar velocity v = dx dτ is assumed to be small, v v H. Furthermore, according to the Helmholtz theorem 3, v can be separated into a part with zero curl (longitudinal part v ) and a part with zero divergence (transverse part v ): v = v + v, with v = 0 and v = 0. (3.12) Dene a scalar potential v and a vector potential Ω such that: v v and v Ω. (3.13) 3.3 Perturbed Newtonian equations Continuity equation The continuity equation in physical coordinates and proper time is Using the Leibniz rule we obtain t ρ + r(ρu) = 0. (3.14) t ρ + u rρ + ρ r u = 0. (3.15) The rst two terms of this equation together form the convective derivative of ρ, d dt ρ + ρ r u = 0. (3.16) 3 The Helmholtz decomposition exists and is unique under the assumption that v 0 as x, see appendix A for a proof. 13

14 Now we go from the (t, r)-system to the (τ, x)-system, 1 d a dτ ρ + 1 ρ u = 0. (3.17) a Multiplying this equation by a and using the denitions of the conformal convective derivative d dτ and the absolute velocity u, we nd: ρ + 3Hρ + (ρv) = 0. (3.18) To the bacground order, this is ρ (0) + 3Hρ (0) = 0, (3.19) which is the well nown mass continuity equation for p = 0 and has the solution ρ (0) a 3. To the linear order we nd the relation ρ (1) + 3Hρ (1) + ρ (0) v (1) = 0. (3.20) Now we use with n = 1 to nd: ρ (n) = (ρ (0) δ (n) ) = ρ (0) δ (n) 3Hρ (0) δ (n) (3.21) δ (1) + v (1) = 0. (3.22) Note that only the scalar part of v survives, because we tae its divergence. Hence, we can also write δ (1) + v (1) = 0, (3.23) where 2 is the Laplace operator in comoving coordinates. To the second order we nd: ρ (2) + 3Hρ (2) + (ρ (1) v (1) ) + ρ (0) v (2) = 0. (3.24) Again, we mae use of eq. (3.21) with n = 2 to nd: δ (2) + (δ (1) v (1) ) + v (2) = 0. (3.25) Poisson equation The Poisson equation connects the gravitational potential with the matter density. In proper coordinates it is r φ = 4πGρ. In comoving coordinates it becomes φ = 4πGa 2 ρ. (3.26) 14

15 To the bacground order, we have φ (0) = 4πGa 2 ρ (0), (3.27) from which we nd the bacground value for φ, φ (0) = 2 3 πga2 ρ (0) x 2 + C(t), (3.28) where C(t) is an arbitrary function of time. To the linear order, we have φ (1) = 4πGa 2 ρ (0) δ (1), (3.29) and to the second order, we nd φ (2) = 4πGa 2 ρ (0) δ (2). (3.30) Euler equation The Euler equation tells us how the velocity eld changes in time given a gravitational potential. In proper coordinates it is Using the convective derivative we can express it in the shorter form t u + u ru = r φ. (3.31) du dt = rφ. (3.32) In the (τ, x)-system the equation becomes after multiplying with the scale factor du dτ = φ. (3.33) Using the denitions of the total velocity and the convective derivative we nd H x + v + Hv + v v = φ. (3.34) To the bacground order, we have, using the solution for φ (0), H = 4 3 πga2 ρ (0), (3.35) which is the Raychaudhuri equation for p = 0. The perturbed order is v + Hv + v v = δφ. (3.36) Note that v v = 1 2 v2 v ω, (3.37) 15

16 where we have dened the vorticity ω v, (3.38) so that eq. (3.36) reads v + Hv v2 v ω = δφ. (3.39) Taing the curl of this equation, we nd the vorticity equation: ω + Hω = (v ω). (3.40) This dierential equation for the vorticity ω has an interesting solution: If ω = 0 everywhere initially, then the vorticity will stay zero at all times. However, the situation changes if we include a pressure gradient in the perturbed part of the Euler equation, v + Hv v2 v ω = δφ 1 p. (3.41) ρ Then the vorticity equation becomes ω + Hω = (v ω) + 1 ρ p. (3.42) ρ2 The second term on the r.h.s. is called the baroclinic contribution. It is zero in the case of a vanishing entropy gradient, S = 0 [9]. Hence, if we consider a model with irrotational, isentropic initial conditions and adiabatic evolution, it follows that ω = 0 always. In the Einstein-de Sitter model, which we consider here, there is no entropy at all. In order to dene entropy, one needs at least two dierent types of uids. Hence, it is reasonable to assume ω = 0, (3.43) which is equivalent to v = 0. (3.44) Hence, in Newtonian theory, the velocity perturbation is a pure potential ow 4, v = v = v. Furthermore, it follows that the vector potential Ω is a solution of the Laplace equation, since 0 = ω = ( Ω) = ( Ω) Ω = Ω, where we set Ω = 0 without any loss of generality. To the linear order, the Euler equation reads v (1) + Hv (1) = φ (1), (3.45) 4 Indeed, a constant non-zero v would also give ω = 0, but this would be in contradiction to the isotropy of the universe. Also, the Helmholtz decomposition is not unique if we allow constant parts in the vector. 16

17 and to the second order, we nd v (2) + Hv (2) + v (1) v (1) = φ (2). (3.46) Summary The important results of this section are the perturbed parts of the continuity, Poisson and Euler equation. In comoving coordinates and conformal time these are: 0. order 1. order 2. order C ρ (0) + 3Hρ (0) = 0 δ (1) + v (1) = 0 δ (2) + (δ (1) v (1) ) + v (2) = 0 P φ (0) = 4πGa 2 ρ (0) φ (1) = 4πGa 2 ρ (0) δ (1) φ (2) = 4πGa 2 ρ (0) δ (2) E H = 4 3 πga2 ρ (0) v (1) + Hv (1) = φ (1) v (2) + Hv (2) + v (1) v (1) = φ (2) Table 3: Continuity equation (C), Poisson equation (P) and Euler equation (E) in bacground, linear and quadratic order perturbation theory. 3.4 Solutions First order solutions In linear order the equations decouple into scalar and vector parts. The decoupled equations are: scalar part vector part continuity equation δ (1) + v (1) = 0 - Poisson equation φ (1) = 4πGa 2 ρ (0) δ (1) - Euler equation v (1) + Hv (1) = φ (1) v (1) + Hv(1) = 0 Table 5: Decoupled rst order Newtonian equations. The vector contribution to v (1) can be neglected because it decays a 1 (this is a special case of the above statement that the vorticity vanishes at all orders in Newtonian theory), so we can set v (1) = v (1). We do a spatial Fourier transformation of the remaining equations. In momentum space these equations become: δ (1) + i v (1) = 0, (3.47) 2 φ (1) = 6 τ 2 δ(1), (3.48) v (1) + 2 τ v(1) = iφ (1), (3.49) where we have used the Friedmann equation to replace the expression 4πGa 2 ρ (0) : 4 τ 2 = H2 = 8πG 3 ρ(0) a 2 4πGρ (0) a 2 = 6 τ 2. 17

18 The solutions can be obtained as follows: First, solve the Poisson equation for δ (1) : The Euler equation can be written as δ (1) = 1 6 τ 2 2 φ (1). (3.50) 1 a (av(1) ) = iφ (1), (3.51) whence v (1) = i 1 a τ dτ aφ (1). (3.52) Now we insert the solutions for δ (1) and v (1) into the the time derivative of the continuity equation, δ (1) + i v (1) = 0 and nd the following equation for φ (1) : φ (1) + τ 6 φ(1) = 0. (3.53) This is a second order dierential equation for φ (1) and has two solutions. Whenever this is the case, we call the dominating solution the growing mode and the subdominant solution the decaying mode. Here, the decaying solution is φ (1) τ 5 and the growing solution is a constant, φ (1) In summary, the growing rst order solutions are: = φ (1) (). (3.54) φ (1) = φ (1) (), (3.55) δ (1) = 1 6 τ 2 2 φ (1) = 2 3 a2 RHφ 2 (1) (), (3.56) v (1) = i 3 τφ(1) = 2 3 ia1/2 R H φ (1) (). (3.57) Note that, in order to convert τ to a, we use the denition of the Hubble parameter: whence where c H 0 2 τ = H = ah = a 2 3t = a 2 t 0 3t 0 t = ah 0a 3/2 = H 0 a 1/2, τ = 2a 1/2 H 1 0 = 2a 1/2 c H 0 c 1 = 2a 1/2 R H c 1, = R H 4300Mpc is the Hubble radius. Here, we use c = 1-units, so that the connection between τ and a is simply τ = 2a 1/2 R H. (3.58) 18

19 The decaying solutions for δ (1) and v (1) can be obtained using the decaying solution for φ (1), φ (1) τ 5, (3.59) δ (1) τ 3, (3.60) v (1) τ 4. (3.61) Initial conditions Now we want to answer the question how big φ is, since all the other quantities depend on it (we drop the superscripts... (1) for the moment). For this we have to go deeper into the theory of the primordial power spectrum. Note that this is not part of Newtonian theory. Ination creates curvature perturbations. Denote the curvature perturbation eld by ζ(x, τ), and its Fourier transform by ζ. The precise denition will be given in subsection An important physical quantity is its power spectrum. It is dened by P ζ () 3 2π 2 ζ 2. (3.62) Measurements of WMAP give a nearly scale-invariant (Harrison-Zel'dovich) power spectrum for the primordial curvature perturbation [10], P ζ () = A 2, A (3.63) Bardeen showed in [11] that in the matter dominated era of the universe the Bardeen potential Φ (we will give the denition later) is related to ζ via Φ = 3 5 ζ, (3.64) As we will show later, the Bardeen potential Φ can be identied with the gravitational potential perturbation φ. Hence, φ has the following power spectrum: P φ () = 3 2π 2 φ 2 = π 2 ζ 2 = 9 25 P ζ() = 9 25 A2. (3.65) Thus, also the spectrum for φ is scale-invariant. However, for the spectrum of δ we nd a scaledependence: or equivalently, P δ () = 3 2π 2 δ 2 = 3 2π a2 4 R 4 H φ 2 = 4 9 a2 4 R 4 HP φ () = 4 25 a2 4 R 4 HA 2 4, (3.66) P δ () = 8 25 π2 R 4 HA 2 a 2. (3.67) This means that there is more power (and therefore more structure) on smaller scales. Measurements show that the -dependence of P δ () is correct on large scales, but at a scale of about eq 0.01 Mpc 1, 19

20 Figure 6: The matter power spectrum today, measured using various techniques [12]. The bend at = eq 0.01 Mpc 1 is a result of the transition between radiation domination and matter domination. which is the scale that enters the horizon at the time of radiation-matter equality, the power spectrum has a bend and decreases roughly as 3 for higher (see gure 6). This bend comes from the fact that the universe was not always matter-dominated, but there was a radiation-dominated era (which is not included so far in our model). In order to explain the bend, we have to nd out how density perturbations grow during radiation domination Transfer function As we discussed above, it is important that in the history of the universe there was a radiationdominated era, where the pressure was not zero. Therefore we consider now matter density perturbations during radiation domination. However, we are going to neglect perturbations in the radiation density, which turns out to be a good approximation. If we consider a universe that contains both matter and radiation, we have to modify the Euler equation, containing both a gravitational potential gradient and a pressure gradient. Then the rst order Newtonian equations are: δ (1) + v (1) = 0, (3.68) φ (1) = 4πGa 2 ρ m δ (1), (3.69) v (1) + Hv (1) = (φ (1) + p(1) ), (3.70) ρ 20

21 where ρ = ρ m + ρ γ is the total bacground energy density. Now we tae the time derivative of eq. (3.68) and subtract the divergence of eq. (3.70), using eq. (3.69) to replace φ (1). This gives the following relation: δ (1) + Hδ (1) = 4πG ρ m a 2 δ (1) + c 2 sδ (1), (3.71) where c 2 s = w = p(1) ρ is the square of the speed of sound. In Fourier space this equation reads: δ (1) + Hδ (1) = 4πG ρ m a 2 δ (1) 2 c 2 sδ (1). (3.72) This equation is nown as the growth equation or Jeans equation for δ (1) [13]. The l.h.s. represents the time change of δ (1), including a friction term Hδ(1), which is also called Hubble damping or Hubble friction (the expansion of the universe slows down the growth of density perturbations). On the r.h.s. there are two competitive sources: gravity, which supports the growth of density perturbations, and pressure, which prevents it. It is convenient to introduce the comoving Jeans wavenumber, and the corresponding comoving Jeans length or Jeans scale, ( 4πG ρm a 2 ) 1/2 J, (3.73) λ J 1 J = Then the Jeans equation can be written as Now we have two cases: c 2 s ( 4πG ρm a 2 ) 1/2. (3.74) c 2 s δ (1) + Hδ (1) = ( 2 J 2 )c 2 sδ (1). (3.75) 1. For scales much smaller than the Jeans length, J, we can approximate the Jeans equation by δ (1) + Hδ (1) = 2 c 2 sδ (1). (3.76) The solutions are oscillating with the frequency c s, and the oscillations are damped by the Hubble friction term, whence the amplitude of the oscillations decays with time. There is no structure growth on sub-jeans scales. 2. For scales much larger than the Jeans length, J, we can approximate the Jeans equation by δ (1) + Hδ (1) = 4πG ρ m a 2 δ (1). (3.77) During matter domination, this gives the growing solution δ (1) During radiation domination, we have a, which we already now. ρ m ρ ρ γ (3.78) 21

22 and hence 4πGa 2 ρ m = 3 2 H2 ρ m ρ γ 1, (3.79) so that we can neglect the source term on the right hand side. Then the Jeans equation becomes δ (1) + Hδ (1) = 0, (3.80) which has the general solution δ (1) = C 1 + C 2 ln a. (3.81) Hence, during radiation domination density perturbations grow at most logarithmically on super- Jeans scales. Now we estimate the Jeans scale. During matter domination we have λ J = 0 since c s = 0, so that 1 perturbations grow on all scales. During radiation domination the speed of sound is c s = 3 and therefore J = 3 4πG ρ m a 2. As we approach the time of radiation-matter equality, we have ρ m ρ γ ρ 2, whence J H. Thus, the Jeans scale grows to the size of the horizon at radiation-matter equality and reduces to zero when matter dominates. Consequently, it maes a dierence if a scale enters the horizon during radiation domination or during matter domination because perturbations on scales that enter during radiation domination experience a suppression in growth. Now we want to quantify this eect. First consider a density perturbation on a scale 1 > 1 eq, which enters the horizon during matter domination. At a later time a f during matter domination it has grown by a factor: δ (1) δ (1) (a f ) (a enter) = H2 enter Hf 2 = 2 Hf 2. (3.82) Now consider a perturbation on a scale 1 < 1 eq. This scale enters the horizon during radiation domination. But then, between the time it enters and the time matter taes over nothing happens to the perturbation. Thus, δ (1) δ (1) (1) (a f ) (a enter) = δ (a f ) δ (1) (a eq) = Heq 2 Hf 2 = 2 eq H 2 f = 2 eq 2 2 Hf 2. (3.83) Hence, the growth on this scale is suppressed by a factor T () 2 eq 2 (3.84) compared to a scale that enters during matter domination. In the last equation we introduced the transfer function T (), which gives the decit in growth that perturbations on scales smaller than 1 eq experience. It should have the following properties: T 2 () = 1 for eq, (3.85) T 2 () 4 eq 4 for eq. (3.86) 22

23 1 0.9 simple transfer function BBKS transfer function T() e [Mpc -1 ] ( ) 1, Figure 7: Two dierent transfer functions. The simple one, T 2 () = cuts away too much eq 4 power on small scales, because it neglects the growth of the CDM density contrast during radiation domination on these scales. A simple choice is T 2 () = eq. (3.87) If we want to mae the transfer function more accurate, we have to account for cold dar matter (CDM) perturbations. Since CDM does not interact with radiation, perturbations do not feel the radiation pressure and grow independently of the Jeans scale (see appendix D). Hence, our simplied transfer function suppresses too much power on small scales. The correction is roughly logarithmic, so that the transfer function should go as: T 2 () = 1 for eq, (3.88) T 2 () 4 eq 4 ln2 ( eq ) for eq. (3.89) The most prominent transfer function that includes this correction and gives power spectra in good agreement with measurements is the BBKS transfer function, introduced in 1986 by Bardeen, Bond, Kaiser and Szalay [14], T BBKS () = ln( q) [ q + (16.1q) 2 + (5.46q) 3 + (6.71q) 4 ] 1/4, (3.90) 2.34q 23

24 where q ΓhMpc 1 and the shape parameter Γ is given by Γ Ω 0 h exp( Ω b 2h Ω b Ω 0 ), (3.91) where Ω 0 is the relative energy density of the universe today and Ω b is the relative baryon density today. Henceforth, we will use the BBKS transfer function with the parameters Ω 0 = 1, Ω b = 0.05 and h = 0.7. We show a plot of the two dierent transfer functions in gure 7. The transfer function connects the real power spectrum and the power spectrum from the Einsteinde Sitter model. Hence, we mae the following transformation: P() T 2 ()P(), (3.92) The transfer function explains the bend in matter power spectrum (see gure 11). Furthermore, now we can answer the question what the initial value for φ (1) is. Including the transfer function, eq. (3.65) becomes P φ () = 3 2π 2 φ(1) 2 = 9 25 A2 T 2 (). (3.93) From this we nd the following initial value for φ (1) : φ (1) = 3 5 A(2π2 ) 1/2 3/2 T (). (3.94) Second order solutions The second order equations do not decouple a priori; we have to neglect the vector parts of the rst order quantities, which can be done since v (1) decays. The decoupled second order equations are: scalar part vector part continuity equation δ (2) + v (2) = (δ (1) v (1) ) - Poisson equation φ (2) 4πGa 2 ρ (0) δ (2) = 0 - Euler equation v (2) + Hv (2) + φ (2) = v (1) v (1) v (2) + Hv(2) = 0 Table 6: Decoupled second order equations. The vector part of the Euler equation gives v (2) a 1 (again, this is a special case of the general result that the vorticity vanishes at all orders in Newtonian theory, as we showed earlier), so we can neglect the vector contribution to v (2) and set v (2) = v (2). Now consider the scalar equations: The sources for the second order perturbations are products of rst order perturbations, which we have written on the r.h.s. of each equation. There are hence two solutions. The homogeneous solutions are obtained by ignoring the source terms, which gives the same time dependence as in rst order perturbation theory. However, the specic solutions grow faster due to the source terms, so that we can neglect the homogeneous solutions. Note that the term v (1) v (1) can also be written as v(1)2 2. Now we need to be careful when transforming to momentum space, because products in x-space become convolutions in -space. 24

25 In momentum space the expression v (1)2 becomes: F[v (1) v (1) ]() = 1 (v v)() (2π) 3 = 1 (2π) 3 d 3 v( ) v( ) = 1 (2π) τ 3 = 1 (2π) τ 3 0 d 3 φ( ) ( )φ( ) π d dϑ 2 sin ϑ ( cos ϑ 2 )φ( )φ( cos ϑ ). 0 The expression (δ (1) v (1) ) becomes: F[ δ (1) v (1) ]() 1 =i (2π) 3 (δ(1) v (1) )() 1 =i (2π) 3 d 3 δ (1) ( )v( ) 1 i =i (2π) 3 18 τ 3 d 3 2 φ( )φ( )( ) = 1 1 (2π) 3 18 τ 3 d 3 2 φ( )φ( ) ( ) = 1 (2π) τ 3 0 π d dϑ 4 sin ϑ ( 2 cos ϑ )φ( )φ( cos ϑ ), 0 0 sin ϑ cos ϕ where we set = 0 and = sin ϑ sin ϕ, so that = cos ϑ. 1 cos ϑ The second order evolution equations in momentum space are: v (2) δ (2) i v (2) = F[ (δ (1) v (1) )], (3.95) 2 φ (2) + 6 τ 2 δ(2) = 0, (3.96) + 2 τ v(2) + iφ (2) = i 1 2 F[v(1) v (1) ]. (3.97) Note that δ (1) v (1) τ 3 and v (1) v (1) τ 2 in leading order. There are other solutions as well, which can be obtained by combining decaying and growing modes of δ (1) and v (1). However, these solutions are all decaying and can be neglected. Hence, it is reasonable to mae the following ansatz by adjusting 25

26 the time dependence of the second order perturbations to the time dependence of the source terms: δ (2) = C τ 4, (3.98) v (2) = D i τ 3, (3.99) φ (2) = E τ 2. (3.100) With this ansatz, we nd the following system of linear equations for the coecients C, D, E : 4C D = F[ δ (1) v (1) ]()τ 3, (3.101) E 2 + 6C = 0, (3.102) 5D + E = 1 2 F[v(1) v (1) ]()τ 2. (3.103) The solutions are obtained by rst solving the convolution integrals and then solving the system of linear equations for the coecients. This is done numerically for three xed scales that we are interested in: the galaxy scale ( 1 = 1 Mpc), the cluster scale ( 1 = 10 Mpc), and the supercluster scale ( 1 = 100 Mpc). Figures 8, 9 and 10 show the plots of the rst and second order perturbations on these scales. Note that we plot the dimensionless quantities (1) δ P δ = 3/2 2π 2 δ(1) etc (a, =1 Mpc -1 ) 1e-04 1e-06 1e-08 Φ 1 1e-10 v 1 δ 1 Φ 2 v 2 δ 2 1e-12 1e a Figure 8: Evolution of rst and second order perturbations for the scale 1 = 1 Mpc. 26

27 (a, =0.1Mpc -1 ) 1e-04 1e-06 1e-08 Φ 1 1e-10 v 1 δ 1 Φ 2 v 2 δ 2 1e-12 1e a Figure 9: Evolution of rst and second order perturbations for the scale 1 = 10 Mpc e-04 (a, =0.01Mpc -1 ) 1e-06 1e-08 1e-10 1e-12 Φ 1 v 1 1e-14 δ 1 Φ 2 v 2 δ 2 1e-16 1e a Figure 10: Evolution of rst and second order perturbations for the scale 1 = 100 Mpc. 27

28 scale 1 Mpc 10 Mpc 100 Mpc a NL a NL Table 7: a NL1 and a NL2 for dierent scales Discussion For early times and large scales linear perturbation theory is valid, since all second order perturbations are substantially smaller than all rst order perturbations. Now a diculty arises to dene the time when a scale goes non-linear. Strictly speaing, linear perturbation theory becomes invalid when the largest second order contribution becomes as important as the smallest rst order contribution. Here this is the crossing of δ (2) and φ (1). We call the scale factor at this crossing a NL1, δ (2) (a NL1) φ (1) (a NL1). (3.104) However, the perturbations series δ = δ (1) + δ (2) still maes sense, since δ (2) is much smaller than at this time. According to the spherical collapse model, collapse happens much later, roughly δ (1) when the density contrast reaches unity [8, 13], which is a commonly accepted indicator for the begin of nonlinearity. A density contrast of 1 means that the density perturbations become as large as the bacground density. In other words, there can be regions which become twice as dense as the bacground. We call the scale factor when this happens a NL2, (1) δ (a NL2) 1. (3.105) The transition between the the linear system and the nonlinear system happens certainly between a NL1 and a NL2 (see table 7). Shortly after a NL2, the system has viralized into a gravitational bound object, where the angular momentum prevents further collapse [8]. This has already happened for galaxies and clusters of galaxies, while perturbations on scales 10 Mpc still behave linear today. The fact that small scales collapse before large scales leads to the bottom-up picture of structure formation: small scale structures form before large scale structures. Figure 11 shows the evaluated matter power spectrum P δ () = δ (1) 2 today. It increases for < eq and decreases 3 ln 2 () for > eq due to the implemented transfer function and shows good correspondence with measurements, as shown in gure 6. However, nonlinear eects become important as P δ () reaches unity, which happens at about = Mpc 1 today. Figure 12 shows the power spectrum for the absolute value of the peculiar velocity P v () = v (1) 2 today. It decreases 1 for < eq and 5 ln 2 () for > eq. 28

29 δ 2 (a=1, ) 1e-04 1e-06 1e-08 1e-10 1e-12 1e [Mpc -1 ] P δ [Mpc 3 ] e [Mpc -1 ] Figure 11: The matter power spectrum today. The top gure shows P δ () and the bottom gure shows P δ (). Linear theory fails when P δ () 1 (vertical line). 29

30 1e-04 1e-05 1e-06 v 2 (a=1, ) 1e-07 1e-08 1e-09 1e-10 1e-11 1e [Mpc -1 ] P v [Mpc 3 ] e-04 1e-05 1e [Mpc -1 ] Figure 12: The power spectrum of v today. The top gure shows P v () and the bottom gure shows P v (). 30

31 3.4.6 Solutions during Λ-domination As we approach a = 1, the cosmological constant becomes more and more important, as matter scales a 3. During Λ-domination the Friedmann and Raychaudhuri equations become H 2 = 8πG 3 a2 ρ Λ, (3.106) H = 8πG 3 a2 ρ Λ. (3.107) However, if we neglect dar energy perturbations, there are no variations in the rst order equations. Hence, the form of the Jeans equation remains unchanged, Note that δ (1) + Hδ (1) = 4πG ρ m a 2 δ (1). (3.108) 4πG ρ m a 2 = 3 2 H2 ρ m ρ Λ 1, (3.109) so that the source term of the Jeans equation becomes negligible. Then we have two solutions, the growing solution δ (1) = const. (3.110) and the decaying solution δ (1) a 2. (3.111) Hence, as we approach Λ-domination, the density perturbations freeze out. A more general solution in the intermediate regime, when dar energy and matter density are equally important, will be discussed later, in section Newtonian cosmological simulations Cosmological N-body simulations use Newton's equation of motion to simulate the behaviour of a xed number of particles under the inuence of their reciprocal gravitational interaction. These simulations are usually run in a box with periodic boundary conditions. In the (t, r)-system the equation of motion can be written as d 2 r i (t) dt 2 = r φ sim, (3.112) where r i (t) is the trajectory of the i-th particle. Transforming to the (τ,x)-system, we nd The l.h.s. is d adτ d adτ (ax i(τ)) = d d adτ adτ (ax i(τ)) = 1 a φ sim. (3.113) d ( Hx i (τ) + dx ) i(τ) = 1 ( H x i (τ) + H dx i(τ) + dx2 i (τ) ) adτ dτ a dτ dτ 2, 31

32 and on the r.h.s. the potential can be split into a bacground part and a perturbed part, 1 a φ sim = 1 a ( φ sim + δφ sim ). Subtracting the bacground (this is the Raychaudhuri equation), we nd d 2 x i (τ) dτ 2 + H dx i(τ) dτ = δφ sim. (3.114) From this equation it can be seen that all freely falling observers (for which δφ sim = 0) will become resting observers after suciently long time, since dxi(τ) dτ 1 a. The gravitational potential perturbation is obtained using the Poisson equation, δφ sim = 4πGa 2 δρ sim, (3.115) which is solved in Fourier space, where the matter density perturbation is calculated by counting particles in cells, δρ sim (x, τ) = a 3 i m i δ D (x x i (τ)). (3.116) 4 The perturbed universe: relativistic treatment 4.1 The perturbed metric In relativistic cosmological perturbation theory we consider small perturbations on the homogenous and isotropic FRW metric tensor and on the energy-momentum tensor and calculate how these perturbations develop in time, using the energy-momentum conservation equation and the Einstein eld equations. For a detailed reference, see [15]. Our conventions are the following: Gree indices will range from 0 to 3 and Latin indices from 1 to 3. µ as well as an index ; µ denote a covariant derivative with respect to the perturbed metric. We are going to drop the superscripts... (1) here, since we only consider linear perturbations. Bacground quantities are denoted with a bar. The FRW metric in a at universe, using comoving coordinates and conformal time, is ( ) ḡ µν a (4.1) 0 δ ij Now consider a small perturbation of the bacground metric: g µν = ḡ µν + δg µν. We dene the metric perturbation ( δg µν a 2 2φ w i w i 2eδ ij + 2h ij ). (4.2) w i and h ij can be further decomposed into scalar, vector and tensor parts, w i = w ;i + wi (4.3) 32

33 type elds constraints degrees of freedom scalar perturbations φ,ψ,w,h - 4 vector perturbations wi, h i i wi = i h i = 0 4 tensor perturbations h T ij i h T ij = (ht ) i i = 0, ht ij = ht ji 2 Table 8: Degrees of freedom in linear perturbation theory. and where D ij is the symmetric traceless double-gradient operator, h ij = D ij h + h (i;j) + h T ij, (4.4) D ij i j 1 3 δ ij. (4.5) Note that w i and h i are pure vector parts with zero divergence ( i w i = i h i = 0) and h T ij is transverse ( i h T ij = 0), traceless ((ht ) i i = 0) and symmetric (ht ij = ht ji ). Thus, in rst order the metric perturbations decouple: δg µν = δgµν S + δgµν V + δgµν T (4.6) ( ) ( ) ( ) = a 2 2φ w ;i + a 2 0 wi + a 2 0 0, (4.7) w ;i 2ψδ ij + 2h ;ij wi 2h (i;j) 0 2h T ij where we have introduced a new variable 5, ψ e + 1 h. (4.8) 3 Now that we have decomposed the metric perturbation, it is instructive to count the degrees of freedom, see table 8. The result is that we have altogether 10 degrees of freedom in the metric perturbation. However, four of them correspond to the freedom of coordinate choice, so that we can put further restrictions on the metric, as will be discussed later. The decomposition into scalar, vector and tensor parts is important for the physical interpretation. The scalar perturbations are the most important, because they couple to density perturbations and give rise to structure formation. The vector perturbations are not very interesting in linear order, because they couple to rotations only, which are decaying in an expanding universe due to angular momentum conservation. Tensor perturbations give rise to gravitational waves. 4.2 Gauge transformations There is no unique mapping between points in the bacground spacetime and points in the perturbed spacetime. Indeed, there are innitely many mappings, all close to each other. Consequently, for 5 Some authors introduce this quantity as the curvature perturbation. However, we will later introduce the curvature perturbation according to the denition of Bardeen [11]. 33

34 .. ~. P ^ P P x α x α α x Figure 13: Illustration of two dierent gauge choices. a given coordinate system of the bacground, there exist innitely many coordinate systems of the perturbed spacetime. A coordinate transformation between these coordinate systems is called a gauge transformation. Consider a point P in the bacground spacetime, whose coordinates are {x α }. Now consider two dierent coordinate systems in the perturbed spacetime, and denote them by { x α } and {ˆx α }. In the x α -coordinates, the bacground point P corresponds to a point P ( x). In the ˆx α - coordinates, this point corresponds to another point ˆP (ˆx), see gure 13 for an illustration. We have by construction x α ( P ) = x α ( P ) = ˆx α ( ˆP ), (4.9) because all coordinates refer to the same point in the manifold. Now one can as the question: what are the ˆx α -coordinates of the point P? These clearly dier from the x α -coordinates and we denote this dierence by ξ α, which is rst-order small due to the fact that all coordinate systems are close to each other: x α ( P ) = ˆx α ( P ) + ξ α, (4.10) x α ( ˆP ) = ˆx α ( ˆP ) + ξ α. (4.11) Note that it does not matter, in which coordinate system we give ξ α, because the dierence is second order. The above equations represent a coordinate transformation, which in cosmological perturbation theory is called gauge transformation. They are equivalent to: ˆx α ( P ) = ˆx α ( ˆP ) ξ α, (4.12) x α ( P ) = x α ( ˆP ) ξ α. (4.13) 34

35 4.2.1 Gauge transformations of scalars, vectors and tensors Using these transformations, we can determine how various geometric objects (scalars, vectors and tensors) transform under a gauge transformation. First, let us determine the transformation of a scalar. Using Taylor expansion, we nd s( P ) = s( ˆP ) + s ˆx α (ˆxα ( P ) ˆx α ( ˆP (1) )) = s( }{{} ˆP ) s x ξ α α ξα (2) = s( ˆP ) s ξ 0. (4.14) Here we have used that (1) the dierence between s ˆx and s α x is rst order small and can be α neglected when multiplied by ξ α and (2) the bacground value s only depends on conformal time τ, because the bacground spacetime is isotropic. The scalar perturbation in a given gauge is dened as δs s( P ) s( P ) (or δs ˆ s( ˆP ) s( P )). The transformation law is δs = s( P ) s( P ) = s( ˆP ) s( P ) s ξ 0 = ˆ δs s ξ 0. (4.15) Hence, any scalar that is constant in the bacground ( s = 0) is gauge-invariant. For a vector wˆµ ( ˆP ) we have (ˆµ refers to the coordinates ˆx µ ): and therefore wˆµ ( P ) = wˆµ ( ˆP ) w µ x α ξα (4.16) w µ ( P ) = ˆxσ x µ wˆσ( P ) = ( xσ ξ σ ) x µ (wˆσ ( ˆP ) w σ x α ξα ) For a vector perturbation we nd the transformation = (δµ σ ξ,µ)(wˆσ σ ( ˆP ) w σ x α ξα ) = wˆµ ( ˆP ) w µ,α ξ α w σ ξ,µ. σ (4.17) δw µ = w µ ( P ) w µ ( P ) = ˆ δw µ w µ,α ξ α w σ ξ σ,µ. (4.18) Obviously, any vector that vanishes in the bacground is gauge-invariant. For a (0,2)-tensor Bˆµˆν ( ˆP ) we have: Bˆµˆν ( P ) = Bˆµˆν ( ˆP ) Bˆµˆν,α ˆ ξ α = Bˆµˆν ( ˆP ) B µν,α ξ α (4.19) 35