Evolution of Cosmic Structure Max Camenzind

Transcription

1 FROM BIG BANG TO BLACK HOLES Evolution of Cosmic Structure Max Camenzind Landessternwarte Königstuhl Heidelberg February 13, 2004

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3 Contents 7 Evolution of Cosmic Structure Evolution of Relativistic Perturbations Homogeneous Universe in Conformal Time Perturbations in the metric and in the energy-momentum tensor Perturbed Einstein s Equations Gauge Transformations Einstein s Equations in Different Gauges Fluid equations in the Newtonian Gauge Fluid equations Gravitational Equations Solutions inside horizon: matter domination Solutions inside horizon: radiation domination Solutions outside horizon Transfer functions On Baryons Role of Scalar Fields Nonlinear Evolution of Cosmic Structure Cosmic Matter Collisionless Components Baryons and Gasdynamics The Spectrum of Primordial Fluctuations Cosmological Simulations Particle Models Progress of Cosmological Simulations The Fractal Structure of Dark Matter Distribution Statistical Measures for Cosmic Density Fields s: Baryons in Simulations Dark Baryons The Lymanα Forest Galaxy Clusters The Cosmic Microwave Background Probes Linear Perturbations Anisotropy Mechanisms The Resulting CMB Power Spectrum More on Acoustic Oscillations Concluding Remarks Bibliography 277

4 4 Contents 7.9 Exercises A Perturbation Calculations 281 A.1 Perturbations of the Einstein Tensor in Newtonian Gauge A Slice A.1.2 Perturbation of Extrinsic Curvature A.1.3 Perturbation of the Spatial Ricci Tensor A.1.4 Perturbed Einstein Tensor A.1.5 Curvature Perturbations

5 7 Evolution of Cosmic Structure The goal of studying cosmological perturbations is to understand the evolution of the structure in the universe, from small ripples generated sometime after the initial quantum state all the way to galaxies, clusters and large scale structure we see today. Relativistic perturbation theory describes this evolution in a general relativistic context. In cosmology such an approach is required, because we want to describe perturbations not just on small scales, where Newtonian laws suffice, but also on scales comparable or larger to the Hubble radius, which can be used as a typical scale of observable universe. This is nowadays a very active field of international research and I will concentrate on a presentation of only the most important aspects. 7.1 Evolution of Relativistic Perturbations Before we introduce the perturbations we must first recapitulate the unperturbed (homogeneous) evolution of the universe Homogeneous Universe in Conformal Time The fundamental assumption of cosmology is that the universe is homogeneous and isotropic on average. To be more precise, the paradigm is that we live in a weakly perturbed Robertson- Walker universe, in which the metric perturbations are small, so the averaging procedure is well defined and the backreaction of the metric fluctuations on the homogeneous equations is negligible. Note that we only require metric perturbations to be small and there is no requirement on the density perturbations, which we know can be large on small scales. One can write the line element in a homogeneous and isotropic universe using conformal time τ and comoving coordinates x i as ds 2 = γ µν dx µ dx ν = a 2 (τ) { dτ 2 + γ ij dx i dx j}. (7.1) This is the Robertson-Walker metric. We will often use greek indices to denote 4-tensors and latin to denote spatial 3-tensors. Here a(τ) is the scale factor expressed in terms of conformal time τ, which is related to the proper time t via dt = a dτ. Similarly, proper coordinates r i are related to the comoving coordinates x i via r i = ax i. We adopt units such that c = 1. The space part of the background metric can be written as γ ij dx i dx j = dχ 2 + r 2 (dθ 2 + sin 2 θdφ 2 ), K 1/2 sin K 1/2 χ, K > 0 r = sin K χ χ, K = 0 ( K) 1/2 sinh( K) 1/2 χ, K < 0 (7.2)

6 212 7 Evolution of Cosmic Structure where K is the curvature term which can be expressed using the density parameter in all components Ω and the Hubble parameter H as K = (Ω 1)H 2. The present day values we usually denote with an additional subscript 0, eg H 0 and Ω 0. The density parameter Ω can have contributions from nonrelativistic matter (Ω m ), such as baryons, cold dark matter (CDM) or massive neutrinos (mixed dark matter MDM, warm dark matter WDM etc.), relativistic matter (Ω r ), such as photons and massless neutrinos and from cosmological constant (Ω Λ ) or some more general form of dark energy (or quintessence, Ω φ ). The advantage of using the conformal time τ is that the metric becomes conformally Euclidean (K = 0), 3-sphere (K > 0) or 3-hyperboloid (K < 0) and leads to a simple geometrical description of light propagation and other processes. The usual starting point in general relativity is the Einstein s equation G ν µ = 8πGT ν µ, (7.3) where G ν µ is the Einstein tensor and T µ ν is the stress-energy tensor. Einstein s tensor is related to the spacetime Ricci tensor R µν by G µν R µν R 2 g µν, R R µ µ, R µν R κ µκν. (7.4) An alternative starting point is the Einstein Hilbert action S = dx 4 ( ) 1 g 16πG R + L, (7.5) where L is the matter Lagrangian density. The action principle δs = 0 leads to equations of motion, which applied to equation (7.5) gives rise to Einstein s equation (7.3) with T µν = 2 L g µν + g µν L. (7.6) This form will be useful when we introduce scalar field degrees of freedom. The Einstein field equations (7.3) show that the stress-energy tensor provides the source for the metric variables. The stress-energy tensor takes the well known form T µν = (ρ + p)u µ u ν + pg µν + pπ µν, (7.7) where ρ and p are the energy density and pressure, u µ = dx µ /dλ (where dλ 2 ds 2 ) is the fluid 4-velocity and pπ µν is the shear stress absent for a perfect fluid. In locally flat coordinates in the fluid frame, T 00 = ρ, T 0i = 0, and T ij = pδ ij for a perfect fluid. Einstein s equations applied to the background metric gives the evolution of the expansion factor a(τ), (ȧ ) 2 H 2 = 8π a 3 G ρa2 K, (7.8) Ḣ = 1 d 2 a a dt 2 = 4π 3 Ga2 ( ρ + 3 p). (7.9) Overdots denote derivatives with respect to the conformal time τ. For convenience we introduced comoving Hubble parameter H = ȧ/a, which will appear often in the equations

7 7.1 Evolution of Relativistic Perturbations 213 below. Its value today is H 0. These are the Friedmann equations applied to the Robertson- Walker metric. The density today ρ 0 is related to the density parameter Ω 0 via 8πG ρ 0 /3 = H 2 0 Ω 0. Remember that for w = p/ ρ < 1/3 the universe is accelerating. The mean density of the universe ρ (and similarly the mean pressure p) can be written as a sum of matter, radiation, cosmological constant or any other dark energy contributions, ρ = ρ m a 3 + ρ r a 4 + ρ Λ + ρ φ a 3(1+w), (7.10) where w is the equation of state of the dark energy. The energy-momentum tensor is required to obey local conservation law T µν ;ν = 0, which gives ρ + 3H(ρ + p) = 0. (7.11) One can also derive the evolution equation for a homogeneous scalar field φ evolving in a potential V (φ). Its Lagrangian is L = [ ] 1 g 2 gµν µ ν φ + V (φ), (7.12) where g µν is the metric. Stress energy tensor is given by which gives T µν = µ φ ν φ + g µν L, (7.13) ρ φ = φ 2 2a 2 + V (φ) p φ = φ 2 V (φ). (7.14) 2a2 Equation of state w = p/ρ is in general a function of time. Continuity equation (7.11) gives φ + 2H φ + a 2 V = 0. (7.15) The scalar field source has to be added to the Friedmann equations above and modifies the expansion of the universe. It obeys the same energy-momentum conservation as the other fluids and can be easily integrated to find energy density as a function of time (or expansion factor as in equation 7.10). In the limit where kinetic term is negligible compared to the potential scalar field reduces to the cosmological constant with w = 1. This case is relevant both for inflation and for the possible late time contribution from the dark energy (Quintessence models) Perturbations in the metric and in the energy-momentum tensor Our universe is not homogeneous: we see inhomogeneities caused by gravity present on all scales, from planets to clusters, superclusters and beyond. We want to describe the deviations from the isotropy and homogeneity of the universe using general relativity. Small perturbations h µν around the Robertson-Walker metric are g µν = a 2 (γ µν + h µν ). (7.16)

8 214 7 Evolution of Cosmic Structure In the most general form one can write the perturbed line element using conformal time τ and comoving coordinates x i as ds 2 = a 2 (τ) { (1 + 2A) dτ 2 2B i d τ dx i + [(1 + 2H L )γ ij + 2h ij ] dx i dx j}. (7.17) The perturbations have been decomposed into time time component 2A (scalar potential), time space component 2B i (shift vector), trace of the space space component 2H L (spatial curvature perturbation) and traceless space space component 2h ij (traceless spatial metric perturbation). In total we have introduced 10 metric perturbations. Vector field B i can be further decomposed into a scalar component, which arises from a gradient of a scalar field, and a pure vector component, which is the remainder of what is left. Similarly we can decompose tensor h ij into a scalar, vector and tensor components. As we show below these perturbations can be decomposed into 4 scalar (m = 0, compressional), 4 vector (m = ±1, vortical) and 2 tensor (m = ±2, gravitational wave) eigenmode components, which differ in their transformation properties under spatial rotations. The advantage of this decomposition is that the linearized equations decouple into separate scalar, vector and tensor components, with no cross coupling between them. Eigenmodes of the Laplacian In linear theory, each eigenmode of the Laplacian for the perturbation evolves independently, and so it is useful to decompose the perturbations via the eigentensor Q (m), where 2 Q (m) γ ij Q (m) ij = k 2 Q (m), (7.18) with representing covariant differentiation with respect to the 3 metric γ ij. Note that to the lowest order one can raise and lower indices on 3-tensors using the three metric γ ij. The eigentensor Q (m) has m indices (suppressed in the above). To obtain a pure vector component, it has to be obtained from a scalar field. In real space this is a gradient of a scalar field. This mean that vector modes satisfy the auxiliary condition Q (±1) i i = 0, (7.19) which represents the divergence free condition for vorticity waves. Similarly, to obtain a pure tensor mode we must subtract out components that can be formed from a scalar and vector field. The auxiliary conditions are γ ij Q (±2) ij = 0 = Q (±2) i, (7.20) ij

9 7.1 Evolution of Relativistic Perturbations 215 which represent the transverse traceles and divergence free conditions respectively, as appropriate for gravity waves. We will often focus on perturbations in flat space, both because they lead to simplified expressions and because they seem to be observationally favored. In this case the eigenmodes are particularly simple. If we assume the direction of the wavevector k is ê 3 then Q (±m) i 1...i m (ê 1 ± iê 2 ) i1... (ê 1 ± iê 2 ) im exp(i k x), (K = 0, m 0), (7.21) where the presence of ê i, which forms a local orthonormal basis with ê 3 = ˆk, ensures the divergenceless and transverse-traceless conditions. One can see now the transformation properties of eigenmodes under rotation in the plane perpendicular to ê 3 is such that Q (±m) = Q (±m) e imψ, (7.22) where ψ is the rotation angle. The eigenmodes are essentially plane waves Q (0) = exp(i k x) (7.23) Q (±1) i = i (ê 1 ± iê 2 ) i exp(i k x) (7.24) 2 Q (±2) 3 ij = 8 (ê 1 ± iê 2 ) i (ê 1 ± iê 2 ) j exp(i k x). (7.25) The vectors e 1 and e 2 span the plane tranverse to k. It is also useful to construct (auxiliary) vector and tensor objects out of the fundamental scalar and vector modes through covariant differentiation Q (0) i = k 1 Q (0) i, (7.26) Q (0) ij = k 2 Q (0) ij γ ijq (0), (7.27) Q (±1) ij = (2k) 1 (Q (±1) i j + Q (±1) j i ). (7.28) For K = 0 this becomes Q (0) i = i(k i /k) Q (0), (7.29) Q (0) ij = (k i k j /k 2 ) Q (0) γ ijq (0), (7.30) Q (±1) ij = i(k j Q (±1) i + k i Q (±1) j )/(2k). (7.31) The eigenmodes form a complete set, so that any perturbation can be expanded in terms of these. The metric perturbations can be broken up into the normal modes of scalar (m = 0), vector (m = ±1) and tensor (m = ±2) type, A = A (0) Q (0), (7.32) H L = H (0) L Q(0), (7.33) 1 B i = B (m) Q (m) i, (7.34) h ij = m= 1 2 m= 2 H (m) T Q (m) ij. (7.35)

10 216 7 Evolution of Cosmic Structure Note that this gives 4 scalar, 4 vector and 2 tensor components. The stress energy tensor can likewise be broken up into scalar, vector, and tensor contributions. The fluctuations can be decomposed into the normal modes as δt 0 0 = δρ (0) Q (0), δt 0 i = 1 m= 1 [(ρ + p)(v(m) B (m) )] Q (m) i, δt i 0 = 1 m= 1 (ρ + p)v(m) Q (m)i, δt i j = δp (0) δ i j Q(0) + 2 m= 2 pπ(m) Q (m)i j. (7.36) Note that the two mixed time space components are not equal. We introduced above the density perturbation δρ (0), pressure perturbation δp (0), velocity perturbation v (m), which has a scalar (m = 0) and vector (±1) component and anistropic stress perturbation pπ (m), which can be scalar (m = 0), vector (m = ±1) or tensor (m = ±2) type. These are in general a sum from all of the species present, δt µν = I δt µν I, (7.37) where the index I stands for baryons, CDM, photons, neutrinos (massive and massless), dark energy etc. A minimally coupled scalar field ϕ also has perturbations, ϕ = φ + δφ. They can be related to the fluid quantities by expanding equation (7.13) to the next order δρ (0) φ = a 2 ( φδ φ A (0) φ2 ) + V δφ, δp (0) φ = a 2 ( φδ φ A (0) φ2 ) V δφ, (ρ φ + p φ )(v (0) φ B(0) ) = a 2 k φδφ, p φ Π (0) φ = 0, (7.38) where V denotes derivative with respect to φ and only the lowest order terms have been kept. This shows that there are no vector or tensor modes associated with the scalar field, as expected in the linear order. Applying energy-momentum conservation equation (7.11) one finds δ φ (0) +2Hδ φ (0) +(k 2 +a 2 V )δφ (0) = ( A (0) 3 H L (0) kb (0) ) φ 2a 2 V A (0). (7.39) Perturbed Einstein s Equations Einstein s equations are also decomposed into various modes (see Appendix and Exercise).

11 7.1 Evolution of Relativistic Perturbations 217 Scalar Equations Einstein s equations for the scalar modes become (suppressing 0 superscripts) [ ( )] (k 2 3K) H L H B T + H k ḢT k 2 = 4πGa 2[ δρ + 3H(ρ + P ) v B ] k [ k 2 (A + H L H T ) + ( τ + 2H)(kB ḢT ) = 8πGa 2 P Π (7.40) 2ä a 2H2 + H τ k2 3 HA ḢL 1 K 3ḢT k 2 (kb ḢT ) = 4πGa 2 (ρ + P ) v B ] k ( A ( τ + H) Ḣ L + kb ) 3 The equations of motion provide two equations = 4πGa 2 ( δp δρ ). ( τ + 3H)δρ + 3HδP = (ρ + P )(kv + 3ḢL) (7.41) ( τ + 4H)(ρ + P ) v B k = δp 2 3 (1 3K/k2 )P Π + (ρ + P )A. (7.42) These are in total 8 variables (4 metric and 4 matter) and 6 equations, leaving two gauge freedoms. Vector Equations The vector modes are determined by two equations (1 2K/k 2 )(kb (±1) Ḣ(±1) T ) [ ] = 16πGa 2 (ρ + P ) v(±1) B (±1) k (7.43) τ + 2H (kb (±1) Ḣ(±1) T ) = 8πGa 2 P Π (±1), (7.44) and one equation from the Euler equations [ τ + 4H ](ρ + P ) v(±1) B (±1) = 1 k 2 (1 2K/k2 )P Π (±1). (7.45) Tensor Equations The Einstein equation for the tensor mode is [ ] τ 2 + 2H τ + (k 2 + 2K) H (±2) T = 8πGa 2 P Π (±2). (7.46) In the absence of anisotropic stresses and K = 0, the tensor equation becomes a source free gravitational wave equation which has solutions in terms of Hankel functions where H (±2) T (kτ) = C 1 H 1 (kτ) + C 2 H 2 (kτ), (7.47) H 1 (x) x m j m (x), H 2 (x) x m n m (x). (7.48) Thereby m = (1 3w)/(1 + 3w). If w > 1/3, then the gravitational wave amplitude is constant above horizon, x 1, and then oscillates and damps.

12 218 7 Evolution of Cosmic Structure Gauge Transformations So far we have described the perturbations in a given coordinate system, but we did not say much about the coordinate system itself. In the absence of perturbations there is a preferred coordinate system, which corresponds to comoving frame, in which the observers see the momentum density to be zero at that point and the comoving observers are free falling. The spatial slices (defined as slices of constant time) are orthogonal to the time threading (defined as a constant coordinate x) and for K = 0 they are spatially flat. For K < 0 they correspond to a 3-hyperboloid and for K > 0 to a 3-sphere. This defines the background SpaceTime. The choice of coordinates becomes nontrivial once we discuss the perturbations and there is no unique choice. One can, for example, choose the coordinates in which the observers are free falling (corresponding to the synchronous gauge), but this is no longer the same choice as if the observers are comoving so that the momentum density vanishes (corresponding to the comoving gauge). One can instead choose a spatially flat gauge so that spatial components of the metric vanish, or a gauge with zero metric shear with threading and slicing being orthogonal (corresponding to conformal Newtonian or longitudinal gauge). The choice of coordinate system has some freedom without affecting anything physical. One can change the coordinates dx µ and the metric perturbations h µν in such a way that the metric distance ds 2 (equation (7.17)) remains invariant. To represent the perturbations we must thus make a gauge choice. A gauge transformation in GR is a change from one coordinate choice to another. The most general form is x µ = x µ + δx µ or explicitly τ = τ + T, x i = x i + L i. T corresponds to a choice in time slicing and L i a choice of spatial coordinates. This can also be decomposed into Fourier modes τ = τ + T (0) Q (0), x i = x i + 1 m= 1 L(m) Q (m) i. Since gauge transformation only uses scalar (m = 0) and vector quantities (m = ±1), it is clear that tensor modes (m = ±2) will not change under gauge transformation. They are thus gauge-invariant. Even though the coordinates can change the metric distance, ds 2 must remain invariant, i.e. g µν dx µ dx ν = g µν d x µ d x ν. Since g µν is a tensor and transforms in the same way as other tensors we can derive the transformation property of a general tensor W µν W µν ( x γ ) = xα x µ x β x ν W αβ ( x γ δx γ ) = W µν W µβ ν δx β W αν µ δx α δx α α W µν. (7.49) A gauge transformation equals a coordinate transformation of the perturbed first order quantities generated by x µ x µ + X µ. (7.50) A symmetric 2 tensor transforms under such a gauge transformation as δw δw + L X W. (7.51)

13 7.1 Evolution of Relativistic Perturbations 219 For the metric this means δg µν δg µν + X σ g µν,σ + g σµ X σ,ν + g σν X σ,µ. (7.52) This transformation law applied to the metric h µν in equation (7.17) gives explicitly à (0) = A (0) T HT (0), B (m) = B (m) + L (m) + kt (m), H (0) L = H (0) L k 3 L(0) HT (0), H (m) T = H (m) T + kl (m), (7.53) where m = 0, ±1. The stress-energy perturbation T µν in different gauges is similarly related by the gauge transformation δρ (0) = δρ (0) ρt (0), δp (0) = δp (0) pt (0), ṽ (m) = v (m) + L (m), Π (m) = Π (m), (7.54) where m = 0, ±1 in the velocity equation and m = 0, ±1, ±2 in anisotropic stress equation. The anisotropic stress is gauge-invariant. Finally, a scalar function transforms as f( τ, x i ) = f(τ, x i ) f τ T f x i L i. (7.55) The spatial gradient of the scalar function is of first order, so the last term in the expression above is of second order and can be dropped. The scalar field thus transforms as δ φ (0) = δφ (0) φt (0). (7.56) Gauge Invariant Bardeen Potentials The most general scalar perturbation of the metric is of the form δg = a 2 [ 2ψ dt 2 2( i B) dx i dt + (2φγ ij i j E) dx i dx j ]. (7.57) One can then derive that the four potentials transform as follows ψ ψ + T + HT (7.58) φ φ HT (7.59) B B + T L (7.60) E E + L. (7.61) With the four potentials ψ, φ, B and E and the two gauge freedoms T and L we can find two gauge invariant quantities, which are not unique. The simplest two potentials have been introduced by Bardeen [1] and are defined as Ψ = ψ 1 d ] [a(ė a dτ + B) (7.62) Φ = φ + H[Ė + B]. (7.63)

14 220 7 Evolution of Cosmic Structure By using the gauge transformations one can show the invariance of these two potentials (see exercises). The Newtonian (or longitudinal) gauge is defined by the choice L = E, T = Ė B (7.64) with the consequence that now B = 0 and E = 0. Obviously, in this gauge E and B will vanish after the gauge transformation, and we are left with the two invariant Bardeen potentials Einstein s Equations in Different Gauges The choice of gauge can be governed by the simplicity of equations, numerical stability of solutions, Newtonian intuition or other considerations. As we discussed above there is no gauge ambiguity for tensor modes. For vector modes the choice can either be B (±1) = 0 or = 0. The latter specifies the gauge completely, since it fixes L (±1), while the former only fixes L (±1) and thus leads to unspecified integration constant in H (±1) T. This however does not lead to any dynamical effects. Since vector modes are unlikely to be generated in the early universe we will not discuss them further. Instead we will look at four popular gauge choices for scalar modes that we will have the chance to use in the rest of the lectures. We will drop the superscripts (0) from now on. H (±1) T Synchronous gauge This gauge was very popular in the early development of perturbation theory. Its main advantage from today s perspective is its numerical stability, which is why it is still the gauge of choice in CMBFAST numerical package [19]. It corresponds to setting A = B = 0, so that only the spatial metric is perturbed. This implies that slicing is orthogonal to the threading and that a set of freely falling observers remains at a fixed coordinate position. To show this one must show that the spatial part of 4-velocity u µ = dx µ /dλ, where λ is affine parameter parametrizing the geodesic, vanishes. Geodesic equation is Du µ Dλ = dx ν uµ ;ν dλ = duµ dλ + Γµ αβ uα u β = 0. (7.65) Since A = B (m) = 0 implies Γ i 00 = 0 it follows that u i = 0 is a geodesic. The property that the fundamental observers follow geodesics means that the coordinates are Lagrangian and this gauge can only be used while δρ/ ρ 1. In the nonlinear regime where this condition is not satisfied one can have orbit crossings where two observers with different Lagrangian coordinates find themselves at the same real (Eulerian space) position. This can only happen if the metric perturbations diverge and the linear perturbation theory is no longer valid. While this limits the use of this gauge at late times it can still be used successfully in the early universe, as long as the density perturbations are small. Another shortcoming of this gauge is that the gauge choice does not fully specify it. One can see this by using a gauge transformation from a general gauge. Imposing à = B = 0 to the equations (7.53) we find T = a 1 aadτ + c 1 a 1 L = (B + kt )dτ + c 2, (7.66)

15 7.1 Evolution of Relativistic Perturbations 221 where c 1 and c 2 are integration constants. These remain unspecified in this gauge. They lead to unphysical gauge mode solutions for the density perturbations outside horizon. While historically this caused some confusion in their interpretation they are not really a problem since these modes do not show up in any observable quantity and are at any rate decaying faster than the physical modes. The remaining two scalar variables in this gauge are H L and H T. Instead of these one often introduces η H L H T /3 and h 6H L, but we will not make this replacement here. Einstein s equations are (k 2 3K)(H L H T ) + 3HḢL = 4πGa 2 δρ, Ḣ L (1 3K/k2 )ḢT = 4πGa 2 ( ρ + p)v/k, Ḧ L + HḢL = 4πGa 2 [ 1 δρ + δp], 3 Ḧ T + HḢT k 2 (H L H T ) = 8πGa 2 pπ. (7.67) Two of these are redundant. The conservation equations are ( τ + 3H) δρ + 3HδP = ( ρ + P )(kv + 3H L ) ( τ + 4H) ( [ ρ + P )v/k ] = δp 2 3 (1 3K/k2 ) P Π. (7.68) Newtonian gauge In Newtonian gauge one sets B = H T = 0. The gauge is also called conformal Newtonian or longitudinal. The remaining two scalar perturbations are renamed into A Ψ and H L Φ, ds 2 = a 2 (τ) { (1 + 2Ψ)dτ 2 + (1 2Φ)γ ij dx i dx j}. (7.69) This gauge is popular since it reduces to Newtonian gravity in the appropriate limit, is suitable for analytical work because of algebraic relations between matter and metric perturbations and does not have gauge ambiguities. It can however be numerically unstable, which is why it is usually not used in numerical codes. Instead one can do numerical computations in synchronous gauge and use the gauge transformation into the Newtonian gauge. A general gauge transformation into Newtonian gauge gives H T = 0 L = H T /k B = 0 T = B/k + ḢT /k 2. (7.70) One can see that there is no remaining gauge freedom, so the gauge is entirely fixed. The main advantage of this gauge is that there is a simple Newtonian correspondence and the equations reduce to Newtonian laws in the limit of small scales. The Einstein s equations are, ( ) (k 2 3K)Φ 3H Φ + HΨ = 4πGa 2 δρ Φ + HΨ = 4πGa 2 [( ρ + p)v/k] Φ KΦ + H( Ψ + 2 Φ) + (2Ḣ + H2 )Ψ k2 (Φ Ψ) = 4πGa 2 δp, k 2 (Ψ Φ) = 8πGa 2 pπ.

16 222 7 Evolution of Cosmic Structure On small scales the second term on the left hand side of first equation above becomes negligible compared to the first one. Similarly we can also neglect K relative to k 2, since curvature scale, if present, is of the order of the Hubble length. The result is a Poisson equation in an expanding universe. This means that we can identify metric perturbation Φ with the Newtonian potential on small scales. From the last of equations above we find Φ = Ψ in the absence of anisotropic stress. This is a good approximation in the matter era where ideal fluids dominate the energy density. We know that for astrophysical sources gravitational potential is roughly Φ v 2 (in units where c = 1), where v is a typical velocity of the object. This implies Φ Ψ 1 almost everywhere in the universe, except near a black hole. So in this gauge the linear perturbation theory is almost always valid and one can use these equations also to describe the late time nonlinear evolution in the density field, as long as the gravitational potential remains small. The conservation equations are ( τ + 3H) δρ + 3Hδp = ( ρ + p)(kv 3 Φ) ( τ + 4H) ([ ρ + p)v/k] = δp 2 3 (1 3K/k2 ) P Π + ( ρ + P (7.71) )Ψ. The Newtonian gauge is useful for analytic treatments, the equations are however numerically unstable. Comoving gauge This gauge is convenient because it gives algebraic relations between matter and metric perturbations. It introduces the curvature perturbation, which is useful when one wants to describe the evolution of perturbations, generated for example by inflation, outside the horizon. It turns out that this quantity is conserved for adiabatic perturbations (see more on this below) and so evolution is particularly simple in this case. On the other hand, this gauge is not particularly intuitive inside the horizon, so one is better off to transform into the Newtonian gauge in this limit. The gauge is defined so that the momentum density T 0 i vanishes. From equation (7.36) this implies B = v. The second constraint can be set to H T = 0. The remaining two scalar perturbations are renamed as A ξ and H L ζ. Gauge transformation from a general gauge into comoving gauge gives ṽ B = 0 T = (v B)/k H T = 0 L = H T /k. (7.72) The gauge is entirely fixed, since these are just algebraic relations between the quantities. Einstein s equations are (k 2 3K) (ζ + Hv/k) = 4πGa 2 δρ Hξ ζ K k v = 0 ( 2 ä a 2H2 + η τ 1 3 k2) ξ ( τ +) ( ζ + k 3 v) = 4πGa2 (δp δρ) k 2 (ξ + ζ) + ( [ τ + 2H) kv = 8πGa 2 pπ. Corresponding energy momentum tensor conservation equations are ( τ + 3H) δρ + 3Hδp = ( ρ + P )(kv + 3 ζ) ( ρ + P )ξ = δp (1 3K/k2 ) P Π. (7.73) (7.74)

17 7.2 Fluid equations in the Newtonian Gauge 223 Note that the gauge condition B = v implies δφ = 0 from 3rd equation (7.38), so the scalar field perturbations vanish in this gauge. Spatially flat gauge While comoving gauge is a useful gauge to describe evolution of perturbations after they cross the horizon, it has the shortcoming that the scalar field perturbations vanish in this gauge. It thus cannot be used to calculate scalar fluctuations from inflation, for example. One way to solve this is to calculate them in another gauge and use the gauge transformation to calculate curvature perturbation in comoving gauge. The simplest gauge to choose is the spatially flat gauge, where H L = H T = 0, (1 3K/k 2 )HkB = 4πGa 2 [δρ + 3H(ρ + p)(v B)/k] HA K k B = 4πGa2 ( ρ + P )(v B)/k ( 2 ä a 2H2 + H τ 1 3 k2) A ( τ + H) k 3 B = 4πGa2 (δp δρ) k 2 A + ( τ + 2H) kb = 8πGa 2 P Π. Corresponding energy-momentum tensor conservation equations are (7.75) Scalar field equation is ( τ + 3H) δρ + 3HδP = ( ρ + P )kv ( τ + 4H) ( [ ρ + P )(v B)/k ] = δp 2 3 (1 3K/k2 ) P Π + ( ρ + P )A. (7.76) δφ + 2Hδφ + (k 2 + a 2 V )δφ = (Ȧ kb) φ 2a 2 V A. (7.77) 7.2 Fluid equations in the Newtonian Gauge For the analysis in this section we use the perturbed metric in the Newtonian gauge [ ds 2 = a 2 (τ) (1 + 2Ψ) dτ 2 (1 2Φ)γ ij dx i dx j], (7.78) where Φ is a measure for the fractional perturbation of the scale factor (curvature perturbation) and Ψ is the effect of the Newtonian potential. As we have seen, when the Universe is filled with an ideal fluid, then Φ = Ψ. τ denotes the conformal time. We will restrict ourselves to the scalar fluctuations, given that only these can lead to growth of perturbations by self-gravity and thus to structure formation in the universe Fluid equations Energy momentum tensor is determined by the matter content in the universe. This can be divided into two classes. In the first class are matter components, which can be described within the fluid approximation. This class includes cold dark matter, baryons and scalar fields. In this section we derive evolution equations for the fluid ingredients that contribute to the

18 224 7 Evolution of Cosmic Structure energy-momentum tensor. Photons and neutrinos that cannot be described as fluids will be discussed in the following Sect. Together with the Einstein s equations these form a closed set of equations that can be evolved forward in time from some given initial conditions. We will focus ourselves to the Newtonian gauge, since the equations are easiest to interpret and the solutions do not suffer from the gauge modes or breakdown in the nonlinear regime. To specify the evolution in fluid approximation one only needs the equations for overdensity δ = δρ/ ρ and velocity v, δu µ = v µ /a, both of which can be obtained from the energymomentum conservation equations. The perturbations of the energy momentum tensor are then given by (in this Newtonian gauge) δt 0 0 = δρ (7.79) δt 0 i = (ρ + P )v i (7.80) δt i 0 = (ρ + P )v i (7.81) δt i j = δp δ i j. (7.82) Matter is given by an equation of state P = wρ (w = 0 for cold dark matter) and the adiabatic sound speed c 2 µν S = P/ ρ. The equations of motion, T ;ν = 0, split into two sets of equations, one equation for the density perturbation δ + 3(c 2 S w) Hδ + (1 w) i v i 3(1 + w) Φ = 0 (7.83) and the Euler equations (1 + w) τ v i + (1 + w)(1 3w) Hv i c 2 S i δ (1 + w) i Ψ = 0. (7.84) The effect of the expansion is seen in the second terms of these equations leading to a kind of damping. When we differentiate the first equation by time and take the divergence of the second equation and neglect the small terms H 2 and ä/a, we arrive at the following expression δ + (1 6w + 3c 2 S) H δ + c 2 S i i δ + (1 + w) i i Ψ + W = 0, (7.85) where W is a pure relativistic term [ W = 3(1 + w) (3w 1) H Φ Φ ]. (7.86) This is a hyperbolic equation for density perturbations (acoustic waves propagating with the sound speed c S ), including some damping term and the gravitational field as a driving source. In the Newtonian limit this leads to the equation (for a direct derivation of this equation from Newtonian hydrodynamics, see Sect ) δ + 2H δ c 2 S 2 δ 2 Φ = 0. (7.87) Thereby, the gravitational potential is given by the Poisson equation 2 Φ = 4πGδρ = 4πGρ δ. (7.88) By assumption cold dark matter (CDM) is cold and its pressure and anisotropic stress are zero. In Fourier space (flat model), one obtains the following evolution equation, δ c + H δ c = k 2 Ψ + 3H Φ + 3 Φ. (7.89)

19 7.2 Fluid equations in the Newtonian Gauge 225 If we further assume the anisotropic stress is negligible, which is almost always a valid approximation except on very large scales in radiation era where neutrinos make a non-negligible contribution, we have Ψ = Φ, which takes the role of Newtonian potential. This equation can be solved analytically for two cases of interest, matter domination and radiation domination. In the first case the souce of potential are CDM fluctuations themselves, while in the second case the souce of potential is coming from radiation and can be treated as external source. We discuss both cases below. For baryons one must also include pressure and Thomson scattering between photons and electrons. Thomson scattering has a well specified angular dependence in the rest frame of the electron and rapid scattering leads to isotropic photon distribution in this frame. The change in photon velocity is proportional to the difference between photon and baryon velocities times the scattering probability and so leads to the exchange of momentum between the photons and baryons (see the lecture on Boltzmann equation for more detail). Momentum conservation requires an opposite term in the baryon momentum conservation equation, δ b = kv b + 3 Φ, v b = Hv b + c 2 skδ b + 4 ρ γ 3 ρ b an e x e σ T (v γ v b ) + kψ, (7.90) where n e is the electron density, x e the ionization fraction and σ T Thomson cross section. We also included the pressure term, relating it to density gradients via adiabatic sound speed c 2 s = ( p/ ρ) S, neglecting entropy gradients. While equation (7.89) was derived for CDM it can also be used to describe baryons on large scales and after the recombination, where the pressure term from baryons and the coupling between baryons and photons can be neglected. We will denote with δ m the matter perturbation when it applies both to CDM and baryons. Oscillations in the Radiation Fluid Let us consider now a radiation fluid (baryons and photons before recombination, where Thomson scattering leads to a rapid momentum exchange between photons and baryons), w = 1/3 and c 2 S = 1/3, resulting in the equation δ δ Ψ 4 Φ = 0. (7.91) This can be written for temperature perturbations δ = 4(δT/T ) = 4Θ in Fourier space Θ k + k2 3 Θ k = k2 3 Ψ k + Φ k. (7.92) This represents a forced acoustic oscillator driven by gravitational forces. When baryons are included, the sound speed is somewhat less than c/ 3. Since pressure is mainly given by radiation and the density is a unique function of temperature, we may write for the sound speed c 2 S = ( P/ T ) ad ( ρ/ T ) Rad + ( ρ/ T ) B, (7.93) which provides the expression c 2 S = 1 3 4ρ Rad 4ρ Rad + 3ρ B = c R, (7.94)

20 226 7 Evolution of Cosmic Structure where R = 3ρ B /4ρ Rad denotes the baryon loading of the radiation fluid. With this we can write the equation for the temperature fluctuation as follows Θ k + 1 k 2 c R Θ k = k2 3 Ψ k + Φ k. (7.95) This is now the fundamnetal equation which governs the evolution of temperature fluctuations before recombination Gravitational Equations For the scalar modes we have only two equations (see Appendix A, K = 0) 3H(HΦ + Φ) + i i Φ = 4πGa 2 δρ (7.96) Φ + 3H Φ + (2Ḣ + H K)Φ = 4πGa2 δp (7.97) Using δp = c 2 Sδρ, we can combine the two equations Φ + 3H(1 + c 2 S) Φ + (2Ḣ + H2 + 3c 2 SH 2 )Φ + c 2 S i i Φ = 0. (7.98) This is a kind of a Klein Gordon equation with signal velocity given by the sound speed and including a damping term. Please note that in the Newtonian limit all the time derivatives vanish and only the Poisson equation survives Solutions inside horizon: matter domination The solutions for perturbations differ depending on whether the mode scale k 1 is larger or smaller than the Hubble radius c/h (we will loosely call this horizon scale, since for both matter and radiation epochs the Hubble radius and causal horizon distance are proportional to each other with a prefactor of order unity). On scales smaller than the Hubble length we can ignore the time derivatives of potential relative to the spatial derivatives. This only works if the solution is not oscillating and must be justified aposteriori from the obtained solution. Under this assumption we drop Φ and Φ terms in equation (7.89). To relate potential to density perturbation we use Poisson s equation, ignoring HΨ and K terms in addition to Φ. We thus obtain a second order differential equation, δ m + H δ m 4πGρ m a 2 δ m = 0. (7.99) The general solution to equation (7.89) consists of a growing and a decaying solution. It is instructive to look at the solutions in some limits. In the matter domination for Ω m = 1 we have a τ 2 from Friedmann equation and so 4πGρ m a 2 = 6/τ 2. Setting the solution as a power law δ = τ α one finds α = 2, 3. Since a τ 2 in matter era the growing mode solution grows as a scale factor, δ m (Ω m = 1, a) = aδ m (Ω m = 1, a = 1). (7.100) From Poisson s equation one finds that gravitational potential remains constant on small scales, Φ = const. It is customary to introduce the growth factor D(a) as a ratio of density fluctuation at a given expansion factor a relative to today. For Ω m = 1 one has D(a) = a.

21 7.2 Fluid equations in the Newtonian Gauge 227 In a universe filled with a cosmological constant, curvature or dark energy the growth will be slowed down relative to Ω m = 1. The effect is only important at late times when the additional component can be dynamically important. No analytic solutions exist in this case, but for a cosmological constant model (w = 1) with or without curvature a very good approximation to the growth factor is [3] D(a) 5Ω m (a)a 2 [ 1 Ω Λ (a) + Ω m (a) 4/ Ω ]. (7.101) m(a) Note that for a given Ω m the growth factor suppression is smaller in a flat universe (Ω Λ > 0) than in an open universe (Ω Λ = 0). This is because the effects of cosmological constant become significant later than those from curvature, as evident from the Friedmann s equation Solutions inside horizon: radiation domination In radiation era the photons are the dominant component to the energy density (we ignore the neutrinos, which contribute 40% of the total energy density). We show first that because of the pressure effects the photon perturbations do not grow. To show this one takes the pressure component of Einstein s equations (??) and subtracts it from one third of the density equation. Since δp = c 2 S δρ with c2 S = c/3 this gives the following equation, Φ + 4H Φ [ + 2Ḣ + 2H2] Φ = c 2 Sk 2 Φ. (7.102) We have again ignored the curvature terms. In radiation epoch one has a τ, from which follows H = 1/τ and so the last term on the left hand side vanishes. Then the equation is Φ + 4 τ Φ = c 2 Sk 2 Φ (7.103) which has the growing mode solution Φ = 3Φ i ( sin z z 3 cos z z 2 ), (7.104) where z = kc s τ and the solution was normalized to the initial potential value Φ i. As expected the photon pressure causes the potential to oscillate and decay away as (kc s τ) 2 inside the horizon. While the inclusion of baryons prior to recombination complicates the equations and changes the sound speed somewhat this conclusion does not change significantly. Gravitational potential therefore decays inside horizon in radiation era. To solve for CDM we take equation (7.89) and evaluate it in radiation epoch, δ c + 1 τ δ c = k 2 Ψ + 3η Φ + 3 Φ. (7.105) The full solution consists of a homogeneous solution (ie solution to the above equation without the source) and a particular solution, which can be written as an integral of Green s function over the source term. Equation (7.104) shows that the latter decays away on small scales and so we can ignore the particular solution. One is left with the homogeneous solution, which has the growing mode δ c = C + ln(τ), (7.106)

22 228 7 Evolution of Cosmic Structure where C is a constant. So the CDM density perturbation grows only logarithmically inside the horizon in radiation era. This has a simple physical interpretation: Prior to horizon crossing both radiation and matter evolve similarly and aquire a velocity term as they cross the horizon. After that CDM decouples from radiation and its velocity decays as τ 1 due to the Hubble drag (damping term). This gives rise to the logarithmic growth of density perturbation Solutions outside horizon Prior to horizon crossing (ck/h 1), the solutions depend on the adopted gauge. As above we will use equations in Newtonian gauge, but we emphasize that the solutions can differ drastically if a different gauge is adopted. For example, while in Newtonian gauge adopted here density perturbations do not grow outside horizon, they in fact do grow in synchronous and conformal gauge. If we ignore the anisotropic stress (making Φ = Ψ), spatial derivatives and curvature K then we again combine first and third of Einstein s equations to obtain Φ + [ 3H(1 + c 2 s,mr) ] ] Φ + [2Ḣ + H2 (1 + 3c 2 s,mr) Φ = 3H2 [ δp c 2 2 ρ mr δρ ], (7.107) where we introduced adiabatic speed c a Here c 2 a = 1 3 [1 + 3y/4]. (7.108) y = ρ m a ( ) 1/2 = x 2 Ωm H 0 + 2x, x = τ, (7.109) ρ r a eq 4a eq c and a eq is expansion factor when matter and radiation densities are equal, (1 + z eq ) = a 1 eq = Ω m h 2 (7.110) for CMB temperature T γ = 2.73K (scaling as Tγ 4 ). This equation can easily be derived from Friedmann s equation (??) if only matter density ρ m (CDM+baryons) and radiation density ρ r (photons and neutrinos) are included. Conformal time at matter-radiation equality is given by τ eq = 2( 2 1)c aeq 16Mpc H 0 Ω m h 2. (7.111) Ω m The right-hand side of equation (7.107) is proportional to the specific entropy fluctuation σ = 3 4 δ r δ m, δp c 2 aδρ = ρc 2 aσ. (7.112) If we restrict ourselves to the adiabatic initial conditions where entropy fluctuations vanish initially they cannot be generated on scales outside the horizon. In that case we can solve equation (7.107) analytically obtaining the growing and decaying solution [17] Φ + = y 8 9y x 9y 3, Φ = 1 + x y 3. (7.113)

23 7.2 Fluid equations in the Newtonian Gauge 229 In the radiation era (y 1) the solution is Φ + = 10/9(1 y/16), while in the matter era Φ + = 1. In general thus Φ = 9 10 Φ +Φ i, (7.114) where Φ i is the initial perturbation. One can see that on scales outside horizon the gravitational potential Φ does not change both in radiation and in matter epochs, but between the two it changes the amplitude by 10% (this gives rise to so called early integrated Sachs-Wolfe effect in CMB). Density perturbations are related to potential Φ through Poisson s equation. On large scales one can ignore spatial derivatives and curvature terms to obtain δ = 2(Φ + Φ/H), (7.115) where δ is the total density perturbation. In the matter and radiation domination limits this gives δ = 2Φ. Since δ = δ γ + yδ m 1 + y (7.116) one finds δ m = 3Φ/2 in radiation era and δ m = 2Φ in the matter era. Density perturbations are therefore also constant on large scales (figure 7.2). We stress again that this is a gauge dependent statement. Of course, for any quantity that is directly observable such as CMB anisotropies the predictions are independent of the gauge choice Transfer functions Any initial power spectrum of density perturbations gets modified because of the different growth of perturbations in different regimes. This can be conveniently expressed in terms of the transfer function, which is defined as how much δ X of a given mode grows (or decays) relative to initial value. For convenience this is divided by k 2 and normalized relative to the amplitude of some low k = k i mode at early times, when the mode is outside the horizon (k i τ 0 1), T X (k) = k2 i δ X(k, τ 0 )δ X (k = k i, τ i ) k 2 δ X (k, τ i )δ X (k = k i, τ 0 ). (7.117) The transfer function on large scales is unity, T (k k i ) = 1. The transfer function can be defined for any species X, such as CDM, baryons, photons, neutrinos and dark energy. Of particular relevance is the transfer function for the total density perturbation δ (equation 7.116), defined as an average over all the species weighted by their mean density, which on small scales determines the gravitational potential Φ. In this case the transfer function is also reflecting the transfer of Newtonian potential from early time until today, T (k) = Φ(k, τ 0)Φ(k = k i, τ i ) Φ(k, τ i )Φ(k = k i, τ 0 ). (7.118) Today δ is dominated by CDM, δ δ c, but it also has contributions from baryons and, if present, massive neutrinos. To obtain the processed power spectrum one multiplies the square of the transfer function with the primordial power spectrum P (k) = P i (k)t 2 (k).

24 230 7 Evolution of Cosmic Structure Outside the horizon the modes do not grow in both matter and radiation era and δ c is constant. When the modes enter horizon in radiation era the density perturbation only grows logarithmically until the matter era as discussed above. If the mode enters in the matter era it begins to grow immediately as τ 2. By definition the modes enter the horizon when k τ 1, so the growth is proportional to k 2 as long as the mode entered in the matter era. Since in our definion of transfer function we divide by k 2 the transfer function remains unity for all the modes entering horizon after the matter-radiation equality. The transfer function for modes entering prior to that will suffer suppression in the transfer function which scales as (kτ eq ) 2 [ln(kτ eq ) + 1] on small scales, where τ eq is the conformal time at matter-radiation equality (equation 7.111). The most important parameter that determines the transfer function shape is conformal time at equality τ eq, which depends on Ω m h when the power spectrum is expressed against k/h (as opposed to Ω m h 2 when k is measured in Mpc 1, equation 7.111). For CDM models the asymptotic slope of the transfer function is the same regardless of the value of cosmological parameters. Below we discuss baryons which can also affect the transfer function. The transfer function for models with the full above list of ingredients was first computed accurately by Bond & Szalay (1983), and is today routinely available via public-domain codes such as CMBFAST (Seljak & Zaldarriaga 1996). These calculations are a technical challenge because we have a mixture of matter (both collisionless dark particles and baryonic plasma) and relativistic particles (collisionless neutrinos and collisional photons), which does not behave as a simple fluid. Particular problems are caused by the change in the photon component from being a fluid tightly coupled to the baryons by Thomson scattering, to being collisionless after recombination. Accurate results require a solution of the Boltzmann equation to follow the evolution in detail. The different shapes of the functions can be understood intuitively in terms of a few special length scales, as follows (Fig. 7.1): Horizon length at matter radiation equality: The main bend visible in all transfer functions is due to the Meszaros effect, which arises because the universe is radiation dominated at early times. Fluctuations in the matter can only grow if dark matter and radiation fall together. This does not happen for perturbations of small wavelength, because photons and matter can separate. Growth only occurs for perturbations of wavelength larger than the horizon distance, where there has been no time for the matter and radiation to separate. The relative diminution in fluctuations at high k is the amount of growth missed out on between horizon entry and z eq, which would be δ DH 2 in the absence of the Meszaros effect. Perturbations with larger k enter the horizon when D H 1/k; they are then frozen until z eq, at which point they can grow again. The missing growth factor is just the square of the change in D H during this period, which is k 2. The approximate limits of the CDM transfer function are therefore T k 1 for kd H (z eq ) 1 and T k [kd H (z eq )] 2, kd H (z eq ) 1. (7.119) This process continues, until z eq = Ω M h 2, where the Universe becomes matter dominated. We therefore expect a characteristic break in the fluctuation spectrum around the comoving horizon length at this time. Using a distance-redshift relation that

25 7.2 Fluid equations in the Newtonian Gauge 231 Figure 7.1: A plot of transfer functions for various adiabatic models, in which T (k) 1 at small k. A number of possible matter contents are illustrated: pure baryons; pure CDM; pure HDM. For dark matter models, the characteristic wavenumber scales proportional to Ω M h 2, marking the break scale corresponding to the horizon length at matter radiation equality. The scaling for baryonic models does not obey this exactly; the plotted case corresponds to Ω M = 1, h = 0.5. ignores vacuum energy at high z, one obtains D H (z eq ) = ( 2 1) 2c (Ω M z eq ) 1/2 16 Mpc H 0 Ω M h 2. (7.120) Free-streaming length: This relatively gentle filtering away of the initial fluctuations is all that applies to a universe dominated by Cold Dark Matter, in which random velocities are negligible. A CDM universe thus contains fluctuations in the dark matter on all scales, and structure formation proceeds via hierarchical process in which nonlinear structures grow via mergers. Examples of CDM would be thermal relic WIMPs with masses of order 100 GeV/c 2. Relic particles that were never in equilibrium, such as axions, also come under this heading, as do more exotic possibilities such as primordial black holes. A more interesting case arises when thermal relics have lower masses. For collisionless dark matter, perturbations can be erased simply by free streaming: random particle velocities cause blobs

26 232 7 Evolution of Cosmic Structure to disperse. At early times (kt > mc 2 ), the particles will travel at c, and so any perturbation that has entered the horizon will be damped. This process switches off when the particles become non relativistic, so that perturbations are erased up to proper lengthscales of ct (kt = mc 2 ). This translates to a comoving horizon scale (2ct/a during the radiation era) at kt = mc 2 of L Free streaming 112 Mpc m/ev. (7.121) A more interesting (and probably practically relevant) case is when the dark matter is a mixture of hot and cold components. The free-streaming length for the hot component can therefore be very large, but within range of observations. The dispersal of HDM fluctuations reduces the CDM growth rate on all scales below L free stream or, relative to small scales, there is an enhancement in large scale power. Acoustic horizon length: The horizon at matter radiation equality also enters in the properties of the baryon component. Since the sound speed is of order c, the largest scales that can undergo a single acoustic oscillation are of order the horizon. The transfer function for a pure baryon universe shows large modulations, reflecting the number of oscillations that have been completed before the universe becomes matter dominated and the pressure support drops. The lack of such large modulations in real data is one of the most generic reasons for believing in collisionless dark matter. Acoustic oscillations persist even when baryons are subdominant, however, and can be detectable as lower level modulations in the transfer function. Silk damping length: Acoustic oscillations are also damped on small scales, where the process is called Silk damping: the mean free path of photons due to scattering by the plasma is non-zero, and so radiation can diffuse out of a perturbation, convecting the plasma with it. The typical distance of a random walk in terms of the diffusion coefficient D is x D t, which gives a damping length of λ S λd H, (7.122) the geometric mean of the horizon size and the mean free path. Since λ = 1/(nσ T ) = 44.3(1 + z) 3 (Ω b h 2 ) 1 proper Gpc, we obtain a comoving damping length of λ S = 16.3 Gpc (1 + z) 5/4 (Ω 2 bω M h 6 ) 1/4. (7.123) This becomes close to the horizon length by the time of last scattering, 1 + z = Fitting formulae: It is invaluable in practice to have some accurate analytic formulae that fit the numerical results for transfer functions. We give below results for some common models of particular interest (illustrated in Fig. 7.1 along with other cases where a fitting formula is impractical). For the models with collisionless dark matter, Ω b Ω M is assumed, so that all lengths scale with the horizon size at matter radiation equality, leading to the definition q k Ωh 2 Mpc 1 (7.124)

27 7.2 Fluid equations in the Newtonian Gauge 233 Figure 7.2: Evolution of δ c (solid) and δ b (dashed). The longer wavelength mode (k = h/mpc) enters the horizon almost entirely in the matter domination and subsequently grows as the scale factor. Higher k mode enters the horizon well in the radiation era and grows slowly until the matter domination. Baryon growth is suppressed until recombination (log(a) 3), after which it catches up with CDM. We consider the following cases: (1) Adiabatic CDM; (2) Adiabatic massive neutrinos (1 massive, 2 massless); (3) Isocurvature CDM; these expressions come from Bardeen et al. (1986; BBKS). Since the characteristic length scale in the transfer function depends on the horizon size at matter radiation equality, the temperature of the CMB enters. In the above formulae, it is assumed to be exactly 2.7 K; for other values, the characteristic wavenumbers scale T 2. For these purposes massless neutrinos count as radiation, and three species of these contribute a total density that is 0.68 that of the photons. T CDM (k) = log( q) 2.34q [ q + (16.1q) 2 +(5.46q) 3 + (6.71q) 4] 1/4 (7.125) T ν (k) = exp( 3.9q 2.1q 2 ) (7.126) ) 1/1.24 T curv (k) = (5.6q) (1 2 + [15.0q + (0.9q) 3/2 + (5.6q) 2] 1.24 (7.127). These expressions are useful for work at a level of 10% precision, but increasingly it is necessary to do better. In particular, these expressions do not include the weak oscillatory features that are expected if the universe has a significant baryon content. Eisenstein & Hu (1998) give an accurate (but long) fitting formula that describes these wiggles for the CDM transfer function. This was extended to cover MDM in Eisenstein & Hu (1999) On Baryons Prior to decoupling baryons are tightly coupled with photons and their effective sound speed is close to c/ 3. In this case baryon perturbations do not grow for the modes inside the horizon. Since decoupling occurs at z 1100, which is typically after matter-radiation equality, CDM modes inside horizon will have grown relative to baryons during this epoch (Fig. 7.2). After decoupling baryon sound speed drops to very low values and the Thomson scattering term in equation (7.90) can be dropped. In this case baryons obey the same equation (7.89) as CDM. If one subtracts baryons from CDM one finds, δ bc + H δ bc = 0, (7.128) where δ bc = δ b δ c. This equation is valid on scales where baryons pressure can be neglected. The growing mode solution for this equation is a constant, so the difference between baryon and CDM density perturbation does not change in time. However, the CDM density perturbation grows as δ c τ 2 a in the matter domination, so the relative difference between baryon and CDM density contrast decreases as a 1 τ 2. Thus δ b catches up with δ c after decoupling and then continues to grow at an equal rate (Fig. 7.2).

28 234 7 Evolution of Cosmic Structure We have so far assumed that CDM grows following the solution to equation (7.99). This is valid if CDM is the dominant component to the matter density. If baryon density is not negligible compared to CDM then baryons will also contribute to the gravitational potential from Poisson s equation. But baryons after decoupling still reflect acoustic oscillations from the epoch before decoupling, when they were tightly coupled with photons (equation 7.104). As a result the total potential will reflect these oscillations and CDM evolution will be modified because of this. Sufficiently large baryon density leads to acoustic oscillations in the transfer function (Fig. 7.1). Another effect of baryons that are dynamically important is that they suppress the transfer function on small scales. This is again expected, since baryons are damped prior to decoupling on small scales due to imperfect coupling with photons (this is the so-called Silk damping, which we will discuss in more detail in later sections). If they are dynamically important this leads to a suppression of the gravitational potential. The latter is the source for the CDM density fluctuations, which are thus suppressed as well (Fig. 7.1) Role of Scalar Fields Scalar fields have many applications in cosmology. One is as an inflaton field, which we discussed in some detail in the Sect. on Inflation. Another is as a candidate for the dark energy (or quintessence), which is responsible for the observed late time acceleration in the universe. Here we analyze its late time dynamics by focusing on the equations of motion. The evolution equation for the scalar field perturbations in Newtonian gauge follows from equation (7.39) Scalar field δ φ + 2ηδ φ + [k 2 + a 2 V ]δφ = ( Ψ + 3 Φ) φ 2a 2 V Ψ. (7.129) In the limit w φ = 1 we have no gravitational source for δφ and there are no perturbations in the scalar field assuming none existed initially. The easiest way to see this is by observing the same evolution equation (7.39) in synchronous gauge, where A (0) = 0. For w φ = 1 one has φ = 0, so all the sources on the right hand side of equation (7.39) vanish (in Newtonian gauge the same conclusion holds, since V = 0, as shown in next section). Cosmological constant and scalar field with w φ = 1 are thus indistinguishable. To solve for perturbations in general we must specify φ, V and V. It is often more convenient to express these in terms of the present density Ω φ0 and the equation of state w φ as a function of time. For w > 1 we must look at the stability analysis of evolution equation. In general gravity will cause the perturbations to grow, but if there is a significant positive pressure to counteract it they will not grow on scales smaller than the Jeans scale, defined roughly as the sound speed times time. For ordinary matter one would expect the sound speed to be given by w φ = p/ ρ, which for w < 0 would even accelerate the collapse of perturbations. However, for the scalar fields the stability of perturbations is not determined by the equation of state. On small scales one has a 2 V H 2 k 2, so ignoring the sources the evolution equation (7.39) is δ φ + 2Hδ φ + k 2 δφ = 0. (7.130) The solution to this equation is a damped oscillator with effective speed of sound c S = c. The sound speed c 2 S = δp φ/δρ φ equal to unity also follows from equations (7.38) in comoving

29 7.3 Nonlinear Evolution of Cosmic Structure 235 gauge, where v = B implies δφ = 0. Note that δρ in comoving gauge is the true density perturbation on small scales, since it satisfies the Poisson s equation (A.19). The scalar field perturbations are thus unimportant except on horizon scale. This justifies our treatment of matter perturbations inside horizon in the presence of dark energy (section 7.2.3), where we assumed it only affects the background evolution. Just as in the case of cosmological constant there are no exact analytical solutions available in the general case. The smooth component leads to a decay of the gravitational potential and the growth of structure is slowed down relative to Ω m = 1 case. For CMB on large scales the scalar field perturbations can make a small difference, since on large scales and at late times scalar field can have a significant contribution to the metric perturbation. These effects are difficult to observe because of the sampling variance. More complicated and interesting are cases of the non-canonic kinetic term (k-essence) and nonminimally coupling of the scalar field (Brans-Dicke type of models), both of which can produce in special cases c 2 S c2. In this case the scalar field perturbations may be more important for the structure evolution even on small scales. 7.3 Nonlinear Evolution of Cosmic Structure The equations of motion are nonlinear, and we have previously only solved them in the limit of linear perturbations. We now discuss evolution beyond the linear regime, first for a few special density models 1, and then considering full numerical solution of the equations of motion. In order to account for the observable Universe, any comprehensive theory or model of cosmology must draw from many disciplines of physics, including gauge theories of strong and weak interactions, the hydrodynamics and microphysics of baryonic matter, electromagnetic fields, and spacetime curvature, for example. Although it is difficult to incorporate all these physical elements into a single complete model of our Universe, advances in computing methods and technologies have contributed significantly towards our understanding of cosmological models, the Universe, and astrophysical processes within them. A sample of numerical calculations addressing specific issues in cosmology are reviewed in the following: from the Big Bang singularity dynamics to the fundamental interactions of gravitational waves; from the quark-hadron phase transition to the large scale structure of the Universe. The emphasis, although not exclusively, is on those calculations designed to test different models of cosmology against the observed Universe. In this Section we provide general notions about the formation and evolutions of cosmic structures, like galaxies and clusters of galaxies, within the standard cosmological scenario, based on a spatially homogeneous and isotropic expanding Universe. For this we introduce the notation for describing the degree of inhomogeneity which characterizes the structure of the Universe on scales 100 Mpc, thus much smaller than the cosmological event horizon ( 4300 Mpc). We explicitely show how the nature (either cold or hot) of the DM affects the formation and evolution of cosmic structures. We finally discuss how modern numerical computational techniques and observations can be joined together to improve our understanding of the nature of our Universe and its evolution. The most successful theory of cosmological structure formation is inflationary cold dark matter (ΛCDM) in which nearly scale invariant adiabatic initial fluctuations are set up by a 1 For the Zeldovich approximation and spherical models, see e.g. Padmanabhan III, pp....; for time constraints, these approximations are not discussed in these lectures

30 236 7 Evolution of Cosmic Structure period of inflation in the early universe. Thus the initial conditions are: the fluctuations in the gravitational potential (which can be related to density fluctuation through Poisson s equation) are almost independent of scale and that the fluctuations in the pressure are proportional to those in the density (which keeps the entropy constant, hence the term adiabatic). A direct consequence of the adiabatic assumption is that a cosmic overdense region contains overdensities of all particle species. The alternative mode, where overdensities of one species counterbalance underdensities in another, is known as isocurvature because the spatial curvature is unchanged. Models incorporating isocurvature initial conditions fare very badly when compared to the observations. Most (though not all) inflation models also predict that the spectrum of fluctuations is gaussian, with zero mean and variance given by the power spectrum. The preponderance of observational evidence suggests that the initial fluctuations which produced the large-scale structure in the universe were gaussian, with non-linear gravitational clustering producing all of the non-gaussianity in the distribution observed today. All the front-running candidates for models of structure formation use gaussian initial conditions. Viable models of structure formation thus differ mostly in what is assumed for the background cosmological parameters, specifically the density in CDM, spatial curvature or a cosmological constant/dark energy component (and its evolution), baryonic component, expansion rate etc. Of the possible models, the only currently viable one is L(ambda)CDM with roughly 1/3 of the energy in the universe being cold dark matter, 2/3 in a cosmological constant or dark energy component, and a few percent being in the form of normal or baryonic matter (most of which is also dark). The initial fluctuations have to be almost exactly Gaussian with a close to scale-invariant spectrum which is nearly power-law in scale. Once the initial spectrum and type of fluctuations is known, the linear evolution (at early times) is determined by the background cosmology and the type of dark matter. Because hot dark matter moves rapidly, it is able to stream out of overdense regions, escaping from the enhanced gravitational potential. This tends to erase fluctuations in the matter distribution on scales smaller than the free-streaming scale (approximately the speed of the HDM particles times the age of the universe). In contrast to this, cold dark matter is able to support perturbations on all scales of cosmological interest. This behaviour is much closer to what is inferred from observations of clustering. The dependence on the background cosmology enters because fluctuations only grow when the universe is matter dominated. Lowering the matter density thus decreases the length of time the perturbations can grow (both at early times, by delaying the dominance of matter over radiation and at late times when the universe becomes curvature or cosmological constant dominated) Cosmic Matter Having indirectly probed the nature of matter in the Universe using the previous estimates, it is now time to turn to direct constraints that have been derived in the past decade. Here, perhaps more than any other area of observational cosmology, new observations have changed the way we think about the Universe. Perhaps the greatest change in cosmological prejudice in the past decade relates to the inferred total abundance of matter in the Universe. Because of the great intellectual attraction Inflation as a mechanism to solve the so called Horizon and Flatness problems in the Universe,

31 7.3 Nonlinear Evolution of Cosmic Structure 237 it is fair to say that most cosmologists, and essentially all particle theorists had implicitly assumed that the Universe is flat, and thus that the density of dark matter around galaxies and clusters of galaxies was sufficient to yield Ω = 1. Over the past decade it became more and more difficult to defend this viewpoint against an increasing number of observations that suggested this was not, in fact, the case in the Universe in which we live. The earliest holes in this picture arose from measurements of galaxy clustering on large scales. The transition from a radiation to matter dominated universe at early times is dependent, of course, on the total abundance of matter. This transition produces a characteristic signature in the spectrum of remnant density fluctuations observed on large scales. Making the assumption that dark matter dominates on large scales, and moreover that the dark matter is cold (i.e. became non-relativistic when the temperature of the Universe was less than about a kev), fits to the two point correlation function of galaxies on large scales yielded Ω M h = Unless h was absurdly small, this would imply that Ω M is substantially less than 1. For the moment, however, perhaps the best overall constraint on the total density of clustered matter in the universe comes from the combination of X-Ray measurements of clusters with large hydrodynamic simulations. The idea is straightforward. A measurement of both the temperature and luminosity of the X-Rays coming from hot gas which dominates the total baryon fraction in clusters can be inverted, under the assumption of hydrostatic equilibrium of the gas in clusters, to obtain the underlying gravitational potential of these systems. In particular the ratio of baryon to total mass of these systems can be derived. Employing the constraint on the total baryon density of the Universe coming from BBN, and assuming that galaxy clusters provide a good mean estimate of the total clustered mass in the Universe, one can then arrive at an allowed range for the total mass density in the Universe. Many of the initial systematic uncertainties in this analysis having to do with cluster modelling have now been dealt with by better observations, and better simulations, so that now a combination of BBN and cluster measurements yields Ω M = 0.35 ± 0.1. The foremost HDM candidate is a particle known as the neutrino. This elusive particle was predicted to exist as early as 1931 by physicist Wolfgang Pauli in order to account for the conservation of momentum and energy in beta decay. One of the most revealing criteria as to whether the Universe is dominated by CDM or HDM is the way that matter, in particular galaxies, are distributed throughout the sky. HDM, as represented primarily by neutrinos, does not account for the pattern of galaxies observed in the Universe. Neutrinos would have emerged from the Big Bang with such highly relativistic velocities (i.e. close to light speed) that they would tend to smooth out any fluctuations in matter density as they streamed out through the Universe. In the early Universe, the neutrino density was enormous, and so most of the matter density could be accounted for by neutrinos. Given their great speeds, neutrinos would tend to free stream out of any overdense regions that is, regions with densities greater than the average density in the Universe. This process implies that density fluctuations could appear only after the neutrinos slowed down considerably. (i.e. As the Universe expanded, its temperature decreased, thereby resulting in neutrino cooling.) With this we obtain the following class of models which are consistent with the idea of inflation: Classical CDM (scdm): it would require a low Hubble constant, H 0 50 km/s/mpc, Λ = 0 und Ω B + Ω CDM = 1. This was the standard model in the 90es. νcdm (MDM): H 0 50 km/s/mpc, with Λ = 0, Ω B = 0.1, Ω HDM = 0.2 and

32 238 7 Evolution of Cosmic Structure Ω CDM = 0.7. Neutrinos can however not provide all the dark matter, since galaxies would form to late in a such a Universe. Such models are now also ruled out. ΛCDM: This is by now the standard model which consists of about two third of dark energy and one third of dark matter. This is often called the concordance model which is defined by the most important parameters following from WMAP. The WMAP satellite has allowed the very precise determination of many cosmological parameters Parameter Value Quantity H 0 71 ± 5 km/s/mpc Hubble constant t ± 0.2 Gyr Age of Universe Ω M 0.27 ± 0.04 Matter density Ω b ± Baryon density Ω Λ 0.73 ± 0.04 Dark energy Ω tot 1.02 ± 0.02 Total density Table 7.1: Parameters of the concordance model following from WMAP Collisionless Components Photonen, Neutrinos und Dunkle Materie verhalten sich heute stoßfrei, d.h. sie entwickeln sich im Universum ohne gegenseitige Wechselwirkung. Diese Formen der Materie sind wie Photonen zu beschreiben, d.h. über ihre Phasenraumfunktionen Photonen werden durch die Planck Funktion beschrieben, Neutrinos über eine Fermi Verteilung. Die DM kann auch im Teilchenbild beschrieben werden, d.h. wir können ihre Dynamik der Teilchentrajektorien berechnen Baryons and Gasdynamics Im Unterschied zu all diesen Teilchen ist die baryonische Materie stoss dominiert. Ihre Entwicklungsgleichungen folgen aus der Gasdynamik für die Dichte ρ B, die Geschwindigkeit v, den Druck P, sowie das Gravitationspotential Φ Continuity equation (mass conservation) ρ B t + (ρ B v) = 0 (7.131) Euler equation (momentum conservation) v 1 + ( v ) v = P Φ (7.132) t ρ B

33 7.3 Nonlinear Evolution of Cosmic Structure 239 Poisson equation (gravitation) 2 Φ = 4πGρ 0 (7.133) Here we have to take into account that all form of matter generates gravity. Der Zusammenhang zwischen Druck und Dichte folgt aus einer Zustandsgleichung, P = P (ρ B, s B ), wenn s B die spezifische Entropie der Baryonen beschreibt. Zur Vereinfachung nehmen wir im folgenden an, dass nur baryonische Materie beteiligt ist und lassen deshalb den Index B entfallen. Wir betrachten nun kleine Störungen um eine Gleichgewichtsverteilung mit Dichte ρ 0, Geschwindigkeit v 0 und Potential Φ 0, die als homogen angenommen wird, ρ 0 = 0 = P 0, ρ = ρ 0 + δρ, v = v 0 + δ v, P = P 0 + δp, Φ = Φ 0 + δφ. (7.134) Durch Subtraktion der Gleichgewichtsgleichungen von den vollen Gleichungen ergeben sich damit folgende Beziehungen. Aus der Kontinuitätsgleichung erhalten wir ( ) d δρ = d = δ v. (7.135) dt dt ρ 0 Dabei bedeutet δρ/ρ 0 den Dichtekontrast, der sich aufgrund der Störung des Geschwindigkeitsfeldes entwickelt. Dann ergibt die Euler Gleichung die Bewegungsgleichung für die Störungen δ v t + (δ v ) v 0 = 1 ρ 0 δp δφ, (7.136) sowie die Poisson Gleichung 2 δφ = 4πGδρ M. (7.137) Da das Universum expandiert, ist es günstig, mitbewegte Koordinaten einzuführen, x = R(t) r (s. Metrik). Damit gilt für eine gestörte Position der Teilchen δ x = r δr(t) + R(t) δ r. (7.138) Die Geschwindigkeit setzt sich deshalb aus zwei Anteilen zusammen v = δ x δt = Ṙ(t) r + R(t) d r dt. (7.139) Der erste Term beschreibt die Hubble Expansion, der zweite Term die Abweichung von der Hubble Expansion, die wir als R(t) u schreiben. Damit kann man die Euler Gleichung wie folgt darstellen d dt (R u) + (R u )Ṙ r 0 = 1 ρ 0 c δp c δφ. (7.140) Wir betrachten jetzt Ableitungen in Raumrichtungen als Ableitung bez. mitbewegter Koordinaten, d/dx = (1/R) d/dr, x = c /R, und damit (R u )Ṙ r = u Ṙ, womit die Euler Gleichung lautet d u dt + 2 Ṙ R u = 1 ρ 0 R 2 cδp 1 δφ. (7.141) R2

34 240 7 Evolution of Cosmic Structure Im folgenden betrachten wir sog. adiabatische Störungen, bei denen δp = c 2 S δρ, d.h. Änderungen des Drucks beruhen nur auf Änderungen der Dichte, und nicht der Entropie. c 2 S ( P/ ρ) s bezeichnet die Schallgeschwindigkeit des Mediums. Dazu bilden wir die Divergenz der gest orten Euler Gleichung c u + 2 Ṙ R ( c u) = 1 ρ 0 2 cδp 2 cδφ. (7.142) Die gestörte Kontinuitätsgleichung leiten wir nach der Zeit ab d 2 dt 2 = c u. (7.143) Durch Erstzen der Euler Gleichung folgt daraus d 2 dt Ṙ d R dt = c2 S ρ 0 R 2 2 cδρ + 4πG δρ M. (7.144) Dies ist eine lineare Wellengleichung in mitbewegten Koordinaten mit einem Quell Term und einem Dämpfungsterm, der durch die kosmische Expansion zustande kommt. Wir machen deshalb den Ansatz ebener Wellen, (t, r) = (t) exp i( k c r), was zu folgender fundamentaler Gleichung für die Amplituden (t) führt d 2 dt Ṙ d R dt + ( kcc 2 2 S 4πGρ B ) = 4πGρ DM DM. (7.145) The proper wave vector is k = R(t) k c. This is the equation of a forced damped oscillator. In the absence of dark matter, the nature of the solution is determined by the sign of the third term. If k 2 > k 2 J, where k 2 J = 4πGρ B c 2 S (7.146) the frequency of the oscillator is real - the perturbation in the baryonic component oscillates as an acoustic vibration. If k 2 < kj 2, then the frequency is imaginary and we get growing and decaying amplitudes. In such a case, perturbations can grow. The term proportional to the Hubble parameter represents the damping that is due to expansion, and the dark matter on the right hand side provides the driving force. When both dark matter and baryons are present, the dark matter source usually outnumbers the baryonic term the growth in dark matter can induce a corresponding growth in baryons at large scales. Modes with λ c < λ J do not grow, whereas long wavelength modes λ c > λ J can grow, where πc 2 S λ J = (7.147) Gρ is the Jeans length. The corresponding Jeans mass is the mass contained within a sphere of radius λ J /2 M J = 4π 3 ρ B ( ) 3 λj (7.148) 2

35 7.4 Cosmological Simulations 241 Putting in numbers, the Jeans mass before recombination is huge M < J Ω B M Ω M Ω M h 2. (7.149) The Jeans mass after decoupling is small M > J Ω B M Ω M Ω M h 2. (7.150) The Spectrum of Primordial Fluctuations The transfer function for models with the full above list of ingredients was first computed accurately by Bond & Szalay (1983), and is today routinely available via public-domain codes such as CMBFAST (Seljak & Zaldarriaga 1996). These calculations are a technical challenge because we have a mixture of matter (both collisionless dark particles and baryonic plasma) and relativistic particles (collisionless neutrinos and collisional photons), which does not behave as a simple fluid. Particular problems are caused by the change in the photon component from being a fluid tightly coupled to the baryons by Thomson scattering, to being collisionless after recombination. Accurate results require a solution of the Boltzmann equation to follow the evolution in detail. The gravitational potential shown in Fig. 7.3 is a statistical fractal at the end of inflation (gaussian random field). During the radiation era, acoustic damping smooths the potential on small scales. Because the density fluctuations are related to the potential by two spatial derivatives, the corresponding linear density field is dominated by high spatial frequency components. The nonlinear evolution of the density field causes the mass to cluster strongly into wispy filaments and dense clumps most of the volume has low density. This intermittency of structure is a sign of strong deviations from gaussian statistics. 7.4 Cosmological Simulations The structures observed in the present Universe cannot be obtained by applying a simple linear perturbation theory. As soon as δ 1, structures will collapse and follow their own graity, thereby deviating away from the common Hubble flow. Such investigations can only be obtained by means of large scale computer simulations. The Universe at Redhsift z = 30 50: In einem CDM Modell werden kleine Skalen schon sehr früh nichtlinear, da das Fluktuationsspektrum zu kleinen Skalen langsam ansteigt. Massenskalen von ( ) M klumpen bereits im Rotverschiebungsbereich z Dies entspricht einer Zeitskala von einigen 100 Mio. Jahren. Die dunkle Materie des Universums zerfällt in eine zellenartige Struktur, die eine Massenverteilung bis zu M aufweist. Das Zentrum jeder Zelle ist ein Massenzentrum dunkler Materie, das von einem Halo umgeben ist. Diese Strukturen bilden die Keimzellen der Sphäroide (Bulges) der Galaxien.

36 242 7 Evolution of Cosmic Structure Figure 7.3: Potential and density for a mixed cold+hot dark matter model, Ω ν = 0.2 (Bertschinger 1995). A slice is shown 50 h 1 Mpc wide. Upper left: potential at the end of inflation; upper right: potential at the end of recombination; lower left: density at the end of recombination; lower right: density at redshift zero Particle Models Um diese Situation auf dem Computer zu simulieren, wird die Bewegung dieser Massenzellen unter ihrer gegenseitigen Gravitationswirkung untersucht. Im allgemeinen beschränkt man sich auf N Massenlemente zur selben Masse (sog. Testmassen). Die Bewegung solcher Testmassen wird am besten in mitbewegten Koordinaten analysiert, i = 1,..., N, N = 10 9 (!) x i = x i /a(t), x (0) = x(0) (7.151) v i = 1 a ( v i H x i ). (7.152)

37 7.4 Cosmological Simulations 243 Die wichtige Geschwindigkeit ist nur die Bewegung relativ zur Raumexpansion H x. Die urspr unglichen Bewegungsgleichungen lauten v = Φ (7.153) x = v (7.154) Φ = 4πGρ(t, x). (7.155) Dies ergibt folgende Gleichungen in mitbewegten Koordinaten v + 2H(t) v = 1 a Φ ä a x. (7.156) Der letzte Term entspricht der Gezeitenkraft. Durch die Transformation Φ = a(t) Φ + a2 ä 2 x 2 erh alt man die Bewegungsgleichung (7.157) v + 2H(t) v = 1 a 3 Φ. (7.158) Die Poisson Gleichung in mitbewegten Koordinaten lautet dann Φ = 4πGa 3 ρ + 3a 2 ä, ρ = ρa 3 (7.159) Da in der materiedominierten Epoche ä = (4/3)4πG ρa(t), bedeutet dies Φ = 4πG(ρ (t, x ) ρ(0)). (7.160) Nur die Differenz ρ ρ(0) tritt als Quelle der Gravitation auf. N Massenelemente werden nun in eine kubische Box verteilt. Die totale Masse verschwindet. Haufen weisen einen positiven Dichteüberschuß auf, ρ ρ > 0, Voids einen negativen. Dies führt dazu, dass sich Haufen und Voids gegenseitig abstoßen. In total, the following system of differential equations has to be integrated v i = x i (i = 1,..., N) (7.161) v i + 2H(t) v i = F i (7.162) F i = 1 a 3 Φ xi. (7.163) Φ = 4πG(ρ (t, x ) ρ(0)) (7.164) ( ) 1 ȧ = H 0 Ω 0 a(t) 1 + Ω Λ (a 2 1) + 1, a(0) = 1 (7.165) H(t) = ȧ (7.166) a ρ(t) = ρ(0)/a 3. (7.167) Mit Hilfe von N Körper Methoden k onnen die Bewegungsgleichungen von einzelnen Massenteilchen auf dem expandierenden Hintergrund integriert werden. Die Poisson Gleichung wird auf einem Gitter gelöst. Heute können Simulationen auf Gittern gefahren werden mit typisch mindestens einzelnen Massen (Abb 7.4). Bei Teilchen und

38 244 7 Evolution of Cosmic Structure einer Boxlänge von 720 Mpc entspricht dies einer Teilchenmasse von M, bei 144 Mpc einer typischen Galaxienmasse M (128 3 /N p ). Im ersten Falle ergibt dies eine räumliche Auflösung von 5.6 Mpc, im zweiten Falle von 1.1 Mpc. Diese Rechnungen ber ucksichtigen nur die dunkle Materie, in neueren Simulationen kann auch die baryonische Materie mitgerechnet werden (Hydrodynamik). Die Anfangsbedingungen bei a = 1 werden nach dem linearen St orungsspektrum generiert mit einer Amplitude = 1/16, normiert auf der Skala von 16 Mpc. Die Simulationen werden zeitlich entwickelt, bis das Powerspektrum die COBE Werte erreicht. Dies entspricht einem Skalenfaktor a 0 = R 0 /R in Progress of Cosmological Simulations Cosmological N boby simulations started in late 1970 s, and since then have played an important part in describing and understanding the nonlinear clustering in the Universe. Such N body calculations in a comoving periodic cube were done for the first time by Miyoshi and Kihara (1975) using N = 400 particles. Suto [22] has recently pointed out that the progress is well fitted by a trend of the form N = (yr 1975), (7.168) where the amplitude is normalized to the first publication by Miyoshi and Kihara. Just for comparison, the total number of CDM particles of mass m CDM in a box of the Universe of one side L is given by N = Ω CDML 3 m CDM Ω CDM 0.23 ( ) 3 hl kev Gpc m CDM h 0.7. (7.169) If we extrapolate this equation, then the number of particles that one can simulate in a (1 Gpc/h) 3 box will reach the real numbers of CDM particles in the box in the year 2348 for m CDM = 1 kev and in the year 2386 for m CDM = 10 5 ev (Fig. 7.4)! After the first simulations, the primary goal of the simulations in 1980 s was to predict observable galaxy distribution from dark matter clustering. So one had to distinguish between galaxies and simulation particles which represent dark matter, i.e. to introduce the notion of galaxy biasing. Davis, Efstathiou, Frenk and White (1985) was one of the most influential and seminal paper in cosmological N body simulations. The most important message they were able to show is the fact that large scale simulations can provide numerous realistic and testable predictions of dark matter scenarios against the observational data. At that time, they only used N = 32 3 particles. Simulations in the 1990 s revealed astonishing progress in understanding the large scale structure. One of the breakthrough was the amazing scaling property in the two point correlation functions. It was found that it can be well approximated by a universal fitting formula which empirically interpolates the linear regime and the non linear stable solution (Peacock & Dodds [?]; Smith et al. [?]). The results for all particles agree well with the theoretical predictions. Only at small scales, where the force resolution is no longer sufficient, deviations occur The Fractal Structure of Dark Matter Distribution Fig. 7.5 illustrates the spatial distribution of dark matter at the present day, in a series of simulations covering a large range of scales. Each panel is a thin slice of the cubical simulation

39 7.4 Cosmological Simulations 245 Figure 7.4: Evolution of the number of particles in cosmological N body simulations. [Source: Suto 2003] volume and shows the slightly smoothed density field defined by the dark matter particles. In all cases, the simulations pertain to the CDM cosmology, a flat cold dark matter model in which Ω M = 0.3, Ω Λ = 0.7 and h = 0.7. The top-left panel illustrates the Hubble volume simulation: on these large scales, the distribution is very smooth. To reveal more interesting structure, the top right panel displays the dark matter distribution in a slice from a volume approximately 2000 times smaller. At this resolution, the characteristic filamentary appearance of the dark matter distribution is clearly visible. In the bottom-right panel, we zoom again, this time by a factor of 5.7 in volume. We can now see individual galactic-size halos which preferentially occur along the filaments, at the intersection of which large halos form that will host galaxy clusters. Finally, the bottom-left panel zooms into an individual galacticsize halo. This shows a large number of small substructures that survive the collapse of the halo and make up about 10% of the total mass (Klypin et al. 1999, Moore et al. 1999). For simulations like the ones illustrated in Figure 7.5, it is possible to characterize the statistical properties of the dark matter distribution with very high accuracy. The statistical error bars in this estimate are actually smaller than the thickness of the line. Similarly, higher order clustering statistics, topological measures, the mass function and clustering of dark matter halos and the time evolution of these quantities can all be determined very precisely from these simulations (e.g. Jenkins et al. 2001, Evrard et al. 2002). In a sense, the problem of the distribution of dark matter in the CDM model can be regarded as largely solved [22].

40 246 7 Evolution of Cosmic Structure Figure 7.5: Slices through 4 different simulations of the dark matter in the CDM cosmology. Denoting the number of particles in each simulation by N, the length of the simulation cube by L, the thickness of the slice by t, and the particle mass by m p, the characteristics of each panel are as follows. Top-left (the Hubble volume simulation, Evrard et al. 2002): N = 10 9, L = 3000 h 1 Mpc, t = 30 h 1 Mpc, m p = h 1 M. Top-right (Jenkins et al. 1998): N = , L = 250 h 1 Mpc, t = 25 h 1 Mpc, m p = h 1 M. Bottom-right (Jenkins et al. 1998): N = , L = 140 h 1 Mpc, t = 14 h 1 Mpc, m p = h 1 M. Bottom-left (Navarro et al. 2002): N = , L = 0.5 h 1 Mpc, t = 1 h 1 Mpc, m p = h 1 M. Biasing of Dark Matter Halos The second progress where N body simulations played a major role in the 1990 s is related to the statistics of dark halos, their mass function and spatial biasing. The standard picture of structure formation predicts that luminous objects form in the gravitational potential of dark

41 7.4 Cosmological Simulations 247 Figure 7.6: Dark matter distribution at redshift z = 6 (top) and in the present Universe (bottom). Parameters: Ω M = 0.30, Ω b = 0.035, σ 8 = 0.90, L = 25 Mpc/h, N cell = [Source: Cen Princeton]

42 248 7 Evolution of Cosmic Structure matter halos. Both the Press Schechter theory and high resolution N body simulations have made significant contributions in constructing a semi analytical framework for halo clustering. Density Profiles for Dark Halos and Cluster Gas One of the most useful result from N body simulations is the discovery of the universal density profile of dark halos. Navarro, Frenk and White (1995, 1997) found that all simulated density profiles can be well fitted by the following simple model (socalled NFW profile) ρ DM (r) = δ c ρ c (z) (r/r S )(1 + r/r S ) 2 (7.170) by an appropriate choice of the scaling radius r S = r S (M) as a function of halo mass M. ρ c (z) = 3H 2 (z)/8πg is the critical density of the Universe at redshift z, r S is the scale radius, and δ c is a characteristic dimensionless density, which takes the form when expressed in terms of the concentration parameter c δ c = c 3 ln(1 + c) c/(1 + c). (7.171) Hence, there are two free parameters for the NFW density profile: r S and c. c is related to the virial radius r vir, defined to be the radius where the enclosed average mass density equals 200ρ c (z), i.e. r vir = cr S. The mass enclosed within radius r is found to be [ ( ) ] M(< r) = 4πρ c (z) δ c rs 3 rs + r ln r. (7.172) r S r + r S The scaling radius found in clusters is typically kpc. For r r S we find M(< r) r 2. Many others have indicated that the inner slope of the density profile is steeper than the NFW value, and current census among N body workers is given by ρ DM 1 (r/r S ) α. (7.173) (1 + r/r S ) 3 α with α , rather than the NFW value α = 1 for r > 0.01 R vir (for recent simulations of halo density profiles see [5]). In the meanwhile, many clusters have been investigated with Chandra and XMM. The gas density is then typically found to obey a β law ρ G (r) = ρ G0 (1 + r 2 /r 2 c) 3β/2 (7.174) with the core radius r c kpc and β The cluster A2589 e.g. (at redshift z = 0.042) has a core radius r c = 40 ± 12 kpc and β = 0.39 ± 0.04 with a central density ρ G0 = g/cm 3 (Buote and Lewis 2003 [2]). This cluster has roughly a constant X ray temperature of 3.16 kev between 15 and 150 kpc.

43 7.4 Cosmological Simulations 249 Figure 7.7: Total gravitational mass and gas mass as a function of radius for Abell 2589 [2]. Also plotted is the mass range expected from the stars (dot dahsed curves): The lower curve assumes M /L V = 1 M /L and the upper curve M /L V = 12 M /L. The upper axis shows the radius in units of the virial radius, R vir = 1.72 Mpc. Cluster Temperature The baryons confined by the gravity of the dark matter in clusters are heated up to virial temperature which is in the range of X ray temperatures. We determine the virial temperature T vir of the system of mass M by equating the gravitational potential (3/5)GM 2 /r of a dark halo to (3M/µm p )k B T vir. Using M = (4π/3)ρ h r 3, µ = and ρ 0 = Ω m h 2 M Mpc 3, we find the virial temperature at redshift z h where the halo has been formed T vir = K ( M M ) 2/3 (f h Ω m h 2 ) 1/3 (1 + z h ), (7.176) 2 If the He fraction is Y = 025 by weight and the gas is fully ionised µ = m Hn H + m He n He 2n H + 3n He = m H Y Y 0.57 m H. (7.175)

44 250 7 Evolution of Cosmic Structure where ρ h = f h ρ 0 (1 + z h ) 3 denotes the mass density in the dark halos and f h 180/Ω 0.7 m. The corresponding virial radius can be written as ( ) 1/3 M R vir = 2.58 Mpc (1 + z h ) 1 h 2/3 50 (7.177) M and the virial velocity (velocity of dark sub halos and baryons in the cluster) as V vir = 1000 km s 1 ( M M ) 1/3 (1 + z h ) 1/2 h 1/3 50. (7.178) This hot cluster gas cools by emission of Bremsstrahlung photons which can be detected by means of X ray telescopes (Einstein, ROSAT, ASCA, Chandra and XMM Newton). The cooling time is typically longer than the Hubble time, except in the very center of massive clusters and galaxies. Simple scaling relations (form the above we have M B M T 3/2 X and L X M B T 1/2 X T X 2 ) predict that the X ray luminosity scales with T X 2. It has been known for many years that the actual relation is steeper (see Fig. 7.8). Figure 7.8: X ray luminosity vs. X-ray temperature derived from ASCA observations of 270 galaxy clusters. The best fit is L X T 3 X. [Source: Mushotzky 2003]

45 7.4 Cosmological Simulations Statistical Measures for Cosmic Density Fields A key question is how to treat the statistical analysis for these data this is how observations and theory are confronted with each other. The concept of an ensemble is used every time we apply probablity theory to such events. The density at a given point in space will have different values in each member of the ensemble, with some overall variance < δ 2 > between members of the ensemble. Statistical homogeneity of the δ field means that this variance must be independent of space points. The actual field found in a given simulation is just one realisation of the statistical process. There are two problems with this line of argument: (i) we have no evidence that the ensemble exists, (ii) in any case we only get to observe one realisation. In order to solve the second question, we just have to look at widely separated parts of space, since the δ fields should be there uncorrelated. This means we have to mesasure the variance < δ 2 > by averaging over large volumes. Fields that satisfy volume average = ensemble average are called ergodic which is a kind of common sense axiom in Cosmology. A Gaussian Random Field We consider the density contrast δ( x) = ρ/ ρ 1 at positions in comoving coordinates. This density field is considered as a stochastic variable, i.e. a random field. The conventional assumption is that the initial perturbations are Gaussian, i.e. i.e. its m point joint probability distribution obeys a multi variate Gaussian P (δ 1,..., δ m ) dδ 1...dδ m = 1 (2π)m detm exp 1 2 m δ i (M 1 ) ij δ j dδ 1...dδ m i,j=1 (7.179) Here, M ij =< δ i δ j > is the covariance matrix and M 1 its inverse. Since M ij = ξ( x i, x j ) is specified by the two point correlation function, this implies that the statistical nature of a Gaussian density filed is completely given by the correlation function. The Gaussian nature of the density field preserves in the linear stage of evolution, but not in the non linear stage. Here, it can be approximated by one point log normal (Fig. 7.9) P (1) (δ) = 1 2πσ 2 1 σ 1 depends on the smoothing scale R exp ( [ln(1 + δ) + σ2 1/2] 2 2σ 2 1 ) δ. (7.180) σ 2 1(R) = ln(1 + σ 2 NL(R)), (7.181) where the variance of the non linear clustering is computed from the non linear power spectrum σ 2 NL(R) = 1 2π 2 0 Ŵ 2 (kr) k 2 P NL (k) dk. (7.182) Ŵ (x) is the window function, e.g. a Gaussian Ŵ (x) = exp( x2 /2) or a top hat Ŵ (x) =

46 252 7 Evolution of Cosmic Structure Figure 7.9: Probability distribution function (PDF) in the non linear regime. Smoothing windows: 2/h Mpc (cyan), 6/h Mpc (red), and 18/h Mpc (green). 3(sin x/x cos x)/x 2. The smoothed density field is then given by δρ( x, R) = d 3 y W ( x y ; R) ρ( y) d 3 k = (2π) 3 Ŵ (kr) ˆρ( k) exp( k x). (7.183) In the case of a top-hat window function we find Ŵ (kr) = = x <R exp(i k x) d 3 x ( ) 3 sin(kr) (kr) 2 cos(kr) (kr) 1, kr 1 0, kr 1. (7.184)

47 7.4 Cosmological Simulations 253 The function W (kr) effectively truncates the sum so that waves with λ 2R will not contribute. Actually in a stochastic sample, we cannot measure the density at a single point x. It is more relevant to consider the total mass M R ( x) in a sphere of radius R centered on the point x x x <R M R ( x) = ρ( x ) d 3 x ( ) = ρv R δ k Ŵ (kr) exp(i x V k). (7.185) To calculate the fluctuations in M R we need [ [M R ( x)] 2 = ρ 2 VR δ k Ŵ (kr) exp(i k x) V + 1 V k k k k δ k δ k Ŵ (kr)ŵ (k R) exp(i( k + k ) x) Taking the average over the volume V, the oscillating terms integrate to zero leaving < M 2 R >= ρ 2 V 2 R [ V With this we obtain the variance of M R ]. (7.186) ] δ k 2 Ŵ 2 (kr). (7.187) k ( ) 2 MR = < M R 2 > < M R > 2 < M R > 2 = 1 V M R Ŵ (kr) 2 δ k 2. (7.188) k Since V R V, we can replace the sum by an integral ( MR M R ) 2 = 1 (2π) 3 d 3 k δ k 2 Ŵ (kr) 2. (7.189) This often written in terms of the variance 2 k isotropic fluctuations) = k3 δ k 2 of the power spectrum as (for ( MR M R ) 2 = 1 dk 2π 2 k k 2 Ŵ (kr) 2. (7.190) The window function cuts off the integral for k > 1/R. Since k is an increasing function of k (see Fig. 7.10), a good approximation for R 8/h Mpc is M R M R k kr (7.191)

48 254 7 Evolution of Cosmic Structure Fourier Transformations If a field were periodic in some large box of side L, then we could just sum over all modes F ( x) = F k exp( i k x) (7.192) The periodicity restricts then the wave numbers to k x = 2πn/L. If we let the box become arbitrarily large, the sum will go to an integral ( ) n L F (x) = F k exp( ikx) d n k (7.193) 2π ( ) n L F k = F (x) exp(ikx) d n x. (7.194) 2π Correlation Function and Power Spectra Of particular importance is the autocorrelation function ξ( r) =< δ( x)δ( x + r) > (7.195) simply referred as the correlation function. Now we use that δ is real ξ( r) =< δkδ k exp(i( k k) x) exp( i k r) > (7.196) k k By periodic boundary conditions, all the cross terms with k k average to zero, and the remaining sum can be written as an integral ξ( r) = V (2π) 3 δ k 2 exp( i k r) d 3 k. (7.197) In short, the correlation function is the Fourier transform of the power spectrum. In Cosmology, the alternative notation is often used P (k) < δ 2 k > (7.198) for the ensemble average power. In an isotropic Universe, the density perturbation spectrum cannot contain a preferred direction, and so we must have an isotropic power spectrum, < δ k ( k) 2 >= δ k 2. The angular part of the k space integral can immediately be done: introduce spherical polars with the polar axis along k and use the reality of ξ so that exp( i k x) cos(kr cos θ). In 3 dimensions this yields then ξ(r) = V (2π) 3 P (k) sin(kr) 4π 2 k 2 dk. (7.199) kr It is now quite common to express the power spectrum in dimensionless form as the variance per logarithmic decade: 2 (k) d ln(< δ2 >) d ln k = V (2π) 3 4πk3 P (k) = 2 π k3 ξ(r) sin(kr) kr r 2 dr. (7.200)

49 7.4 Cosmological Simulations 255 The function is dimensionless and 2 (k) = 1 means that there are order unity fluctuations from modes in a logarithmic bin around k (Fig. 7.10). Density fluctuations of order unity are achieved on scales of about 8/h Mpc (the scale of large clusters). Large volume redshift surveys (2dFGRS and SDSS) allow to determine the cosmic variance even beyond this scale. In CDM models, the shape of this spectrum only depends on the parameter Ω m h. These Figure 7.10: Cosmic variance as a function of wavenumber, as observed from galaxy clustering in 2dFGRS. At small scales, approaches a constant value for a primordial spectrum P (k) k. The solid points with error bars show the power estimate. The window function correlates the results at different k values, and also distorts the large-scale shape of the power spectrum. The solid and dashed lines show various CDM models, all assuming n = 1. For the case with non negligible baryon content, a big bang nucleosynthesis value of Ω B h 2 = 0.02 is assumed, together with h = 0.7. A good fit is clearly obtained for Ω m h = 0.2 (in CDM models, the cosmic variance is only a function of Ω m h). The observed power at large k will be boosted by nonlinear effects. [Source: Peacock] observations therefore clearly rule out models with Ω m 1. It is also customary to normalize the spectrum in terms of σ 8 M R /M R at the scale of 8h 1 Mpc. This is one of the essential parameters appearing in cosmological simulations. Using the halo model, it is then possible to calculate the correlations of the nonlinear density field, neglecting only the large scale correlations in halo positions. The power spectrum determined in this way is shown in Fig. 7.11, and turns out to agree very well with the exact nonlinear result on small and intermediate scales. The lesson here is that a good deal of the nonlinear correlations of the dark matter field can be understood as a distribution of random clumps, provided these are given the correct distribution of masses and mass-dependent density profiles.

50 256 7 Evolution of Cosmic Structure Figure 7.11: The decomposition of the mass power spectrum according to the halo model. The dashed line shows linear theory, and the open circles show the predicted 1 halo contribution. The full lines show the contribution of different mass ranges to the 1-halo term: bins of width a factor 10 in width, starting at h 1 M and ending at h 1 M. [Source: Peacock] By counting galaxies, one is not necessarily counting the distribution of the total mass. This introduces a kind of bias n G n G = b ρ ρ. (7.201) b is called the bias parameter. Bias independent information can be obtained from counts of galaxy clusters. The largest clusters have masses of the order of the mean mass contained in spheres of 8/h Mpc. Recent measurements give σ 8 Ω 0.56 m 0.5. So for Ω m = 0.3 this corresponds to a value of σ Topological Measures Turning next to statistics for continuous density fields, we note first that N-point correlation functions and power spectra are naturally defined and very useful in this case much as they are for point sets. A family of new statistics, reviewed by Melott (1990), is based on the two-dimensional surfaces of constant density (isodensity surfaces). The best known of these is the genus per unit volume g(ν) as a function of the standardized density contour level ν = δ/σ (Gott et al 1986). The total genus G of a surface is a topological invariant measuring the number of holes (as in a doughnut) minus the number of isolated regions. One of its attractions is the fact that an exact prediction exists for the shape of G(ν) for Gaussian random fields (Doroshkevich 1970, Bardeen et al 1986, Hamilton et al 1986), enabling a test

51 7.4 Cosmological Simulations 257 of Gaussianity on large scales where the matter distribution is expected to still approximate the linearly growing initial conditions. When the smoothing scale is varied, the amplitude of the genus curve varies in a spectrum-dependent way, providing another useful clustering statistic. Computer programs for computing the genus are given by Melott (1990); an alternative and simpler algorithm is provided by Coles et al (1996). Figure 7.12: Minkowski functionals as a function of ν f for R g = 5/h Mpc in two regions of SDSS. The results favour the LCDM model with initial Gaussian perturbations. Minkowski functionals (Mecke et al 1994) have recently been introduced in cosmology as a very powerful descriptor of the topology of isodensity surfaces. In three dimensions, there are four Minkowski functionals (V 0, V 1, V 2, V 3 ); two of them are the genus (actually, its relative the Euler characteristic) and surface area statistics discussed above, and the other two are the covered volume (related to the void probability function) and integral mean curvature. Analytical results for Gaussian random fields have been provided by Schmalzing & Buchert (1997), who also have made available a computer program for computing these statistics from a point process. The insight and unification provided by these recent results suggests

52 258 7 Evolution of Cosmic Structure a promising future for Minkowski functionals as a statistic for both cosmological simulations and redshift surveys s: Baryons in Simulations Understanding galaxy formation is a much more difficult problem than understanding the evolution of the dark matter distribution. In the CDM theory, galaxies form when gas, initially well mixed with the dark matter, cools and condenses into emerging dark matter halos. In addition to gravity, a non-exhaustive list of the processes that now need to be taken into account includes: the shock heating and cooling of gas into dark halos, the formation of stars from cold gas and the evolution of the resulting stellar population, the feedback processes generated by the ejection of mass and energy from evolving stars, the production and mixing of heavy elements, the extinction and reradiation of stellar light by dust particles, the formation of black holes at the centres of galaxies and the influence of the associated quasar emission. These processes span an enormous range of length and mass scales. For example, the parsec scale relevant to star formation is a factor of 10 8 smaller than the scale of a galaxy supercluster. The best that can be done with current computing techniques is to model the evolution of dark matter and gas in a cosmological volume with resolution comparable to a single galaxy. Subgalactic scales must then be regarded as subgrid scales and followed by means of phenomenological models based either on our current physical understanding or on observations. In the approach known as semi-analytic modelling (White & Frenk 1991), even the gas dynamics is treated phenomenologically using a simple, spherically symmetric model to describe the accretion and cooling of gas into dark matter halos. It turns out that this simple model works suprisingly well as judged by the good agreement with results of full N-body/gasdynamical simulations (Benson et al. 2001b, Helly et al. 2002, Yoshida et al. 2002). The main difficulty encountered in cosmological gas dynamical simulations arises from the need to suppress a cooling instability present in hierarchical clustering models like CDM. The building blocks of galaxies are small clumps that condense at early times. The gas that cools within them has very high density, reflecting the mean density of the Universe at that epoch. Since the cooling rate is proportional to the square of the gas density, in the absence of heat sources, most of the gas would cool in the highest levels of the mass hierarchy leaving no gas to power star formation today or even to provide the hot, X-ray emitting plasma detected in galaxy clusters. Known heat sources are photoionisation by early generations of stars and quasars and the injection of energy from supernovae and active galactic nuclei. These processes, which undoubtedly happened in our Universe, belong to the realm of subgrid physics which cosmological simulations cannot resolve. Different treatments of this feedback result in different amounts of cool gas and can lead to very different predictions for the properties of the galaxy population. This is a fundamental problem that afflicts cosmological simulations even when they are complemented by the inclusion of semi-analytic techniques. In this case, the resolution of the calculation can be extended to arbitrarily small mass halos, perhaps allowing a more realistic treatment of feedback. Although they are less general than full gasdynamical simulations, simulations in which the evolution of gas is treated semi-analytically make experimentation with different prescriptions relatively simple and efficient (Kauffmann, White & Guiderdoni 1993, Somerville & Primack 1999, Cole et al. 2000) Serious attempts to create galaxies from hydrodynamical simulations have been initiated in early 1990 s (Cen & Ostriker 1992; Katz, Hernquist & Weinberg 1992, Navarro et al. 2002), those simulated galaxies are still far from realistic.

53 7.5 Dark Baryons Dark Baryons The Lymanα Forest The Lyα forest represents the optically thin (at the Lyman edge) component of Quasar Ab- Figure 7.13: All quasars at high redshift exhibit huge numbers of narrow Absorption Systems. DLA: damped Lymanα lines, LLS: Lyman limit system. sorption Systems (QAS), a collection of absorption features in QSO spectra extending back to high redshifts (Fig. 7.13). QAS are effective probes of the matter distribution and the physical state of the Universe at early epochs when structures such as galaxies are still forming and evolving. Although stringent observational constraints have been placed on competing cosmological models at large scales by the COBE satellite and over the smaller scales of our local Universe by observations of galaxies and clusters, there remains sufficient flexibility in the cosmological parameters that no single model has been established conclusively. The relative lack of constraining observational data at the intermediate to high redshifts (0 < z < 6), where differences between competing cosmological models are more pronounced, suggests that QAS can potentially yield valuable and discriminating observational data. 7.6 Galaxy Clusters Clusters of galaxies are the largest gravitationally bound systems known to be in quasi equilibrium. This allows for reliable estimates to be made of their mass as well as their dynamical and thermal attributes. The richest clusters, arising from 3σ density fluctuations, can be as massive as solar masses, and the environment in these structures is composed of shock heated gas with temperatures of order degrees Kelvin which emits thermal bremsstrahlung and line radiation at X ray energies. Also, because of their spatial size 1/h Mpc and separations of order 50/h Mpc, they provide a measure of nonlinearity on scales close to the perturbation normalization scale 8/h Mpc. Observations of the substructure, distribution, luminosity, and evolution of galaxy clusters are therefore likely to provide signatures of the underlying cosmology of our Universe, and can be used as cosmological probes in the easily observable redshift range 0 > z > 2.

54 260 7 Evolution of Cosmic Structure Figure 7.14: All quasars at high redshift exhibit huge numbers of narrow absorption lines starting at the wavelength of the quasar s own Lyman alpha emission line and extending blueward. These are Lyman alpha absorption from foreground structures, in which the quasar light probes an otherwise invisible component of cosmic gas. This component evolves strongly with cosmic time, since we see dramatically more absorbers (be they clouds, filaments, or even crowding in velocity rather than in space) toward higher redshifts. Internal Structure Thomas et al. have investigated the internal structure of galaxy clusters formed in high resolution N body simulations of four different cosmological models, including standard, open, and flat but low density universes. They find that the structure of relaxed clusters is similar in the critical and low density universes, although the critical density models contain relatively more disordered clusters due to the freeze-out of fluctuations in open universes at late times. The profiles of relaxed clusters are very similar in the different simulations, since most clusters are in a quasi-equilibrium state inside the virial radius and follow the universal density profile of Navarro et al. [1997]. There does not appear to be a strong cosmological dependence in the

55 7.6 Galaxy Clusters 261 Figure 7.15: Distribution of the gas density at redshift z = 3 from a numerical hydrodynamics simulation of the Lyα forest. The simulation adopted a CDM spectrum of primordial density fluctuations, normalized to the second year COBE observations, a Hubble parameter of h = 0.5, a comoving box size of 9.6 Mpc, and baryonic density of composed of 76% hydrogen and 24% helium. The region shown is 2.4 Mpc (proper) on a side. The isosurfaces represent baryons at ten times the mean cosmic density (characteristic of typical filamentary structures) and are color coded to the gas temperature (red = K, yellow = K). The higher density contours trace out isolated spherical structures typically found at the intersections of the filaments. A single random slice through the cube is also shown, with the baryonic overdensity represented by a rainbow like color map changing from black (minimum) to red (maximum). The He + mass fraction is shown with a wire mesh in this same slice. Notice that there is fine structure everywhere. To emphasize fine structure in the minivoids, the mass fraction in the overdense regions has been rescaled by the gas overdensity wherever it exceeds unity. [Source: Peter Aninos]

56 262 7 Evolution of Cosmic Structure profiles as suggested by previous studies of clusters formed from pure power law initial density fluctuations. However, because more young and dynamically evolving clusters are found in critical density universes, Thomas et al. suggest that it may be possible to discriminate among the density parameters by looking for multiple cores in the substructure of the dynamic cluster population. They note that a statistical population of 20 clusters can distinguish between open and critically closed universes. Number Density Evolution The evolution of the number density of rich clusters of galaxies can be used to compute and Ω M and σ 8 (the power spectrum normalization on scales of 8/h Mpc) when numerical simulation results are combined with the constraint σ 8 ΩM 0.5, derived from observed present-day abundances of rich clusters. Bahcall et al. computed the evolution of the cluster mass function in five different cosmological model simulations and find that the number of high mass (Coma-like) clusters in flat, low models decreases dramatically by a factor of approximately 10 3 from z = 0 to z = 0.5. For low Ω M, high σ 8 models, the data results in a much slower decrease in the number density of clusters over the same redshift interval. Comparing these results to observations of rich clusters in the real Universe, which indicate only a slight evolution of cluster abundances to redshifts z 1, they conclude that critically closed standard CDM and Mixed Dark Matter (MDM) models are not consistent with the observed data. The models which best fit the data are the open models with low bias (Ω M = 0.3 ± 0.1 and σ 8 = 0.85 ± 0.5), and flat low density models with a cosmological constant (Ω M = 0.34 ± 0.13). X Ray Luminosity Function The evolution of the X-ray luminosity function, and the size and temperature distribution of rich clusters of galaxies are all potentially important discriminants of cosmological models. Bryan et al. investigated these properties in a high resolution numerical simulation of a standard CDM model normalized to COBE. Although the results are highly sensitive to grid resolution, their primary conclusion, that the standard CDM model predicts too many bright X-ray emitting clusters and too much integrated X-ray intensity, is robust since an increase in resolution will only exaggerate these problems. Evrard et al. (2002) The Cosmic Microwave Background Probes Linear Perturbations The cosmic microwave background (CMB) is the afterglow radiation left over from the hot Big Bang. Its temperature is extremely uniform all over the sky. However, tiny temperature variations or fluctuations (at the part per million level) can offer great insight into the origin, evolution, and content of the universe. Perturbations at that time were still in the linear regime - a fact which considerably simplifies the analysis. 3 Even in the early Universe, all material was not distributed entirely uniformly. Observations of the CMB suggest that slightly more dense regions of ionized Hydrogen gas and pho- 3 A rigorous treatment of the anisotropies of the CMB is given in [11], [12], [7] and [8].

57 7.7 The Cosmic Microwave Background Probes Linear Perturbations 263 tons existed, which gradually collapsed because of their own gravitation. At about 300,000 years of age, the Universe had cooled sufficiently so that the material became un-ionized, allowing electrons and protons to combine and form ordinary Hydrogen gas (a process referred to as recombination). This allowed the Universe to become transparent and photons were able to travel freely. It is at this point when condensed regions of gas and photons appear as bright spots when viewed through today s microwave telescopes. Measuring the size of these spots with Viper or CBI, for example, can reveal their distance from Earth. After releasing the photons, the cloud of un ionized Hydrogen gas continued to collapse to form galaxy clusters Anisotropy Mechanisms The evolution of first-order perturbations in the various energy density components and the metric are described with the following sets of equations: The photons and neutrinos are described by distribution functions f(t, x, p). A fundamental simplifying assumption is that the energy dependence of both is given by the blackbody distribution. The space dependence is generally Fourier transformed, so the distribution functions can be written as Θ(t, k, n), where the function has been normalized to the temperature of the blackbody distribution and n represents the direction in which the radiation propagates. The time evolution of each is given by the Boltzmann equation. For neutrinos, collisions are unimportant so the Boltzmann collision term on the right side is zero; for photons, Thomson scattering off electrons must be included. The dark matter and baryons are in principle described by Boltzmann equations as well, but a fluid description incorporating only the lowest two velocity moments of the distribution functions is adequate. Thus each is described by the Euler and continuity equations for their densities and velocities. The baryon Euler equation must include the coupling to photons via Thomson scattering. Metric perturbation evolution and the connection of the metric perturbations to the matter perturbations are both contained in the Einstein equations. This is where the subtleties arise. A general metric perturbation has 10 degrees of freedom, but four of these are unphysical gauge modes. The physical perturbations include two degrees of freedom constructed from scalar functions (the Bardeen potenitals Φ and Ψ), two from a vector, and two remaining tensor perturbations (see previous Sect.). Physically, the scalar perturbations correspond to gravitational potential and anisotropic stress perturbations; the vector perturbations correspond to vorticity and shear perturbations; and the tensor perturbations are two polarizations of gravitational radiation. Tensor and vector perturbations do not couple to matter evolving only under gravitation; in the absence of a stiff source of stress energy, like cosmic defects or magnetic fields, the tensor and vector perturbations decouple from the linear perturbations in the matter. A variety of different variable choices and methods for eliminating the gauge freedom have been developed. The subject can be fairly complicated. A detailed discussion and comparison between the Newtonian and synchronous gauges, along with a complete set of equations, has been given previously (also see Hu et al. [13]). An elegant and physically appealing formalism based on an entirely covariant and gauge invariant description of all physical quantities has been developed for the microwave background by Challinor and Lasenby (1999) and Gebbie et al. (2000), based on earlier work by Ehlers (1993) and Ellis and Bruni (1989).

58 264 7 Evolution of Cosmic Structure The Boltzmann equations are partial differential equations, 4 which can be converted to hierarchies of ordinary differential equations by expanding their directional dependence in Legendre polynomials. The result is a large set of coupled, first order linear ordinary differential equations which form a well-posed initial value problem. Initial conditions must be specified. Generally they are taken to be so-called adiabatic perturbations: initial curvature perturbations with equal fractional perturbations in each matter species. Such perturbations arise naturally from the simplest inflationary scenarios. Alternatively, isocurvature perturbations can also be considered: these initial conditions have fractional density perturbations in two or more matter species whose total spatial curvature perturbation cancels. Figure 7.16: Illustrating the physical mechanisms that cause CMB anisotropies. The shaded arc on the right represents the last scattering shell; an inhomogeneity on this shell affects the CMB through its potential, adiabatic and Doppler perturbations. Further perturbations are added along the line of sight by time-varying potentials (ISW) and by electron scattering from hot gas (Sunyaev-Zeldovich effect). The density field at last scattering can be Fourier analysed into modes of wavevector k. These spatial perturbation modes have a contribution that is in general damped by averaging over the shell of last scattering. Short wavelength modes are more heavily affected (i) because more of them fit inside the scattering shell, and (ii) because their wavevectors point more nearly radially for a given projected wavelength. [Illustration by Peacock] 4 The Boltzmann equation for a phase space distribution is of the form, p = ɛ n for photons, d dτ f(τ, x, p) = τ f + ( n )f Γ i αβ pα p β f = C[f], (7.202) pi where C[f] is the collision integral (given by Thomson scattering for photons). By using the geodesics equations, this can also be written in the form τ f + ( n )f + ɛ f ɛ + f ṅi n i = C 1 C 2. (7.203) C 1 is the amount of photons scattered into the beam of radiation travelling in direction n and C 2 is the amount of photons scattered out of the beam.

59 7.7 The Cosmic Microwave Background Probes Linear Perturbations 265 Tight Coupling Two basic time scales enter into the evolution of the microwave background. The first is the photon scattering time scale t T = 1/(σ T n e c) = s (T/eV ) 3 (x e Ω B h 2 ) 1, (7.204) the mean time between Thomson scatterings. The other is the expansion time scale of the universe, 1/H, where H is the Hubble parameter. At temperatures significantly greater than 0.5 ev, hydrogen and helium are completely ionized and t T 1/H. The Thomson scatterings which couple the electrons and photons occur much more rapidly than the expansion of the universe; as a result, the baryons and photons behave as a single tightly coupled fluid. During this period, the fluctuations in the photons mirror the fluctuations in the baryons. (Remember that recombination occurs at around 0.5 ev rather than 13.6 ev because of the huge photonbaryon ratio; the universe contains somewhere around 10 9 photons for each baryon, as we know from primordial nucleosynthesis.) The photon distribution function for scalar perturbations can be written as Θ(τ, k, µ), where µ = k n, and the scalar character of the fluctuations guarantees the distribution cannot have any azimuthal directional dependence. (The azimuthal dependence for vector and tensor perturbations can also be included in a similar decomposition). The moments of the distribution are defined as Θ(τ, k, µ) = ( i) l Θ l (τ, k)p l (µ). (7.205) l=0 Tight coupling implies that Θ l = 0 for l > 1. Physically, the l = 0 moment corresponds to the photon energy density perturbation, while l = 1 corresponds to the bulk velocity. During tight coupling, these two moments must match the baryon density and velocity perturbations. Any higher moments rapidly decay due to the isotropizing effect of Thomson scattering; this follows immediately from the photon Boltzmann equation with a source from Thomson scattering [13], [12] Θ + iµ(θ + Ψ) = Φ + ṫ T [Θ 0 Θ 1 10 Θ 2P 2 (µ) iµv B ]. (7.206) ṫ T = σ T x e n e (a/a 0 ) is the differential optical depth to Thomson scattering, and the scale factor a/a 0 = 1/(1 + z). The appearance of the photon quadrupole Θ 2 represents the angular dependence of the Thomson scattering. 5 The gravitational potentials Φ and Ψ have two effects on the temperature fluctuations, both introduced already in the paper by Sachs and Wolfe in The gradient of the Newtonian 5 For a derivation using the perturbed Boltzmann equation, see e.g. Durrer [7]. The equation can be derived from the following arguments: The intensity distribution is expected to be a Planckian with redshifted temperature T = (T 0 /a)(1 + Θ), where Θ denotes the fractional perturbation in temperature of the form Θ(τ, x, n). The distribution function for the photons is f = f P (aɛ/(1 + Θ)). The total variation is given by df dτ = aɛ [ d ln(aɛ) 1 + Θ f P dτ The time derivative of Θ follows from dθ dτ = τ Θ + d xi dτ dθ dτ Θ + Θ ɛ ] = df dτ ɛ + d ni dτ. (7.207) c nθ, (7.208)

60 266 7 Evolution of Cosmic Structure potential induces a gravitational redshift on the photons as they travel through the potential well. Since the potential difference merely induces a fractional temperature shift, the combination Θ + Ψ is the resultant temperature perturbation after the photon climbs out of the potential with negative value for Ψ. This is called the ordinary Sachs Wolfe effect. The time dependence of the metric term Φ causes its own time dilation effect known as the integrated Sachs Wolfe (ISW) effect. The baryons evolve under the continuity and Euler equations B = k(v B Θ 1 ) γ (7.212) V B = HV B + kψ + ṫ T (Θ 1 V B )/R. (7.213) The Newtonian potential acts as a source for the velocity evolution through infall and gives rise to the adiabatic growth of perturbations. Free Streaming In the other regime, for temperatures significantly lower than 0.5 ev, t T 1/H and photons on average never scatter again until the present time. This is known as the free streaming epoch. Since the radiation is no longer tightly coupled to the electrons, all higher moments in the radiation field develop as the photons propagate. In a flat background spacetime, the exact solution is simple to derive [11] (Θ + Ψ)(τ 0, µ) = τ0 τ { [Θ + Ψ iµv B ]ṫ T + Φ + Ψ } exp[ τ T (τ, τ 0 )] exp[ikµ(τ τ 0 )] dτ. (7.214) The optical depth is measured from the moment τ to the present epoch τ 0, τ T (τ, τ 0 ) = τ0 ṫ τ T dτ. We have dropped the quadrupole term, since it vanishes in the tight coupling limit. The combination ṫ T exp( τ) is called the conformal time visibility function, it is the probability that a photon scattered within dτ of τ. It has a sharp peak at the last scattering epoch. After scattering ceases, the photons evolve according to the Liouville equation Θ + ikµθ = 0 (7.215) where the last two terms vanish. For a metric given by Bardeen potentials, the geodesic equations give d ln(aɛ) dτ = Ψ + Φ. (7.209) Usin gthis expression in the Boltzmann equation, we get the evolution equation for temperature fluctuations τ Θ + ( n )Θ = Ψ + Φ + The collisional term can be expressed in terms of the Thomson scattering rate as dθ dτ c dθ dτ. (7.210) c = an e σ T [ Θ Rad + n V e ]. (7.211) The first two terms represent absorption and reemission of radiation. The third term is due to the Doppler effect from the velocity fields of the electrons.

61 7.7 The Cosmic Microwave Background Probes Linear Perturbations 267 with the trivial solution 6 Θ(τ, k, µ) = exp( ikµ(τ τ ) Θ(τ, k, µ). (7.218) τ is the conformal time when free streaming begins. Taking moments on both sides Θ l (τ, [ k) = (2l + 1) Θ 0 (τ, k)j l (kτ kτ ) + Θ 1 (τ, ] k)j l(kτ kτ ), (7.219) with j l as the spherical Bessel functions. The process of free streaming essentially maps spatial variations in the photon distribution at the last scattering surface (wavenumber k) into angular variations on the sky today (moment l). A more accurate expression follows from the solution (7.214) [13], l 2, Θ l (τ 0, k) [Θ 0 + Ψ](τ ) (2l + 1)j l (k(τ 0 τ )) + Θ 1 (τ ) [lj l 1 (k(τ 0 τ )) (l + 1)j l+1 (k(τ 0 τ ))] + (2l + 1) τ0 τ [ Ψ + Φ]j l (k(τ 0 τ)) dτ. (7.220) This shows how the integrated Sachs Wolfe effect contributes to the temperature fluctuations. Diffusion Damping In the intermediate regime during recombination, t T 1/H, photons propagate a characteristic distance L D during this time. Since some scattering is still occurring, baryons experience a drag from the photons as long as the ionization fraction is appreciable. A second order perturbation analysis shows that the result is damping of baryon fluctuations on scales below L D, known as Silk damping or diffusion damping. This effect can be modelled by the replacement Θ 0 (τ, k) Θ 0 (τ, k) exp[ (k/k D ) 2 ]. (7.221) The Resulting CMB Power Spectrum The fluctuations in the universe are assumed to arise from some random statistical process. We are not interested in the exact pattern of fluctuations we see from our vantage point, since this is only a single realization of the process. Rather, a theory of cosmology predicts an underlying distribution, of which our visible sky is a single statistical realization. The most basic statistic describing fluctuations is their power spectrum. A temperature map on the sky T ( n) is conventionally expanded in spherical harmonics, T ( n) T 0 1 = a lm Ym( n) l (7.222) l=1 6 Remember from quantum mechanics the identity with exp(iµτ) = (2l + 1)i l j l (kτ) P l (µ) (7.216) l=0 (2l + 1)j l = lj l 1 (l + 1)j l+1. (7.217)

62 268 7 Evolution of Cosmic Structure where a lm = 1 T 0 dωt ( n)y lm( n) (7.223) are the temperature multipole coefficients and T 0 is the mean CMB temperature. The l = 1 term is indistinguishable from the kinematic dipole and is normally ignored. The temperature angular power spectrum C l is then given by < a lma l m >= C l δ ll δ mm. (7.224) where the angled brackets represent an average over statistical realizations of the underlying distribution. The parameters C l determine the correlation function for temperature fluctuations < T T ( n) T T ( n ) > = < a lm a l m > Y lm( n) Y l m ( n ) ll mm = l C l Y lm ( n)y lm( n ) = 1 (2l + 1)C l P l (µ = n n ). (7.225) 4π l m= l l Here we have used the addition theorem for spherical harmonics. Since we have only a single sky to observe, an unbiased estimator of Ĉl is constructed as Ĉ l = 1 2l + 1 m=l m= l a lma lm. (7.226) The statistical uncertainty in estimating C l by a sum of 2l + 1 terms is known as cosmic variance. The constraints l = l and m = m follow from the assumption of statistical isotropy: C l must be independent of the orientation of the coordinate system used for the harmonic expansion. These conditions can be verified via an explicit rotation of the coordinate system. The measurements of the parameters C l is one of the most important issue in observational cosmology. The results prior to WMAP are shown in Fig The data obtained with WMAP in the first year (Fig. 7.21) demonstrate the superb power of this instrument. Integrating over all k modes in the perturbation we get the following equality 2l + 1 4π C l = V 2π 2 dk k k 3 Θ l (τ 0, k) 2. (7.227) 2l + 1 A given cosmological theory will predict C l as a function of l, which can be obtained from evolving the temperature distribution function as described above. This prediction can then be compared with data from measured temperature differences on the sky. Figure 7.20 shows a typical temperature power spectrum from the inflationary class of models. The distinctive sequence of peaks arise from coherent acoustic oscillations in the fluid during the tight coupling epoch and are of great importance in precision tests of cosmological models. The effect of diffusion damping is clearly visible in the decreasing power above l = When viewing angular power spectrum plots in multipole space, keep in mind that l = 200 corresponds approximately to fluctuations on angular scales of one degree, and the angular scale is inversely proportional to l. The vertical axis is conventionally plotted as l(l + 1)C l, because the Sachs Wolfe temperature fluctuations from a scale invariant spectrum of density perturbations appears as a horizontal line on such a plot, l(l + 1) Cl SW const (see Exercise 4).

63 7.7 The Cosmic Microwave Background Probes Linear Perturbations 269 Figure 7.17: CMB power spectrum prior to WMAP. [Source: Tegmark] More on Acoustic Oscillations Before decoupling, the matter in the universe has significant pressure because it is tightly coupled to radiation. This pressure counteracts any tendency for matter to collapse gravitationally. Formally, the Jeans mass is greater than the mass within a horizon volume for times earlier than decoupling. During this epoch, density perturbations will set up standing acoustic waves in the plasma. Under certain conditions, these waves leave a distinctive imprint on the power spectrum of the microwave background, which in turn provides the basis for precision constraints on cosmological parameters. This section reviews the basics of the acoustic oscillations.

64 270 7 Evolution of Cosmic Structure The Oscillator Equation In their classic 1996 paper, Hu and Sugiyama transformed the basic equations describing the evolution of perturbations into an oscillator equation (see also discussion in Sect. /.2.1). Combining the zeroth moment of the photon Boltzmann equation with the baryon Euler equation for a given k-mode in the tight coupling approximation (mean baryon velocity equals mean radiation velocity) gives (in Newtonian gauge upto first order in ṫ T ) Θ 0 + Ṙ 1 + R Θ 0 + kc 2 SΘ 0 = Φ Ṙ R Φ 1 3 k2 Ψ F (τ), (7.228) where Θ 0 is the zeroth moment of the temperature distribution function (proportional to the Figure 7.18: Evolution of temperature fluctations before recombination. The tight coupling limit is compared to a full numerical solution for the monopole and dipole component. photon density perturbation), R = 3ρ B /4ρ is proportional to the scale factor a, H = ȧ/a is

65 7.7 The Cosmic Microwave Background Probes Linear Perturbations 271 the conformal Hubble parameter, and the sound speed is given by c 2 S = 1/(3 + 3R). (All overdots are derivatives with respect to conformal time.) Ψ and Φ are the scalar metric perturbations in the Newtonian gauge; if we neglect the anisotropic stress, which is generally small in conventional cosmological scenarios, then Φ = Ψ. But the details are not very important. The equation represents damped, driven oscillations of the radiation density, and the various physical effects are easily identified. The second term on the left side is the damping of oscillations due to the expansion of the universe. The third term on the left side is the restoring force due to the pressure, since c 2 S = dp/dρ. On the right side, the first two terms depend on the time variation of the gravitational potentials, so these two are the source of the Integrated Sachs-Wolfe effect. The final term on the right side is the driving term due to the gravitational potential perturbations. As Hu and Sugiyama emphasized, these damped, driven acoustic oscillations account for all of the structure in the microwave background power spectrum. This is simply the equation of a forced damped oscillator. The homogeneous equation can be solved by the WKB method in the limit where the frequency is slowly varying. The solutions are oscillatory functions with phases kr S. The particular solution is then found by Figure 7.19: Temperature fluctuation spectrum at recombination. Fluctuations on the last scattering surface free stream to the observer creating thereby anisotropies. The phase relation between the monopole and the dipole determine the Doppler peak structure. The dipole is significantly smaller than the monopole. At large scales, the Sachs Wolf effect dominates. At small scales, the oscillations are damped by the Silk damping.

66 272 7 Evolution of Cosmic Structure the Green s method [ ] 1/4 1 + R(τ) Θ0 (τ) = 3 + k τ 0 where r S (τ) is the sound horizon r S (τ) = Θ 0 (0) cos[kr S (τ)] + 3 k [ Θ0 (0) + 1 4ṘΘ 0(0)] sin[kr S (τ)] [1 + R(τ )] 3/4 sin[kr S (τ) kr S (τ )] F (τ ) dτ, (7.229) = 2 3 τ 0 c S (τ ) dτ = [ ] R + R + Req log R eq 1 +. (7.230) R eq k eq R eq is the value of R at z eq Ω M h 2 and k eq (14 Mpc) 1 Ω M h 2. The dipole solution can be obtained from the monopole by means of kθ 1 = 3(Θ 0 Φ). The dipole oscillates π/2 out of phase with the monopole. The cos(kr S ) solution is the dominant adiabatic contribution. Therefore, the peaks in the power spectrum will be located at the scale k m which satisfies k m r S (τ ) = mπ where m is an integer > 1. This locates the first Doppler peak roughly at the sound horizon, which is close to, but conceptually distinct from the Jeans scale (causal horizon). The last significant feature in the power spectrum is the diffusion cutoff. The current data [21] are contrasted with some CDM models in Fig The key feature that is picked out is the peak at l 220, together with harmonics of this scale at higher l. Beyond l 1000, the spectrum is clearly damped, in a manner consistent with the expected effects of photons diffusing away from baryons (Silk damping), plus smearing of modes with wavelength comparable to the thickness of the last scattering shell. This last effect arises because recombination is not instantaneous, so the redshift of last scattering shows a scatter around the mean, with a thickness corresponding to approximately σ r = 7(Ω m h 2 ) 1/2 Mpc. On scales larger than this, we see essentially an instantaneous imprint of the pattern of potential perturbations and the acoustic baryon/photon oscillations. The significance of the main acoustic peak scale is that it picks out the (sound) horizon at last scattering. The redshift of last scattering is almost independent of cosmological parameters at z LS If we assume that the universe is matter dominated at last scattering, the horizon size is D LS H = 184/ Ω m h 2 Mpc. (7.231) The angle this subtends is discussed in Sect The above Figure shows that heavily open universes yield a main CMB peak at scales much smaller than the observed l 220, and these can be ruled out. Indeed, open models were disfavoured for this reason long before any useful data existed near the peak, simply because of strict upper limits at l However, once a non-zero vacuum energy is allowed, the story becomes more complicated, and it turns out that large degrees of spatial curvature cannot be excluded using the CMB alone. Evolution of CMB Data The pace of progress in CMB experiments has maintained an astonishing rate for a decade. Following the 1992 COBE detection of fluctuations, 5 years of effort yielded the unclear

67 7.7 The Cosmic Microwave Background Probes Linear Perturbations 273 Figure 7.20: CMB spectrum. At largest scales (l < 30), the monopole from the ordinary Sachs Wolf effect dominates. The monopole is clearly important for the overall peak structure. Diffusion damping significantly reduces fluctuations beyond the first peak and cuts the anisotropies beyond l > picture of the first panel in Fig. 7.22, in which of order 10 experiments gave only vague evidence for a peak in l 2 C l. By the year 2000, this had been transformed to a clear picture of a peak at l 200, although there was no model independent evidence for higher harmonics. The present situation is much more satisfactory, with 3 peaks established in a way that does not require any knowledge of the CDM model. The WMAP results (Spergel et al. 2003) measure the power spectrum about as well as possible (i.e. hitting the limit of cosmic variance from a finite sky) up to the second peak. At smaller scales, however, there is still much scope for improvement, and the rate of advance is unlikely to drop in the future ( whu/cmbex.html lists 14 ongoing experiments). Polarisation in CMB Thomson scattering can produce polarisation provided the incident radiation is not isotropic, induced e.g. by velocity gradients in the baryon photon fluid. Before recombination, successive scatterings destroy the build up of any polarisation. One therefore expects a small degree of polarisation created a recombination, partially correlated with the temperature anisotropies. Conveniently, the polarisation field is decomposed into two scalar fields denoted by E and B (in analogy to electromagnetic fields). The power spectrum of the E part is expected to be about ten times smaller than for the temperature field Θ, the B-part is only generated by ten-

68 274 7 Evolution of Cosmic Structure Figure 7.21: CMB power spectrum for two different world models compared to the first year WMAP data. In an open world model the first acoustic peak would occur at smaller angular separations. sor fields. The WMAP team did not yet release a measurement of the EE power spectrum, but did a measurement of the Θ E cross power spectrum. This quantifies the expected correlation of the temperature and E field. These Θ E spectra turned out to be unexpectly large at low l values. This gives strong evidence that the optical depth to the last scattering surface is rather important, τ es Based on these observations, it has been suggested that reionisation happened rather early, at around redshift 17. Future: WMAP vs. PLANCK The WMAP and PLANCK experiments have the following similarities and differences: Both map the full sky, from an orbit around the Lagrangian point L2 of the Sun Earth system, to minimise secondary radiation from the Earth. Both are based on the use of off axis Gregorian telescopes in the 1.5 m class. Both aim at making polarisation measurements. The American WMAP has been designed for rapid implementation and is based on fully explored techniques. Its observational strategy uses differential scheme. Two telescopes

69 7.7 The Cosmic Microwave Background Probes Linear Perturbations 275 Figure 7.22: Dramatic change took place in CMB power spectrum measurements around the turn of the 21st century. Although some rise from the COBE level was arguably known even by 1997, a clear peak around l 200 only became established in 2000, whereas by 2003 definitive measurements of the spectrum at l 800, limited mainly by cosmic variance, had been made by the WMAP satellite.