Part-III Cosmology Michaelmas Term 2009

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1 Part-III Cosmology Michaelmas Term 2009 Anthony Challinor Office hour (B2.08): Monday 14:00 15:00 Part II Large-scale structure formation The real universe is far from homogeneous and isotropic. 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) is starts to look much more homogeneous. Furthermore, we know from the CMB that the universe was smooth to around 1 part in 10 5 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. This final part of the course is structured as follows: Statistics of random fields Newtonian structure formation Introduction to relativistic perturbation theory The matter power spectrum Fluctuations from inflation CMB temperature anisotropies (probably non-examinable) 1.1 Statistics of random fields 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.

2 Part-III Cosmology 2 Figure 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. Credit: 2dF. Figure 2: Fluctuations in the CMB temperature, as determined from five years of WMAP data, about the average temperature of K. The fluctuations are at the level of only a few parts in 105. Credit: WMAP science team. 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. 1 For a recent review on the question of why quantum fluctuations from inflation can be treated as

3 Part-III Cosmology 3 To keep the Fourier analysis simple, we shall only consider flat (K = 0) background models and we denote comoving spatial positions by x 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, f(x) = 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 ξ(x,y) f(x)f(y) = Df Pr[f]f(x)f(y), (1.1.1) where the integral is a functional integral (or path integral) over field configurations. Statistical homogeneity means that the statistical properties of the translated field, ˆT a f(x) f(x a), (1.1.2) are the same as the original field, i.e. Pr[f(x)] = Pr[ ˆT a f(x)]. For the two-point correlation, this means that ξ(x,y) = ξ(x a,y a) ξ(x,y) = ξ(x y), (1.1.3) 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, a ˆRf(x) f(r 1 x), (1.1.4) where R is a rotation matrix, are the same as the original field, i.e. Pr[f(x)] = Pr[ ˆRf(x)]. For the two-point correlator, we must have Combining statistical homogeneity and isotropy gives ξ(x,y) = ξ ( R 1 (x y) ) ξ(x,y) = ξ(r 1 x, R 1 y) R. (1.1.5) ξ(x,y) = ξ( x y ), (1.1.6) 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. classical, see Keifer & Polarski (2008), available online at R

4 Part-III Cosmology 4 We can repeat these arguments to constrain the form of the correlators in Fourier space. We adopt the symmetric Fourier convention, so that d 3 x f(k) = 3/2f(x)e ik x d 3 k and f(x) = 3/2f(k)eik x. (1.1.7) (2π) (2π) Note that for real fields, f(k) = f ( k). Under translations, the Fourier transform acquires a phase factor: d ˆT 3 x a f(k) = (2π) 3/2f(x a)e ik x d 3 x = )e ik x e ik a (2π) 3/2f(x = f(k)e ik a. (1.1.8) Invariance of the two-point correlator in Fourier space is then f(k)f (k ) = f(k)f (k ) e i(k k ) a f(k)f (k ) = F(k)δ(k k ), (1.1.9) for some (real) function F(k). We see that different Fourier modes are uncorrelated. Under rotations, d ˆRf(k) 3 x = x)e ik x (2π) 3/2f(R 1 d 3 x = x)e i(r 1k) (R 1 x) (2π) 3/2f(R 1 = f(r 1 k), (1.1.10) so, additionally demanding invariance of the two-point correlator under rotations implies ˆRf(k)[ ˆRf(k )] = f(r 1 k)f (R 1 k ) = F(R 1 k)δ(k k ) = F(k)δ(k k ) R. (1.1.11) (We have used δ(r 1 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, P f (k), of a homogeneous and isotropic field, f(x), by f(k)f (k ) = 2π2 k 3 P f(k)δ(k k ). (1.1.12) The normalisation factor 2π 2 /k 3 in the definition of the power spectrum is conventional and has the virtue of making P f (k) dimensionless if f(x) is. a

5 Part-III Cosmology 5 The correlation function is the Fourier transform of the power spectrum: d 3 k d 3 k f(x)f(y) = (2π) 3/2 (2π) f(k)f (k ) e ik x e ik y 3/2 }{{} 2π 2 k 3 P f(k)δ(k k ) = 1 dk 4π k P f(k) dω k e ik (x y). (1.1.13) 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 2π 1 1 dµ e ik x y µ = 4πj 0 (k x y ), (1.1.14) where j 0 (x) = sin(x)/x is a spherical Bessel function of order zero. It follows that dk ξ(x,y) = k P f(k)j 0 (k x y ). (1.1.15) Note that this only depends on x y as required by Eq. (1.1.6). The variance of the field is ξ(0) = d lnk P f (k). A scale-invariant spectrum has P(k) = const. and its variance receives equal contributions from every decade in k 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(x 1 ), f(x 2 ),...,f(x N )] T, the probability density function for f is a multi-variate Gaussian fully specified by the correlation function where f i f(x i ), so that f i f j = ξ( x i x j ) ξ ij, (1.1.16) 1 Pr(f) e f iξij f j det(ξij ). (1.1.17) 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 ), 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.

6 Part-III Cosmology Random fields on the sphere Spherical harmonics form a basis for (square-integrable) functions on the sphere: f(ˆn) = l l=0 m= l f lm Y lm (ˆn). (1.1.18) The Y lm are familiar from quantum mechanics as the position-space representation of the eigenstates of ˆL 2 = 2 and ˆL z = i φ : 2 Y lm = l(l + 1)Y lm φ Y lm = imy lm, (1.1.19) with l an integer 0 and m an integer with m l. The spherical harmonics are orthonormal over the sphere, dˆn Y lm (ˆn)Y l m (ˆn) = δ ll δ mm, (1.1.20) so that the spherical multipole coefficients of f(ˆn) are f lm = dˆn f(ˆn)y lm (ˆn). (1.1.21) There are various phase conventions for the Y lm ; here we adopt Ylm = ( 1)m Y l m so that flm = ( 1)m f l m for a real field. What is the implication of statistical isotropy for the correlators of f lm? For the twopoint correlator, it turns out that we must have 2 f lm f l m = C lδ ll δ mm, (1.1.22) where C l is the angular power spectrum of f. What does this imply for the two-point correlation function? We have f(ˆn)f(ˆn ) = f lm fl m Y }{{} lm (ˆn)Y l m (ˆn ) lm l m C l δ ll δ mm = C l Y lm (ˆn)Y lm(ˆn ) = C(θ), (1.1.23) l m }{{} 2l+1 4π P l(ˆn ˆn ) 2 A plausibility argument is as follows. Under rotations, the subset of the Y lm with a given l (so 2l + 1 elements) transforms irreducibly so the δ ll form of the correlator is preserved under rotation. For rotation through γ about the z-axis, Under rotations, Y lm (θ, φ) Y lm (θ, φ γ) = e imγ Y lm (θ, φ) f lm e imγ f lm. so invariance requires the correlator be δ mm. f lm f l m e imγ e im γ f lm f l m,

7 Part-III Cosmology 7 where ˆn ˆn = cos θ and we used the addition theorem for spherical harmonics to express the sum of products of the Y lm in terms of the Legendre polynomials P l (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 C(0) = 2l + 1 4π C l d lnl l(l + 1)C l. (1.1.24) 2π l It is conventional to plot l(l+1)c l /(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: 1 C l = 2π d cosθ C(θ)P l (cosθ). (1.1.25) 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 (1.2.1) Euler t u + u r u = 1 ρ rp r Φ (1.2.2) Poisson 2 rφ = 4πGρ, (1.2.3) 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 2 rφ = 4πGρ Λ. (1.2.4)

8 Part-III Cosmology 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 first part of the course, but we include it again here from the perspective of the fluid equations. Taking Φ = 0 at r = 0, the Poisson equation (1.2.4) integrates as ( r 2 Φ ) = (4πGρ Λ)r 2 r r Φ r = 1 (4πGρ Λ)r 3 The Euler equation then becomes Φ = 1 6 (4πGρ Λ)r2. (1.2.5) H t r + H2 r r r = 1 (4πGρ Λ)r }{{} 3 r H t + H2 = 1 (Λ 4πGρ). (1.2.6) 3 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. (1.2.7) 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 ρ t + 3 a a t = 0 ρ a 3, (1.2.8) which describes the dilution of the mass density by expansion. Equations (1.2.6) and (1.2.7) have a first integral ( K = a 2 H 2 8πG 3 ρ 1 ) 3 Λ. (1.2.9)

9 Part-III Cosmology 9 This is easily checked by differentiating: K ( t = 2a2 H H 2 8πG 3 ρ 1 ) ( 3 Λ + a 2 2H H t 8πG ) ρ 3 t [ = a 2 2H 3 16πG Hρ 2 ( 3 3 HΛ + 2H H 2 4πG 3 ρ + 1 ) 3 Λ = 0. It follows that ] + 8πGHρ (1.2.10) H 2 + K a = 1 (8πGρ + Λ). (1.2.11) 2 3 In general relativity, K/a 2 is 1/6 of the intrinsic curvature of the surfaces of homogeneity 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: ( ) t r ( ) = t ( ) = t x x + ( ) x x t r H(t)x, (1.2.12) where we use to denote the gradient with respect to x at fixed t; and r = a 1. (1.2.13) In what follows, t should be understood as being taken at fixed x Perturbation analysis We now perturb ρ, u and Φ about their background values: ρ ρ(t) + δρ ρ(t)(1 + δ) (1.2.14) P P(t) + δp (1.2.15) u a(t)h(t)x + v (1.2.16) Φ Φ(x, t) + φ. (1.2.17)

10 Part-III Cosmology 10 Here, δ is the fractional overdensity in the fluid and φ the perturbed gravitational potential. Since, for a particle in the fluid, dr dt = d(ax) dt = ahx + a dx dt = u, (1.2.18) 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). The continuity equation (1.2.1) becomes (on using Eq ) (1 + δ) t ρ H ρx δ + ρ t δ + ρ [(1 + δ)(ahx + v)] = 0. (1.2.19) a Gathering terms that are zeroth, first and second-order in products of perturbed quantities gives t ρ + 3 ρh + ( }{{} t ρ + 3 ρh)δ + ρ t δ + ρ a v + ρ (v δ + δ v) = 0. (1.2.20) }{{}} a {{} 0th order 1st order 2nd order The background equation (1.2.7) sets the zero-order part to zero. In linear perturbation theory, we assume the perturbations are small enough (and their spatial derivatives) that we can ignore the second-order part, so that t δ + 1 a v = 0. (1.2.21) Exercise: show that Eqs (1.2.2) and (1.2.3) linearise to t v + Hv = 1 a ρ δp 1 φ a (1.2.22) 2 φ = 4πGa 2 ρδ. (1.2.23) Scalar/vector decomposition We can always decompose the vector v as v = v }{{} scalar part + v }{{}, (1.2.24) vector part where v = 0. It follows from Eq. (1.2.21) that the vector part of v does not lead to clumping of the matter. Since v = v, v describes the vorticity of the fluid

11 Part-III Cosmology 11 recalling that r = a 1, the physical vorticity r u = a 1 v. In linear theory, the scalar and vector parts decouple. For example, consider the (comoving) curl of the perturbed Euler equation (1.2.22), t v = t ( v ) = H v. (1.2.25) It follows that v decays as 1/a in an expanding universe so the vorticity falls as 1/a 2. This decay of the vorticity is consistent with the circulation theorem, u dr = const. for a path comoving with the fluid. For general initial conditions, the peculiar velocity approaches a gradient at late times and the vector modes can be neglected. For initial conditions from inflation, vector modes are not excited in the first place. They are, however, important in models with continual sourcing of perturbations by cosmic defects Jeans length The time derivative of the perturbed continuity equation (1.2.21) gives 2 t δ 1 a H v + 1 a tv = 0. (1.2.26) Combining with the perturbed Euler equation (1.2.22) and the Poisson equation (1.2.23), we find t 2 δ 1 a H v 1 ( a Hv + 1 a ρ δp + 1 ) a φ = 0 2 t δ 2 a H v 1 a 2 ρ 2 δp 1 a 2 2 φ = 0 2 t δ + 2H tδ 4πG ρδ 1 a 2 ρ 2 δp = 0. (1.2.27) This is the fundamental equation for the growth of structure in Newtonian theory. It shows the general competition between infall by gravitational attraction the 4πG ρδ term and pressure support, 2 δp. Consider a barotropic fluid such that P = P(ρ); then δp = P ρ ρδ c2 s ρδ (1.2.28) where c 2 s is the sound speed. Using this in Eq. (1.2.27), and Fourier expanding so that 2 k 2, gives ( ) c t 2 2 δ + 2H tδ + s k 2 4πG ρ δ = 0. (1.2.29) a 2

12 Part-III Cosmology 12 This is the equation for a damped (in an expanding universe) oscillator provided that c 2 sk 2 a 2 > 4πG ρ, (1.2.30) and, in this case, the pressure support gives rise to acoustic oscillations (sound waves) in the fluid. However, for c 2 sk 2 /a 2 < 4πG ρ, the system is unstable to gravitational accretion. Perturbations with proper wavelength 2πa/k exceeding the (proper) Jeans wavelength, λ J c s π G ρ, (1.2.31) are gravitationally unstable, while on smaller scales pressure supports oscillations. For a fluid with equation of state P/ρ > 1/3, the proper Jeans length in an expanding universe grows faster than a comoving scale ( a). This has the consequence that Fourier modes of the perturbations that start off outside the Jeans length, where they evolve by gravitational accretion, later come inside the Jeans length and subsequently undergo acoustic oscillations. The Jeans length is roughly the radius R of a region of background density ρ such the free-fall time, t ff, equals the sound-crossing time, t sound. To see this, note that the free-fall time is the time to collapse under gravity. Consider a shell of matter on the edge of a collapsing mass M of intitial radius R, such that at time t later the shell is at radius r. The mass enclosed is always M so that the gravitational acceleration is GM/r 2. Equating this to 2 t r, and solving for a particle initially at rest, the shell collapses to r = 0 in the free-fall time where t ff R3/2 GM 1 G ρ. (1.2.32) The sound-crossing time is simply t sound = R/c s and this equals the free-fall time for R c s / G ρ λ J. Fluctuations larger than the Jeans length do not have time for pressure to resist gravitational infall since the time to infall is less than the time it takes to propagate a pressure disturbance (i.e. a sound wave) across the perturbation. Note, finally, that the free-fall time is roughly the Hubble time, 1/H, when curvature and dark energy are negligible Applications to cold dark matter Solutions in an Einstein-de Sitter phase After matter-radiation equality, but well before dark energy comes to dominate, our universe is well described by an Einstein-de Sitter model having P 0 and zero

13 Part-III Cosmology 13 curvature or Λ. Scales of cosmological interest are much larger than the Jeans scale for the baryons and so both CDM fluctuations and those for the baryons have the same dynamical equations. We shall show shortly that quickly after recombination, the fractional overdensity in the baryons, δ b, approaches that in the CDM, δ c, and the matter behaves like a single pressure-free fluid with total density contrast δ m = ρ bδ b + ρ c δ c ρ b + ρ c δ c. (1.2.33) Since H 2 ρ a 3, we have a t 2/3 and so H = 2/(3t) and 4πG ρ = 2/(3t 2 ). Equation (1.2.27) then gives the evolution of the density fluctuations in the pressurefree matter as 2 t δ m + 4 3t tδ m 2 3t 2δ m = 0. (1.2.34) Trying solutions like t p gives independent solutions δ m t 1 and δ m t 2/3 a. The growing-mode solution of the density contrast therefore grows like the scale factor. Note that here, in an expanding universe, gravitational attraction has given rise to power-law growth of δ to be compared to the exponential growth predicted in a nonexpanding model. The Poisson equation (1.2.23) tells us that the gravitational potential is constant since, in Fourier space, k 2 φ = 4πGa 2 ρ }{{}}{{} δ a 3 a = const. (1.2.35) The Meszaros effect The Meszaros effect describes the way that CDM grows only logarithmically on scales inside the sound horizon during radiation domination. Generally, CDM (or anything else) feels the gravity of all clustered components so Eq. (1.2.27) generalises to the ith component of a set of non-interacting (except through gravity) fluids as 2 t δ i + 2H t δ i 4πG j ρ j δ j 1 a 2 ρ i 2 δp i = 0. (1.2.36) Specialising to pressure-free CDM we have 2 t δ c + 2H t δ c 4πG j ρ j δ j = 0. (1.2.37) Our Newtonian treatment at least makes it plausible that the Jeans length for perturbations in the radiation fluid (for which c s = 1/ 3) during radiation domination is of the order of the Hubble radius. Radiation fluctuations on scales smaller than this therefore oscillate as sound waves and their time-averaged density contrast vanishes

14 Part-III Cosmology 14 (we shall show this properly when we develop relativistic perturbation theory). It follows that the CDM is essentially the only clustered component during the acoustic oscillations of the radiation, and so 2 t δ c + 1 t tδ c 4πG ρ c δ c = 0, (1.2.38) where we used a t 1/2 and so H = 1/(2t). Since δ c evolves only on cosmological timescales (it has no pressure support for it to do otherwise), 2 t δ c H 2 δ c 4πG ρ c δ c (1.2.39) during radiation domination, as ρ r ρ c. We can therefore ignore the last term in Eq. (1.2.38) compared to the others and we have solutions with δ c = const. and δ c ln t. We see that the rapid expansion due to the effectively unclustered radiation reduces the growth of δ c to only logarithmic. Late-time suppression of structure formation by Λ At late times, the dominant clustered component is the matter and we have 2 t δ m + 2H t δ m 4πG ρ m δ m = 0. (1.2.40) In matter domination, this reduces to Eq. (1.2.34) and δ m grows like a, but when Λ comes to dominate a e t Λ/3 and H const. It follows that 4πG ρ m H 2 (currently 4πG ρ m /H ) and 2 t δ m + 2H t δ m 0. (1.2.41) The solutions of this are δ m = const. or δ m e 2t Λ/3 a 2 and Λ suppresses the growth of structure. Note also that a constant density contrast implies that the gravitational potential decays as a 2 ρ m a 1. This leaves an imprint in the CMB called the integrated Sachs-Wolfe effect (see later). Evolution of baryon fluctuations after decoupling Before decoupling, the baryon dynamics is linked to that of the radiation by efficient (Compton) scattering. On sub-hubble scales, δ b oscillates like the radiation but, after matter-radiation equality, δ c grows like a. It follows that just after decoupling, δ c δ b. Subsequently, the baryons fall into the potential wells sourced mainly by the CDM and δ b δ c as we shall now show. Ignoring baryon pressure and Λ, the coupled dynamics of the baryon and CDM fluids

15 Part-III Cosmology 15 Figure 3: Ratio of the matter power spectrum to a smooth spectrum (i.e. a model with no baryons) showing the expected baryon acoustic oscillations. Credit: Percival et al. after decoupling is approximately given by 2 t δ b + 4 3t tδ b = 4πG( ρ b δ b + ρ c δ c ) (1.2.42) 2 t δ c + 4 3t tδ c = 4πG( ρ b δ b + ρ c δ c ). (1.2.43) We can decouple these equations by using normal coordinates δ m (see equation ) and δ c δ b. Then 2 t + 4 3t t = 0 = const. or t 1/3, (1.2.44) while δ m follows Eq. (1.2.34) and has solutions t 1 and t 2/3. Since we see that δ b approaches δ c. δ c = ρ mδ m + ρ b δ b ρ m δ m ρ c δ m = 1, (1.2.45) δ m The non-zero initial value of δ b at decoupling, and, more importantly t δ b, leaves a small imprint in the late-time δ m that oscillates with scale. These baryon acoustic oscillations have recently been detected in the clustering of galaxies (see Fig. 3). 1.3 Introduction to relativistic perturbation theory The Newtonian treatment of Section 1.2 is inadequate on scales larger than the Hubble radius, and for relativistic fluids (like tightly-coupled radiation). The correct description requires a full general-relativistic treatment.

16 Part-III Cosmology 16 The basic idea of relativistic perturbation theory is straightforward: perturb the metric and stress-energy tensor in the Einstein equations about their Friedmann-Robertson- Walker (FRW) forms, and, for linear perturbations, drop products of small quantities. We then solve the coupled system of equations δg µν = 8πGδT µν + Λδg µν. (1.3.1) Even in linear theory there are some technical complexities due to the gauge freedom inherent in general relativity (i.e. the freedom over choice of coordinates). For this reason we shall only give an introduction to relativistic perturbation theory here; for a full treatment see the recommended textbooks. We begin by recalling the background metric and dynamics. The spatially-flat FRW metric is ds 2 = a 2 (η)(dη 2 δ ij dx i dx j ) = a 2 η µν dx µ dx ν. (1.3.2) To avoid unnecessary complications, we shall only consider flat (K = 0) universes here. The Friedmann equations for such models are, in conformal time, H 2 = 1 3 a2 (8πGρ + Λ) (1.3.3) Ḣ = 1 6 a2 [2Λ 8πG(ρ + 3P)], (1.3.4) where H is the conformal Hubble parameter, H η a/a = ah, and overdots denote differentiation with respect to conformal time Metric perturbations The most general perturbation to the background metric is ds 2 = a 2 (η) { (1 + 2ψ)dη 2 2B i dx i dη [(1 2φ)δ ij + 2E ij ] dx i dx j}. (1.3.5) The following points should be noted. The quantities ψ and φ are scalar functions of η and x i though, as we shall see shortly, they are not Lorentz scalars. We have chosen Cartesian spatial coordinates in the background, but, had we chosen differently (e.g. spherical-polar coordinates), the perturbation B i would have transformed like a three-vector. (However, ψ and φ would not change and hence are three-scalars.) To be specific, under η-independent relabelling of spatial coordinates, x i x i, B i xj x B i j. (1.3.6)

17 Part-III Cosmology 17 The perturbation E ij is a symmetric (E ij = E ji ) and trace-free (δ ij E ij = 0) three-tensor, i.e. under x i x i, E ij xk x l x i x E j kl. (1.3.7) The background spatial metric a 2 δ ij transforms similarly. We shall adopt the useful convention that Latin indices on spatial vectors and tensors are raised and lowered with δ ij. For example, B i δ ij B j and E i j = δ ik E kj. Scalar, vector and tensor decomposition In Section we decomposed the peculiar velocity into scalar and vector parts. It is convenient to use the same decomposition here, so, for example, B i = i B }{{} scalar part + Bi T }{{}, (1.3.8) vector part where the vector part is transverse (divergence-free), δ ij j B T i. A similar decomposition works for symmetric, trace-free 3-tensors: E ij = i j E + (i Ej) T + Eij T, (1.3.9) }{{}}{{}}{{} scalar part vector part tensor part where i j E i j E 1 3 δ ij 2 E (1.3.10) is trace-free; E T i is transverse; and E T ij is symmetric, trace-free and transverse, δ ik k E T ij = 0. This decomposition is unique in Euclidean space for smooth, bounded E ij that decay at infinity. A formal proof can be found in J.M. Stewart, Class. Quantum Grav. 7 (1990) Here we shall just give a simple non-examinable demonstration by constructing the scalar, vector and tensor parts in Fourier space. Non-examinable demonstration of scalar, vector and tensor decomposition: Assuming the Fourier transform of E ij exists, we have E ij (k) = d 3 x (2π) 3/2E ij(x)e ik x. (1.3.11) Consider decomposing E ij (k) onto a basis of trace-free tensors. We start with a basis of vectors (ˆk,m,m ) where m is a complex vector perpendicular to ˆk satisfying m m = 0

18 Part-III Cosmology 18 and m m = 1, and m is the complex conjugate of m. For example, if ˆk were along the z-axis, we could take m = (ˆx+iŷ)/ 2. Taking all possible trace-free, symmetric tensor products of pairs of this basis, we obtain a basis on which E ij (k) can be expanded: Note that the sixth possible pairing m i m j = 1 2 M(0) ij M (0) ij (ˆk) = ˆk iˆkj M (1) ij (ˆk) = ˆk (i m j) M ( 1) ij (ˆk) = ˆk (i m j) M (2) ij (ˆk) = m i m j M ( 2) ij (ˆk) = m i m j. (1.3.12) 1 2 M(2) ij M( 2) ij, (1.3.13) and so is not linearly independent. The five-dimensional basis is as expected since their are five functional degrees of freedom in the trace-free, symmetric E ij. We can now expand E ij (k) as E ij (k) = n E (n) (k)m (n) ij (ˆk). (1.3.14) Consider the n = 0 term: d 3 k (2π) ˆk iˆkj E (0) (k)e ik x = ( 3/2 i j 13 ) 2 d 3 k E (0) (k) (2π) 3/2 k 2 e ik x = i j E, (1.3.15) which gives the scalar part of E ij. For the n = 1 term, d 3 k (2π) ˆk 3/2 (i m j) E (1) (k)e ik x d 3 k = i (i (2π) 3/2m j)(ˆk)e (1) (k)e ik x, (1.3.16) so, combining the n = ±1 terms we find the vector-mode part of E ij : d 3 k [ ] i (i m (2π) 3/2 j) (ˆk)E (1) (k) + m j)(ˆk)e ( 1) (k) e ik x = (i Ej) T. (1.3.17) The vector Ei T is transverse since ˆk m(ˆk) = 0. Finally, for the n = 2 terms we have d 3 k [ ] m (2π) 3/2 i (ˆk)m j (ˆk)E (2) (k) + m i(ˆk)m j(ˆk)e ( 2) (k) = Eij T, (1.3.18) which is symmetric and trace-free by construction, and is transverse since, again, ˆk m(ˆk) = 0. If we count the functional degrees of freedom in the scalar, vector and tensor parts, the original five degrees of freedom in E ij are re-distributed as follows. Scalar one degree of freedom [E(x, η)].

19 Part-III Cosmology 19 has three components, but once con- Vector two degrees of freedom since Ei T straint (the divergence vanishes). Tensor two degrees of freedom since E T ij has five components but δik k E T ij is three constraints (one for each component). Given the uniqueness of the decomposition, in any 3-tensor equation the scalar, vector and tensor parts of the left and right-hand sides must be equal and, in linear theory, this means that the perturbation types decouple completely. As in Newtonian theory, scalar perturbations describe clumping of matter while vector modes describe vortical motions. The new feature in the relativistic treatment here is the tensor modes describing gravitational waves. We shall mostly be concerned with scalar perturbations here. Orthonormal frame vectors Finally, it will prove useful to construct explicitly an orthonormal frame of 4-vectors, (E 0 ) µ and (E i ) µ, in the perturbed metric. Take the timelike (E 0 ) µ to be the 4-velocity u µ of an observer at rest relative to the coordinate system. It follows that (E 0 ) µ must be parallel to δ µ 0 and normalising gives, at linear order, since then (E 0 ) µ = a 1 (1 ψ)δ µ 0, (1.3.19) g µν (E 0 ) µ (E 0 ) ν = a 2 (1 2ψ)g 00 = a 2 (1 2ψ)a 2 (1 + 2ψ) = 1. (1.3.20) Note how we have dropped second-order terms (i.e. products of perturbations) here. The spacelike (E i ) µ are a little more involved since, generally, the coordinate vectors δ µ i are not orthogonal to u µ unless B i = 0. The following construction has the required properties: (E i ) µ = a [ 1 B i δ µ 0 + (1 + φ)δµ i E ] i j δ µ j. (1.3.21) We can easily check that these are orthogonal to (E 0 ) µ : g µν (E 0 ) µ (E i ) ν = a 1 (1 ψ)g 0ν (E i ) ν = a 2 (1 ψ) [ g 00 B i + g 0i (1 + φ) g 0j E i j ] = B i B i = 0. (1.3.22) Exercise: show that g µν (E i ) µ (E j ) ν = δ ij.

20 Part-III Cosmology Matter perturbations Recall that the energy and momentum of the matter is described by the stress-energy tensor. In an orthonormal frame, the components of T µν are T ˆ0ˆ0 = ρ(η) + δρ T ˆ0î = q i T îĵ = [ P(η) + δp]δ ij Π ij energy density momentum density flux of ith component of 3-momentum along jth direction, (1.3.23) where Π ij is the trace-free anisotropic stress. We shall use the perturbations δρ ρ(η)δ, δp, q i and Π ij to describe the perturbations to the matter and fields present, and their components are defined on the orthonormal frame in Eqs (1.3.19) and (1.3.21). However, it will be convenient to construct the coordinate components of T µν which are given by T µν = (E α ) µ (E β ) ν T ˆαˆβ. (1.3.24) We start with T 00 : For T 0i, we have Finally, for T ij, T 00 = (E 0 ) 0 (E 0 ) 0 T ˆ0ˆ0 + 2(E 0 ) 0 (E i ) 0 T ˆ0î + (E i ) 0 (E j ) 0 T îĵ = a 2 (1 2ψ) ρ(1 + δ) + O(2) + O(2) = a 2 ρ(1 + δ 2ψ). (1.3.25) T 0i = (E 0 ) 0 (E 0 ) i T ˆ0ˆ0 + (E 0 ) 0 (E j ) i T ˆ0ĵ + (E j ) 0 (E 0 ) i T ˆ0ĵ + (E j ) 0 (E k ) i T ĵˆk = 0 + a 2 δjq i j a 2 B j δk i jk Pδ = a 2 (q i + PB i ). (1.3.26) T ij = (E 0 ) i (E 0 ) j T ˆ0ˆ0 + (E 0 ) i (E k ) j T ˆ0ˆk + (E k ) i (E 0 ) j T ˆkˆ0 + (E k ) i (E l ) j T ˆkˆl = a [ 2 (1 + φ)δk i E i][ k (1 + φ)δ j l E j][ l ( P + δp)δ kl Π kl] = a [ 2 Pδ ij + (2 Pφ + δp)δ ij 2 PE ij Π ij]. (1.3.27) Note how these components mix the perturbations to the stress-energy tensor with the metric perturbations. Things look neater in term of the mixed coordinate components: T 0 0 = g µ0 T 0µ = g 00 T 00 + g 0i T 0i = a 2 (1 + 2ψ)a 2 ρ(1 2ψ + δ) + O(2) = ρ(1 + δ). (1.3.28)

21 Part-III Cosmology 21 Exercise: Show that the other mixed components are T i 0 = q i (1.3.29) T i j = ( P + δp)δj i + Πi j, (1.3.30) where Π i j δ jk Π ik. We can gain some insight into the momentum density by considering a perfect fluid with 4-velocity u µ which is a small perturbation to the 4-velocity of observers at rest in our coordinates. The stress-energy tensor for a perfect fluid is T µ ν = (ρ + P)u µ u ν Pδ µ ν. (1.3.31) Since the 4-velocity u µ = dx u /dτ, where τ is the proper time, and g µν u µ u ν = 1, we have dx µ dx ν 1 = g µν dτ dτ ( ) 2 dη dx µ dx ν = g µν dτ dη dη ( ) 2 dη dx = (g i g 0i dτ dη + g ij dx i dη ) dx j. (1.3.32) dη If we write the coordinate velocity dx i /dη = v i and assume this is a small perturbation, then Eq. (1.3.32) reduces to 1 = ( ) 2 dη g 00 = a 2 (1 + 2ψ) dτ ( ) 2 dη dτ dη dτ = 1 (1 ψ) (1.3.33) a at linear order. The components of the fluid s 4-velocity are then u µ = a 1 [1 ψ, v i ], (1.3.34) and u 0 = g 00 u 0 + g 0i u i = a 2 (1 + 2ψ)a 1 (1 ψ) + O(2) = a(1 + ψ) (1.3.35) u i = g i0 u 0 + g ij u j = a 2 B i a 1 a 2 δ ij a 1 v j = a(b i + v i ). (1.3.36) Using these expressions for the components u µ and u µ in Eq. (1.3.31), we find q i = T i 0 = (ρ + P)a 1 v i a(1 + ψ) = (ρ + P)v i. (1.3.37)

22 Part-III Cosmology 22 Generally, we can define an effective peculiar velocity of the matter (for any component, or the total) by q i ( ρ + P)v i. (1.3.38) We end this section by noting that the scalar, vector and tensor decomposition can also be applied to the perturbations to the stress-energy tensor: δρ and δp have scalar parts only, q i has scalar and vector parts, and Π ij has scalar, vector and tensor parts Gauge transformations The value of the perturbation to a quantity, even one that is a Lorentz scalar, will change under a coordinate transformation. For a given event in the real, clumpy universe, changing coordinates alters the point in the background model that is associated with that event and, if a quantity evolves in the background, the perturbation to that quantity will change. We need to be aware of this gauge freedom and develop strategies for handling it. Consider small changes in coordinates η = η + T(η, x i ), x i = x i + L i (η, x j ). (1.3.39) For a (Lorentz) scalar field Φ (such as the inflaton field driving inflation), at some event originally labelled by η and x i, but by η and x i in the new coordinates, the new perturbation at the same event is δ Φ = Φ Φ( η) = Φ Φ(η + T) (since Φ is homogeneous) = Φ }{{ Φ(η) T Φ, (1.3.40) } δφ to first order in the perturbation to the coordinates. This gives δ Φ = δφ T Φ. (1.3.41) Things are a little more complicated for the metric perturbations since g µν is not a Lorentz scalar. If ḡ µν is the metric in the background, then δ g µν = g µν ḡ µν ( η, x i ) = xα x µ x β x ν g αβ ḡ µν ( η, x i ) = xα x µ x β = δg µν + [ δgαβ + ḡ x ν αβ (η, x i ) ] ḡ µν ( η, x i ) ( x α x β x µ x ν δα µ δβ ν ) ḡ αβ (η, x i ) T ḡ µν (η, x i ) L i i ḡ µν (η, x i ) (1.3.42)

23 Part-III Cosmology 23 to linear order. To evaluate this, we require x α / x µ which, considered as a matrix, is the inverse of x µ / x α since x α x µ x µ x = β δα β. (1.3.43) The matrix of derivatives is, from Eq. (1.3.39), x α x µ = ( η/ η η/ x i x i / η x i / x j ) ( ) 1 + T = i T L i δj i + jl i, (1.3.44) where α labels the rows and µ the columns. The inverse of a matrix of the form I +, were I is the identity and is a small perturbation, is I to first-order in. It follows that x α x µ = ( η/ η η/ x i x i / η x i / x j ) ( 1 T = L i ) i T δj i jl i. (1.3.45) We can now substitute into Eq. (1.3.42), and make use of Eq. (1.3.5), to find how the metric perturbations transform. We shall do one case in detail here; the others are left as an exercise. For the 00 component, ( ) x α x β δ g 00 = δg 00 + ḡ αβ T ḡ 00 L i i ḡ 00 so we have 2a 2 ψ = 2a 2 ψ + η ( η η η η 1 η δα 0 δβ 0 ) ḡ η x i η η ḡ0i + xi x j η η ḡij T η a 2 = 2a 2 ψ 2 Ta 2 + O(2) + O(2) 2T Ha 2, (1.3.46) ψ = ψ T HT. (1.3.47) Exercise: By considering the other metric components, show that φ = φ + HT il i (1.3.48) B i = B i + i T L i (1.3.49) Ẽ ij = E ij i L j. (1.3.50) We can also repeat this analysis for the stress-energy tensor. Here, it is more convenient to work with the mixed components which transform as δ T µ ν = xµ x β x α x T α ν β T µ ν(η + T, x i + L i ) ( ) x = δt µ µ x β ν + x α x ν δµ α δβ ν T α β T T µ ν L i µ i T ν, (1.3.51)

24 Part-III Cosmology 24 where T µ ν is the stress-energy tensor in the background. For T 0 0 = ρ + δρ, we have ( ) η x β δ ρ = δρ + x α η δ0 αδ β 0 T α β T ρ L i i ρ ( ) η η = δρ + η η 1 ρ η x j x i η Pδ j i T ρ [ = δρ + (1 + T)(1 T) ] 1 ρ O(2) T ρ, (1.3.52) so that The remaining transformations are left as an exercise. δ ρ = δρ T ρ. (1.3.53) Exercise: Show that the pressure perturbation, momentum density and anisotropic stress transform as δ P = δp T P (1.3.54) q i = q i + ( ρ + P) L i (1.3.55) Π i j = Π i j. (1.3.56) Generally, the perturbations to the stress-energy tensor, δρ, δp, q i and Π ij, change under gauge transformations for two reasons: (i) the transformation changes the 4- velocity of the observer; and (ii) the identified point in the background changes. The density and pressure only change at second-order due to changes in the observer s velocity, but their perturbation changes linearly under gauge-transformations since the density and pressure do not vanish (and evolve) in the background. The momentum density vanishes in the background so its transformation comes solely from the change in the observer s velocity. Indeed, an observer at rest in the new coordinates originally had coordinate 3-velocity L i and we see from Eq. (1.3.55) that the peculiar velocity of the matter changes by minus this amount (to linear order). The anisotropic stress is invariant under (linear) gauge transformations. Gauge-invariant variables Consider scalar perturbations in some gauge, and apply a gauge transformation that has only scalar modes, i.e. η = η + T, x i = x i + δ ij j L. (1.3.57)

25 Part-III Cosmology 25 The original perturbations to the metric are ψ, φ, i B and i j E. We see from Eqs (1.3.47) (1.3.50) that the new metric perturbations are also scalar in character with potentials ψ = ψ T HT (1.3.58) φ = φ + HT L (1.3.59) B = B + T L (1.3.60) Ẽ = E L. (1.3.61) There are four functional degrees of freedom in the scalar perturbations to the metric and two gauge functions (T and L) so we expect to be able to construct two gaugeinvariant quantities (i.e. ones that do not change under gauge transformations). One possibility is the Bardeen variables 3 Ψ ψ + H(B Ė) + Ḃ Ë (1.3.62) Φ φ H(B Ė) E. (1.3.63) It is also possible to combine metric and matter variables to form gauge-invariant quantities. One useful combination that we shall use repeatedly is where q i = ( ρ + P) i v. ρ δρ 3H( ρ + P)(B + v), (1.3.64) Unless we fix the gauge completely, gauge modes will appear when we solve the perturbed field equations. Gauge modes are apparent perturbations that arise from gauge transformations of the unperturbed background model. For example, a foolish choice of constant-time hypersurfaces in an FRW universe will produce an apparent density perturbation of the form δρ = T ρ. Such a perturbation is not physical and can be made to vanish by a gauge transformation. By construction, the gauge-invariant variables receive no contributions from gauge modes. Generally, we can use the gauge freedom inherent in relativistic perturbation theory to simplify the form of our equations, but different choices are convenient for different problems. The choice of gauge is mostly an issue on super-hubble scales. As we shall illustrate later (Sec ), for scales small compared to the Hubble radius, the perturbations are essential equal in all (physical) gauges. 3 These were first written down in the classic paper of J.M. Bardeen, Phys. Rev. D 22 (1980) 1882.

26 Part-III Cosmology Field equations for scalar perturbations in the conformal Newtonian gauge We shall consider only scalar perturbations here. We can use the two gauge functions T and L to set the metric perturbations E and B to zero. This defines the conformal Newtonian gauge ds 2 = a 2 (η) [ (1 + 2ψ)dη 2 (1 2φ)δ ij dx i dx j]. (1.3.65) For perturbations that decay at spatial infinity, the conformal Newtonian gauge is unique (i.e. the gauge is fixed) 4. In this gauge, the physics appears rather simple since the hypersurfaces of constant time are orthogonal to the worldlines of observers at rest in the coordinates (since g 0i = 0) and the induced geometry of the constant-time hypersurfaces is isotropic 5. Note the similarity of the metric to the usual weak-field limit of general relativity about Minkowski space; we shall see that φ plays the role of the gravitational potential. Perturbed connection coefficients To derive the field equations, we first require the perturbed connection coefficients. Generally, Γ µ νρ = 1 2 gµκ ( ν g κρ + ρ g κν κ g νρ ). (1.3.66) The metric is diagonal so simple to invert: g µν = 1 a 2 ( 1 2ψ 0 0 (1 + 2φ)δ ij ). (1.3.67) Here we shall work out Γ 0 00 left as an exercise: to first order in the perturbations; the remaining terms are Γ 0 00 = 1 2 g00 (2 η g 00 η g 00 ) = 1 2 g00 η g 00 = 1 2a 2(1 2ψ) η[a 2 (1 + 2ψ)] = H + ψ. (1.3.68) 4 More generally, a gauge transformation that corresponds to a small, time-dependent but spatially constant boost i.e. L i (η) and a compensating time translation with i T = L i (η) to keep the constanttime hypersurfaces orthogonal will preserve E ij = 0 and B i = 0 and hence the form of the metric in Eq. (1.3.65). However, such a transformation would not preserve the decay of the perturbations at infinity. 5 It can be shown that the relative motion of the coordinate observers is shear and vorticity free.

27 Part-III Cosmology 27 Exercise: Show that the other components of the connection are Γ 0 0i = iψ (1.3.69) Γ i 00 = δ ij j ψ (1.3.70) [ ] Γ 0 ij = Hδ ij φ + 2H(φ + ψ) δ ij (1.3.71) Γ i j0 = Hδ i j φδ i j (1.3.72) Γ i jk = 2δ i (j k) φ + δ jk δ il l φ. (1.3.73) Stress-energy conservation The field equations G µν = 8πGT µν +Λg µν imply conservation of energy and momentum via the contracted Bianchi identity: µ G µν = 0 µ T µν = 0. (1.3.74) This conservation law is the relativistic version of the continuity and Euler equations (1.2.1) and (1.2.2) in Newtonian dynamics. It is more convenient to work with the mixed components µ T µ ν = 0 or, explicitly, µ T µ ν + Γ µ µρt ρ ν Γ ρ µνt µ ρ = 0. (1.3.75) Consider first the 0 component which will give the conservation of energy. We have 0 T i T i 0 + Γ µ µ0 T Γ µ µi T i 0 Γ 0 00 }{{} T 0 0 Γ 0 i0 T i 0 Γ i 00 }{{} T 0 i Γ i j0 }{{} T j i = 0. (1.3.76) O(2) O(2) O(2) Substituting for the perturbed stress-energy tensor and the connection coefficients gives η ( ρ + δρ) + i q i + (H + ψ + 3H 3 φ)( ρ + δρ) (H + ψ)( ρ + δρ) (H φ)δ [ j i ( P + δp)δ j i + i] Πj = 0 ρ + η δρ + i q i + 3H( ρ + δρ) 3 ρ φ + 3H( P + δp) 3 P φ = 0. (1.3.77) The zero-order part gives the conservation of energy in the background, and the first-order part gives ρ + 3H( ρ + P) = 0, (1.3.78) η δρ + i q i 3( ρ + P) φ + 3H(δρ + δp) = 0. (1.3.79)

28 Part-III Cosmology 28 The term i q i describes changes in energy in a volume due to flow through the surfaces; the remaining two terms describe the perturbation to the dilution of energy density by expansion (including PV work) with 3 φ arising from the conformal time derivative of the perturbation to the spatial volume element (1 3φ)d 3 x. If we write δρ in terms of the fractional overdensity, δρ = ρδ, and q i in terms of the peculiar velocity, q i = ( ρ + P)v i, Eq. (1.3.79) becomes δ + ( 1 + P ρ ) ( i v i 3 φ) + 3H ( δp ρ P ) ρ δ = 0. (1.3.80) In the limit P ρ, we recover the Newtonian continuity equation in conformal time, δ + i v i 3 φ = 0, but with a general-relativistic correction due to the perturbation to the rate of expansion of space. If we now consider the ith component of Eq. (1.3.75), this will give the conservation of 3-momentum. We start with µ T µ i + Γ µ µρ T ρ i Γ ρ µit µ ρ = 0 η T 0 i + j T j i + Γ µ µ0t 0 i + Γ µ µj T j i Γ 0 0i T 0 0 Γ 0 ji T j 0 Γ j 0i T 0 j Γ j ki T k j = 0. (1.3.81) Note that this involves a mixed component of the stress-energy tensor, T 0 i that we have not written down explicitly before we can find it from Eqs (1.3.25) and (1.3.29) as follows: Equation (1.3.81) then becomes T 0 i = g iµ T 0µ = g i0 T 00 + g ij T 0j = 0 a 2 (1 2φ)δ ij a 2 q i = q i. (1.3.82) [ q i + j ( P + δp)δ j i + i] Πj 4Hqi ( j ψ 3 j φ) Pδ j i iψ ρ ( ) Hδ ji q j + Hδ j i q j + 2δ j (i k)φ + δ ki δ jl l φ Pδ k j = 0 }{{} 3 i φ P q i i δp + j Π j i 4Hq i ( ρ + P) i ψ = 0. (1.3.83) Writing q i = ( ρ + P)v i, and using Eq. (1.3.78), we get the relativistic version of the Euler equation: v i + 1 ρ + P iδp 1 ρ + P jπ j i + Hv i + P ρ + P v i + i ψ = 0. (1.3.84) This is like the Euler equation for a viscous fluid [c.f. Eq. (1.2.22)], with pressure gradients, the divergence of the anisotropic stress and gravitational infall driving v i, but with corrections due to redshifting of peculiar velocities (Hv i ) and P/ρ effects.

29 Part-III Cosmology 29 Once an equation of state of the matter (and other constitutive relations) are specified, we just need the gravitational potentials ψ and φ to close the system of equations of energy conservation. Perturbed Einstein equation We require the perturbation to the Einstein tensor, G µν R µν g µν R/2, so we first need to calculate the perturbed Ricci tensor R µν and scalar R. The Ricci tensor is a contraction of the Riemann tensor and can be expressed in terms of the connection as R µν = ρ Γ ρ µν νγ ρ µρ + Γα µν Γρ αρ Γα µρ Γρ να. (1.3.85) We shall derive the 00 component here with the others left as an exercise. We have R 00 = ρ Γ ρ 00 η Γ ρ 0ρ + Γ α 00 Γρ αρ Γα 0ρ Γρ 0α. (1.3.86) When we sum over ρ, the terms with ρ = 0 cancel so we need only consider summing over ρ = 1, 2, 3, i.e. R 00 = i Γ i 00 η Γ i 0i + Γ α 00Γ i αi Γ α 0iΓ i 0α = i Γ i 00 ηγ i 0i + Γ0 00 Γi 0i + Γj 00Γ i ji Γ 0 0i }{{} Γi 00 }{{} O(2) O(2) Γ j 0i Γi 0j = 2 ψ 3 η (H φ) 3(H + ψ)(h φ) (H φ) 2 δ j i δi j = 3Ḣ + 2 ψ + 3H( φ + ψ) + 3 φ. (1.3.87) Exercise: Show that the remaining components of the perturbed Ricci tensor are R 0i = 2 i φ + 2H i ψ (1.3.88) R ij = [Ḣ + 2H 2 φ + 2 φ 2(Ḣ + 2H2 )(φ + ψ) H ψ φ] 5H δ ij + i j (φ ψ). (1.3.89) We now form the Ricci tensor g µν R µν using It follows that R = g 00 R g 0i R }{{ 0i +g ij R } ij. (1.3.90) 0 a 2 R = (1 2ψ)R 00 (1 + 2φ)δ ij R ij [ = (1 2ψ) 3Ḣ + 2 ψ + 3H( φ + ψ) ] + 3 φ 3(1 + 2φ) [Ḣ + 2H 2 φ + 2 φ 2(Ḣ + 2H2 )(φ + ψ) H ψ 5H φ ] (1 + 2φ) 2 (φ ψ). (1.3.91)

30 Part-III Cosmology 30 Simplifying, we find, to linear order, a 2 R = 6(Ḣ + H2 ) ψ 4 2 φ + 12(Ḣ + H2 )ψ + 6 φ + 6H( ψ + 3 φ). (1.3.92) Finally, we can form the Einstein tensor. The 00 component is G 00 = R g 00R = 3Ḣ + 2 ψ + 3H( φ + ψ) + 3 φ + 3(1 + 2ψ)(Ḣ + H2 ) 1 [ 2 2 ψ 4 2 φ + 12(Ḣ 2 + H2 )ψ + 6 φ + 6H( ψ + 3 φ) ]. (1.3.93) Most of the terms cancel leaving the simple result G 00 = 3H φ 6H φ. (1.3.94) The 0i component of the Einstein tensor is simply R 0i since g 0i = 0 in the conformal Newtonian gauge: G 0i = 2 i φ + 2H i ψ. (1.3.95) The remaining components are G ij = R ij 1 2 g ijr = [Ḣ + 2H 2 φ + 2 φ 2(Ḣ + 2H2 )(φ + ψ) H ψ φ] 5H δ ij + i j (φ ψ) 3(1 2φ)(Ḣ + H2 )δ ij + 1 [ 2 2 ψ 4 2 φ + 12(Ḣ 2 + H2 )ψ + 6 φ + 6H( ψ + 3 φ) ] δ ij. (1.3.96) This neatens up (only a little!) to give [ G ij = (2Ḣ + H2 )δ ij + 2 (ψ φ) + 2 φ + 2(2Ḣ + H2 )(φ + ψ) + 2H ψ + 4H φ ] + i j (φ ψ). (1.3.97) δ ij Substituting the perturbed Einstein tensor, metric and stress-energy tensor into the Einstein equation gives the equations of motion for the metric perturbations and the zero-order Friedmann equations. For example, G 00 = 8πGT 00 + Λg 00 3H φ 6H φ = 8πG ( g 00 T g 0i T i 0) + Λa 2 (1 + 2ψ) = 8πGa 2 ρ(1 + 2ψ)(1 + δ) + Λa 2 (1 + 2ψ). (1.3.98) The zero-order part gives H 2 = 8πG 3 a2 ρ Λa2, (1.3.99)

31 Part-III Cosmology 31 which is just the Friedmann equation (1.3.3). The first-order part of Eq. (1.3.98) gives 2 φ = (8πGa 2 ρ + Λa 2 )ψ + 3H φ + 4πGa 2 ρδ. ( ) which, on using Eq. (1.3.99), reduces to 2 φ = 3H( φ + Hψ) + 4πGa 2 ρδ. ( ) The 0i Einstein equation is G 0i = 8πGT 0i + Λg 0i. ( ) The last term on the right vanishes in the conformal Newtonian gauge, and It follows that T 0i = g 0µ g iν T µν = g 00 g ij T 0j (since g 0i = 0) = a 2 (1 + 2ψ)a 2 (1 2φ)δ ij a 2 q j = a 2 q i. ( ) i φ + H i ψ = 4πGa 2 q i. ( ) If we write q i = ( ρ + P) i v and assume the perturbations decay at infinity, we can integrate Eq. ( ) to get Substituting this in the 00 Einstein equation gives φ + Hψ = 4πGa 2 ( ρ + P)v. ( ) 2 φ = 4πGa 2 [ ρδ 3H( ρ + P)v ]. ( ) This is of the form of a Poisson equation but with source density ρδ 3H( ρ+ P)v. What is the physical meaning of this term? It is numerically equal to the gauge-invariant variable ρ of Eq. (1.3.64) since B = 0 in the conformal Newtonian gauge. Let us introduce comoving hypersurfaces as those that are orthogonal to the worldlines of a set of observers comoving with the total matter (i.e. they see q i = 0) and are the constant-time hypersurfaces in the comoving gauge for which q i = 0 and B i = 0. It follows that is the fractional overdensity in the comoving gauge and we see from Eq. ( ) that this is the source term for the gravitational potential φ. The final content of the Einstein equation is contained in its ij components: G ij = 8πGT ij + Λg ij = 8πGg ik g jl T kl a 2 Λ(1 2φ)δ ij = 8πGa 4 (1 2φ) 2 δ ik δ jl a 2 [ Pδ kl + (2 Pφ + δp)δ kl Π kl] a 2 Λ(1 2φ)δ ij = 8πGa 2 (1 4φ) [ Pδij + (2 Pφ + δp)δ ij Π ij ] a 2 Λ(1 2φ)δ ij = a 2 (8πG P Λ)δ ij + a 2 [ 8πG(δP 2 Pφ) + 2Λφ ] δ ij 8πGa 2 Π ij. ( )

32 Part-III Cosmology 32 Using Eq. (1.3.97) in the left-hand side, the zero-order part is 2Ḣ + H2 = a 2 (8πG P Λ), ( ) which, with Eq. (1.3.3) recover the second Friedmann equation (1.3.4). The first-order part of Eq. ( ) is [ 2 (ψ φ) + 2 φ + 2(2Ḣ + H2 )(φ + ψ) + 2H ψ + 4H φ ] δ ij + i j (φ ψ) = a 2 [ 8πG(δP 2 Pφ) + 2Λφ ] δ ij 8πGa 2 Π ij. ( ) We can further manipulate this by considering the trace and trace-free parts separately. The latter is i j (φ ψ) = 8πGa 2 Π ij. ( ) This shows that, in the absence of anisotropic stress (and assuming appropriate decay at infinity), φ = ψ so there is then only one gauge-invariant degree of freedom in the metric. The remaining content of Eq. ( ) is contained in the trace part; contracting with δ ij /3 gives 2 (ψ φ) + 2 φ + 2(2Ḣ + H2 )(φ + ψ) + 2H ψ + 4H φ (φ ψ) = 8πGa 2 δp 2a 2 (8πG P Λ)φ. ( ) Rearranging, and using Eq. ( ) in the right-hand side we get φ (ψ φ) + (2Ḣ + H2 )ψ + H ψ + 2H φ = 4πGa 2 δp. ( ) Of course, the Einstein equations and the energy and momentum conservation equations form a redundant (but consistent!) set because of the Bianchi identity. We can use whichever subsets are most convenient for the particular problem at hand Density perturbations of a radiation fluid We shall first consider the evolution of density perturbations in a universe containing only a radiation fluid, i.e. an ideal fluid with P = ρ/3. We shall later argue how this applies to the photons and neutrinos during radiation domination in our multicomponent universe. The basic idea is to get a closed equation for the potential φ (noting that φ = ψ since Π ij = 0 by assumption) and then use this to determine the density fluctuations in the radiation. We start from Eq. ( ) with δp = δρ/3. In a universe dominated by radiation, H 2 a 2 hence a η and H = 1/η; it follows that φ + 3 η φ 1 4πGa2 ρ η2φ = δ r. ( ) 3

33 Part-III Cosmology 33 We can eliminate δ r using the 00 Einstein equation in the form of Eq. ( ) to find φ + 3 η φ 1 η 2φ = φ 1 ( φ + 1 ) η η φ φ + 4 η φ φ = 0. ( ) This is a damped wave equation with propagation speed 1/ 3 as expected for a radiation fluid where the square of the sound speed is P/ ρ = 1/3. On Fourier expanding ( 2 φ k 2 φ), Eq. ( ) gives φ + 4 η φ + k2 3 φ = 0. ( ) Finally, writing φ = u(x)/x where x kη/ 3, we have u + 2 x u + (1 2x ) u = 0. ( ) 2 This has independent solutions which are just the spherical Bessel functions j 1 (x) and n 1 (x) 6. These can be written explicitly as j 1 (x) = sin x x 2 n 1 (x) = cosx x 2 cosx x = x 3 + O(x3 ) The regular (growing-mode) solution for φ is thus sin x x = 1 x 2 + O(x0 ). ( ) φ j 1(kη/ 3) kη, ( ) and is constant outside the sound horizon, i.e. for k 1 η/ 3. The asymptotic form of the spherical Bessel functions are j l (x) 1 ( x sin x lπ ), ( ) 2 hence φ cos(kη/ 3)/(kη) 2 well inside the sound horizon. As expected from Eq. ( ), we have oscillations at frequency k/ 3 with a slow damping of the amplitude (over the order of a Hubble time). 6 Generally, spherical Bessel functions j l and n l satisfy y l + 2 ( ) x y l + l(l + 1) 1 x 2 y l = 0, where y l is j l or n l.

34 Part-III Cosmology 34 The Poisson equation ( ) relates the potential to the density perturbation in the comoving gauge. Here, this is the gauge that is comoving with the radiation fluid. Fourier expanding, we have so that k 2 φ = 3 2 H2 r = 3 2η 2 r, ( ) r = 2 3 (kη)2 φ kηj 1 (kη/ 3). ( ) This grows like η 2 a 2 outside the sound horizon and oscillates with constant amplitude inside the sound horizon. These sub-horizon oscillations are what give rise to the acoustic peaks in the power spectrum of the CMB anisotropies (see later). We have to work a little harder to find the density perturbation in the conformal Newtonian gauge. Using Eq. ( ), we have k 2 φ 3 ( φ + 1 ) η η φ = 3 2η 2δ r δ r = 2 3 (kη)2 φ 2η φ 2φ. ( ) On large scales, kη 1, φ is of the form A + B(kη) 2 so η φ = 2B(kη) 2 φ and hence δ r 2φ (kη 1) ( ) and is constant. On small scales, kη 1, φ oscillates and so φ kφ and we find δ r 2 3 (kη)2 φ = r. ( ) We see that well inside the horizon, the density perturbations in the comoving and Newtonian gauge coincide. This is indicative of the general result that there are no gauge ambiguities inside the horizon. Finally, from Eq. ( ) we see that v i = η2 2 ( φ + 1 ) x i η φ, ( ) so the Newtonian-gauge peculiar velocity of the radiation fluid vanishes like kηφ/2 on large scales. Perturbations in radiation during radiation domination How does the discussion above apply to our real universe with its multiple components (e.g. baryons, CDM, radiation and neutrinos)? Well before matter-radiation equality

35 Part-III Cosmology 35 (but well after nucleosynthesis), photons and neutrinos are the dominant components and the photons are kept isotropic in the rest frame of the baryons (here taken to include leptons, i.e. electrons) because scattering is very efficient. After neutrino decoupling, the neutrinos free-stream and their distribution function will generally not remain isotropic in the presence of fluctuations in their energy and momentum density and in the spacetime metric. However, neutrino anisotropic stress is produced causally it requires a particle to free-stream of the order of the wavelength of the perturbation so, for perturbations that are outside the Hubble radius, anisotropic stress can be ignored and φ = ψ. On these large scales, the dynamical equations of the photons and the neutrinos are identical while the baryon density is much smaller than the photon energy density 7. As we shall describe shortly, for an important class of initial conditions (called adiabatic) the photon and neutrino fractional overdensities and peculiar velocities are equal and, to the extent that neutrino anisotropic stress can be ignored, this condition is preserved in time since their dynamical equations are the same. On super-hubble scales in radiation domination for adiabatic initial conditions, the photons and neutrinos effectively behave like a single radiation fluid whose fractional overdensity equals that of the photons (and neutrinos), δ r = δ γ, and is what we calculated above. On sub-hubble scales in radiation domination, the above treatment is an approximate description of the fluctuations in the photons; a more careful treatment properly requires small corrections be made to account for differences in the clustering properties of the photons and free-streaming neutrinos. Simple inflation models predict initial fluctuations that are adiabatic. This means that the energy densities of all species are constant on hypersurfaces for which the total energy density is constant, and all species have the same peculiar velocities. The statement that the peculiar velocities are equal is gauge invariant since v i v i + L i under a gauge transformation and the L i term cancels when differencing a pair of peculiar velocities. (A simpler way to see this is to note that relative velocities are observable; they do not require a choice of coordinates to define them.) For the fluctuations in the energy densities, the quantity δρ I ρ I + P I δρ J ρ J + P J, ( ) where I and J label the species, is also gauge-invariant since δρ I δρ I T ρ I and ρ I = 3H( ρ I + P I ). Moreover, it vanishes for adiabatic fluctuations since then all δρ I = 0 in a gauge for which the constant-time hypersurfaces coincide with those of uniform total density. It follows that for adiabatic fluctuations, the total density perturbation in any gauge, δρ tot = ρ tot δ tot = I ρ I δ I, ( ) 7 This qualification of negligible baryon density is needed since the photons and baryons exchange momentum through scattering and this alters the evolution of their (common) peculiar velocity from that for a photon-only fluid if the baryons have comparable energy density to the photons.

36 Part-III Cosmology 36 is dominated by the species that is dominant in the background since all the δ I are comparable; deep in the radiation era, this is the radiation. If we now consider the (total) comoving gauge, this is necessarily comoving with the radiation in the radiation era and the density perturbation in this gauge, which is the source of the gravitational potential via the Poisson equation ( ), is just the perturbation to the radiation energy density on its comoving hypersurfaces, r. We noted above that the adiabatic conditions are preserved between photons and neutrinos on super-hubble scales. This actually holds true more generally, and in particular for mixtures of barotropic fluids [i.e. P I = P I (ρ I )]. For the densities, this follows directly from Eq. (1.3.80) and the fact that relative velocities vanish initially (we shall show this explicitly in the next section for the case of radiation and CDM); for the velocities it follows from causality since velocities can only separate via causal processes (stress gradients here) Evolution of CDM and baryon perturbations in radiation domination on large scales We discussed the evolution of CDM perturbations on sub-hubble scales using Newtonian theory in Section However, to describe the evolution on large scales we require a relativistic treatment. For adiabatic initial conditions, the CDM, baryon and radiation perturbations are related by 3δ r /4 = δ c = δ b, ( ) and they co-move with each other. As advertised above, we can show that the condition Eq. ( ) is preserved on large scales by considering their respective continuity equations. Equation (1.3.80) gives δ r + 4 i v i r/3 4 φ = 0 δ c + i v i c 3 φ = 0 η (3δ r /4 δ c ) + i (v i r vi c ) = 0, ( ) and, since the relative velocities coincide on large scales, 3δ r /4 remains equal to δ c outside the horizon for adiabatic initial conditions. A similar treatment holds for the baryons and radiation, but now vr i = vb i holds on all scales larger than the mean-free path to (Thomson) scattering for any initial condition. For adiabatic initial conditions, this implies δ b = 3δ r /4 on all scales where tight-coupling holds. We see from this discussion that, for adiabatic initial conditions, δ c = δ b = 3φ/2 on large scales in radiation domination and are constant. Once the mode enters the sound horizon, the radiation and baryons begin to oscillate acoustically and we can use

37 Part-III Cosmology 37 the Newtonian treatment of the Meszaros effect in Section to describe the CDM evolution Evolution of matter fluctuations in the matter era In the matter era, well after recombination, the comoving frame moves with the matter (CDM and baryons). The total density perturbation on comoving hypersurfaces is then essentially m and this is related to φ via 2 φ = 4πGa 2 ρ m m. ( ) The pressure fluctuation is negligible and so, from Eq. ( ), the potential evolves as φ + 3H φ + (2Ḣ + H2 )φ = 0. ( ) In matter domination, a η 2 and H = 2/η so 2Ḣ + H2 = 0 and we find φ + 6 η φ = 0. ( ) The solutions of this are φ = const. and a decaying solution φ η 5 a 5/2, just as in Newtonian theory (see equation ) but now valid on all scales. The comoving density contrast therefore evolves as m a 2 a 3 }{{} from ρ ( 1 a 5/2 } {{ } from φ ) ( a a 3/2 ). ( ) These also agree with the Newtonian treatment, but note that this is for the comoving density contrast and the Newtonian-gauge result differs (the growing mode is constant) outside the horizon. More generally, the late-time evolution of m follows from combining Eqs ( ) and ( ). Since a 2 ρ m a 1, we have which rearranges to (exercise!) 2 η( m /a) + 3H η ( /a) + (2Ḣ + H2 )( m /a) = 0, ( ) m + H m + (Ḣ H2 ) m = 0. ( ) The Friedmann equations (1.3.3) and (1.3.4) for matter and Λ give Ḣ H 2 = 1 3 a2 Λ 4πGa2 ρ m a2 Λ 8πGa2 ρ m 3 = 4πGa 2 ρ m, ( )

38 Part-III Cosmology 38 Figure 4: The variation of the density contrast in photons (red), baryons (green) and CDM (blue) with scale factor for k = 0.01 Mpc 1 (top) and k = 1.0 Mpc 1 (bottom). The density contrasts are plotted in the conformal Newtonian gauge. so, generally, the comoving density contrast evolves as m + H m 4πGa 2 ρ m m = 0. ( ) This is just the conformal-time version of the Newtonian equation (1.2.40) and so we recover the usual suppression of the growth of structure by Λ but now on all scales Summary of perturbation growth from adiabatic initial conditions The evolution of the perturbations to the energy densities of photons, baryons and CDM in the Newtonian gauge are illustrated in Fig. 4 for adiabatic initial conditions. The main points to note are as follows. The evolution is adiabatic (i.e. δρ I /( ρ I + P I ) = δρ J /( ρ J + P J ) for all species I and J) until modes enter the Hubble radius. The Newtonian-gauge densities are

39 Part-III Cosmology 39 constant outside the horizon, but the comoving gauge densities grow. Radiation and baryons undergo acoustic oscillations for modes that enter the sound horizon prior to decoupling. If modes additionally enter the Hubble radius during radiation domination, the CDM perturbation only grows logarithmically (Meszaros effect) until matterradiation equality, after which it grows like a. Baryons quickly fall into the CDM potential wells after decoupling and δ b δ c. Once dark energy comes to dominate the expansion, the density perturbation growth slows down and eventually the perturbations freeze out Comoving curvature perturbation There is an important quantity that is conserved on super-hubble scales for adiabatic, scalar fluctuations irrespective of the equation of state of the matter: the comoving curvature perturbation. This is the perturbation to the intrinsic curvature scalar of comoving hypersurfaces, i.e. those hypersurfaces orthogonal to the worldlines that comove with the total matter (i.e. for which q i = 0). Its importance is that it allows us to match the perturbations from inflation to those in the radiation-dominated universe on large scales without needing to know the (uncertain) details of the reheating phase at the end of inflation. In some arbitrary gauge, let us work out the intrinsic curvature of surfaces of constant time. The induced metric, γ ij, on these surfaces is just the spatial part of Eq. (1.3.5), i.e. γ ij a 2 [(1 2φ)δ ij + 2E ij ], ( ) and is not Euclidean because of the perturbations φ and E ij. The 3D metric has an associated (metric-compatible) connection (3) Γ i jk = 1 2 γil ( j γ kl + k γ jl l γ jk ), ( ) where γ ij is the inverse of the induced metric, γ ij = a 2 [ (1 + 2φ)δ ij 2E ij]. ( ) We actually only need the inverse metric to zero-order to compute the connection to first-order since the spatial derivatives of the γ ij are all first-order in the perturbations. We have (3) Γ i jk = δ il j ( φδ kl + E kl ) + δ il k ( φδ jl + E jl ) δ il l ( φδ jk + E jk ) = ( 2δ i (j k) φ δ il δ jk l φ ) + ( 2 (j E k) i δ il l E jk ). ( )

40 Part-III Cosmology 40 We can form the intrinsic curvature tensor in the same way as in 4D spacetime. The intrinsic curvature is the associated Ricci scalar, given by (3) R = γ ik l (3) Γ l ik γ ik k (3) Γ l il + γ ik(3) Γ l ik (3) Γ m lm γ ik(3) Γ m il (3) Γ l km. ( ) To first-order, we therefore have a 2(3) R = δ ik l (3) Γ l ik δ ik k (3) Γ l il. ( ) This involves two contractions of the connection. The first is The second is δ ik(3) Γ l ik = δ ik ( 2δ l (i k) φ δ jl δ ik j φ ) + δ ik ( 2 (i E k) l δ jl j E ik ) = 2δ kl k φ + 3δ jl j φ + 2 i E il δ jl j (δ ik E ik ) }{{} 0 = δ kl k φ + 2 k E kl. ( ) (3) Γ l il = δ l l i φ δ l i l φ + i φ + l E i l + i E l l l E i l Using these results in Eq. ( ) gives = 3 i φ. ( ) a 2(3) R = l ( δ kl k φ + 2 k E kl) + 3δ ik k i φ = 2 φ + 2 i j E ij φ = 4 2 φ + 2 i j E ij. ( ) Note that this vanishes for vector and tensor perturbations (as do all perturbed scalars) since then φ = 0 and i j E ij = 0. For scalar perturbations, E ij = i j E so i j E ij = δ il δ jm i j ( l m E 1 ) 3 δ lm 2 E = 2 2 E E = E,. ( ) Finally, we find a 2(3) R = 4 (φ ) 3 2 E. ( ) We define the curvature perturbation as (φ + 2 E/3) and the comoving curvature perturbation R is this quantity evaluated in the comoving gauge (B i = 0 = q i ). It will prove convenient to have a gauge-invariant expression for R so that we can evaluate it from the perturbations in any gauge. Since B and v vanish in the comoving gauge, we can always add linear combinations of these to (φ+ 2 E/3) to form a gauge-invariant combination that equals R.

41 Part-III Cosmology 41 Exercise: Use Eqs (1.3.55) and (1.3.59) (1.3.61), to show that R = φ E + H(B + v) ( ) is a gauge-invariant expression for the comoving curvature perturbation. We now want to prove that R is conserved on large scales for adiabatic perturbations. We shall do so by working in the conformal Newtonian gauge, in which case R = φ + Hv (since B and E vanish). We can now use the 0i Einstein equation ( ) to eliminate the peculiar velocity in favour of the potentials: We now differentiate to find R = φ H( φ + Hψ). ( ) 4πGa 2 ( ρ + P) 4πGa 2 ( ρ + P)Ṙ = 4πGa2 ( ρ + P) φ + Ḣ( φ + Hψ) + H( φ + Ḣψ + H ψ) H( φ + Hψ) ( ( ρ + P) 2H( ρ + P) + ρ + P ) = 4πGa 2 ( ρ + P) φ + Ḣ( φ + Hψ) + H( φ + Ḣψ + H ψ) [ ( )] H( φ P + Hψ) 2H 3H 1 +, ( ) ρ where we used ρ = 3H( ρ + P) in the second equality. In the term involving P, we now use the Poisson equation in the form of Eq. ( ) to replace φ + Hψ with a combination of the potential and the density contrast, and in the first term on the right we use H 2 Ḣ = 4πGa2 ( ρ + P) to find 4πGa 2 ( ρ + P)Ṙ = (H2 Ḣ) φ + Ḣ( φ + Hψ) + H( φ + Ḣψ + H ψ) + H 2 ( φ + Hψ) + H P ρ ( 2 φ 4πGa 2 ρδ ). ( ) Adding and subtracting 4πGa 2 HδP on the right-hand side and simplifying gives 4πGa 2 ( ρ + P)Ṙ [ φ = H + H ψ + 2H φ ] + (2Ḣ + H2 )ψ 4πGa 2 δp ( ) P + 4πGa 2 H δp ρ ρδ + H P ρ 2 φ. ( ) }{{} δp nad

42 Part-III Cosmology 42 Here, we have introduced the non-adiabatic pressure perturbation. It is gauge-invariant, since δp δp T P and δρ δρ T ρ, and it vanishes for a barotropic equation of state, P = P(ρ) such as in a fluid with no entropy perturbations. More generally, it vanishes for adiabatic fluctuations (see Section 1.3.5) in a mixture of barotropic fluids (since then, there exist hypersurfaces on which all the δρ I vanish, and, if all species are barotropic, the δp I will also vanish on these hypersurfaces). Finally, if we use the trace-part of the ij Einstein equation ( ) we find 4πGa 2 ( ρ + P)Ṙ = 1 3 H 2 (φ ψ) + 4πGa 2 HδP nad + H P ρ 2 φ. ( ) If the non-adiabatic pressure vanishes, the right-hand side Hk 2 φ Hk 2 R at scale k (where we have used equation ), so that Ṙ/H (k/h)2 R and vanishes on super-hubble scales 8. We saw earlier that for adiabatic fluctuations the potential is constant during radiation domination on scales larger than the sound horizon, and on all (linear) scales during matter domination. How does the potential change during the matter-radiation transition on large scales? During radiation domination, a constant potential is related to the (super-hubble) R by R = φ During matter domination, we have instead R = φ Since R is constant, we have H 2 φ 16πG ρa 2 /3 = φ H2 φ 2H 2 = 3 2 φ. ( ) H2 φ 4πG ρa = φ H2 φ 2 3H 2 /2 = 5 3 φ. ( ) 5 3 φ matter = 3 2 φ radiation φ matter = 9 10 φ radiation = 3 R. ( ) 5 This result is useful for relating the amplitude of the large-angle fluctuations in the CMB to the amplitude of primordial fluctuations. In Section 1.4, we shall compute the statistics of R at the end of inflation in Fourier space. These statistics are related to those of any (linear) observable, since, in linear perturbation theory, different scales decouple so the Fourier transform of some observable, such as δ(k), will be linear in the primordial R(k). 8 Note that the first term on the right-hand side of Eq. ( ) vanishes on all scales if the anisotropic stress vanishes.

43 Part-III Cosmology The matter power spectrum The power spectrum of the late-time distribution of matter is a key cosmological observable. It can be estimated in galaxy surveys by assuming that the fractional fluctuations in the number density of galaxies traces (generally in a biased manner) the fractional fluctuations in the matter. Consider the fractional matter overdensity in the comoving gauge well after recombination, m (η,k). The matter power spectrum P m (η; k) is then defined by m (η,k) m(η,k ) 2π2 k 3 P m (η; k)δ(k k ). ( ) The power spectrum P m (η; k) is dimensionless, but frequently a dimensional spectrum P m (η; k) is used, where P m (η; k) 2π2 k 3 P m (η; k). ( ) One may also meet real-space measures of matter clustering, such a σ R. This is defined to be the (real-space) variance of m averaged in spheres of radius R. This is equivalent to the variance of m convolved with 3Θ(R x )/(4πR 3 ), i.e. a normalised spherical top-hat of radius R. The Fourier transform of such a top-hat is W(kR), where W(x) 3 x3(sin x x cosx). ( ) From the convolution theorem, the variance of the convolved field is d σr 2 3 = k (2π) W 2 (kr)p 3 m (k). ( ) Historically, a scale R = 8h 1 Mpc is chosen, where the Hubble constant H 0 = 100h km s 1 Mpc 1, since σ 8 1. The smoothing scale at which the variance of a low-pass-filtered field exceeds unity here 8h 1 Mpc marks the scale at which perturbation theory breaks down and non-linear effects become important. In the best-fit model to the WMAP CMB data, σ The theoretical matter power spectrum at z = 0 is plotted in Fig. 5 based on linear perturbation theory. Also plotted is a fit to the results of N-body simulations that takes account of the non-linear growth of the density field on small scales. The main features of this plot are as follows. On large scales, P m (k) grows as k. The power spectrum turns over around k 0.01 Mpc 1 corresponding to the horizon size at mattter-radiation equality.

44 Part-III Cosmology 44 Figure 5: The matter power spectrum P m (k) at z = 0 in linear theory (solid) and with non-linear corrections (dashed). Beyond the peak, the power falls as k 3 ln 2 (k/k eq ), where k eq is the wavenumber of a mode that enters the horizon at the matter-radiation transition. There are small amplitude baryon acoustic oscillations in the spectrum (see also Fig. 3). Linear theory applies on scales k 1 > 10 Mpc at z = 0. The shape of the matter power spectrum as inferred from galaxy clustering agrees well with the theoretical prediction in Fig. 5. Inferring the amplitude is more problematic due to the issue of galaxy bias noted above. To understand the shape of the matter power spectrum, we note that on scales where linear perturbation theory holds we can write (for adiabatic initial conditions) m (η,k) = T(η, k)r(0,k), ( ) where T(η, k) is the transfer function which relates the primordial curvature perturbation to the comoving matter perturbation. The primordial curvature power spectrum is almost scale-free (P m (0; k) const.; see Sec. 1.4) so it contributes a factor of k 3 to P m (k). This primordial shape is modified by the transfer function whose scale dependence results from the physics of structure formation. First consider modes with k < k eq. These were outside the horizon throughout radiation domination. For these, it turns out to be simplest to consider the Newtonian-gauge

45 Part-III Cosmology 45 potential φ since, as we showed in Sec , during matter domination φ(η,k) = 3R(0,k)/5 for k < k eq. The potential is suppressed uniformly on all scales when Λ comes to dominate, so that φ(η,k)/r(0,k) remains independent of k. We can relate m to φ via the Poisson equation ( ) which, in Fourier space, gives m k 2 φ. It follows that T(η, k) k 2 k < k eq, ( ) and so P m (k) k 4 /k 3 k on large scales. For k > k eq, the mode entered the horizon during radiation domination. Generally, on sub-hubble scales, the density contrast in the comoving and Newtonian gauges are the same (see the discussion in Sec ) so we can work with δ m rather than m. Since CDM dominates the total matter, we shall simplify further by considering the evolution of δ c. The Newtonian-gauge δ c is constant in time until horizon entry and, moreover, δ c (0,k) φ(0,k) R(0,k) so all modes have approximately the same variance at horizon entry. The Meszaros effect operates inside the horizon during radiation domination and δ c then grows logarithmically with proper time t. After matter-radiation equality, δ c grows as a. Shorter wavelength modes enter the horizon earlier and have had more logarithmic growth by the end of the radiation era than longer modes. The ratio δ c (t eq,k) δ c (0,k) 1 + ln(t eq/t k ) k > k eq ( ) due to the Meszaros effect, where t eq is the time of matter-radiation equality and t k is the time of horizon entry. Since a(t k )H(t k ) = k, and a t 1/2 in radiation domination, t k 1/k 2. The Meszaros enhancement is thus ln(k/k eq ) for k > k eq. After the end of the radiation era, δ c grows uniformly on all scales (expect for the small scaledependent effect of the baryons which gives rise to the baryon acoustic oscillations in P m (k)) and so at late times the k-dependence of the transfer function is Finally, this gives T(η, k) ln(k/k eq ) k k eq. ( ) P m (k) k 3 ln 2 (k/k eq ) k k eq. ( ) 1.4 Fluctuations from inflation Inflation was originally proposed to solve shortcomings of the big bang model, but it was quickly realised that inflation should also naturally produce fluctuations in the geometry of spacetime and the post-inflationary matter distribution. The key to seeing how this arises is to note that the comoving Hubble radius, H 1, decreases in time during inflation since dh 1 dt = d dt ( ) 1 da = dt ( ) 2 da d 2 a dt dt < 0 (1.4.1) 2

46 Part-III Cosmology 46 during accelerated expansion. This means that fluctuations of fixed comoving scale exit the Hubble radius during inflation. Equivalently, the physical Hubble radius, H 1, is almost constant during slow-roll inflation but the physical wavelength of fluctuations is stretched quasi-exponentially and so pass outside the Hubble radius. Critically, this means that quantum fluctuations on (physically) microscopic scales are stretched up to cosmologically significant scales during inflation. Consider a fluctuation with comoving wavenumber k. It exits the Hubble radius during inflation at some time t k when a = a k and H = H k such that k = a k H k. During the post-inflationary deceleration of the universe, this mode will re-enter the Hubble radius. The ratio of the current Hubble radius, H0 1, to the current physical wavelength of the fluctuation is k = a kh k = a k a e H k, (1.4.2) a 0 H 0 a 0 H 0 a e a 0 H 0 where a e is the scale factor at the end of inflation. The quantity N(k) ln(a e /a k ) is the number of e-folds of inflation that occur after the mode of wavenumber k exits the Hubble radius. It follows that k a 0 H 0 = e N(k)a e a 0 e N(k)a e a 0 ( H 2 k H 2 0 ( V ρ crit,0 ) 1/2 ) 1/2, (1.4.3) where we have used the Friedmann equation in a flat universe (appropriate once inflation is appreciably underway) dominated by the potential energy, V, of a scalar field 3Hk 2 = V/M2 Pl and introduced the critical density today via 3H0 2 = 8πGρ crit,0 = ρ crit,0 /MPl 2. If we assume that the energy density at the end of inflation is approximately V (i.e. that the Hubble rate changes little between Hubble exit and the end of inflation), and that all of this energy converts instantaneously (or at least over very few e-folds) into relativistic species (i.e. radiation), then a e /a 0 (ρ rad,0 /V ) 1/4 where ρ rad,0 is the current radiation density. We then have, from Eq. (1.4.3), that ( ) k ( ρrad,0 ) ( ) 1/4 1/2 V N(k) = ln + ln + ln a 0 H 0 V ρ crit,0 ( ) ( ) k V 1/4 = ln + ln (1.4.4) a 0 H GeV So, for inflation near the GUT scale ( GeV), fluctuations that are currently at the Hubble scale exited the Hubble radius 55 e-folds before the end of inflation Quantisation of a light scalar field in slow-roll inflation During slow-roll inflation, the Hubble parameter is almost constant and the spatial sections are flat. The spacetime is therefore almost de Sitter. We shall begin by con-

47 Part-III Cosmology 47 sidering the quantum fluctuations of a light scalar field in a quasi-de Sitter Robertson- Walker universe ignoring the fluctuations in the spacetime geometry. This is clearly inconsistent with having fluctuations in the scalar field but we shall explain later how our result relates to a full calculation of the coupled fluctuations in the matter and the metric. The action for a scalar field, denoted here by Φ to avoid confusion with the metric fluctuation φ, is, in conformal time and comoving coordinates, [ 1 ( ) ] S = dηd 3 x 2 a2 Φ2 ( Φ) 2 a 4 V (Φ). (1.4.5) This follows from Eq. (1.9.9). In particular, the metric is a 2 η µν so g = a 4 and g µν µ Φ ν Φ = a 2 η µν µ Φ ν Φ. In the background cosmology, Φ = Φ(η) is homogeneous. Its equation of motion follows from varying the action in Eq. (1.4.5): Φ + 2H Φ + a 2 V = 0, (1.4.6) where V dv/dφ. Note that Eq. (1.4.6) is the conformal-time version of Eq. (1.9.11). Consider now fluctuations in Φ. It is convenient to write Φ = Φ + u/a where u = aδφ. Expanding the action to second-order in u (which is what is required to get the field equations to linear-order in the fluctuation u) we get the sum of the zero-order action, a term S (1) that is first-order in u, and a term S (2) that is second-order in u. The first-order part is [ S (1) = dηd 3 x a Φ ] u ȧ Φu a 3 V ( Φ)u dηd 3 x[ η (a Φ) + Φȧ + a 3 V ( Φ) ]u, (1.4.7) }{{} a[ Φ+2H Φ+a 2 V ( Φ)] where we have integrated by parts in the second line and dropped the boundary term since this does not contribute to the variation of the action. We see that the first-order action vanishes by virtue of the background field equation, as expected. This leaves the second-order action, S (2) = 1 dηd 3 x [ u 2 2Hu u + ( H 2 a 2 V ( Φ) ) u 2 ( u) 2]. (1.4.8) 2 Integrating the u term by parts, and dropping the boundary term, gives S (2) = 1 [ ] dηd 3 x u 2 + (Ḣ + H 2 a 2 V ( Φ) )u 2 ( u) 2 2 = 1 dηd 3 x [ u 2 + ( ä/a a 2 V ( Φ) ) u 2 ( u) 2]. (1.4.9) 2

48 Part-III Cosmology 48 During slow-roll inflation, the effective squared mass of the inflaton, m 2 eff V ( Φ) is much less than the Hubble rate squared. This follows since m 2 eff H = V ( Φ) 2 H 2 3M2 Pl V ( Φ) V ( Φ) 3η V, (1.4.10) where, recall from Eq. (1.9.20) that 3H 2 V/M 2 Pl, and the slow-roll parameter η V was introduced in Eq. (1.9.18). Since η V 1 during slow-roll inflation, we see that m 2 eff H2. Comparing the typical size of ä/a and a 2 m 2 eff that multiply u2 in the second-order action, we see that for quasi-de Sitter space, ȧ = a 2 H ä a 2ȧH = 2a2 H 2 a 2 m 2 eff. (1.4.11) We can therefore drop the effective mass term from the action, working instead with [ u 2 + (ä/a)u 2 ( u) 2] S (2) dηd 3 x 1 2 } {{ } L. (1.4.12) Apart from the (important!) term (ä/a)u 2, this looks like the action for a free field in Minkowski space with quadratic Lagrangian L. The equation of motion for u follows from varying the action in Eq. (1.4.12): Canonical quantization L u = L x µ ( µ u) ä a u = η u i (δ ij j u) 0 = ü ä a u 2 u. (1.4.13) We now aim to quantize the field u following the standard methods of canonical quantization from quantum field theory. We first define the momentum conjugate to u by π u L = u. (1.4.14) u We then promote π and u to operator-valued, Hermitian fields, ˆπ(η, x) and û(η, x) that satisfy the equal-time commutation relations [û(η,x), ˆπ(η,x )] = iδ(x x ) (1.4.15) (setting h = 1). We shall work in the Heisenberg picture where the quantum state of the system is constant in time but the operators evolve satisfying the quantum analogues of their classical equations of motion. For û(η,x), we have 2 û η 2 ä aû 2 û = 0. (1.4.16)

49 Part-III Cosmology 49 The general solution of this equation is of the form d 3 k [â(k)uk û(η,x) = (η)e ik x + â (k)u (η)e ik x] (2π) 3/2 k, (1.4.17) where â(k) is a time-independent operator, â (k) is its Hermitian conjugate, and u k (η) is a complex-valued solution of the Fourier-space version of the classical equation of motion: ü k (η) + (k 2 ä/a)u k (η) = 0. (1.4.18) Note that u k (η) should be chosen so that u k and u k are linearly independent for Eq. (1.4.17) to be the general Hermitian solution. Note further that the mode functions only depend on k = k. The momentum ˆπ(η,x) = η û(η,x) is then determined by differentiation. The operators â(k) are constrained by the requirement that the fields û and ˆπ satisfy the equal-time commutation relations in Eq. (1.4.15). The Fourier transform of û(η, x) is, from Eq. (1.4.17), and, since ˆπ(η,x) = η û(η,x), we have û(η,k) = â(k)u k (η) + â ( k)u k (η), (1.4.19) ˆπ(η,k) = â(k) u k (η) + â ( k) u k (η). (1.4.20) The fields û(η,x) and ˆπ(η,x) are Hermitian so that, in Fourier space, û (η,k) = û(η, k) and ˆπ (η,k) = ˆπ(η, k). We can solve Eqs (1.4.19) and (1.4.20) for â(k) and â ( k) to find â(k) = w 1 k [ u k (η)û(η,k) u k (η)ˆπ(η,k)], (1.4.21) where w k u k (η) u k (η) u k(η)u k (η) is the Wronskian of the mode functions. The Wronskian is time-independent since Eq. (1.4.18) for the mode functions has no firstderivative term; it is also imaginary. To form commutation relations for the â(k), we require the non-zero equal-time commutator [û(η,k), ˆπ(η,k )] = = i d 3 x d 3 x (2π) 3/2 (2π) [û(η,x), 3/2 ˆπ(η,x )] }{{} d 3 x (2π) 3e i(k+k ) x iδ(x x ) e ik x e ik x = iδ(k + k ). (1.4.22)

50 Part-III Cosmology 50 We then have [â(k), â (k )] = 1 w k w k { u k (η)u k (η)[û(η,k), ˆπ (η,k )] u k (η) u k (η)[ˆπ(η,k), û (η,k )] } We can also show that = 1 w k w k { u k(η)u k (η)[û(η,k), ˆπ(η, k )] u k(η) u k (η)[û(η, k ), ˆπ(η,k)]} = i w k w k { u k(η)u k (η) u k(η) u k (η)}δ(k k ) = iw 1 k δ(k k ). (1.4.23) [â(k), â(k )] = 0. (1.4.24) It is conventional to choose the mode functions so that the Wronskian w k = i, in which case Eq. (1.4.23) reduces to the canonical form [â(k), â (k )] = δ(k k ). In Minkowski space, the â (k) and â(k) are the usual particle creation and annihilation operators, respectively. Exercise: Verify the commutation relation in Eq. (1.4.24). Well inside the Hubble radius during inflation, k 2 ä/a and the canonically-normalised solutions of Eq. (1.4.18) are u k (η) = e ikη 2k (k H). (1.4.25) (This solution can always be multiplied by a k-dependent phase factor but this can then be absorbed into â(k) without altering the canonical commutation relations.) These mode functions are the same as in Minkowksi space and so we recover the usual quantum field theory of a free field in Minkowski space on scales well inside the Hubble radius. We shall assume that the fluctuations of the inflaton are in their ground state, 0. This state is annihilated by all the â(k): â(k) 0 = 0. (1.4.26) In the Heisenberg picture, the field remains in this quantum state as inflation proceeds. Power spectrum We can now find the power spectrum of u(η,x) and hence of the inflaton fluctuations. We get the power spectrum, P u (k) by computing the two-point correlator of the the

51 Part-III Cosmology 51 field u in Fourier space (see Eq ). The correlator is here defined as the quantum expectation value so, in the vacuum state, Noting that we have 0 û(η,k)û (η,k ) 0 = 2π2 k 3 P u(k)δ(k k ). (1.4.27) û (η,k) 0 = u k (η)â (k) 0, (1.4.28) The power spectrum of u is thus 0 û(η,k)û (η,k ) 0 = u k (η)u k (η) 0 â(k)â (k ) 0 = u k (η)u k (η) 0 [â(k), â (k )] 0 = u k (η) 2 δ(k k ). (1.4.29) P u (k) = k3 2π 2 u k(η) 2. (1.4.30) Finally, since u is related to the inflaton fluctuation by u = aδφ, the power spectrum of δφ is P δφ (k) = k3 2 u k (η) 2π 2 a(η). (1.4.31) The equations of motion for the mode functions, Eq. (1.4.18), has a growing-mode solution u a well after Hubble exit (k H) and so the power spectrum of δφ becomes constant outside the Hubble radius. We require a more detailed solution for the mode functions to set the amplitude of the power spectrum. Well inside the Hubble radius, we have seen u k = e ikη / 2k, while well outside u a. We only need the evolution through Hubble exit to interpolate between these two limits. During slow-roll inflation the fractional variation in the Hubble rate per Hubble time is small so we shall treat H as a constant, H k, for the few e-folds either side of Hubble exit. Then, in this interval ȧ/a = H = ah k a = 1 H k η. (1.4.32) Here, η at the start of inflation, kη = 1 at Hubble exit (since k = H = ah then) and η 0 in the infinite future. It follows that The mode equation is then ä = 2 H k η 3 ä a = 2 η 2. (1.4.33) ü k + (k 2 2/η 2 )u k = 0. (1.4.34)

52 Part-III Cosmology 52 The exact solution with the required behaviour well inside the Hubble radius (kη ) is ( u k (η) = e ikη 1 i ). (1.4.35) 2k kη A few e-folds after Hubble exit u k (η) ie ikη η 2k 3 so u k (η) a(η) ih ke ikη. (1.4.36) 2k 3 It follows that, for modes that have exited the Hubble radius when the Hubble rate is H k, the power spectrum of the inflaton fluctuations is constant in time and equal to ( ) 2 Hk P δφ (k) =. (1.4.37) 2π Since H varies only slowly during inflation, H k and hence P δφ (k) depend only weakly on k. This simple result is very important; it states that a light scalar field in quaside Sitter spacetime acquires an almost-scale-invariant spectrum of fluctuations with amplitude (H k /2π) 2. The spectrum would be exactly scale-invariant were it not for the small decrease in the Hubble rate during inflation The comoving curvature perturbation from slow-roll inflation At the end of inflation, the inflaton decays into other particles reheating the universe. Treating the decay as instantaneous on a hypersurface of uniform Φ we shall see shortly that such hypersurfaces are comoving and, by the Poisson equation ( ), have uniform energy density all the decay products of the inflaton will also be uniformly distributed across this surface. This means that the primordial fluctuations in the reheated universe will be adiabatic and are fully characterised by their (conserved) comoving curvature perturbation, R. So, our aim is to calculate R but the calculation of the previous section ignored the very metric fluctuations on which R depends. A more careful calculation, computing the action to second-order in perturbations for the coupled system of scalar metric and inflaton field fluctuations shows that our previous analysis is still valid provided that we interpret δφ as the field fluctuations on hypersurfaces with zero intrinsic curvature 9. We can relate δφ on zero-curvature hypersurfaces to the comoving curvature perturbation as follows. The gauge-invariant expression for R is, from Eq. ( ), R = φ E + H(B + v). (1.4.38) 9 The action for the coupled system is equivalent to Eq. (1.4.9) up to terms that vanish with the slow-roll parameters. In detail, the term ä/a V ( Φ) gets replaced with z/z where z a Φ/H. To leading-order in slow-roll, both terms reduce to ä/a.

53 Part-III Cosmology 53 We can eliminate the peculiar velocity v in terms of the inflaton field fluctuation by noting that, in any gauge, T i 0 = q i = ( ρ + P)δ ij j v. (1.4.39) The stress-energy tensor for a scalar field was given in Eq. (1.9.2), which we write here as ( ) 1 T µ ν = µ Φ ν Φ δ µ ν 2 ( ρφ) 2 V. (1.4.40) It follows that q i = g iµ µ Φ η Φ = g i0 Φ2 + g ij Φ j Φ = a 2 B i Φ2 a 2 Φδ ij j δφ = a 2 Φ2 δ ij j (B + δφ/ Φ ). (1.4.41) (Here, we have used δg µν = ḡ µα δg αβ ḡ βν which follows from the general result for the perturbation of the inverse of a matrix.) Since ρ+ P = a 2 Φ2 (see Eqs and 1.9.4), we have a 2 Φ2 δ ij j v = a 2 Φ2 δ ij j (B + δφ/ Φ ) B + v = δφ/ Φ. (1.4.42) We can thus write the comoving curvature perturbation for a universe dominated by a scalar field in gauge-invariant form as R = φ E HδΦ Φ. (1.4.43) It is straightforward to verify that this is gauge invariant using Eqs (1.3.59) and (1.3.61) and noting that Φ is a Lorentz scalar so its perturbation transforms as δφ δφ T Φ. Note that the comoving gauge has q i = B i = 0 which imply δφ = 0, as previously advertised. We have computed the quantum fluctuations of Φ in the zero-curvature gauge, in which the hypersurfaces of constant time have zero intrinsic curvature. From Eq. ( ), φ + 2 E/3 = 0 in this gauge and so R = HδΦ/ Φ (zero-curvature gauge). (1.4.44) It follows from Eq. (1.4.37) that the power spectrum of R after horizon crossing is ( ) 2 ( ) Ḣ H 2 2 P R (k) = P δφ(k) =. (1.4.45) Φ 2π t Φ

54 Part-III Cosmology 54 The right-hand side should be evaluated at Hubble exit for the given k 10. Slow-roll expansion For a slowly-rolling scalar field, we can express P R (k) directly in terms of the inflaton potential. During slow roll, the potential energy of the field dominates over the kinetic energy and so H 2 1 V ( Φ), (1.4.46) 3MPl 2 and the field evolution is friction limited: 3H t Φ = V ( Φ). (1.4.47) (See Eqs and ). It follows that ( ) H 2 2 ( ) 3H 3 2 P R (k) = (V/M2 Pl )3 2π t Φ 2πV 3(2π) 2 (V ) 2 = 8 ( ) V 1/4 4 1, (1.4.48) 3 8πMPl ǫ V where the slow-roll parameter ǫ V M2 Pl 2 ( ) V 2 (1.4.49) was introduced in Eq. (1.9.17). The large-angle CMB observations constrain P R (k) on current Hubble scales [via Eq. (1.5.33)]. It follows that V V 1/ ǫ 1/4 V GeV. (1.4.50) The quantity V 1/4 describes the energy scale of inflation and, since ǫ V 1, the energy scale is at least two orders of magnitude below the Planck scale ( GeV). It is, however, plausible, that inflation occurred around the GUT scale, GeV. Spectral index of the primordial power spectrum We have already noted that slow-roll inflation produces a spectrum of curvature perturbations that is almost scale-invariant. We can quantify the small departures from scaleinvariance by forming the spectral index n s (k). Generally, this is a scale-dependent quantity defined by n s (k) 1 d ln P R(k), (1.4.51) d lnk 10 Our earlier calculation showed that δφ was constant outside the horizon. This is not quite correct when we include the effect of metric perturbations and the effective mass of the field. The correct result is that δφ Φ/H outside the Hubble radius, so that R is constant. This is why it is correct to evaluate the H/ t Φ term in Eq. (1.4.45) at Hubble exit.

55 Part-III Cosmology 55 where the 1 is conventional and (unfortunately!) means that a scale-free spectrum has n s = 1. For a constant n s, this definition implies a power-law spectrum for some pivot scale k pivot. We can evaluate n s by noting that and, since k = ah at Hubble exit, P R (k) = A s (k/k pivot ) ns 1 (1.4.52) d d lnk = d ln k dt = H dt d lnk d Φ dt d d Φ, (1.4.53) ( 1 + ) th. (1.4.54) H 2 The Friedmann equation, and the slow-roll approximation then gives ( ) t H ρ + H = 3 P 2 2 ρ 3 2 ( t Φ) 2 V = 1 2 (3H t Φ) 2 3H 2 V ( ) M2 Pl V 2 = ǫ V, (1.4.55) 2 V so that d lnk/dt H(1 ǫ V ). We thus have, to leading-order in the slow-roll parameters, d d lnk 1 d Φ d H dt d Φ V d 3H 2 d Φ M 2 Pl We can now differentiate Eq. (1.4.48) to find V V d d Φ M Pl 2ǫV d d Φ. (1.4.56) d n s 1 = M Pl 2ǫV d Φ (ln V ln ǫ V ) ( ) V = M Pl 2ǫV V ǫ V. (1.4.57) ǫ V

56 Part-III Cosmology 56 The derivative of ǫ V is ( d ln ǫ V V d Φ = 2 V ) V V ( 2 ηv 2 ) ǫ V, (1.4.58) ǫv M Pl where η V is the slow-roll parameter related to the curvature of the potential; see Eq. (1.4.10). This gives the final, simple result n s (k) 1 = 2η V ( Φ) 6ǫ V ( Φ). (1.4.59) We see that departures from scale-invariance are first order in the slow-roll parameters. It can be shown that dn s /d lnk is second-order in slow roll so a power-law primordial power spectrum is a very good approximation for slow-roll inflation Gravitational waves from inflation Gravitational waves (tensor modes) are also excited during inflation. Recall, from Section 1.3.1, that these are described by a metric of the form ds 2 = a 2 (η) [ dη 2 (δ ij + 2E T ij )dxi dx j], (1.4.60) where Eij T is trace-free and δik k Eij T = 0. There are two degrees of freedom associated with Eij T and these behave like massless scalar fields during inflation. Their quantum fluctuations are stretched up to cosmological scales during inflation and they acquire a power spectrum P h (k) = 8 ( ) 2 Hk, (1.4.61) MPl 2 2π outside the Hubble radius, where the power spectrum conventions are such that (2Eij T )(2Eij T ) = d ln k P h (k). (1.4.62) Gravitational waves are thus a direct probe of the Hubble rate during inflation, or, using the slow-roll approximation, of the energy scale. Note that P h (k) ( V 1/4 8πMPl ) 4, (1.4.63) r P h(k) P R (k) 16ǫ V, (1.4.64)

57 Part-III Cosmology 57 which defines the dimensionless tensor-to-scalar ratio r. The spectrum of gravitational waves is almost scale-invariant with a spectral index n t d ln P h(k) d lnk d lnv d lnk = M V Pl 2ǫV V = M Pl 2ǫV 2ǫV M Pl = 2ǫ V. (1.4.65) Note that this is always negative (the spectrum is said to be red) which is a direct consequence of the Hubble parameter falling as inflation proceeds. It follows that r 8n t in slow-roll inflation which is an example of a slow-roll consistency relation between the spectra of curvature perturbations and gravitational waves Current observational constraints on r and n s In slow-roll inflation, the quantities r and n s constrain the fractional gradient and curvature of the inflaton potential and so constrain its shape. The effect of increasing n s is to enhance the power in cosmological observables, such as the CMB anisotropy power spectrum, on small scales over that on large scales. The CMB can also be used to constrain primordial gravitational waves, though future direct detection of gravitational waves from space may ultimately prove to be a more sensitive probe. Gravitational waves produce an anisotropic expansion of space (a shear) once they enter the Hubble radius which causes the redshift of a CMB photon to depend on its direction of propagation. This gives rise to temperature anisotropies but they are confined to large angular scales since the amplitude of gravitational waves falls away asymptotically as 1/a inside the Hubble radius. Since there are few independent modes in the CMB to measure on large angular scales, the accuracy of any determination of r via this route is limited 11. Current constraints in the r-n s plane are plotted in Fig. 6. These parameters are rather degenerate in current data: increasing n s and r boosts power on all scales and so, if the amplitude A s is reduced, produces little net effect on the CMB angular power spectrum. The constraints on the individual parameters are r < 0.22 (95% confidence) and n s = 0.960±0.013 (with 68% error interval). The result for n s has been interpreted as supportive evidence that cosmological inflation really occurred. Other observations supporting inflation include: The curvature of the universe is measured to be close to zero: < Ω K < ; 11 Polarization of the CMB is a more promising method for detecting primordial gravitational waves.

58 Part-III Cosmology 58 Figure 6: Constraints in the r-n s plane from five years of WMAP CMB data (blue) and WMAP plus distance information from supernovae and baryon acoustic oscillations (red). In each case the 68% and 95% contours are plotted. Note the degeneracy between r and n s. Credit: WMAP science team. There is no strong evidence for non-gaussianity of the primordial fluctuations 12 ; There are measured correlations between the CMB temperature anisotropies and polarization on large angular scales that can only have arisen from apparently acausal (i.e. super-hubble) fluctuations around the time of recombination; There is no evidence that the primordial fluctuations were not adiabatic. For many, detecting a background of primordial gravitational waves consistent with a red power spectrum would be conclusive evidence of inflation. The ultimate test of single-field, slow-roll inflation would be verification of the slow-roll consistency relation, r = 8n t, but this is extremely challenging observationally. 12 We have not had time to discuss the Gaussianity or otherwise of the primordial fluctuations here. Slow-roll inflation implies that the fluctuations are essentially free fields and, for the vacuum state 0, must therefore be Gaussian distributed. (The ground state wavefunction of the harmonic oscillator is a Gaussian function for the same reason.)