3136cosmology 2 - PHAS 3136 Cosmology Part 2: The Perturbed Universe PDF

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PHAS 3136 Cosmology Part 2: The Perturbed Universe...

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Cosmology Part II: The Perturbed Universe Hiranya V. Peiris1, ∗ 1

Department of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT, U.K.

Classes 13/1 20/1 27/1 03/2 10/2 17/2 24/2 03/3 10/3 17/3 24/3

Thursdays, Spring Term Cruciform B.3.01 10.00 − 13.00 Physics E1 10.00 − 13.00 Physics E7 10.00 − 13.00 Physics E1 10.00 − 13.00 Cruciform B.3.01 09.00 − 12.00 (note different time) NO LECTURE Physics E1 10.00 − 13.00 Physics E1 10.00 − 13.00 Physics E7 10.00 − 13.00 Physics E1 10.00 − 13.00 Physics E1 10.00 − 13.00 Office G04, Kathleen Lonsdale Building. Course Website

http://zuserver2.star.ucl.ac.uk/∼hiranya/PHASM336/

Introductory Reading: 1. Liddle, A. An Introduction to Modern Cosmology. Wiley (2003) Complementary Reading 1. Dodelson, S. Modern Cosmology. Academic Press (2003) ∗ 2. Carroll, S.M. Spacetime and Geometry. Addison-Wesley (2004) ∗ 3. Liddle, A.R. and Lyth, D.H. Cosmological Inflation and Large-Scale Structure. Cambridge (2000) 4. Kolb, E.W. and Turner, M.S. The Early Universe. Addison-Wesley (1990) 5. Weinberg, S. Gravitation and Cosmology. Wiley (1972) 6. Peacock, J.A. Cosmological Physics. Cambridge (2000) 7. Mukhanov, V. Physical Foundations of Cosmology. Cambridge (2005) Books denoted with a ∗ are particularly recommended for this course. Acknowledgements These notes borrow with gratitude from excellent notes by (in no particular order) Richard Battye, Anthony Challinor and Wayne Hu. It most closely parallels the treatment found in Dodelson. I am supported by STFC, the European Commission, and the Leverhulme Trust. Errata Any errata contained in the following notes are solely my own. Reports of any typos or unclear explanations in the notes will be gratefully received at the email address below. The notes are evolving, and the most up-to-date version at any given time will be found on the website above.

∗ Electronic

address: [email protected]

2 Contents I. LARGE SCALE STRUCTURE FORMATION A. Overview of structure formation II. STATISTICS OF RANDOM FIELDS A. Random fields in 3D Euclidean space B. Gaussian random fields C. Random fields on the sphere

3 3 4 4 6 6

III. NEWTONIAN STRUCTURE FORMATION A. Background cosmology B. Comoving coordinates C. Perturbation analysis D. Jeans’ length E. Applications to cold dark matter 1. Solutions in an Einstein-de Sitter phase 2. The Meszaros effect 3. Late-time suppression of structure formation by Λ 4. Evolution of baryon fluctuations after decoupling

8 8 9 9 10 11 11 12 12 13

IV. RELATIVISTIC STRUCTURE FORMATION

14

V. INFLATION AND THE ORIGIN OF STRUCTURE A. Schematic overview of origin of structure in the inflationary paradigm B. Quantizing the harmonic oscillator C. Tensor perturbations D. Scalar perturbations E. Slow-roll expansion F. Spectral index of the primordial power spectrum G. Observable predictions and current observational constraints VI. THE COSMIC MICROWAVE BACKGROUND VII. THE MATTER POWER SPECTRUM

14 14 14 15 18 20 20 22 23 24

3 I. LARGE SCALE STRUCTURE FORMATION

The real universe is far from homogeneous and isotropic except on the largest scales. Figure 1 shows slices through the 3D distribution of galaxy positions from the 2dF galaxy redshift survey out to a comoving distance of 600 Mpc. The distribution of galaxies is clearly not random; instead they are arranged into a delicate cosmic web with galaxies strung out along dense filaments and clustering at their intersections leaving huge empty voids. However, if we smooth the picture on large scales (∼ 100 Mpc) it starts to look much more homogeneous. Furthermore, we know from the CMB that the universe was smooth to around 1 part in 105 at the time of recombination; see Fig. 2. The aim of this part of the course is to study the growth of large-scale structure in an expanding universe through gravitational instability acting on small initial perturbations. We shall then learn how these initial perturbations were likely produced by quantum effects during cosmological inflation.

FIG. 1 Slices through the 3D map of galaxy positions from the 2dF galaxy redshift survey. Note that redshift 0.15 is at a comoving distance of 600 Mpc. Figure credit: 2dF.

FIG. 2 Fluctuations in the CMB temperature, as determined from five years of WMAP data, about the average temperature of 2.725 K. The fluctuations are at the level of only a few parts in 105 . Credit: WMAP science team.

A. Overview of structure formation

We will compute Pi (k), the initial power spectrum of density fluctuations, e.g. from inflation. The aim of this section is to understand how this initial spectrum is processed by the evolution of the universe, using linear perturbation theory. This processing is often quantified in terms of the transfer function: δk (t0 ) = T (k)δk (ti ) ⇒ P (k) = T 2 (k)Pi (k) .

(2.1.1)

4 II. STATISTICS OF RANDOM FIELDS

READING: This section, which will not be covered during class, gives the precise mathematical definition of some key concepts: the power spectrum, correlation function and angular power spectrum, that you will need later in the course. These derivations are NON-EXAMINABLE, although of course the use of the physical concepts specified above is examinable. You are invited to read the following material to deepen your understanding of the subsequent material. When we come to the relevant concepts later in the course, you will be expected to have a grasp of where they came from and what they mean.

Theory (e.g. quantum mechanics during inflation) only allows us to predict the statistical properties of cosmological fields (such as the matter overdensity δρ). Here, we explore the basic statistical properties enforced on such fields by assuming the physics that generates the initial fluctuations, and subsequently processes them, respects the symmetries of the background cosmology, i.e. isotropy and homogeneity. Throughout, we denote expectation values with angle brackets, e.g. hδρi; you should think of this as a quantum expectation value or an average over a classical ensemble1 . To keep the Fourier analysis simple, we shall only consider flat (K = 0) background models and we denote comoving spatial positions by x.

A. Random fields in 3D Euclidean space

Consider a random field f (x) – i.e. at each point f (x) is some random number – with zero mean, hf (x)i = 0. The probability of realising some field configuration is a functional Pr[f (x)]. Correlators of fields are expectation values of products of fields at different spatial points (and, generally, times). The two point correlator is Z ξ(x, y) ≡ hf (x)f (y)i = Df Pr[f ]f (x)f (y) , (2.2.1) where the integral is a functional integral (or path integral) over field configurations. Statistical homogeneity means that the statistical properties of the translated field, Tˆa f (x) ≡ f (x − a) ,

(2.2.2)

ˆ a f (x)]. For the two-point correlation, this means that are the same as the original field, i.e. Pr[f (x)] = Pr[ T ⇒

ξ (x, y) = ξ (x − a, y − a) ∀a ξ (x, y) = ξ (x − y) ,

(2.2.3)

so the two-point correlator only depends on the separation of the two points. Statistical isotropy mean that the statistical properties of the rotated field, ˆ (x) ≡ f (R−1 x) , Rf

(2.2.4)

ˆ (x)]. For the two-point correlator, where R is a rotation matrix, are the same as the original field, i.e. Pr[f (x)] = Pr[ Rf we must have ξ(x, y) = ξ(R−1 x, R−1 y) ∀R .

(2.2.5)

Combining statistical homogeneity and isotropy gives

⇒

1

  ξ(x, y) = ξ R−1 (x − y) ξ(x, y) = ξ(|x − y|) ,

∀R

(2.2.6)

For a recent review on the question of why quantum fluctuations from inflation can be treated as classical, see Keifer & Polarski (2008), available online at http://arxiv.org/abs/0810.0087.

5 so the two-point correlator depends only on the distance between the two points. Note that this holds even if correlating fields at different times, or correlating different fields. We can repeat these arguments to constrain the form of the correlators in Fourier space. We adopt the symmetric Fourier convention, so that Z Z d3 x d3 k −ik·x f (k)eik·x . (2.2.7) f (k) = f ( x)e and f ( x) = (2π)3/2 (2π)3/2 Note that for real fields, f (k) = f ∗ (−k). Under translations, the Fourier transform acquires a phase factor: Z d3 x Tˆa f (k) = f (x − a)e−ik·x (2π)3/2 Z ′ d 3 x′ f (x′ )e−ik·x e−ik·a = 3/2 (2π) = f (k)e−ik·a .

(2.2.8)

Invariance of the two-point correlator in Fourier space is then ⇒

hf (k)f ∗ (k′ )i = hf (k)f ∗ (k′ )ie−i(k−k hf (k)f ∗ (k′ )i = F (k)δ (k − k′ ) ,

′

)·a

∀a

for some (real) function F (k). We see that different Fourier modes are uncorrelated. Under rotations, Z d3 x ˆ f (R−1 x)e−ik·x Rf (k) = (2π)3/2 Z d3 x −1 −1 = f (R−1 x)e−i(R k)·(R x) (2π)3/2 = f (R−1 k) ,

(2.2.9)

(2.2.10)

so, additionally demanding invariance of the two-point correlator under rotations implies ˆ (k)[Rf ˆ (k′ )]∗ i = hf (R−1 k)f ∗ (R−1 k′ )i = F (R−1 k)δ(k − k′ ) = F (k)δ(k − k′ ) ∀R . hRf

(2.2.11)

−1

(We have used δ(R k) = detRδ(k) = δ(k) here.) This is only possible if F (k) = F (k) where k ≡ |k|. We can therefore define the power spectrum, Pf (k), of a homogeneous and isotropic field, f (x), by hf (k)f ∗ (k′ )i =

2π 2 Pf (k)δ (k − k′ ) . k3

(2.2.12)

The normalisation factor 2π 2 /k 3 in the definition of the power spectrum is conventional and has the virtue of making Pf (k) dimensionless if f (x) is. The correlation function is the Fourier transform of the power spectrum: Z ′ d3 k d 3 k′ hf (k)f ∗ (k′ )i eik·x e−ik ·y hf (x)f (y)i = {z } (2π)3/2 (2π)3/2 | 2π2 k3

=

1 4π

Z

dk Pf (k) k

Z

Pf (k)δ(k −k ′ )

dΩk eik·(x−y) .

(2.2.13)

We can evaluate the angular integral by taking x − y along the z-axis in Fourier space. Setting k · (x − y) = k|x − y|µ, the integral reduces to Z 1 dµ eik|x−y|µ = 4πj0 (k|x − y|) , (2.2.14) 2π −1

where j0 (x) = sin(x)/x is a spherical Bessel function of order zero. It follows that Z dk ξ(x, y) = Pf (k )j0 (k |x − y|) . k

(2.2.15)

Note that this only depends on |x − y|R as required by Eq. (2.2.6). The variance of the field is ξ(0) = d ln k Pf (k). A scale-invariant spectrum has P(k) = const. and its variance receives equal contributions from every decade in k .

6 B. Gaussian random fields

For a Gaussian (homogeneous and isotropic) random field, Pr[f (x)] is a Gaussian functional of f (x). If we think of discretising the field in N pixels, so it is represented by a N -dimensional vector f = [f (x1 ), f (x2 ), . . . , f (xN )]T , the probability density function for f is a multi-variate Gaussian fully specified by the correlation function hfi fj i = ξ(|xi − xj |) ≡ ξij ,

(2.2.16)

where fi ≡ f (xi ), so that −1

e−fi ξij fj . Pr(f ) ∝ p det(ξij )

(2.2.17)

Since f (k) is linear in f (x), the probability distribution for f (k) is also a multi-variate Gaussian. Since different Fourier modes are uncorrelated (see Eq. 2.2.9), they are statistically independent for Gaussian fields. Inflation predicts fluctuations that are very nearly Gaussian and this property is preserved by linear evolution. The cosmic microwave background probes fluctuations mostly in the linear regime and so the fluctuations look very Gaussian (see Fig. 2). Non-linear structure formation at late times destroys Gaussianity and gives the filamentary cosmic web (see Fig. 1). Searching for primordial non-Gaussianity to probe departures from simple inflation is a very hot topic but no convincing evidence for primordial non-Gaussianity has yet been found. C. Random fields on the sphere

Spherical harmonics form a basis for (square-integrable) functions on the sphere: f (ˆ n) =

∞ l X X

flm Ylm (ˆ n) .

(2.2.18)

l=0 m=−l

The Ylm are familiar from quantum mechanics as the position-space representation of the eigenstates of ˆL2 = −∇2 ˆ = −i∂ : and L z φ ∇2 Ylm = −l(l + 1)Ylm ∂φ Ylm = imYlm ,

(2.2.19)

with l an integer ≥ 0 and m an integer with |m| ≤ l. The spherical harmonics are orthonormal over the sphere, Z n)Yl∗′ m′ (ˆ n) = δll′ δmm′ , (2.2.20) d ˆn Ylm (ˆ so that the spherical multipole coefficients of f (ˆ n) are Z ∗ n)Y lm flm = d n ˆ f (ˆ (ˆ n) .

(2.2.21)

∗ m ∗ = (−1)m Y There are various phase conventions for the Ylm ; here we adopt Y lm l −m so that f lm = (−1) fl −m for a real field. What is the implication of statistical isotropy for the correlators of flm ? For the two-point correlator, it turns out that we must have2

hflm fl∗′ m′ i = Cl δll′ δmm′ ,

2

(2.2.22)

A plausibility argument is as follows. Under rotations, the subset of the Ylm with a given l (so 2l + 1 elements) transforms irreducibly so the δll′ form of the correlator is preserved under rotation. For rotation through γ about the z-axis, Ylm (θ, φ) → Ylm (θ, φ − γ) = e−imγ Ylm (θ, φ)

⇒

Under rotations, ′

hflm fl∗′ m′ i → e−imγ eim γ hflm fl∗′ m′ i , so invariance requires the correlator be ∝ δmm′ .

flm → e−imγ flm .

7 where Cl is the angular power spectrum of f . What does this imply for the two-point correlation function? We have XX hflm fl∗′ m′ iYlm (ˆn)Yl∗′ m′ (ˆ n′ ) n)f (ˆ n′ )i = hf (ˆ | {z } l′ m′ lm

=

X

Cl δll′ δmm′

Cl

X

∗ n)Ylm n′ ) = C(θ) , Ylm (ˆ (ˆ

(2.2.23)

m

l

|

2l+1 4π

{z

n·ˆ Pl (ˆ n′ )

}

where n ˆ·n ˆ ′ = cos θ and we used the addition theorem for spherical harmonics to express the sum of products of the Ylm in terms of the Legendre polynomials Pl (x). It follows that the two-point correlation function depends only on the angle between the two points, as required by statistical isotropy. Note that the variance of the field is Z X 2l + 1 l(l + 1)Cl C(0) = Cl ≈ d ln l . (2.2.24) 4π 2π l

It is conventional to plot l(l + 1)Cl /(2π) which we see is the contribution to the variance per log range in l. Finally, we note that we can invert the correlation function to get the power spectrum by using orthogonality of the Legendre polynomials: Cl = 2π

Z

1

d cos θ C(θ)Pl (cos θ) . −1

(2.2.25)

8 III. NEWTONIAN STRUCTURE FORMATION

Newtonian gravity is an adequate approximation of general relativity in cosmology on scales well inside the Hubble radius and when describing non-relativistic matter (for which the pressure P is much less than the energy density ρ). Newtonian gravity underlies all cosmological N -body simulations of the non-linear growth of structure and is much more intuitive than the full linearised treatment of general relativity (to be introduced later). In particular, in cosmology we can use the Newtonian treatment to describe sub-Hubble fluctuations in the cold dark matter (CDM) and baryons after decoupling. Consider an ideal, self-gravitating non-relativistic fluid with density (for this section only, the mass density which, given our assumptions is essentially the total energy density) ρ, pressure P ≪ ρ and velocity u. Denote the usual Newtonian position vector by r and time by t. The equations of motion of the fluid are as follows: Continuity

∂t ρ + ∇r · (ρu) = 0

1 ∂t u + u · ∇r u = − ∇r P − ∇r Φ ρ 2 ∇ r Φ = 4πGρ ,

Euler Poisson

(2.3.1) (2.3.2) (2.3.3)

where the gravitational potential Φ determines the gravitational acceleration by g = −∇r Φ. We can fudge the Poisson equation to get the correct Friedmann equations (see later) including the cosmological constant Λ by taking ∇2r Φ = 4πGρ − Λ .

(2.3.4)

A. Background cosmology

To recover the background dynamics (described by the Friedmann equations), we consider a uniform expanding ball of fluid satisfying Hubble’s law u = H (t)r. (Note the velocity goes to the speed of light at the Hubble radius!) This was covered in the third year cosmology course, but we include it again here from the perspective of the fluid equations. Taking Φ = 0 at r = 0, the Poisson equation (2.3.4) integrates as   ∂ 2 ∂Φ = (4πGρ − Λ)r2 r ∂r ∂r 1 ∂Φ = (4πGρ − Λ)r ⇒ ∂r 3 1 (2.3.5) ⇒ Φ = (4πGρ − Λ)r2 . 6 The Euler equation then becomes ∂H 1 r + H 2 r · ∇r r = − (4πGρ − Λ)r | {z } ∂t 3 r

⇒

∂H 1 + H 2 = (Λ − 4πGρ) . ∂t 3

(2.3.6)

This is the Newtonian limit of one of the Friedmann equations (the relativistic result replaces ρ with the sum of the energy density and three times the pressure, ρ + 3P ). The continuity equation becomes ∂t ρ + ∇r · [ρ(t)H (t)r] = 0 ⇒ ∂t ρ + 3ρH = 0 .

(2.3.7)

This is the usual Friedmann statement of energy conservation for ρ ≪ P . Introducing the scale factor a via ∂t a/a = H , we have 1 ∂ρ 3 ∂a + =0 a ∂t ρ ∂t

⇒

which describes the dilution of the mass density by expansion.

ρ ∝ a−3 ,

(2.3.8)

9 Equations (2.3.6) and (2.3.7) have a first integral   8πG 1 −K = a2 H 2 − ρ− Λ . 3 3 This is easily checked by differentiating:    1 ∂K 8πG ∂H 2 2 2 − ρ − − Λ + a 2H = 2a H H − ∂t 3 3 ∂t   2 16πG Hρ − H Λ + 2H −H 2 − = a2 2H 3 − 3 3 = 0.

 8πG ∂ρ 3 ∂t   1 4πG ρ + Λ + 8πGHρ 3 3

(2.3.9)

(2.3.10)

It follows that H2 +

K 1 = (8πGρ + Λ) . a2 3

(2.3.11)

In general relativity, K/a2 is 1/6 of the intrinsic curvature of the surfaces of homogeneity. B. Comoving coordinates

A comoving observer in the background (i.e. unperturbed) cosmology has velocity dr/dt = H (t)r hence position r = a(t)x where x is a constant. Rather than labelling events by t and r, it is convenient to use t and x, where x are comoving spatial coordinates: x = r/a(t). Note these are Lagrangian coordinates in the background but not in the perturbed model. Derivatives transform as follows:       ∂x ∂ ∂ + = · ∇x ∂t r ∂t x ∂t r   ∂ − H (t)x · ∇ , (2.3.12) = ∂t x where we use ∇ to denote the gradient with respect to x at fixed t; and ∇r = a−1 ∇ .

(2.3.13)

In what follows, ∂t should be understood as being taken at fixed x. C. Perturbation analysis

We now perturb ρ, u and Φ about their background values: ρ → ρ¯(t) + δρ ≡ ρ¯(t)(1 + δ) P → P¯ (t) + δP u → a(t)H (t)x + v ¯ x, t) + φ . Φ → Φ(

(2.3.14) (2.3.15) (2.3.16) (2.3.17)

Here, δ is the fractional overdensity in the fluid and φ the perturbed gravitational potential. Since, for a particle in the fluid, dr dx d(ax) = aHx + a = = u, dt dt dt

(2.3.18)

we see that adx/dt = v, so the peculiar velocity v describes changes in the comoving coordinates of fluid elements in time (i.e. departures from the background Hubble flow).

10 The continuity equation (2.3.1) becomes (on using Eq. 2.3.12) ρ¯ (1 + δ)∂t ρ¯ − H ρ¯x · ∇δ + ρ¯∂t δ + ∇ · [(1 + δ)(aHx + v)] = 0 . a Gathering terms that are zeroth, first and second-order in products of perturbed quantities gives ρ¯ ρ¯ ρH )δ + ρ¯∂t δ + ∇ · v + (v · ∇δ + δ∇ · v) = 0 . ∂t ρ¯ + 3¯ ρH + (∂t ρ¯ + 3¯ | {z } a a | {z } | {z } 0th−order

1st−order

(2.3.19)

(2.3.20)

2nd−order

The background equation (2.3.7) sets the zero-order part to zero. In linear perturbation theory, we assume the perturbation...