Theoretical Cosmology and Astrophysics Lecture notes - Chapter 7 A. Refregier April 24, 2017 7 Cosmological Perturbations 1 In this chapter, we will consider perturbations to the FRW smooth model of the universe. For simplicity, we will consider the flat FRW spacetime. 7.1 Metric perturbations Let us consider perturbations h µν to the FRW metric defined through where ḡ µν (x) = g µν = ḡ µν + h µν, (1) 1 0 0 0 0 a 2 (t) 0 0 0 0 a 2 (t) 0 0 0 0 a 2 (t), (2) denotes the flat, unperturbed FRW metric. We further require that h µν ḡ µν, i.e. that the perturbations are small. In order to decompose the metric perturbation h µν, recall that we can decompose any vector field v( x) in 3-dimensional Euclidean space into a parallel and perpendicular part as v( x) = v ( x) + v ( x), (3) where v = v = 0. We can further write these two parts as v = v and v = A, where v is a scalar and A is a vector. We can check that these indeed satisfy the conditions specified above: v = ( v) = 0 and ( A) = 0. Note that the fields v and A are not uniquely defined by these relations. We can indeed add the quantities v v + const. (4) A A + f, (5) Notes taken by A. Nicola. 1 See chapter 5 in Dodelson, S., Modern Cosmology, 2003, Academic Press and chapter 5 in Weinberg, S., Cosmology, 2008, Oxford University Press. 1
where f is a scalar, without affecting either v or v. This property is called gauge freedom. Similarily we can decompose the metric perturbation h µν (x) into h 00 = E, ] F h i0 = a x i + G i, (6) h ij = a 2 Aδ ij + 2 B x i x j + C i x j + C ] j x i + D ij, where A = A(x), B = B(x), etc. and A, B, E, F are scalars C i, G i are divergenceless vectors D ij is a traceless, symmetric and divergenceless tensor, and the perturbations satisfy the conditions C i x i = G i x i = 0, D ij x i = 0, D ii = 0, D ij = D ji. (7) Since h µν is a symmetric 4 4 tensor, it has 10 independent degrees of freedom. It is easy to check that the new perturbation fields amount to the same number of degrees of freedom: no. of components divergenceless A, B, E, F 4 1 = 4 C i, G i 2 3 = 6 2 ( 1) D ij 9-3 - 1 = 5-3 15-5 = 10 The decomposition theorem states that scalar, vector and tensor perturbation modes do not couple to first order, i.e. they evolve independently. This means that Einstein s field equations can be solved for each perturbation mode separately. It can be shown that the amplitude of the vector modes decays as a function of time. The tensor modes correspond to gravitational waves, which are only important for CMB polarisation. Here, we will only consider scalar modes. 7.2 Gauge transformations Let us consider the spacetime coordinate transformation (see Fig.1) x µ x µ = x µ + ɛ µ (x), (8) where ɛ µ (x) is small, just as h µν is small compared to ḡ µν. Under such a coordinate transformation, the metric transforms as g µν(x ) = g λκ (x) xλ x κ x µ x ν. (9) 2
Inserting Eq. 8 into Eq. 9 leads to ) ) g µν(x ) = (δ λµ (δ ɛλ κν x µ ɛκ x ν g λκ (x). (10) Writing g λκ (x) = ḡ λκ (x) + h λκ (x) and expanding to first order finally leads to g µν(x ) ḡ µν (x) + h µν (x) ɛλ x µ ḡλν(x) ɛκ x ν ḡµκ(x). (11) M 0 x µ x 0 µ Figure 1: Illustration of coordinate transformations. Such a coordinate transformation affects the coordinates and unperturbed fields as well as the perturbations to the fields. In order to derive the transformations to the perturbed fields which leave the physics invariant we consider gauge transformations. Under a gauge transformation, the metric transforms as g µν (x) g µν(x). (12) From the generic metric transformation law given by Eq. 9 we get to first order in ɛ(x) and h µν (x) g µν(x ) = g µν(x + ɛ) = g µν(x) + g µν(x ) x λ ɛ λ. (13) Solving for g µν(x) and expressing g µν(x ) in terms of g µν (x) = ḡ µν (x) + h µν (x) through Eq. 9, gives us µν(x ) g µν(x) = g µν(x ) g x λ ɛ λ = ḡ µν (x) + h µν (x) ɛλ x µ ḡλν(x) ɛκ x ν ḡµκ(x) ḡ µν(x) x λ ɛ λ, (14) Letting g µν(x) = ḡ µν (x) + h µν(x) and h µν(x) = h µν (x) + h µν (x) finally gives us h µν (x) = h µν(x) h µν (x) = ɛλ x µ ḡλν(x) ɛκ x ν ḡµκ(x) ḡ µν(x) x λ ɛ λ. (15) We thus see that gauge transformations correspond to changes in the metric perturbations. 3
The FRW metric is given by 1 0 0 0 ḡ µν (x) = Therefore we obtain h 00 = 2 ɛ0 t, 0 a 2 (t) 0 0 0 0 a 2 (t) 0 0 0 0 a 2 (t). (16) h i0 = ɛi t a2 + ɛ0 x i, (17) h ij = ɛi x j a2 ɛj x i a2 2a da dt δ ijɛ 0. We can simplify these equations by noting that ɛ µ = g µν ɛ ν = (ḡ µν + h µν )ɛ ν ḡ µν ɛ ν. Therefore we can use the smooth FRW metric to raise and lower indices on ɛ ν, i.e. ɛ 0 = ɛ 0 and ɛ i = a 2 ɛ i, and we obtain the transformations of the metric perturbations under gauge transformations: h 00 = 2 ɛ 0 t, h i0 = ɛ i t + 2 a h ij = ɛ i x j ɛ j x i + 2ada dt δ ijɛ 0. da dt ɛ i ɛ 0 x i, (18) In order to study how the scalar-vector-tensor components of the metric transform under a gauge transformation, we decompose the spatial part of the 4- vector ɛ µ into the gradient of a scalar plus a divergenceless vector ɛ i = ɛs x i + ɛv i, with ɛv i = 0. (19) xi Inserting this decomposition into Eq. 18, using Eq. 6 and considering only scalar modes gives us A = 2 da a dt ɛ 0, B = 2 a 2 ɛs, E = 2 dɛ 0 dt, (20) F = 1 ( ɛ 0 dɛs a dt + 2 ) da a dt ɛs. Similar identities hold for the vector and tensor perturbations. 7.2.1 The choice of gauge For the Newtonian gauge we choose ɛ S such that B = 0 and ɛ 0 such that F = 0. We are therefore left with the scalar perturbations A and E, which are relabelled 4
as E = 2Ψ, (21) A = 2Φ. (22) The perturbed metric in Newtonian gauge then becomes, keeping only scalar perturbations g 00 = 1 2Ψ, g 0i = 0, g ij = a 2 δ ij 1 + 2Φ]. (23) For the synchronous gauge we choose ɛ 0 such that E = 0 and ɛ S such that F = 0. The perturbed metric in synchronous gauge then becomes, keeping only scalar perturbations ] g 00 = 1, g 0i = 0, g ij = a 2 (1 + A)δ ij + 2 B x i x j. (24) There are other possible gauge choices. An example for a further gauge choice would be the co-moving gauge. It is also possible to perform cosmological perturbation theory solely in terms of gauge-invariant variables; this is the so-called gauge-invariant perturbation theory. In the following, we will choose the Newtonian gauge, which has the advantage that it is the easiest to relate to the Newtonian limit. 7.2.2 Geometrical interpretation of gauge transformations 2 To define perturbations, we need to compare two manifolds to each other (see Fig. 2): M: perturbed spacetime manifold with metric g µν M: background (unperturbed) spacetime manifold with metric ḡ µν. For this purpose, we need a mapping (diffeomorphism) φ between M and M. Then we can define the metric perturbation h µν as h µν = (φ 1 g) µν ḡ µν, (25) where everything is defined on M. Gauge freedom arises because there are many permissible (i.e. when perturbations are small) mappings φ betwen M and M. M M ḡ µ g µ 1 Figure 2: Illustration of mapping φ between manifolds M and M. For example, consider a mapping Λ ɛ of M onto itself, induced by a vector field ɛ µ on M, i.e. x µ x µ + ɛ µ (x) in a given coordinate system; as illustrated in 2 See chapter 7.1 in Carroll, S. M., Spacetime and Geometry: An Introduction to General Relativity, 2004, Addison Wesley. 5
Fig. 3. Then (φ Λ ɛ ) 1 = Λ 1 ɛ φ 1 is a new mapping between M to M, which induces new metric perturbations h (ɛ) µν = (φ Λ ɛ ) 1 g] µν ḡ µν = Λ 1 ɛ (φ 1 g)] µν ḡ µν. (26) All possible infinitesimal Λ ɛ s correspond to the gauge transformations we considered above (see Eq. 12). M M 1 Figure 3: Illustration of gauge freedom. 7.2.3 Number of degrees of freedom The metric g µν is a symmetric rank 2 tensor, i.e. it has 10 independent components. However, we saw above that there is gauge freedom with gauge transformations generated by vector fields ɛ µ which have 4 components. Therefore the number of physical degrees of freedom in GR is 10 4 = 6. 7.3 Einstein equations Recall that in Newtonian gauge, the flat perturbed FRW metric, keeping only scalar perturbations, is given by g 00 = 1 2Ψ, g 0i = 0, g ij = a 2 δ ij 1 + 2Φ]. (27) Let us compute the Christoffel symbols for this metric. In general, the Christoffel symbols are defined as Γ µ αβ = gµν gαν 2 x β + g βγ x α g ] αβ x ν. (28) The Christoffel symbols for the perturbed FRW metric in Newtonian gauge are therefore given by Γ 0 00 = Ψ,0, Γ 0 0i = Γ 0 i0 = Ψ,i, Γ 0 ij = δ ij a 2 H + 2H(Φ Ψ) Φ,0 ], Γ i 00 = 1 a 2 Ψ,i, (29) Γ i 0j = Γ i j0 = δ ij (H + Φ,0 ), Γ i jk = δ ij Φ,k + δ ik Φ,j δ jk Φ,i, 6
where H = 1 da a dt denotes the Hubble parameter. We can check that we recover the smooth universe results when we only look at the 0 th order terms in the Christoffel symbols: Γ 0 00 = Γ 0 0i = Γ 0 i0 = 0, Γ 0 ij = δ ij a 2 H = δ ij a da dt, Γ i 00 = Γ i jk = 0, (30) Γ i 0j = Γ i 1 da j0 = δ ij a dt. From the Christoffel symbols we can calculate the Ricci tensor for the perturbed FRW metric. The Ricci tensor is given by R µν = Γ α µν,α Γ α µα,ν + Γ α βαγ β µν Γ α βνγ β µα. (31) For the perturbed FRW metric we therefore obtain R 00 = 3 1 d 2 a a dt 2 + 1 a 2 Ψ,ii 3Φ,00 + 3H(Ψ,0 2Φ,0 ), R ij = δ ij ( 2a 2 H 2 + a d2 a dt 2 + a 2 Φ,00 Φ,ii ] (Φ,ij + Ψ ij ). We can again check the 0 th order terms R 00 = 3 1 d 2 a a dt 2, ] R ij = δ ij 2a 2 H 2 + a d2 a dt 2 = δ ij 2 ) (1 + 2Φ 2Ψ) + a 2 H(6Φ,0 Ψ,0 ) (32) ( da dt ) ] 2 + a d2 a dt 2. (33) Note that we will not need R 0i, because the perturbed FRW metric is diagonal. The Ricci scalar is given by R = g µν R µν. (34) Inserting the explicit components of both the metric and the Ricci tensor therefore leads to ( R = 6 H 2 + 1 d 2 ) a a dt 2 (1 2Ψ) 2 a 2 Ψ,ii +6Φ,00 6H(Ψ,0 4Φ,0 ) 4 a 2 Φ,ii. (35) We can again check that the 0 th order term agrees with that for the FRW metric ( R = 6 H 2 + 1 d 2 ) a a dt 2. (36) Using the Ricci tensor and the Ricci scalar we can finally compute the Einstein tensor, which is given by G µν = R µν 1 2 g µνr. (37) 7
It is more convenient to work with the Einstein tensor with one index raised, i.e. G µ ν = g µρ G ρν = R µ ν 1 2 gµ νr. (38) For the perturbed FRW metric we therefore obtain the time-time component of the Einstein tensor as G 0 1 d 2 a 0 = 3 a dt 2 1 ( ) ] 2 ( ) 2 da da a 2 + 3 6 1 ( ) 2 da 1 dt dt a dt Φ da,0 + 6 Ψ + 2 a dt a 2 Φ,ii. (39) We can again check the 0 th order component G 0 0 = ḡ 00 G 00 = 3 1 d 2 a a dt 2 1 a 2 The space-space component of the Einstein tensor is given by ( ) ] 2 da. (40) dt G i j = Aδ ij 1 a 2 (Φ,ij + Ψ,ij ), (41) where A contains almost a dozen terms which are all proportional to δ ij and therefore contribute only to the trace of G i j. Since we will only need to consider the longitudinal and traceless part of the Einstein tensor, we will not need to compute the explicit form of A. Einstein s field equations are then given by G µ ν = 8πGT µ ν. (42) Decomposing the metric into the background and the perturbations, we can write these equations as (Ḡµ ν + G µ ) ( ν = 8πG T µ ν + T µ ) ν. (43) In order to compute the equations governing the evolution of perturbations in the universe, we therefore need to compute the traceless part of T µ ν from the Boltzmann equations. 8