Cosmological Perturbation Theory, Part 1 Hannu Kurki-Suonio 17.3.2020 About These Lecture Notes

Home , Newtonian gauge, Perturbation (astronomy), Primordial fluctuations

Cosmological Perturbation Theory, part 1 Hannu Kurki-Suonio 17.3.2020 About these lecture notes:

I lectured a course on cosmological perturbation theory at the University of Helsinki in the spring of 2003, in the fall of 2010, in the fall of 2015; and now I am lecturing it in spring 2020. These notes are modified from the 2015 notes for 2020; work in progress. This course cover similar topics as Cosmology II: structure formation, inflation, and a bit of CMB; but on a deeper level and more widely. In Cosmology II we accepted many results as given; now they are derived rigorously from general relativity. Topics not covered: Quantum field theory is not used, except for a brief discussion of generation of primordial perturbations during inflation in Section 6.6. Non-linear effects or higher- order perturbation theory are not covered. Open and closed cosmologies are not covered, i.e., the background (unperturbed) spacetime is assumed flat. Of CMB we discuss only the Sachs–Wolfe effect, in Section 26. I have lectured a couple of times a separate course on CMB Physics, but it is not currently in the teaching program. There is an unfortunate variety in the notation employed by various authors. My notation is the result of first learning perturbation theory from Mukhanov, Feldman, and Brandenberger (Phys. Rep. 215, 213 (1992)) and then from Liddle and Lyth (Cosmological Inflation and Large-Scale Structure, Cambridge University Press 2000, Chapters 14 and 15), and represents a compromise between their notations.

1 Contents

1 Perturbative General Relativity 1

2 The Background Universe 2

3 The Perturbed Universe 4

4 Gauge Transformations 5 4.1 Gauge Transformation of the Metric Perturbations ...... 8 4.2 Perturbation theory as gauge theory ...... 10

5 Separation into Scalar, Vector, and Tensor Perturbations 10

6 Perturbations in Fourier Space 13 6.1 Gauge Transformation in Fourier Space ...... 15 6.2 About the Fourier Convention ...... 16

7 Scalar Perturbations 17 7.1 BardeenPotentials ...... 18

8 Conformal–Newtonian Gauge 19 8.1 Perturbation in the Curvature Tensors ...... 19

9 Perturbation in the Energy Tensor 21 9.1 Separation into Scalar, Vector, and Tensor Parts ...... 23 9.2 Gauge Transformation of the Energy Tensor Perturbations...... 24 9.2.1 GeneralRule ...... 24 9.2.2 ScalarPerturbations ...... 24 9.2.3 Conformal-Newtonian Gauge ...... 24 9.3 Scalar Perturbations in the Conformal-Newtonian Gauge ...... 24

10 Field Equations for Scalar Perturbations in the Newtonian Gauge 25

11 Energy-Momentum Continuity Equations 26

12 Perfect Fluid Scalar Perturbations in the Newtonian Gauge 27 12.1 FieldEquations...... 27 12.2 Adiabaticperturbations ...... 28 12.3 FluidEquations...... 29 12.4 Adiabatic perturbations at superhorizon scales ...... 29

13 Scalar Perturbations for a Barotropic Perfect Fluid 29

14 Scalar Perturbations in the Matter-Dominated Universe 30

15 Scalar Perturbations in the Radiation-Dominated Universe 33

16 Other Gauges 35 16.1 SlicingandThreading ...... 35 16.2ComovingGauge ...... 36 16.3MixingGauges ...... 38 16.4 ComovingCurvaturePerturbation ...... 38

2 16.5 Perfect Fluid Scalar Perturbations, Again ...... 40 16.5.1 Adiabatic Perfect Fluid Perturbations at Superhorizon Scales ...... 41 16.6 UniformEnergyDensityGauge ...... 41 16.7 SpatiallyFlatGauge ...... 42

17 Synchronous Gauge 43

18 Fluid Components 47 18.1 DivisionintoComponents ...... 47 18.2 Gauge Transformations and Entropy Perturbations ...... 48 18.3Equations ...... 49

19 Simpliﬁed Matter+Radiation Universe 51 19.1 Background solution for radiation+matter ...... 51 19.2Perturbations ...... 52 19.3InitialEpoch ...... 55 19.3.1 Adiabaticmodes ...... 55 19.3.2 IsocurvatureModes ...... 56 19.3.3 OtherPerturbations ...... 57 19.4 Full evolution for large scales ...... 58 19.5 Initial Conditions in Terms of Conformal Time ...... 59

20 Eﬀect of a Homogeneous Component 62 20.1 GeneralConsiderations...... 62 20.2 Λ+PerturbedComponent ...... 64 20.3Λ+PerfectFluid...... 65 20.4Λ+Matter ...... 66

21 The Real Universe 67

22 Primordial Era 69 22.1 NeutrinoAdiabaticity ...... 71 22.2Matter...... 72 22.2.1 The Completely Adiabatic Solution ...... 73 22.2.2 Baryon and CDM Entropy Perturbations ...... 73 22.3 Neutrinoperturbations...... 74 22.3.1 Neutrinos in the early universe ...... 75 22.3.2 Primordial neutrino density isocurvature mode ...... 75 22.4 Tight-Coupling Approximation ...... 77 22.5 Real-Universe Initial Conditions as Expansion in ConformalTime ...... 77

23 Superhorizon Evolution and Relation to Original Perturbations 82

24 Gaussian Initial Conditions 83

25 Large Scales 87 25.1 Superhorizonevolution...... 88 25.1.1 Adiabaticmode...... 88 25.1.2 Isocurvaturemodes...... 89 25.2 Throughthehorizon ...... 89

3 26 Sachs–Wolfe Eﬀect 91 26.1 Ordinary Sachs-Wolfe Eﬀect and Primordial Correlations ...... 94

27 Approximate Treatment of the Smaller Scales 95 27.1 Small Scales during Radiation-Dominated Epoch ...... 96 27.1.1 Radiation ...... 97 27.1.2 Matter...... 98 27.2 Small Scales to the Matter-Dominated Epoch ...... 100 27.3 TransferFunction...... 101

28 Acoustic Oscillations of the Baryon-Photon Fluid 105 28.1 Insidethehorizon ...... 106

29 Modiﬁed Gravity 108

A General Perturbation 111

B Rules to Convert between Diﬀerent Time Coordinates 114

C Equations 115

4 1 PERTURBATIVE GENERAL RELATIVITY 1

1 Perturbative General Relativity

In the perturbation theory of general relativity one considers a spacetime, the perturbed spacetime, that is close to a simple, symmetric, spacetime, the background spacetime, that we already know. In the development of perturbation theory we keep referring to these two diﬀerent spacetimes (see Fig. 1).

Figure 1: The background spacetime and the perturbed spacetime.

This means that there exists a coordinate system on the perturbed spacetime, where its metric can be written as gµν =g ¯µν + δgµν , (1.1) whereg ¯µν is the metric of the background spacetime (we shall refer to the background quantities 1 with the overbar ), and δgµν is small. We also require that the ﬁrst and second partial derivatives, 2 δgµν,ρ and δgµν,ρσ are small . The curvature tensors and the energy tensor of the perturbed spacetime can then be written as

µ ¯µ µ Gν = Gν + δGν (1.2) µ ¯µ µ Tν = Tν + δTν , (1.3)

µ µ where δGν and δTν are small. Subtracting the Einstein equations of the two spacetimes,

µ µ ¯µ ¯µ Gν = 8πGTν and Gν = 8πGTν , (1.4) from each other, we get the ﬁeld equation for the perturbations

µ µ δGν = 8πGδTν . (1.5)

The above discussion requires a pointwise correspondence between the two spacetimes, so that we can perform the comparisons and the subtractions. This correspondence is given by the coordinate system (x0,x1,x2,x3): the point P¯ in the background spacetime and the point P in the perturbed spacetime which have the same coordinate values, correspond to each other. Now, given a coordinate system on the background spacetime, there exist many coordinate systems,

1Beware: the overbars will be dropped eventually. 2Actually it is not always necessary to require the second derivatives of the metric perturbation to be small; but then our development would require more care. 2 THE BACKGROUND UNIVERSE 2 all close to each other, for the perturbed spacetime, for which (1.1) holds. The choice among these coordinate systems is called the gauge choice, to be discussed a little later. In first-order (or linear) perturbation theory, we drop all terms from our equations which contain products of the small quantities δgµν , δgµν,ρ and δgµν,ρσ . The field equation (1.5) becomes then a linear differential equation for δgµν , making things much easier than in full GR. In second-order perturbation theory, one keeps also those terms with a product of two (but no more) small quantities. In these lectures we only discuss first-order perturbation theory. The simplest case is the one where the background is the Minkowski space. Theng ¯µν = ηµν , µ µ and G¯ν = T¯ν = 0. In cosmological perturbation theory the background spacetime is the Friedmann–Robertson– Walker universe. Now the background spacetime is curved, and is not empty. While it is homogeneous and isotropic, it is time-dependent. In these lectures we shall only consider the case where the background is the flat FRW universe. This case is much simpler than the open and closed ones3, since now the t = const time slices (“space at time t”) have Euclidean geometry. This will allow us to do 3-dimensional Fourier transformations in space. The separation into the background and perturbation is always done so that the spatial average of the perturbation is zero, i.e., the background value (at time t) is the spatial average of the full quantity over the time slice t = const.

2 The Background Universe

Our background spacetime is the ﬂat Friedmann–Robertson–Walker universe FRW(0). The background metric in comoving coordinates (t,x,y,z) is

ds2 =g ¯ dxµdxν = dt2 + a2(t)δ dxidxj = dt2 + a2(t)(dx2 + dy2 + dz2) , (2.1) µν − ij − where a(t) is the scale factor, to be solved from the (ﬂat universe) Friedmann equations,

a˙ 2 8πG H2 = = ρ¯ (2.2) a 3 a¨ 4πG = (¯ρ + 3¯p) (2.3) a − 3 where H a/a˙ is the Hubble parameter, ≡ d , ·≡ dt and theρ ¯ andp ¯ are the homogeneous background energy density and pressure. Another version of the second Friedmann equation is

a¨ a˙ 2 H˙ = 4πG(¯ρ +p ¯) . (2.4) ≡ a − a − We shall ﬁnd it more convenient to use as a time coordinate the conformal time η, deﬁned by dt dη = (2.5) a(t) so that the background metric is

ds2 =g ¯ dxµdxν = a2(η) dη2 + δ dxidxj = a2(η)( dη2 + dx2 + dy2 + dz2) . (2.6) µν − ij − 3For cosmological perturbation theory for open or closed FRW universe, see e.g. Mukhanov, Feldman, Bran- denberger, Phys. Rep. 215, 203 (1992). 2 THE BACKGROUND UNIVERSE 3

That is, 2 µν 2 µν g¯ = a (η)η g¯ = a− (η)η , (2.7) µν µν ⇒ where 1 − µν 1 [ηµν ] [η ]   (2.8) ≡ ≡ 1  1      (the “Minkowski metric”). Using the conformal time, the Friedmann equations are (exercise)

a 2 8πG 2 = ′ = ρa¯ 2 (2.9) H a 3 4πG 2 ′ = (¯ρ + 3¯p)a , (2.10) H − 3 where d d ′ = a = a( )˙ (2.11) ≡ dη dt and a ′ = aH =a ˙ (2.12) H ≡ a is the conformal, or comoving, Hubble parameter. Note that

2 a′ ′ a′′ a′ 2 2 a¨ 2 2 ′ = = = (aa˙ )˙ a˙ = aa¨ = a = a (H˙ + H ) . (2.13) H a a − a − a The energy continuity equation

ρ¯˙ = 3H(¯ρ +p ¯) (2.14) − becomes just ρ¯′ = 3 (¯ρ +p ¯) 3 (1 + w)¯ρ . (2.15) − H ≡ − H For later convenience we deﬁne the equation-of-state parameter p¯ w (2.16) ≡ ρ¯ and the speed-of-sound parameter4 2 p¯˙ p¯′ cs . (2.17) ≡ ρ¯˙ ≡ ρ¯′ These two quantities always refer to the background values. From the Friedmann equations (2.9,2.10) and the continuity equation (2.15) one easily derives additional useful background relations, like (exercise)

1 2 ′ = (1 + 3w) , (2.18) H − 2 H w ′ = 3 (w c2) , (2.19) 1+ w H − s and 2 p¯′ = wρ¯′ + w′ρ¯ = 3 (1 + w)c ρ¯ . (2.20) − H s 4Its square root turns out to be the speed of sound if our ρ and p describe ordinary ﬂuid. Even if they do not, we nevertheless deﬁne this quantity, although the name may then be misleading. 3 THE PERTURBED UNIVERSE 4

1 1 Eq. (2.18) shows that w = 3 corresponds to constant comoving Hubble length − = const. 1 − H 1 For w < 3 the comoving Hubble length shrinks with time (“inﬂation”), whereas for w > 3 − 2 − it grows with time (“normal” expansion). When w = const., we have cs = w. From these one can derive (exercise) further background relations, useful5 for converting 2 from equation-of-state quantities w and cs to Hubble quantities:

1 2 2H˙ 2 2H˙ w = 1+ H′ = 1 1+ w = 1 H′ = −3 2 − − 3H2 ⇒ 3 − 2 −3H2 H H 2 ˙ 2 2 1 ′′ 2( ′) 1 ¨ 2H cs w = HH − H = H − 3 2 2 −3HH˙ − H H H −H′ ! 3 ¨ 2 1 ′′ ′ H cs = H − HH −H = 1 3 2 − − 3HH˙ H H −H′ 4 + 2 3 H¨ 1+ c2 = H′′ − HH′ H = . (2.21) s 3 ( 2 ) −3HH˙ H H −H′ We see that the Hubble parameter H decreases with time (H˙ < 0) for normal forms of matter and energy (w> 1) and is constant for a universe with nothing but vacuum energy (w = 1). − − The case w < 1 is called phantom energy. We shall always assume w > 1 (unless otherwise − − speciﬁed).

3 The Perturbed Universe

We write the metric of the perturbed (around FRW(0)) universe as

2 gµν =g ¯µν + δgµν = a (ηµν + hµν ) , (3.1) where hµν , as well as hµν,ρ and hµν,ρσ are assumed small. Since we are doing first-order perturbation theory, we shall drop from all equations all terms with are of order (h2) or higher, and O just write “=” to signify equality to first order in hµν . Here the perturbation hµν is not a tensor in the perturbed universe, neither is ηµν, but we define

hµ ηµρh , hµν ηµρηνσh . (3.2) ν ≡ ρν ≡ ρσ One easily ﬁnds (exercise), that the inverse metric of the perturbed spacetime is

µν 2 µν µν g = a− (η h ) (3.3) − (to first order). We shall now give different names for the time and space components of the perturbed metric, defining6 2A Bi [hµν ] − − (3.4) ≡ Bi 2Dδij + 2Eij − − where D 1 hi 1 h (3.5) ≡− 6 i ≡− 6 carries the trace h of the spatial metric perturbation hij, and Eij is traceless,

δijE Ei E = 0 . (3.6) ij ≡ i ≡ ii 5Well, some of the H forms do not look very useful. I list them here for completeness. 6Liddle and Lyth ([3], Eq. (14.95); [7], Eq. (8.8)) have the opposite sign convention for D. 4 GAUGE TRANSFORMATIONS 5

Since indices on hµν are raised and lowered with ηµν , we immediately have 2A +B [hµν ] − i (3.7) ≡ +Bi 2Dδij + 2Eij − i ij On Bi and Eij we do not raise/lower indices (or if we do, it is just the same thing, B = δ Bj = Bi). The line element is thus

ds2 = a2(η) (1 + 2A)dη2 2B dηdxi + [(1 2D)δ + 2E ] dxidxj . (3.8) − − i − ij ij i i The function A(η,x ) is called the lapse function, and Bi(η,x ) the shift vector.

4 Gauge Transformations

The association between points in the background spacetime and the perturbed spacetime is via the coordinate system xα . As we noted earlier, for a given coordinate system in the { } background, there are many possible coordinate systems in the perturbed spacetime, all close to each other, that we could use. In GR perturbation theory, a gauge transformation means a coordinate transformation between such coordinate systems in the perturbed spacetime. (It may be helpful to temporarily forget at this point what you have learned about gauge transformations in other ﬁeld theories, e.g. electrodynamics, so that you can learn the properties of this concept here with a fresh mind, without preconceptions.) In this section, we denote the coordinates of the background by xα, and two diﬀerent coordinate systems in the perturbed spacetime (corresponding to two “gauges”) byx ˆα andx ˜α.7 The coordinatesx ˆα andx ˜α are related by a coordinate transformation

x˜α =x ˆα + ξα , (4.1)

α α ∂ξα ∂ξα where ξ and the derivatives ξ ,β are ﬁrst-order small. The diﬀerence between ∂xˆβ and ∂x˜β is α α second-order small, and thus ignored, so we can write just ξ ,β. In fact, we shall think of ξ as living on the background spacetime. The situation is illustrated in Fig. 2. The coordinate system xˆα associates point P¯ in the { } background with Pˆ, whereas x˜α associates the same background point P¯ with another point { } P˜. The association is by x˜α(P˜) =x ˆα(Pˆ)= xα(P¯) . (4.2) The coordinate transformation relates the coordinates of the same point in the perturbed spacetime, i.e.,

x˜α(P˜) =x ˆα(P˜)+ ξα x˜α(Pˆ) =x ˆα(Pˆ)+ ξα . (4.3)

Now the diﬀerence ξα(P˜) ξα(Pˆ) is second-order small. Thus we write just ξα and associate it − with the background point: ξα = ξα(P¯)= ξα(xβ) . Using Eqs. (4.2) and (4.3), we get the relation between the coordinates of the two diﬀerent points in a given coordinate system,

xˆα(P˜) =x ˆα(Pˆ) ξα − x˜α(P˜) =x ˜α(Pˆ) ξα . (4.4) − 7Ordinarily (when not doing gauge transformations) we write just xα for both the background and perturbed spacetime coordinates. 4 GAUGE TRANSFORMATIONS 6

Figure 2: Gauge transformation. The background spacetime is on the left and the perturbed spacetime, with two diﬀerent coordinate systems, is on the right. The “const” for the coordinate lines refer to the same constant for the two coordinate systems, i.e.,η ˜ =η ˆ = const η(P¯) andx ˜i =x ˆi = const xi(P¯). ≡ ≡

Let us now consider how various tensors transform in the gauge transformation. We have, of course, the usual GR transformation rules for scalars (s), vectors (wα), and other tensors,

s = s ˆ wα˜ = Xα˜wβ βˆ ˆ Aα˜ = Xα˜Xδ Aγˆ β˜ γˆ β˜ δ˜ ˆ B = XγˆXδ B (4.5) α˜β˜ α˜ β˜ γˆδˆ where ∂x˜α Xα˜ = δα + ξα βˆ ≡ ∂xˆβ β ,β δ ˆ ∂xˆ Xδ = δδ ξδ . (4.6) β˜ ≡ ∂x˜β β − ,β These rules refer to the values of these quantities at a given point in the perturbed spacetime. However, this is not what we want now! Please, pay attention, since the following is central to understanding GR perturbation theory! We shall be interested in perturbations of various quantities. In the background spacetime α ¯α ¯ we may have various 4-scalar fieldss ¯, 4-vector fieldsw ¯ and tensor fields A β, Bαβ. In the perturbed spacetime we have corresponding perturbed quantities,

s =s ¯ + δs wα =w ¯α + δwα α ¯α α A β = A β + δA β Bαβ = B¯αβ + δBαβ . (4.7) Consider first the 4-scalar s. The full quantity s =s ¯ + δs lives on the perturbed spacetime. However, we cannot assign a unique background quantitys ¯ to a point in the perturbed spacetime, 4 GAUGE TRANSFORMATIONS 7 because in different gauges this point is associated with different points in the background, with different values ofs ¯. Therefore there is also no unique perturbation δs, but the perturbation is gauge-dependent. The perturbations in different gauges are defined as δs(xα) s(Pˆ) s¯(P¯) ≡ − δs(xα) s(P˜) s¯(P¯) . (4.8) b ≡ − The perturbation δs is obtained frome a subtraction between two spacetimes, and we consider it as living on the background spacetime. It changes in the gauge transformation. Relate now δs to δs: ∂s ∂s ∂s¯ b s(P˜)= s(Pˆ)+ (Pˆ) xˆα(P˜) xˆα(Pˆ) = s(Pˆ) (Pˆ)ξα = s(Pˆ) (P¯)ξα , e ∂xˆα − − ∂xˆα − ∂xα h i where we approximated ∂s (Pˆ) ∂s¯ (P¯), since the difference8 between them is a first order ∂xˆα ≈ ∂xα perturbation, and multiplication by ξα makes it second order. Since our background is homogeneous (but time dependent!) ,s ¯ =s ¯(η,xi) =s ¯(η) only, and

∂s¯ α ∂s¯ 0 0 (P¯)ξ = (P¯)ξ =s ¯′ξ . ∂xα ∂η Thus we get 0 s(P˜)= s(Pˆ) s¯′ξ , (4.9) − and our ﬁnal result for the gauge transformation of δs is

α 0 α 0 δs(x ) = s(Pˆ) s¯′ξ s¯(P¯) = δs(x ) s¯′ξ . (4.10) − − − In analogy with (4.8),e the perturbations in vector andb tensor fields in the two gauges are defined α δw (xβ) wαˆ(Pˆ) w¯α(P¯) α ≡ − δw (xβ) wα˜(P˜) w¯α(P¯) . (4.11) c ≡ − and f α δA (xγ ) Aαˆ (Pˆ) A¯α (P¯) β ≡ βˆ − β α δA (xγ ) Aα˜ (P˜) A¯α (P¯) c β ≡ β˜ − β δB (xγ ) B (Pˆ) B¯ (P¯) fαβ ≡ αˆβˆ − αβ δB (xγ ) B (P˜) B¯ (P¯) . (4.12) c αβ ≡ α˜β˜ − αβ Consider the case of a typef (0, 2) 4-tensor field. We have ∂B ∂B¯ B (P˜) = B (Pˆ)+ µˆνˆ xˆα(P˜) xˆα(Pˆ) = B (Pˆ) µν (P¯)ξα (4.13) µˆνˆ µˆνˆ ∂xˆα − µˆνˆ − ∂xα h i and ∂B¯ B (P˜) = XρˆXσˆB (P˜) = (δρ ξρ )(δσ ξσ ) B (Pˆ) ρσ (P¯)ξα µ˜ν˜ µ˜ ν˜ ρˆσˆ µ − ,µ ν − ,ν ρˆσˆ − ∂xα ∂B¯ = B (Pˆ) ξρ B (Pˆ) ξσ B (Pˆ) µν (P¯)ξα µˆνˆ − ,µ ρˆνˆ − ,ν µˆσˆ − ∂xα ∂B¯ = B (Pˆ) ξρ B¯ (P¯) ξσ B¯ (P¯) µν (P¯)ξα , (4.14) µˆνˆ − ,µ ρν − ,ν µσ − ∂xα 8 This difference is the perturbation of the covariant vector s,α. We are assuming perturbations are first order small also in quantities derived by covariant derivation from the “primary” quantities. 4 GAUGE TRANSFORMATIONS 8

where we can replace Bµˆσˆ(Pˆ) with B¯µσ(P¯) in the two middle terms, since it is multiplied by a σ ﬁrst-order quantity ξ ,ν and we can thus ignore the perturbation part, which becomes second order. Subtracting the background value at P¯ we get the gauge transformation rule for the tensor perturbation δBµν ,

δB B (P˜) B¯ (P¯) µν ≡ µ˜ν˜ − µν ∂B¯ = B (Pˆ) B¯ (P¯) ξρ B¯ (P¯) ξσ B¯ (P¯) µν (P¯)ξα f µˆνˆ − µν − ,µ ρν − ,ν µσ − ∂xα = δB ξρ B¯ ξσ B¯ B¯ ξα . (4.15) µν − ,µ ρν − ,ν µσ − µν,α In a similar mannerc we obtain the gauge transformation rules for 4-vector perturbations, α α δw = δw + ξα w¯β w¯α ξβ (4.16) ,β − ,β and perturbations of type (1, 1) 4-tensorsf c (exercise), µ µ δA = δA + ξµ A¯ρ ξσ A¯µ A¯µ ξα . (4.17) ν ν ,ρ ν − ,ν σ − ν,α Since the background is isotropicf c and homogeneous, and our background coordinate system fully respects these properties, the background 4-vectors and tensors must be of the form

A¯0 0 w¯α = (w ¯0,~0) A¯α = 0 , (4.18) β 0 1 δi A¯k 3 j k and they depend only on the (conformal) time coordinate η. Using these properties we can write (exercise) the gauge transformation rules for the individual components of 4-scalar, 4-vector and type (1,1) 4-tensor perturbations (we now drop the hats from the ﬁrst gauge) as

0 δs = δs s¯′ξ 0 − δw = δw0 + ξ0 w¯0 w¯0 ξ0 e ,0 − ,0 i i i 0 fδw = δw + ξ ,0w¯ 0 δA = δA0 A¯0 ξ0 (4.19) f0 0 − 0,0 0 δA = δA0 + 1 ξ0 A¯k ξ0 A¯0 f i i 3 ,i k − ,i 0 i δA = δAi + ξi A¯0 1 ξi A¯k f 0 0 ,0 0 − 3 ,0 k i δA = δAi 1 δi A¯k ξ0 . (4.20) f j j − 3 j k,0 i The following combinations (thef trace and the traceless part of δAj) are also useful:

k δA = δAk A¯k ξ0 f k k − k,0 i k δA 1 δi δA = δAi 1 δi δAk . (4.21) j − 3 j f k j − 3 j k i Thus the traceless part of δAj fis gauge-invariant!f

4.1 Gauge Transformation of the Metric Perturbations Applying the gauge transformation equation (4.15) to the metric perturbation, we have

δg = δg ξρ g¯ ξσ g¯ g¯ ξ0 , (4.22) µν µν − ,µ ρν − ,ν µσ − µν,0 e 4 GAUGE TRANSFORMATIONS 9

α 0 where we have replaced the sumg ¯µν,αξ withg ¯µν,0ξ , since the background metric depends only on the time coordinate x0 = η, and dropped the hats from the ﬁrst gauge. Remembering 2 g¯µν = a (η)ηµν from Eq. (2.7), we have

g¯µν,0 = 2a′aηµν (4.23) and a δg = δg + a2 ξρ η ξσ η 2 ′ η ξ0 . (4.24) µν µν − ,µ ρν − ,ν µσ − a µν From Eqs. (3.1) ande (3.4) we have

2 2A Bi [δgµν ]= a − − (4.25) B 2Dδ + 2E − i − ij ij Applying the gauge transformation law (4.24) now separately to the diﬀerent metric perturbation components, we get ﬁrst

δg 2a2A˜ 00 ≡ − a = δg + a2 ξρ η ξσ η 2 ′ η ξ0 e 00 − ,0 ρ0 − ,0 0σ − a 00 a = 2a2A + a2 +ξ0 + ξ0 + 2 ′ ξ0 , (4.26) − ,0 ,0 a from which we obtain the gauge transformation law

a A˜ = A ξ0 ′ ξ0 . (4.27) − ,0 − a

Similarly, from δg0i we obtain B˜ = B + ξi ξ0 , (4.28) i i ,0 − ,i and from δgij, a Dδ˜ + E˜ = Dδ + E 1 (ξi + ξj ) ′ ξ0δ . (4.29) − ij ij − ij ij − 2 ,j ,i − a ij 1 i j k The trace of 2 (ξ ,j + ξ ,i) is ξ ,k, so we can write

1 (ξi + ξj )= 1 δ ξk + 1 (ξi + ξj ) 1 δ ξk , (4.30) 2 ,j ,i 3 ij ,k 2 ,j ,i − 3 ij ,k where the last two terms are the traceless part, and we can separate Eq. (4.29) into

a D˜ = D + 1 ξk + ′ ξ0 3 ,k a E˜ = E 1 (ξi + ξj )+ 1 δ ξk . (4.31) ij ij − 2 ,j ,i 3 ij ,k Sometimes it may turn out that a gauge transformation causes all the perturbations to vanish. This means that the perturbations were not real (or physical)—the perturbed spacetime diﬀered from the background spacetime only by having a perturbed coordinate system! We say that such perturbations are “pure gauge”. 5 SEPARATION INTO SCALAR, VECTOR, AND TENSOR PERTURBATIONS 10

4.2 Perturbation theory as gauge theory The perturbations are defined as differences between the perturbed spacetime and the background spacetime. So in which spacetime do they “live”? What we are doing in perturbation theory, is that we are building a description of the perturbed spacetime as a set of functions, or fields, the perturbations, which live on the background spacetime9. The background spacetime is the canvas on which we are painting the picture of the perturbed spacetime. For the same perturbed spacetime there exists many equivalent descriptions of this kind and these are related by gauge transformations. The formalism of this theory of fields in the background spacetime is similar to gauge theories of particle physics.

5 Separation into Scalar, Vector, and Tensor Perturbations

In GR perturbation theory there are two kinds of coordinate transformations of interest. One is the gauge transformation just discussed, where the coordinates of the background are kept ﬁxed, but the coordinates in the perturbed spacetime are changed, changing the correspondence between the points in the background and the perturbed spacetime. The other kind is one where we keep the gauge, i.e, the correspondence between the background and perturbed spacetime points, ﬁxed, but do a coordinate transformation in the background spacetime. This then induces a corresponding coordinate transformation in the perturbed spacetime. Our background coordinate system was chosen to respect the symmetries of the background, and we do not want to lose this property. In cosmological perturbation theory we have chosen the background coordinates to respect its homogeneity property, which gives us a unique slicing of the spacetime into homogeneous t = const. spacelike slices. Thus we do not want to change this slicing. This leaves us: 1. homogeneous transformations of the time coordinate, i.e., reparameterizations of time, of which we already had an example, when we switched from cosmic time t to conformal time η, 2. and transformations in the space coordinates10

i′ i′ k x = X kx , (5.1) i′ where Xk is independent of time; which is the case we consider in this section. We had chosen the coordinates for our background, FRW(0), so that the 3-metric was Eu- clidean, 2 gij = a δij , (5.2) and we want to keep this property. This leaves us rotations. The full transformation matrices are then µ′ 1 0 1 0 µ 1 0 ′ ′ X ρ = i = i and X ρ′ = i , (5.3) 0 X 0 R 0 R ′ k k k i′ 11 T i′ i′ T where R k is a rotation matrix , with the property R R = I, or R kR l = (R R)kl = δkl. Thus T 1 i′ k R = R− so that R k = R i′ . 9In practice, they are just functions of the four coordinates, η, x, y, z, of the background spacetime, and since in cosmological perturbation theory the time slicing of the background spacetime is fixed, and since we here just consider flat background spacetimes, the perturbations become just fields in Euclidean space that evolve in time. 10In this section we use ′ to denote the other coordinate system. Do not confuse with ′ ≡ d/dη in the other sections. ′ 11 i i In this notation R j and R j′ are two different matrices, corresponding to opposite rotations; the position of the ′ indicates which way we are rotating. We have put the first index upstairs to follow the Einstein summation convention—but we could have written Ri′j and Rij′ just as well. 5 SEPARATION INTO SCALAR, VECTOR, AND TENSOR PERTURBATIONS 11

This coordinate transformation in the background induces the corresponding transformation,

µ′ µ′ ρ x = X ρ x , (5.4) into the perturbed spacetime. Here the metric is

1 2A B 2A B g = a2 − − − i = a2η + a2 − − i . (5.5) µν B (1 2D)δ + 2E µν B 2Dδ + 2E − i − ij ij − i − ij ij Transforming the metric, µ ′ ′ ν gρ σ = X ρ′ X σ′ gµν , (5.6) we get for the diﬀerent components

µ ν 0 0 2 g ′ ′ = X ′ X ′ g = X ′ X ′ g = g = a ( 1 2A) 0 0 0 0 µν 0 0 00 00 − − µ ν 0 j 2 j g ′ ′ = X ′ X ′ g = X ′ X ′ g = a R ′ B 0 l 0 l µν 0 l 0j − l j i j 2 i j i j g ′ ′ = X ′ X ′ g = a (1 2D)δ R ′ R ′ + 2E R ′ R ′ k l k l ij − ij k l ij k l 2 h i j i = a (1 2D)δ + 2E R ′ R ′ , (5.7) − kl ij k l h i from which we identify the perturbations in the new coordinates,

A′ = A

D′ = D j ′ Bl = R l′ Bj j ′ ′ i Ek l = R k′ R l′ Eij . (5.8) Thus A and D transform as scalars under rotations in the background spacetime coordinates, Bi transforms as a 3-vector, and Eij as a 3D tensor. While staying in a fixed gauge, we can thus think of them as scalar, vector, and tensor fields on the 3D Euclidean background space. We are, however, not yet satisfied. We can extract two more scalar quantities and one more vector quantity from Bi and Eij. We know from Euclidean 3D vector calculus, that a vector field can be divided into two parts, the first one with zero curl, the second one with zero divergence,

B~ = B~ S + B~ V , with B~ S =0 and B~ V = 0 , (5.9) ∇× ∇ · and that the ﬁrst one can be expressed as (minus) a gradient of some scalar ﬁeld12

B~ S = B . (5.10) −∇ In component notation,

B = B + BV , where δijBV = 0 . (5.11) i − ,i i i,j

In like manner, the symmetric traceless tensor ﬁeld Eij can be divided into three parts,

S V T Eij = Eij + Eij + Eij , (5.12)

12This sign convention corresponds to thinking of the scalar function B as a “potential”, where the vector field S B~ “flows downhill”. We use the same letter B here for both the original vector field Bi and this scalar potential B. There should be no confusion since the vector field always has an index (or~ ). Same goes for the E. 5 SEPARATION INTO SCALAR, VECTOR, AND TENSOR PERTURBATIONS 12

S V 13 where Eij and Eij can be expressed in terms of a scalar ﬁeld E and a vector ﬁeld Ei,

ES = ∂ ∂ 1 δ 2 E = E 1 δ δklE (5.13) ij i j − 3 ij∇ ,ij − 3 ij ,kl V 1 ij ~ E = (Ei,j + Ej,i), where δ Ei,j = E = 0 (5.14) ij − 2 ∇ · ik T ij T and δ Eij,k = 0, δ Eij = 0 . (5.15)

S V We see that Eij is symmetric and traceless by construction. Eij is symmetric by construction, T and the condition on Ei makes it traceless. The tensor Eij is assumed symmetric, and the two conditions on it make it transverse and traceless. The meaning of “transverse” and the nature of the above construction becomes clearer in the next section when we do this in Fourier space. Under rotations in background space,

A′ = A, B′ = B , D′ = D , E′ = E , j j V V ′ Bl′ = R l′ Bj , El = R l′ Ej , T i j T Ek′l′ = R k′ R l′ Eij . (5.16) The metric perturbation can thus be divided into

1. a scalar part, consisting of A, B, D, and E,

V 2. a vector part, consisting of Bi and Ei, T 3. and a tensor part Eij. The names “scalar”, “vector”, and “tensor” refer to their transformation properties under rotations in the background space.14 µ µ The Einstein tensor perturbation δGν and the energy tensor perturbation δTν can likewise µ be divided into scalar+vector+tensor; the scalar part of δGν coming only from the scalar part of δgµν and so on. The important thing about this division is that the scalar, vector, and tensor parts do not couple to each other (in ﬁrst-order perturbation theory), but they evolve independently. This allows us to treat them separately: We can study, e.g., scalar perturbations as if the vector and tensor perturbations were absent. The total evolution of the full perturbation is just a linear superposition of the independent evolution of the scalar, vector, and tensor part of the perturbation. V We imposed one constraint on each of the 3-vectors Bi and Ei, and 3 + 1 = 4 constraints T on the symmetric 3-d tensor Eij leaving each of them 2 independent components. Thus the 10 degrees of freedom corresponding to the 10 components of the metric perturbation hµν are divided into

1 + 1 + 1 + 1 = 4 scalar (2 physical, 2 gauge) 2 + 2 = 4 vector (2 physical, 2 gauge) 2 = 2 tensor (5.17) degrees of freedom.

13 kl ik T I have a tendency to write things like δ E,kl and (later) δ kkEij , to respect the traditional Einstein summation convention, where the indices to be summed over are one up, one down; but since here the location of the T space index has no eﬀect, one could as well right these in the shorter forms E,kk and kiEij . 14Thus “scalar” does not mean, e.g., that the perturbation would be invariant under gauge transformations— scalar perturbations are not gauge-invariant, as we have already seen, e.g. in Eqs. (4.27) and (4.31). 6 PERTURBATIONS IN FOURIER SPACE 13

The scalar perturbations are the most important. They couple to density and pressure perturbations and exhibit gravitational instability: overdense regions grow more overdense. They are responsible for the formation of structure in the universe from small initial perturbations. Vector perturbations couple to rotational velocity perturbations in the cosmic fluid. They tend to decay in an expanding universe, and are therefore probably not important in cosmology. We have done all of the above in a fixed gauge. It turns out that gauge transformations affect scalar and vector perturbations, but tensor perturbations are gauge-invariant (shown in Sec. 6.1). Tensor perturbations are gravitational waves, this time15 in an expanding universe. When they are extracted from the total perturbation by the above separation procedure, they are automatically in the “transverse traceless gauge”. (This expression is clarified in the next section.) Tensor perturbations have cosmological importance since, if strong enough, they have an observable effect on the anisotropy of the cosmic microwave background.

6 Perturbations in Fourier Space

Because our background space is ﬂat we can Fourier expand the perturbations. For an arbitrary perturbation f = f(η,xi)= f(η, ~x), we write

i~k ~x f(η, ~x)= f~k(η)e · . (6.1) X~k (Using a Fourier sum implies using a ﬁducial box with some volume V . At the end of the day we can let V , and replace remaining Fourier sums with integrals.) In ﬁrst-order perturbation →∞ theory each Fourier component evolves independently. We can thus just study the evolution of ~ a single Fourier component, with some arbitrary wave vector k, and we drop the subscript ~k from the Fourier amplitudes. Since ~x = (x1,x2,x3) is a comoving coordinate, ~k is a comoving wave vector. The comoving (or coordinate) wave number k ~k and wavelength λ = 2π/k are related to the physical ≡ | | wavelength and wave number of the Fourier mode by

2π 2π 1 kphys = = = a− k . (6.2) λphys aλ

Thus the wavelength λphys of the Fourier mode ~k grows in time as the universe expands. In the separation into scalar+vector+tensor, we follow Liddle&Lyth and include an additional factor k ~k in the Fourier components of B and E , and a factor k2 in E, so that we ≡ | | i have, e.g.,

B~k(η) i~k ~x B(η, ~x)= e · k X~k E~k(η) i~k ~x E(η, ~x)= e · . (6.3) k2 X~k S S The purpose of this is to make them have the same dimension and magnitude as Bi , Eij and V 16 Eij . That is, S V S V T Bi = Bi + Bi , and Eij = Eij + Eij + Eij , (6.4)

15Contrasted to perturbation theory around Minkowski space, which is the way gravitational waves are usually introduced in GR. 16 V T Powers of k cancel in Eqs. (6.5). The metric perturbations, A, B, D, E, Bi , Ei, and Eij will then all have the same dimension in Fourier space, which facilitates comparison of their magnitudes. This unorthodox convention is actually very confusing and you have to learn to watch for it. Additional reasons for this convention are given in Sec. 6.2. 6 PERTURBATIONS IN FOURIER SPACE 14 where k BS = B becomes BS = i i B i − ,i i − k k k ES = ∂ ∂ 1 δ 2 E becomes ES = i j + 1 δ E , ij i j − 3 ij∇ ij − k2 3 ij i EV = 1 (E + E ) becomes EV = (k E + k E ) , (6.5) ij − 2 i,j j,i ij −2k i j j i and the conditions

ij V ij ik T ij T δ Bi,j = 0 , δ Ei,j = 0 , and δ Eij,k = δ Eij = 0 (6.6) become

δij k BV = ~k B~ V = 0 , δijk E = ~k E~ = 0 , and δikk ET = δij ET = 0 . (6.7) j i · j i · k ij ij To make the separation into scalar+vector+tensor parts as clear as possible, rotate the background coordinates so that the z axis becomes parallel to ~k,

~k = kzˆ = (0, 0, k) (6.8)

(ˆz denoting the unit vector in z direction.) Then

BS = (0, 0, iB) (6.9) i − and 1 1 0 3 E 3 E ES = 0 + 1 E = 1 E (6.10) ij    3   3  E 1 E 2 E − 3 − 3       and we can write the scalar part of δgµν as

2A +iB − 2( D + 1 E) δgS = a2  − 3  (6.11) µν 2( D + 1 E) − 3  +iB 2( D 2 E)   − − 3    For the vector part we have then

~k B~ V = 0 B~ V = (B ,B , 0) (6.12) · ⇒ 1 2 ~k E~ = 0 E~ = (E , E , 0) (6.13) · ⇒ 1 2 and E i i 1 EV = − (k E + k E )= E , (6.14) ij 2k i j j i −2  2  E1 E2   so that the vector part of δgµν is

B B − 1 − 2 B iE δgV = a2  − 1 − 1  . (6.15) µν B iE − 2 − 2  iE iE   − 1 − 2    6 PERTURBATIONS IN FOURIER SPACE 15

For the tensor part,

δikk ET k ET = 0 ET ET = 0 (6.16) k ij ≡ i ij ⇒ 3j ≡ j3 Xi so that T T E11 E12 ET = ET ET . (6.17) ij  12 − 11    The tensor part of δgµν becomes

T T T 2 2E11 2E12 2 h+ h δgµν = a  T T  = a  ×  , (6.18) 2E12 2E11 h h+ − × −             T 1 where we have denoted the two gravitational wave polarization amplitudes by E11 = 2 h+ and T 1 E12 = h . 2 × We see how the scalar part of the perturbation is associated with the time direction, the wave direction ~k and the trace. The vector part is associated with the two remaining space directions, those transverse to the wave vector. Thus the vectors have only two independent components. The tensor part is also associated with these two transverse directions; being also symmetric and traceless, it thus has only two independent components. Putting all together, the full metric perturbation (for a Fourier mode in the z direction) is

2A B1 B2 +iB − −1 − 2 B1 2( D + 3 E)+ h+ h iE1 δgµν = a  − − 1× −  . (6.19) B2 h 2( D + E) h+ iE2 − × − 3 − −  +iB iE iE 2( D 2 E)   − 1 − 2 − − 3    6.1 Gauge Transformation in Fourier Space Consider then how the gauge transformation appears in Fourier space. We need now the Fourier transform of the gauge transformation vector

µ µ i~k ~x ξ (η, ~x)= ξ (η)e · . (6.20) ~k X~k For a single Fourier mode, the gauge transformation Eqs. (4.27,4.28,4.31) become

0 a′ 0 A˜ = A (ξ )′ ξ (6.21) − − a i 0 B˜ = B + (ξ )′ ik ξ (6.22) i i − i a D˜ = D + 1 ik ξi + ′ ξ0 (6.23) 3 i a E˜ = E 1 i k ξj + k ξi + 1 iδ k ξk . (6.24) ij ij − 2 i j 3 ij k For illustration, consider again a mode in the z direction, ~k = (0, 0, k). Now the new part added into the matrix of Eq. (6.19) is

0 a′ 0 1 2 3 0 2(ξ )′ + 2 ξ (ξ )′ (ξ )′ (ξ )′ + ikξ a ′ 1 − a 0 − − 1 2  (ξ )′ 2 a ξ ikξ  a − 2 − a′ 0 − 2 (6.25) (ξ )′ 2 a ξ ikξ  − 3 0 1 − 2 − a′ 0 3   (ξ )′ + ikξ ikξ ikξ 2 ξ 2ikξ   − − − − a −    6 PERTURBATIONS IN FOURIER SPACE 16

(note the cancelations on the diagonal from the D and Eij parts). We see that no new tensor part is introduced. Thus the tensor part of the metric perturbation is gauge-invariant17. But we see that the components ξ0 and ξ3 are responsible for a new scalar part and the components ξ1 and ξ2 are responsible for a new vector part. For Fourier modes in an arbitrary direction, the above means that the time component ξ0 and the component of the space part ξ~ parallel to ~k are responsible for a change in the scalar perturbation and the transverse part of ξ~ is responsible for a change in the vector perturbation. This freedom of doing gauge transformations can be used, e.g., to set 2 of the 4 scalar quantities of scalar perturbations and 2 of the 4 independent components of vector perturbations to zero. Thus only two of the degrees of freedom are real physical degrees of freedom in each case. Thus there are in total 6 physical degrees of freedom, 2 scalar, 2 vector, and 2 tensor. The other 4 (of the total 10) are just gauge degrees of freedom, representing perturbing just the coordinates, not the spacetime. If the perturbation can be completely eliminated by a gauge transformation, we say the perturbation is “pure gauge”, i.e., it is not a real perturbation of spacetime, just a perturbation in the coordinates.

6.2 About the Fourier Convention Let us call the preceding method of introducing the Fourier components of scalar, vector, and tensor parts method A. Consider a vector ﬁeld

i~k ~x ~v(~x)= ~v~ke · . (6.26) X~k Its scalar part is ~vS(~x) = v(~x), where the potential v(~x) is Fourier expanded according to −∇ Eq. (6.3), so that

v(A) ~k i~k ~x v(~x)= e · k X~k ~ S ik (A) i~k ~x ~v (~x)= − v e · (6.27) k ~k X~k Another way (method B) to introduce the separation into scalar, vector, and tensor parts is to do it in Fourier space. We divide ~v = ~vS + ~vV into scalar and vector parts so that ~vS is the ~k ~k ~k ~k component parallel to ~k, with v(B) ~vS (6.28) ~k ≡ | ~k | and ~vV the component orthogonal to ~k. Thus ~vS = (~k/k)v(B) and ~k ~k ~k ~ S k (B) i~k ~x ~v (~x)= v e · . (6.29) k ~k X~k In this method there is no “unconventional” division of Fourier components with k (or k2 for scalar parts of tensor ﬁelds). In both methods the ﬁnal quantity to represent the scalar

17To make the meaning of this statement clear: Consider a perturbation that is initially purely tensor, and do an arbitrary gauge transformation. The parts that get added to the perturbation are of scalar and/or vector nature. Thus this perturbation is not gauge-invariant; but its tensor part—because of the way we have deﬁned the tensor part of a perturbation—is. The scalar and vector parts that appeared in the gauge transformation are pure gauge. 7 SCALAR PERTURBATIONS 17

perturbation in Fourier space is v~k. In both methods, or conventions, v~k has the same magnitude, corresponding to the magnitude of ~vS, which is important later when we compare terms in ~k equations based on powers of (k/ ); but they have different phase: H v(B) = iv(A) . (6.30) ~k − ~k Convention A has the advantage that the coefficients in all perturbation equations stay real, whereas in convention B the imaginary unit i appears here and there. (Note that in Fourier space the perturbations themselves have complex values: their relative phases are related to how the maxima and minima of different quantities are located in different places in the plane waves. It could also be considered an advantage of convention B that the i in the equations reminds us of this.) Liddle and Lyth[3] use convention A. Dodelson[5] uses convention B. We will keep using convention A. We will introduce a number of scalar quantities for which this applies. These include the scalar parts of vector fields: B, v, ξ; and the scalar parts of tensor fields: E, Π.

7 Scalar Perturbations

From here on (except for the beginning of Sec. 9, where we discuss perturbations in the energy tensor) we shall consider scalar perturbations only. They are the ones responsible for the structure of the universe (i.e., the deviation from the homogeneous and isotropic FRW universe). The metric is now

ds2 = a(η)2 (1 + 2A)dη2 + 2B dηdxi + [(1 2ψ)δ + 2E ] dxidxj , (7.1) − ,i − ij ,ij where we have deﬁned1819 the curvature perturbation

ψ D + 1 2E . (7.2) ≡ 3 ∇ In Fourier space this reads ψ = D 1 E . (7.3) ~k ~k − 3 ~k The components of hµν are

2A B h = − ,i . (7.4) µν B 2ψδ + 2E ,i − ij ,ij

Exercise: Curvature of the spatial hypersurface. The hypersurface η = const. is a 3- (3) i dimensional curved manifold. Calculate the connection coeﬃcients Γjk and the scalar curvature (3) ij (3) R g Rij of this 3-space for a scalar perturbation in terms of ψ and E. ≡ Now if we start from a pure scalar perturbation and do an arbitrary gauge transformation, represented by the ﬁeld ξµ = (ξ0,ξi), we may introduce also a vector perturbation. This vector perturbation is however, pure gauge, and thus of no interest. Just like we did for the shift vector i Bi earlier, we can divide ξ into a part with zero divergence (a transverse part) and a part with zero curl, expressible as a gradient of some function ξ,

ξi = ξi δij ξ = ξ~ ξ where ξi = ξ~ = 0 . (7.5) tr − ,j tr −∇ tr,i ∇ · tr 18In my spring 2003 lecture notes (CMB Physics / Cosmological Perturbation Theory) I was using the symbol ψ for what I am now denoting D. The present notation is better in line with common usage. 19My sign convention for ψ is that of MFB[1]. The opposite sign convention is common. 7 SCALAR PERTURBATIONS 18

i 0 The part ξtr is responsible for the spurious vector perturbation, whereas ξ and ξ,j change the scalar perturbation. For our discussion of scalar perturbations we thus lose nothing, if we decide i that we only consider gauge transformations, where the ξtr part is absent. These “scalar gauge transformations” are fully speciﬁed by two functions, ξ0 and ξ,

η˜ = η + ξ0(η, ~x) x˜i = xi δijξ (η, ~x) (7.6) − ,j and they preserve the scalar nature of the perturbation. Also for ξ we use the Fourier convention (6.3a). Applied to scalar perturbations and gauge transformations, our transformation equations (4.27,4.28,4.31) become

a′ A˜ = A ξ0′ ξ0 − − a 0 B˜ = B + ξ′ + ξ a D˜ = D 1 2ξ + ′ ξ0 − 3 ∇ a E˜ = E + ξ , (7.7) where we use the notation ∂/∂η for quantities which depend on both η and ~x. The quantity ′ ≡ ψ deﬁned in Eq. (7.2) is often used as the fourth scalar variable instead of D. For it, we get

a ψ˜ = ψ + ′ ξ0 = ψ + ξ0 . (7.8) a H In Fourier space the last three equations in (7.7) become

0 B˜ = B + ξ′ + kξ D˜ = D + 1 kξ + ξ0 3 H E˜ = E + kξ . (7.9)

7.1 Bardeen Potentials We now deﬁne the following two quantities, called the Bardeen potentials,2021

Φ A + (B E′) + (B E′)′ ≡ H1 2 − − Ψ D + E (B E′) = ψ (B E′) . (7.10) ≡ 3 ∇ −H − −H − These quantities are invariant under gauge transformations (exercise).

20These may not appear well-motivated deﬁnitions just now, but wait until you see (8.3) in Sec. 8. 21For the second Bardeen potential, the opposite sign convention is common. Often Ψ is used to denote my Φ, and Φ to denote my Ψ or −Ψ. My sign and naming convention here is that of MFB[1]. Bardeen[2] originally called them ΦA and −ΦH . 8 CONFORMAL–NEWTONIAN GAUGE 19

8 Conformal–Newtonian Gauge

We can use the gauge freedom to set the scalar perturbations B and E equal to zero. From Eq. (7.7) we see that this is accomplished by choosing

ξ = E 0 − ξ = B + E′ . (8.1) − Doing this gauge transformation we arrive at a commonly used gauge, which has many names: the conformal-Newtonian gauge (or sometimes, for short, just the Newtonian gauge), the longi- tudinal gauge, Poisson gauge, and the zero-shear gauge. We shall denote quantities in this gauge with the sub- or superscript N. Thus BN = EN = 0, whereas you immediately see that

AN = Φ DN = ψN = Ψ . (8.2)

Thus the Bardeen potentials are equal to the two nonzero metric perturbations in the conformal- Newtonian gauge. In the Newtonian limit both Bardeen potentials become equal to the Newto- nian gravitational potential perturbation. From here on, until otherwise noted, we shall calculate in the conformal-Newtonian gauge. The metric is thus just22

ds2 = a(η)2 (1 + 2Φ)dη2 + (1 2Ψ)δ dxidxj , (8.3) − − ij or

2 1 2Φ µν 2 1+2Φ g = a − − and g = a− − , (8.4) µν (1 2Ψ)δ (1 + 2Ψ)δ − ij ij or 2Φ 2Φ h = − and hµν = − . (8.5) µν 2Ψδ 2Ψδ − ij − ij 8.1 Perturbation in the Curvature Tensors From the conformal-Newtonian metric (8.3) we get the connection coeﬃcients

Γρ = 1 gρσ (g + g g ) (8.6) µν 2 σµ,ν σν,µ − µν,σ as (exercise)

′ 0 a 0 0 a′ a′ Γ00 = a +Φ′ Γ0k =Φ,k Γij = a δij 2 a (Φ+Ψ)+Ψ′ δij − (8.7) ′ h i Γi =Φ Γi = a δi Ψ δi Γi = Ψ δi +Ψ δi +Ψ δ 00 ,i 0j a j − ′ j kl − ,l k ,k l ,i kl and the sums

α a′ Γ = 4 +Φ′ 3Ψ′ 0α a − Γα = Φ 3Ψ (8.8) iα ,i − ,i 22 2 2 2 i j Dodelson ([5], Eq. (4.9)) has ds = a(η) −(1 + 2Ψ)dη +(1+2Φ)δij dx dx , so he has both the opposite 2 2 2 i j sign and naming convention. Lyth&Liddle ([7], Eq. (8.32)) has ds = a(η) −(1 + 2Ψ)dη + (1 − 2Φ)δij dx dx , so they have my sign convention but opposite naming convention. 8 CONFORMAL–NEWTONIAN GAUGE 20 where we have dropped all terms higher than ﬁrst order in the small quantities Φ and Ψ. Thus these expressions contain only 0th and 1st order terms, and separate into the background and perturbation, accordingly: α ¯α α Γβγ = Γβγ + δΓβγ , (8.9) where ¯0 ¯0 ¯0 Γ00 = Γ0k = 0 Γij = δij H H (8.10) Γ¯i = 0 Γ¯i = δi Γ¯i = 0 00 0j H j kl and 0 0 0 δΓ00 =Φ′ δΓ0k =Φ,k δΓij = [2 (Φ+Ψ)+Ψ′] δij − H (8.11) δΓi =Φ δΓi = Ψ δi δΓi = Ψ δi +Ψ δi +Ψ δ . 00 ,i 0j − ′ j kl − ,l k ,k l ,i kl The Ricci tensor is

R = Γα Γα +Γα Γβ Γα Γβ µν νµ,α − αµ,ν αβ νµ − νβ αµ = R¯ + δΓα δΓα + Γ¯α δΓβ + Γ¯β δΓα Γ¯α δΓβ Γ¯β δΓα . (8.12) µν νµ,α − αµ,ν αβ νµ νµ αβ − νβ αµ − αµ νβ Calculation gives (exercise)

2 R = 3 ′ + 3Ψ′′ + Φ + 3 (Φ′ +Ψ′) 00 − H ∇ H R0i = 2(Ψ′ + Φ),i H2 Rij = ( ′ + 2 )δij H H 2 2 + Ψ′′ + Ψ (Φ′ + 5Ψ′) (2 ′ + 4 )(Φ + Ψ) δ − ∇ −H − H H ij + (Ψ Φ) (8.13) − ,ij µ Next we raise an index to get Rν . Note that we can not just raise the index of the background and perturbation parts separately, since

µ µα µα µα ¯ ¯µ µα ¯ µα Rν = g Rαν = (¯g + δg )(Rαν + δRαν )= Rν + δg Rαν +g ¯ δRαν . (8.14)

We get

0 2 2 2 R0 = 3a− ′ + a− 3Ψ′′ Φ 3 (Φ′ +Ψ′) 6 ′Φ 0 H2 − −∇ − H − H R = 2a− Ψ′ + Φ i − H ,i i 0 2 R = R = 2a− Ψ′ + Φ 0 − i H ,i i 2 2 i R = a− ( ′ + 2 )δ j H H j 2 2 2 + a− Ψ′′ + Ψ (Φ′ + 5Ψ′) (2 ′ + 4 )Φ δij 2 − ∇ −H − H H + a− (Ψ Φ) . (8.15) − ,ij and summing for the curvature scalar

0 i R = R0 + Ri 2 2 = 6a− ( ′ + ) 2H H 2 2 + a− 6Ψ′′ + 2 (2Ψ Φ) 6 (Φ′ + 3Ψ′) 12( ′ + )Φ . (8.16) − ∇ − − H − H H 9 PERTURBATION IN THE ENERGY TENSOR 21

And, ﬁnally, the Einstein tensor

0 0 1 G0 = R0 2 R − 2 2 2 2 2 = 3a− + a− 2 Ψ + 6 Ψ′ + 6 Φ 0 −0 H − ∇ H H Gi = Ri Gi = Ri = R0 = G0 0 0 − i − i Gi = Ri 1 δi R j j − 2 j 2 2 i = a− ( 2 ′ )δ − H −H j 2 2 2 i + a− 2Ψ′′ + (Φ Ψ) + (2Φ′ + 4Ψ′) + (4 ′ + 2 )Φ δ ∇ − H H H j 2 + a− (Ψ Φ) . (8.17) − ,ij Note the background (written ﬁrst) and perturbation parts in all these quantities. Since the µ µ background R¯ν and G¯ν are diagonal, the oﬀ-diagonals contain just the perturbation, and we have 0 0 0 0 Ri = Gi = δRi = δGi . (8.18)

9 Perturbation in the Energy Tensor

Consider then the energy tensor23. The background energy tensor is necessarily of the perfect ﬂuid form24

T¯µν = (¯ρ +p ¯)¯uµu¯ν +p ¯g¯µν ¯µ µ µ Tν = (¯ρ +p ¯)¯u u¯ν +pδ ¯ ν . (9.1) Because of homogeneity,ρ ¯ =ρ ¯(η) andp ¯ =p ¯(η). Because of isotropy, the fluid is at rest, u¯i = 0 u¯µ = (¯u0, 0, 0, 0) in the background universe. Since ⇒ u¯ u¯µ =g ¯ u¯µu¯ν = a2η u¯µu¯ν = a2(¯u0)2 = 1 , (9.2) µ µν µν − − we have 1 u¯µ = (1,~0) andu ¯ = a( 1,~0) . (9.3) a µ − The energy tensor of the perturbed universe is µ ¯µ µ Tν = Tν + δTν . (9.4) Just like the metric perturbation, the energy tensor perturbation has 10 degrees of freedom, of which 6 are physical and 4 are gauge. It can likewise be divided into scalar+vector+tensor, with 4+4+2 degrees of freedom, of which 2+2+2 are physical. The perturbation can also be divided into perfect fluid + non-perfect, with 5+5 degrees of freedom. µ µ The perfect fluid degrees of freedom in δTν are those which keep Tν in the perfect fluid form

µ µ µ Tν = (ρ + p)u uν + pδν . (9.5) Thus they can be taken as the density perturbation, pressure perturbation, and velocity perturbation 1 ρ =ρ ¯ + δρ, p =p ¯ + δp , and ui =u ¯i + δui = δui v . (9.6) ≡ a i 23This section could actually have been earlier. We do not specify a gauge here, and the restriction to scalar perturbations is done only in the end. 24The “imperfections” can only show up in the energy tensor if there is inhomogeneity or anisotropy. Whether an observer would “feel” thep ¯ as pressure is another matter, which depends on the interactions of the ﬂuid particles. But gravity only cares about the energy tensor. 9 PERTURBATION IN THE ENERGY TENSOR 22

The δu0 is not an independent degree of freedom, because of the constraint u uµ = 1. We µ − shall call v aui (9.7) i ≡ the velocity perturbation. It is equal to the coordinate velocity, since (to ﬁrst order)

dxi ui ui = = = aui = v . (9.8) dη u0 u¯0 i

It is also equal to the ﬂuid velocity observed by a comoving (i.e., one whose xi = const.) observer, since the ratio of change in comoving coordinate dxi to change in conformal time dη equals the ratio of the corresponding physical distance adxi to the change in cosmic time dt = adη. We also deﬁne the relative energy density perturbation δρ δ , (9.9) ≡ ρ¯ which is a dimensionless quantity in coordinate space (but not in Fourier space). µ To express u and uν in terms of vi, write them as

µ µ µ 1 0 1 1 1 u =u ¯ + δu a− + δu , a− v , a− v , a− v ≡ 1 2 3 u =u ¯ + δu ( a + δu , δu , δu , δu ) . (9.10) ν ν ν ≡ − 0 1 2 3 These are related by u = g uν and u uµ = 1. Using ν µν µ − 1 2A B g = a2 − − − i , (9.11) µν B (1 2D)δ + 2E − i − ij ij we get

µ 2 1 0 ij 2 1 u = g u = a ( 1 2A)(a− + δu ) δ a B a− v 0 0µ − − − i j = a a2δu0 2aA (9.12) − − − st (where we dropped higher than 1 order quantities, like Bivj), from which follows

δu = a2δu0 2aA . (9.13) 0 − − Likewise δu = u = g uµ = aB + av . (9.14) i i iµ − i i We solve the remaining unknown, δu0 from 1 u uµ = . . . = 1 2aδu0 2A = 1 δu0 = A (9.15) µ − − − − ⇒ −a Thus we have for the 4-velocity 1 uµ = (1 A, v ) and u = a( 1 A, v B ) . (9.16) a − i µ − − i − i Inserting this into Eq. (9.5) we get

µ ¯µ µ Tν = Tν + δTν ρ¯ 0 δρ (¯ρ +p ¯)(v B ) = − + − i − i . (9.17) 0pδ ¯ i (¯ρ +p ¯)v δpδi j − i j 9 PERTURBATION IN THE ENERGY TENSOR 23

i There are 5 remaining degrees of freedom in the space part, δTj , corresponding to perturbations away from the perfect ﬂuid from. We write them as

δp δT i = δpδi +Σ p¯ + Π . (9.18) j j ij ≡ p¯ ij Here Σ and Π Σ /p¯ are symmetric and traceless, which makes the separation into ij ij ≡ ij δp δT k (9.19) ≡ k and Σ δT i 1 δi δT k (9.20) ij ≡ j − 3 j k i unique (the trace and the traceless part of δTj ). Σij is called anisotropic stress or anisotropic pressure. Πij is its dimensionless version. For a perfect ﬂuid Σij = Πij = 0.

9.1 Separation into Scalar, Vector, and Tensor Parts µ The energy tensor perturbation δTν is built out of the scalar perturbations δρ, δp, the 3-vector ~v = vi and the traceless 3-tensor Πij. Just like for the metric perturbations, we can extract a scalar perturbation out of ~v:

v = vS + vV , where vS = v i i i i − ,i and ~vV = 0 . (9.21) ∇ · and a scalar + a vector perturbation out of Πij:

S V T Πij = Πij + Πij + Πij , (9.22) where

ΠS = ∂ ∂ 1 δ 2 Π ij i j − 3 ij∇ ΠV = 1 (Π + Π ) and (9.23) ij − 2 i,j j,i ik T δ Πij,k = 0 . (9.24)

We see that perfect fluid perturbations (Πij = 0) do not have a tensor perturbation component. For Fourier components of vi and Πij we use the same (Liddle&Lyth) convention as for Bi and Eij (see Sec. (6)), so that k vS = i i v i − k k k ΠS = i j + 1 δ Π . (9.25) ij − k2 3 ij In the early universe, we have anisotropic pressure from the cosmic neutrino background during and after neutrino decoupling, and from the cosmic microwave background during and after photon decoupling. Perturbations in the metric will make the momentum distribution of noninteracting particles anisotropic (this is anisotropic pressure). If there are sufficient interactions among the particles, these will isotropize the momentum distribution. Decoupling means that the interactions become too weak for this. If we aim for high precision in our calculations, we need to take this anisotropic pressure into account. For a more approximate treatment, the perfect fluid approximation can be made, which simplifies the calculations significantly. 9 PERTURBATION IN THE ENERGY TENSOR 24

9.2 Gauge Transformation of the Energy Tensor Perturbations 9.2.1 General Rule Using the gauge transformation rules from Sect. 4 we have

0 0 0 0 0 δT = δρ = δT T¯ ξ = δρ +ρ ¯′ξ 0 − 0 − 0,0 − i δT = (¯ρ +p ¯)˜v = δT i + ξi (T¯0 1 T¯k) f 0 − e i 0 ,0 0 − 3 k i = (¯ρ +p ¯)vi ξ (¯ρ +p ¯) f − − ,0 1 k 1 k k 0 0 δT = δp = (δT T¯ ξ ) = δp p¯′ξ 3 k 3 k − k,0 − i k δT 1 δi δT =p ¯Π˜ = δT i 1 δi δT k =p ¯Π (9.26) j − 3 j f k e ij j − 3 j k ij and we get the gaugef transformationf laws for the diﬀerent parts of the energy tensor perturbation:

0 δρ = δρ ρ¯′ξ (9.27) − 0 δp = δp p¯′ξ (9.28) e − i v˜i = vi + ξ,0 (9.29) e Π˜ ij = Πij (9.30) ρ¯ δ˜ = δ ′ ξ0 = δ + 3 (1 + w)ξ0 . (9.31) − ρ¯ H

i Thus the anisotropic stress is gauge-invariant (being the traceless part of δTj ). Note that the δρ and δp equations are those of a perturbation of a 4-scalar, as they should be, as ρ and p are, indeed, 4-scalars.

9.2.2 Scalar Perturbations For scalar perturbations, v = v and ξi = ξ , so that we have i − ,i − ,i

v˜ = v + ξ′ Π˜ = Π . (9.32)

These hold both in coordinate space and Fourier space (we use the same Fourier convention for ξ as for v and B).

9.2.3 Conformal-Newtonian Gauge We get to the conformal-Newtonian gauge by ξ0 = B + E and ξ = E. Thus − ′ − N δρ = δρ +ρ ¯′(B E′) = δρ 3 (1 + w)¯ρ(B E′) N − − H 2 − δp = δp +p ¯′(B E′) = δp 3 (1 + w)csρ¯(B E′) N − − H − v = v E′ − ΠN = Π . (9.33)

9.3 Scalar Perturbations in the Conformal-Newtonian Gauge From here on we shall (unless otherwise noted)

1. consider scalar perturbations only, so that v = v and B = B i − ,i i − ,i 10 FIELD EQUATIONS FOR SCALAR PERTURBATIONS IN THE NEWTONIAN GAUGE25

2. use the conformal-Newtonian gauge, so that B = 0.

Thus the energy tensor perturbation has the form

δρN (¯ρ +p ¯)vN δT µ = − − ,i . (9.34) ν (¯ρ +p ¯)vN δpN δi +p ¯(Π 1 δ 2Π) ,i j ,ij − 3 ij∇ 10 Field Equations for Scalar Perturbations in the Newtonian Gauge

We can now write the Einstein equations

µ µ δGν = 8πGδTν (10.1)

µ for scalar perturbations in the conformal-Newtonian gauge. We have the left-hand side δGν µ from Sect. 8.1 and the right-hand side δTν from Sect. 9.3:

0 2 2 N δG = a− 2 Ψ + 6 (Ψ′ + Φ) = 8πGδρ 0 − ∇ H H − 0 2 N δG = 2a− Ψ′ + Φ = 8πG(¯ρ +p ¯)v i − H ,i − ,i i 2 N δG = 2a− Ψ′ + Φ = 8πG(¯ρ +p ¯)v 0 H ,i ,i i 2 2 2 i δG = a− 2Ψ ′′ + (Φ Ψ) + (2Φ′ + 4Ψ′) + (4 ′ + 2 )Φ δ j ∇ − H H H j 2 N i 1 2 + a− (Ψ Φ) = 8πG δp δ +p ¯(Π δ Π) . (10.2) − ,ij j ,ij − 3 ij∇ i i Separating the δGj equation into its trace and traceless part (the trace of δj is 3, and the trace of (Ψ Φ) is 2(Ψ Φ)) the full set of Einstein equations is − ,ij ∇ − 2 2 N 3 (Ψ′ + Φ) Ψ = 4πGa δρ (10.3) H H −∇ − 2 N (Ψ′ + Φ) = 4πGa (¯ρ +p ¯)v (10.4) H ,i ,i 2 1 2 2 N Ψ′′ + (Φ′ + 2Ψ′) + (2 ′ + )Φ + (Φ Ψ) = 4πGa δp (10.5) H H H 3 ∇ − (∂ ∂ 1 δi 2)(Ψ Φ) = 8πGa2p¯(∂ ∂ 1 δi 2)Π . (10.6) i j − 3 j∇ − i j − 3 j∇ The oﬀ-diagonal part of the last equation gives

(Ψ Φ) = 8πGa2p¯Π for i = j . (10.7) − ,ij ,ij 6 In Fourier space this reads k k k k (Ψ Φ )= i j 8πGa2p¯Π for i = j . (10.8) − i j ~k − ~k − k2 ~k 6 (with the Liddle&Lyth Fourier convention for Π). Since we can always rotate the background coordinate system so that more than one of the components of ~k are non-zero, this means that

k2(Ψ Φ ) = 8πGa2p¯Π for ~k = ~0 . (10.9) ~k − ~k ~k 6 The 0th Fourier component represents a constant oﬀset. But the split into a background and a perturbation is always chosen so that the spatial average of the perturbation vanishes (and the background value thus represents the spatial average of the full perturbed quantity). Thus we have (going back to ~x-space)

Ψ Φ = 8πGa2p¯Π . (10.10) − 11 ENERGY-MOMENTUM CONTINUITY EQUATIONS 26

Likewise, since the spatial average of a perturbation is always zero, the equality of gradients of two perturbations means the equality of those perturbations themselves. Thus Eq. (10.4) says that 2 N Ψ′ + Φ = 4πGa (¯ρ +p ¯)v . (10.11) H Inserting this into Eq. (10.3) gives

2Ψ = 4πGa2ρ¯ δN + 3 (1 + w)vN . (10.12) ∇ H where we have deﬁned the relative energy density perturbation δρ δ . (10.13) ≡ ρ¯

(and w p/¯ ρ¯). ≡ The ﬁnal form of the Einstein equations can be divided into two constraint equations

2Ψ = 3 2 δN + 3 (1 + w)vN (10.14) ∇ 2 H H Ψ Φ = 3 2wΠ (10.15) − H that apply to any given time slice; and to two evolution equations

3 2 N Ψ′ + Φ = (1 + w)v (10.16) H 2 H 2 1 2 3 2 N Ψ′′ + (Φ′ + 2Ψ′) + (2 ′ + )Φ + (Φ Ψ) = δp /ρ¯ (10.17) H H H 3 ∇ − 2 H that determine how the metric perturbation evolves in time. In the above we used the background relation 4πGa2ρ¯ = 3 2 , (10.18) 2 H which follows directly from the Friedmann equation (2.2). In Fourier space the Einstein equations can be written as

k 2 Ψ = 3 δN + 3(1 + w)HvN (10.19) − 2 k H k 2 (Ψ Φ) = 3wΠ (10.20) − H 1 3 N − Ψ′ +Φ = (1 + w)H v (10.21) H 2 k 2 N 2 1 2 ′ 1 k 3 δp − Ψ′′ + − Φ′ + 2Ψ′ + 1+ H Φ (Φ Ψ) = , (10.22) H H 2 − 3 − 2 ρ¯ H H where the powers of are arranged so that the distance scales k 1 and time scales dη are always H − related to the conformal Hubble scale 1. H− 11 Energy-Momentum Continuity Equations

µ µ We know that from the Einstein equation, Gν = 8πGTν , the energy-momentum continuity equations, µ Tν;µ = 0 , (11.1) follow. Just like for the background universe, we may use the energy-momentum continuity equations instead of some of the Einstein equations. 12 PERFECT FLUID SCALAR PERTURBATIONS IN THE NEWTONIAN GAUGE 27

Calculating T µ = T µ +Γµ T α Γα T µ = 0 (11.2) ν;µ ν,µ αµ ν − νµ α to ﬁrst order in perturbations, one obtains the 0thorder (background) equation

ρ¯′ = 3 (¯ρ +p ¯) (11.3) − H and the 1storder (perturbation) equations, which for scalar perturbations in the conformal- Newtonian gauge are (exercise) N N N 2 N (δρ )′ = 3 (δρ + δp )+(¯ρ +p ¯)( v + 3Ψ′) (11.4) N − H N N ∇ N 2 2 (¯ρ +p ¯)(v )′ = (¯ρ +p ¯)′v 4 (¯ρ +p ¯)v + δp + p¯ Π+(¯ρ +p ¯)Φ (11.5) − − H 3 ∇ Note that vN is the velocity potential, ~vN = vN . It is easy to interpret the various terms in −∇ these equations. In the energy perturbation equation (11.4), we have first the effect of the background expansion, then the effect of velocity divergence (local fluid expansion) and then the effect of the expansion/contraction in the metric perturbation. In the momentum perturbation equation (11.5), the lhs and the first term on the right represent the change in inertia velocity. The second on the right is the effect of background ex- × pansion. The third and last terms represent forces due to gradients in pressure and gravitational potential. With manipulations involving background relations, these can be worked (exercise) into the form N N 2 N N δp (δ )′ = (1+ w) v + 3Ψ′ + 3 wδ (11.6) ∇ H − ρ¯ N N N w′ N δp 2 w 2 (v )′ = (1 3w)v v + + Π+Φ . (11.7) −H − − 1+ w ρ¯ +p ¯ 3 1+ w ∇ In Fourier space, N N N N δp (δ )′ = (1 + w) kv 3Ψ′ + 3 wδ (11.8) − − H − ρ¯ N N N w′ N δp 2 w (v )′ = (1 3w)v v + k k Π+ kΦ . (11.9) −H − − 1+ w ρ¯ +p ¯ − 3 1+ w Equations (11.4), (11.6), (11.8), and related equations may be referred to as the energy continuity equation; and (11.5), (11.7), (11.9), and related equations as the Euler equation. These two fluid evolution equations are not independent of the Einstein equations, but they can be used instead of the two Einstein evolution equations (10.16) and (10.17).

12 Perfect Fluid Scalar Perturbations in the Newtonian Gauge

12.1 Field Equations For a perfect fluid, things simplify a lot, since now Π = 0 and thus for a perfect fluid Ψ=Φ , (12.1) and we have only one degree of freedom in the scalar metric perturbation. We can now replace Ψ with Φ in the field equations. The original set becomes 2 2 N Φ 3 (Φ′ + Φ) = 4πGa δρ (12.2) ∇ − H H 2 N Φ′ + Φ = 4πGa (¯ρ +p ¯)v (12.3) H ,i ,i 2 2 N Φ′′ + 3 Φ′ + (2 ′ + )Φ = 4πGa δp , (12.4) H H H 12 PERFECT FLUID SCALAR PERTURBATIONS IN THE NEWTONIAN GAUGE 28 and the reworked set becomes25

2Φ = 4πGa2ρ¯ δN + 3 (1 + w)vN ∇ H = 3 2 δN + 3 (1 + w)vN (12.8) 2 H H 2 N 3 2 N Φ′ + Φ = 4πGa (¯ρ +p ¯)v = (1 + w)v (12.9) H 2 H 2 2 N 3 2 N Φ′′ + 3 Φ′ + (2 ′ + )Φ = 4πGa δp = δp /ρ¯ , (12.10) H H H 2 H where we have used Eq. (2.9). Note how (12.8) resembles the Newtonian perturbation theory result 2Φ = 4πGa2ρδ¯ – indeed, we expect to recover the Newtonian perturbation theory results ∇ in the appropriate limit! We deﬁne the total entropy perturbation as

δp δρ δp δρ H . (12.11) S≡H p¯ − ρ¯ ≡ p¯˙ − ρ¯˙ ′ ′ From the gauge transformation equations (9.27,9.28) we see that it is gauge invariant. Using the background relationsρ ¯ = 3 (1 + w)¯ρ andp ¯ = c2ρ¯ we can also write ′ − H ′ s ′ 1 δρ 1 δp = , (12.12) S 3(1 + w) ρ¯ − c2 ρ¯ s from which we get δp = c2 [δρ 3(¯ρ +p ¯) ] , (12.13) s − S which holds in any gauge. Using the entropy perturbation, we can now write in the rhs of Eq. (12.10)

δpN /ρ¯ = c2 δN 3(1 + w) , (12.14) s − S where, from (12.8) and (12.10),

N N 2 2 2 2 2 δ = 3 (1 + w)v + Φ = (Φ′ + Φ) + Φ . (12.15) − H 3 2 ∇ − H 3 2 ∇ H H H With some use of background relations, the evolution equation (12.10) becomes

2 2 1 2 2 2 2 9 2 − Φ′′ + 3(1 + c ) − Φ′ + 3(c w)Φ = c − Φ c (1 + w) . (12.16) H s H s − sH ∇ − 2 s S 12.2 Adiabatic perturbations Perturbations where the total entropy perturbation vanishes,26

= 0 δp = c2δρ (12.17) S ⇔ s 25If we change the time variable from conformal time η to cosmic time t, they read

∇2Φ = 4πGa2ρ¯ δN + 3aH(1+ w)vN (12.5) h i Φ+˙ HΦ = 4πGa(¯ρ +p ¯)vN (12.6) Φ¨ + 4HΦ+˙ 2H˙ + 3H2 Φ = 4πGδpN . (12.7)

26 2 2 For pressureless matter, where δp =p ¯ = w = cs = 0, Eq. (12.11) is not deﬁned, but δp = csδρ holds always (0 = 0). We use then this latter deﬁnition for adiabaticity; and the perturbations are necessarily adiabatic. 13 SCALAR PERTURBATIONS FOR A BAROTROPIC PERFECT FLUID 29 are called adiabatic perturbations.27 For adiabatic perturbations, Eq. (12.16) becomes (going to Fourier space)

2 2 2 1 2 csk − Φ′′ + 3(1 + c ) − Φ′ + 3(c w)Φ = Φ , (12.18) H ~k s H ~k s − ~k − ~k H from which Φ (η) can be solved, given the initial conditions. From Eq. (12.9) we then get vN (η), ~k ~k and after that, from Eq. (12.8), δN (η). ~k In terms of ordinary cosmic time, Eq. (12.18) becomes

2 2 2 1 2 csk H− Φ¨ + (4 + 3c )H− Φ˙ + 3(c w)Φ = Φ , (12.19) ~k s ~k s − ~k − H ~k 12.3 Fluid Equations For a perfect ﬂuid, Eqs. (11.6) and (11.7) become

N N 2 N N δp (δ )′ = (1+ w) v + 3Φ′ + 3 wδ (12.20) ∇ H − ρ¯ N N N w′ N δp (v )′ = (1 3w)v v + +Φ . (12.21) −H − − 1+ w ρ¯ +p ¯

12.4 Adiabatic perturbations at superhorizon scales For superhorizon scales, k , we can drop the rhs of Eq. (12.19). Using background relations ≪H to rewrite the equation-of-state parameters in terms of the Hubble parameter, it becomes 1 1 Φ¨ (H¨ HH˙ )Φ˙ (HH¨ 2H˙ 2)Φ = 0 . (12.22) − H˙ − − H˙ − The general solution is (exercise)

H t H Φ (t) = A 1 adt + B . (12.23) ~k ~k − a ~k a Z0 The lower integration limit (written as t = 0) is arbitrary, since the eﬀect of changing it can be absorbed in the constant B~k. The second term is a decaying mode, it decays at least as fast as 1/a (unless w< 1). − 13 Scalar Perturbations for a Barotropic Perfect Fluid

An equation of state of the form p = p(ρ) , (13.1) i.e., there are no other thermodynamic variables than ρ that the pressure would depend on, is called barotropic.28 In this case the perturbations are guaranteed to be adiabatic, since now

δp p¯′ dp 2 = = = cs . (13.2) δρ ρ¯′ dρ 27Warning: Sometimes people may say “adiabatic perturbations”, when they mean perturbations which were initially adiabatic. Such perturbations do not usually stay adiabatic as the universe evolves. The proper expression for these is “adiabatic mode”. 28This barotopic equation of state is not the same concept as the baryotropic equation of state p =(γ − 1)ρ one encounters in some other branches of physics. 14 SCALAR PERTURBATIONS IN THE MATTER-DOMINATED UNIVERSE 30

(Also, we keep assuming the fluid is perfect.) This discussion will also apply to adiabatic perturbations of general perfect fluids.29 That is, when the fluid in principle may have more state variables, but these other degrees of freedom are not “used”. Let us show that the adiabaticity of perturbations,

2 p¯′ δp = csδρ δρ , (13.3) ≡ ρ¯′ implies that a unique relation (13.1) holds everywhere and -when: Note first, that in the background solution, wherep ¯ =p ¯(t) andρ ¯ =ρ ¯(t), we can (assuming ρ¯ decreases monotonously with time) invert for t(¯ρ), and thusp ¯ =p ¯(t(¯ρ)), defining a function 2 p¯(¯ρ), whose derivative is cs. Now Eq. (13.3) guarantees that also p =p ¯ + δp and ρ =ρ ¯ + δρ satisfy this same relation. We note a property, which illuminates the nature of adiabatic perturbations: A small region of the perturbed universe is just like the background universe at a slightly earlier or later time. We can thus think of adiabatic perturbations as a perturbation in the “timing” of the different parts of the universe. (In adiabatic oscillations, this “corresponding background solution time” may oscillate back and forth.)

Once we have solved Φ~k(η) from (12.18),

2 2 2 2 Φ′′ + 3(1 + c ) Φ′ + 3(c w) Φ + (c k) Φ = 0 , (13.4) s H s − H s we get vN (η) and δN (η) from (12.9) and (12.8), which read as ~k ~k

N 2k 2 1 v = − Φ′ + − Φ 3(1 + w) H H 2 2 N 2 k N 2 k 1 δ = Φ 3(1 + w) H v = Φ 2 − Φ′ +Φ (13.5) −3 − k −3 − H H H in Fourier space.

14 Scalar Perturbations in the Matter-Dominated Universe

Let us now consider density perturbations in the simplest case, the matter-dominated universe. By “matter” we mean here non-relativistic matter, whose pressure is so small compared to energy density, that we can ignore it here.3031 In general relativity this is often called “dust”. According to our present understanding, the universe was radiation-dominated for the ﬁrst few ten thousand years, after which it became matter-dominated. For our present discussion, we take “matter-dominated” to mean that matter dominates energy density to the extent that we can ignore the other components. This approximation becomes valid after the ﬁrst few million years. Until late 1990’s it was believed that this matter-dominated state persists until (and beyond) the present time. But new observational data points towards another component in the energy density of the universe, with a large negative pressure, resembling vacuum energy, or

29Thus we could have called this section “Adiabatic perfect fluid scalar perturbations”. The reason we did not, is that mentioned in the earlier footnote. In this section we require the perturbations stay adiabatic the whole time. Perturbations of general fluids, which are initially (when they are at superhorizon scales) adiabatic, acquire entropy perturbations when they approach and enter the horizon. 30Likewise, we can ignore its anisotropic stress. Thus nonrelativistic matter is a perfect fluid for our purposes. 31Note that there are situations where we can make the matter-dominated approximation at the background level, but for the perturbations pressure gradients are still important at small distance scales (large k). Here we, however, we make the approximation that also pressure perturbations can be ignored. 14 SCALAR PERTURBATIONS IN THE MATTER-DOMINATED UNIVERSE 31 a cosmological constant. This component is called “dark energy”. The dark energy seems to have become dominant a few billion (109) years ago. Thus the validity of the matter-dominated approximation is not as extensive as was thought before; but anyway there was a significant period in the history of the universe, when it holds good. We now make the matter-dominated approximation, i.e., we ignore pressure,

2 p¯ = w = cs =0 and δp =Π=0 . (14.1)

This is our ﬁrst example of solving a perturbation theory problem. The order of work is always:

1. Solve the background problem.

2. Using the background quantities as known functions of time, solve the perturbation problem.

In the present case, the background equations are

a 2 8πG 2 = ′ = ρa¯ 2 (14.2) H a 3 4πG 2 ′ = ρa¯ , (14.3) H − 3 from which we have 2 2 ′ + = 0 . (14.4) H H The background solution is the familiar k = 0 matter-dominated Friedmann model, a t2/3. ∝ But let us review the solution in terms of conformal time. Sinceρ ¯ a 3(1+w) a 3, (14.2) says ∝ − ∝ − that a a1/2, which gives ′ ∝ a(η) η2 . (14.5) ∝ Since dt = adη, or dt/dη = a, we get t(η) η3 a3/2 or a t2/3. ∝ ∝ ∝ From a η2 we get ∝ a′ 2 2 = and ′ = (14.6) H≡ a η H −η2 Thus, from Eq. (14.2), 3 6 4πGa2ρ¯ = 2 = . (14.7) 2H η2 The perturbation equations (12.8), (12.9), and (12.10) withp ¯ = w = δp = 0, are

2Φ = 4πGa2ρ¯ δN + 3 vN (14.8) ∇ H 2 N Φ′ + Φ = 4πGa ρv¯ (14.9) H2 Φ′′ + 3 Φ′ + (2 ′ + )Φ = 0 . (14.10) H H H Using Eq. (14.4), Eq. (14.10) becomes (or directly from (13.4)) 6 Φ′′ + 3 Φ′ =Φ′′ + Φ′ = 0 , (14.11) H η whose solution is 5 Φ(η, ~x) = C1(~x)+ C2(~x)η− . (14.12) 14 SCALAR PERTURBATIONS IN THE MATTER-DOMINATED UNIVERSE 32

The second term, η 5 is the decaying part. We get C (~x) and C (~x) from the initial values ∝ − 1 2 Φin(~x), Φin′ (~x) at some initial time η = ηin,

5 Φin(~x) = C1(~x)+ C2(~x)ηin− (14.13) 6 Φ′ (~x) = 5C (~x)η− (14.14) in − 2 in as

1 C1(~x) = Φin(~x)+ 5 ηinΦin′ (~x) (14.15) 1 6 C (~x) = η Φ′ (~x) (14.16) 2 − 5 in in

Unless we have very special initial conditions, conspiring to make C1(~x) vanishingly small, the decaying part soon becomes C (~x) and can be ignored. Thus we have the important result ≪ 1 Φ(η, ~x)=Φ(~x) , (14.17) i.e., the Bardeen potential Φ is constant in time for perturbations in the ﬂat matter-dominated universe. Ignoring the decaying part, we have Φ′ = 0 and we get for the velocity perturbation from Eq. (14.9) Φ 2Φ vN = H = = 1 Φη η a1/2 t1/3 . (14.18) 4πGa2ρ¯ 3 3 ∝ ∝ ∝ H and Eq. (14.8) becomes 3 2Φ = 4πGa2ρ¯ δN + 2Φ = 2 δN + 2Φ (14.19) ∇ 2H or 2 δN = 2Φ + 2Φ . (14.20) − 3 2 ∇ H In Fourier space this reads

2 k 2 δN (η) = 2+ Φ . (14.21) ~k − 3 ~k " H # Thus we see that for superhorizon scales, k , or k H, the density perturbation stays ≪H phys ≪ constant, δN = 2Φ = const. (14.22) ~k − ~k whereas for subhorizon scales, k , or k H, they grow proportional to the scale factor, ≫H phys ≫ 2 k 2 δ = Φ η2 a t2/3 . (14.23) ~k −3 ~k ∝ ∝ ∝ H 1 Since the comoving Hubble scale − grows with time, various scales k are superhorizon H 1 1 to begin with, but later become subhorizon as − grows past k− . (In physical terms, in the 1 H 1 3/2 expanding universe λ /2π = k− a grows slower than the Hubble scale H a .) We phys phys ∝ − ∝ say that the scale in question “enters the horizon”. (The word “horizon” in this context refers just to the Hubble scale 1/ , and not to other definitions of “horizon”.) We see that density H perturbations begin to grow when they enter the horizon, and after that they grow proportional 15 SCALAR PERTURBATIONS IN THE RADIATION-DOMINATED UNIVERSE 33 to the scale factor. Thus the present magnitude of the density perturbation at comoving scale 32 k should be a0/ak times its primordial value a δ (t ) 0 δN , (14.25) ~k 0 ~k,pr ∼ ak where a0 is the present value of the scale factor, and ak is its value at the time the scale k “entered horizon”. The “primordial” density perturbation δN refers to the constant value it ~k,pr had at superhorizon scales, after the decaying part of Φ had died out. Of course Eq. (14.25) is valid only for those (large) scales where the perturbation is still small today. Once the perturbation becomes large, δ 1, perturbation theory is no more valid. We say the scale in ∼ question “goes nonlinear”33. One has to remember that these results refer to the density and velocity perturbations in the conformal-Newtonian gauge only. In some other gauge these perturbations, and their growth laws would be different. However, for subhorizon scales general relativistic effects become unimportant and a Newtonian description becomes valid. In this limit, the issue of gauge choice becomes irrelevant as all “sensible gauges” approach each other, and the conformal- Newtonian density and velocity perturbations become those of a Newtonian description. The Bardeen potential can then be understood as a Newtonian gravitational potential due to density perturbations. (Eq. (14.8) acquires the form of the Newtonian law of gravity as the second term on the right becomes small compared to the first term, δN . The factor a2 appears on the right since 2 on the left refers to comoving coordinates.) ∇ 15 Scalar Perturbations in the Radiation-Dominated Universe

For radiation, 1 2 1 4 1 p = ρ w = c = , ρ¯ a− and δp = δρ . (15.1) 3 ⇒ s 3 ∝ 3 From the Friedmann equation,

2 a′ 8πG 2 2 = ρa¯ a− a′ = const , (15.2) a 3 ∝ ⇒ we get the background solution

a′ 1 1 a η = and ′ = . (15.3) ∝ ⇒ H≡ a η H −η2

From (13.4),

4 1 2 Φ′′ + Φ′ + k Φ = 0 (15.4) η 3

32If we take into account the presence of “dark energy”, the above result is modiﬁed to

aDE N δ~k(t0) ∼ δ~k,pr , (14.24) ak where aDE & a0/2 is the value of a when dark energy became dominant, since that stops the growth of density perturbations. For ΛCDM, this modification was done accurately in Cosmology II, Sec. 8.3.5. 33Since in our universe this happens only at subhorizon scales, the nonlinear growth of perturbations can be treated with Newtonian physics. First the growth of the density perturbation (for overdensities) becomes much faster than in linear perturbation theory, but then the system “virializes”, settling into a relatively stable structure, a galaxy or a galaxy cluster, where further collapse is prevented by the conservation of angular momentum, as the different parts of the system begin to orbit the center of mass of the system. For underdensities, we have of course ρ ≥ 0 ⇒ δ ≥−1 always, so the underdensity cannot “grow” beyond that. 15 SCALAR PERTURBATIONS IN THE RADIATION-DOMINATED UNIVERSE 34 or 2 1 2 η Φ′′ + 4ηΦ′ + 3 (kη) Φ = 0 . (15.5) The obvious differences from the corresponding matter-dominated equation, Eq. (14.11), is that now we have k-dependence and that Φ = const. = 0 is not a solution. 6 Equation (15.4) can be converted to a Bessel equation for u ηΦ: ≡ 2 k2 2 u′′ + u′ + u = 0 . (15.6) η 3 − η2 The equation for spherical Bessel functions is

d2j 2 dj l(l + 1) l + l + 1 j = 0 . (15.7) dx2 x dx − x2 l Writing kη x = cskη , (15.8) ≡ √3 where cs is the speed of sound, Eq. (15.6) becomes Eq. (15.7) with l = 1. Thus the solutions are

kη kη u(η) = Aj + Bn . (15.9) 1 √ 1 √ 3 3 Spherical Bessel functions can be written in terms of trigonometric functions. In particular, sin x x cos x j (x) = − 1 x2 cos x x sin x n (x) = − − (15.10) 1 x2 Consider the limiting behavior at superhorizon scales x 0. Since → sin x x 1 x3 and cos x 1 1 x2 , (15.11) ∼ − 6 ∼ − 2 we have x 1 j (x) and n (x) (15.12) 1 ∼ 3 1 ∼ −x2 →∞ Thus the n1 solution diverges at early times, i.e., it is a decaying solution, so we discard it. The ﬁnal solution is

sin kη kη cos kη 1 √3 − √3 √3 √3 kη Φ(η) = u = A~k 3 = A~k j1 (15.13) η kη kη √3 √3 √ (where A~k is k/ 3 times the earlier A). From (13.5)

N 1 2 v = 2 (kη Φ′ + kηΦ) N 2 2 4 N 2 2 δ = (kη) Φ v = (kη) Φ 2(ηΦ′ + Φ) . (15.14) − 3 − kη − 3 −

For superhorizon scales (kη 1, i.e., at early times η k 1), we have Φ(η) 1 A = const. ≪ ≪ − ≈ 3 ~k Likewise δN 2Φ const, but vN 1 kηΦ grows η. ≈− ≈ ≈ 2 ∝ 16 OTHER GAUGES 35

For subhorizon scales, i.e., at later times, after horizon entry (η k 1 so that kη 1), the ≫ − ≫ cosine part dominates, so that for the gravitational potential we have cos kη/√3 Φ(η) 3A~k 2 , (15.15) ≈ − (kη) which oscillates with frequency ω = 2πf = k/√3= csk and decaying amplitude

3A~ k . (15.16) (kη)2 The ﬂuid quantities δN and vN N 1 2 1 2 3 v = (kη Φ′ + kηΦ) kη Φ′ A c sin(c kη) 2 ≈ 2 ≈ 2 ~k s s δN 2 (kη)2Φ 2A cos(c kη) , (15.17) ≈ − 3 ≈ ~k s oscillate with constant amplitude.

16 Other Gauges

Different gauges are good for different purposes, and therefore it is useful to be able to work in different gauges, and to switch from one gauge to another in the course of a calculation. When one uses more than one gauge it is important to be clear about which gauge each quantity refers to. One useful gauge is the comoving gauge. Particularly useful quantities which refer to the comoving gauge, are the comoving density perturbation δC and the comoving curvature perturbation ψC . (16.1) R≡− This is often just called the curvature perturbation, however that term is also used to refer to ψ in any other gauge, so beware! (There are different sign conventions for and ψ. In my sign R conventions, positive , but negative ψ , correspond to positive curvature of the 3D η = const R~k ~k slice. Gauges are usually specified by giving gauge conditions. These may involve the metric perturbation variables A, D, B, E (e.g., the Newtonian gauge condition E = B = 0), the energy-momentum perturbation variables δρ, δp, v, or both kinds.

16.1 Slicing and Threading The gauge corresponds to the coordinate system xµ = η,xi in the perturbed spacetime. { } { } The conformal time η gives the slicing of the perturbed spacetime into η = const time slices (3D spacelike hypersurfaces). The spatial coordinates xi give the threading of the perturbed spacetime into xi = const threads (1D timelike curves). See Fig. 3. Slicing and threading are orthogonal to each other if and only if the shift vector vanishes, Bi = 0. In the gauge transformation, x˜α = xα + ξα , (16.2) or η˜ = η + ξ0 xi = xi + ξi , (16.3) ξ0 changes the slicing, and ξi changes the threading. From the 4-scalar transformation law, δs˜ = δs s¯ ξ0, we see that perturbations in 4-scalars, e.g., δρ and δp depend only on the slicing. − ′ Thus slicing is a more important property of the gauge than its threading. In some cases, a gauge is specified by defining the slicing only, leaving the threading unspecified. 16 OTHER GAUGES 36

Figure 3: Slicing and threading the perturbed spacetime.

16.2 Comoving Gauge We say that the slicing is comoving, if the time slices are orthogonal to the ﬂuid 4-velocity. For scalar perturbations such a slicing always exists. This condition turns out to be equivalent to the condition that the ﬂuid velocity perturbation vi equals the shift vector Bi. For scalar perturbations, Comoving slicing v = B . (16.4) ⇔ From the gauge transformation equations (7.7) and (9.32),

v˜ = v + ξ′ 0 B˜ = B + ξ′ + ξ (16.5) we see that we get to comoving slicing by ξ0 = v B. − We say that the threading is comoving, if the threads are world lines of ﬂuid elements, i.e., the velocity perturbation vanishes, vi = 0.34 For scalar perturbations,

Comoving threading v = 0 . (16.6) ⇔ We get to comoving threading by the gauge transformation ξ = v. Note that comoving threads ′ − are usually not geodesics, since pressure gradients cause the fluid flow to deviate from free fall. The comoving gauge is defined by requiring both comoving slicing and comoving threading. Thus Comoving gauge v = B = 0 . (16.7) ⇔ (We assume that we are working with scalar perturbations.) The threading is now orthogonal to the slicing. We denote the comoving gauge by the sub- or superscript C. Thus the statement vC = BC = 0 is generally true, whereas the statement v = B = 0 holds only in the comoving gauge.

34Thus here “comoving” means with respect to the fluid: the coordinate system is comoving with the fluid flow. 16 OTHER GAUGES 37

We get to the comoving gauge from an arbitrary gauge by the gauge transformation

ξ′ = v − ξ0 = v B . (16.8) − This does not fully specify the coordinate system in the perturbed spacetime, since only ξ′ is speciﬁed, not ξ. Thus we remain free to do time-independent transformations x˜i = xi ξ(~x) , (16.9) − ,i while staying in the comoving gauge. This, however, does not change the way the spacetime is sliced and threaded by the coordinate system, it just relabels the threads with diﬀerent coordinate values xi. Applying Eq. (16.8) to the general scalar gauge transformation Eqs. (7.7,7.8,9.27,9.28,9.32), we get the rules to relate the comoving gauge perturbations to perturbations in an arbitrary gauge. For the metric:

C A = A (v B)′ (v B) (16.10) − − −H − BC = B v + (v B) = 0 − − DC = 1 2ξ + (v B) − 3 ∇ H − EC = E + ξ ψC = ψ + (v B) . ≡ −R H − For the energy tensor:

C δρ = δρ ρ¯′(v B) = δρ + 3 (1 + w)¯ρ(v B) (16.11) C − − H 2 − δp = δp p¯′(v B) = δp + 3 (1 + w)c ρ¯(v B) − − H s − δC = δ + 3 (1 + w)(v B) H − vC = v v = 0 − ΠC = Π . Because of the remaining gauge freedom (relabeling the threading) left by the comoving gauge C C C condition (only ξ′ is ﬁxed, not ξ), D and E are not fully ﬁxed. However, ψ is, and likewise

C E ′ = E′ v . (16.12) − In particular, we get the transformation rule from the Newtonian gauge (where A = Φ, B = 0, ψ = D = Ψ, E = 0) to the comoving gauge. For the metric:

AC = Φ vN ′ vN (16.13) − −H = Ψ vN R − −H EC ′ = vN . − For the energy tensor: δρC = δρN + 3 (1 + w)¯ρvN (16.14) H δpC = δpN + 3 (1 + w)c2ρv¯ N H s δC = δN + 3 (1 + w)vN . H Thus, for example, we see that in the matter-dominated universe discussed in Sect. 14, Eqs. (14.18,14.20) lead to

C N N 2 2 2 2 2 δ = δ + 3 v = 2Φ + Φ+2Φ = Φ − a (16.15) H − 3 2 ∇ 3 2 ∇ ∝ H ∝ H H 16 OTHER GAUGES 38 both inside and outside (and through) the horizon. Note that in Fourier space we have included an extra factor of k in v, so that, e.g., Eq. (16.14) reads in Fourier space as C N 1 N δ = δ + 3 (1 + w)k− v . (16.16) H 16.3 Mixing Gauges We see that Eq. (16.14c) is the rhs of Eq. (10.14), which we can thus write in the shorter form:

2Ψ = 4πGa2ρδ¯ C . (16.17) ∇ This is an equation which has a Newtonian gauge metric perturbation on the lhs, but a comoving gauge density perturbation on the rhs. Is it dangerous to mix gauges like this? No, if we know what we are doing, i.e., in which gauge each quantity is, and the equations were derived correctly for this combination of quantities. We could also say that, instead of working in any particular gauge, we work with gauge- invariant quantities, that the quantity that we denote by δC is the gauge-invariant quantity deﬁned by δC = δ + 3 (1 + w)(v B) H − which just happens to coincide with the density perturbation in the comoving gauge. Just like Ψ happens equal the metric perturbation ψ in the Newtonian gauge. Switching to the comoving gauge for δ and δp, but keeping the velocity perturbation in the Newtonian gauge, the Einstein and continuity equations can be rewritten as (exercise): 2Ψ = 3 2δC (16.18) ∇ 2 H Ψ Φ = 3 2wΠ (16.19) − 3H 2 N Ψ′ + Φ = (1 + w)v (16.20) H 2 H C 2 2 2 1 2 3 2 δp Ψ′′ + (2 + 3c ) Ψ′ + Φ′ + 3(c w) Φ+ (Φ Ψ) = (16.21) s H H s − H 3 ∇ − 2 H ρ¯

2 N 2 δC′ 3 wδC = (1+ w) v + 2 w Π (16.22) − H C ∇ H ∇ δp 2 w 2 v′ + v = + Π+Φ . (16.23) N H N ρ¯ +p ¯ 3 1+ w ∇

16.4 Comoving Curvature Perturbation The comoving curvature perturbation ψC = ψ (v B) (16.24) R ≡ − − −H − turns out to be a useful quantity for discussing superhorizon perturbations. From (16.20) we have that

N 2 v = (Ψ′ + Φ) , (16.25) 3 2(1 + w) H H so that the relation of the comoving curvature perturbation and the Bardeen potentials is35

2 1 = Ψ − Ψ′ +Φ . (16.27) R − − 3(1 + w) H 35In terms of ordinary cosmic time t, 2 H R = −Ψ − Ψ+˙ HΦ = −Ψ+ Ψ+˙ HΦ , (16.26) 3(1 + w)H H˙ where we used the background equation (2.21a). 16 OTHER GAUGES 39

Derivating Eq. (16.27),

2w′ 1 2 ′ 1 ′ = Ψ′ + − Ψ′ +Φ H Ψ′ + − Ψ′′ +Φ′ R − 3(1 + w)2 H − 3(1 + w) − 2 H H 1 2 2 2 − 4 + 6cs 2 w cs = H Ψ′′ Ψ′ Φ′ + 2 − Φ , −3(1 + w) − 3(1 + w) − 3(1 + w) H 1+ w (where we used some background relations), and using the Einstein equations (10.22) and (10.19), we get an evolution equation for , R 3 1 2 1 2 1 2 (1 + w) − ′ = − Ψ′′ + − Φ′ + 2Ψ′ + 3cs − Ψ′ +Φ 3wΦ − H R H 2 H 2 1 1H 2 − = − Ψ′′ + (2 + 3c ) − Ψ′ + − Φ′ + 3(c w)Φ H s H H s − 2 N 2 1 1 k 3 δp = 3c − Ψ′ +Φ + (Φ Ψ) + s H 3 − 2 ρ¯ H k 2 k 2 δpN = c2 Ψ+ 1 (Φ Ψ) + 3 c2δN . (16.28) − s 3 − 2 ρ¯ − s H H From Sec. 12 we have 1 1 δp = δ δp = c2[δρ 3(¯ρ +p ¯) ] (16.29) S 3(1 + w) − c2 ρ¯ ⇒ s − S s (valid in any gauge), so that Eq. (16.28) becomes the important result (exercise)

2 3 1 k 2 1 9 2 (1 + w) − ′ = c Ψ+ (Ψ Φ) + c (1 + w) . (16.30) 2 H R s 3 − 2 s S H or 2 1 2 k 2 1 2 − ′ = c Ψ+ (Ψ Φ) + 3c . (16.31) H R 3(1 + w) s 3 − sS H As an evolution equation, this does not appear very useful, since it mixes metric perturbations from two different gauges. However, the importance of this equation comes from the two observations we can now make immediately: 1) For adiabatic perturbations, = 0, the S second term on the rhs vanishes. 2) For superhorizon perturbations, i.e., for Fourier modes whose wavelength is much larger than the Hubble distance, k , we can ignore the first term ≪H on the rhs. Thus: For adiabatic perturbations, the comoving curvature perturbation stays constant outside the horizon. Adiabatic scalar perturbations have only one degree of freedom, and thus their full evolution is captured in the evolution of . R A general perturbation at a given time can be decomposed into an adiabatic component, which has = 0, and an isocurvature component, which has = 0. Because of the linearity S R of first order perturbation theory, these components evolve independently, and the evolution of the general perturbation is just the superposition of the evolution of these two components, and thus they can be studied separately. Beware, however, that the ”adiabatic” component does not necessarily remain adiabatic in its evolution, and the ”isocurvature” component does not necessarily maintain zero comoving curvature. We later show (in Sec. 18.3) that adiabatic perturbations stay adiabatic while they are well outside the horizon. Adiabatic perturbations are important, since the simplest theory for the origin of structure of the universe, single-field inflation, produces adiabatic perturbations. Using the constancy of , it is easy to follow the evolution of these perturbations while they are well outside the R horizon. 16 OTHER GAUGES 40

Using Eq. (16.21) we can write the second line of Eq. (16.28) as

3 2 2 2 (1 + w) ′ = Ψ′′ + (2 + 3c ) Ψ′ + Φ′ + 3(c w) Φ − 2 HR s H H s − H δp = 3 2 C 1 2(Φ Ψ) (16.32) 2 H ρ¯ − 3 ∇ − or (using Eq. 16.19)

C C 1 δp 2 w 2 2 δ 2 w 2 − ′ = Π = c 3 Π (16.33) H R −ρ¯ +p ¯ − 3 1+ w ∇ − s 1+ w − S − 3 1+ w ∇ which is completely in the comoving gauge.

16.5 Perfect Fluid Scalar Perturbations, Again With Π = 0 Ψ = Φ, Eqs. (16.18,16.19,16.20,16.21,16.22,16.23) become ⇒ 2 3 2 Φ = 2 δC (16.34) ∇ 3 H2 Φ′ + Φ = (1 + w)v (16.35) H 2 H N

2 2 2 3 2 δpC Φ′′ + 3(1 + c ) Φ′ + 3(c w) Φ = (16.36) s H s − H 2 H ρ¯

2 δ′ 3 wδ = (1+ w) v (16.37) C − H C ∇ N δpC v′ + v = +Φ . (16.38) N H N ρ¯ +p ¯ and Eqs. (16.27,16.33) become 2 = Φ Φ′ + Φ (16.39) R − − 3(1 + w) H H C C 1 δp 2 δ − ′ = = c 3 (16.40) H R −ρ¯ +p ¯ − s 1+ w − S and Eq. (16.36) as

2 2 1 2 3 2 C − Φ′′ + 3(1 + c ) − Φ′ + 3(c w)Φ = c [δ 3(1 + w) ] . (16.41) H s H s − 2 s − S which is Eq. (12.16), with just 2Φ replaced by δC using Eq. (16.34). Our aim is to get a ∇ diﬀerential equation with preferably just one perturbation quantity to solve from, so this might seem a step backwards, replacing one of the Φ with δC , but we can now go ahead and replace also all the other Φ with δC : Taking the Laplacian of this (exercise), we get the Bardeen equation

2 2 1 3 2 2 2 2 2 − δ ′′ + 1 6w + 3c − δ ′ 1 + 8w 6c 3w δ = c − [δ 3(1 + w) ] , H C − s H C − 2 − s − C sH ∇ C − S (16.42) a diﬀerential equation from which we can solve the evolution of the comoving density perturbation for superhorizon scales (when one can ignore the rhs) and for adiabatic perturbations at all scales (when = 0). S For the general case we need also an equation for . To be able to do this, we need to take S a closer look at the ﬂuid, see Sec. 18 and beyond. 16 OTHER GAUGES 41

16.5.1 Adiabatic Perfect Fluid Perturbations at Superhorizon Scales Rewrite Eq. (16.39) as 2 1 5 + 3w − Φ′ + Φ = (1 + w) . (16.43) 3 H 3 − R If we have a period in the history of the universe, where we can approximate w = const, then, for adiabatic perturbations at superhorizon scales, Eq. (16.43) is a diﬀerential equation for Φ with w = = const for that period. This equation has a special solution R 3 + 3w Φ= . −5 + 3w R The corresponding homogeneous equation is

1 5 + 3w − Φ′ + Φ = 0 H 2 dΦ 5 + 3w a = Φ ⇒ da − 2 5+3w Φ = Ca− 2 ⇒ so that the general solution to Eq. (16.43) is

3 + 3w 5+3w Φ = + Ca− 2 . (16.44) −5 + 3w R If w const for a long enough time, the second part becomes negligible, and we have ≈ 3 + 3w Φ=Ψ= = const (16.45) −5 + 3w R In particular, we have the relations

2 1 Φk = k (adiab., rad.dom, w = , k ) (16.46) −3R 3 ≪H 3 Φk = k (adiab., mat.dom, w = 0, k ) . (16.47) −5R ≪H While the universe goes from radiation domination to matter domination, w is not a constant, 2 so Eq. (16.45) does not apply, but we know from Eq. (16.47) that, Φk changes from 3 k to 3 − R k, i.e, changes by a factor 9/10 (assuming k the whole time, so that k stays constant). − 5 R ≪H R 16.6 Uniform Energy Density Gauge The uniform energy density gauge (time slicing)36 U is deﬁned by the condition

δρU = 0 . (16.48)

Since 0 δρ = δρ ρ¯′ξ , (16.49) − we get to this gauge by e δρ ξ0 = . (16.50) ρ¯′ 36Sometimes we refer to a gauge just by the condition on time slicing, i.e., ξ0, since that is more important in CPT, leaving ξ unspeciﬁed. 16 OTHER GAUGES 42

Thus δρ ψU = ψ + ξ0 = ψ + . (16.51) H H ρ¯′ The uniform energy density curvature perturbation ζ, deﬁned as δρ δ ζ ψU = ψ = ψ + (16.52) ≡ − − −H ρ¯′ − 3(1 + w) is a widely used quantity in inﬂation literature. Going to U from C, C C δ 2 2 2 ζ = ψ + = + − Ψ . (16.53) − 3(1 + w) R 9(1 + w)H ∇ In Fourier space, 2 k 2 ζk = k Ψk , (16.54) R − 9(1 + w) H so that at superhorizon scales ζ (k ) . (16.55) ≈ R ≪H Therefore, ζ has the same useful property as : For adiabatic perturbations, ζ remains R constant outside the horizon.

16.7 Spatially Flat Gauge The spatially flat gauge, denoted by the sub/superscript Q, is defined by the condition that the curvature perturbation ψ vanishes, i.e., ψQ = 0 . (16.56) Since the metric perturbation ψ transforms as ψ˜ = ψ + ξ0 . (16.57) H we get to the spatially flat gauge by the gauge transformation 0 1 ξ = − ψ . (16.58) −H From the gauge transformation rule of the relative density perturbation (9.31), δ˜ = δ + 3 (1 + w)ξ0 , (16.59) H we get that δQ = δ 3(1 + w)ψ . (16.60) − In particular, δQ = δC + 3(1 + w) = 3(1 + w)ζ , (16.61) R so that 1 ζ = δQ (16.62) 3(1 + w) and the comoving curvature perturbation is proportional to the difference between the relative R density perturbations in the spatially flat and comoving gauges, 1 1 2 k 2 = δQ δC = δQ + Ψ , (16.63) R 3(1 + w) − 3(1 + w) 3 " H # where we used Eq. (16.18) for the latter equality. Thus for superhorizon scales we have the correspondence 1 δQ (16.64) R ≈ 3(1 + w) between the comoving gauge curvature perturbation and the flat gauge density perturbation. We shall later use the spatially flat gauge to solve perturbation equations for scalar fields. 17 SYNCHRONOUS GAUGE 43

17 Synchronous Gauge

The synchronous gauge was the ﬁrst one to be used in cosmological perturbation theory (by Lifshitz in 1946) and it is often used in numerical work. The synchronous gauge is deﬁned by the requirement A = Bi = 0. Synchronous gauge can be used both for scalar and vector perturbations. In this section we consider only scalar perturbations, where it means that

AZ = BZ = 0 . (17.1)

(We use Z instead of S to denote synchronous gauge, so as not to confuse it with the scalar part of a perturbation.) One gets to synchronous gauge from an arbitrary gauge with a gauge transformation ξµ that satisﬁes

ξ0′ + ξ0 = A (17.2) H 0 ξ′ = ξ B . (17.3) − − This is a diﬀerential equation from which to solve ξ0. Thus it is not easy to switch from another gauge to synchronous gauge. We see that, like for comoving gauge, only the time derivative of ξ is determined. In addition, ξ0 is determined only up to solutions of the homogeneous equation ξ0′ + ξ0 = 0. Thus the gauge is not fully speciﬁed by the synchronous condition, H which historically led to confusion until theorists learned to separate gauge modes within the synchronous gauge from physical ones. In the synchronous gauge the threads are orthogonal to the slices, since B = 0, and the metric perturbation is only in the space part of the metric,

h = 2DZ δ + 2 EZ 1 2EZ δ ij − ij ,ij − 3 ∇ ij = 2ψZ δ + 2EZ . (17.4) − ij ,ij The line element is

ds2 = a2 dη2 + [(1 2ψ)δ + 2E ] dxidxj . (17.5) − − ij ,ij For a comoving observer, i.e., one moving along a thread (a space coordinate line xi = const.), the proper time dτ is given by dτ 2 = ds2 = a2dη2, so that dτ = adη = dt. Thus her clock − shows the coordinate time (t, not η). Moreover, the threads are geodesics: From Eq. (A.2), setting A = B = 0, the Christoﬀel symbols are

0 Γ00 = (17.6) 0 Hi Γ0i = Γ00 = 0 0 Γ = [(1 2ψ)δ + 2E ] δ ψ′ + E′ ij H − ij ,ij − ij ,ij i Γ = δ δ ψ′ + E′ 0j H ij − ij ,ij Γi = δ ψ δ ψ + δ ψ + E . jk − ij ,k − ik ,j jk ,i ,ijk 0 i From Γ0i = Γ00 = 0 follows that the space coordinate lines are geodesics. Namely, the geodesic equations

x¨0 +Γ0 x˙ αx˙ β =x ¨0 + x˙ 0x˙ 0 +Γ0 x˙ ix˙ j = 0 (17.7) αβ H ij i i α β i i 0 j i j k x¨ +Γαβx˙ x˙ =x ¨ + 2Γ0jx˙ x˙ +Γjkx˙ x˙ = 0 , (17.8) where now d/dτ, not d/dt, are satisfied by xi = const. x˙ i = 0 andx ˙ 0 dη/dτ = a 1. · ≡ ⇒ ≡ − Thus one can construct the synchronous gauge coordinate system by choosing an initial spacelike 17 SYNCHRONOUS GAUGE 44 hypersurface, distributing observers carrying space coordinate values on that hypersurface, with their initial 4-velocities orthogonal to the hypersurface (i.e., they are at rest with respect to that surface), synchronizing the clocks of the observers, and letting the observers then fall freely. The world lines of these observers are then the space coordinate lines and their clocks show the time coordinate. Note that the synchronous gauge leaves the choice of this initial hypersurface free (this is the remaining gauge freedom in ξ0 already mentioned above). In the case of dust (p = 0), the fluid flow lines are geodesics, and we can choose them to be the threads (and the initial hypersurface orthogonal to them), in which case our synchronous gauge becomes equal to the comoving gauge. Pressure makes the comoving gauge different from the synchronous gauge, since pressure gradients accelerate the fluid away from geodesics. We use Ma & Bertschinger[4] (hereafter MB) as our main reference for synchronous gauge and adopt from their notation

h 6DZ hiZ ≡ − ≡ i η ψZ = DZ + 1 2EZ ≡ 3 ∇ µ 2EZ (17.9) ≡ (only two of these are independent), so that

h = 1 hδ + (∂ ∂ 1 δ 2)µ ij 3 ij i j − 3 ij∇ = 2ηδ + µ . (17.10) − ij ,ij Unfortunately, the MB notation for the synchronous curvature perturbation is the same as our notation for conformal time (MB use τ for the latter). Let’s hope this does not lead to confusion. (In this Section, from here on, η stands always for the synchronous metric perturbation (17.9b); we avoid writing a symbol for conformal time.) MB discuss the synchronous metric perturbation in terms of the variables h and η, instead of the pair h,µ or η,µ. In coordinate space this appears impractical, since to solve µ from h and η requires integration (η = 1 ( h + 2µ)). However, 6 − ∇ MB work entirely in Fourier space, where η = 1 (h + µ) or µ = h 6η. − 6 − − From this point on, in this Section, we work in Fourier space. The metric perturbation µ Follows the Fourier convention for E, so, from (17.9),

η = 1 (h + µ) or µ = h 6η . (17.11) − 6 − − The metric is k k h = 2DZ δ + 2 i j + 1 δ EZ ij − ij − k2 3 ij = 2DZ δ 2kˆ kˆ EZ + 2 EZ δ − ij − i j 3 ij = 1 hδ kˆ kˆ µ + 1 µδ 3 ij − i j 3 ij = kˆ kˆ h + (kˆ kˆ 1 δ )6η , (17.12) i j i j − 3 ij which is MB Eq. (4), where kˆ k /k. i ≡ i From Eq. (8.1), we get from the synchronous gauge to the Newtonian gauge by

1 Z 1 1 ξZ N = E = µ = (h + 6η) (17.13) → −k −2k 2k 0 1 Z 1 1 ξZ N = + E ′ = ξZ′ N = + µ′ → k2 −k → 2k2 17 SYNCHRONOUS GAUGE 45

From Eq. (7.10), the Bardeen potentials are

1 Z Z 1 1 Φ = E ′ + E ′′ = µ′ µ′′ = h′′ + 6η′′ + (h′ + 6η′) −k2 H 2k2 −H − 2k2 H Z 1 Z 1 Ψ = ψ + E ′ = η (h′ + 6η′) , (17.14) k2 H − 2k2 H which is MB Eq. (18). The opposite transformation, from Newtonian to synchronous gauge is, of course, just

0 0 ξN Z = ξZ N ξN Z = ξZ N , (17.15) → − → → − → so we can easily express it in synchronous gauge quantities, which may actually be what we want. We get the Einstein tensor perturbations from Eq. (A.7), setting A = B = 0,

0 2 2 δG0 = a− 2k ψ + 6 D′ (17.16) 0 2 H δG = a− 2ik ψ′ i − i i 2 δG0 = a− 2ikiψ′ i 2 2 1 2 1 2 i δG = a− 2D′′ + k D + 4 D′ k E + E′′ + E′ δ j H − 3 3 3 H j 2 1 1 k k a− D E + (E′′ + 2 E′) − i j − 3 k2 H i 2 2 δG = a− 6D′′ + 2k ψ + 12 D′ . i H In the MB η, h notation,

0 2 2 δG0 = a− 2k η h′ (17.17) 0 2 −H δG = a− 2ik η′ i − i i 2 δG0 = a− 2ikiη′ i 2 1 2 i δG = a− h′′ η′′ (h′ + 2η′)+ k η δ j − 2 − −H j 2 1 1 k k a− η ( h′′ + 3η′′ + h′ + 6 η′) − i j − k2 2 H H i 2 2 δG = a− h′′ + 2k η 2 h′ . i − − H The Einstein equations are thus

2 1 2 Z 3 2 Z k η h′ = 4πGa δρ = δ (17.18) − 2 H − − 2 H 2 2 Z 3 2 Z k η′ = 4πGa (¯ρ +p ¯)kv = (1 + w)kv 2 H Z 2 2 Z 2 δp h′′ + 2 h′ 2k η = 24πGa δp = 9 H − − − H ρ¯ 2 2 2 h′′ + 6η′′ + 2 h′ + 12 η′ 2k η = 16πGa p¯Π = 6 wΠ , H H − − − H which is MB Eq. (21). Note that MB uses the notation 2 p¯ 2 w θ ~v = 2v = kv and σ Π= Π . (17.19) ≡ ∇ · −∇ ≡ 3 (¯ρ +p ¯) 3 1+ w 17 SYNCHRONOUS GAUGE 46

We get the continuity equations from Eq. (A.15), setting A = B = 0 and D = 1 h, − 6 Z Z Z 1 Z δρ ′ = 3 (δρ + δp ) (¯ρ +p ¯)( h′ + kv ) (17.20) − H − 2 Z Z Z Z 2 (¯ρ +p ¯)v ′ = (¯ρ +p ¯)′v 4 (¯ρ +p ¯)v + kδp kp¯Π − − H − 3 Z Z Z 1 Z δp δ ′ = (1 + w)(kv + h′) + 3 wδ − 2 H − ρ¯ Z w′ kδp w vZ ′ = (1 3w)vZ vZ + 2 kΠ . −H − − 1+ w ρ¯ +p ¯ − 3 1+ w

The two last equations are MB Eq. (29).

Exercise: Derive the synchronous gauge Einstein equations and continuity equations from the corresponding Newtonian gauge equations by gauge transformation. 18 FLUID COMPONENTS 47

18 Fluid Components

18.1 Division into Components In practice, the cosmological ﬂuid consists of many components (e.g., particle species: photons, baryons, CDM, neutrinos, . . . ). It is useful to divide the energy tensor into such components:

µ µ Tν = Tν(i) , (18.1) Xi which have their corresponding background and perturbation parts ¯µ ¯µ µ µ Tν = Tν(i) and δTν = δTν(i) . (18.2) Xi Xi Here i labels the diﬀerent components (so I’ll use l and m for space indices). Often the component pressure is a unique function of the component energy density, pi = pi(ρi), or can be so approximated (common cases are pi = 0 and pi = ρi/3), although this is not true for the total pressure and energy density. The total ﬂuid and component quantities for the background are related as

ρ¯ = ρ¯i

p¯ = X p¯i = wiρ¯i p¯ ρ¯ w X= Xi w ≡ ρ¯ ρ¯ i X 2 p¯′ p¯i′ ρ¯i′ 2 cs = = ci , (18.3) ≡ ρ¯′ ρ¯′ ρ¯′ P X where p¯i 2 p¯i′ wi and ci . (18.4) ≡ ρ¯i ≡ ρ¯i′ The total ﬂuid and component quantities for the perturbations are related as

δρ = δρi = ρ¯iδi

δp = X δpi X

Xδρ ρ¯i δ = δ , (18.5) ≡ ρ¯ ρ¯ i X where δ δρ /ρ¯ . From δT l, i ≡ i i 0

(¯ρ +p ¯)vl = (¯ρi +p ¯i)vl(i) (18.6) we get that X ρ¯ +p ¯ 1+ w ρ¯ v = i i v = i i v (18.7) l ρ¯ +p ¯ l(i) 1+ w ρ¯ l(i) and from X X Σlm = Σlm(i) = p¯iΠlm(i) , where Π Σ /p¯ , we get that X X lm(i) ≡ lm(i) i Σ p¯ w ρ¯ Π lm = i Π = i i Π . (18.8) lm ≡ p¯ p¯ lm(i) wρ¯ lm(i) X X From here on, we consider scalar perturbations only. 18 FLUID COMPONENTS 48

18.2 Gauge Transformations and Entropy Perturbations The gauge transformations for the ﬂuid component perturbations are 0 δρi = δρi ρ¯i′ ξ (18.9) − 0 δpi = δpi p¯i′ ξ (18.10) e − v˜i = vi + ξ′ e Π˜ i = Πi (18.11)

ρ¯i′ 0 δ˜i = δi ξ (18.12) − ρ¯i Note that the ξ0 and ξ are the same for all fluid components. Those gauge conditions on ξ0 and ξ that refer to fluid perturbations that we have discussed so far refer to the total fluid. Thus, e.g., in the comoving gauge the total velocity perturbation v vanishes, but the component velocities vi do not vanish (unless they happen to be all equal). Thus, the velocity v that appears in the gauge transformation equations to comoving gauge refers to the total velocity perturbation.37 For example the component gauge tranformation equations that correspond to Eq. (16.14) read C N N δρi = δρi ρ¯i′ v (18.13) C N − N δp = δp p¯′ v i i − i C N ρ¯i′ N δi = δi v . − ρ¯i Since the gauge transformations are the same for all components, we find some gauge invari- ances. The relative velocity perturbation between two components i and j, v v is gauge invariant. (18.14) i − j Like the total anisotropic stress, also the component anisotropic stresses

Πi are gauge invariant. (18.15) We can also define a kind of entropy perturbation (different from the total entropy perturbation defined earlier!) S δρi δρj Sij 3 (18.16) ≡− H ρ¯i′ − ρ¯j′ ! between two fluid components i and j which turns out to be gauge invariant due to the way the density perturbations δρi transform. This is actually the most common type of quantity called “entropy perturbation” in the literature. It is a special case of a generalized entropy perturbation δx δy S (18.17) xy ≡H x¯ − y¯ ′ ′ between any two 4-scalar quantities x and y, which is gauge invariant due to the way 4-scalar perturbations transform. Beware of the many different quantities called “entropy perturbation”! Some of them can be interpreted as perturbations in some entropy/particle ratio. What is common to all of them, is that they all vanish in the case of adiabatic perturbations. (Entropy is a useful quantity in cosmology, since for most of the evolution of the universe, entropy is conserved to high accuracy. However, in these notes we are not using it, so we do not need the possible connection between the concepts “entropy” and “entropy perturbation”.)

37In some situations we may want to assign special status to one fluid component, and define a gauge which is N comoving with that fluid component, so then the vi of that component would appear in the gauge transformation equations for all fluid components. 18 FLUID COMPONENTS 49

18.3 Equations The Einstein equations (both background and perturbation) involve the total fluid quantities. The metric perturbations are not divided into components38 due to different fluid components! There is a single gravity, due to the total fluid, which each fluid component obeys. If there is no energy transfer between the fluid components in the background universe, the background energy continuity equation is satisfied separately by such independent components,

ρ¯′ = 3 (¯ρ +p ¯ ) , (18.18) i − H i i and in that case we can write Eq. (18.3) as ρ¯ +p ¯ c2 = i i c2 , (18.19) s ρ¯ +p ¯ i X and Eq. (18.16) as

δρi δρj δi δj Sij = = . (18.20) (1 + wi)¯ρi − (1 + wj)¯ρj 1+ wi − 1+ wj

We also get (2.19) componentwise,

wi′ 2 = 3 (wi ci ) . (18.21) 1+ wi H − Likewise, the gauge transformation equations (18.13) become

δρC = δρN + 3 (1 + w )¯ρ vN (18.22) i i H i i δpC = δpN + 3 (1 + w )c2ρ¯ vN i i H i i i δC = δN + 3 (1 + w )vN . i i H i N N Note that this transformation involves the total v , not the component vi , which makes the comoving gauge less practical for a many-component fluid. But if there is energy transfer between two fluid components, then their component energy continuity equations acquire an interaction term. Even if there is no energy transfer between fluid components at the background level, the perturbations often introduce energy and momentum transfer between the components. It may also be the case that the energy transfer can be neglected in practice, but the momentum transfer remains important. In the case of noninteracting fluid components (no energy or momentum transfer), we have the perturbation energy and momentum continuity equations separately for each such fluid component, µ Tν(i);µ = 0 . (18.23) For the case of scalar perturbations in the conformal-Newtonian gauge, they read

N N 2 N N δpi (δ )′ = (1+ w ) v + 3Ψ′ + 3 w δ (18.24) i i ∇ i H i i − ρ¯ i N N N wi′ N δpi 2 wi 2 (vi )′ = (1 3wi)vi vi + + Πi +Φ . (18.25) −H − − 1+ wi ρ¯i +p ¯i 3 1+ wi ∇

38There are exceptions to this in the literature, where contributions to some metric perturbation quantities due to different fluid components are defined. 18 FLUID COMPONENTS 50

In Fourier space they become

N N N N δpi (δ )′ = (1+ w ) kv + 3Ψ′ + 3 w δ (18.26) i i − i H i i − ρ¯ i N N N wi′ N kδpi 2 wi (vi )′ = (1 3wi)vi vi + kΠi + kΦ . (18.27) −H − − 1+ wi ρ¯i +p ¯i − 3 1+ wi If there are interactions between the fluid components, there will be interaction terms (“collision terms”) in these equations. Note that one cannot write the mixed gauge equations for fluid components by just replacing the fluid quantities in equations (16.22,16.23) with component quantities, since these equations were derived by making a gauge transformation between the comoving and Newtonian gauges, which involved the total fluid velocity. For the case where energy transfer between components can be neglected both at the background and perturbation level, we can use Eqs. (18.18) and (18.24) to find the time derivative of the entropy perturbation Sij,

2 N N 2 N N 2 N N ci δρi δpi cj δρj δpj Sij′ = vi vj + 3 − − (18.28) ∇ − H ρ¯i +p ¯i − ρ¯j +p ¯j ! = 2(v v ) 9 c2 c2 , ∇ i − j − H i Si − j Sj where δp δρ i i (18.29) Si ≡H p¯ − ρ¯ i′ i′ is the (gauge invariant) internal entropy perturbation of component i. If we have in addition, that for both components the equation of state has the form pi = fi(ρi), then the internal entropy perturbations vanish39, and we have simply

2 S′ = (v v ) . (18.30) ij ∇ i − j In Fourier space Eq. (18.30) reads

1 k S′ = k(v v ) or − S′ = (v v ) , (18.31) ij − i − j H ij − i − j H showing that entropy perturbations remain constant at superhorizon scales k . From this ≪ H follows that adiabatic perturbations (Sij = 0) stay adiabatic while outside the horizon. We can also obtain an equation for Sij′′ from Eq. (18.25), if also momentum transfer between components (at the perturbation level) can be neglected (and we also keep the assumption pi = fi(ρi)):

N N S′′ = k(v′ v′ ) = k (v )′ (v )′ ij − i − j − i − j w w′ = k (1 3w )vN (1 3w )vN + k i′ vN j vN H − i i − − j j 1+ w i − 1+ w j i j 2 N c2δN c δ w w k2 i i j j + 2 k2 i Π j Π . (18.32) − 1+ w − 1+ w 3 1+ w i − 1+ w j " i j # i j If one further assumes that both components are perfect ﬂuid, then one can drop the last term; and if one assumes that the equation-of-state parameter is constant for both components, then one can drop the second term.

39 For baryons this may require that we ignore baryon pressure, since pb = pb(nb,T )= nbT , and ρb = ρb(nb,T )= 3 nb(mb + 2 T ). 19 SIMPLIFIED MATTER+RADIATION UNIVERSE 51

In the synchronous gauge, Eqs. (18.26,18.27) become

Z Z Z 1 Z δpi (δ )′ = (1 + w ) kv + h′ + 3 w δ (18.33) i − i i 2 H i i − ρ¯ i Z Z Z wi′ Z kδpi 2 wi (vi )′ = (1 3wi)vi vi + kΠi . (18.34) −H − − 1+ wi ρ¯i +p ¯i − 3 1+ wi Since E does not appear in the general gauge scalar continuity equations (A.15), the only diﬀerence in them when going from Newtonian to synchronous gauge (as both have B = 0) is that Ψ = DN is replaced by 1 h = DZ and Φ = AN is dropped as AZ = 0. − 6 19 Simpliﬁed Matter+Radiation Universe

Consider the case where the energy tensor consists of two perfect fluid components, matter with 1 p = 0 and radiation with p = 3 ρ, that do not interact with each other, i.e., there is no energy or momentum transfer between them. (Compared to the real universe, this is a simplification since, while cold dark matter does not much interact with the other fluid components, the baryonic matter does interact with photons. Also, the radiation components of the real universe, neutrinos and photons, behave like a perfect fluid only until their decoupling. We also ignore dark energy.)

19.1 Background solution for radiation+matter We have now two ﬂuid components, ρ = ρr + ρm , where 4 3 ρ a− and ρ a− , r ∝ m ∝ and 1 pm =0 and pr = 3 ρr . The equation of state and sound speed parameters are

2 2 1 wm = cm =0 and wr = cr = 3 . (19.1) We deﬁne a ρ y = m , (19.2) ≡ aeq ρr where Ωr aeq = (19.3) Ωm is the scale factor at matter-radiation equality, so that ρ 1 ρ y ρ + p 4 ρ + p 3y r = m = r r = m m = (19.4) ρ 1+ y ρ 1+ y ρ + p 4 + 3y ρ + p 4 + 3y and 1 4 + 3y 4 3y w = 1+ w = c2 = 1 3c2 = . (19.5) 3(1 + y) 3(1 + y) s 3(4 + 3y) − s 4 + 3y The Friedmann equation is 2 2 1 da 8πG 2 8πG 2 = ρa = (1 + y)ρ a− H ≡ a2 dη 3 3 r0 dy = 2Cdη , ⇒ √1+ y 19 SIMPLIFIED MATTER+RADIATION UNIVERSE 52 where 2πG 1 1 H0 1 Ωm C ρr0 = Ωr = H0 . (19.6) ≡ r 3 aeq 2 aeq 2 √Ωr p The solution is

2 2 1 2 2 y = 2Cη + C η or a(η)= ΩrH0η + 4 ΩmH0 η . (19.7) At the time of matter-radiation equality, p

y = y = C2η2 + 2Cη = 1 Cη = √2 1 , (19.8) eq eq eq ⇒ eq − so we can write the solution as η η 2 y = 2 + , (19.9) η η 3 3 where ηeq 1 2 √Ωr η3 = √2 + 1 ηeq = = (19.10) ≡ √2 1 C H0 Ωm − is the time when y = 3 (ρm = 3ρr). The Hubble parameter is a y η + η ′ = ′ = 3 . (19.11) 1 2 H≡ a y η3η + 2 η At matter-radiation equality it has the value

2√2 1 4 2√2 2√2 Ωm eq = = − = = √2 H0 . (19.12) H √2 + 1 ηeq ηeq η3 √Ωr At early times, η η , the universe is radiation dominated, so that ≪ 3 2η 1 1 y 1 a η = a− . (19.13) ≈ η3 ≪ ⇒ ∝ ⇒ H η ∝ At late times, η η , the universe is matter dominated, so that ≫ 3 2 η 2 2 1/2 y 1 a η = a− . (19.14) ≈ η ≫ ⇒ ∝ ⇒ H η ∝ 3 When solving for perturbations it turns out to be more convenient to use y (or log y) as time coordinate instead of η. Inverting Eq. (19.9), we have that

√1+ y 1 η = 1+ y 1 η3 = − ηeq , (19.15) − √2 1 p − and √1+ y 2 √1+ y = = Heq . (19.16) H y η3 y √2

19.2 Perturbations In terms of the component perturbations the total perturbations are now 1 y δ = δ + δ (19.17) 1+ y r 1+ y m 4 3y v = v + v (19.18) 4 + 3y r 4 + 3y m 19 SIMPLIFIED MATTER+RADIATION UNIVERSE 53 and the relative entropy perturbation is

S S = δ 3 δ . (19.19) ≡ mr m − 4 r

From the pair (19.17,19.19) we can solve δm and δr in terms of δ and S: 3 + 3y 4 δ = δ + S m 4 + 3y 4 + 3y 4 + 4y 4y δ = δ S . (19.20) r 4 + 3y − 4 + 3y

Likewise we can express vm and vr in terms of the total and relative velocity perturbations, v and v v : m − r 4 v = v + (v v ) m 4 + 3y m − r 3y v = v (v v ) . (19.21) r − 4 + 3y m − r We can now also relate the total entropy perturbation to S: S 1 δρ 1 δp y = = . . . = S = 1 (1 3c2)S . (19.22) S 3(1 + w) ρ¯ − c2 ρ¯ 4 + 3y 3 − s s The Bardeen equation (16.42) becomes now

2 2 1 3 2 2 − δ ′′ + 1 6w + 3c − δ ′ 1 + 8w 6c 3w δ H C − s H C − 2 − s − C k 2 = c2 δ (1 + w)(1 3c2)S − s C − − s H k 2 y = c2 δ S . (19.23) − s C − 1+ y H We get the entropy evolution equation by derivating (18.30),

S′ = k(v v ) S′′ = k(v′ v′ ) . (19.24) − m − r ⇒ − m − r N 2 Using the vi evolution equations (18.27), or equivalently, using (18.32) with wm = cm = 0 and 2 1 wr = cr = 3 (and Πi = 0), this becomes (exercise)

N 1 2 N S′′ = k(v v )+ kv + k δ . (19.25) H m − r H r 4 r N Here the δr is converted to the comoving gauge using the total ﬂuid velocity (see Eq. 18.22),

N C 1 N δ = δ 3 (1 + w )k− v (19.26) r r − H r so that Eq. (19.24) becomes

4 1 2 C S′′ = k (v v )+ k δ . (19.27) H 4 + 3y m − r 4 r Replacing v v by k 1S we get (Exercise) the Kodama-Sasaki equation [8] m − r − − ′ 2 2 4 1 k 1+ y C y − S′′ + − S′ = δ S . (19.28) H 4 + 3y H 4 + 3y − 4 + 3y H 19 SIMPLIFIED MATTER+RADIATION UNIVERSE 54 or 2 2 2 1 1 k 1 C 2 − S′′ + 3c − S′ = δ (1 3c )S . H sH 3 1+ w − − s H The two equations (19.23) and (19.28) form a pair of ordinary diﬀerential equations, from which we can solve the evolution of the perturbations δC (η) and S (η). Since the coeﬃcient ~k ~k functions of these equations can more easily be expressed in terms of the scale factor y, it may be more convenient to use y as the time coordinate instead of η. The time derivatives are converted with (exercise)

2 1 df 2 2 d f 1 df − f ′ = y and − f ′′ = y + (1 3w)y (19.29) H dy H dy2 2 − dy and the equations become40

d2δC dδC k 2 y y2 + 3 (1 5w +2c2)y 3 (1+8w 6c2 3w2)δC = c2 δC S (19.30) dy2 2 − s dy − 2 − s − − s − 1+ y H d2S dS k 2 1+ y y y2 + 1 (1 3w +6c2)y = δC S . dy2 2 − s dy 4+3y − 4+3y H

For solving the other perturbation quantities, we collect here the relevant equations:

3 2 Φ = H δC (19.31) −2 k N 2 k 1 v = − Φ′ +Φ 3(1 + w) H H δN = δC 3 H (1 + w)vN − k 2 = Φ Φ′ + Φ R − − 3(1 + w) H H When judging which quantities are negligible at superhorizon scales (k ), one has to exercise ≪H some care, and not just look blindly at powers of k/ in equations which contain diﬀerent per- H turbation quantities. From Eq. (19.31a) one sees that at superhorizon scales, a small comoving δC can still be important and cause a large gravitational potential perturbation Φ. Eq. (16.40),

C 1 2 δ − ′ = c 3 , H R − s 1+ w − S may seem to contradict the statement that for adiabatic perturbations stays constant at R superhorizon scales, but the explanation is that is then of the same order of magnitude as Φ, R whereas δC is suppressed by two powers of k/ compared to Φ. Using Eqs. (19.31a,19.22) we H rewrite (16.40) as 2 1 2 1 2 k 2 − ′ = c Φ + (1 3c )S . (19.32) H R s 1+ w 3 − s " H # 40These are Eqs. (2.12a) and (2.12b) in Kodama&Sasaki[8], when one replaces the photon+baryon ﬂuid in [8] 2 1 2 with just radiation with wr = cr = 3 . The w and cs in these expressions could be written in terms of y, or the 2 y expressions on the rhs could be written in terms of w, cs, andρ ¯m/ρ¯; but in this form I found them easy to compare to [8]. 19 SIMPLIFIED MATTER+RADIATION UNIVERSE 55

19.3 Initial Epoch For y 1, ≪ 2 2 Heq (19.33) H ≈ 2y2 and the equations (19.23,19.28,19.30) can be approximated by

2 2 1 k − δ′′ 2δ = (δ yS) (19.34) H C − C − 3 C − H2 2 1 1 k − S′′ + − S′ = (δ yS) (19.35) H H 4 C − H or

d2δC k 2 2 k 2 y2 2δC = 1 (δC yS) = y2δC y3S (19.36) dy2 − − 3 − −3 − H Heq d2S dS k 2 1 k 2 y2 + y = 1 (δC yS) = y2δC y3S . (19.37) dy2 dy 4 − 2 − H Heq At early times, all cosmologically intersecting scales are outside the horizon, and we start by making the approximation, that the rhs of these equations can be ignored. Then the evolution of δC and S decouple and the general solutions are

δC = A y2 + B y 1 (19.38) ~k ~k ~k − S~k = C~k + D~k ln y . (19.39)

Thus, for each Fourier component ~k, there are four independent modes. We identify the adiabatic 41 growing (A~k) and decaying (B~k) modes, and the isocurvature “growing” (C~k) and decaying (D ) modes. The D-mode is indeed decaying, since 0 < y 1, so ln y is large and negative, ~k ≪ and gets smaller as y grows. The decaying modes diverge as y 0, but we can suppose that our description of the → universe is not valid all the way to y = 0, but at very early times there is some process that is responsible for fixing the initial values of δC , δC ′, S , and S at some early time y > 0, and ~k ~k ~k ~k′ init fixing thus A~k, B~k, C~k, and D~k. Let us now consider the effect of the rhs of the eqs. These couple the δ and S. Thus it is not consistent to assume that the other is exactly zero.

19.3.1 Adiabatic modes

Consider ﬁrst the adiabatic modes (C~k = D~k = 0). The coupling forces the existence of a small nonzero S, which at ﬁrst is δC . We can thus keep ignoring S on the rhs, and for the δC ≪ equation we can keep ignoring the whole rhs. But for the S equation we now have very small S on the lhs, and therefore we can not ignore the large δ on the rhs, even though it is suppressed by (k/ )2, or y2. Thus for adiabatic modes the pair of equations can be approximated as H 2 − δC′′ 2δC = 0 (19.40) H − 2 2 1 1 k − S′′ + − S′ = δ (19.41) H H 4 C H 41The term “isocurvature” will be explained later. 19 SIMPLIFIED MATTER+RADIATION UNIVERSE 56 or d2δC y2 2δC = 0 (19.42) dy2 − d2S dS 1 k 2 y2 + y = y2δC . (19.43) dy2 dy 2 Heq The solution for δ remains Eq. (19.38), but we also get that 1 k 2 1 k 2 S = A y4 + B y . (19.44) ~k 32 ~k 2 ~k Heq Heq Thus for the growing adiabatic mode 1 k 2 1 k 2 S = δC y2 = δC (19.45) ~k 32 ~k 64 ~k Heq H and for the decaying adiabatic mode 1 k 2 1 k 2 S = δC y2 = δC . (19.46) ~k 2 ~k 4 ~k Heq H So you see that, although these are called adiabatic modes, the perturbations are not exactly adiabatic! The name just means that S 0 as y 0. The entropy perturbations remain small → → compared to the density perturbation while the Fourier mode is outside the horizon, but can become large near and after horizon entry.

19.3.2 Isocurvature Modes Consider then the isocurvature modes (A = B = 0). Now the coupling causes a small δC S. ~k ~k ≪ We can ignore the rhs of the S equation, but the large S cannot be ignored on the rhs of the δC equation. Thus for isocurvature modes the pair of equations can be approximated by 2 2 1 k − δ′′ 2δ = + yS (19.47) H C − C 3 2 1 H − S′′ + − S′ = 0 (19.48) H H or d2δC 2 k 2 y2 2δC = + y3S (19.49) dy2 − 3 Heq d2S dS y2 + y = 0 . (19.50) dy2 dy The solution for S remains Eq. (19.39), but for the density perturbation we get 1 k 2 1 k 2 5 δC = C y3 + D y3 ln y y3 . (19.51) ~k 6 ~k 6 ~k − 4 Heq Heq (For the decaying isocurvature mode the diﬀerential equation for δ is a little more diﬃcult, but you can check (Exercise) that the above is the correct solution.) For the growing isocurvature mode we have thus that 1 k 2 1 k 2 δC = S y3 = yS . (19.52) ~k 6 ~k 12 ~k Heq H So although the relative entropy perturbation stays constant at early times, the density perturbation is growing in this mode. For the decaying isocurvature mode 1 k 2 5 y 1 k 2 δC = y S yS . (19.53) ~k 12 − 4 ln y ~k ≈ 12 ~k H H 19 SIMPLIFIED MATTER+RADIATION UNIVERSE 57

19.3.3 Other Perturbations

1 For the initial epoch, w = 3 , and Eqs. (19.31) become

3 2 Φ = H δC (19.54) ~k −2 k ~k N 1 k 1 v = − Φ′ +Φ ~k 2 H ~k ~k H δN = δC 4 H vN ~k ~k − k ~k 3 1 1 = Φ − Φ′ R~k − 2 ~k − 2 H ~k For the growing adiabatic mode, we have

3 2 3 2 Φ = H A y2 = Heq A = const. (19.55) ~k −2 k ~k −4 k ~k 9 2 = 3 Φ = Heq A = const. (19.56) R~k − 2 ~k 8 k ~k 1 k √2 k vN = = y (19.57) ~k −3 R~k − 3 R~k H Heq δN = 2Φ = 4 . (19.58) ~k − ~k 3 R~k For the growing isocurvature mode, where S = const. we have

1 eq Φ~ = yS~ = H ηS~ (19.59) k − 8 k −8√2 k 1 1 ~ = eqS~ η = S~ y = 2Φ~ (19.60) Rk 4√2H k 4 k − k 1 k vN = S y2 (19.61) ~k ~k −4√2 eq H 1 k 2 δN = S y3 + 1 S y 1 S y = 4Φ = 2 . (19.62) ~k 6 ~k 2 ~k ≈ 2 ~k − ~k R~k Heq Thus the growing adiabatic mode is characterized by a constant and the growing isocur- R vature mode by a constant S. If both modes are present, and these constants are of a similar magnitude, then the growing S associated with the adiabatic mode is negligible as long as k , and the growing associated with the isocurvature mode is negligible (hence the name ≪H R “isocurvature”) as long as y 1. ≪ Thus, after the decaying modes have died out, the general (adiabatic+isocurvature) mode is characterized by these two constants (for each Fourier mode), which we denote by (rad) and R~k S~k(rad). Including the two small growing contributions we have, during the initial epoch,

9 2 = Heq A + 1 yC = (rad) + 1 yS (rad) (19.63) R~k 8 k ~k 4 ~k R~k 4 ~k 1 k 2 1 k 4 S = A y4 + C = S (rad) + (rad)y4 ~k 32 ~k ~k ~k 36 R~k Heq Heq 1 k 4 = S (rad) + (rad) . (19.64) ~k 9 R~k H 19 SIMPLIFIED MATTER+RADIATION UNIVERSE 58

19.4 Full evolution for large scales The full evolution in the general case is not amenable to analytic solution, and has to be solved numerically. This is not too diﬃcult since we just have a pair of ordinary diﬀerential equations to solve for each k. The results from Sect. 19.3 can be used to set initial values at some small y. For large scales (k k ) we can, however, solve the evolution analytically. We now ≪ eq ≡Heq drop the decaying modes and consider what happens to the growing modes after the radiation- dominated epoch. Our basic equations are (19.32) and (19.28):

2 1 2 1 2 k 2 − ′ = c Φ + (1 3c )S (19.65) H R~k s 1+ w 3 ~k − s ~k " H # 2 2 2 1 1 k 1 2 − S′′ + 3c − S′ = δ (1 3c )S . (19.66) H ~k sH ~k 3 1+ w ~k − − s ~k H

We see that for superhorizon scales, S~k = const. remains a solution even when the universe is no longer radiation dominated. Since the other solution has decayed away, we conclude that

S = const. = S (rad) for k . (19.67) ~k ~k ≪H We can also see that, for the adiabatic mode, stays constant for superhorizon scales, R~k since on the rhs, S remains negligible for k . For scales that enter the horizon during the ~k ≪H matter-dominated epoch, the of adiabatic perturbations stays constant even through and R~k after horizon entry, since c2 becomes negligibly small, before k/ and S become large. We can s H ~k thus conclude that

= const. = (rad) for adiabatic modes with k k . (19.68) R~k R~k ≪ eq

If the isocurvature mode is present, however, we cannot assume that S~k is negligible for superhorizon scales; it’s just constant, and we have, for k , ≪H 1 2 2 − ′ = c (1 3c )S , (19.69) H R~k s − s ~k or d ~ 4 Rk = S . (19.70) dy (4 + 3y)2 ~k Integrating this, we get y = (rad) + S (rad) for k . (19.71) R~k R~k 4 + 3y ~k ≪H

For k k , this has reached the ﬁnal value (rad) + 1 S (rad) when the universe has become ≪ eq R~k 3 ~k matter dominated, before horizon entry. After that, stays constant even through and after R~k horizon entry by the same argument as for the adiabatic mode. Thus we conclude that

= const. = (rad) + 1 S (rad) for k k and η η . (19.72) R~k R~k 3 ~k ≪ eq ≫ eq For smaller scales, the exact solution has to be obtained numerically. It is customary to represent the solution in terms of transfer functions TRR(k) and TRS(k), which we can deﬁne by

= const. T (k) (rad) + 1 T (k)S (rad) for η η , (19.73) R~k ≡ RR R~k 3 RS ~k ≫ eq so that T (k)= T (k)=1 for k k . RR RS ≪ eq Finally, we can ask what happens to S~k after horizon entry (Exercise). . . . 19 SIMPLIFIED MATTER+RADIATION UNIVERSE 59

19.5 Initial Conditions in Terms of Conformal Time

The full (linear) evolution of perturbations at all scales k needs to be calculated numerically with codes such as CAMB. These codes require theoretically derived initial conditions. CAMB uses conformal time as time variable and works in synchronous gauge. The initial conditions are given as truncated series in powers of conformal time η. Our goal here is to derive the CAMB initial conditions as they are given in “CAMB Notes” by Antony Lewis [9]. These are for the real universe with baryons, CDM, photons, and neutrinos. Our simpliﬁed universe of this Section corresponds to the case without neutrinos (who are not perfect) and baryons (who interact with photons). Thus we should compare to the CAMB equations setting fν = fb = 0 (in Lewis’ notation Rν ad Rb), so that photons (γ) represent radiation (r) and CDM (c) represents matter (m). Our strategy is to derive the density perturbations in comoving gauge and then do a gauge transformation to synchronous gauge. We start from the Bardeen (19.23) and Kosama–Sasaki (19.28) equations

′′ 2 ′ 3 2 2 2 2 2 2 δ + 1 6w +3c δ 1+8w 6c 3w δC = c k δC (1 + w)(1 3c )S C − s H C − 2 − s − H − s − − s ′′ 2 ′ 1 2 1 2 S +3c S = k δC (1 3c )S . sH 3 1+ w − − s Writing the coeﬃcients in terms of y, they read (exercise)

2 3 2 2 ′′ 2 5+3y ′ 16+38y + 30y +9y 2 (kη3) y η δ + η3δ δC = δC S 3 C √1+ y 4+3y C − (1 + y)(4 + 3y) y2 −3(1 + 3 y) − 1+ y 4 2 2 ′′ √1+ y 2 ′ (kη3) η S + η3S = (1 + y)δC yS (19.74) 3 1+ 3 y y 4(1 + 3 y) − 4 4 h i Deﬁning η x we have y =2x + x2 , (19.75) ≡ η3 and (19.74) becomes (exercise)

6 3 2 19 79 2 3 69 4 27 5 9 6 ′′ 5 1+ x + x ′ 1+ x + x + 12x + x + x + x η2x2δ + 5 5 η x2δ 2 4 8 8 8 16 δ 3 C 5 9 2 3 3 3 C 9 17 2 69 3 79 4 3 5 3 6 C 2 1+ 2 x + 4 x + 4 x − 1+ 2 x + 2 x + 8 x + 16 x + 2 x + 16 x 1 x2 2 1+ 1 x = (kη )2 δ + (kη )2 2 x3S 3 3 3 2 C 3 7 19 2 3 3 4 −3 1+ 2 x + 4 x 3 1+ 2 x + 4 x +3x + 4 x 2 1 ′′ 1+ x ′ 1 1+2x + x 1 1+ x η2xS + η S = (kη )2 xδ (kη )2 2 x2S (19.76) 3 3 2 3 3 3 3 3 3 2 C 3 3 3 2 1+2x + 2 x + 8 x 4 1+ 2 x + 4 x − 2 1+ 2 x + 4 x

We are working in the limit kη = kη3x 1 and η η3 x 1. We search for solutions as series in x, which we will truncate at fourth and ﬁfth≪ order,42≪i.e., ⇒ ≪

2 3 4 2 3 4 5 δC A + Bx + Cx + Dx + Jx and S E + F x + Gx + Hx + Kx + Lx . (19.77) ≈ ≈ We do not allow for negative powers of x in this expansion, since a solution containing a negative power is a decaying mode, and we are not interested in those. To solve all the 11 constants from A to L we need to expand each coeﬃcient expression in (19.76) to suﬃcient order in x. It turns out that we need all terms in (19.76) up to x4. This expansion converts (19.76) into43 (exercise)

2 2 ′′ 5 13 8 2 2 ′ 1 1 2 1 3 37 4 x η δ + 1 x + x x η δ 2 1+ x + x + x x δC 3 C 2 − 10 5 3 C − 4 4 8 − 32 1 2 3 3 2 2 2 2 3 3(kη3) 1 2 x + 2 x x δC + 3 (kη3) (1 3x) x S ′′ ≈ − − ′ − xη2S + 1 x + 1 x2 + 1 x3 5 x4 η S 3 − 2 8 − 8 3 1 2 1 1 2 3 3 1 2 3 2 2 (kη ) 1+ x x + x xδC (kη ) 1 x + x x S. (19.78) ≈ 4 3 2 − 2 8 − 2 3 − 4 42 The fourth order in δC is overkill, but we will need Sto ﬁfth order to do the gauge transformation for the adiabatic mode. I originally did this to third order, thus the odd naming of the fourth order coeﬃcients. 43I have not checked all the highest order terms, so that the result for J (which we do not use) may not be trusted. The step from (19.76) to (19.78) I have now checked with WolframAlpha. 19 SIMPLIFIED MATTER+RADIATION UNIVERSE 60

Substituting (19.77) into (19.78) and requiring each power of x up to x4 to agree in (19.78) gives 10 equations for 11 unknowns, but it turns out that one of these equations does not give additional information, leaving two degrees of freedom. We get (exercise) A = B = F = G = 0, so that C gives the lowest order for δC and E gives the lowest order for S. Expressing the other constants in terms of these, we get44 (exercise)

D = 9 C + 1 (kη )2E − 8 6 3 J = 119 C 1 (kη )2C 19 (kη )2E 80 − 30 3 − 60 3 H = 1 (kη )2E − 18 3 1 2 1 2 K = 64 (kη3) C + 48 (kη3) E L = 3 (kη )2C 1 (kη )2E + 1 (kη )4E, (19.79) − 800 3 − 120 3 600 3 so that

C 2 9 3 119 1 2 4 1 2 3 19 2 4 δ = C~ x x + (kη ) x + E~ (kη ) x (kη ) x ~k k − 8 80 − 30 3 k 6 3 − 60 3 1 2 3 1 2 4 1 2 1 4 5 S~ = E~ 1 (kη3)x + (kη3) x + (kη 3) + (kη3) x k k − 18 48 − 120 600 1 2 4 3 2 5 +C~ (kη ) x (kη ) x . (19.80) k 64 3 − 800 3 We identify an adiabatic mode (δC = (x2) and S = (x4)), whose initial amplitude is given by 0 O C 3O C~k and an isocurvature mode (S = (x ) and δ = (x )), whose initial amplitude is given by E~k. Dropping some higher powers that weO will not need, theO adiabatic mode has

δC x2 9 x3 + (x4) ∝ − 8 O S 1 (kη )2x4 3 (kη )2x5 + (x6) (19.81) ∝ 64 3 − 800 3 O and the isocurvature mode has

δC 1 (kη )2x3 + (x4) ∝ 6 3 O S 1 1 (kη )2x3 + 1 (kη )2x4 + (x5) . (19.82) ∝ − 18 3 48 3 O For comparison to [9], deﬁne

Ωm 2 ω ω H0 η3 = and x = η . (19.83) ≡ √Ωr ⇒ ω 2 To get the other perturbations, using results from Sec. 19.2, we need the expansions (exercise) 1 1 1+ 1 x 1 x2 = 1+ 1 ωη 1 ω2η2 H ≈ 2 − 4 η 4 − 16 η 1+ w 4 1 1 x + 3 x2 = 4 1 1 ωη + 3 ω2η2 . (19.84) ≈ 3 − 2 4 3 − 4 16 Isocurvature mode. Dropping the (x4) part of δC and the (x5) part of S, and not writing the O O initial amplitude E~k (i.e., all perturbations below are to be multiplied by the value of E~k, i.e., by the initial value of S~k), the isocurvature mode is δC = 1 ωk2η3 + (η4) 12 O S = 1 1 ωk2η3 + 1 ω2k2η4 + (η5) . (19.85) − 36 192 O Using (19.31) (exercise), we get

Φ = 1 ωη + (η2) − 8 O vN = 1 ωkη2 + (η3) − 8 O δN = 1 ωη + (η2) 2 O = 1 ωη + (η2) . (19.86) R 4 O 44I have not checked the E part of L, which we will not use. 19 SIMPLIFIED MATTER+RADIATION UNIVERSE 61

Note that we got δN entirely from the vN part of (19.31b); δC contributes to it at (η3), but to calculate δN to (η3) we would need vN to (η4), which requires Φ to (η3), which requiresO δC to (η5). From (19.20)O we get (exercise) O O O

δC = 1 3 ωη + 3 ω2η2 9 ω3η3 + 5 ωk2η3 + (η4) m − 4 8 − 64 144 O δC = ωη + 1 ω2η2 3 ω3η3 + 1 ωk2η3 + (η4) . (19.87) r − 2 − 16 12 O

The entropy perturbation S and the velocity diﬀerence vm vr are gauge invariant, and we have −

1 ′ 1 2 1 2 3 4 vm vr = S = ωkη ω kη + (η ) . (19.88) − −k 12 − 48 O Using (19.21) we get (exercise)

vN = 1 ωkη2 + 1 ω2kη3 + (η4) m − 24 60 O vN = 1 ωkη2 + 3 ω2kη3 + (η4) . (19.89) r − 8 80 O Z Now we want to change to synchronous gauge. We use the remaining gauge freedom to set vm 0, i.e., the threads, which in synchronous gauge must be geodesics, follow the matter world lines (which≡ are geodesics, since matter here is assumed pressureless and not interacting with the other ﬂuid component). This means that (exercise)

vZ = vZ vZ = 1 ωkη2 + 1 ω2kη3 + (η4) r r − m − 12 48 O 4 vZ = vZ = 1 ωkη2 + 1 ω2kη3 + (η4) . (19.90) 4+3y r − 12 12 O We get from synchronous gauge to comoving gauge by

δC = δZ +3H (1 + w )vZ (19.91) i i k i (since BZ = 0), so we have (exercise)

δZ = δC 3H vZ = 1 1 ωη + 3 ω2η2 + (η3) m m − k − 2 16 O δZ = δC 4H vZ = 2 ωη + 1 ω2η2 + (η3) . (19.92) r r − k − 3 4 O

Z Z Z These results for Φ, δm, δr , and vr agree with [9] (setting there Rc = 1 and Rν = 0).

Adiabatic mode. Calculation of ~ for the adiabatic mode gives Rk 2 9 ω 2 ~ = C~ + (η ) . (19.93) Rk 16 k2 k O

2 2 Let us normalize the adiabatic mode so that ~ = 1 initially, i.e, set C~ = (16/9)k /ω . This means that Rk k all perturbations below are to be multiplied by the true initial value of ~ . Thus the adiabatic mode is Rk δC = 4 k2η2 1 ωk2η3 + (η4) 9 − 4 O S = 1 k4η4 1 ωk4η5 + (η6) . (19.94) 144 − 1200 O N C C 3 4 N Exercise: Calculate Φ to (η), v to (η), to (η), δm and δr to (η ), vm vr to (η ), vm and vN to (η), vZ to (η4), OvZ to (η4),O and δRZ andOδZ to (η3). O − O r O r O O m r O 20 EFFECT OF A HOMOGENEOUS COMPONENT 62

20 Eﬀect of a Homogeneous Component

Consider the effect of a fluid component, whose perturbations we ignore, so that it affects only the background solution. The main application is dark energy. In case dark energy is just a cosmological constant, it has no perturbations. If the dark energy is close to a cosmological constant (w 1), its perturbations should be small. ∼− Note: Sec. 20.1 is quite raw, since the assumption of ignoring the perturbations of the h component is an approximation, unless it is a cosmological constant; and proper considerations on what approximations in the equations are valid in what conditions are missing. Thus Sec. 20.1 really just serves now as an intermediate step to Sec. 20.2.

20.1 General Considerations For simplicity, consider a two-component ﬂuid: a homogeneous (h) component, whose perturbations vanish, δh = vh = δph = Πh = 0 (20.1) and a perturbed (p) component, so that for the background quantities (we now drop the bars from background quantities) ρ = ρp + ρh and p = pp + ph. We deﬁne the density parameter for the perturbed component ρ Ω p . (0 < Ω < 1) (20.2) ≡ ρ From Sec. 18.1,

w = Ωw + (1 Ω)w 1+ w = Ω(1+ w ) + (1 Ω)(1 + w ) p − h ⇒ p − h 1+ w 1+ w c2 = p Ωc2 + h (1 Ω)c2 s 1+ w p 1+ w − h δ = Ωδp

δp = δpp 1+ w v = p Ωv 1+ w p w Π = p ΩΠ . (20.3) w p From (10.14)–(10.17) the Einstein equations in Newtonian gauge become

2Ψ = 3 2Ω δN + 3 (1 + w )vN ∇ 2 H p H p p 2 Ψ Φ = 3 ΩwpΠp − 3H 2 N Ψ′ + Φ = Ω(1 + w )v H 2 H p p N 2 1 2 3 2 δpp Ψ′′ + (Φ′ + 2Ψ′) + (2 ′ + )Φ + 3 (Φ Ψ) = 2 Ω . (20.4) H H H ∇ − H ρp

From (18.24) and (18.25), the ﬂuid continuity equations for the p component are

N N 2 N N δpp (δp )′ = (1+ wp) vp + 3Ψ′ + 3 wpδp ∇ H − ρp ! N N N wp′ N δpp 2 wp 2 (vp )′ = (1 3wp)vp vp + + Πp +Φ . (20.5) −H − − 1+ wp ρp + pp 3 1+ wp ∇ If we wanted to remain exact, similar equations should hold also for the h component, and we N see that even if we started with δh = vh = δph = Πh = 0, they could not remain zero, since δh 20 EFFECT OF A HOMOGENEOUS COMPONENT 63

is sourced by 3(1 + w )Ψ and vN is sourced by Φ, unless w = 1, which is the case of vacuum h ′ h h − energy (or Λ). For vacuum energy, v is not defined, since it comes from δT i (ρ +p )v and h 0h ≡ h h h,i ρh + ph = 0 for vacuum energy. For other than vacuum energy for the homogeneous component 45 we are making an approximation. The smaller 1 + wh is, the better the approximation should be. Mixed gauge. (Compare to Sec. 16.3.) There are now two possibilities for defining the comoving C C gauge, we can either require v =0 or vp = 0. Since only the p fluid is flowing, the second choice seems more promising, and we choose it here, defining

C N N δ δ +3 (1 + wp)v p ≡ p H p C N 2 N δp δp +3 (1 + wp)c ρpv , (20.6) p ≡ p H p p N N i.e., using vp instead of v , so that the gauge is comoving with the p fluid flow. Beware that this is a different gauge than the comoving gauge used earlier, and thus earlier results for the comoving gauge do not apply as such. With the help of (20.6), (20.5) becomes

C ′ C 2 N 2 9 2 N (δ ) 3 wpδ = (1+ wp) v +2 wp Πp (1 + wp)(1 + wh)(1 Ω)v p − H p ∇ p H ∇ − 2 H − p C N ′ N δpp 2 wp 2 (vp ) + vp = + Πp +Φ . (20.7) H ρp + pp 3 1+ wp ∇

Since the homogeneity approximation for h should only be good when (1+ wh)(1 Ω) is small, we can probably ignore the last term on the rhs of Eq. (20.7a). − The Einstein equations that involve density and pressure perturbations can now be rewritten (exercise)

2Ψ = 3 2ΩδC ∇ 2 H p C ′′ 2 ′ ′ 2 2 1 2 3 2 δpp Ψ +(2+3cp) Ψ + Φ + 3(cp w) Φ+ 3 (Φ Ψ) = 2 Ω . (20.8) H H − H ∇ − H ρp Note that the second equation involves the p ﬂuid sound speed but total w.

2 Matter. If the perturbed component is matter, δpm = Πm = wm = cm = 0, we have

w = (1 Ω)wh 1+ w = 1+ wh Ωwh − ⇒ − 1+ w c2 = h (1 Ω)c2 s 1+ w − h Ω v = v . (20.9) 1+ w p The Einstein equations become

Ψ = Φ 2Φ = 3 2Ω δN +3 vN = 3 2ΩδC ∇ 2 H m H m 2 H m ′ 3 2 N Φ + Φ = 2 Ωvm ′′ ′ ′ H H Φ +3 Φ + (2 + 2)Φ = 0 (20.10) H H H and the ﬂuid continuity equations become

N ′ 2 N ′ (δm) = vm + 3Φ N ′ N ∇ (vm ) + vm = Φ HC ′ 2 N 9 2 N (δ ) = v (1 + wh)(1 Ω)v , (20.11) m ∇ m − 2 H − m 45There are possibilities for making the approximation of ignoring h perturbations in diﬀerent ways, leading to ′ equations that diﬀer from each other by, e.g., 3(1 − Ω)(1 + wh)Ψ (contribution of the ignored source term in the h energy continuity equation). 20 EFFECT OF A HOMOGENEOUS COMPONENT 64 where we can probably ignore the last term on the rhs of Eq. (20.11c).

Exercise: Ω derivatives. Show that

′ Ω = 3 Ω(w wp) ′′ H2 15− 27 2 2 2 2 Ω = Ω (w wp)+ w(w wp) 9(c c ) 9(wc wpc ) , (20.12) H 2 − 2 − − s − p − s − p and that, if ρh is vacuum energy,

′ Ω = 3 Ω(1 Ω)(1 + wp) ′′ − H2 − 15 27 2 Ω = Ω(1 Ω)(1 + wp) + w 9c , (20.13) −H − 2 2 − s and that, if ρh is vacuum energy and ρp is matter,

′ Ω = 3 Ω(1 Ω) ′′ − H − Ω = 2Ω(1 Ω) 15 27 (1 Ω) . (20.14) −H − 2 − 2 −

20.2 Λ + Perturbed Component

When the homogeneous component is vacuum energy, ρh = const, 1+ wh = 0, the preceding equations become exact (and there is no ambiguity about how to do the approximation, as no approximation is made). Now

w = Ωw (1 Ω) = Ω(1+ w ) 1 1+ w = Ω(1+ w ) p − − p − ⇒ p δ = Ωδp

δp = δpp 1+ wp v = Ωvp = vp Ω(1 + wp) Ωw Π = p Π . (20.15) Ωw (1 Ω) p p − − The speed of sound is undeﬁned for vacuum energy but

2 p′ pp′ 2 cs = = cp . (20.16) ≡ ρ′ ρp′

Since now v = vp, the ‘special’ comoving gauge we introduced in Sec. 20.1 becomes the usual one,

δC δN + 3 (1 + w )vN p ≡ p H p δpC δpN + 3 (1 + w )c2ρ vN . (20.17) p ≡ p H p s p The Einstein and ﬂuid equations in the Newtonian gauge are the same as in Sec. 20.1, but N N with vp = v , and now they are exact:

2Ψ = 3 2Ω δN + 3 (1 + w )vN ∇ 2 H p H p 2 Ψ Φ = 3 ΩwpΠp − 3H 2 N Ψ′ + Φ = Ω(1 + w )v H 2 H p N 2 1 2 3 2 δp Ψ′′ + (Φ′ + 2Ψ′) + (2 ′ + )Φ + 3 (Φ Ψ) = 2 Ω (20.18) H H H ∇ − H ρp 20 EFFECT OF A HOMOGENEOUS COMPONENT 65 and

N N 2 N N δpp (δp )′ = (1+ wp) v + 3Ψ′ + 3 wpδp ∇ H − ρp ! N N N wp′ N δpp 2 wp 2 (v )′ = (1 3wp)v v + + Πp +Φ . (20.19) −H − − 1+ wp ρp + pp 3 1+ wp ∇

In the mixed gauge (using 2 + 2 = 3w 2) H′ H − H 2Ψ = 3 2ΩδC ∇ 2 H p C 2 2 2 1 2 3 2 δpp Ψ′′ + (2 + 3cs) Ψ′ + Φ′ + 3(cs w) Φ+ 3 (Φ Ψ) = 2 Ω (20.20) H H − H ∇ − H ρp and

C C 2 N 2 (δ )′ 3 w δ = (1+ w ) v + 2 w Π p − H p p p ∇ p H p∇ p C N N δpp 2 wp 2 (v )′ + v = + Πp +Φ . (20.21) H ρp + pp 3 1+ wp ∇

20.3 Λ + Perfect Fluid (This section generalizes Sec. 16.5 to the presence of Λ.) For a perfect ﬂuid, Πp = 0 and Ψ = Φ. Equations (20.20) and (20.18c) become

2Φ = 3 2ΩδC (20.22) ∇ 2 H p 3 2 N Φ′ + Φ = (1 + w )Ωv H 2 H p C 2 2 2 3 2 δpp 3 2 2 C Φ′′ + 3(1 + cs) Φ′ + 3(cs w) Φ = 2 Ω = 2 Ωcs δp 3(1 + wp) , H − H H ρp H − S where δp δρ δp δρ 1 δρ 1 δp = = p p = p p (20.23) S H p − ρ H p − ρ 3(1 + w ) ρ − c2 ρ ′ ′ p′ p′ p p s p (in any gauge), and (20.21) becomes

C C 2 N (δ )′ 3 w δ = (1+ w ) v p − H p p p ∇ p C N N δpp (v )′ + v = +Φ . (20.24) H ρp + pp

In Sec. 16.5 we derived the Bardeen equation (16.42) by taking the Laplacian of the second Einstein evolution equation in the mixed gauge, here (20.22c). Here we can also proceed by C C inserting δ =Ωδp in (16.42) to get (exercise)

2 C 2 1 C 3 2 2 C − δ ′′ + 1 6w + 3c − δ ′ Ω + (10 2Ω)w 6c 3Ωw δ H p − p s H p − 2 − p − s − p p 2 2 2 C = c − δ 3(1 + w ) , (20.25) sH ∇ p − p S the Λ version of the Bardeen equation. 20 EFFECT OF A HOMOGENEOUS COMPONENT 66

20.4 Λ + Matter

2 2 Finally, assume the perturbed component is just matter, δpp = wp = cp = 0, which makes cs = 0 and w =Ω 1. The Einstein equations become − 2Φ = 3 2Ω δN + 3 vN = 3 2ΩδC ∇ 2 H m H 2 H m 3 2 N Φ′ + Φ = 2 Ωv H2 H Φ′′ + 3 Φ′ + 3(1 Ω) Φ = 0 , (20.26) H − H and the ﬂuid equations become

N 2 δ ′ = v + 3Φ′ m ∇ N δC ′ = 2v m ∇ N v ′ + v = Φ . (20.27) N H N The Bardeen equation (20.25) becomes

2 C 1 C 3 C − δ ′′ + − δ ′ = Ωδ , (20.28) H m H p 2 m which can also be easily derived (exercise) from the ﬂuid equations (20.27) with the help of the constraint equation (20.26a).46 Using t, a, or ln a as time variable instead of η, (20.28) can be written

2 ¨C 1 ˙C H− δm + 2H− δm d2δC 1 dH 3 dδC = m + + m da2 H da a da C C 3 C = (δm)∗∗ + [2 + (ln H)∗] (δm)∗ = 2 Ωδm , (20.29) where I deﬁned the notation d/d ln a.47 This is the same equation (“Jeans equation”) ∗ ≡ we derived in Cosmology II for ΛCDM from Newtonian perturbation theory (the ﬁrst and second forms), except that the Newtonian density perturbation δm has been replaced with the C comoving density perturbation δm and the equation is now valid for all distance scales (whereas the Newtonian result was only for subhorizon scales). The third form is Eq. (I.3.20) in [10].

46Whereas the alternative way, which starts by taking the Laplacian of the evolution equation (20.26c), is more cumbersome. This, however, was the way we derived the general form of the Bardeen equation (16.42). 47This is my own notation. In some modern literature, ′ is used for this, but I am using ′ ≡ d/dη (with some exceptions). 21 THE REAL UNIVERSE 67

21 The Real Universe

Note: From here on, the density and velocity perturbations will be in the Newtonian gauge unless otherwise speciﬁed. To avoid a clutter of indices we drop the sub- or supercript N. We also drop the bars from the background quantities as is standard practice after the perturbation equation have been derived. 48 According to present understanding, the universe contains 5 major “ﬂuid” components: “baryons” (including electrons), cold dark matter, photons, neutrinos, and the mysterious dark energy, ρ = ρb + ρc + ργ + ρν + ρDE . (21.1) We shall here assume there are no perturbations in the dark energy. We make the approximation that the pressure of baryons and cold dark matter can be 49 ignored. Thus pb = pc = 0 (both for background and perturbations). For photons, pγ = ργ/3. We assume massless neutrinos, so the same relation holds for them. Thus we have

2 2 wb = wc = cb = cc = 0 2 2 1 wγ = wν = cγ = cν = 3 . (21.2) Moreover,

δpb = δpc = 0 1 δpγ = 3 δργ 1 δpν = 3 δρν . (21.3) Thus we have the happy situation, that for each component we have a unique relation pi = pi(ρi), which moreover is very simple, either pi =0 or pi = ρi/3. (Also, the simplest kind of dark energy, vacuum energy, has p = ρ ). DE − DE The components are, however, not all independent. Cold dark matter does not interact with the other components. We can ignore the interactions of neutrinos, since we are now only interested in times much after neutrino decoupling. But the baryons and photons interact via Thomson scattering. Our continuity equations for perturbations are thus (for scalar perturbations in the conformal- Newtonian gauge)

2 δ′ = v + 3Ψ′ (21.4) c ∇ c vc′ = vc +Φ −H2 δ′ = v + 3Ψ′ + (collision term) b ∇ b vb′ = vb + Φ + collision term −H4 2 δ′ = v + 4Ψ′ + (collision term) γ 3 ∇ γ 1 1 2 v′ = δ + Π + Φ + collision term γ 4 γ 6 ∇ γ 4 2 δν′ = 3 vν + 4Ψ′ 1 ∇ 1 2 v′ = δ + Π +Φ . ν 4 ν 6 ∇ ν

We have put the collision terms for δb′ and δγ′ in parenthesis, and we drop them from here on, since it will turn out that they can be neglected, and only momentum transfer between photons and baryons is important.

48In fact, it would not matter if they were confused with the full (background + perturbation) quantities, since after multiplying with a perturbation, the diﬀerence is of second order. 49For small distance scales the baryon pressure is important after photon decoupling. If we were interested in small-scale structure formation, we should include it. For the cosmic microwave background it is not needed. 21 THE REAL UNIVERSE 68

In Fourier space these equations read

δ′ = kv + 3Ψ′ (21.5) c − c v′ = v + kΦ c −H c δ′ = kv + 3Ψ′ b − b vb′ = vb + kΦ + collision term −H4 δ′ = kv + 4Ψ′ γ − 3 γ 1 1 v′ = kδ kΠ + kΦ + collision term γ 4 γ − 6 γ 4 δν′ = 3 kvν + 4Ψ′ −1 1 v′ = kδ kΠ + kΦ . ν 4 ν − 6 ν (Remember our Fourier convention for v and Π.) These equations are supplemented by 2 Einstein equations (there are 4 Einstein equations for perturbations, but since we are also using continuity equations, only two of them remain independent). Thus we have 10 perturbation equations, but there are 12 perturbation quantities to solve. (If we think of the Einstein equations as the equations for Φ and Ψ, the “extra” quantities lacking an equation of their own are the anisotropic stresses Πγ and Πν. In the perfect fluid approximation these vanish, and the number of quantities equals the number of equations.) Also, we do not yet have the collision terms. Thus more work is needed. This will lead us to the Boltzmann equations which employ a more detailed description of the fluid components, in terms of distribution functions.50 This is done in CMB Physics, from which we will pick a couple of results in the following. In synchronous gauge one can simplify the cold dark matter equations, since cold dark matter falls freely (in our approximation). Thus we can use cold dark matter particles as the freely falling observers that define the synchronous space coordinate, so that

Z vc = 0 . (21.6)

In synchronous gauge Eqs. (21.5) become thus

1 δ′ = h′ (21.7) c − 2 vc = 0 1 δ′ = kv h′ b − b − 2 vb′ = vb + collision term −H4 2 δ′ = kv h′ γ − 3 γ − 3 1 1 v′ = kδ kΠ + collision term γ 4 γ − 6 γ 4 2 δν′ = 3 kvν 3 h′ −1 −1 v′ = kδ kΠ . ν 4 ν − 6 ν 50 The distribution functions fi(η, ~x, ~p) give the distribution of particles (of ﬂuid component i) in the six- dimensional phase space {(~x, ~p)}, i.e., position ~x and momentum ~p. In perturbation theory the distributions are assumed to be close to thermal equilibrium, so that they can be given in terms of a temperature perturbation Θi(η, ~x, nˆ), which depends on the momentum direction nˆ. That the temperature perturbation may be considered independent of |~p| is an important result derived in CMB Physics. In Fourier space we have the temperature perturbation Θi(η,~k, nˆ), and the direction dependence can be expanded in spherical harmonics to be represented m ~ ~ by the coeﬃcients Θℓ (η, k). When the spherical harmonic expansion is done with respect to the wave vector k 0 ~ direction, it turns out that for scalar perturbations only the Θℓ (η, k) are nonzero (vector perturbations excite ±1 ±2 Θℓ and tensor perturbations Θℓ ), and we write them as Θℓ (with some normalization factors that we don’t spell out here—leaving such detail to CMB Physics.) The lowest multipoles or moments correspond to familiar i 1 i 1 i 1 quantities: Θ0 = 4 δi, Θ1 = 3 vi, and Θ2 = 12 Πi, where i = γ or ν. 22 PRIMORDIAL ERA 69

22 Primordial Era

The initial conditions for the evolution of the large scale structure and the cosmic microwave background can be specified during the radiation-dominated epoch, sufficiently early that all scales k of interest are outside the horizon. We do not want to deal with the electron-positron annihilation or big-bang nucleosynthesis (BBN),51 so we limit this time period to start after BBN. We assume it is sufficiently far after whatever event created the perturbations in the first place, so that we can assume that all decaying modes have already died out. The comoving horizon at the BBN epoch is 1 1 T − − 1 kpc (22.1) H ≈ 100 keV · so all cosmological scales are still well outside horizon then. We are not just specifying “initial values” at some particular instant of time. Rather, we solve the perturbation equations for this particular epoch, and find that the solutions are characterized by quantities that remain constant for the whole epoch. Other perturbation quantities are related to these constant quantities by some powers of kη. These perturbations during this epoch we call the “primordial perturbations”. There are different modes of primordial perturbations. In adiabatic perturbations all fluid perturbations are determined by the metric perturbations. However, the metric perturbations depend only on the total fluid perturbations δ, δp, v, and Π. Thus there are additional degrees of freedom in the component fluids: entropy perturbations. For adiabatic perturbations, all component velocity perturbations are equal, vi = v, and the density perturbations are related δ δ i = . (22.2) 1+ wi 1+ w The (relative) entropy perturbations are defined

δρi δρj δi δj Sij 3 = (22.3) ≡ − H ρi′ − ρj′ ! 1+ wi − 1+ wj (we assume we can ignore energy transfer between fluid components). For N fluid components, there are N 1 independent entropy perturbations. Often photons − are taken as the reference fluid component for entropy perturbations, so that the independent entropy perturbations are taken to be52

δi 3 Si δγ , i = γ . (22.4) ≡ 1+ wi − 4 6 For this section, we shall work mainly in the Newtonian gauge. Unless otherwise specified, δ δN and v = vN (!!!). The relevant equations are the Einstein equations (10.19-10.22): ≡ 2 1 1 k 1 − Ψ′ +Φ+ Ψ = δ (22.5) H 3 − 2 H 1 3 − Ψ′ +Φ = (1 + w)H v (22.6) H 2 k 2 2 1 2 ′ 1 k 3 δp − Ψ′′ + − Φ′ + 2Ψ′ + 1+ H Φ (Φ Ψ) = (22.7) H H 2 − 3 − 2 ρ H H k 2 (Ψ Φ) = 3wΠ , (22.8) − H 51It is really the electron-positron annihilation we want to avoid. In BBN the energy transfer from baryons to photons is small from the baryon point of view and negligible from the photon point of view. 52Another possibility is to use the total radiation as the reference fluid, and we well actually do so later. 22 PRIMORDIAL ERA 70 the fluid equations (21.5), and we also want to refer to the comoving curvature perturbation

2 2 1 = Ψ Φ − Ψ′ R − − 3(1 + w) − 3(1 + w)H 2 1 5 + 3w 2 − Ψ′ + Ψ = (1 + w) + (Ψ Φ) (22.9) ⇒ 3H 3 − R 3 − (from Eq. 16.27). The early radiation-dominated era (the primordial era) has 4 properties which simplify the solution of the perturbation equations:

1. All scales of interest are outside the horizon, k . This allows us to drop some (but not ≪H necessarily all) of the gradient terms (those with k) from the perturbation equations.53

2. Radiation domination, ρ , ρ ρ , ρ , ρ γ ν ≫ b c DE 1 w = c2 = 1 the background solution is = , (22.10) ⇒ s 3 ⇒ H η and we can ignore the baryon, CDM, and DE contributions to the total fluid perturbation. We can also ignore the collision term in the photon velocity equation (but not in the baryon velocity equation) since the momentum the baryonic fluid can transfer to the photon fluid is negligible compared to the inertia density of the photon fluid.

3. This era is before recombination and photon decoupling, so baryons and photons are tightly coupled v = v (22.11) ⇒ b γ and the continuous interaction with baryons (really the electrons) keeps the photon distribution isotropic Π = 0 . (22.12) ⇒ γ 4. m T We can approximate neutrinos to be massless. This helps in solving ν ≪ ⇒ the evolution of the neutrino momentum distribution. We do not discuss this here; this belongs to the course of CMB Physics, and we need to take one result, Eq. (22.34), from there.

Thus we have

δ = δ = (1 f )δ + f δ (22.13) r − ν γ ν ν Π = fνΠν . (22.14) where54 ρν fν = const = 0.4089 . (22.16) ≡ ργ + ρν

53For example, if an equation has terms α +(k/H)2α, we can drop the second term, since it is ≪ the first. But if we have α +(k/H)2β, we cannot drop the second term unless we know that β is not ≫ α. If an evolution equation for α contains H−1α′ +(k/H)2α, we can drop the second term, since its contribution to the change of α over a cosmological timescale H−1 is negligible. 54From Cosmology I, 4/3 ρν 7 4 = Neff = 0.6918 , where Neff = 3.046 . (22.15) ργ 8 11 22 PRIMORDIAL ERA 71

The relevant equations thus become

1 4 k 1 − δ′ = v + 4 − Ψ′ (22.17) H γ −3 γ H H 1 1 k k − v′ = δ + Φ (22.18) H γ 4 γ H H 1 4 k 1 − δ′ = v + 4 − Ψ′ (22.19) H ν −3 ν H H 1 1 k 1 k k − v′ = δ Π + Φ . (22.20) H ν 4 ν − 6 ν H H H and

1 1 − Ψ′ +Φ = δ (22.21) H − 2 1 − Ψ′ +Φ = 2Hv (22.22) H k 2 1 1 − Ψ′′ + − Φ′ + 2Ψ′ Φ = δ (22.23) H H − 2 k 2 (Ψ Φ) = f Π (22.24) − ν ν H 2 1 4 2 − Ψ′ +2Ψ = + (Ψ Φ) , (22.25) 3H −3R 3 − 22.1 Neutrino Adiabaticity We consider now the simpler case, when there are no neutrino entropy perturbations,

S = 0 δ = δ = δ and v = v = v . (22.26) ν ⇒ ν γ ν γ We allow for the presence of baryon and CDM entropy perturbations. However, during the radiation-dominated epoch their eﬀect on metric perturbations is negligible. Thus the evolution of metric perturbations are as if the perturbations were adiabatic. In the simpler case discussed in Sec. 19, where we had Φ = Ψ, we found that the growing adiabatic mode had Φ = Ψ = const Guided by that, we now try the ansatz

Φ= const and Ψ= const (22.27) and check that it is a solution. The Einstein and equations are satisﬁed with R δ = δ = δ = 2Φ = const (22.28) ν γ − 1 k k v = v = v = Φ = 1 δ = 1 kηΦ (22.29) ν γ 2 − 4 2 H H 1 k 2 Π = (Ψ Φ) (22.30) ν f − ν H = (Ψ + 1 Φ) = const (22.31) R − 2 In the δ and δ equations the (k/ ) terms become (1/3)(k/ )2δ δ and can be ignored, and γ′ ν′ H H ≪ we see that the equations are satisﬁed (the implied change in δi over a Hubble time is negligible). Using = 1/η the velocity equations become H 1 ηv′ = kηδ + kηΦ γ 4 γ 1 1 ηv′ = kηδ kηΠ + kηΦ . ν 4 ν − 6 ν 22 PRIMORDIAL ERA 72

Here the Π term can be ignored since Π is suppressed by (k/ )2 compared to Ψ and Φ (we ν ν H assume that Ψ is not Φ), and we then see that these equations are also satisfied. ≫ We are still missing a piece of information that would tell us what Φ Ψ, or, in other words, − what Πν is. The neutrino anisotropy Πν depends on the neutrino momentum distribution. Before neutrino decoupling, interactions kept Πν = 0. After neutrinos decoupled, neutrinos have been “freely streaming”, i.e., moving without collisions through the perturbed universe. In CMB Physics we derive a so-called Boltzmann hierarchy of equations ℓ ℓ + 1 Θℓ′ = kΘℓ 1 lΘℓ+1 , (22.32) 2ℓ + 1 − − 2ℓ + 1 which relates the evolution of the different moments of the momentum distribution of freely ν 1 ν 1 ν 1 streaming particles to each other. With Θ0 = 4 δν ,Θ1 = 3 vν, and Θ2 = 12 Πν, these give 4 δν′ = 3 kvν −1 1 vν′ = 4 kδν 6 kΠν 8 − 36 ν Πν′ = 5 kvν 5 kΘ3 ν 1 − 4 ν (Θ )′ = kΠ kΘ . (22.33) 3 28 ν − 7 4 The first two equations are familiar, except one still has to add the effect of gravity (the metric perturbations Ψ′, Φ). Metric perturbations do not affect the higher equations in the hierarchy. The moments Θ are gauge invariant for ℓ 2. ℓ ≥ Before decoupling all the higher moments, Θ for ℓ 2, vanish. This leads to a “decreasing ℓ ≥ hierarchy”, where the lower moments seed the higher moments, so that a higher moment Θℓ 55 is one order higher (smaller) in time than the previous moment Θℓ 1. Thus for early times (superhorizon scales) we can truncate the hierarchy by ignoring moments− higher than some ℓ depending on how accurate we want to be (to how high order in η we want to have the results.) Truncating at ℓ = 2, the “second moment” Πν depends then only on the “first moment” vν, and the relevant equation is 1 8 k − Π′ = v . (22.34) H ν 5 ν H This finally allows us to solve (exercise):

Ψ = (1+ 2 f )Φ 1.164Φ (22.35) 5 ν ≈ 2 Π = (kη)2Φ (22.36) ν 5 3 4 = 1+ f Φ = const (22.37) R −2 15 ν 2 1 Φ = 4 0.6011 (22.38) −3 1+ 15 fν R ≈ − R 2 2 1+ 5 fν Ψ = 4 0.6994 . (22.39) −3 1+ 15 fν R ≈ − R (We see that the perfect ﬂuid approximation, which gave Ψ = Φ = 2 , led to a 10% error − 3 R in Φ and to a 5% error in Ψ.)

22.2 Matter During the early radiation-dominated era, the metric and the radiation perturbations do not care about matter perturbations, but matter perturbations will become important later, and

55CMB Physics 2004, p. I1.8 22 PRIMORDIAL ERA 73 therefore we are interested in their “primordial” behavior in the radiation-dominated era. The continuity equations for baryons and CDM are

1 k 1 − δ′ + v 3 − Ψ′ = 0 (22.40) H c c − H H 1 k 1 − δ′ + v 3 − Ψ′ = 0 H b b − H H 1 k − v′ + v Φ = 0 H c c − H 1 k 4ργ − v′ + v Φ = an σ (v v ) , H b b − e T 3ρ γ − b H b where the collision term in the last equation is derived in CMB Physics, σT is the Thomson cross section for photon-electron scattering, and ne is the free electron number density. Well before photon decoupling, aneσT (4ργ )/(3ρb) is very large, and the collision term forces vb = vγ (baryons are tightly coupled to photons). For the above photon+neutrino adiabatic growing mode solution, these become

ηδc′ + kηvc = 0 (22.41)

ηδb′ + kηvb = 0 ηv′ + v kηΦ = 0 c c − 4ργ ηvb′ + vb kηΦ = aneσT (v vb) . − 3ρb −

22.2.1 The Completely Adiabatic Solution One solution for Eq. (22.41) is the completely adiabatic solution:

3 3 δc = δb = 4 δ = 2 Φ = const. (22.42) 1− vc = vb = v = 2 kηΦ . To check this, substitute Eq. (22.42) into Eq. (22.41). This gives

1 2 0+ 2 (kη) Φ = 0 for the δ equations, and 1 kηΦ+ 1 kηΦ kηΦ = 0 , 2 2 − for the v equations, so the latter equations are indeed satisﬁed. The δ equations are satisﬁed to accuracy (kη)2 1, i.e., in a Hubble time, δ will deviate from its initial value 3 Φ by about ≪ i − 2 1 (kη)2Φ, a negligible change. − 2 22.2.2 Baryon and CDM Entropy Perturbations There are three independent entropy perturbations: the neutrino, baryon, and CDM entropy perturbations, S 3 (δ δ ) S δ 3 δ S δ 3 δ . (22.43) ν ≡ 4 ν − γ b ≡ b − 4 γ c ≡ c − 4 γ Their evolution equations are

S′ = k(v v ) S′ = k(v v ) S′ = k(v v ) . (22.44) ν − ν − γ b − b − γ c − c − γ The relative entropy perturbation stays constant unless there is a corresponding relative velocity perturbation. 22 PRIMORDIAL ERA 74

Entropy perturbations also tend to stay constant at superhorizon scales56 , as

1 k − S′ = (v v ) . (22.45) H i − i − γ H Assume now the neutrino-adiabatic growing mode solution of Sec. 22.1. This assumes Sν = 0, but we may still have baryon and CDM entropy perturbations. The baryon and neutrino density perturbations are

3 3 δb = 4 δ + Sb and δc = 4 δ + Sc . (22.46)

Since baryons are tightly coupled to photons, vb = vγ = v, we have

S′ = 0 S = const. δ = const. (22.47) b ⇒ b ⇒ b For CDM we do not have this constraint. Write

v v v v = 1 kηΦ+ v . (22.48) rel ≡ c − ⇒ c 2 rel Eq. (22.41c) becomes

1 1 η 2 kΦ+ ηvrel′ + 2 kηΦ+ vrel kηΦ = 0 − 1 ηv′ = v v η− . ⇒ rel − rel ⇒ rel ∝ Thus we have 1 1 vc = 2 (kη)Φ + Cη− . (22.49) from which we identify a growing mode and a decaying mode. As time goes on, the decaying mode decays away, and v v. Ignoring the decaying mode, we have c → v = v S = const. δ = const. (22.50) c ⇒ c ⇒ c Thus (assuming neutrino adiabaticity), the “initial conditions” at the early radiation-dominated ~ epoch can be speciﬁed by giving three constants for each Fourier mode k:Φk(rad), Sc~k(rad), and Sb~k(rad). The general perturbation is a superposition of three modes, where two of these constants are zero:

(Φ,Sc,Sb) = (Φ, 0, 0) adiabatic mode (ADI) (22.51)

(Φ,Sc,Sb) = (0,Sc, 0) CDM density isocurvature mode (CDI)

(Φ,Sc,Sb) = (0, 0,Sb) baryon density isocurvature mode (BDI) .

In the following we shall use instead of Φ (see Eq. 22.37) as the ﬁrst initial value constant R (since it is better in staying constant also later).

22.3 Neutrino perturbations Bucher, Moodley, and Turok [6] identified 5 different possible modes of primordial perturbations: Adiabatic growing mode (ADI), Baryon density isocurvature mode (BDI), CDM density isocurvature mode (CDI), Neutrino density isocurvature mode (NDI), and Neutrino velocity isocurvature mode (NVI). The three first ones we discussed above; now we shall discuss the NDI mode. The NVI mode, where there is an initial neutrino velocity perturbation, is difficult to motivate, so we do not discuss that.

56This property is not as general as the constancy of R at superhorizon scales: The result (22.45) relies on two assumptions: 1) no interaction terms in the component energy continuity equations, and 2) the component ﬂuids have a unique relation pi = pi(ρi). Note also that this does not hold for the “total entropy perturbation” S. 22 PRIMORDIAL ERA 75

22.3.1 Neutrinos in the early universe Before neutrino decoupling (T 1 MeV), neutrinos were interacting with electrons, positrons, ≥ and nucleons, which were interacting with photons. These interactions kept these particles in thermal and kinetic equilibrium, eliminating any neutrino anisotropic pressure and forcing vν = vγ . They did not necessarily force δν = δγ . However, they forced Tν = Tγ. The other degrees of freedom for neutrino density in thermal equilibrium are given by the neutrino chemical potentials µνe , µνµ , and µντ . Thus neutrino entropy perturbations require nonzero neutrino chemical potentials. The usual assumption is that these chemical potentials are small, just like for electrons and baryons, µ /T = (10 9), so that any neutrino entropy perturbations would be negligible, but ν ν O − we do not know this, observational upper limits are much weaker. Thus we could have a primordial neutrino density isocurvature mode. To have a primordial neutrino velocity isocurvature mode, would require that it was generated after neutrino decoupling.

22.3.2 Primordial neutrino density isocurvature mode We now find the primordial neutrino density isocurvature mode during the early radiation dominated era. The basic equations are Eqs. (22.13-22.25) and (22.34) with = 1/η. H We defined earlier S S 3 (δ δ ), which is common in literature. Lyth and Liddle ν ≡ νγ ≡ 4 ν − γ [7] argue that it would be better to use the total radiation perturbation δr here instead of δγ . This makes no difference when we have neutrino adiabaticity, since then δγ = δν = δr. But in case of the neutrino isocurvature mode these differ. For clarity, I will use below the notations S and S 3 (δ δ ) instead of S . From (22.13), they are related by νγ νr ≡ 4 ν − r ν S = (1 f )S . (22.52) νr − ν νγ Lyth and Liddle thus want to define the neutrino isocurvature density mode by the initial condition = S = S = 0 instead of = S = S = 0, arguing that this is the form in R br cr R bγ cγ which this mode is more likely to be generated. This distinction will matter only when we get to discuss the matter perturbations in this mode; until that the discussion below applies to both cases. We search for a solution for the primordial perturbation with = 0 but S = 0. Thus we R νr 6 expect nonzero δν and δγ. The difference from the previous isocurvature modes is that now the entropy perturbation is between the dominating fluid quantities. This means that it is likely to affect the metric. Although we require = 0, Φ and Ψ may be nonzero. From (22.18) R and (22.20) we expect nonzero v and v , suppressed by kη 1, and from (22.34) a nonzero γ ν ≪ Π , suppressed by (kη)2, which by (22.24) implies a nonzero Ψ Φ. Note that because of the ν − (kη)2 in Eq. (22.34) there is no superhorizon suppression in Ψ Φ, although there is in Π , and − ν therefore we cannot ignore this effect of Πν . Thus at least one of Φ and Ψ must be nonzero, and there is indeed a metric perturbation in this isocurvature mode. Guided by the previous cases, we look for a solution with δγ, δν , Φ, and Ψ constant, now with = 0. The terms with derivatives in Eqs. (22.21-22.25) vanish, and we get immediately R 2Ψ = 2 (Ψ Φ) Ψ = 1 Φ (22.53) 3 − ⇒ − 2 Φ = 1 δ δ = 2Φ (22.54) − 2 ⇒ − v = 1 kηΦ = 1 kηδ (22.55) 2 − 4 f Π = (kη)2(Ψ Φ) = 3 (kη)2Φ = 3 (kη)2δ . (22.56) ν ν − − 2 4 In (22.20), 1 1 ηv′ = kηδ kηΠ + kηΦ , (22.57) ν 4 ν − 6 ν 22 PRIMORDIAL ERA 76

2 the Πν term is suppressed by (kη) compared to the Φ term, so we drop it, and solve, with initial condition vν = 0, that v = 1 kηδ + kηΦ = 1 kηδ 1 kηδ . (22.58) ν 4 ν 4 ν − 2 Likewise we get v = 1 kηδ + kηΦ = 1 kηδ 1 kηδ. (22.59) γ 4 γ 4 γ − 2 2 (Eqs. (22.17) and (22.19) would give time derivatives to δγ and δν, that are suppressed by (kη) so that we can ignore them. Same applies to S , so S = const, although v = v .) νγ νγ ν 6 γ Now Eq. (22.34)

8 2 2 4 2 ηΠ′ = kηv becomes ηΠ′ = (kη) δ (kη) δ (22.60) ν 5 ν ν 5 ν − 5 whose solution (with initial condition Πν = 0) is Π = (kη)2 1 δ 2 δ . (22.61) ν 5 ν − 5 Combining the two Eqs., (22.56) and (22.61), relating the neutrino anisotropic pressure Πν to density perturbations, one coming from Einstein equations (gravity) and the other from the Boltzmann hierarchy (kinematics), we have

3 δ = f ( 1 δ 2 δ) , (22.62) 4 ν 5 ν − 5 from which we solve

15 + 8fν δν = δ (22.63) 4fν 3 45 + 12fν Sνr 4 (δν δ) = δ (22.64) ≡ − 16fν 16fν 4 4fν δ = Sνr = Sνr 0.1311Sνr (22.65) 45 + 12fν 3 15 + 4fν ≈ 60 + 32fν δν = Sνr 1.464Sνr (22.66) 45 + 12fν ≈ 4 S 60 + 32f 4 δ = δ νr = ν S γ ν − 3 1 f 45 + 12f − 3 3f νr − ν ν − ν 16 f 4f = ν ν S 0.7913S (22.67) 3 4f + 15 − 3(1 f ) νr ≈ − νr ν − ν 1 8fν Φ = 2 δ = Sνr (22.68) − −45 + 12fν 1 4fν 4 fν Ψ = 4 δ = Sνr = Sνr (22.69) 45 + 12fν 3 15 + 4fν 5 vν = kηSνr (22.70) 15 + 4fν 19f v = v 1 kηS = ν kηS (22.71) γ ν − 3 νγ −3(1 f )(15 + 4f ) νr − ν ν 4 2 Πν = (kη) Sνr . (22.72) 15 + 4fν

Thus ρν and ργ have perturbations that are in the opposite direction to each other, but do not 57 cancel, leaving a total δ in the same direction as δν . 57Eq. (22.67) disagrees with Eq. (12.26) of Lyth&Liddle[7]. Eq. (22.69) agrees with Eq. (12.22) of [7], except for the sign. In [7] the sign of δ appears to be the same as δγ , opposite to δν and Sνr. Note that in [7] Ψ and Φ have interchanged meaning but have the same sign convention (see their Eq. (8.32) for the metric). 22 PRIMORDIAL ERA 77

In the early radiation-dominated era, the CDM and baryon perturbations have no effect 1 on the above solution, and since it shares the results Ψ′ = Φ′ = 0 and v = 2 kηΦ with the neutrino adiabatic case, we can repeat the arguments in Sec. 22.2.2 so that the CDM and baryon perturbations follow the same equations even if the neutrino adiabaticity condition is relaxed by allowing the presence of the NDI mode, except that now v = v = v and = v, b γ 6 ν 6 because of the tight coupling between baryons and photons (but this difference will not affect the entropy perturbations, since the effect of velocity differences is suppressed by k/ ). H In the NDI mode, the primordial Sbr and Scr are zero, so that

3 3 4fν δb = δc = 4 δr = 4 δ = Sνr (22.73) 15 + 4fν during the primordial epoch.

22.4 Tight-Coupling Approximation From (21.5), the baryon and photon velocity equations are 1 vb′ = vb + kΦ+ (vγ vb) (22.74) −H Rηcoll − 1 1 1 vγ′ = 4 kδγ 6 kΠγ + kΦ+ (vb vγ) , (22.75) − ηcoll − where 1 3ρb ηcoll and R . (22.76) ≡ aneσT ≡ 4ργ 2 Here ne is the number density of free electrons and σT = 66.5fm is the Thomson cross section. After electron-position annihilation but before recombination, n a 3. Before photon e ∝ − decoupling, η 1, so that (22.74) and (22.75) force v v . coll ≪H− b ≈ γ Earlier we ignored both the collision term and the photon anisotropy Πγ in (22.75). One can go one level better by going to ﬁrst order in ηcoll. From (22.74),

2 v = v Rη (v′ + v kΦ) = v Rη (v′ + v kΦ) + (Rη ) . (22.77) b γ − coll b H b − γ − coll γ H γ − O coll Inserting (22.77), ignoring its (Rη )2 part, to (22.75), it becomes, ignoring also the Π , O coll γ 1 (1 + R)v′ kδ Rv +(1+ R)kΦ . (22.78) γ ≈ 4 γ −H γ

In the tight-coupling approximation, we solve vγ from (22.78), but keep still

vb = vγ and Πγ = 0 (22.79)

(missing piece: should show that (22.79) holds to (η ); now it is just based on the statement O coll in [9]).

22.5 Real-Universe Initial Conditions as Expansion in Conformal Time

To obtain suﬃciently accurate initial conditions for a numerical solution of the full linear evolution of perturbations in the real universe, we want to do a similar thing as we did for the simpliﬁed universe in Sec. 19.5. The background equations are

2 1 1+ x 1 y =2x + x , = 1 , w = (22.80) H x 1+ 2 x 3(1 + y) 22 PRIMORDIAL ERA 78

Now we use the Newtonian gauge equations. We expand

Φ = A + Bx + Cx2 Ψ = D + Ex + F x2 N 2 δi = Gi + Hix + Jix N 2 3 vi = Lix + Mix + Nix 2 3 4 Πν = Qx + Sx + T x , (22.81)

N 2 3 except that for baryons we set vb = vγ , so that vb = Lγx + Mγx + Nγx . From (22.38) and (22.68),

2 1 4 A = 4 ~k(rad) + 15 fν Sνr,~k(rad) (22.82) −3 1+ fν R 15 h i (here “rad” refers to the constant primordial value during the early radiation-dominated epoch). From ′ 8 N 36 ν ν ′ 1 4 ν 3 4 (22.33cd), Π = kv kΘ and (Θ ) = kΠν + (x ), and calling Θ = Ux + (x ), one gets ν 5 ν − 5 3 3 28 O 3 O 1 4 8 2 9 U = kη Q,Q = kη Lν , S = kη Mν ,T = kη Nν kη U. (22.83) 84 3 15 3 15 3 5 3 − 5 3 The total densities and velocities are given by 1 y δ = (f δ + f δ )+ (f δ + f δ ) 1+ y γ γ ν ν 1+ y b b c c 4 y v = (f v + f v )+ (f v + f v ) . (22.84) 4+3y γ γ ν ν 1+ y b b c c

The equations to solve to (x) are the ﬂuid density equations O

Hb +2Jbx = kη Lγx +3E +6F x − 3 Hc +2Jcx = kη3Lcx +3E +6F x − 4 Hγ +2Jγx = 3 kη3Lγx +4E +8F x − 4 Hν +2Jν x = kη Lν x +4E +8F x (22.85) − 3 3 and the equations to solve to (x2) are the ﬂuid velocity equations and the Einstein constraint equations O 2 N Lc +2Mcx +3Ncx = η3vc + kη3Φ 2 −H1 N N (1 + R)(Lγ +2Mγx +3Nγx = kη δ R η v +(1+ R)kη Φ 4 3 γ − H 3 γ 3 2 1 N 1 2 Lν c +2Mνx +3Nνx = kη δ kη Qx + kη Φ 4 3 ν − 6 3 3 2 2 k (Φ Ψ) = 3 wfν Πν − H x2k2Ψ = x2 3 2(δN +3(1+ w)( /k)vN . (22.86) − 2 H H

For comparison to [9], we want δi and vi also in the synchronous gauge,

Z Z Z Z 2 δi = Gi + Hi x + Ji x Z Z Z 2 Z 3 vi = Li x + Mi x + Ni x . (22.87)

for which we solve to (x2) the gauge transformation equations O 1 1 vZ = (vN vN ) x i x i − c Z N N Z δ = δ +3(1+ wi)H (v v ) (22.88) i i k −

Z (where we used vc = 0).

The above set seemed too complicated to solve by hand, so I wrote the following Mathematica script to solve them: 22 PRIMORDIAL ERA 79

Clear["Global‘,*"] y = 2x + x^2 hub = (1/x)((1+x)/(1+(1/2)x)) w = 1/(3(1+y)) rcr = 1 semr = 0 senr = 0 a = (-2/3)(1/(1+(4/15)fn))(rcr+(4/15)fn senr) gb = gm gc = gm fc = 1 - fb fg = 1 - fn rbg = (3/4)(fb/fg)y ph = a + b x + c x^2 ps = d + ee x + f x^2 db = gb + hb x + jb x^2 dc = gc + hc x + jc x^2 dg = gg + hg x + jg x^2 dn = gn + hn x + jn x^2 dbz = gbz + hbz x + jbz x^2 dcz = gcz + hcz x + jcz x^2 dgz = ggz + hgz x + jgz x^2 dnz = gnz + hnz x + jnz x^2 vb = lg x + mg x^2 + ng x^3 vc = lc x + mc x^2 + nc x^3 vg = lg x + mg x^2 + ng x^3 vn = ln x + mn x^2 + nn x^3 vgz = lgz x + mgz x^2 + ngz x^3 vnz = lnz x + mnz x^2 + nnz x^3 u = (1/84) k q q = (4/5)k ln s = (8/15)k mn t = (2/5)k nn - (9/5)k u pin = q x^2 + s x^3 + t x^4 delta = (1/(1+y))(fg dg + fn dn) + (y/(1+y))(fb db + fc dc) vee = (4/(4+3y))(fg vg + fn vn) + (3y/(4+3y))(fb vb + fc vc) veez = (4/(4+3y))(fg vgz + fn vnz) + (3y/(4+3y))fb vgz Solve[{ Series[hb + 2jb x, {x,0,1}] == Series[-k lg x + 3 ee + 6f x, {x,0,1}], Series[hc + 2jc x, {x,0,1}] == Series[-k lc x + 3 ee + 6f x, {x,0,1}], Series[hg + 2jg x, {x,0,1}] == Series[-(4/3)k lg x + 4 ee + 8f x, {x,0,1}], Series[hn + 2jn x, {x,0,1}] == Series[-(4/3)k ln x + 4 ee + 8f x, {x,0,1}], Series[lc + 2mc x + 3nc x^2, {x,0,2}] == Series[-hub vc + k ph, {x,0,2}], Series[(1+rbg)(lg + 2mg x + 3ng x^2), {x,0,2}] == Series[(1/4)k dg - rbg hub vg + (1+rbg)k ph, {x,0,2}], Series[ln + 2mn x + 3nn x^2, {x,0,2}] == Series[(1/4)k dn - (1/6)k q x^2 + k ph, {x,0,2}], semr == gm - (3/4)(fg gg + fn gn), senr == (3/4)(1-fn)(gn-gg), Series[k^2(ps-ph), {x,0,2}] == Series[3hub^2 w fn pin, {x,0,2}], Series[x^2 k^2 ps, {x,0,2}] == Series[-x^2(3/2)hub^2(delta+3(1+w)(hub/k)vee), {x,0,2}], Series[vgz/x, {x,0,2}] == Series[(vg - vc)/x, {x,0,2}], Series[vnz/x, {x,0,2}] == Series[(vn - vc)/x, {x,0,2}], Series[dbz, {x,0,2}] == Series[db + 3(hub/k)(vee-veez), {x,0,2}], Series[dcz, {x,0,2}] == Series[dc + 3(hub/k)(vee-veez), {x,0,2}], Series[dgz, {x,0,2}] == Series[dg + 4(hub/k)(vee-veez), {x,0,2}], Series[dnz, {x,0,2}] == Series[dn + 4(hub/k)(vee-veez), {x,0,2}] }, {b,c,d,ee,f,gm,gg,gn,hb,hc,hg,hn,jb,jc,jg,jn,lc,lg,ln,mc,mg,mn,nc,ng,nn,gbz,gcz,ggz,gnz, hbz,hcz,hgz,hnz,jbz,jcz,jgz,jnz,lgz,lnz,mgz,mnz,ngz,nnz}] Here k stands for kη and hub for η . The initial conditions are given by setting the values of rcr 3 H 3 ~ (rad), senr S ~ (rad), and semr S ~ (rad) fbS ~ (rad) + fcS ~ (rad). An initial relative ≡ Rk ≡ νr,k ≡ mr,k ≡ br,k cr,k entropy perturbation between baryons and CDM, S ~ (rad) S ~ (rad) = Gb Gc has no dynamical br,k − cr,k − 22 PRIMORDIAL ERA 80

eﬀect (it aﬀects only the values of Gb and Gc), so for simplicity we set it to zero, Gb = Gc.

Adiabatic mode. We get the adiabatic mode by setting ~k(rad) = 1, Smr,~k(rad) = Sνr,~k(rad) = 0. This gives the results R

2 1 25 8fν 3 2 Φ = 4 − ωη + (η ) −3 1+ 15 fν − 8 (15+2fν)(15 + 4fν ) O 2 2 1+ 5 fν 5 15+16fν 2 Ψ = 4 + ωη + (η ) −3 1+ 15 fν 8 (15+2fν )(15 + 4fν) O 16 N N 1 1 1+ 15 fν 2 δb = δc = 4 + 2 4 ωη + (η ) 1+ 15 fν 8 (1 + 15 fν )(1 + 15 fν) O 16 N N 4 1 1 1+ 15 fν 2 δγ = δν = 4 + 2 4 ωη + (η ) 3 1+ 15 fν 6 (1 + 15 fν )(1 + 15 fν ) O N 1 1 1 5 4fν 2 3 vc = 4 kη + − 4 ωkη + (η ) −3 1+ 15 fν 8 (15+2fν )(1 + 15 fν ) O vN = vN = vN + (η3) b γ c O vN = vN + (η3) ν c O δZ = δZ = 1 (kη)2 + (η3) b c 4 O δZ = δZ = 1 (kη)2 + (η3) γ ν 3 O Z vc = 0 vZ = vZ = 1 (kη)3 + (η4) b γ 36 O Z 1 23+4fν 3 4 vν = (kη) + (η ) 36 15+4fν O 4 4 1 2 1 1 5 fν 2 3 Πν = 4 (kη) + 4 − 2 ωk η . (22.89) −15 1+ 15 45 (1 + 15 fν )(1 + 15 fν ) Equality here means equality to the order included in the expansion in (22.81). Notice how the results 2 N 3 N are much simpler in synchronous gauge. The (η ) terms in Φ, Ψ, and δi ; and the (η ) terms in vi looked so complicated that I did not write themO here. O 22 PRIMORDIAL ERA 81

Matter density isocurvature mode. We get the matter density isocurvature mode by setting ~ (rad) = 0, S (rad) = 1, S (rad) = 0. To get the baryon density isocurvature mode (BDI), Rk mr,~k νr,~k multiply all quantities by fb, except the constant term in δb which stays equal to 1, and the constant term in δc, which becomes zero. To get the CDM density isocurvature mode (CDI), multiply all quantities by fc, except the constant term in δc which stays equal to 1, and the constant term in δb, which becomes zero. We get the results

4 1 1 15 fν 2 Φ = − 2 ωη + (η ) −8 1+ 15 fν O 4 1 1+ 15 fν 2 Ψ = 2 ωη + (η ) −8 1+ 15 fν O 4 N N 3 1+ 15 fν 2 δb = δc = 1 2 ωη + (η ) − 8 1+ 15 fν O 4 N N 1 1+ 15 fν 2 δγ = δν = 2 ωη + (η ) 2 1+ 15 fν O 4 N 1 1 15 fν 2 3 vc = − 2 ωkη + (η ) −24 1+ 15 fν O N 1 1 2 3 vν = 2 ωkη + (η ) −8 1+ 15 fν O vN = vN = vN + (η3) b γ ν O δZ = δZ = 1 1 ωη + 3 ω2η2 + (η3) b c − 2 16 O δZ = δZ = 2 ωη + 1 ω2η2 + (η3) γ ν − 3 4 O Z vc = 0

Z Z 1 2 1 1+3fb fν 2 3 4 vb = vγ = 12 ωkη + 48 − ω kη + (η ) − 1 fν O − vZ = 1 ωkη2 + 1 ω2kη3 + (η4) ν − 12 48 O 1 1 2 3 Πν = 2 ωk η . (22.90) −15 1+ 15 fν 23 SUPERHORIZON EVOLUTION AND RELATION TO ORIGINAL PERTURBATIONS82

Netrino density isocurvature mode. We get the neutrino density isocurvature (NDI) mode by setting ~ (rad) = 0, S (rad) = 0, S (rad) = 3/4 (to match with the normalization used in CAMB Rk mr,~k νr,~k and [9]). This gives the results

2 2 fν 1 (1 75 )fν 2 Φ = 4 2 − 4 ωη + (η ) −15 1+ 15 fν − 12 (1 + 15 fν )(1 + 15 fν ) O 2 1 fν 1 (1 15 )fν 2 Ψ = 4 2 − 4 ωη + (η ) 15 1+ 15 fν − 60 (1 + 15 fν )(1 + 15 fν ) O 2 N 3 fν fν 1 15 fν 2 δb = 4 2 − 4 ωη + (η ) 15 1+ 15 fν − 20 (1 + 15 fν )(1 + 15 fν ) O N N 3 δc = δb + (η ) O 8 2 N 11 (1 + 11 fν )fν 1 (1 15 fν )fν 2 δγ = 4 2− 4 ωη + (η ) −15 (1 + fν )(1 fν) − 15 (1 + )(1 + fν )fν O 15 − 15 15 8 2 N 1+ 15 fν 1 (1 15 fν )fν 2 δν = 4 2− 4 ωη + (η ) −1+ 15 fν − 15 (1 + 15 )(1 + 15 fν )fν O N 1 fν 1 fν 2 3 vc = 4 kη + 2 4 ωkη + (η ) −15 1+ 15 fν 30 (1 + 15 fν )(1 + 15 fν ) O N 1 1 1 fν 2 3 vν = 4 kη + 2 4 ωkη + (η ) 4 1+ 15 fν 30 (1 + 15 fν )(1 + 15 fν ) O N N 19 fν 2 vb = vγ = 4 kη + (η ) −60 (1 fν )(1 + fν) O − 15 Z Z fν 2 2 3 δb = δc = k η + (η ) 8(1 fν ) O − Z fν fν 2 2 3 δγ = + k η + (η ) −1 fν 6(1 fν ) O − − δZ = 1 1 k2η2 + (η3) ν − 6 O Z vc = 0 Z Z fν 3 fbfν 2 3 vb = vγ = kη + 2 ωkη + (η ) −4(1 fν ) 16 (1 fν ) O − 4 − Z 1 1 1+ 27 fν 3 4 vν = 4 kη 40 4 (kη) + (η ) − 1+ 15 fν O 1 4 2 3 Πν = (1 + fν )(kη) + (η ) . (22.91) 5 15 O

23 Superhorizon Evolution and Relation to Original Perturba- tions

The usual thinking nowadays is that the perturbations were generated much, much earlier than neutrino decoupling, e.g., during inflation. Thus we are justified in ignoring the decaying solutions and the NVI mode. While the radiation-dominated period is convenient for specifying the ”initial” conditions Φ~k(rad), Scr,~k(rad), Sbr,~k(rad), Sνr,~k(rad), these values are NOT “truly” initial—instead we call them the “primordial” values. We would like to relate them to the “original” perturbations, i.e., to the situation immediately after the perturbations were generated. We shall denote the generation time by t or just . It may be different for different scales k (for ∗ ∗ inflation it is around the horizon exit time). This relation depends on the assumed generating mechanism (e.g., inflation) and what happened between generation ( ) and the primordial epoch ∗ (rad) (e.g. reheating after inflation). 24 GAUSSIAN INITIAL CONDITIONS 83

In this first part of the Cosmological Perturbation Theory course we do not assume a particular mechanism for the generation of the perturbations (we just assume it was some Gaus- sian random process)—the second part of the course will discuss inflation as the generating mechanism—so now we make just some more general observations. In the usual scenarios the perturbations are outside the horizon during the interval between and rad; but some other properties of the radiation-dominated “primordial” era may not ∗ hold. Let us thus collect what general properties follow just from the superhorizon (k ) ≪ H assumption. From the general equation (16.30) we see that = const. for adiabatic perturbations R′ R ( = 0) at superhorizon scales. The Bardeen potentials Φ and Ψ do not necessarily stay constant. S Thus is a better quantity to describe the adiabatic mode, and we have R (rad) = ( ) (ADI) . (23.1) R~k R~k ∗ On the other hand, if = 0, will evolve, and thus (rad) may have received a contribution S 6 R R~k from the “original” entropy perturbations at . ∗ While during the primordial era Scr, Sbr, and Sνr remain constant, this is not necessarily true at earlier times. The energy components corresponding to CDM, baryons, and neutrinos may have been in some other form (some fields) before they became these particles, and the properties assumed in the derivation of Eq.(18.31), i.e., no energy transfer between these components and no internal entropy perturbation within them, may not have held. Maybe it is not even clear how to separate out the energy components corresponding to the later CDM, baryons, and neutrinos. But to arrive at nonzero primordial Scr, Sbr, and Sνr we need to have had some original entropy degrees of freedom responsible for them, which we, for now, denote with S ( ), S ( ), and cr,~k ∗ br,~k ∗ S ( ). νr,~k ∗ Thus, in general, Scr, Sbr, and Sνr may have evolved. However, at superhorizon scales they are not affected by . (We have not shown this; but a rough argument is that changes in entropy R perturbations are “local” physics that are not affected by curvature at superhorizon scales.) We can thus represent the relation between the primordial (rad) and original ( ) values ∗ formally as (rad) ( ) ~k 1 T Scr (k) T S (k) T Sνr (k) ~k R R R br R R ∗ S ~ (rad) 0 T (k) T (k) T (k) S ~ ( )  cr,k  =  ScrScr ScrSbr ScrSνr   cr,k ∗  , (23.2) Sbr,~k(rad) 0 TSbrScr (k) TSbrSbr (k) TSbrSνr (k) Sbr,~k( )    ∗   S ~ (rad)   0 TSνrScr (k) TSνrSbr (k) TSνrSνr (k)   S ~ ( )   νr,k     νr,k ∗        where the Tij(k)= Tij(rad,t , k) are transfer functions giving the relation between quantities at ∗ the primordial (rad) era and at the time t near origin. They do not depend on the direction of ∗ the Fourier mode wave vector ~k since we assume that laws of physics are isotropic. In words, at superhorizon scales: 1. Curvature perturbations remain constant for adiabatic perturbations. R~k 2. Curvature perturbations may be seeded by entropy perturbations. 3. Entropy perturbations are not affected by curvature perturbations. 4. Entropy perturbations may evolve by themselves.

24 Gaussian Initial Conditions

In this Section, by “initial conditions” we mean the primordial (rad) values of quantities. At the end of this Section we comment on their relation to the “original” ( ) values. ∗ 24 GAUSSIAN INITIAL CONDITIONS 84

If we knew the initial values ( (rad), S (rad), S (rad), S (rad)) for all Fourier modes R~k cr,~k br,~k νr,~k ~k, we could calculate from them, using our perturbation equations, the evolution of the universe to obtain the present universe (to the extent first-order perturbation theory holds). Or vice versa, from the observed present universe we could calculate backwards to obtain these initial values.58 But this is not a realistic program—how could we know all these initial values! And the reverse program puts way too much demands on observations; and anyway, what use would we have for this huge collection of information ( (rad),S (rad),S (rad),S (rad)) ? { R~k cr,~k br,~k νr,~k } Instead, we proceed in a statistical manner. Present theories for the origin of the perturbations generate them by a random process. Thus they do not specify initial values; they specify only their probability distributions. Using our perturbation theory we can then calculate probability distributions for observables in the present universe. Comparing then various statistical measures of the observed universe to these probability distributions we can compare theory with observation. We shall assume that the initial conditions (rad), S (rad), S (rad), S (rad) are the R~k cr,~k br,~k νr,~k results of some statistically homogeneous and isotropic Gaussian random process.59 “Gaussian” means that the probability distribution of an individual Fourier component is Gaussian: R2 I2 1 ~k 1 ~k Prob( ~ ) = exp exp Rk √2πσ −2σ2 × √2πσ −2σ2 ~k ~k ! ~k ~k ! R R R R 2 1 ~ = exp |Rk| , (24.1) 2πσ2 − 2σ2 ~k ~k ! R R and likewise for Scr,~k(rad), Sbr,~k(rad), Sνr,~k(rad). Note that the Fourier coefficients are complex quantities. Here R and I denote their real and imaginary parts. The reality of (~x) implies ~k ~k R that = . From this distribution we get for the mean and the variance: ~k ~k∗ R− R = 0 hR~ki 2 = 2σ2 . (24.2) ~k ~k h|R |i R From statistical homogeneity follows (as shown in Cosmology II) that the probabilities of different Fourier modes are independent (i.e., not correlated): ~ ~ ∗′ = S∗′ = S S∗′ =0 for k = k′ , (24.3) hR~kR~k i hR~k ~k i h ~k ~k i 6 where S is S , S , or S . Because of the complex conjugate this will hold also for ~k = ~k cr br νr ∗ ′ − although these two Fourier modes are not independent of each other. Statistical isotropy means that these probability distributions are independent of the direction of ~k, depending only on its magnitude:

σ ~k = σ (k) . (24.4) R R 58This is just a rhetorical point. If taken seriously, it has some fundamental problems: First order perturbation theory remains reasonably accurate to present times only for the largest scales. For smaller scales, nonlinear evolution has hopelessly messed up the information about the initial values. Also, our observations are not about the present (t = t0) universe. Rather, they are about our past light cone. 59Actually, we are not really using the Gaussianity assumption for anything here. We just use the statistical homogeneity and isotropy. It is common in the literature to include the independence of the Fourier components in the concept of “Gaussian perturbations”, and thus I ended up originally calling this section “Gaussian Initial Conditions”. The point of Gaussianity is that the covariances (24.5) give a full statistical description. Full description of non-Gaussian perturbations requires also higher than second-order correlations. 24 GAUSSIAN INITIAL CONDITIONS 85

When there are several independent perturbation quantities (we have now four: ~k(rad), ~ R Scr,~k(rad), Sbr,~k(rad), Sνr,~k(rad)) per Fourier mode k, they may be correlated (but the correlation is independent of the direction of ~k). Thus we have N = 4 variances and N(N 1)/2 = 6 − correlations (covariances):

2 ~ ~∗′ = 2σ (k) δ~~ ′ P (k)δ~~ ′ hRkRk i R kk ≡ R kk 2 S S∗ ′ = 2σ (k) δ ′ P (k)δ ′ h cr,~k cr,~k i Scr ~k~k ≡ Sc ~k~k 2 S S∗ ′ = 2σ (k) δ ′ P (k)δ ′ h br,~k br,~k i Sbr ~k~k ≡ Sb ~k~k 2 S S∗ ′ = 2σ (k) δ ′ P (k)δ ′ h νr,~k νr,~k i Sνr ~k~k ≡ Sν ~k~k ~ S∗ ~ ′ C Sc (k)δ~~ ′ hRk cr,k i ≡ R kk

~ S∗ ~ ′ C Sb (k)δ~~ ′ hRk br,k i ≡ R kk ~ S∗ ~ ′ C Sν (k)δ~~ ′ hRk νr,k i ≡ R kk S S∗ ′ C (k)δ ′ h cr,~k br,~k i ≡ ScSb ~k~k S S∗ ′ C (k)δ ′ h cr,~k νr,~k i ≡ ScSν ~k~k S S∗ ′ C (k)δ ′ . (24.5) h br,~k νr,~k i ≡ SbSν ~k~k We can write this more compactly as

A A∗ ′ C (k)δ ′ , (24.6) h i~k j~k i ≡ ij ~k~k where A = (rad),S (rad),S (rad),S (rad) for i = 1, 2, 3, 4. The diagonal elements of i~k R~k cr,~k br,~k νr,~k this matrix, P (k) C (k), are called primordial power spectra. i ≡ ii The first property, (24.1), is not as important as the other two, (24.3) and (24.4). If it holds, then all statistical information is contained in these 4 power spectra Pi(k) and 6 covariances Cij(k). But if we only consider “quadratic estimators” (e.g., the angular power spectra Cℓ of the cosmic microwave background or the present matter power spectrum Pm(t0, k)) we can drop this first assumption and these estimators are still determined by just the primordial power spectra and covariances (we do still require that the mean of the probability distribution is zero). Thus the full (or required) statistical information about the initial conditions is specified by the k-dependent symmetric real 4 4 covariance matrix C (k). The matrix is symmetric and × ij real since, e.g.,

C (k) = C ( ~k )= S = S = S = C ( ~k )= C (k)= S S ~k ~k∗ ∗ ~k ~k ~k ∗ ~k S S R R | | hR i hR− − i h − R− i R |− | R = S~ ~∗ = ~∗S~ = C S(k)∗ (24.7) h kRki hRk ki R Consider now some quantity f(~x) in the present universe (or, more generally, some time later than the primordial time). For as long as linear perturbation theory holds, all Fourier modes ~k evolve independently, and also the ADI, CDI, BDI, and NDI modes evolve independently60 of each other. Because of the linearity of the theory, the Fourier component f~k depends linearly on the initial values Ai~k. This dependence can be express as a transfer function ~ f~k = Tfi(k)Ai~k = Tfi(k)Ai~k (sum over i) . (24.8)

Because our physical laws are isotropic, Tfi cannot depend on the direction of ~k.

60 Note that the perturbations R~k, Scr,~k, Sbr,~k, and Sνr,~k do not evolve independently. Do not confuse pertu- bation modes with perturbation quantities. 24 GAUSSIAN INITIAL CONDITIONS 86

The variance of f~k is now

f f ∗′ = T (k)A T ∗ (k′)A∗ ′ = T (k)T ∗ (k′)C (k)δ ′ = h ~k ~k i h fi i~k fj j~k i fi fj ij ~k~k P (k)δ ′ , (24.9) ≡ f ~k~k where the power spectrum of f is

P (k)= T (k)T ∗ (k′)C (k)= T (k)T (k′)∗ C (k) . (24.10) f fi fj ij ℜ fi fj ij (The symmetry of C was used for the second equality. stands for the real part.) ij ℜ The covariance of two quantities f and g is likewise

f g∗′ = T (k)A T ∗ (k′)A∗ ′ = T (k)T ∗ (k′)C (k)δ ′ = h ~k ~k i h fi i~k gj j~k i fi gj ij ~k~k C (k)δ ′ . (24.11) ≡ fg ~k~k

From this follows that Cgf = Cfg∗ . However, f(~x) and g(~x) are presumably real quantities, so that f = T (k)A = T (k)A is equal to f = T (k)A . Thus the transfer functions ~k∗ fi∗ i∗~k fi∗ i, ~k ~k fi i, ~k 61 − − − Tfi(k) (and Tgi(k)) are real. To summarize,

f f ∗′ = T (k)T (k′)C (k)δ ′ P (k)δ ′ h ~k ~k i fi fj ij ~k~k ≡ f ~k~k f g∗′ = T (k)T (k′)C (k)δ ′ C (k)δ ′ . (24.12) h ~k ~k i fi gj ij ~k~k ≡ fg ~k~k

The primordial power spectrum Cij(k) is given by the theory of the origin of primordial perturbations (e.g., inﬂation, the topic of the second part of this course). Our task (the ﬁrst part of this course) is to calculate the transfer functions Tfi(k) for all quantities f we are interested in. We obtain them from the linear perturbation theory equations we have already discussed. In the preceding, we assumed for simplicity that f(~x) referred to some quantity in the present universe, so that the transfer function Tfi(k) represented the transfer from the primordial epoch to the present time. More generally, we can consider f at some other time t, writing

f~k(t)= Tfi(t, k)Ai~k . (24.13)

Even more generally, we can specify transfer functions taking us from some earlier time t1 to some later time t2, writing

f~k(t2)= Tfi(t2,t1, k)Ai~k(t1) . (24.14)

In the preceding discussion we had t2 = t0 and “t1 = trad” and we did not write these times explicitly in the transfer functions. Likewise, in Sec. 23 we discussed transfer functions with t2 = trad and t1 = t (perturbation generation time). Linear evolution∗ preserves (or transmits) the Gaussian nature of the perturbations: If the

Ai~k have the statistical properties of statistically isotropic Gaussian random variables, then f~k and g~k will have them also. Let us now finally try to justify our initial assumption that the primordial perturbations were Gaussian: Quantum fluctuations are known to have this Gaussian property. If the original perturbations were produced by quantum fluctuations (as is the case in inflation) and the physics and evolution leading from these quantum fluctuations to the primordial perturbations at trad is linear, then the primordial perturbations are Gaussian. This linear process can be represented by the transfer

61Different Fourier conventions for, e.g., scalar parts of vector quantities may lead to complex transfer functions in some cases. 25 LARGE SCALES 87 functions T (rad, , k), where α indexes the relevant variables of the generation process. Thus iα ∗ “our” Gaussian random process = original perturbation generation process + subsequent linear evolution from generation time to the primordial epoch. It may be that the relevant variables of the first part are uncorrelated, but correlations Cij(k) appear from the second part via the T (rad, , k) transfer function. iα ∗ If one wants to stay agnostic about the original perturbation generation process, one could still motivate the Gaussianity assumption as being a natural one. The Gaussian distribution is particularly simple, since the probability distribution of each Fourier component is fully specified by just a single number, its variance (whose square root is the typical size (the rms value of the distribution) of the perturbation). Many natural processes produce distributions that are close to Gaussian. This is explained by the Central limit theorem, which states that the probability distribution of the mean of a set of independent random variables approaches the Gaussian distribution in the limit where the number of these variables becomes large, regardless of what is the probability distribution of each random variable. In cosmology, we are not really able to measure separately each Fourier mode ~k, but rather our observations always average over a large number of modes. Thus the assumption of the independence of the initial values of each Fourier mode already implies that the Gaussianity assumption is likely to work fairly well. Finally, we can refer to observations. Even if the original perturbations were perfectly Gaus- sian, we can expect the primordial perturbations to deviate from Gaussianity to the extent that first order perturbation theory gives only an approximation for the relation between the primordial and original perturbations. From observations, the typical magnitude of the primordial 5 4 perturbations is 10− . . . 10− . Thus a natural expectation for the magnitude of the second order 10 8 corrections and the error in the linearity assumption would be the square of this, 10− . . . 10− , 5 4 or a relative error of 10− . . . 10− . While there is only “one kind” of Gaussianity, possibilities for a deviation from Gaussianity are infinite. A very simple form of non-Gaussianity of primordial perturbations is one where the perturbation is related to a Gaussian perturbation via a simple transformation

Φ(~x) =Φ (~x) f Φ (~x)2 , (24.15) G − NL G 62 where fNL is called the (local) non-linearity parameter. Its value gives the level of non- Gaussianity. It is customary to define it in terms of the primordial Bardeen potential Φ. Here Φ is the true Bardeen potential, and ΦG is a quantity with a Gaussian distribution related to Φ in the above way. According to the preceding argument a natural expectation for fNL is that it would be of order 1. From the simplest inflation models we actually get fNL 1, whereas ≪ 5 there are some interesting models where fNL 1. Since the values of Φ are of the order of 10− 4 ≫ to 10− , an fNL of the order 1 has a very small effect, unobservable with current methods. A non-Gaussianity of a similar magnitude than the primordial perturbation itself requires fNL of the order 104. The result from Planck is [11]

f = 0.8 5.0 68% CL , (24.16) NL ± which is in agreement with naive expectations and most (but not all) inﬂation models.

25 Large Scales

In this section we consider the scales k k , i.e., scales that enter the horizon during ≪ eq ≡ Heq the matter-dominated epoch. Their evolution is much easier to calculate than those of smaller

62 Note that in the literature there is a lot of confusion about the sign of fNL. 25 LARGE SCALES 88 scales. We ignore dark energy, so the calculation will not extend beyond the matter-dominated epoch, when dark energy begins to have an eﬀect. We also assume that neutrino masses are small enough to be ignored. We can take a number of results from Sec. 19: the background solution

η 2 η η + η 1 4 y = + 2 = 3 w = c2 = (25.1) η η H η η + 1 η2 3(1 + y) s 3(4 + 3y) 3 3 3 2 and the relation y = S (25.2) S 4 + 3y mr where y a/a = ρ /ρ . The differences from Sec. 19 are that now the matter component is ≡ eq m r further divided into baryons and cold dark matter and the radiation component to photons and neutrinos. The radiation part is not a perfect fluid, since neutrinos, and after photon decoupling also photons, have anisotropic pressure, so that Φ = Ψ. However, you will notice that the 6 following discussion will be almost the same as in Sec. 19.4, since once the universe becomes mater-dominated, Ψ becomes equal to Φ, and the nature of radiation becomes irrelevant. We define f ρ /ρ (25.3) b ≡ b m so that we have

δ = f δ + (1 f )δ and δ = f δ + (1 f )δ (25.4) m b b − b c r ν ν − ν γ and S = f S + (1 f )S . (25.5) mr b br − b cr 25.1 Superhorizon evolution Consider ﬁrst the part of the evolution, when k is still outside the horizon, k . Compared ≪H to Sec. 22 we are giving up the assumptions of radiation domination and tight coupling between photons and baryons. Instead we calculate the evolution of the perturbations from the radiation- dominated epoch to the matter-dominated epoch and the discussion applies during and after photon decoupling. Since Φ = Ψ, we don’t get the Bardeen equation we used in Sec. 19. Instead 6 we use the evolution equation (16.31 or C.8) R 2 1 d 2 k 2 1 2 2 − ′ = y R = c Ψ+ (Ψ Φ) + 3c 3c H R dy 3(1 + w) s 3 − sS ≈ sS H 4 y = S , (25.6) 4 + 3y 4 + 3y mr (same as 19.69) where S = const, since k (from 18.31 or C.6). mr ≪H 25.1.1 Adiabatic mode For the adiabatic mode, S = 0, so = const. The Bardeen potentials evolve, so none of the mr R terms in (C.7), which follows from (16.27),

2 1 5 + 3w 2 − Ψ′ + Ψ = (1 + w) + (Ψ Φ) , (25.7) 3H 3 − R 3 − is constant, but once the universe becomes matter dominated,Ψ Φ, since the diﬀerence came ≈ from the radiation anisotropic pressure, w 0, and 2/η, so we are left with ≈ H≈ 1 5 ηΨ′ + Ψ = , (25.8) 3 3 −R 25 LARGE SCALES 89 whose solution is 3 5 Ψ= + Cη− . (25.9) − 5 R 5 Once the decaying part Cη− has died out, we have

Φ=Ψ= 3 = const = 3 (rad) , (mat.dom) (25.10) − 5 R − 5 R and, from (10.19 or C.1)

δ = δ = δ = δ = 2Φ = 6 (rad) . (mat.dom, k ) (25.11) c b m − 5 R ≪H 25.1.2 Isocurvature modes (This repeats Sec. 19.4, since the equation to integrate, (25.6) = (19.69), is the same.) For the isocurvature modes, initially = 0, but S = 0. Integrating (25.6), R mr 6 y y 4 dy 4 1 3 y (y) = d = 9 Smr 2 = 9 Smr 4 = Smr . (25.12) R 0 R 0 4 − y + − 4 3y + 4 Z Z y + 3 " 3 # As the universe becomes matter dominated (y ), →∞ 1 S = const = 1 S (rad) . (25.13) R → 3 mr 3 mr We have then

Φ = Ψ= 3 = const = 1 S (rad) , (mat.dom) (25.14) − 5 R − 5 mr and δ = δ = 2Φ = 2 S (rad) . (mat.dom, k ) (25.15) m − 5 mr ≪H The individual density perturbations δb and δc will depend on the Sbr and Scr (exercise). The S will not have an eﬀect on them, since it does not aﬀect . νr S Exercise: Solve the matter-dominated era superhorizon δb and δc in terms of the primordial , S and S . R br mr 25.2 Through the horizon When the (k k ) perturbations approach the horizon, the universe is already matter dom- ≪ eq inated. This means that c2 0 and Φ Ψ, so that from Eq. (25.6) we have that = const s ≈ ≈ R even inside the horizon. We now have Eq. (25.9) for Ψ with = const both for the adiabatic R and isocurvature modes. Since Ψ had already settled to a constant before horizon entry, it will stay at this constant solution, and we have

Φ=Ψ= 3 = 3 (rad) 1 (1 f )S (rad) 1 f S (rad) . (mat.dom) (25.16) − 5 R − 5 R − 5 − b cr − 5 b br From (14.21 or C.1) we have that

1 k 2 δ = δ = 2 1+ Φ . (25.17) m − 3 " H #

Through and inside the horizon the entropy perturbations Sij do not necessarily stay constant any more: they could be changed by velocity differences. The baryon and CDM velocity equations differ by the baryon-photon collision term. In the NDI mode we had a velocity difference between neutrinos and photons, and the collision term forced vb = vγ while vc = v in 25 LARGE SCALES 90 the primordial era. In the other three modes all velocities were equal in the primordial era. We now consider only these modes: In ADI, BDI, and CDI modes the collision term disappears in the primordial era, since vb = vγ. The velocity equations depend on the density perturbations only through the common potential. For as long as vb = vγ , the velocity equations for the different fluid components differ only due to the anisotropic pressure terms for photons and neutrinos. These terms are suppressed by k/ so they will not cause velocity differences at superhorizon scales. When the H scales approach the horizon these terms begin to affect v and v . However, for k k the γ ν ≪ eq universe is then already matter dominated, so the baryons and CDM no longer care about the neutrinos and photons (for baryons the collision term is suppressed by ργ/ρb). Thus we will still have vb = vc = v. (Here and elsewhere we are ignoring baryon pressure; at these large scales, k k , its effect is negligible.) ≪ eq Therefore, for the adiabatic mode, we still have δb = δc = δm. For the BDI and CDI modes these density perturbations differ, but since vb = vc, the relative entropy perturbation S = δ δ = S S stays constant. (The S and S no longer stay bc b − c br − cr br cr constant but we do not care about them any more.) Thus in the matter-dominated era we have

δ = δ + f S = δ + f [S (rad) S (rad)] b c bc c br − cr δ = δ f S = δ f [S (rad) S (rad)] (25.18) c − b bc − b br − cr where 1 k 2 δ = 1+ 6 (rad) + 2 f S (rad) + 2 f S (rad) . (25.19) 3 × 5 R 5 c cr 5 b br " H # and f = 1 f . Since δ grows as (k/ )2 whereas the diﬀerences between δ, δ and δ stay c − b H c b constant, the relative diﬀerences will eventually become negligible, so that at late times δ b ≈ δ δ δ. c ≈ m ≈ Thus we have that density perturbation transfer functions from the primordial era to the matter-dominated era are

2 6 1 k 6 1 2 Tδ (η, k) = 1+ = + (kη) (25.20) R 5 3 5 10 " H # 2 2 1 k 2 fc 2 fc TδScr (η, k) = fc 1+ = fc + (kη) = Tδ (η, k) 5 3 5 30 3 R " H # 2 2 1 k 2 fb 2 fb TδS (η, k) = fb 1+ = fb + (kη) = Tδ (η, k) br 5 3 5 30 3 R " H # Isocurvature perturbations lead to matter perturbations that are a factor 1/3 smaller than those from adiabatic perturbations (from initial and S of equal size). In the matter power spectrum R mr Pδ this becomes a factor 1/9. If we have both primordial and S their correlations are also important. The density R mr power spectrum is

Pδ(η, k) = Tδi(k)Tδj (k)Cij(k) (25.21) 6 1 2 = + (kη)2 5 10 × 2 2 fc fb 2fc 2fb 2fcfb P (k)+ PScr (k)+ PS (k)+ C Scr (k)+ C S (k)+ CScrS (k) . × R 9 9 br 3 R 3 R br 9 br 26 SACHS–WOLFE EFFECT 91

The power spectra P (k) are necessarily nonnegative, but the correlations C (k) for i = j may i ij 6 be positive or negative indicating correlation or anticorrelation. We see that (positively) correlated curvature and entropy perturbations enhance the density power spectrum Pδ(k) whereas anticorrelated63 curvature and entropy perturbations weaken it.

26 Sachs–Wolfe Eﬀect

Consider photon travel in the perturbed universe. The geodesic equation is

d2xµ dxα dxβ +Γµ = 0 , (26.1) du2 αβ du du where u is an aﬃne parameter of the geodesic. For photons, we choose u so that the photon 4-momentum is dxµ pµ = , (26.2) du which allows us to write the geodesic equation as dpµ +Γµ pαpβ = 0 . (26.3) du αβ Dividing by p0 = dη/du, this becomes

dpµ pαpβ +Γµ = 0 . (26.4) dη αβ p0 In the following, we need only the time component of this equation,

dp0 pipj +Γ0 p0 + 2Γ0 pk +Γ0 = 0 . (26.5) dη 00 0k ij p0

µ Assuming scalar perturbations and using the Newtonian gauge (the Γαβ from Eq. (8.7)), this becomes 0 i j dp 0 k δijp p + +Φ′ p + 2Φ p + 2 (Φ + Ψ) Ψ′ = 0 . (26.6) dη H ,k H− H − p0 These 4-momentum components pµ are in the coordinate frame. What the observer interprets as the photon energy and momentum are the components pµˆ in his local orthonormal frame. Since the metric is diagonal, the conversion is easy, pµˆ = g pµ (for a comoving observer): | µµ| ˆ p E p0 = a√1+2Φ p0 = a(1 + Φ)p0 ≡ ˆ pi = a√1 2Ψ pi = a(1 Ψ)pi . (26.7) − − 2 î ˆj Since photons are massless, E = δijp p . In the background universe, the photon energy redshifts as E¯ a 1 q¯ aE¯ = const. In ∝ − ⇔ ≡ the presence of perturbations, q aE = const. Thus we define q and ~q, ≡ 6 2 0 0 2 q aE = a (1 + Φ)p p = a− (1 Φ)q ≡ ⇒ − i î 2 i i 2 i q ap = a (1 Ψ)p p = a− (1 + Ψ)q (26.8) ≡ − ⇒ 63Beware: Some authors have the opposite sign conventions for entropy and/or curvature perturbations, changing our correlation to anticorrelation and vice versa (unless both sign conventions are opposite!). One actually gets this opposite sign convention if you take the “entropy perturbation” Siγ to literally mean “relative perturbation in photon entropy per i particle”. Our sign convention for relative entropy perturbations seems to be the more common in literature. 26 SACHS–WOLFE EFFECT 92

2 i j where q = δijq q , as suitable quantities to track the perturbation in the redshift. Rewriting Eq. (26.6) in terms of q and ~q (and dropping 2nd order terms) gives (exercise)

dq dΦ k (1 Φ) = q qΦ′ 2q Φ + qΨ′ . (26.9) − dη dη − − ,k Here the rhs is 1st order small, therefore dq/dη is also 1st order small, and we can drop the factor (1 Φ). Dividing by q we get − 1 dq dΦ ~q Φ = Φ′ +Ψ′ 2 ·∇ . (26.10) q dη dη − − q Here the total derivative along the photon geodesic is

d ∂ dxk ∂ = + (26.11) dη ∂η dη ∂xk and ~q (1 Ψ)pk pk dxk = − = to 0th order (26.12) q (1 + Φ)p0 ≈ p0 dη so that ~q Φ dxk ∂Φ dΦ ∂Φ 2 ·∇ = 2 = 2 , (26.13) − q − dη ∂xk − dη − ∂η so that Eq. (26.10) becomes

1 dq dΦ 1 dq = +Φ′ +Ψ′ . (26.14) q dη − dη ≈ q¯ dη

The relative perturbation in the photon energy, δE/E¯, that the photon accumulates when traveling from x to xobs in the perturbed universe is thus ∗ δE δq dq = = = dΦ+ Φ′ +Ψ′ dη E¯ q¯ q − Z Z Z ηobs ∂Φ ∂Ψ = Φ(x ) Φ(xobs)+ + dη , (26.15) ∗ − ∂η ∂η Zη∗ where the integrals are along the photon path. Here x = (η , ~x ) denotes the location ~x at the ∗ ∗ ∗ ∗ last scattering surface from where the photon originated at the time η of photon decoupling. ∗ For a thermal distribution of photons, a uniform relative photon energy perturbation corresponds to a temperature perturbation of the same amount:

δT δE = . (26.16) T E¯ jour Here “jour” refers to the temperature perturbation the photon distribution accumulates on the journey between x and xobs. ∗ The other contributions to the observed CMB temperature anisotropy are due to the local photon energy density perturbation and photon velocity perturbation at the origin of the photon, on the last scattering surface:

δT 1 = δγ(x ) ~v(x ) nˆ , (26.17) T 4 ∗ − ∗ · intr wheren ˆ is the direction the observer is looking at. 26 SACHS–WOLFE EFFECT 93

For a given observer, the Φ(xobs) part is common to photons from all directions, and the observer interprets it as part of the mean (background) photon temperature. Thus the observed CMB temperature anisotropy is

ηobs δT 1 N N ∂Φ ∂Ψ = δγ (x ) ~v (x ) nˆ + Φ(x )+ + dη . (26.18) T 4 ∗ − ∗ · ∗ ∂η ∂η Zη∗ (There is also a contribution from the motion of the observer that causes a dipole pattern in the observed anisotropy. To get rid of that, the dipole of the observed anisotropy is subtracted away from the observations before any cosmological analysis.) For large scales, much larger than the horizon size at photon decoupling, the Doppler eﬀect ~vN (x ) nˆ is small compared to the other terms. The contribution − ∗ ·

δT 1 N = δγ (x )+Φ(x ) (26.19) T 4 ∗ ∗ SW is called the ordinary Sachs–Wolfe eﬀect, and the contribution

δT ηobs ∂Φ ∂Ψ = + dη (26.20) T ∂η ∂η ISW Zη∗ is called the integrated Sachs–Wolfe effect (ISW). During the matter-dominated epoch Φ = Ψ = const., so there are two contributions to ISW: the early Sachs-Wolfe effect (ESW) from the time after photon decoupling when the universe was not yet completely matter dominated, and the late Sachs-Wolfe effect (LSW) from the time when dark energy began to have an effect on the expansion of the universe. LSW becomes important at the largest scales and ESW at scales that are comparable to the horizon at photon decoupling. For k k < k (or ℓ ℓ < ℓ ) the ordinary Sachs-Wolfe effect is the dominant ≪ dec eq ≪ dec eq contribution. We now give the result for this in the (not very good) approximation that the universe was already matter-dominated at the time of photon decoupling. We need Φ and δγ at η , and we consider only scales that were superhorizon at that time. For these scales the ∗ equations that we need have no k-dependence, so we can apply them directly at coordinate space (assuming that we ignore all smaller-scale contributions, e.g., by making observations with a coarse resolution or smoothing the observations afterwards). 4 4 For the adiabatic mode, δγ = 3 δm = 3 δ (since entropy perturbations have remained zero at superhorizon scales), and

δ = 2Φ = 6 (rad) − 5 R δT = 1 δ +Φ = 1 Φ = 1 (rad) . (26.21) ⇒ T 3 3 − 5 R For the BDI and CDI modes, the entropy perturbations have remained constant at superhorizon scales, and δ = 4 (δ S )= 4 (δ S ), where δ = δ +f (S S ) and δ = δ +f (S S ). γ 3 b − b 3 c − c b m c b − c c m b c− b We write just Si for Sir = Siγ, since we do not consider the NDI mode. From Sec. 25.1.2 we have δ = δ = 2Φ = 2 (f S + f S ) . (26.22) m − 5 c c b b From these we get (exercise) that

δ = 4 (f S + f S ) , (26.23) γ − 5 c c b b so that δT = 1 δ +Φ = 2 (f S + f S ) , (26.24) T 4 γ − 5 c c b b 26 SACHS–WOLFE EFFECT 94 both terms contributing equally. In total, the ordinary Sachs-Wolfe eﬀect becomes δT 1 (rad) 2 S (rad) (26.25) T ≈ − 5 R − 5 m Primordial isocurvature perturbations lead to an ordinary Sachs-Wolfe eﬀect that is twice as large as that from primordial adiabatic perturbations (from initial S and of equal size). In m R the angular power spectrum Cℓ this becomes a factor 4.

26.1 Ordinary Sachs-Wolfe Eﬀect and Primordial Correlations

Proper CMB analysis is done in terms of the angular power spectrum Cℓ (which is the analog of power spectrum on the celestial sphere). But for now, let us do this in coordinate space, as the ordinary Sachs-Wolfe eﬀect refers to just the position on the last scattering (photon decoupling) surface, from which the CMB photons we observe originate. For this purpose we deﬁne a covariance matrix of primordial perturbations

C A (~x)A (~x) (26.26) ij ≡ h i j i where the Ai(~x) are “smoothed out to leave only the large scales”. The Cij are expectation values, so they are the same for all ~x. Deﬁne also the transfer functions TSW,i so that

δT (~x) = T A (~x) . (26.27) T SW,i i Thus we have (in our approximation)

1 2 2 TSW, = TSW,Sc = fc TSW,S = fb . (26.28) R − 5 − 5 b − 5 The expectation value of the variance of δT/T due to the SW eﬀect is

δT/T )2 = T T A A (26.29) SW SW,i SW,jh i ji 1 4f 2 4f 2 4f 4f 8f f = 2 + c S2 + b S2 + c S + b S + b c S S 25hR i 25 h c i 25 h b i 25 hR ci 25 hR bi 25 h c bi Uncorrelated curvature and entropy perturbations add up in the rms sense,

2 2 2 (δT/T ) ) SW = (δT/T ) ) + (δT/T ) ) Sm , h i h iR h i q q whereas positively correlated ones lead to a stronger effect and anticorrelated ones lead to a weaker effect. Perfectly anticorrelated perturbations could in principle cancel the SW effect completely, i.e., if (rad) = 2S (rad) (26.30) R − m everywhere, then (δT/T )SW = 0 everywhere. These would still give rise to matter density perturbations. For them to cancel (at large scales), we need

(rad) = 1 S (rad) . (26.31) R − 3 m 27 APPROXIMATE TREATMENT OF THE SMALLER SCALES 95

27 Approximate Treatment of the Smaller Scales

(I follow Dodelson[5], Secs. 7.1.2, 7.3, and 7.4.) Smaller scales are more diﬃcult to solve than the large k k scales. To be able to obtain ≪ eq some relatively easy results we now make some approximations:

1. We ignore neutrino and photon anisotropy, so that Π = 0 and Ψ = Φ, which we call the gravitational potential; and assume neutrino adiabaticity (i.e., no NDI mode). In reality neutrino anisotropy has about a 10 % effect, as we saw in Sec. 22.1, so ignoring it is not an excellent approximation. On the other hand, the photon distribution remains isotropic until we approach photon decoupling, which happens after matter-radiation equality, so the photon anisotropy will have a smaller effect on the metric. Together these assumptions and approximations mean that there is no difference in the initial values or the evolution equations of photons and neutrinos, so that δγ = δν = δr = δ and vγ = vν = vr = v. 2. We ignore baryons completely, and treat matter as if it consisted of CDM only. Since in reality, baryons make about one sixth of all matter, this approximation should be good to about 20 %. (What happens with the baryons is that, because of their tight coupling with photons, baryon perturbations do not grow during the radiation-dominated epoch, instead they oscillate with the radiation; but after photon decoupling they fall into the gravitational wells of the CDM, so that the baryon perturbation becomes almost the same as the CDM perturbation).

Together these approximations mean that there is no difference between photons and neutrinos so that we have a single radiation component. Thus we are back to the simplified universe of Sec. 19, so this can be seen as a continuation of that section. With the above approximations the two first Einstein equations (10.19 or C.1) and (10.21 or C.2) become

2 1 1 k 1 − Φ′ +Φ+ Φ = δ (27.1) H 3 − 2 H 1 3 − Φ′ +Φ = (1 + w)Hv , (27.2) H 2 k and the ﬂuid equations (C.5) become

δm′ + kvm = 3Φ′ v′ + v = kΦ m H m 4 δ′ + kv = 4Φ′ r 3 r 1 v′ kδ = kΦ . (27.3) r − 4 r Multiplying the first Einstein equation with 3 2 = 8πGρa2 we get H 2 2 2 k Φ + 3 Φ′ + Φ = 4πGa δρ = 4πGa (ρ δ + ρ δ ) . (27.4) H H − − m m r r Here (27.3 and 27.4) we have five differential equations and five unknowns: δm, vm, δr, vr, and Φ. It is easy to write a simple computer code to solve these equations numerically (exercise). For background quantities use the analytic solution from Sec. 19. Because of the approximations, the late time evolution of the radiation component we get is not useful; but we get a reasonable (better than 20%) approximation of the dark matter evolution. We did not 27 APPROXIMATE TREATMENT OF THE SMALLER SCALES 96

Figure 4: Output from a numerical solution of (27.3) and (27.4) for adiabatic initial conditions. Left: N N The density perturbations δm and δr . Right: The gravitational potential Φ.

make any assumption about scales, so this will apply for all scales—it is just that because of the approximations we made, the earlier analytic treatment of the large scales is more accurate. Subtracting the second Einstein equation from the ﬁrst one we get

3 3 k2Φ = 4πGa2 δρ + H(ρ + p)v = 4πGa2 ρ δ + ρ δ + H (ρ v + 4 ρ v ) (27.5) − k − m m r r k m m 3 r r which is not an evolution equation (no time derivatives), but a constraint equation. It can be used instead of the Φ evolution equation (27.4) for some purposes.

Numerical solution. I wrote a python script to solve the set of diﬀerential equations (27.3 and 27.4) and ran it with adiabatic and isocurvature initial conditions for k = 50/η3, a scale which enters −4 −4 during the radiation-dominated era. I started at an initial time η = 10 η3, i.e., x = 10 . In Sec. 19.5 N we didn’t do the δi , but we can get the initial conditions from Sec. 22.5 by setting fb = fν = 0. Thus, to get the adiabatic mode I set the initial condition

1 1 δm = 1+ 8 ωη = 1+ 4 x 1 1 2 1 1 2 vm = 3 kη + 24 ωkη = 3 kη3x + 12 kη3x −4 − δr = 3 δm vr = vm Φ = 2 + 1 ωη = 2 + 1 x (27.6) − 3 24 − 3 12 and to get the isocurvature mode I set the initial condition

3 3 δm = 1 ωη = 1 x − 8 − 4 1 2 1 2 vm = 24 ωkη = 12 kη3x −1 − δr = 2 ωη = x 1 2 1 2 vr = ωkη = kη x − 8 − 4 3 Φ = 1 ωη = 1 x . (27.7) − 8 − 4 Output from these two runs is shown in Figs. 4 and 5. 27 APPROXIMATE TREATMENT OF THE SMALLER SCALES 97

Figure 5: Same as Fig. 4, but for isocurvature initial conditions. Note how, compared to the adiabatic mode, the matter density perturbation misses the initial growth by about a factor of 10 during the radiation dominated era.

27.1 Small Scales during Radiation-Dominated Epoch Next we consider the small scales, k k , i.e., scales that enter the horizon during the radiation- ≫ eq dominated epoch. Now we can make the approximation of radiation domination while we follow the perturbations through the horizon. In this subsection we consider only the part of the evolution during the radiation-dominated epoch. Radiation domination means that we can ignore the matter components while we calculate the evolution of the radiation components and the metric perturbations Φ.64 For the background we have now

a η 1 2 1 y = 2 1 = and w = cs = 3 (27.8) ≡ aeq η3 ≪ H η and for the perturbations (of the total ﬂuid)

1 δp = 3 δρ . (27.9)

27.1.1 Radiation The ﬂuid perturbation equations (C.5) for radiation become

4 ηδ′ + 3 kηv = 4ηΦ′ (27.10) 1 ηv′ kηδ = kηΦ . (27.11) − 4 The Einstein equations (27.1) and (27.2) become

1 2 1 ηΦ′ +Φ+ (kη) Φ = δ (27.12) 3 − 2 2 ηΦ′ +Φ = v , (27.13) kη which immediately give 4 2 δ = v (kη)2Φ , (27.14) −kη − 3 64In this section we follow Chapter 7 of Dodelson[5]. Note that the multipole moments of the photon brightness m function Θℓ in Dodelson’s notation are related to the Θℓ of CMB Physics 2007 and to perturbations of the photon 0 1 1 0 1 energy tensor by Θ0 ≡ Θ0 ≡ 4 δγ and Θ1 ≡ 3 Θ1 ≡ 3 vγ . 27 APPROXIMATE TREATMENT OF THE SMALLER SCALES 98 which we use to eliminate δ from (27.11) so that it becomes

1 2 ηv′ + v = 1 (kη) kηΦ . (27.15) − 6 We then derivate (27.13) to arrive at

2 2 2 4 1 2 2 Φ′′ + Φ′ = ηv′ v = ηv′ + v v = k Φ Φ′ , (27.16) η kη3 − kη3 − kη3 − 3 − η so that the ﬁnal diﬀerential equation for Φ in the radiation-dominated era is

4 1 2 Φ′′ + Φ′ + k Φ = 0 . (27.17) η 3

This is the radiation-dominated universe case, which we already did in Sec. 15, where we got the growing mode solution

sin kη kη cos kη 1 √3 − √3 √3 Φ(η) = u = 3Φ(rad) 3 (27.18) η kη √3 to (27.17). For superhorizon scales (kη 1, i.e., at early times η k 1), we have Φ(η) Φ(rad) = ≪ ≪ − ≈ const. For subhorizon scales, i.e., at later times, after horizon entry (η k 1 so that kη 1), the ≫ − ≫ cosine part dominates, so that for the gravitational potential we have

cos kη/√3 Φ(η) 9Φ(rad) 2 , (27.19) ≈ − (kη) which oscillates with frequency ω = 2πf = (1/√3)k = csk and a decaying amplitude 9Φ(rad) . (27.20) (kη)2

The radiation density δ and velocity v perturbations at these late (but still radiation-dominated) times oscillate with constant amplitude:

1 2 9 v = (kη Φ′ + kηΦ) Φ(rad)c sin(c kη) 2 ≈ 2 s s δ 2 (kη)2Φ 6Φ(rad) cos(c kη) . ≈ − 3 ≈ s 27.1.2 Matter While matter is subdominant during the radiation-dominated epoch, we still want to know what happens to it during that epoch, since it becomes important later. The matter ﬂuid equations (from 27.3ab) are thus

ηδm′ + kηvm = 3ηΦ′ (27.21)

ηvm′ + vm = kηΦ . (27.22)

Derivating the ﬁrst one and then using the second to get rid of vm gives the matter perturbation equation 1 3 2 δ′′ + δ′ = 3Φ′′ + Φ′ k Φ F (k, η) , (27.23) m η m η − ≡ 27 APPROXIMATE TREATMENT OF THE SMALLER SCALES 99 where the source function F (k, η) is a known function that we get from the solution (27.18). The general solution will be the solution

δm = C1 + C2 ln kη (27.24) of the homogeneous equation plus a special solution of the full equation. A special solution (exercise) is

η δ (η) = dη′F (k, η′)η′(ln kη′ ln kη) (27.25) m − − Z0 (constructed from the homogeneous equation solutions and the source function). Thus the general solution will be

η δ (η) = C + C ln kη dη′F (k, η′)η′(ln kη′ ln kη) . (27.26) m 1 2 − − Z0 Note ﬁrst that the source function and its integral are proportional to the initial value of Φ, i.e., Φ (rad). Otherwise we care now only about their limiting behavior as kη 1 and kη 1. ~k ≪ ≫ As η 0 the integral 0 (exercise). For the solution to stay ﬁnite, we thus must have → → C2 = 0 (i.e., we reject the decaying mode). Thus

C = δ (0) = δ (rad) = 3 Φ (rad) + S (rad) , (27.27) 1 m m − 2 ~k m,~k where S = δ 4 δ is the matter entropy perturbation. For the rest of this subsection we m m − 3 r consider only the adiabatic mode, so Sm,~k(rad) = 0 and

C = δ (0) = 3 Φ (rad) . (27.28) 1 m − 2 ~k After horizon entry Φ, and therefore also F (k, η), decays. Thus, if we rewrite the integral term in (27.26) as

η η dη′F (k, η′)η′ ln kη′ + ln kη dη′F (k, η′)η′ B Φ (rad) + B Φ (rad) ln kη , (27.29) − → 1 ~k 2 ~k Z0 Z0 both integrals stop changing as a function of the upper limit η for kη 1, and we can capture ≫ the eﬀect of F (k, η) in these two constants

∞ ∞ B Φ (rad) dη′F (k, η′)η′ ln kη′ and B Φ (rad) dη′F (k, η′)η′ , (27.30) 1 ~k ≡− 2 ~k ≡ Z0 Z0 i.e., δ ( 3 + B + B ln kη)Φ (rad) for kη 1 . (27.31) m ≈ − 2 1 2 ~k ≫ Rewrite 3 + B + B ln kη A ln(Bkη) = A ln B A ln(kη) , (27.32) − 2 1 2 ≡ − − − so that

A = B 9.0 − 2 ≈ B = exp[(B 3 )/B ) 0.62 (27.33) 1 − 2 2 ≈ where the numerical results can be obtained by numerical integration of the solution (27.18). Thus we have the ﬁnal result

δ AΦ (rad) ln(Bkη) 9.0Φ (rad) ln(0.62kη) (kη 1) (27.34) m ≈ − ~k ≈ − ~k ≫ 27 APPROXIMATE TREATMENT OF THE SMALLER SCALES 100 for the growth of matter perturbations inside the horizon during the radiation-dominated epoch. Since in this epoch = 1/η and y η we can write the kη also as H ∝ y kη = (rad.dom) (27.35) yk where y 1 is the value of y at horizon entry (k = ). k ≪ H After horizon entry (as a η), ∝ dδ dδ m = m AΦ (rad) = 9.0Φ (rad) , (27.36) d ln a d ln(kη) → − ~k − ~k i.e., for each e-folding δ picks another 9Φ (rad) (for each 10-folding another 20.3Φ (rad)). m − ~k − ~k A more detailed look at the evolution of δm through the horizon would show that the oscillations in Φ (see Eq. 27.18) are reflected in F (k, η) and cause small oscillations around (27.34) that gradually fade into insignificance. For radiation perturbations we have gravity contending with pressure, causing the radiation density perturbation to oscillate around zero with a constant amplitude. Cold dark matter sees only the gravity (caused by the radiation): it begins to fall towards the initial gravity wells; when the gravity wells begin to oscillate, these will alternately slow down and accelerate this fall, but since this oscillation amplitude is decreasing it is not able to stop the fall—the initial kick the CDM got before the first reversal of Φ carries it on towards the bottom of the initial potential wells. Thus the CDM density perturbation keeps growing, albeit only logarithmically.

27.2 Small Scales to the Matter-Dominated Epoch In Sec. 27.1 we found that during the radiation-dominated epoch the radiation (and baryon) density perturbation δr oscillates without growing, whereas the CDM perturbation δc grows logarithmically. This will have the eﬀect that δρm = δmρm will become larger than δρr = δrρr while ρm is still smaller than ρr, i.e., the perturbation becomes matter dominated at some y = y 1 while the background is still radiation dominated. The gravitational potential δeq ≪ Φ will then be determined by matter perturbations. In this section we solve the evolution of matter perturbations in the approximation that we ignore the contribution of the radiation perturbation, while we use the background solution that includes both matter and radiation contributions. This leaves a gap between Sec. 27.1 and this section for going from δ δ to m ≪ r δ δ . We will bridge the gap at the end of this section by matching the two solutions. m ≫ r We work now in the combined 1) subhorizon k and 2) perturbations matter-dominated ≫H δρ δρ (y y ) limit but with the full matter+radiation background solution. The m ≫ r ≫ δeq relevant perturbation equations, from Eqs. (27.3) and (27.5), are now

δm′ + kvm = 3Φ′ (27.37)

vm′ + vm = kΦ (27.38) H y y k2Φ = 4πGa2ρ δ = 4πGa2ρ δ = 3 2 δ (27.39) − m m y + 1 m 2 H y + 1 m

Derivating (27.37) and using (27.38) and (27.37) we get

2 δ′′ + δ′ = 3Φ′′ + 3 Φ′ k Φ (27.40) m H m H − or 2 2 1 2 1 k − δ′′ + − δ′ = 3 − Φ′′ + 3 − Φ′ Φ , (27.41) H m H m H H − H 27 APPROXIMATE TREATMENT OF THE SMALLER SCALES 101 where on the rhs the last term dominates as k . We thus have ≫H 2 1 3 y − δ′′ + − δ′ = δ . (27.42) H m H m 2 y + 1 m We now change to using y as the time coordinate (see Appendix B) to get the Meszaros equation

d2δ 3y + 2 dδ 3 y y2 m + m = δ . (27.43) dy2 2(y + 1) dy 2 y + 1 m

This equation applies when y y and y y and will remain valid until dark energy begins ≫ δeq ≫ k to aﬀect the expansion of the universe. It is a second order diﬀerential equation so it will have two independent solutions. They are (exercise) 2 D (y) = y + 1 3 2 √1+ y + 1 D (y) = y + ln 2 1+ y . (27.44) 2 3 √1+ y 1 − − p Their late-time (y 1) behavior is (exercise) ≫ D1(y) y ∝ 3/2 D (y) y− , (27.45) 2 ∝ so D1(y) is a growing mode and D2(y) is a decaying mode. The general solution is thus

δm(y) = C1D1(y)+ C2D2(y) . (27.46) To find C and C we match this solution with the y y y solution (27.34) 1 2 k ≪ ≪ δeq y δ = AΦ (rad) ln B . (27.47) m − ~k y k at some time y = y which is near y , or at least y y 1. Neither solution is valid near m δeq k ≪ m ≪ yδeq, but we expect that the behavior near yδeq is not dramatically different. (We want to check how much the matching depends on ym so therefore we do not fix it to be, e.g., equal to yδeq.) By matching we mean that we require both solutions to give the same value for δm(ym) and dδm/dy(ym), i.e., we solve (exercise) C1 and C2 from the two matching conditions: y AΦ (rad) ln B m = C D (y )+ C D (y ) − ~k y 1 1 m 2 2 m k AΦ~k(rad) dD1 dD2 = C1 (ym)+ C2 (ym) . (27.48) − ym dy dy

Actually, if we are only interested in the late-time behavior, we only need to solve C1, since the decaying mode will not be important later. The matching will initially give a combination of the growing mode and the decaying mode, but the decaying mode will decay away after some time. C will pick up some y -dependence, but this disappears in the limit y 1 and we get 1 m m ≪ the approximate result (exercise) 3 4B C = 3 ln AΦ (rad) . (27.49) 1 2 − y ~k k Thus our ﬁnal result, the density perturbation growing mode at y y , is ≫ δeq 4Be 3 δ C D (y) 3 AΦ (rad) ln − (y + 2 ) . (27.50) m ≈ 1 1 ≈ − 2 ~k y 3 k 27 APPROXIMATE TREATMENT OF THE SMALLER SCALES 102

27.3 Transfer Function

The transfer function TΦ(k) relates the gravitational potential during the matter-dominated epoch to its primordial value. For small scales, we get it from Eq. (27.50). Since now k ≪ H the last term on the lhs of (27.1) dominates and using y δρ δρ δ δ (27.51) ≈ m ⇒ ≈ y + 1 m we have 3 2 9 2 4Be 3 y(y + 2 ) Φ(y) = H δ = A H Φ (rad) ln − 3 . (27.52) −2 k m 4 k ~k y y + 1 k This equation has both time-dependent and scale-dependent quantities. We want to express these in terms of y a/a and k/k . (Note that we have not ﬁxed the normalization of the ≡ eq eq scale factor a and the normalization of k depends on the normalization of a, but these ratios are dimensionless quantities independent of this normalization.) So what are and y ? Use the result (19.16) from Sec. 19, H k 1+ y 2 1+ y k2 2 = Heq = eq (27.53) H y2 2 y2 2 (the deﬁnition of k is k = .) This gives eq eq Heq 2 1+ y k 2 H = eq (27.54) k 2y2 k and for yk, i.e., y at the time the scale k enters the horizon,

2 2 2 2 1+ yk keq keq 1 keq k = k = 2 2 yk = (27.55) H yk 2 ≈ 2yk ⇒ √2 k (since this happens during the radiation-dominated epoch, y 1). k ≪ Thus Eq. (27.52) is

2 2 9 y + 3 keq 3 k Φ(y) = A ln 4√2Be− Φ (rad) , (27.56) 8 y k k ~k eq where in the matter-dominated epoch (y + 2 )/y 1, so that Φ becomes time-independent. The 3 ≈ small-scale behavior of the transfer function TΦ(k) is thus

2 2 9 keq 3 k keq k T (k) = A ln 4√2Be− = 12.6 ln 0.17 . (k k ) Φ 8 k k k k ≫ eq eq eq (27.57) (Note that the logarithm is negative if k . 6keq; Eq. (27.57) is not supposed to apply for this small k.) In the literature this result is usually given as

2 2 5 keq 3 k keq k T (k) = A ln 4√2Be− = 11.3 ln 0.17 , (k k ) (27.58) 4 k k k k ≫ eq eq eq since in this simpliﬁed universe the large-scale transfer function is 9 T (k) = (k k ) (27.59) Φ 10 ≪ eq 27 APPROXIMATE TREATMENT OF THE SMALLER SCALES 103

1 1 1 1 11 1 11

Figure 6: Transfer function T (k) for CDM, adiabatic primordial ﬂuctuations. The black curve is the BBKS transfer function (27.61), the red curve is the small-scale approximate analytical result (27.58) (the dotted red curve is Dodelson (7.69), which for some reason has slightly diﬀerent numerical factors), the two black dotted lines correspond to T (k) 1 and T (k) = (k/k )2, and the green vertical line gives ≡ eq k = keq. The k scale is for a reference model with Ωm = 0.3, h = 0.7, for which keq = 0.0153 h/Mpc = 1/(65 h−1Mpc).

( stays constant but Φ changes from its radiation-dominated value of 2 to the matter- R − 3 R dominated value 3 , see Sec. 16.5.1) and this transfer function is customarily normalized to − 5 R unity at large scales, i.e. T (k) T (k) Φ . (27.60) ≡ T (k k ) Φ ≪ eq At scales closer to k the transfer function changes smoothly from the k k solution to eq ≪ eq the k k solution. To ﬁnd how, one has to solve the equations (27.3 and 27.4) numerically. ≫ eq (The logarithm in Eq. (27.57) changes sign at k = 5.9k , but the k k result is supposed eq ≫ eq to apply only at larger k than this, so the transfer function is everywhere positive.) Bardeen, Bond, Kaiser, and Szalay [12] gives a ﬁtting formula, the BBKS transfer function

ln(1 + 2.34q) 1 T (k) = , (27.61) 2.34q [1 + 3.89q + (16.1q)2 + (5.64q)3 + (6.71q)4]1/4 where q = 0.073(k/keq), to such numerical results. See Fig. 6 for these results. The slope of the BBKS transfer function is d ln T 2.34q 1 3.89q + 2(16.1q)2 + 3(5.64q)3 + 4(6.71q)4 = 1 . (27.62) d ln q (1 + 2.34q) ln(1 + 2.34q) − 4 1 + 3.89q + (16.1q)2 + (5.64q)3 + (6.71q)4 − For an accurate (linear perturbation theory) calculation of the true transfer function of the real universe, there are publicly available computer programs, such as CMBFAST, CAMB65, and CLASS66; you give your favorite values for the cosmological parameters as input. They represent the current state of the art. The exact result can be given in form of the transfer

65https://camb.info/ 66http://class-code.net/ 27 APPROXIMATE TREATMENT OF THE SMALLER SCALES 104

1 1

1 1 1 1 11 1 1 1 1 11 1

Figure 7: Left: Transfer function T (k) calculated with CAMB (blue curve) for adiabatic primordial fluctuations in the flat ΛCDM model with ωb = 0.023, ωc = 0.124, h = 0.7 (so that Ωm = 0.3), and massless neutrinos (a neutrino mass 0.06 eV for one neutrino species changes the transfer function by less than the width of the curve). The black curve is the BBKS transfer function (27.61), the black dotted lines and the green vertical line are as in Fig. 6. The main difference from BBKS is due to baryons. Right: The ratio (blue) of the T (k) from CAMB to the BBKS transfer function.

function T (k) we defined above. We show in Fig. 7 a transfer function calculated with CAMB. The effect of baryon acoustic oscillations (i.e., the oscillations of δbγ before decoupling, which leave a trace in δb) shows up as a small-amplitude wavy pattern in the k > keq part of the transfer function, since different modes k were at a different phase of the oscillation when that ended around tdec. Here we studied only the adiabatic mode. The logarithmic part is due to the growth of the CDM perturbation while the universe is still radiation dominated, i.e., δm was growing due to the potential Φ due to the radiation perturbation. In the isocurvature mode this effect will be missing as the potential (now due to δm) will only appear once the universe becomes matter dominated. Should add here also the calculation of the isocurvature mode. This should be much easier than the adiabatic mode. Is it too much for a homework problem? Exercise: Find the

isocurvature mode transfer function TΦSm for small scales. 28 ACOUSTIC OSCILLATIONS OF THE BARYON-PHOTON FLUID 105

28 Acoustic Oscillations of the Baryon-Photon Fluid

In the preceding discussion of subhorizon evolution of small scales (those that enter during radiation domination) our main focus was on the CDM, and we ignored baryons, so that the effect on the CDM of the baryon-photon fluid oscillations before photon decoupling was obtained in the approximation where we had just the photon fluid oscillating. Let us discuss the baryon-photon fluid without ignoring baryons. Now our main interest is in what happens to photons until photons decouple from baryons, since this is essential for the CMB. Thus we now care about the effect of the baryons on the photons, but not so much about what happens to the baryons, since after decoupling the baryons will anyway fall into the CDM potential wells, so that δ δ . b → c From (21.5) we have the fluid equations for baryons and photons,

δ′ = kv + 3Ψ′ (28.1) b − b 1 vb′ = vb + kΦ+ (vγ vb) (28.2) −H Rηcoll − 4 δ′ = kv + 4Ψ′ (28.3) γ − 3 γ 1 1 1 vγ′ = 4 kδγ 6 kΠγ + kΦ+ (vb vγ ) , (28.4) − ηcoll − where 3ρb R R′ = R (28.5) ≡ 4ργ ⇒ H and 1 ηcoll = (28.6) aneσT is the mean (conformal) time between collisions with electrons for a photon. We work in the tight coupling limit: η 1, which will keep v v and Π small. We coll ≪H− γ − b γ ignore the latter, but keep the former, but only to lowest order in the tight coupling limit. This allows us to write (28.2) as

v = v Rη v′ + v kΦ v Rη v′ + v kΦ (28.7) b γ − coll b H b − ≈ γ − coll γ H γ − and (28.4) becomes 1 v′ kδ + kΦ R v′ + v kΦ (28.8) γ ≈ 4 γ − γ H γ − or 1 [(1 + R)vγ ]′ = k 4 δγ +(1+ R)Φ . (28.9) From here on, we write the above tight-coupling approximations as equalities. The tight-coupling approximation fails when photons decouple at tdec. Multiplying (28.3) by 1 + R and derivating gives

R′ 2 2 4 2 4R′ δ′′ + δ′ + c k δ = k Φ+4Ψ′′ + Ψ′ , (28.10) γ 1+ R γ s γ −3 1+ R where 1 c2 = (28.11) s 3(1 + R) is the speed of sound in the baryon-photon ﬂuid. Deﬁning the temperature perturbation

Θ 1 δ , (28.12) ≡ 4 γ 28 ACOUSTIC OSCILLATIONS OF THE BARYON-PHOTON FLUID 106

(28.10) becomes67

R′ 2 2 Θ′′ + Θ′ + c k Θ = F (η) , (28.14) 1+ R s ~k where 1 2 R′ F (η) k Φ+Ψ′′ + Ψ′ . (28.15) ~k ≡ 3 1+ R Equation (28.14) has the form of harmonic oscillator with a friction term due to change of

R and a forcing term F~k(η) due to the potentials. Without the friction and forcing terms, and ignoring the time dependence of c we would get sinusoidal oscillations Θ exp(ikc η), giving s ~k ∝ s an oscillation period 2π/(kcs) > 1/k, i.e., larger than the time of horizon entry. The friction term damps the amplitude of the oscillations. The forcing term comes from the gravitational potentials to which all forms of matter (neutrinos, CDM, baryons, photons) contribute. After neutrino decoupling the neutrino density becomes practically homogeneous (for as long as neutrinos remain relativistic), so their contribution is negligible. When the universe becomes matter dominated (and actually already a bit earlier) CDM dominates the Φ part, but since the CDM perturbation grows slowly compared to the time scale of the baryon-photon oscillation, once the perturbation is inside the horizon, baryons and photons remain important for Ψ′ and Ψ′′. 3 Approximating vb vγ, we have δb′ = 4 δγ′ , so if initially we have the adiabatic relation 3 ≈ δb = 4 δγ , this relation is maintained in the tight-coupling approximation.

28.1 Inside the horizon Inside the horizon, the solution of (28.14) will be oscillating. When we are well inside the horizon, k , the oscillation will be rapid compared to the change of the background quantities. Let ≫H us ﬁrst ignore the forcing term F~k(η). Write

Θ = AeiB , (28.16) where A and B are real, A(η) representing the slowly changing amplitude and B(η) giving the rapid oscillation. The real and imaginary parts of the homogeneous (no F~k) version of (28.14) become

2 2 A′′ R′ A′ k (B′) = − A − 1+ R A 3(1 + R) A 1 B 1 R ′ = ′′ ′ . (28.17) A −2 B′ − 2 1+ R Neglecting the derivatives of A in the ﬁrst equation gives

η B′ = c k B(η)= k c (η)dη = kr (η)+ φ , (28.18) s ⇒ s s Z η where rs(η) is the sound horizon 0 csdη and φ is an integration constant. Inserting this in the second equation gives R A′ 1 R′ 1/4 = A (1 + R)− . (28.19) A −4 1+ R ⇒ ∝

67 Noting the symmetry between the derivatives of δγ and Ψ, this can be written, e.g., in the form

′ ′ 1 2 (1 + R)(Θ − Ψ) = − 3 k [Θ+(1+ R)Φ] . (28.13) 28 ACOUSTIC OSCILLATIONS OF THE BARYON-PHOTON FLUID 107

Thus

1/4 i(krs+φ) Θ~k(η) = A~k(1 + R)− e 1/4 1/4 = B~k(1 + R)− cos krs + C~k(1 + R)− sin krs . (28.20)

Ignoring Φ′ in (28.3) gives

3 3 1 5/4 ik i(krs+φ) v = Θ′ = A (1 + R)− R′ + e γ −k −k ~k −4 √ 3/4 3(1 + R) 3/4 i(krs+φ) i√3A (1 + R)− e . (28.21) ≈ − ~k Let is then consider the effect of the forcing term (28.15). From the constraint equation (10.19), 3 2 Ψ = H δ + 3H(1 + w)v (28.22) −2 k k and Φ is not very different from Ψ. The potentials receive contributions from CDM, baryons, and photons. For k the δ contribution dominates. Once the universe becomes matter- ≫ H dominated, CDM dominates the contribution to the Φ term in (28.15); but well inside the horizon, the derivatives of Ψ will be dominated by the rapid oscillation of the baryon-photon contributions to δ′, ρb ργ ργ δ′ δ′ + δ′ (1 + R) δ′ . (28.23) ≈ ρ b ρ γ ≈ ρ γ Divide (28.14) by 2 to make it dimensionless and facilitate comparison of magnitudes: H 2 2 2 R 1 1 k 1 k 2 R 1 − Θ′′ + − Θ′ + Θ = Φ+ − Ψ′′ + − Ψ′ . (28.24) H 1+ RH 3(1 + R) −3 H 1+ RH H H The time derivatives come mainly from the oscillation eikrs eikcsη, so ∝ ∼ 2 2 1 d k 2 d 2 k − c and − c . (28.25) H dη ∼ s H dη2 ∼ s H H Thus both the Φ and Ψ contribution have the same power of k/ ; but the Φ term is more ′′ H important by the factor (ρc/ρ)δc is larger than (1 + R)(ργ /ρ)δγ . This factor also compensates the ( /k)2 suppression of the potentials compared to the Θ terms in (28.24). Thus for the end H stages of the acoustic oscillation, when CDM perturbations have become dominant, but photons have not yet decoupled, we ignore the less important Ψ′ and Ψ′′ contributions to the forcing, and (28.14) becomes 2 R′ k Θ′′ + Θ′ + [Θ + (1 + R)Φ] = 0. (28.26) 1+ R 3(1 + R) We can likewise ignore derivatives of Φ or (1 + R)Φ, so we can rewrite this as 2 R′ k [Θ + (1 + R)Φ]′′ + [Θ + (1 + R)Φ]′ + [Θ + (1 + R)Φ] = 0, (28.27) 1+ R 3(1 + R) which is the equation we already solved, except now for Θ + (1 + R)Φ instead of Θ. Thus the solution is 1/4 i(krs+φ) Θ (η) = (1 + R)Φ + A (1 + R)− e , (28.28) ~k − ~k ~k where A~k and φ are constants. The oscillation ends when photons decouple at tdec. What happens in detail to the photons is discussed in CMB Physics; but from the point of view of considering the possible effect of the photons on matter perturbations, we can summarize it by saying that the photon fluid becomes homogeneous but anisotropic. 29 MODIFIED GRAVITY 108

29 Modiﬁed Gravity

The greatest mystery in cosmology is the cause of the acceleration of the expansion of the universe. The two main classes of proposed explanations are dark energy and modified gravity.68 The dividing line between these two classes of explanations is not sharp; there are suggested models that could be interpreted to belong into either one or as a mixture of both; but in simple terms a dark energy model is one where GR is valid but there is a new energy component with an equation of state with negative pressure (a total p/ρ < 1/3 is needed for acceleration) at − least since z 1; whereas a modified gravity model can not be expressed this way. ∼ Both a modified gravity explanation and a dark energy explanation should modify the expansion law a(t) to get this acceleration (ä> 0 since z 1). At the background level modified ∼ gravity and dark energy cannot be distinguished from each other observationally, since we have only one function a(t) to explain, and in principle any expansion history a(t) can be explained with a suitable behavior of the dark energy equation of state wde(t). The two classes of explanations are distinguished by their effect on perturbations; since in a dark energy model wde(t) determines also the perturbations according to the GR-based perturbation equations we have discussed; whereas in a modified gravity model the perturbations will behave differently. If there is no effect on perturbations then the two explanations cannot be distinguished: if gravity is modified by adding the cosmological constant Λ to the Einstein equation, the effect is the same as introducing dark energy with w 1, i.e., vacuum energy. de ≡− A conservative approach to modifying gravity (general relativity) is to assume that the nature of gravity is still curvature of spacetime, which can be expressed with a metric; and that the equivalence principle is still valid, i.e., freely falling test particles follow geodesics and energy-momentum is locally “covariantly conserved”, i.e., the equation µν T ;ν = 0 (29.1) is still valid. The same arguments as in standard GR lead then to the assumption that at large scales and early times the metric is the FRW metric with perturbations. From (29.1) then follows that our fluid perturbation equations remain valid. With these assumptions the modification of gravity is a modification of the Einstein equation Gµν = 8πGT µν . (29.2) At the background level this modification will lead to a modification of the Friedmann equations, which should lead to the observed acceleration. At the linear perturbation level, we got from the Einstein equation four equations for scalar perturbations; but these are not independent of the fluid equations, so that if we use the fluid equations we need to supplement them with just two of the Einstein equations for the metric perturbations. We pick the two constraint equations k2Ψ = 4πGa2ρδC − Ψ Φ = 8πGpΠ . (29.3) − Based on observations the acceleration issue becames relevant only when the universe is already matter dominated. Thus we can ignore the effects of radiation and consider the cosmic fluid to be matter, so that p = 0. The second constraint equation then becomes Ψ = Φ, or Ψ/Φ = 1.

68It has also been suggested that this could be a backreaction effect, i.e., due to nonlinear effects of inhomogeneities in standard GR; or an effect of inhomogeneities on observations that has caused them to be misinter- preted. Both of these explanations rely on effects that are difficult to calculate but are usually expected to be small. For them to explain the (real or apparent) acceleration, the effect would have to be large enough to explain away ΩΛ ∼ 0.7. As observations become more accurate while remaining reasonably consistent with ΛCDM, it becomes less plausible that such effects could conspire to mimic ΛCDM to such accuracy; whereas one can think of modified gravity or dark energy models that are close to, but not exactly, ΛCDM. 29 MODIFIED GRAVITY 109

Figure 8: The 68% and 95% confidence constraints on modified gravity parameters Σ0 and µ0 from DES first-year data (blue), other cosmological data (brown), and their combination (red), assuming a ΛCDM background. GR corresponds to Σ0 = µ0 = 0 (the intersection of the dashed lines). Note that the two datasets constrain the ΛCDM parameters differently and therefore the red region differs significantly from the intersection of the blue and brown regions. From [14].

There are many suggested models for modified gravity and the power of observational data to distinguish between them will be limited, especially in the near term. It makes sense to approach the application of observations to theory in a phenomenological way: to search if there is evidence for modification of (29.3) and how should they be modified. One can accomplish this by introducing modified constraint equations [13, 10]

k2Ψ = 4πGQ(a, k)a2ρδC − Ψ = η(a, k)Φ , (29.4) where Q(a, k) and η(a, k) are two time- and scale-dependent functions to be determined from observations. These functions can be derived from different modified gravity theories and models and compared to observations. The simplest approach is to use ΛCDM as the background model and just modify the perturbations this way. In GR, Q η 1. ≡ ≡ Different kinds of observations are more sensitive to Φ (growth of structure and peculiar velocities) or Φ + Ψ (ISW and weak lensing), i.e., to the combinations Q/η or Q/η + Q, so we define Q µ ≡ η Σ 1 (Q + Q/η) = 1 (µη + µ) (29.5) ≡ 2 2 so that we can write (29.4) as

k2Φ = 4πGa2µ(a, k)ρδC − k2(Φ+Ψ) = 8πGa2Σ(a, k)ρδC , (29.6) − and one can use the pair of functions (µ, Σ) instead of (Q, η). Of course the data will never be suﬃcient to determine these two functions exactly, so one must parametrize them with a small number of parameters to be ﬁt to observations. The simplest 29 MODIFIED GRAVITY 110 parametrization is to assume them to be constants over both the observationally relevant times and scales. The Dark Energy Survey (DES) collaboration used the parametrization

ΩΛ(z) ΩΛ(z) µ(z) = 1+ µ0 , Σ(z) = 1+Σ0 (29.7) ΩΛ ΩΛ

(with ΛCDM as the background universe, giving the ΩΛ(z)). The motivation was to let the modification become larger closer to present times when a modification is needed to account for the acceleration. Using the DES first-year data combined with other cosmological date they obtained the observational constraints[14]

Σ = 0.06 0.08 0 ± µ = 0.1 0.5 (29.8) 0 − ± (see Fig. 8), consistent with GR. A GENERAL PERTURBATION 111

A General Perturbation

From Eq. (3.8) we have that the general perturbed metric (around the ﬂat Friedmann model) is

ds2 = a2(η) (1 + 2A)dη2 2B dηdxi + [(1 2D)δ + 2E ] dxidxj . (A.1) − − i − ij ij The Christoﬀel symbols are

0 Γ00 = + A′ (A.2) 0 H Γ0i = Bi + A,i 0 −H 1 Γ = [(1 2A 2D)δ + 2E ]+ (B + B ) δ D′ + E′ ij H − − ij ij 2 i,j j,i − ij ij i Γ00 = Bi Bi′ + A,i i −H −1 Γ = δ + (B B ) D′δ + E′ 0j H ij 2 j,i − i,j − ij ij Γi = δ B δi D δi D + δ D + E + E E . jk H jk i − j ,k − k ,j jk ,i ij,k ik,j − jk,i and we have the Christoﬀel sums

µ Γ = 4 + A′ 3D′ (A.3) 0µ H − Γµ = A 3D . iµ ,i − ,i Note that the Christoﬀel sums contain only scalar perturbations. Thus for vector and tensor perturbations, these sums contain only the background value Γµ = 4 . 0µ H The Einstein tensor is

0 2 2 2 2 2 G0 = 3a− + a− 2 D + 6 D′ + 6 A 2 Bi,i Eik,ik (A.4) 0 − 2 H − ∇ 1 H H − H − G = a− 2D′ 2 A (B B ) E′ i − ,i − H ,i − 2 i,kk − k,ik − ik,k i 2 1 2 G = a− 2D′ + 2 A + (B B ) + 2 ′B 2 B + E′ 0 ,i H ,i 2 i,kk − k,ik H i − H i ik,k i 2 2 G = a− 2 ′ δ j − H −H ij 2 2 2 i +a− 2D′′ (D A)+ (2A′ + 4D′) + (4 ′ + 2 )A B′ 2 B E δ −∇ − H H H − k,k − H k,k − kl,kl j 2 1 2 +a− (D A) + (B′ + B′ )+ (B + B )+ E′′ E + E + E + 2 E′ − ,ij 2 i,j j,i H i,j j,i ij −∇ ij ik,jk jk,ik H ij A GENERAL PERTURBATION 112

For scalar perturbations, the perturbation in the Einstein tensor becomes

0 2 2 2 2 δG0 = a− 2 ψ + 6 D′ + 6 A + 2 B (A.5) 0 2 − ∇ H H H∇ δG = a− 2ψ′ 2 A i − ,i − H ,i i 2 2 δG = a− 2ψ′ + 2 A 2 ′B + 2 B 0 ,i H ,i − H ,i H ,i i 2 2 2 2 2 i δG = a− 2D′′ (D A)+ (2A′ + 4D′) + (4 ′ + 2 )A + B′ + 2 B δ j −∇ − H H H ∇ H∇ j 2 1 2 2 1 2 2 2 i +a− ( E) E′′ E′ δ − 3 ∇ ∇ − 3 ∇ − 3 H∇ j 2 1 2 +a− D A B′ 2 B + E′′ + E + 2 E′ . − − − H 3 ∇ H ,ij The trace of the space part is

i 2 2 2 2 2 2 δG = a− 6D′′ 2 ψ + 2 A + 3 (2A′ + 4D′) + (12 ′ + 6 )A + 2 B′ + 4 B . i − ∇ ∇ H H H ∇ H∇ (A.6) In Fourier space these read as

0 2 2 2 δG0 = a− 2k ψ + 6 D′ + 6 A 2 kB (A.7) 0 2 H H − H δG = a− 2ik ψ′ 2i k A i − i − H i i 2 2 δG = a− 2ik ψ′ + 2i k A 2i ′k B + 2i k B 0 i H i − H i H i i 2 2 2 i δG = a− 2D′′ + k (D A)+ (2A′ + 4D′) + (4 ′ + 2 )A kB′ 2 kB δ j − H H H − − H j 2 1 2 1 2 i +a− k E + E′′ + E′ δ − 3 3 3 H j 2 1 1 1 k k a− D A B′ 2H B E + (E′′ + 2 E′) − i j − − k − k − 3 k2 H i 2 2 2 2 δG = a− 6D′′ + 2k ψ 2k A + (6A′ + 12D′) + (12 ′ + 6 )A 2kB′ 4 kB . i − H H H − − H A GENERAL PERTURBATION 113

The general perturbed energy tensor is

0 T0 = ρ¯ δρ (A.8) 0 − − Ti = (¯ρ +p ¯)(vi Bi) i − T0 = (¯ρ +p ¯)vi i − i i Tj =pδ ¯ j + δpδj +p ¯Πij

The energy continuity equations

T µ T µ +Γµ T α Γα T µ = 0 (A.9) ν;µ ≡ ν,µ αν ν − νµ α become the background equation ρ¯′ + 3 (¯ρ +p ¯) = 0 (A.10) H and the ﬂuid perturbation equations

δρ′ = 3 (δρ + δp)+(¯ρ +p ¯)(3D′ ~v) (A.11) − H − ∇ · (¯ρ +p ¯)(v B )′ = (¯ρ +p ¯)′(v B ) 4 (¯ρ +p ¯)(v B ) i − i − i − i − H i − i δp p¯Π (¯ρ +p ¯)A − ,i − ij,j − ,i For scalar perturbations, the ﬂuid perturbation equations become

2 δρ′ = 3 (δρ + δp)+(¯ρ +p ¯)(3D′ + v) (A.12) − H ∇ (¯ρ +p ¯)(v B)′ = (¯ρ +p ¯)′(v B) 4 (¯ρ +p ¯)(v B) − − − − H − +δp + 2 p¯ 2Π+(¯ρ +p ¯)A . 3 ∇ Using δ δρ/ρ¯ and background relations, these can be written ≡

2 δp δ′ = (1+ w) v + 3D′ + 3 wρ (A.13) ∇ H − ρ¯ 2 δp 2 w 2 (v B)′ = (1 3c )(v B)+ + Π+ A . − −H − s − ρ¯ +p ¯ 3 1+ w ∇

The factor (1 3c2) can also be written as (1 3w) + w /(1 + w). H − s − H ′ In Fourier space these are

δρ′ = 3 (δρ + δp)+(¯ρ +p ¯)(3D′ kv) (A.14) − H − (¯ρ +p ¯)(v B)′ = (¯ρ +p ¯)′(v B) 4 (¯ρ +p ¯)(v B) − − − − H − +kδp 2 kp¯Π+ k(¯ρ +p ¯)A − 3 δp δ′ = (1+ w)( kv + 3D′) + 3 wδ − H − ρ¯ w′ kδp 2 w (v B)′ = (1 3w)(v B) (v B)+ kΠ+ kA . − −H − − − 1+ w − ρ¯ +p ¯ − 3 1+ w B RULES TO CONVERT BETWEEN DIFFERENT TIME COORDINATES 114

B Rules to Convert between Diﬀerent Time Coordinates

The cosmic time t and the conformal time η are related by d 1 d 1 dt = adη = ( )˙ = ( )′ or ( )′ = a( )˙ (B.1) ⇒ dt a dη ⇒ a For any two functions of time, f and g, we have

df f˙ f = = ′ (B.2) dg g˙ g′ The ordinary Hubble parameter H and the conformal (or comoving) Hubble parameter are related by a = aH =a ˙ = ′ (B.3) H a Sometimes it’s convenient to use the scale factor a or its logarithm ln a as the time coordinate. We have the following relations between the derivatives wrt these time coordinates:

1 1 df df − f ′ = H− f˙ = a = (B.4) H da d ln a 2 2 2 2 1 2 d f 1 3w df d 1 + 3w df − f ′′ = H− f¨+ H− f˙ = a + − a = f (B.5) H da2 2 da d ln a − 2 d ln a In many equations the combination

2 1 2 1 − f ′′ + 2 − f ′ = H− f¨+ 3H− f˙ (B.6) H H appears. We also have that 2 2 ′ = aa¨ = a H˙ + H . (B.7) H C EQUATIONS 115

C Equations

All ﬂuid quantities below are in the Newtonian gauge. All equations are in Fourier space. The Einstein equations, from Sec. 10, are

2 1 1 k 1 − Ψ′ +Φ+ Ψ = δ (C.1) H 3 − 2 H 1 3 − Ψ′ +Φ = (1 + w)Hv (C.2) H 2 k 2 2 1 1 k 3 δp − Ψ′′ + − Φ′ + 2Ψ′ 3wΦ (Φ Ψ) = (C.3) H H − − 3 − 2 ρ H k 2 (Ψ Φ) = 3wΠ , (C.4) − H The ﬂuid equations for the “real universe”, from Secs. 21 and 22, are

1 k 1 − δ′ + v 3 − Ψ′ = 0 H c c − H H 1 k − v′ + v Φ = 0 H c c − H 1 k 1 − δ′ + v 3 − Ψ′ = 0 H b b − H H 1 k 4ργ − v′ + v Φ = an σ (v v ) H b b − e T 3ρ γ − b H b 1 4 k 1 − δ′ + v 4 − Ψ′ = 0 H γ 3 γ − H H 1 1 k k 1 k − v′ δ Φ = Π + an σ (v v ) H γ − 4 γ − −6 γ e T b − γ H H H 1 4 k 1 − δ′ + v 4 − Ψ′ = 0 H ν 3 ν − H H 1 1 k k 1 k − v′ δ Φ = Π , (C.5) H ν − 4 ν − −6 ν H H H and, from Sec. 18,

1 k δi δj − S′ = (v v ) where S = . (C.6) H ij − i − j ij 1+ w − 1+ w H i j From Sec. 16 we have the relation between the Bardeen potentials and the comoving curvature perturbation, 2 1 5 + 3w 2 − Ψ′ + Ψ = (1 + w) + (Ψ Φ) , (C.7) 3H 3 − R 3 − and the evolution equation

2 1 2 k 2 1 2 − ′ = c Ψ+ (Ψ Φ) + 3c . (C.8) H R 3(1 + w) s 3 − sS H REFERENCES 116

References

[1] V.F. Mukhanov, H.A. Feldman, and R.H. Brandenberger, Theory of Cosmological Pertur- bations, Phys. Rep. 215, 213 (1992)

[2] J. Bardeen, Gauge-invariant cosmological perturbations, Phys. Rev. D 22, 1789 (1980)

[3] A.R. Liddle and D.H. Lyth: Cosmological Inﬂation and Large-Scale Structure (Cambridge University Press 2000)

[4] C.-P. Ma and E. Bertschinger, Cosmological perturbation theory in the synchronous and conformal Newtonian gauges, ApJ 455, 7 (1995)

[5] S. Dodelson: Modern Cosmology (Academic Press 2003)

[6] M. Bucher, K. Moodley, and N. Turok, General primordial cosmic perturbation, Phys. Rev. D 62, 083508 (2000), astro-ph/9904231; Phys. Rev. Lett. 87, 191301 (2001), astro- ph/0012141

[7] D.H. Lyth and A.R. Liddle: The Primordial Density Perturbation (Cambridge University Press 2009), errata at http://astronomy.sussex.ac.uk/~andrewl/pdp.html

[8] H. Kodama and M. Sasaki, Evolution of isocurvature perturbations II. Radiation-dust universe, Int. J. Mod. Phys. bf A2, 491 (1987)

[9] A.Lewis, CAMB Notes (Dated: April 2, 2014), http://cosmologist.info/notes/CAMB.ps.gz

[10] L. Amendola et al., Cosmology and fundamental physics with the Euclid satellite, Living Rev. Relativ. 21, 2 (2018)

[11] Planck Collaboration, Planck 2015 results. XVII. Constraints on primordial non- Gaussianity, arXiv:1502.01592, Astronomy & Astrophysics 594, A16 (2016)

[12] J.M. Bardeen, J.R. Bond, N. Kaiser, A.S. Szalay, The statistics of peaks in Gaussian ran- domn ﬁelds, Astrophys. J. 304, 15 (1986), Appendix G

[13] L. Pogosian, A. Silvestri, K. Koyama, and G.-B. Zhao, How to optimally parametrize devi- ations from general relativity in the evolution of cosmological perturbations, Phys. Rev. D 81, 104023 (2010)

[14] DES Collaboration (T.M.C. Abbott et al.), Dark Energy Survey year 1 results: Constraints on extended cosmological models from galaxy clustering and weak lensing, Phys. Rev. D 99, 123505 (2019), arXiv:1810.02499