3 Relativistic Perturbation Theory

3.1 Metric Decomposition and Gauge Transformation

3.1.1 FRW Metric and its Perturbations We describe the background for a spatially homogeneous and isotropic universe with the FRW metric

ds2 = g dxµdxb = a2(⌘) d⌘2 + a2(⌘)¯g dx↵dx , (3.1) µ⌫ ↵ where a(⌘) is the scale factor and g¯↵ is the metric tensor for a three-space with a constant spatial curvature K = H2 (1 ⌦ ). We use the Greek indices ↵,, for 3D spatial components and µ, ⌫, for 4D spacetime compo- 0 tot ··· ··· nents, respectively. To describe the real (inhomogeneous) universe, we parametrize the perturbations to the homogeneous background metric as

g := a2 (1 + 2A) ,g:= a2B ,g:= a2 (¯g +2C ) , (3.2) 00 0↵ ↵ ↵ ↵ ↵ where 3-tensor A, B↵ and C↵ are perturbation variables and they are based on the 3-metric g¯↵. Due to the symmetry of the metric tensor, we have ten components, capturing the deviation from the background. The inverse metric tensor can µ⇢ µ be obtained by using g g⇢⌫ = ⌫ and expanding to the linear order as 1 1 1 g00 = ( 1+2A) ,g0↵ = B↵ ,g↵ = g¯↵ 2C↵ . (3.3) a2 a2 a2 ⇣ ⌘ For later convenience, we also introduce a time-like four-vector, describing the motion of an observer ( 1=u uµ): µ 1 1 u0 = (1 A) ,u↵ := U ↵ ,u= a (1 + A) , (3.4) a a 0 u = a (U B ):=av := a( v + v(v)) ,v:= U + , v(v) = U (v) B(v) , (3.5) ↵ ↵ ↵ ↵ ,↵ ↵ ↵ ↵ ↵ ↵ where U is again based on g¯↵.

3.1.2 Scalar-Vector-Tensor Decomposition Given the splitting of the spatial hypersurface and the symmetry associated with it, we decompose the perturbation variables (to all orders) as

(v) (v) (t) ↵ ,↵ (v)↵ A := ↵,B↵ := ,↵ + B↵ ,C↵ := 'g¯↵ + ,↵ + C(↵ ) + C↵ ,U:= U + U , (3.6) | | subject to the transverse and traceless conditions:

(v)↵ (v)↵ (v)↵ (t)↵ (t) (v)↵ B ↵ 0 ,C↵ 0 ,v↵ 0 ,C↵ 0 ,C↵ 0 ,U↵ =0, (3.7) | ⌘ | ⌘ | ⌘ ⌘ | ⌘ | where the vertical bar represents the covariant derivative with respect to the 3-metric g¯↵: ↵ ↵ ¯↵ ¯ X = X , + X ,X↵ = X↵, ↵X . (3.8) | | This simply implies that the scalar perturbations describe the longitudinal modes and the vector (v) and the tensor (t) perturbations describe the transverse modes. Furthermore, the tensor perturbation is traceless. The decomposed scalar perturbations can be obtained as

1 1 ↵ 1 1 1 ↵ ↵ = B ,= + R¯ 3 C C , (3.9) r ↵ 2 2 r r ↵ ↵ ✓ ◆ 1 ⇣ ⌘ 1 ↵ 1 1 1 ↵ ↵ ' = C + R¯ 3 C C , 3 ↵ 6 2 r r ↵ ↵ ✓ ◆ ⇣ ⌘ AST513 Theoretical Cosmology JAIYUL YOO where is the covariant derivative based on g¯ (i.e., vertical bar) and = ↵ is the Laplacian operator. The r↵ ↵ r r↵ presence of the Ricci scalar (R¯ =6K) for the three-space indicates that covariant derivatives are non-commutative. ¯ R↵ =2K g¯↵[g¯] . (3.10)

The decomposed vector and tensor components are computed in a similar manner as

1 (v) 1 (v) 1 1 B = B B ,C=2 + R¯ C C , (3.11) ↵ ↵ r↵ r ↵ 3 r ↵ r↵ r r ✓ ◆ 1h i (t) 1 1 1 1 1 C = C C g¯ g¯ + R¯ 3 C C ↵ ↵ 3 ↵ 2 r↵r 3 ↵ 2 r r ✓ ◆✓ ◆ 1 h i 1 1 2 + R¯ C C , r(↵ 3 r ) r) r r ✓ ◆ h i (v)↵ (v)↵ (t) (t)↵ and they satisfy the transverse condition B ↵ = C ↵ = C↵ =0and the traceless condition C↵ =0. | | | 3.1.3 Comparison in Notation Convention Bardeen convention: 1 A ↵,B(0) k,H ' + ,H , (3.12) ! ! L ! 3 T ! B(1)Q(1) B ,H(1)Q kC ,H(2)Q C . (3.13) ↵ ! ↵ T ↵ ! ↵ T ↵ ! ↵ Weinberg convention:

↵ , ' ,uav ' ,⇣' , (3.14) ! ! ! R! v ! 1 ⇧ 1 p p ⇧ ,⇡S := ,⇡V ⇧ ,⇡T ⇧(t) . (3.15) ! 3a2 ! a2 i ! 2a ↵ ij ! ↵ Dodelson convention: ↵ ,' ,ikvk2v ,v ikv . (3.16) ! ! ! ! 3.1.4 Gauge Transformation The general covariance of general relativity guarantees that any coordinate system can be used to describe the physics and it has to be independent of coordinate systems. This is known as the diffeomorphism symmetry in general relativity. However, when we split the metric into the background and the perturbations around it by choosing a coordinate system, we explicitly change the correspondence of the physical Universe to the background homogeneous and isotropic Universe. Hence, the metric perturbations transform non-trivially (or gauge transform), and the diffeomorphism invariance implies that the physics should be gauge-invariant. The gauge group of general relativity is the group of diffeomorphisms. A diffeomorphism corresponds to a differen- tiable coordinate transformation. The coordinate transformation on the manifold can be considered as one generated M by a smooth vector field ⇣µ. Given the vector field ⇣µ, consider the solution of the differential equation dµ() d = ⇣µ [⌫ ()] ,µ ( = 0) = xµ , = ⇣µ@ , (3.17) d P P P d µ P µ µ µ defines the parametrized integral curve x ()=P () with the tangent vector ⇣ (xP ) at P . Therefore, given the vector field ⇣µ on we can define an associated coordinate transformation on as xµ x˜µ = µ ( = 1) for any given P . M M P ! P P Assuming that ⇣µ is small one can use the perturbative expansion for the solution of equation to obtain

2 d 1 d 1 ⌫ x˜µ = µ ( = 1) = µ ( = 0) + µ + µ + = xµ + ⇣µ(x )+ ⇣µ ⇣⌫ + (⇣3)=e⇣ @⌫ xµ . P P P d P 2 d2 P ··· P P 2 ,⌫ O =0 =0 (3.18) This parametrization corresponds to the gauge-transformation with ⇣µ.

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In general, any gauge-transformation of tensor T for an inﬁnitesimal change ⇣ can be expressed in terms of the Lie derivative (valid to all orders of T) T := T˜ T = £ T+ (⇣2) , £ Aµ = Aµ ⇣⌫ ⇣µ A⌫ , £ T = T ⇣⇢ +T ⇣⇢ +T ⇣⇢ , (3.19) ⇣ ⇣ O ⇣ ,⌫ ,⌫ ⇣ µ⌫ µ⌫,⇢ ⇢⌫ ,µ µ⇢ ,⌫ where they are all evaluated at the same coordinate and the derivatives in the Lie derivatives can be replaced with covariant derivatives (Lie derivatives are tensorial). To all orders in ⇣, we have 1 T˜ (x)=T(x) £ T + £2T + =exp[ £ ] T . (3.20) ⇣ 2 ⇣ ··· ⇣ Therefore, the gauge-transformation in perturbation theory is simply (1) (n) (n 1) T¯ =0,T = £ T¯ ,T = £ T , (3.21) ⇣ ⇣ ⇣ ⇣ ⇣ where we used that ⇣ is also a perturbation. In fact, there are two ways of looking at the transformation in perturbation theory. For example, the metric tensor has to transform as a tensor. But once we split it into the background and the perturbations, there exist two ways g =¯g + h ,g = £ g : (1) g¯ = £ g,¯ h = £ h, (2) g¯ =0,h = £ h £ g,¯ µ⌫ µ⌫ µ⌫ ⇣ ⇣ ⇣ ⇣ ⇣ ⇣ ⇣ ⇣ ⇣ ⇣ (3.22) where we suppressed the tensor indices. In (1), the background and the perturbation transform altogether like tensors (at the same coordinates), such that the sum transforms like a tensor. In perturbation theory, we do not use this, because the inﬁnitesimal transformation ⇣ is always considered as a perturbation. However, for example we can consider some general spatial rotation ⇣, such that the background metric also changes.1

3.1.5 Linear-Order Gauge Transformation At the linear order, the Lie derivative is trivial, and the the most general coordinate transformation in Eq. (3.18) becomes x˜µ = xµ + ⇠µ ,⇠µ := (T, ↵) , ↵ := L,↵ + L(v)↵ , (3.23) L L where we now use ⇠µ = ⇣µ. The transformation of the metric tensor at the leading order in ⇠ is then g (x)=˜g (x) g (x)= (⇠ + ⇠ )= £ g , (3.24) ⇠ µ⌫ µ⌫ µ⌫ µ;⌫ ⌫;µ ⇠ µ⌫ where the semi-colon represents the covariant derivative with respect to the full metric gµ⌫. Under the coordinate transformation, the scalar quantities gauge-transform as 1 ↵˜ = ↵ (aT )0 , ˜ = T + L0 , '˜ = ' T, ˜ = L, (3.25) a H U˜ = U L0 , v˜ = v T, ˜ = aT , ˜ =  +3HaT˙ + T, (3.26) a the vector metric perturbations gauge-transform as (v) (v) (v) (v) (v) (v) (v) B˜ = B + L0 , C˜ = C L , U˜ = U + L 0 , (3.27) ↵ ↵ ↵ ↵ ↵ ↵ ↵ ↵ ↵ and the tensor perturbations are gauge-invariant at the linear order, where we deﬁned the scalar shear := a ( + 0) of the normal observer (n =0) and the extrinsic 3-curvature K := 3H +  and its perturbation  := 3H↵ 3˙' . ↵ a2 Based on the above gauge transformation properties, we can construct linear-order gauge-invariant quantities. The gauge-invariant variables are 1 ⇢ ⇢˙ ' := ' aHv , ' := ' H, v := v ,' := ' + ,:= a v, v a 3(⇢ + p) v ⇢ ˙ 1 (v) (v) (v) (v) (v) (v) ↵ := ↵ 0 ,:= ', := B + C 0 ,v:= U B . (3.28) a ' H ↵ ↵ ↵ ↵ ↵ ↵ (v) (0) These gauge-invariant variables (↵,',v, ↵,v↵ ) correspond to A, H , vs , , and vc in Bardeen (1980).

1 In FRW, we use the spatial metric g¯↵ unspecified, implying we can do a further spatial transformation to e.g., spherical coordinate and so on and change the background metric (while it remains covariant). The time component is fixed, otherwise it ruins the FRW symmetry (g0↵ component in the background or different coefficient in time component for example.

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3.1.6 Popular Choices of Gauge Condition By a suitable choice of coordinates, we can set T = L =0, simplifying the metric. For simplicity, we only consider the scalar perturbations in the following two cases.

The conformal Newtonian Gauge.— in which we choose the spatial and the temporal gauge conditions: • ˜ = =0 L =0, ˜ = =0 T =0,=0. (3.29) ! ! All the gauge modes are ﬁxed, and the metric in this gauge condition is ds2 = a2 (1 + 2 ) d⌘2 + a2 (1 + 2)¯g dx↵dx , := ↵ = ↵ ,:= ' = ' , (3.30) ↵ and the velocity vector is then U = v , v = U. (3.31) r The metric and its equations appear more like the Newtonian equations, and hence the name. We will use this gauge condition to illustrate and simplify the problems.

Synchronous-Comoving Gauge.— in which we choose the spatial and the temporal gauge conditions: • ↵˜ = ↵ =0 (aT )0 =0, ˜ = =0 T = L0 , (3.32) ! ! such that the metric becomes ds2 = a2d⌘2 + a2 (¯g +2C ) dx↵dx . (3.33) ↵ ↵ All the metric perturbations in this gauge condition are included in the spatial metric tensor. However, as apparent from the above gauge condition, the gauge freedoms are not completely fixed: 1 F (x) T = L0 = F (x) ,L= d⌘ + G(x) , (3.34) a a Z where F and G are two arbitrary functions of spatial coordinates. Typically, this issue is resolved by assuming additional condition at the initial epoch v =0 F (x)=0,T=0. (3.35) ! This condition is indeed the temporal comoving gauge condition, and hence the whole choice is often referred to as the comoving-synchronous gauge (or synchronous-comoving). The comoving gauge is often chosen with a different spatial gauge condition ( =0). Note, however, that the spatial function G(x) is still left unspecified, and hence is a gauge mode, whereas U for example is physical. Due to this deficiency, we will not use this gauge condition in the following. The notation convention in Ma and Bertschinger (1995): 1 h := kˆ kˆ h + kˆ kˆ 6⌘ 2C ,h6' + 2,⌘ '. (3.36) ij i j i j 2 ij ! ij ! ! ✓ ◆ 3.2 Energy-Momentum Tensor

3.2.1 Formal Definition We will consider a simple action of the matter sector, in addition to the gravity described by the Einstein-Hilbert action: R S = p gd4x + , (3.37) 16⇡G Lm Z  where the matter Lagrangian includes the cosmological fluids and other matter fields such as scalars and so on. The energy-momentum tensor defined by the action

m µ⌫ µ⌫ m Tµ⌫ = gµ⌫ m 2 Lµ⌫ ,T= g m +2 L , (3.38) L g L gµ⌫ is indeed the conserved current (tensor) of the action under the space-time translation invariance. The Noether theorem says that when there exists a (global) symmetry, there exists a conserved current. The space-time translation invariance is the symmetry of general relativity, and the Noether current associated with this symmetry is the energy-momentum tensor: Tµ⌫;⌫ =0. (3.39)

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3.2.2 General Decomposition for Cosmological Fluids For our purposes, we are not interested in the microscopic states of the systems, but interested in their macroscopic states, often described by the density, the pressure, the temperature, and so on. The energy-momentum tensor for a ﬂuid can be expressed in terms of the ﬂuid quantities measured by an observer with four velocity uµ as (the most general decomposition) T := ⇢u u + p + q u + q u + ⇡ , (3.40) µ⌫ µ ⌫ Hµ⌫ µ ⌫ ⌫ µ µ⌫ where is the projection tensor and Hµ⌫ = g +u u , µ =3,uµq =0=uµ⇡ ,⇡= ⇡ ,⇡µ =0. (3.41) Hµ⌫ µ⌫ µ ⌫ Hµ µ µ⌫ µ⌫ ⌫µ µ

The variables ⇢, p, qµ and ⇡µ⌫ are the energy density, the isotropic pressure (including the entropic one), the (spatial) energy flux and the anisotropic pressure measured by the observer with uµ, respectively, i.e., 1 ⇢ = T uµu⌫ ,p= T ab ,q= T u⇢ ,⇡= T ⇢ p . (3.42) µ⌫ 3 µ⌫H µ ⇢ Hµ µ⌫ ⇢HµH⌫ Hµ⌫ Remember that these fluid quantities are observer-dependent. We decompose the fluid quantities into the background and the perturbations: p˙ ⇢ :=⇢ ¯ + ⇢ , p :=p ¯ + p, p := c2⇢ + e, c2 := , (3.43) s s µ˙ p ⇢ q := aQ ,⇡:= a2⇧ ,e:=p ˙ , = , (3.44) ↵ ↵ ↵ ↵ p˙ ⇢˙ where Q↵ and ⇧↵ are based on g¯↵. At the background level, all the above fluid quantities vanish, except ⇢ =¯⇢ and 2 p =¯p. For the adiabatic perturbations, we have e =0and the sound speed is cs = p/⇢ =˙p/⇢˙. For multiple fluids, we can add up the individual energy-momentum tensor to derive the total energy-momentum tensor. In the case of multiple fluids, their fluid velocities are not necessarily identical, and there exist non-vanishing energy flux. Non-vanishing e and parametrize the entropic perturbations of the fluids. Though these relations are exact, we will be concerned with linear-order perturbations. Raising the index of the energy momentum tensor, we derive

T 0 = ⇢ + (2) ,T0 =(¯⇢ +¯p)(U B )+Q + (2) , (3.45) 0 O ↵ ↵ ↵ ↵ O T ↵ = p↵ +⇧↵ + (2) ,T↵ = (¯⇢ +¯p)U ↵ Q↵ , (3.46) O 0 and the (spatial) energy ﬂux and the anisotropic pressure satisfy

q =0+ (2) ,⇡=0+ (2) . (3.47) 0 O 0µ O 3.2.3 Tetrad Approach Given the observer with uµ, one can deﬁne a local Lorentz frame (where the metric is Minkowski) by constructing three µ µ µ µ spacelike orthonormal vectors [ei] . For example, one can construct three rectangular basis vectors [ex] , [ey] , [ez] and of course [e ]µ = uµ = [et]µ, where the component index a of [e ]µ represents their coordinates. The orthonormality t a condition and the spacelike normalization is

⌘ = g [e ]µ[e ]⌫ = g [e ]µ[e ]⌫ , 0=g [e ]µ[e ]⌫ , 1=g [e ]µ[e ]⌫ , (3.48) ab µ⌫ a b ! ij µ⌫ i j µ⌫ i t µ⌫ t t where a, b, = t, x, y, z represent the local Lorentz indices. ··· Assuming that the four velocity of the observer is indeed the ﬂuid velocity, we can go to the rest-frame of the ﬂuid by using the tetrad expressions as

3 3 1 ⇢ = T [e ]µ[e ]⌫ ,p= T [e ]µ[e ]⌫ , = [e ] [e ] . (3.49) µ⌫ t t 3 µ⌫ i i Hµ⌫ i µ i ⌫ Xi=1 Xi=1

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The other fluid quantities can be readily expressed in terms of the tetrad basis, and the energy-momentum tensor in the rest-frame of the fluid is then ⇢ q q q x y z qx p + ⇡xx ⇡xy ⇡xz Tab = 0 1 . (3.50) q ⇡ p + ⇡ ⇡ y yx yy yz B q ⇡ ⇡ p + ⇡ C B z zx zy zz C @ A The orthogonality condition for the energy-flux and the anisotropic stress implies

µ µ µ t i 0=u qµ = qt , 0=u ⇡µ⌫ = ⇡ta , 0=⇡µ = ⇡t + ⇡i = ⇡ii . (3.51)

However, we should pay attention to the difference in the quantities expressed in the rest-frame and in the FRW coordinate:

q := q [e ]µ = Q(1)↵ + (2) ,⇡:= ⇡ [e ]µ[e ]⌫ =⇧(1)↵ + (2) . (3.52) i µ i ↵ i O ij µ⌫ i i ↵ i j O As noted, the fluid quantities are observer-dependent. Given the energy momentum tensor in terms of the fluid rest- µ µ frame quantities (qµ =0), if the observer is moving with et relative to the fluid velocity u , the observer measures different fluid quantities from those defined in the rest frame. The energy-momentum tensor can be projected into the observer rest-frame as

T =(⇢ + p)u u + p⌘ + ⇡ ,u:= u eµ , := eµe⌫ = , (3.53) ab a b ab ab a µ a Hij Hµ⌫ i j ij where the ﬂuid velocity and the anisotropic pressure satisfy

1=uau = u2 + u2 , 0=⇡µ ⇡ = ⇡ . (3.54) a t i µ ! tt ii Furthermore, the anisotropic pressure is perpendicular to the ﬂuid velocity:

uj 0=ua⇡ = ut⇡ + ui⇡ ,⇡= ⇡ . (3.55) ab ta ia ti ij ut Therefore, the energy density and the pressure measured by the observer are 1 1 1 ⇢˜ =(⇢ + p)u2 p + ⇡ , p˜ = (⇢ + p)u u + p + ⇡ = p + (⇢ + p)(u2 1) + ⇡ , (3.56) t tt 3 i i 3 ii 3 t tt ⇥ ⇤ and the anisotropic pressure is 1 ⇡˜ =(⇢ + p)uiuj + ⇡ (⇢ + p)(u2 1) + ⇡ , 0=˜⇡ =˜⇡ . (3.57) ij ij 3 ij t tt ti tt ⇥ ⇤ Since the velocities of the ﬂuid and the observer are different, the observer measures the non-vanishing spatial energy ﬂux

t t t q˜i = T i =(⇢ + p)u ui + ⇡ i . (3.58)

At the linear order in perturbations, the velocity of the ﬂuid measured by the observer is

ua = ea uµ = 1,Ui U i + (2) . (3.59) µ obs O Therefore, the ﬂuid quantities measured by the observer are

0=⇡ = ⇡ , ⇢˜ = ⇢, p˜ = p, ⇡˜ = ⇡ , q˜ =(¯⇢ +¯p) U i U i . (3.60) tt ti ij ij i obs In other words, whoever the observer is, the fluid quantities the observer measures are identical to those at the fluid rest frame, except the spatial energy flux.

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3.2.4 Distribution Function In cosmology, photons and neutrinos are the most important radiation components, and they are not described by the ﬂuid approximation. Their statistical properties are captured by the distribution function F :

F := f¯+ f, (3.61) where the background distribution f¯often follows the equilibrium distribution and the perturbation f describes the deviation from the equilibrium. The equilibrium distribution for massless particles is fully described by the physical momentum and the temperature, and it is independent of position and time.2 In the rest-frame of an observer, the physical energy E and the momentum P a can be measured, and the energy-momentum tensor can be re-constructed as

d3P T ab = g P aP bF, (3.62) E Z where the four momentum satisﬁes the on-shell condition m2 = P P a and E = P t and g is the spin-degeneracy of the a particle, equal to two for photons and one for left-handed neutrinos. The ﬂuid elements can be readily computed as

P iP j ⇢ = g d3PEF, qi = Q ↵ = g d3PPiF, pij + ⇡ij = g d3P F. (3.63) ↵ i E Z Z Z For later convenience, we introduce an angular decomposition

l 4⇡ l 2l +1 2 f(P, nˆ):= ( i) f (P )Y (ˆn) ,f(P ) i d nYˆ ⇤ (ˆn)f(P, nˆ) , (3.64) 2l +1 lm lm lm ⌘ 4⇡ lm lm r r X Z where P i = Pni and nˆ is the unit directional vector. The normalization convention may differ in literature. The perturbations in the ﬂuid quantities are then related to the distribution function as

4 1 4⇡g 1 P ⇢ =4⇡g dP P 2Ef ,p=TrT ij = dP f , (3.65) 00 3 E 00 Z0 Z0 where we performed the angular integration. Higher moments of the ﬂuid elements will be related to the higher-moments of the distribution function. At the linear order, the spatial energy ﬂux from the distribution function is related to the relative velocity as

q =(¯⇢ +¯p) U i U i = Q ↵ , (3.66) i f obs ↵ i and the off-diagonal part of the energy-momentum tensor is

T 0 =(¯⇢ +¯p) U obs B + Q =(¯⇢ +¯p) U f B . (3.67) ↵ ↵ ↵ ↵ ↵ ↵ ⇣ ⌘ ⇣ ⌘ 3.3 Einstein Equations

3.3.1 Christoffel Symbols In the absence of the torsion, the Christoffel symbols are uniquely determined by the metric tensor as

@xµ @2⇠ 1 d2⇠µ µ = = gµ(g + g g ) , 0= ,d⌧2 = ⌘ d⇠µd⇠⌫ , (3.68) ⌫⇢ @⇠ @x⌫@x⇢ 2 ⌫,⇢ ⇢,⌫ ⌫⇢, d⌧ 2 µ⌫

2In the background, the physical momentum and the temperature redshift in the same way.

22 AST513 Theoretical Cosmology JAIYUL YOO where ⇠µ is a freely falling coordinate and d⌧ is the proper time. To the linear order in perturbations, we derive

0 a0 0 a0 = + A0 + 0, = A B , (3.69) 00 a !H 0↵ ,↵ a ↵ ! ,↵ ↵ ↵ ↵ a0 ↵ ,↵ = A| B 0 B , (3.70) 00 a ! 0 a0 a0 a0 ↵ = g¯↵ 2 g¯↵A + B(↵ ) + C↵0 +2 C↵ g¯↵ (1 2 )+ 0 +2 g¯↵ , (3.71) a a | a !H H ↵ a0 ↵ 1 ↵ ↵ ↵ ↵ ↵ 0 = + B| B + C 0 + 0 , (3.72) a 2 | !H ⇣ ⌘ ↵ ¯↵ a0 ↵ ↵ ↵ ¯↵ ↵ ,↵ = + g¯B +2C( ) C| +2,() g¯ , (3.73) a | ! where the conformal Hubble parameter is = a /a and ¯↵ is the Christoffel symbols based on 3-metric g¯ . H 0 ↵ 3.3.2 Riemann Tensor The Riemann tensor can then be constructed in terms of the Christoffel symbols as

Rµ := µ µ +✏ µ ✏ µ =µ µ ✏ µ +✏ µ , (3.74) ⌫⇢ ⌫,⇢ ⌫⇢, ⌫ ⇢✏ ⌫⇢ ✏ ⌫;⇢ ⌫⇢; ⌫ ⇢✏ ⌫⇢ ✏ and the Riemann tensor has all the information of the geometry, such that how any four vector changes locally is fully determined by the Riemann tensor 2uµ;[⌫⇢] = uR µ⌫⇢ . (3.75) Out of the Riemann tensor, we can construct the Ricci tensor (and Ricci scalar) by contracting the Riemann tensor as

⇢ µ Rµ⌫ := Rµ⇢⌫ ,R= Rµ , (3.76) and construct the (conformal) Weyl tensor as 1 R C := R (g R + g R g R g R )+ (g g g g ) . (3.77) µ⌫⇢ µ⌫⇢ 2 µ⇢ ⌫ ⌫ µ⇢ ⌫⇢ µ µ ⌫⇢ 6 µ⇢ ⌫ µ ⌫⇢ The Riemann tensor has the symmetry:

Rµ⌫⇢ = R[µ⌫][⇢] = R⇢µ⌫ ,Rµ⌫⇢ + Rµ⇢⌫ + Rµ⌫⇢ = Rµ[⌫⇢] =0, (3.78) such that its 20 independent components can be separated into the Ricci tensor (10 components) and the (traceless) Weyl curvature tensor (also 10 components with all the symmetry of the Riemann tensor)

µ⌫ C µ⌫ =0,Cµ⌫⇢ = C[µ⌫][⇢] = C⇢µ⌫ ,Cµ⌫⇢ + Cµ⇢⌫ + Cµ⌫⇢ = Cµ[⌫⇢] =0. (3.79)

The Ricci tensor is algebraically set by matter distribution through the Einstein equation, but the Weyl tensor is determined by differential equations with suitable boundary conditions. To the background, we derive

¯↵ ↵ ↵ 2 ↵ R =2K[g¯] ,R =2 K + [g¯] ,C↵ [] = K g¯↵[C] +¯g[C]↵ , (3.80) H | ¯↵ where R is the Riemann tensor based on 3-metric g¯↵. To the linear order in perturbations, we derive the Riemann

23 AST513 Theoretical Cosmology JAIYUL YOO tensor:

Rµ =0,R0 = B ,R0 =0, (3.81) ⌫00 00↵ H ↵ 0↵ 0 R ↵0 = 0g¯↵ A0 +2 0A g¯↵ A,↵ + B(0↵ ) + B(↵ ) + C↵00 + C↵0 +2 0C↵ , (3.82) H H H | | H | H H 0 ⇥ ⇤1 R ↵ =2g¯↵[A,] B↵ [] + (B ↵ B ↵) 2C↵0 [ ] , (3.83) H | 2 | | | ↵ ↵ ↵ ↵ 1 ↵ ↵ 0 1 ↵ ↵ ↵ ↵ R 00 = 0 A0 A| + B| + B + B| + B + C 00 + C 0 , (3.84) H H 2 | 2H | H ↵ ↵ ↵ ↵ ⇣ 2 ↵ ⌘ ↵ ⇣ ⌘ | R 0 =2[A,] B[] + B [] 2 [B] 2C[0 ] , (3.85) H | H | ↵ ,↵ ↵ ↵ 2 ↵ ↵ 1 ↵ ↵ ↵ ↵ R 0 = g¯A A, + 0g¯B g¯B B B| B + C 0 C0| ,(3.86) H H H 2 | | ⇣ ⌘| R↵ = R¯↵ + 2 ↵g¯ ↵g¯ (1 2A) (3.87) H 1 ↵ ↵ ↵ ↵ ↵ ↵ + g¯ B| + B g¯ B | + B +2 B( ) 2 B( ) 2H | | | | ↵ ↵ ↵ ↵ ↵ ↵ + g¯h C ⇣0 g¯ C 0 +⌘ C0 ⇣ C0 +2 ⌘ C C i H H ↵ ↵ ↵ ↵ +2C⇥( ) 2C( ) + C| C| , ⇤ | | and by contracting the Riemann tensor we derive the Ricci tensor and the Ricci scalar:

↵ ↵ ↵ ↵ R00 = 3 0 +3 A0 +A B 0↵ B ↵ C↵00 C↵0 , (3.88) H H | H | H 2 1 1 0| R0↵ =2A,↵ 0B↵ 2 B↵ + B↵ B ↵ C 0↵ + C↵ , (3.89) H H H 2 2 | | 2 R↵ =2Kg¯↵ + 0 +2 g¯↵(1 2A) A0g¯↵ A,↵ + B(0↵ ) +2 B(↵ ) + g¯↵B (3.90) H H H | | H | H | a0 2 +C↵00 +2 C↵0 +2 0 +2 C↵ + g¯↵C 0 +2C(↵ ) C ↵ C↵ , a H H H | | 1 2 2 R = 6 0 + + K 6 A0 12 0 + A 2A (3.91) a2 H H H H H ⇥ ↵ ↵ ↵ ↵ ↵ ↵ ↵ +2B 0↵ +6 B ↵ +2C↵00 +6 C↵0 4KC↵ 2C↵ +2C ↵ . | H | H | i HW: derive the Riemann tensor and the Ricci tensor in the conformal Newtonian gauge • 3.3.3 Einstein Equation and Background Equation

The Einstein equation is that the Einstein tensor Gµ⌫ is proportional to the energy-momentum tensor Tµ⌫: 1 G := R g R +⇤g =8⇡GT , (3.92) µ⌫ µ⌫ 2 µ⌫ µ⌫ µ⌫ where G is the Newton’s constant and ⇤ is the cosmological constant. The cosmological constant can be put in the right-hand side as a part of the energy-momentum tensor: ⇤ ⇢ = p = . (3.93) ⇤ ⇤ 8⇡G The trace of the Einstein equation gives

T = ⇢ +3p, R=8⇡G(⇢ 3p) + 4⇤ , (3.94) and the Ricci tensor is completely set by the trace of the energy-momentum tensor. To the background, the Einstein equation yields the Friedmann equation give 8⇡G K ⇤ a¨ 4⇡G ⇤ H2 = ⇢ + ,H2 + H˙ = = (⇢ +3p)+ , (3.95) 3 a2 3 a 3 3

24 AST513 Theoretical Cosmology JAIYUL YOO and the energy-momentum conservation yields

⇢˙ +3H (⇢ + p)=0. (3.96)

In terms of the conformal time, the Friedmann equation becomes

2 2 4⇡G 2 2 a00 4⇡G 2 0 = a (H + H˙ )= a (⇢ +3p) , + 0 = = a (⇢ 3p) K. (3.97) H 3 H H a 3 In a ﬂat Universe (K =0) dominated by an energy component ⇢ a n, we can derive the analytic solutions to the / Friedmann equation:

2 2/n n to 2 ⌘o 2 1 a ⌘ n 2 t ,t⌘ n 2 ,H= H = , = = . (3.98) / / / o t nt H Ho ⌘ n 2 ⌘ ✓ ◆ ✓ ◆ 3.3.4 Linear-Order Einstein Equation The Einstein equation can be expanded up to the linear order in perturbations, and decomposed into the evolution equations describing the scalar, the vector, and the tensor perturbations. At the linear order, they do not mix.

Scalar perturbations.— • +3K G0 : H + ' = 4⇡G⇢, (3.99) 0 a2 +3K G0 :  + = 12⇡G(⇢ + p)av , (3.100) ↵ a2 G↵ G0 :˙ +2H + 3H˙ + ↵ =4⇡G (⇢ +3p) , (3.101) ↵ 0 a2 ✓ ◆ 1 G↵ ↵G :˙ + H ' ↵ =8⇡G⇧ , (3.102) 3 where we deﬁned

1 1 1 (t)  := 3H↵ 3˙' ,:= a+a0 , ⇧↵ := ⇧,↵ g¯↵⇧ + ⇧(↵ )+⇧↵ . (3.103) a2 a2 | 3 a | ✓ ◆ The energy-momentum conservation yields

1 T ⌫ : ⇢˙ +3H (⇢ + p) (⇢ + p)  3H↵ + v =0, (3.104) 0;⌫ a ✓ ◆ [a4(⇢ + p)v] 1 1 2 +3K T ⌫ : · ↵ p + ⇧ =0. (3.105) ↵;⌫ a4(⇢ + p) a a(⇢ + p) 3 a2 ✓ ◆ Vector perturbations.— • +2K G0 : (v) +8⇡G(⇢ + p)v(v) =0, (3.106) ↵ 2a2 ↵ ↵ ↵ ˙ (v) (v) (v) G : ↵ +2H ↵ =8⇡G⇧↵ , (3.107) 4 (v) (v) [a (⇢ + p)v↵ ] +2K ⇧↵ T ⌫ : · + =0. (3.108) ↵;⌫ a4(⇢ + p) 2a2 ⇢ + p

Tensor perturbations.— • 2K G : C¨(t) +3HC˙ (t) C(t) =8⇡G⇧(t) . (3.109) ↵ ↵ ↵ a2 ↵ ↵

25 AST513 Theoretical Cosmology JAIYUL YOO

3.3.5 Einstein Equation in the conformal Newtonian Gauge In the conformal Newtonian gauge we have

 =3H 3˙ ,=0,U= v , v = U. (3.110) r By substituting into the Einstein equation, we derive +3K H + = 4⇡G⇢ (+ 3K) + a 3H 3˙ = 4⇡Ga2⇢ , (3.111) a2 ! H  = 12⇡G(⇢ + p)av , + = 8⇡⇣G⇧ , ⌘ (3.112) ˙ +2H + 3H˙ + =4⇡G (⇢ +3p) . (3.113) a2 ✓ ◆ Equation (3.111) can be further manipulated as

(k2 3K) =4⇡Ga2⇢¯ + a  =4⇡Ga2⇢¯[ +3 v(1 + w)] . (3.114) H H Assuming a ﬂat Universe (K =0) and a pressureless medium (p = p =0), we can further simplify the equation as

3 2 + =0,= (a )· = 12⇡G⇢av , = 4⇡Ga ⇢ ,v0 + v = , (3.115) a v H where v is the density ﬂuctuation in the comoving gauge:

= +3 v. (3.116) v H

As we will see, the comoving-gauge curvature perturbation 'v is conserved on large scales throughout the evolution. So, it is convenient to derive the relation of the conformal Newtonian gauge quantities to the comoving-gauge curvature (sometimes denoted as ⇣ or ) R 3H 3˙ ˙ H 2 /H 5+3w 'v = v = = , (3.117) H 12⇣⇡Ga(⇢ + p)⌘ 3 1+w !k=0 3(1 + w) where we assume ⇧=0and ˙ =0in the super horizon limit.