Astro 448 Cosmic Microwave Background Wayne Hu Astro 448 Acoustic Kinematics Recombination • Equilibrium Number Density Distribution of a Non-Relativistic Species

Home , Thomson scattering

Astro 448 Cosmic Microwave Background Wayne Hu Astro 448 Acoustic Kinematics Recombination • Equilibrium number density distribution of a non-relativistic species

3/2 miT n = g e−mi/T i i 2π • Apply to the e− + p ↔ H system: Saha Equation

n n x2 e p = e nH nb 1 − xe 1 m T 3/2 = e e−B/T nb 2π

where B = me + mp − mH = 13.6eV

• Naive guess of T = B for recombination would put z∗ ≈ 45000. Recombination • But the photon-baryon ratio is very low

−8 2 ηbγ ≡ nb/nγ ≈ 3 × 10 Ωbh

• Eliminate in favor of ηbγ and B/T through m n = 0.244T 3 , e = 3.76 × 104 γ B • Big coefﬁcient

x 2 B 3/2 e = 3.16 × 1015 e−B/T 1 − xe T

T = 1/3eV → xe = 0.7, T = 0.3eV → xe = 0.2 • Further delayed by inability to maintain equilibrium since net is through 2γ process and redshifting out of line Thomson Scattering • Thomson scattering of photons off of free electrons is the most important CMB process with a cross section (averaged over polarization states) of

2 8πα −25 2 σT = 2 = 6.65 × 10 cm 3me

• Density of free electrons in a fully ionized xe = 1 universe

−5 2 3 −3 ne = (1 − Yp/2)xenb ≈ 10 Ωbh (1 + z) cm ,

where Yp ≈ 0.24 is the Helium mass fraction, creates a high (comoving) Thomson opacity

τ˙ ≡ neσT a where dots are conformal time η ≡ R dt/a derivatives and τ is the optical depth. Temperature Fluctuations • Observe blackbody radiation with a temperature that differs at 10−5 coming from the surface of last scattering, with distribution 3 function (speciﬁc intensity Iν = 4πν f(ν) each polarization)

f(ν) = [exp(2πν/T (nˆ)) − 1]−1 • Decompose the temperature perturbation in spherical harmonics X T (nˆ) = T`mY`m(nˆ) `m • For Gaussian random ﬂuctuations, the statistical properties of the temperature ﬁeld are determined by the power spectrum

∗ hT`mT`0m0 i = δ``0 δmm0 C` where the δ-function comes from statistical isotropy Spatial vs Angular Power • Take the radiation distribution at last scattering to also be described by an isotropic temperature ﬁeld T (x) and recombination to be instantaneous Z T (nˆ) = dD T (x)δ(D − D∗)

where D is the comoving distance and D∗ denotes recombination. • Describe the temperature ﬁeld by its Fourier moments

Z d3k T (x) = T (k)eik·x (2π)3 with a power spectrum

∗ 0 3 0 hT (k) T (k )i = (2π) δ(k − k )PT (k) Spatial vs Angular Power • Note that the variance of the ﬁeld Z d3k hT (x)T (x)i = P (k) (2π)3 Z k3P (k) Z = d ln k ≡ d ln k ∆2 (k) 2π2 T 2 so it is more convenient to think in the log power spectrum ∆T (k) • Temperature ﬁeld Z d3k T (nˆ) = T (k)eik·D∗nˆ (2π)3 • Expand out plane wave in spherical coordinates

ikD∗·nˆ X ` ∗ e = 4π i j`(kD∗)Y`m(k)Y`m(nˆ) `m Spatial vs Angular Power • Multipole moments

Z d3k T = T (k)4πi`j (kD )Y (k) `m (2π)3 ` ∗ `m • Power spectrum

Z 3 ∗ d k 2 `−`0 ∗ hT T 0 0 i = (4π) (i) j (kD )j 0 (kD )Y (k)Y 0 0 (k)P (k) `m ` m (2π)3 ` ∗ ` ∗ `m ` m T Z 2 2 = δ``0 δmm0 4π d ln k j` (kD∗)∆T (k)

R ∞ 2 2 with 0 j` (x)d ln x = 1/(2`(` + 1)), slowly varying ∆T 4π∆2 (`/D ) 2π C = T ∗ = ∆2 (`/D ) ` 2`(` + 1) `(` + 1) T ∗ 2 so `(` + 1)C`/2π = ∆T is commonly used log power Tight Coupling Approximation 3 2 • Near recombination z ≈ 10 and Ωbh ≈ 0.02, the (comoving) mean free path of a photon

1 λ ≡ ∼ 2.5Mpc C τ˙ small by cosmological standards!

• On scales λ λC photons are tightly coupled to the electrons by Thomson scattering which in turn are tightly coupled to the baryons by Coulomb interactions • Specifically, their bulk velocities are defined by a single fluid

velocity vγ = vb and the photons carry no anisotropy in the rest frame of the baryons • → No heat conduction or viscosity (anisotropic stress) in ﬂuid Zeroth Order Approximation • Momentum density of a ﬂuid is (ρ + p)v, where p is the pressure • Neglect the momentum density of the baryons

(ρ + p )v ρ + p 3ρ R ≡ b b b = b b = b (ργ + pγ)vγ ργ + pγ 4ργ Ω h2 a ≈ 0.6 b 0.02 10−3

4 since ργ ∝ T is ﬁxed by the CMB temperature T = 2.73(1 + z)K – OK substantially before recombination • Neglect radiation in the expansion

2 ρm Ωmh a = 3.6 −3 ρr 0.15 10 Number Continuity • Photons are not created or destroyed. Without expansion

n˙ γ + ∇ · (nγvγ) = 0

−3 but the expansion or Hubble ﬂow causes nγ ∝ a or

a˙ n˙ + 3n + ∇ · (n v ) = 0 γ γ a γ γ

• Linearize δnγ = nγ − n¯γ

a˙ (δn )· = −3δn − n ∇ · v γ γ a γ γ · δnγ = −∇ · vγ nγ Continuity Equation 3 • Number density nγ ∝ T so deﬁne temperature ﬂuctuation Θ

δn δT γ = 3 ≡ 3Θ nγ T • Real space continuity equation

1 Θ˙ = − ∇ · v 3 γ • Fourier space

1 Θ˙ = − ik · v 3 γ Momentum Conservation • No expansion: q˙ = F • De Broglie wavelength stretches with the expansion a˙ q˙ + q = F a for photons this the redshift, for non-relativistic particles expansion drag on peculiar velocities • Collection of particles: momentum → momentum density

(ργ + pγ)vγ and force → pressure gradient a˙ [(ρ + p )v ]· = −4 (ρ + p )v − ∇p γ γ γ a γ γ γ γ 4 1 ρ v˙ = ∇ρ 3 γ γ 3 γ v˙ γ = −∇Θ Euler Equation • Fourier space

v˙ γ = −ikΘ • Pressure gradients (any gradient of a scalar field) generates a curl-free flow • For convenience define velocity amplitude: ˆ vγ ≡ −ivγk • Euler Equation:

v˙γ = kΘ • Continuity Equation: 1 Θ˙ = − kv 3 γ Oscillator: Take One • Combine these to form the simple harmonic oscillator equation

¨ 2 2 Θ + csk Θ = 0 where the adiabatic sound speed is deﬁned through

2 p˙γ cs ≡ ρ˙γ

2 here cs = 1/3 since we are photon-dominated • General solution:

Θ(0)˙ Θ(η) = Θ(0) cos(ks) + sin(ks) kcs R where the sound horizon is deﬁned as s ≡ csdη Harmonic Extrema • All modes are frozen in at recombination (denoted with a subscript ∗) yielding temperature perturbations of different amplitude for different modes. For the adiabatic (curvature mode) Θ(0)˙ = 0

Θ(η∗) = Θ(0) cos(ks∗) • Modes caught in the extrema of their oscillation will have enhanced ﬂuctuations

kns∗ = nπ yielding a fundamental scale or frequency, related to the inverse sound horizon

kA = π/s∗ and a harmonic relationship to the other extrema as 1 : 2 : 3... Peak Location • The fundmental physical scale is translated into a fundamental angular scale by simple projection according to the angular

diameter distance DA

θA = λA/DA

`A = kADA • In a ﬂat universe, the distance is simply D = D ≡ η − η ≈ η , √ A 0 ∗ 0 the horizon distance, and kA = π/s∗ = 3π/η∗ so

η∗ θA ≈ η0 1/2 ◦ • In a matter-dominated universe η ∝ a so θA ≈ 1/30 ≈ 2 or

`A ≈ 200 Curvature • In a curved universe, the apparent or angular diameter distance is

no longer the conformal distance DA = R sin(D/R) 6= D • Objects in a closed universe are further than they appear! gravitational lensing of the background... • Curvature scale of the universe must be substantially larger than current horizon • Flat universe indicates critical density and implies missing energy given local measures of the matter density “dark energy”

• D also depends on dark energy density ΩDE and equation of state

w = pDE/ρDE. • Expansion rate at recombination or matter-radiation ratio enters

into calculation of kA. Doppler Effect • Bulk motion of ﬂuid changes the observed temperature via Doppler shifts

∆T = nˆ · vγ T dop • Averaged over directions

∆T v = √γ T rms 3 • Acoustic solution √ √ vγ 3 ˙ 3 √ = − Θ = kcs Θ(0)sin(ks) 3 k k = Θ(0)sin(ks) Doppler Peaks? • Doppler effect for the photon dominated system is of equal amplitude and π/2 out of phase: extrema of temperature are turning points of velocity • Effects add in quadrature:

∆T 2 = Θ2(0)[cos2(ks) + sin2(ks)] = Θ2(0) T • No peaks in k spectrum! However the Doppler effect carries an angular dependence that changes its projection on the sky ˆ nˆ · vγ ∝ nˆ · k • Coordinates where zˆ k kˆ

Y10Y`0 → Y`±1 0

0 recoupling j`Y`0: no peaks in Doppler effect Restoring Gravity • Take a simple photon dominated system with gravity • Continuity altered since a gravitational potential represents a stretching of the spatial fabric that dilutes number densities – formally a spatial curvature perturbation • Think of this as a perturbation to the scale factor a → a(1 + Φ) so that the cosmogical redshift is generalized to a˙ a˙ → + Φ˙ a a a˙ (δn )· = −3δn + Φ˙ − n ∇ · v γ γ a γ γ so that the continuity equation becomes 1 Θ˙ = − kv − Φ˙ 3 γ Restoring Gravity • Gravitational force in momentum conservation F = −m∇Ψ generalized to momentum density modiﬁes the Euler equation to

v˙γ = k(Θ + Ψ) • General relativity says that Φ and Ψ are the relativistic analogues of the Newtonian potential and that Φ ≈ −Ψ. • In our matter-dominated approximation, Φ represents matter density ﬂuctuations through the cosmological Poisson equation

2 2 k Φ = 4πGa ρm∆m where the difference comes from the use of comoving coordinates for k (a2 factor), the removal of the background density into the

background expansion (ρ∆m) and ﬁnally a coordinate subtlety that

enters into the deﬁnition of ∆m Astro 448 Acoustic Dynamics Constant Potentials • In the matter dominated epoch potentials are constant because

infall generates velocities as vm ∼ kηΨ • Velocity divergence generates density perturbations as 2 ∆m ∼ −kηvm ∼ −(kη) Ψ • And density perturbations generate potential ﬂuctuations as 2 Φ ∼ ∆m/(kη) ∼ −Ψ, keeping them constant. Note that because of the expansion, density perturbations must grow to keep potentials constant. 2 • Here we have used the Friedman equation H = 8πGρm/3 and η = R d ln a/(aH) ∼ 1/(aH) • More generally, if stress perturbations are negligible compared with density perturbations ( δp δρ ) then potential will remain roughly constant – more speciﬁcally a variant called the Bardeen or comoving curvature ζ is constant Oscillator: Take Two • Combine these to form the simple harmonic oscillator equation

k2 Θ¨ + c2k2Θ = − Ψ − Φ¨ s 3 • In a CDM dominated expansion Φ˙ = Ψ˙ = 0. Also for photon 2 domination cs = 1/3 so the oscillator equation becomes

¨ ¨ 2 2 Θ + Ψ + csk (Θ + Ψ) = 0 • Solution is just an offset version of the original

[Θ + Ψ](η) = [Θ + Ψ](0) cos(ks)

• Θ + Ψ is also the observed temperature ﬂuctuation since photons lose energy climbing out of gravitational potentials at recombination Effective Temperature • Photons climb out of potential wells at last scattering • Lose energy to gravitational redshifts • Observed or effective temperature

Θ + Ψ

• Effective temperature oscillates around zero with amplitude given by the initial conditions • Note: initial conditions are set when the perturbation is outside of horizon, need inflation or other modification to matter-radiation FRW universe. • GR says that initial temperature is given by initial potential Sachs-Wolfe Effect and the Magic 1/3 • A gravitational potential is a perturbation to the temporal coordinate [formally a gauge transformation] δt = Ψ t • Convert this to a perturbation in the scale factor, Z da Z da t = ∝ ∝ a3(1+w)/2 aH aρ1/2 where w ≡ p/ρ so that during matter domination δa 2 δt = a 3 t • CMB temperature is cooling as T ∝ a−1 so δT δa 1 Θ + Ψ ≡ + Ψ = − + Ψ = Ψ T a 3 Baryon Loading • Baryons add extra mass to the photon-baryon fluid • Controlling parameter is the momentum density ratio:

pb + ρb 2 a R ≡ ≈ 30Ωbh −3 pγ + ργ 10 of order unity at recombination • Momentum density of the joint system is conserved

(ργ + pγ)vγ + (ρb + pb)vb ≈ (pγ + pγ + ρb + ργ)vγ

= (1 + R)(ργ + pγ)vγb where the controlling parameter is the momentum density ratio:

pb + ρb 2 a R ≡ ≈ 30Ωbh −3 pγ + ργ 10 of order unity at recombination New Euler Equation • Momentum density ratio enters as a˙ [(1 + R)(ρ + p )v ]· = −4 (1 + R)(ρ + p )v γ γ γb a γ γ γb − ∇pγ − (1 + R)(ργ + pγ)∇Ψ same as before except for (1 + R) terms so

· [(1 + R)vγb] = kΘ + (1 + R)kΨ • Photon continuity remains the same k Θ˙ = − v − Φ˙ 3 γb • Modiﬁcation of oscillator equation 1 1 [(1 + R)Θ˙ ]· + k2Θ = − k2(1 + R)Ψ − [(1 + R)Φ˙ ]· 3 3 Oscillator: Take Three • Combine these to form the not-quite-so simple harmonic oscillator equation

d k2 d c2 (c−2Θ˙ ) + c2k2Θ = − Ψ − c2 (c−2Φ˙ ) s dη s s 3 s dη s

2 where cs ≡ p˙γb/ρ˙γb 1 1 c2 = s 3 1 + R • In a CDM dominated expansion Φ˙ = Ψ˙ = 0 and the adiabatic ˙ approximation R/R ω = kcs

[Θ + (1 + R)Ψ](η) = [Θ + (1 + R)Ψ](0) cos(ks) Baryon Peak Phenomenology • Photon-baryon ratio enters in three ways • Overall larger amplitude: 1 [Θ + (1 + R)Ψ](0) = (1 + 3R)Ψ(0) 3 • Even-odd peak modulation of effective temperature 1 [Θ + Ψ] = [±(1 + 3R) − 3R] Ψ(0) peaks 3 1 [Θ + Ψ] − [Θ + Ψ] = [−6R] Ψ(0) 1 2 3

• Shifting of the sound horizon down or `A up √ `A ∝ 1 + R • Actual effects smaller since R evolves Photon Baryon Ratio Evolution • Oscillator equation has time evolving mass

d c2 (c−2Θ˙ ) + c2k2Θ = 0 s dη s s

−2 • Effective mass is is meﬀ = 3cs = (1 + R) • Adiabatic invariant E 1 1 = m ωA2 = 3c−2kc A2 ∝ A2(1 + R)1/2 = const. ω 2 eﬀ 2 s s • Amplitude of oscillation A ∝ (1 + R)−1/4 decays adiabatically as the photon-baryon ratio changes Oscillator: Take Three and a Half • The not-quite-so simple harmonic oscillator equation is a forced harmonic oscillator d k2 d c2 (c−2Θ˙ ) + c2k2Θ = − Ψ − c2 (c−2Φ) s dη s s 3 s dη s changes in the gravitational potentials alter the form of the acoustic oscillations • If the forcing term has a temporal structure that is related to the frequency of the oscillation, this becomes a driven harmonic oscillator • Term involving Ψ is the ordinary gravitational force • Term involving Φ involves the Φ˙ term in the continuity equation as a (curvature) perturbation to the scale factor Potential Decay • Matter-to-radiation ratio

ρm 2 a ≈ 24Ωmh −3 ρr 10

of order unity at recombination in a low Ωm universe • Radiation is not stress free and so impedes the growth of structure

2 2 k Φ = 4πGa ρr∆r

−4 ∆r ∼ 4Θ oscillates around a constant value, ρr ∝ a so the Netwonian curvature decays. • General rule: potential decays if the dominant energy component has substantial stress ﬂuctuations, i.e. below the generalized sound horizon or Jeans scale Radiation Driving • Decay is timed precisely to drive the oscillator - close to fully coherent

[Θ + Ψ](η) = [Θ + Ψ](0) + ∆Ψ − ∆Φ 1 5 = Ψ(0) − 2Ψ(0) = Ψ(0) 3 3 • 5× the amplitude of the Sachs-Wolfe effect! • Coherent approximation is exact for a photon-baryon ﬂuid but reality is reduced to ∼ 4× because of neutrino contribution to radiation • Actual initial conditions are Θ + Ψ = Ψ/2 for radiation domination but comparison to matter dominated SW correct External Potential Approach • Solution to homogeneous equation

(1 + R)−1/4cos(ks) , (1 + R)−1/4sin(ks) • Give the general solution for an external potential by propagating impulsive forces √ 3 1 (1 + R)1/4Θ(η) = Θ(0)cos(ks) + Θ(0)˙ + R˙ (0)Θ(0) sin ks k 4 √ 3 Z η + dη0(1 + R0)3/4sin[ks − ks0]F (η0) k 0 where R˙ k2 F = −Φ¨ − Φ˙ − Ψ 1 + R 3 • Useful if general form of potential evolution is known Damping • Tight coupling equations assume a perfect ﬂuid: no viscosity, no heat conduction • Fluid imperfections are related to the mean free path of the photons in the baryons

−1 λC = τ˙ where τ˙ = neσT a is the conformal opacity to Thompson scattering • Dissipation is related to the diffusion length: random walk approximation √ p p λD = NλC = η/λC λC = ηλC the geometric mean between the horizon and mean free path

• λD/η∗ ∼ few %, so expect the peaks :> 3 to be affected by dissipation Equations of Motion • Continuity

k Θ˙ = − v − Φ˙ , δ˙ = −kv − 3Φ˙ 3 γ b b where the photon equation remains unchanged and the baryons

follow number conservation with ρb = mbnb • Euler k v˙ = k(Θ + Ψ) − π − τ˙(v − v ) γ 6 γ γ b a˙ v˙ = − v + kΨ +τ ˙(v − v )/R b a b γ b

where the photons gain an anisotropic stress term πγ from radiation viscosity and a momentum exchange term with the baryons and are compensated by the opposite term in the baryon Euler equation Viscosity • Viscosity is generated from radiation streaming from hot to cold regions • Expect

k π ∼ v γ γ τ˙ generated by streaming, suppressed by scattering in a wavelength of the ﬂuctuation. Radiative transfer says

k π ≈ 2A v γ v γ τ˙

where Av = 16/15 k k v˙ = k(Θ + Ψ) − A v γ 3 v τ˙ γ Oscillator: Penultimate Take • Adiabatic approximation ( ω a/a˙ ) k Θ˙ ≈ − v 3 γ • Oscillator equation contains a Θ˙ damping term d k2c2 k2 d c2 (c−2Θ˙ ) + s A Θ˙ + k2c2Θ = − Ψ − c2 (c−2Φ˙ ) s dη s τ˙ v s 3 s dη s

• Heat conduction term similar in that it is proportional to vγ and is suppressed by scattering k/τ˙. Expansion of Euler equations to leading order in kτ˙ gives R2 A = h 1 + R since the effects are only signiﬁcant if the baryons are dynamically important Oscillator: Final Take • Final oscillator equation

d k2c2 k2 d c2 (c−2Θ˙ ) + s [A + A ]Θ˙ + k2c2Θ = − Ψ − c2 (c−2Φ˙ ) s dη s τ˙ v h s 3 s dη s • Solve in the adiabatic approximation Z Θ ∝ exp(i ωdη)

k2c2 −ω2 + s (A + A )iω + k2c2 = 0 (1) τ˙ v h s Dispersion Relation • Solve h ω i ω2 = k2c2 1 + i (A + A ) s τ˙ v h i ω ω = ±kc 1 + (A + A ) s 2 τ˙ v h i kc = ±kc 1 ± s (A + A ) s 2 τ˙ v h • Exponentiate

Z Z 1 c2 exp(i ωdη) = e±iks exp[−k2 dη s (A + A )] 2 τ˙ v h ±iks 2 = e exp[−(k/kD) ] (2)

• Damping is exponential under the scale kD Diffusion Scale • Diffusion wavenumber Z 1 1 16 R2 k−2 = dη + D τ˙ 6(1 + R) 15 (1 + R) • Limiting forms Z −2 1 16 1 lim kD = dη R→0 6 15 τ˙ Z −2 1 1 lim kD = dη R→∞ 6 τ˙ • Geometric mean between horizon and mean free path as expected from a random walk

2π 2π −1 1/2 λD = ∼ √ (ητ˙ ) kD 6 Astro 448 Polarization Stokes Parameters • Polarization state of radiation in direction nˆ described by the ∗ intensity matrix Ei(nˆ)Ej (nˆ) , where E is the electric ﬁeld vector and the brackets denote time averaging. • As a hermitian matrix, it can be decomposed into the Pauli basis

P = C E(nˆ) E†(nˆ)

= Θ(nˆ)σ0 + Q(nˆ) σ3 + U(nˆ) σ1 + V (nˆ) σ2 , where ! ! ! ! 1 0 0 1 0 −i 1 0 σ0 = σ1 = σ2 = σ3 = 0 1 1 0 i 0 0 −1

• Stokes parameters recovered as Tr(σiP)/2 Linear Polarization ∗ ∗ ∗ ∗ • Q ∝ hE1E1 i − hE2E2 i, U ∝ hE1E2 i + hE2E1 i. • Counterclockwise rotation of axes by θ = 45◦ √ √ 0 0 0 0 E1 = (E1 − E2)/ 2 ,E2 = (E1 + E2)/ 2

0 0∗ 0 0∗ ◦ 0 • U ∝ hE1E1 i − hE2E2 i, difference of intensities at 45 or Q • More generally, P transforms as a tensor under rotations and

Q0 = cos(2θ)Q + sin(2θ)U U 0 = − sin(2θ)Q + cos(2θ)U • or

Q0 ± iU 0 = e∓2iθ[Q ± iU] acquires a phase under rotation and is a spin ±2 object Coordinate Independent Representation • Two directions: orientation of polarization and change in amplitude, i.e. Q and U in the basis of the Fourier wavevector for small sections of sky are called E and B components Z E(l) ± iB(l) = − dnˆ[Q0(nˆ) ± iU 0(nˆ)]e−il·nˆ Z = −e∓2iφl dnˆ[Q(nˆ) ± iU(nˆ)]eil·nˆ

• For the B-mode to not vanish, the polarization must point in a direction not related to the wavevector - not possible for density ﬂuctuations in linear theory • Generalize to all-sky: plane waves are eigenmodes of the Laplace operator on the tensor P. Spin Harmonics • Laplace Eigenfunctions

2 ∇ ±2Y`m[σ3 ∓ iσ1] = −[l(l + 1) − 4]±2Y`m[σ3 ∓ iσ1] • Spin s spherical harmonics: orthogonal and complete Z ∗ dnˆsY`m(nˆ)sY`m(nˆ) = δ``0 δmm0

X ∗ 0 0 0 sY`m(nˆ)sY`m(nˆ ) = δ(φ − φ )δ(cos θ − cos θ ) `m

where the ordinary spherical harmonics are Y`m = 0Y`m • Given in terms of the rotation matrix r 2` + 1 Y (βα) = (−1)m D` (αβ0) s `m 4π −ms Statistical Representation • All-sky decomposition X [Q(nˆ) ± iU(nˆ)] = [E`m ± iB`m]±2Y`m(nˆ) `m • Power spectra

∗ EE hE`mE`mi = δ``0 δmm0 C` ∗ BB hB`mB`mi = δ``0 δmm0 C` • Cross correlation

∗ ΘE hE`mE`mi = δ``0 δmm0 C` others vanish if parity is conserved Thomson Scattering • Differential cross section

dσ 3 = |Eˆ 0 · Eˆ|2σ , dΩ 8π T 2 ˆ 0 ˆ where σT = 8πα /3me is the Thomson cross section, E and E denote the incoming and outgoing directions of the electric ﬁeld or polarization vector. • Summed over angle and incoming polarization

X Z dσ dnˆ0 = σ dΩ T i=1,2 Polarization Generation • Heuristic: incoming radiation shakes an electron in direction of electric field vector Eˆ 0 • Radiates photon with polarization also in direction Eˆ 0 • But photon cannot be longitudinally polarized so that scattering into 90◦ can only pass one polarization • Linearly polarized radiation like polarization by reflection • Unlike reflection of sunlight, incoming radiation is nearly isotropic • Missing linear polarization supplied by scattering from direction orthogonal to original incoming direction • Only quadrupole anisotropy generates polarization by Thomson scattering Acoustic Polarization • Break down of tight-coupling leads to quadrupole anisotropy of

k π ≈ v γ τ˙ γ 1/2 2 • Scaling kD = (τ/η ˙ ∗) → τ˙ = kDη∗

• Know: kDs∗ ≈ kDη∗ ≈ 10 • So:

k 1 πγ ≈ vγ kD 10

` 1 ∆P ≈ ∆T `D 10 Acoustic Polarization • Gradient of velocity is along direction of wavevector, so polarization is pure E-mode • Velocity is 90◦ out of phase with temperature – turning points of oscillator are zero points of velocity:

Θ + Ψ ∝ cos(ks); vγ ∝ sin(ks) • Polarization peaks are at troughs of temperature power Cross Correlation • Cross correlation of temperature and polarization

(Θ + Ψ)(vγ) ∝ cos(ks) sin(ks) ∝ sin(2ks) • Oscillation at twice the frequency • Correlation: radial or tangential around hot spots • Partial correlation: easier to measure if polarization data is noisy, harder to measure if polarization data is high S/N or if bands do not resolve oscillations • Good check for systematics and foregrounds • Comparison of temperature and polarization is proof against features in initial conditions mimicking acoustic features Reionization • Ionization depth during reionization

Z Z n σ τ(z) = dηn σ a = d ln a e T ∝ (Ω h2)(Ω h2)−1/2(1 + z)3/2 e T H(a) b m Ω h2 Ω h2 −1/2 1 + z 3/2 = b m 0.02 0.15 61

• Quasars say zri ≥ 7 so τ > 0.04 • During reionization, cosmic quadrupole of ∼ 30µK from the Sachs-Wolfe effect scatters into E-polarization • Few percent optical depth leads to fraction of a µK signal • Peaks at horizon scale at recombination: quadrupole source

j2(kD∗) maximal at kD∗ ≈ kη ≈ 2 Breaking degeneracies • First objects, breaking degeneracy of initial amplitude vs optical depth in the peak heights

−2τ C` ∝ e only below horizon scale at reionization • Breaks degeneracies in angular diameter distance by removing an

ambiguity for ISW-dark energy measure, helps in ΩDE − wDE plane Gravitational Wave • Gravitational waves produce a quadrupolar distortion in the temperature of the CMB like effect on a ring of test particles • Like ISW effect, source is a metric perturbation with time dependent amplitude • After recombination, is a source of observable temperature anisotropy – but is therefore confined to low order multipoles • Generated during inflation by quandum fluctuations Gravitational Wave Polarization • In the tight coupling regime, quadrupole anisotropy suppressed by scattering

h˙ π ≈ γ τ˙ • Since gravitational waves oscillate and decay at horizon crossing, the polarization peaks at the horizon scale at recombination not the damping scale • More distinct signature in the B-mode polarization since symmetry of plane wave is broken by the transverse nature of gravity wave polarization Astro 448 Linear Perturbation Theory Covariant Perturbation Theory • Covariant = takes same form in all coordinate systems • Invariant = takes the same value in all coordinate systems • Fundamental equations: Einstein equations, covariant conservation of stress-energy tensor:

Gµν = 8πGTµν µν ∇µT = 0 • Preserve general covariance by keeping all degrees of freedom: 10 for each symmetric 4×4 tensor

1 2 3 4 5 6 7 8 9 10 Metric Tensor • Expand the metric tensor around the general FRW metric

2 2 g00 = −a , gij = a γij .

where the “0” component is conformal time η = dt/a and γij is a 2 spatial metric of constant curvature K = H0 (Ωtot − 1). • Add in a general perturbation (Bardeen 1980)

g00 = −a−2(1 − 2A) , g0i = −a−2Bi , ij −2 ij ij ij g = a (γ − 2HLγ − 2HT ) .

i • (1) A ≡ a scalar potential; (3) B a vector shift, (1) HL a ij perturbation to the spatial curvature; (5) HT a trace-free distortion to spatial metric = (10) Matter Tensor • Likewise expand the matter stress energy tensor around a homogeneous density ρ and pressure p:

0 T 0 = −ρ − δρ , 0 T i = (ρ + p)(vi − Bi) , i i T0 = −(ρ + p)v , i i i T j = (p + δp)δ j + pΠ j ,

• (1) δρ a density perturbation; (3) vi a vector velocity, (1) δp a

pressure perturbation; (5) Πij an anisotropic stress perturbation • So far this is fully general and applies to any type of matter or coordinate choice including non-linearities in the matter, e.g. cosmological defects. Counting DOF’s

20 Variables (10 metric; 10 matter) −10 Einstein equations −4 Conservation equations +4 Bianchi identities −4 Gauge (coordinate choice 1 time, 3 space) —— 6 Degrees of freedom • Without loss of generality these can be taken to be the 6 components of the matter stress tensor • For the background, specify p(a) or equivalently w(a) ≡ p(a)/ρ(a) the equation of state parameter. Scalar, Vector, Tensor • In linear perturbation theory, perturbations may be separated by their transformation properties under rotation and translation. • The eigenfunctions of the Laplacian operator form a complete set

∇2Q(0) = −k2Q(0) S , 2 (±1) 2 (±1) ∇ Qi = −k Qi V , 2 (±2) 2 (±2) ∇ Qij = −k Qij T , • Vector and tensor modes satisfy divergence-free and transverse-traceless conditions

i (±1) ∇ Qi = 0 i (±2) ∇ Qij = 0 ij (±2) γ Qij = 0 Vector and Tensor Modes vs. Vector and Tensor Quantities • A scalar mode carries with it associated vector (curl-free) and tensor (longitudinal) quantities • A vector mode carries and associated tensor (neither longitudinal or transverse) quantities • These are built from the mode basis out of covariant derivatives and the metric

(0) −1 (0) Qi = −k ∇iQ , 1 Q(0) = (k−2∇ ∇ + γ )Q(0) , ij i j 3 ij 1 Q(±1) = − [∇ Q(±1) + ∇ Q(±1)] , ij 2k i j j i Spatially Flat Case • For a spatially ﬂat background metric, harmonics are related to plane waves:

Q(0) = exp(ik · x)

(±1) −i Q = √ (eˆ1 ± ieˆ2)iexp(ik · x) i 2 r3 Q(±2) = − (eˆ ± ieˆ ) (eˆ ± ieˆ ) exp(ik · x) ij 8 1 2 i 1 2 j

where eˆ3 k k. Chosen as spin states, c.f. polarization. • For vectors, the harmonic points in a direction orthogonal to k suitable for the vortical component of a vector • For tensors, the harmonic is transverse and traceless as appropriate for the decompositon of gravitational waves Perturbation k-Modes • For the kth eigenmode, the scalar components become

(0) (0) A(x) = A(k) Q , HL(x) = HL(k) Q , δρ(x) = δρ(k) Q(0) , δp(x) = δp(k) Q(0) ,

the vectors components become

1 1 X (m) (m) X (m) (m) Bi(x) = B (k) Qi , vi(x) = v (k) Qi , m=−1 m=−1 and the tensors components

2 2 X (m) (m) X (m) (m) HT ij(x) = HT (k) Qij , Πij(x) = Π (k) Qij , m=−2 m=−2 Homogeneous Einstein Equations • Einstein (Friedmann) equations:

1 da2 8πG = ρ a dt 3 1 d2a 4πG = − (ρ + 3p) a dt2 3 so that w ≡ p/ρ < −1/3 for acceleration µ • Conservation equation ∇ Tµν = 0 implies

ρ˙ a˙ = −3(1 + w) ρ a Homogeneous Einstein Equations • Counting exercise:

20 Variables (10 metric; 10 matter) −17 Homogeneity and Isotropy −2 Einstein equations −1 Conservation equations +1 Bianchi identities —— 1 Degree of freedom

• without loss of generality choose ratio of homogeneous & isotropic component of the stress tensor to the density w(a) = p(a)/ρ(a). Acceleration Implies Negative Pressure • Role of stresses in the background cosmology

• Homogeneous Einstein equations Gµν = 8πGTµν imply the two Friedman equations (ﬂat universe, or associating curvature 2 ρK = −3K/8πGa )

1 da2 8πG = ρ a dt 3 1 d2a 4πG = − (ρ + 3p) a dt2 3 so that the total equation of state w ≡ p/ρ < −1/3 for acceleration µ • Conservation equation ∇ Tµν = 0 implies ρ˙ a˙ = −3(1 + w) ρ a • so that ρ must scale more slowly than a−2 Questions regarding Dark Energy • Coincidence: given the very different scalings of matter and dark energy with a, why are they comparable now? • Stability: why doesn’t negative pressure imply accelerated collapse? or why doesn’t the vacuum suck? • Answer: stability is associated with stress (pressure) gradients not stress (pressure) itself.

• Example: the cosmological constant wΛ = −1, a constant in time and space – no gradients. • Example: a scalar ﬁeld where w = p/ρ 6= δp/δρ = sound speed. Covariant Scalar Equations • Einstein equations (suppressing 0) superscripts (Hu & Eisenstein 1999):

1 a˙ 1 (k2 − 3K)[H + H + (kB − H˙ )] L 3 T a k2 T a˙ = 4πGa2 δρ + 3 (ρ + p)(v − B)/k , Poisson Equation a 1 d a˙ k2(A + H + H ) + + 2 (kB − H˙ ) L 3 T dη a T = 8πGa2pΠ , a˙ 1 K A − H˙ − H˙ − (kB − H˙ ) a L 3 T k2 T = 4πGa2(ρ + p)(v − B)/k , " # a¨ a˙ 2 a˙ d k2 d a˙ 1 2 − 2 + − A − + (H˙ + kB) a a a dη 3 dη a L 3 1 = 4πGa2(δp + δρ) . 3 Covariant Scalar Equations • Conservation equations: continuity and Navier Stokes

d a˙ a˙ + 3 δρ + 3 δp = −(ρ + p)(kv + 3H˙ ) , dη a a L d a˙ (v − B) 2 K + 4 (ρ + p) = δp − (1 − 3 )pΠ + (ρ + p)A , dη a k 3 k2 µν • Equations are not independent since ∇µG = 0 via the Bianchi identities. • Related to the ability to choose a coordinate system or “gauge” to represent the perturbations. Covariant Scalar Equations • DOF counting exercise

8 Variables (4 metric; 4 matter) −4 Einstein equations −2 Conservation equations +2 Bianchi identities −2 Gauge (coordinate choice 1 time, 1 space) —— 2 Degrees of freedom

• without loss of generality choose scalar components of the stress tensor δp, Π . Covariant Vector Equations • Einstein equations

2 (±1) ˙ (±1) (1 − 2K/k )(kB − HT ) = 16πGa2(ρ + p)(v(±1) − B(±1))/k , d a˙ + 2 (kB(±1) − H˙ (±1)) dη a T = −8πGa2pΠ(±1) .

• Conservation Equations

d a˙ + 4 [(ρ + p)(v(±1) − B(±1))/k] dη a 1 = − (1 − 2K/k2)pΠ(±1) , 2 • Gravity provides no source to vorticity → decay Covariant Vector Equations • DOF counting exercise

8 Variables (4 metric; 4 matter) −4 Einstein equations −2 Conservation equations +2 Bianchi identities −2 Gauge (coordinate choice 1 time, 1 space) —— 2 Degrees of freedom

• without loss of generality choose vector components of the stress tensor Π(±1). Covariant Tensor Equation • Einstein equation

d2 a˙ d + 2 + (k2 + 2K) H(±2) = 8πGa2pΠ(±2) . dη2 a dη T • DOF counting exercise

4 Variables (2 metric; 2 matter) −2 Einstein equations −0 Conservation equations +0 Bianchi identities −0 Gauge (coordinate choice 1 time, 1 space) —— 2 Degrees of freedom • wlog choose tensor components of the stress tensor Π(±2). Arbitrary Dark Components • Total stress energy tensor can be broken up into individual pieces • Dark components interact only through gravity and so satisfy separate conservation equations • Einstein equation source remains the sum of components. • To specify an arbitrary dark component, give the behavior of the stress tensor: 6 components: δp, Π(i), where i = −2, ..., 2. • Many types of dark components (dark matter, scalar fields, massive neutrinos,..) have simple forms for their stress tensor in terms of the energy density, i.e. described by equations of state. • An equation of state for the background w = p/ρ is not sufficient to determine the behavior of the perturbations. Gauge • Metric and matter fluctuations take on different values in different coordinate system • No such thing as a “gauge invariant” density perturbation! • General coordinate transformation:

η˜ = η + T x˜i = xi + Li

free to choose (T,Li) to simplify equations or physics. Decompose these into scalar and vector harmonics.

• Gµν and Tµν transform as tensors, so components in different frames can be related Gauge Transformation • Scalar Metric: a˙ A˜ = A − T˙ − T , a B˜ = B + L˙ + kT , k a˙ H˜ = H − L − T , L L 3 a

H˜T = HT + kL , • Scalar Matter (Jth component):

δρ˜J = δρJ − ρ˙J T ,

δp˜J = δpJ − p˙J T ,

v˜J = vJ + L˙ , • Vector:

˜(±1) (±1) ˙ (±1) ˜ (±1) (±1) (±1) (±1) (±1) ˙ (±1) B = B + L , HT = HT + kL , v˜J = vJ + L , Common Scalar Gauge Choices • A coordinate system is fully speciﬁed if there is an explicit prescription for (T,Li) or for scalars (T,L) • Newtonian:

B˜ = H˜T = 0 Ψ ≡ A˜ (Newtonian potential)

Φ ≡ H˜L (Newtonian curvature)

L = −HT /k 2 T = −B/k + H˙ T /k

Good: intuitive Newtonian like gravity; matter and metric algebraically related; commonly chosen for analytic CMB and lensing work Bad: numerically unstable Example: Newtonian Reduction

• In the general equations, set B = HT = 0:

a˙ (k2 − 3K)Φ = 4πGa2 δρ + 3 (ρ + p)v/k a k2(Ψ + Φ) = 8πGa2pΠ

so Ψ = −Φ if anisotropic stress Π = 0 and

d a˙ a˙ + 3 δρ + 3 δp = −(ρ + p)(kv + 3Φ˙ ) , dη a a d a˙ 2 K + 4 (ρ + p)v = kδp − (1 − 3 )p kΠ + (ρ + p) kΨ , dη a 3 k2 • Competition between stress (pressure and viscosity) and potential gradients Common Scalar Gauge Choices • Comoving:

˜ 0 B =v ˜ (Ti = 0)

HT = 0 ξ = A˜

ζ = H˜L (Bardeen curvature) ∆ = δ˜ (comoving density pert) T = (v − B)/k

L = −HT /k Good: Algebraic relations between matter and metric • Euler equation becomes an algebraic relation between stress and potential 2 3K (ρ + p)ξ = −δp + 1 − pΠ 3 k Common Scalar Gauge Choices • Einstein equation lacks momentum density source

a˙ K ξ − ζ˙ − kv = 0 a k2 • Combine: ζ is conserved if stress ﬂuctuations negligible, e.g. above the horizon if |K| H2

a˙ δp 2 3K p ζ˙ + Kv/k = − + 1 − Π → 0 a ρ + p 3 k2 ρ + p Bad: explicitly relativistic choice Common Scalar Gauge Choices • Synchronous:

A˜ = B˜ = 0 1 η ≡ −H˜ − H˜ L L 3 T

hT = H˜T or h = 6HL Z −1 −1 T = a dηaA + c1a Z L = − dη(B + kT ) + c2

Good: stable, the choice of numerical codes

Bad: residual gauge freedom in constants c1, c2 must be speciﬁed as an initial condition, intrinsically relativistic. Common Scalar Gauge Choices • Spatially Unperturbed:

H˜L = H˜T = 0

L = −HT /k A˜ , B˜ = metric perturbations a˙ −1 1 T = H + H a L 3 T Good: eliminates spatial metric in evolution equations; useful in inflationary calculations (Mukhanov et al) Bad: intrinsically relativistic. • Caution: perturbation evolution is governed by the behavior of stress fluctuations and an isotropic stress fluctuation δp is gauge dependent. Hybrid “Gauge Invariant” Approach • With the gauge transformation relations, express variables of one gauge in terms of those in another – allows a mixture in the equations of motion • Example: Newtonian curvature and comoving density

(k2 − 3K)Φ = 4πGa2ρ∆ ordinary Poisson equation then implies Φ approximately constant if stresses negligible. • Example: Exact Newtonian curvature above the horizon derived through Bardeen curvature conservation Gauge transformation a˙ v Φ = ζ + a k Hybrid “Gauge Invariant” Approach Einstein equation to eliminate velocity

a˙ Ψ − Φ˙ = 4πGa2(ρ + p)v/k a Friedman equation with no spatial curvature

a˙ 2 8πG = a2ρ a 3

With Φ˙ = 0 and Ψ ≈ −Φ a˙ v 2 = − Φ a k 3(1 + w) Hybrid “Gauge Invariant” Approach Combining gauge transformation with velocity relation

3 + 3w Φ = ζ 5 + 3w Usage: calculate ζ from inflation determines Φ for any choice of matter content or causal evolution. • Example: Scalar field (“quintessence” dark energy) equations in comoving gauge imply a sound speed δp/δρ = 1 independent of potential V (φ). Solve in synchronous gauge (Hu 1998). Astro 448 Inflationary Perturbations Scalar Fields • Stress-energy tensor of a scalar field

1 T µ = ∇µϕ ∇ ϕ − (∇αϕ ∇ ϕ + 2V )δµ . ν ν 2 α ν

• For the background hφi ≡ φ0 1 1 ρ = a−2φ˙2 + V p = a−2φ˙2 + −V φ 2 0 φ 2 0

• So for kinetic dominated wφ = pφ/ρφ → 1

• And potential dominated wφ = pφ/ρφ → −1 • A slowly rolling (potential dominated) scalar ﬁeld can accelerate the expansion and so solve the horizon problem or act as a dark energy candidate Equation of Motion • Can use general equations of motion of dictated by stress energy conservation a˙ ρ˙ = −3(ρ + p ) , φ φ φ a to obtain the equation of motion of the background ﬁeld φ a˙ φ¨ + 2 φ˙ + a2V 0 = 0 , 0 a 0

• Likewise for the perturbations φ = φ0 + φ1

−2 ˙ ˙ ˙2 0 δρφ = a (φ0φ1 − φ0A) + V φ1 , −2 ˙ ˙ ˙2 0 δpφ = a (φ0φ1 − φ0A) − V φ1 , −2 ˙ (ρφ + pφ)(vφ − B) = a kφ0φ1 ,

pφπφ = 0 , Equation of Motion • The stress of the perturbations is deﬁned through

v − B a˙ δp = δρ + 3(ρ + p ) φ (1 − c2 ) φ φ φ φ k a φ 2 where cφ ≡ p˙φ/ρ˙φ is the “adiabatic” sound speed

• So for the comoving gauge where vφ = B, δpφ = δρφ so the sound

speed relevant for stability is δpφ/δρφ = 1. Very useful for solving system since in this gauge everything is specified by w(a) • Scalar field fluctuations are stable inside the horizon and are a good candidate for the smooth dark energy • More generally, continuity and Euler equations imply a˙ φ¨ = −2 φ˙ − (k2 + a2V 00)φ + (A˙ − 3H˙ − kB)φ˙ − 2Aa2V 0 . 1 a 1 1 L 0 Inflationary Perturbations • Classical equations of motion for scalar field inflaton determine the evolution of scalar field fluctuations generated by quantum fluctuations • Since the Bardeen or comoving curvature ζ is conserved in the absence of stress fluctuations (i.e. outside the apparent horizon, calculate this and we’re done no matter what happens in between inflation and the late universe (reheating etc.)

• But in the comoving gauge φ1 = 0! no scalar-ﬁeld perturbation • Solution: solve the scalar ﬁeld equation in the dual gauge where

the curvature HL = 0 (and HT = 0 to fix the gauge completely, as the “spatially unperturbed” or “spatially flat” gauge) and transform the result to the comoving gauge Transformation to Comoving Gauge • Scalar field transforms as scalar field ˜ ˙ φ1 = φ1 − φ0T ˜ ˙ • To get to comoving frame φ1 = 0, T = φ1/φ0, and ˜ HT = HT + kL so k a˙ ζ = H − L − T, L 3 a H a˙ φ = H + T − 1 L ˙ 3 a φ0 • Transformation particularly simple from a gauge with

HT = HL = 0, i.e. spatially unperturbed metric a˙ φ ζ = − 1 ˙ a φ0 Scalar Field Eqn of Motion • Scalar field perturbation in spatially unperturbed gauge is simply proportional to resulting Bardeen curvature with the proportionality constant as the expansion rate over roll rate – enhanced • Scalar field fluctuation satisfies classical equation of motion

a˙ φ¨ = −2 φ˙ − (k2 + a2V 00)φ + (A˙ − kB)φ˙ − 2Aa2V 0 . 1 a 1 1 0 • Metric terms may be eliminated through Einstein equations

a˙ −1 A = 4πGa2 (ρ + p )(v − B)/k a φ φ φ a˙ −1 = 4πG φ˙ φ a 0 1 Scalar Field Eqn of Motion • And a˙ −1 a˙ kB = 4πGa2[ δρ + 3 (ρ + p )(v − B)/k] a φ a φ φ φ a˙ −1 a˙ −2 = 4πG[ (φ˙ φ˙ + a2V 0φ ) − (4πGφ˙ )2φ˙ φ + 3φ˙ φ ] a 0 1 1 a 0 0 1 0 1 ˙ • So A − kB ∝ φ1 with proportionality that depends only on the background evolution – Einstein & scalar ﬁeld equations reduce to a single second order diff eq! • Equation resembles a damped oscillator equation with a particular dispersion relation a˙ φ¨ + 2 φ˙ + [k2 + f(η)]φ 1 a 1 1 Exact Equation • Rewrite equations of motion in terms of slow roll parameters but do not require them to be small or constant. • Deviation from de Sitter expansion

3 ≡ (1 + w ) 2 φ 3 φ˙2/a2V = 2 0 1 ˙2 2 1 + 2 φ0/a V 2 2 • Deviation from overdamped limit of d φ0/dt = 0

φ¨ a˙ −1 δ ≡ 0 − 1 φ˙ a Exact Equation • Friedman equations:

a˙ 2 = 4πGφ˙2−1 a 0 d a˙ a˙ 2 = (1 − ) dη a a • Homogenous scalar ﬁeld equation

a˙ φ˙ (3 + δ) = −a2V 0 0 a • Combination a˙ ˙ = 2(δ + ) a Exact equation • Rewrite in u ≡ aφ to remove expansion damping

u¨ + [k2 + g(η)]u = 0

where Mukhanov

g(η) ≡ f(η) + − 2 a˙ 2 a˙ = − [2 + 3δ + 2 + (δ + )(δ + 2)] − δ˙ a a z¨ = − z and

a˙ −1 z ≡ a φ˙ a 0 Slow Roll Limit ˙ a˙ • Slow roll 1, δ 1, δ a

a˙ 2 u¨ + [k2 − 2 ]u = 0 a • or for conformal time measured from the end of inﬂation

η˜ = η − ηend Z a da 1 η˜ = 2 ≈ − aend Ha aH • Compact, slow-roll equation:

2 u¨ + [k2 − ]u = 0 η˜2 Slow Roll Limit • Slow roll equation has the exact solution:

i u = A(k ± )e∓ikη˜ η˜ • For |kη˜| 1 (early times, inside Hubble length) behaves as free oscillator

lim u = Ake∓ikη˜ |kη˜|→∞ • Normalization A will be set by origin in quantum fluctuations of free field Slow Roll Limit • For |kη˜| 1 (late times, Hubble length) fluctuation freezes in i lim u = ± A = ±iHaA |kη˜|→0 η˜

φ1 = ±iHA a˙ 1 ζ = ∓iHA ˙ a φ0 • Slow roll replacement

a˙ 2 1 8πGa2V 3 4π = = 4πG = ˙2 2 2 a φ0 3 2a V mpl • Bardeen curvature power spectrum

3 2 3 2 2 k |ζ| 2k H 2 ∆ζ ≡ 2 = 2 A 2π π mpl Quantum Fluctuations • Simple harmonic oscillator Hubble length

u¨ + k2u = 0 • Quantize the simple harmonic oscillator

uˆ = u(k, η)â + u∗(k, η)â† where u(k, η) satisfies classical equation of motion and the creation and annihilation operators satisfy

[a, a†] = 1, a|0i = 0 • Normalize wavefunction [ˆu, du/dηˆ ] = i 1 u(k, η) = √ e−ikη˜ 2k Quantum Fluctuations • Zero point ﬂuctuations of ground state

hu2i = h0|u†u|0i = h0|(u∗aˆ† + uaˆ)(uaˆ + u∗aˆ†)|0i = h0|aâˆ†|0i|u(k, η˜)|2 = h0|[â, aˆ†] +a ˆ†aˆ|0i|u(k, η˜)|2 1 = |u(k, η˜)|2 = 2k • Classical equation of motion take this quantum fluctuation outside horizon where it freezes in. Slow roll equation • So A = (2k3)1/2 and curvature power spectrum

2 2 1 H ∆ζ ≡ 2 π mpl Tilt • Curvature power spectrum is scale invariant to the extent that H is constant • Scalar spectral index

d ln ∆2 ζ ≡ n − 1 d ln k S d ln H d ln = 2 − d ln k d ln k • Evaluate at horizon crossing where ﬂuctuation freezes

d ln H k dH dη˜ = d ln k −kη˜=1 H dη˜ −kη˜=1 dk −kη˜=1

k 2 1 = (−aH ) = − H −kη˜=1 k2 where aH = −1/η˜ = k Tilt • Evolution of d ln d ln a˙ = − = −2(δ + ) η˜ = 2(δ + ) d ln k d lnη ˜ a • Tilt in the slow-roll approximation

nS = 1 − 4 − 2δ Relationship to Potential • To leading order in slow roll parameters

3 φ˙2/a2V = 2 0 1 ˙2 2 1 + 2 φ0/a V 3 ≈ φ˙2/a2V 2 0 3 a4V 02 a˙ ≈ , (3φ˙ = −a2V 0) a2V 9(˙a/a)2 0 a 1 3 V 0 2 a˙ 2 8πG ≈ , = a2V 6 8πG V a 3 1 V 0 2 ≈ 16πG V so 1 is related to the ﬁrst derivative of potential being small Relationship to Potential • And

φ¨ a˙ −1 δ = 0 − 1 ˙ φ0 a a˙ −1 V 0 (φ˙ ≈ −a2 ) 0 a 3 a2V 0 a˙ −2 V 0V 00 (φ¨ ≈ − (1 + ) + a4 ) 0 3 a 9 1 a2V 0 a2 3 V 0V 00 1 V 00 ≈ − − (1 + ) + − 1 ≈ − a2V 0/3 3 9 8πG V 8πG V so δ is related to second derivative of potential being small. Very flat potential. Gravitational Waves • Gravitational wave amplitude satisfies Klein-Gordon equation (K = 0), same as scalar field a˙ H¨ (±2) + 2 H˙ (±2) + k2H(±2) = 0 . T a T T • Acquires quantum fluctuations in same manner as φ. Lagrangian sets the normalization

r 3 φ → H(±2) 1 T 16πG • Scale-invariant gravitational wave amplitude (each component: (±2) √ NB more traditional notation HT = (h+ ± ih×)/ 6)

2 2 2 16πG H 4 H ∆H = 2 = 2 3 · 2π 2 3π mpl Gravitational Waves 2 4 • Gravitational wave power ∝ H ∝ V ∝ Ei where Ei is the energy scale of inﬂation • Tensor tilt: d ln ∆2 d ln H H ≡ n = 2 = −2 d ln k T d ln k • Consistency relation between tensor-scalar ratio and tensor tilt

2 ∆H 4 2 2 = = − ∆ζ 3 3 • Measurement of scalar tilt and gravitational wave amplitude constrains inﬂationary model in the slow roll context • Comparision of tensor-scalar ratio and tensor tilt tests the idea of slow roll itself Gravitational Wave Phenomenology • Equation of motion a˙ H¨ (±2) + 2 H˙ (±2) + k2H(±2) = 0 . T a T T • has solutions

(±2) HT = C1H1(kη) + C2H2(kη) −m H1 ∝ x jm(x) −m H2 ∝ x nm(x) where m = (1 − 3w)/(1 + 3w) • If w > −1/3 then gravity wave is constant above horizon x 1 and then oscillates and damps • If w < −1/3 then gravity wave oscillates and freezes into some value, just like scalar ﬁeld Gravitational Wave Phenomenology • A gravitational wave makes a quadrupolar (transverse-traceless) distortion to metric • Just like the scale factor or spatial curvature, a temporal variation in its amplitude leaves a residual temperature variation in CMB photons – here anisotropic • Before recombination, anisotropic variation is eliminated by scattering • Gravitational wave temperature effect drops sharply at the horizon scale at recombination ˙ • Source to polarization goes as τ/˙ HT and peaks at the horizon not damping scale • B modes formed as photons propagate – the spatial variation in the plane waves modulate the signal: described by Boltzmann eqn. Astro 448 Boltzmann Formalism Boltzmann Equation • CMB radiation is generally described by the phase space

distribution function for each polarization state fa(x, q, η), where x is the comoving position and q is the photon momentum • Boltzmann equation describes the evolution of the distribution function under gravity and collisions • Low order moments of the Boltzmann equation are simply the covariant conservation equations • Higher moments provide the closure condition to the conservation law (speciﬁcation of stress tensor) and the CMB observable – ﬁne scale anisotropy • Higher moments mainly describe the simple geometry of source projection Liouville Equation • In absence of scattering, the phase space distribution of photons is conserved along the propagation path • Rewrite variables in terms of the photon propagation direction

q = qnˆ, so fa(x, nˆ, q, η) and

d f (x, nˆ, q, η) = 0 dη a ∂ dx ∂ dnˆ ∂ dq ∂ = + · + · + · f ∂η dη ∂x dη ∂nˆ dη ∂q a • For simplicity, assume spatially ﬂat universe K = 0 then dnˆ/dη = 0 and dx = nˆdη

∂ f˙ + nˆ · ∇f +q ˙ f = 0 a a ∂q a Correspondence to Einstein Eqn. • Geodesic equation gives the redshifting term

q˙ a˙ 1 = − − ninjH˙ − H˙ + niB˙ − nˆ · ∇A q a 2 T ij L i • which is incorporated in the conservation and gauge transformation equations • Stress energy tensor involves integrals over the distribution function the two polarization states

Z d3q qµqν T µν = (f + f ) (2π)3 E a b • Components are simply the low order angular moments of the distribution function Angular Moments • Deﬁne the angularly dependent temperature perturbation

1 Z q3dq ˆ Θ(x, n, η) = 2 (fa + fb) − 1 4ργ 2π and likewise for the linear polarization states Q and U • Decompose into normal modes: plane waves for spatial part and spherical harmonics for angular part

r 4π Gm(k, x, nˆ) ≡ (−i)` Y m(nˆ) exp(ik · x) ` 2` + 1 ` r 4π Gm(k, x, nˆ) ≡ (−i)` Y m(nˆ) exp(ik · x) ±2 ` 2` + 1 ±2 ` • In a spatially curved universe generalize the plane wave part Normal Modes • Temperature and polarization ﬁelds

Z 3 d k X (m) Θ(x, nˆ, η) = Θ Gm (2π)3 ` ` `m Z 3 d k X (m) (m) [Q ± iU](x, nˆ, η) = [E ± iB ] Gm (2π)3 ` ` ±2 ` `m • For each k mode, work in coordinates where k k z and so m = 0 represents scalar modes, m = ±1 vector modes, m = ±2 tensor modes, |m| > 2 vanishes. Since modes add incoherently and Q ± iU is invariant up to a phase, rotation back to a ﬁxed coordinate system is trivial. Scalar, Vector, Tensor • Normalization of modes is chosen so that the lowest angular mode for scalars, vectors and tensors are normalized in the same way as the mode function

0 (0) 0 i (0) 0 i j (0) G0 = Q G1 = n Qi G2 ∝ n n Qij ±1 i (±1) ±1 i j (±1) G1 = n Qi G2 ∝ n n Qij ±2 i j (±2) G2 = n n Qij where recall

Q(0) = exp(ik · x)

(±1) −i Q = √ (eˆ1 ± ieˆ2)iexp(ik · x) i 2 r3 Q(±2) = − (eˆ ± ieˆ ) (eˆ ± ieˆ ) exp(ik · x) ij 8 1 2 i 1 2 j Geometrical Projection • Main content of Liouville equation is purely geometrical and describes the projection of inhomogeneities into anisotropies • Spatial gradient term hits plane wave:

r4π nˆ · ∇eik·x = inˆ · keik·x = i kY 0(nˆ)eik·x 3 1 • Dipole term adds to angular dependence through the addition of angular momentum r4π κm κm Y 0Y m = ` Y m + `+1 Y m 3 1 ` p(2` + 1)(2` − 1) `−1 p(2` + 1)(2` + 3) `+1 √ m 2 2 where κ` = ` − m is given by Clebsch-Gordon coefﬁcients. Temperature Hierarchy • Absorb recoupling of angular momentum into evolution equation for normal modes

κm κm Θ˙ (m) = k ` Θ(m) − `+1 Θ(m) − τ˙Θ(m) + S(m) ` 2` + 1 `−1 2` + 3 `+1 ` `

(m) where S` are the gravitational (and later scattering sources; added scattering suppression of anisotropy) • An originally isotropic ` = 0 temperature perturbation will eventually become a high order anisotropy by “free streaming” or simple projection • Original CMB codes solved the full hierarchy equations out to the ` of interest. Integral Solution • Hierarchy equation simply represents geometric projection, exactly as we have seen before in the projection of temperature perturbations on the last scattering surface • In general, the solution describes the decomposition of the source (m) S` with its local angular dependence as seen at a distance x = Dnˆ. • Proceed by decomposing the angular dependence of the plane wave

ik·x X `p 0 e = (−i) 4π(2` + 1)j`(kD)Y` (nˆ) ` m • Recouple to the local angular dependence of G`

X (m) Gm = (−i)`p4π(2` + 1)α (kD)Y m(nˆ) `s `s` ` ` Integral Solution • Projection kernels:

(0) `s = 0, m = 0 α0` ≡ j` (0) 0 `s = 1, m = 0 α1` ≡ j` • Integral solution:

(m) Z η0 Θ (k, η0) X (m) (m) ` = dηe−τ S α (k(η − η)) `s `s` 0 2` + 1 0 `s • Power spectrum:

(m)∗ (m) 2 Z dk X k3hΘ Θ i C = ` ` ` π k (2` + 1)2 m

• Solving for C` reduces to solving for the behavior of a handful of sources Polarization Hiearchy • In the same way, the coupling of a gradient or dipole angular momentum to the spin harmonics leads to the polarization hiearchy:

κm 2m κm E˙ (m) = k 2 ` E(m) − B(m) − 2 `+1 − τE˙ (m) + E (m) ` 2` − 1 `−1 `(` + 1) ` 2` + 3 ` ` κm 2m κm B˙ (m) = k 2 ` B(m) + B(m) − 2 `+1 − τE˙ (m) + B(m) ` 2` − 1 `−1 `(` + 1) ` 2` + 3 ` `

m p 2 2 2 2 where 2κ` = (` − m )(` − 4)/` is given by the Clebsch-Gordon coefﬁcients and E, B are the sources (scattering only). • Note that for vectors and tensors |m| > 0 and B modes may be (m) generated from E modes by projection. Cosmologically B` = 0 Polarization Integral Solution • Again, we can recouple the plane wave angular momentum of the source inhomogeneity to its local angular dependence directly

E(m)(k, η ) Z η0 ` 0 = dηe−τ E (m)(m)(k(η − η)) `s `s` 0 2` + 1 0 B(m)(k, η ) Z η0 ` 0 = dηe−τ E (m)β(m)(k(η − η)) `s `s` 0 2` + 1 0 • The only source to the polarization is from the quadrupole

anisotropy so we only need `s = 2, e.g. for scalars s 3 (` + 2)! j (x) (0)(x) = ` β(0) = 0 2` 8 (` − 2)! x2 2` Truncated Hierarchy

• CMBFast uses the integral solution and relies on a fast j` generator • However sources are not external to system and are deﬁned through the Boltzmann hierarchy itself • Solution: recall that we used this technique in the tight coupling regime by applying a closure condition from tight coupling • CMBFast extends this idea by solving a truncated hierarchy of equations, e.g. out to ` = 25 with non-reﬂecting boundary conditions Thomson Collision Term • Full Boltzmann equation

d f = C[f , f ] dη a,b a b • Collision term describes the scattering out of and into a phase space element • Thomson collision based on differential cross section dσ 3 = |Eˆ 0 · Eˆ|2σ , dΩ 8π T where Eˆ 0 and Eˆ denote the incoming and outgoing directions of the electric ﬁeld or polarization vector. Scattering Calculation • Start in the electron rest frame and in a coordinate system ﬁxed by the scattering plane, spanned by incoming and outgoing directional vectors −nˆ0 · nˆ = cos β, where β is the scattering angle

• Θk: in-plane polarization state; Θ⊥: ⊥-plane polarization state • Transfer probability (constant set by τ˙)

2 0 0 Θk ∝ cos β Θk, Θ⊥ ∝ Θ⊥ • and with the 45◦ axes as

ˆ 1 ˆ ˆ ˆ 1 ˆ ˆ E1 = √ (Ek + E⊥), E2 = √ (Ek − E⊥) 2 2 Stokes Parameters • Deﬁne the temperature in this basis

ˆ ˆ 2 0 ˆ ˆ 2 0 Θ1 ∝ |E1 · E1| Θ1 + |E1 · E2| Θ2 1 1 ∝ (cos β + 1)2Θ0 + (cos β − 1)2Θ0 4 1 4 2 ˆ ˆ 2 0 ˆ ˆ 2 0 Θ2 ∝ |E2 · E2| Θ2 + |E2 · E1| Θ1 1 1 ∝ (cos β + 1)2Θ0 + (cos β − 1)2Θ0 4 2 4 1 0 0 or Θ1 − Θ2 ∝ cos β(Θ1 − Θ2) • Deﬁne Θ, Q, U in the scattering coordinates

1 1 1 Θ ≡ (Θ + Θ ),Q ≡ (Θ − Θ ),U ≡ (Θ − Θ ) 2 k ⊥ 2 k ⊥ 2 1 2 Scattering Matrix • Transfer of Stokes states, e.g.

1 1 1 Θ = (Θ + Θ ) ∝ (cos2 β + 1)Θ0 + (cos2 β − 1)Q0 2 k ⊥ 4 4 • Transfer matrix of Stokes state T ≡ (Θ, Q + iU, Q − iU)

T ∝ S(β)T0

 2 1 2 1 2  cos β + 1 − 2 sin β − 2 sin β 3   S(β) =  − 1 sin2 β 1 (cos β + 1)2 1 (cos β − 1)2  4  2 2 2  1 2 1 2 1 2 − 2 sin β 2 (cos β − 1) 2 (cos β + 1) normalization factor of 3 is set by photon conservation in scattering Scattering Matrix • Transform to a ﬁxed basis, by a rotation of the incoming and outgoing states T = R(ψ)T where   1 0 0   R(ψ) =  0 e−2iψ 0    0 0 e2iψ

giving the scattering matrix

R(−γ)S(β)R(α) = √  0 0 q 3 −2 q 3 2  Y2 (β, α) + 2 5Y0 (β, α) − 2 Y2 (β, α) − 2 Y2 (β, α) r √ 1 4π  0 2iγ −2 2iγ 2 2iγ   − 6 2Y2 (β, α)e 3 2Y2 (β, α)e 3 2Y2 (β, α)e  2 5  √  0 −2iγ −2 −2iγ 2 −2iγ − 6 −2Y2 (β, α)e 3 −2Y2 (β, α)e 3 −2Y2 (β, α)e Addition Theorem for Spin Harmonics • Spin harmonics are related to rotation matrices as r 2` + 1 Y m(θ, φ) = D` (φ, θ, 0) s ` 4π −ms Note: for explicit evaluation sign convention differs from usual (e.g. Jackson) by (−1)m • Multiplication of rotations

X ` ` ` Dmm00 (α2, β2, γ2)Dm00m(α1, β1, γ1) = Dmm0 (α, β, γ) m00 • Implies r X 2` + 1 Y m∗(θ0, φ0) Y m(θ, φ) = (−1)s1−s2 Y −s1 (β, α)eis2γ s1 ` s2 ` 4π s2 ` m Sky Basis • Scattering into the state (rest frame)

Z dnˆ0 C [T] =τ ˙ R(−γ)S(β)R(α)T(nˆ0) , in 4π 2 Z dnˆ0 1 Z X =τ ˙ (Θ0, 0, 0) + τ˙ dnˆ0 P(m)(nˆ, nˆ0)T(nˆ0) . 4π 10 m=−2 where the quadrupole coupling term is P(m)(nˆ, nˆ0) =

 m∗ 0 m q 3 m∗ 0 m q 3 m∗ 0 m  Y2 (nˆ ) Y2 (nˆ) − 2 2Y2 (nˆ ) Y2 (nˆ) − 2 −2Y2 (nˆ ) Y2 (nˆ) √  m∗ 0 m m∗ 0 m m∗ 0 m   − 6Y2 (nˆ ) 2Y2 (nˆ) 3 2Y2 (nˆ ) 2Y2 (nˆ) 3 −2Y2 (nˆ ) 2Y2 (nˆ)  ,  √  m∗ 0 m m∗ 0 m m∗ 0 m − 6Y2 (nˆ ) −2Y2 (nˆ) 3 2Y2 (nˆ ) −2Y2 (nˆ) 3 −2Y2 (nˆ ) −2Y2 (nˆ) expression uses angle addition relation above. We call this term

CQ. Scattering Matrix • Full scattering matrix involves difference of scattering into and out of state

C[T] = Cin[T] − Cout[T] • In the electron rest frame Z dnˆ0 C[T] =τ ˙ (Θ0, 0, 0) − τ˙T + C [T] 4π Q which describes isotropization in the rest frame. All moments have −τ e suppression except for isotropic temperature Θ0. Transformation into the background frame simply induces a dipole term Z dnˆ0 C[T] =τ ˙ nˆ · v + Θ0, 0, 0 − τ˙T + C [T] b 4π Q Source Terms (m) • Temperature source terms Sl (rows ±|m|; ﬂat assumption

 (0) ˙ (0) (0) ˙ (0) (0) 2 ˙ (0)  τ˙Θ0 − HL τv˙ b + B τP˙ − 3 HT √  (±1) (±1)   0τv ˙ + B˙ (±1) τP˙ (±1) − 3 H˙   b 3 T  (±2) ˙ (±2) 0 0τP ˙ − HT where 1 √ P (m) ≡ (Θ(m) − 6E(m)) 10 2 2 • Polarization source term √ (m) (m) E` = −τ˙ 6P δ`,2 (m) B` = 0 Astro 448 Secondary Anisotropy Secondary Anisotropy • CMB photons traverse the large-scale structure of the universe from z = 1000 to the present. • With the nearly scale-invariant adiabatic fluctuations observed in the CMB, structures form from the bottom up, i.e. small scales first, a.k.a. hierarchical structure formation. • First objects reionize the universe between z ∼ 7 − 30 • Main sources of secondary anisotropy • Gravitational: Integrated Sachs-Wolfe effect (gravitational redshift) and gravitational lensing • Scattering: peak suppression, large-angle polarization, Doppler effect(s), inverse Compton scattering Transfer Function • Transfer function transfers the initial Newtonian curvature to its value today (linear response theory) Φ(k, a = 1) Φ(k , a ) T (k) = norm init ) Φ(k, ainit) Φ(knorm, a = 1 • Conservation of Bardeen curvature: Newtonian curvature is a constant when stress perturbations are negligible: above the horizon during radiation and dark energy domination, on all scales during matter domination • When stress fluctuations dominate, perturbations are stabilized by the Jeans mechanism • Hybrid Poisson equation: Newtonian curvature, comoving density

perturbation ∆ ≡ (δρ/ρ)com implies Φ decays

(k2 − 3K)Φ = 4πGρ∆ ∼ η−2∆ Transfer Function • Matter-radiation example: Jeans scale is horizon scale and ∆

freezes into its value at horizon crossing ∆H ≈ Φinit

• Freezing of ∆ stops at ηeq

−2 −2 Φ ∼ (kηeq) ∆H ∼ (kηeq) Φinit

• Conventionally knorm is chosen as a scale between the horizon at matter radiation equality and dark energy domination. • Small correction since growth with a smooth radiation component is logarithmic not frozen • Run CMBfast to get transfer function or use ﬁts Transfer Function −2 −1 • Transfer function has a k fall-off beyond keq ∼ ηeq

k–2

T ( k ) 0.1

wiggles

0.01 0.0001 0.001 0.01 0.1 1 k (h–1 Mpc)

• Additional baryon wiggles are due to acoustic oscillations at recombination – an interesting means of measuring distances Growth Function • Same physics applies to the dark energy dominated universe • Under the dark energy sound horizon, dark energy density frozen. Potential decays at the same rate for all scales

Φ(k , a) g(a) = norm Φ(knorm, ainit)

• Pressure growth suppression: δ ≡ δρm/ρm ∝ aφ

d2g 5 3 dg 3 + − w(z)Ω (z) + [1 − w(z)]Ω (z)g = 0 , d ln a2 2 2 DE d ln a 2 DE

where w ≡ pDE/ρDE and ΩDE ≡ ρDE/(ρm + ρDE) with initial conditions g = 1, dg/d ln a = 0

• As ΩDE → 0 g =const. is a solution. The other solution is the decaying mode, elimated by initial conditions ISW effect • Potential decay leads to gravitational redshifts through the integrated Sachs-Wolfe effect

• Intrinsically a large effect since 2∆Φ = 6Ψinit/3 • But net redshift is integral along along line of sight

Z η0 Θ`(k, η0) −τ ˙ = dηe [2Φ(k, η)]j`(k(η0 − η)) 2` + 1 0 Z η0 −τ = 2Φ(k, ηMD) dηe g˙(D)j`(kD) 0 • On small scales where k g/g˙ , can pull source out of the integral

Z η0 1r π dηg˙(D)j`(kD) ≈ g˙(D = `/k) 0 k 2` R p evaluated at peak, where we have used dxj`(x) = π/2` ISW effect • Power spectrum 2 Z dk k3hΘ∗(k, η )Θ (k, η )i C = ` 0 ` 0 ` π k (2` + 1)2 2π2 Z = dηDg˙ 2(η)∆2 (`/D, η ) l3 Φ MD 2 • Or l Cl/2π ∝ 1/` for scale invariant potential. This is the Limber equation in spherical coordinates. Projection of 3D power retains only the transverse piece. For a general dark energy model, add in the scale dependence of growth rate on large scales. • Cancellation of redshifts and blueshifts as the photon traverses many crests and troughs of a small scale ﬂuctuation during decay. Enhancement of the ` < 10 multipoles. Difﬁcult to extract from cosmic variance and galaxy. Current ideas: cross correlation with other tracers of structure Gravitational Lensing • Lensing is a surface brightness conserving remapping of source to image planes by the gradient of the projected potential

Z η0 (D − D) φ(nˆ) = 2 dη ∗ Φ(Dnˆ, η) . η∗ DD∗ such that the ﬁelds are remapped as

x(nˆ) → x(nˆ + ∇φ) ,

where x ∈ {Θ, Q, U} temperature and polarization. • Taylor expansion leads to product of ﬁelds and Fourier mode-coupling Flat-sky Treatment • Talyor expand

Θ(nˆ) = Θ(˜ nˆ + ∇φ) 1 = Θ(˜ nˆ) + ∇ φ(nˆ)∇iΘ(˜ nˆ) + ∇ φ(nˆ)∇ φ(nˆ)∇i∇jΘ(˜ nˆ) + ... i 2 i j • Fourier decomposition

Z d2l φ(nˆ) = φ(l)eil·nˆ (2π)2 Z d2l Θ(˜ nˆ) = Θ(˜ l)eil·nˆ (2π)2 Flat-sky Treatment • Mode coupling of harmonics Z Θ(l) = dnˆ Θ(nˆ)e−il·nˆ

Z d2l = Θ(˜ l) − 1 Θ(˜ l )L(l, l ) , (2π)2 1 1 where

L(l, l1) = φ(l − l1)(l − l1) · l1 1 Z d2l + 2 φ(l )φ∗(l + l − l)(l · l )(l + l − l) · l . 2 (2π)2 2 2 1 2 1 2 1 1 • Represents a coupling of harmonics separated by L ≈ 60 peak of deﬂection power Power Spectrum • Power spectra

∗ 0 2 0 ΘΘ hΘ (l)Θ(l )i = (2π) δ(l − l ) Cl , ∗ 0 2 0 φφ hφ (l)φ(l )i = (2π) δ(l − l ) Cl ,

becomes

Z 2 ΘΘ 2 ˜ΘΘ d l1 ˜ΘΘ φφ 2 C = 1 − l R C + C C [(l − l1) · l1] , l l (2π)2 |l−l1| l1 where 1 Z dl R = l4Cφφ . (3) 4π l l Smoothing Power Spectrum ˜ΘΘ • If Cl slowly varying then two term cancel

Z d2l C˜ΘΘ 1 Cφφ(l · l )2 ≈ l2RC˜ΘΘ . l (2π)2 l 1 l • So lensing acts to smooth features in the power spectrum. Smoothing kernel is L ∼ 60 the peak of deﬂection power spectrum

• Because acoustic feature appear on a scale lA ∼ 300, smoothing is a subtle effect in the power spectrum. • Lensing generates power below the damping scale which directly reflect power in deflections on the same scale Polarization Lensing • Polarization field harmonics lensed similarly

Z d2l [Q ± iU](nˆ) = − [E ± iB](l)e±2iφl el·nˆ (2π)2 so that

[Q ± iU](nˆ) = [Q˜ ± iU˜](nˆ + ∇φ) ˜ ˜ i ˜ ˜ ≈ [Q ± iU](nˆ) + ∇iφ(nˆ)∇ [Q ± iU](nˆ) 1 + ∇ φ(nˆ)∇ φ(nˆ)∇i∇j[Q˜ ± iU˜](nˆ) 2 i j Polarization Power Spectra • Carrying through the algebra

Z 2 EE 2 ˜EE 1 d l1 2 φφ C = 1 − l R C + [(l − l1) · l1] C l l 2 (2π)2 |l−l1| ˜EE ˜BB ˜EE ˜BB × [(Cl1 + Cl1 ) + cos(4ϕl1 )(Cl1 − Cl1 )] , Z 2 BB 2 ˜BB 1 d l1 2 φφ C = 1 − l R C + [(l − l1) · l1] C l l 2 (2π)2 |l−l1| ˜EE ˜BB ˜EE ˜BB × [(Cl1 + Cl1 ) − cos(4ϕl1 )(Cl1 − Cl1 )] , Z 2 ΘE 2 ˜ΘE d l1 2 φφ C = 1 − l R C + [(l − l1) · l1] C l l (2π)2 |l−l1| ˜ΘE × Cl1 cos(2ϕl1 ) , • Lensing generates B-modes out of the acoustic polaraization E-modes contaminates gravitational wave signature if 16 Ei < 10 GeV. Reconstruction from the CMB • Correlation between Fourier moments reﬂect lensing potential

0 0 0 0 hx(l)x (l )iCMB = fα(l, l )φ(l + l ) ,

where x ∈ temperature, polarization fields and fα is a fixed weight that reflects geometry

• Each pair forms a noisy estimate of the potential or projected mass - just like a pair of galaxy shears • Minimum variance weight all pairs to form an estimator of the lensing mass Scattering Secondaries • Optical depth during reionization

Ω h2 Ω h2 −1/2 1 + z 3/2 τ ≈ 0.066 b m 0.02 0.15 10 • Anisotropy suppressed as e−τ . Integral solution

Z η0 Θ`(k, η0) −τ (0) = dηe S0 j`(k(η0 − η)) + ... 2` + 1 0

• Isotropic (lare scale) fluctuations not supressed since suppression represents isotropization by scattering • Quadrupole from the Sachs-Wolfe effect scatters into a large angle polarization bump Doppler Effects • Velocity fields of 10−3 and optical depths of 10−2 would imply large Doppler effect due to reionization • Limber approximation says only fluctuations transverse to line of sight survive • In linear theory, transverse fluctuations have no line of sight velocity and so Doppler effect is highly suppressed. • Beyond linear theory: modulate the optical depth in the transverse direction using density fluctuations or ionization fraction fluctuations. Generate a modulated Doppler effect • Linear fluctuations: Vishniac effect; Clusters: kinetic SZ effect; ionization patches: inhomogeneous reionization effect Thermal SZ Effect • Thermal velocities also lead to Doppler effect but first order contribution cancels because of random directions 2 • Residual effect is of order v τ ≈ Te/me τ and can reach a sizeable

level for clusters with Te ≈ 10keV. • Raleigh-Jeans decrement and Wien enhancement described by second order collision term in Boltzmann equation: Kompaneets equation • Clusters are rare objects so contribution to power spectrum suppressed, but may have been detected by CBI/BIMA: extremely

sensitive to power spectrum normalization σ8 • White noise on large-scales (l < 2000), turnover as cluster proﬁle is resolved Astro 448 Data Pipeline Gaussian Statistics • Statistical isotropy says two-point correlation depends only on the power spectrum X Θ(nˆ) = Θ`mY`m(nˆ) `m ∗ ΘΘ hΘ`mΘ`0m0 i = δ``0 δmm0 C`

m • Reality of field says Θ`m = (−1) Θ`(−m) • For a Gaussian random field, power spectrum defines all higher order statistics, e.g.

hΘ`1m1 Θ`2m2 Θ`3m3 Θ`4m4 i

m1+m2 ΘΘ ΘΘ = (−1) δ`1`3 δm1(−m3)δ`2`4 δm2(−m4)C`1 C`2 + all pairs Idealized Statistical Errors • Take a noisy estimator of the multipoles in the map ˆ Θ`m = Θ`m + N`m and take the noise to be statistically isotropic

∗ NN hN`mN`0m0 i = δ``0 δmm0 C` • Construct an unbiased estimator of the power spectrum ˆΘΘ ΘΘ hC` i = C`

l 1 X CˆΘΘ = Θˆ ∗ Θˆ − CNN ` 2` + 1 `m `m ` m=−l • Variance in estimator 2 hCˆΘΘCˆΘΘi − hCˆΘΘi2 = (CΘΘ + CNN )2 ` ` ` 2` + 1 ` ` Cosmic and Noise Variance • RMS in estimator is simply the total power spectrum reduced by p 2/Nmodes where Nmodes is the number of m-mode measurements NN • Even a perfect experiment where C` = 0 has statistical variance due to the Gaussian random realizations of the ﬁeld. This cosmic variance is the result of having only one realization to measure. • Noise variance is often approximated as white detector noise. Removing the beam to place the measurement on the sky

2 2 ΘΘ T `(`+1)σ2 T `(`+1)FWHM2/8 ln 2 N` = e = e dT dT

where dT can be thought of as a noise level per steradian of the temperature measurement, σ is the Gaussian beam width, FWHM is the full width at half maximum of the beam Idealized Parameter Forecasts • A crude propagation of errors is often useful for estimation purposes.

• Suppose Cαβ describes the covariance matrix of the estimators for

a given parameter set πα. • Deﬁne F = C−1 [formalized as the Fisher matrix later]. Making

an inﬁnitesimal transformation to a new set of parameters pµ

X ∂πα ∂πβ Fµν = Fαβ ∂pµ ∂pν αβ

• In our case πα are the C` the covariance is diagonal and pµ are cosmological parameters

X 2` + 1 ∂CΘΘ ∂CΘΘ F = ` ` µν ΘΘ NN 2 2(C + C ) ∂pµ ∂pν ` ` ` Idealized Parameter Forecasts • Polarization handled in same way (requires covariance) • Fisher matrix represents a local approximation to the transformation of the covariance and hence is only accurate for well constrained directions in parameter space • Derivatives evaluated by ﬁnite difference • Fisher matrix identiﬁes parameter degeneracies but only the local direction – i.e. all errors are ellipses not bananas Beyond Idealizations: Time Ordered Data • For the data analyst the starting point is a string of “time ordered” data coming out of the instrument (post removal of systematic errors!) • Begin with a model of the time ordered data as

dt = PtiΘi + nt

where i denotes pixelized positions indexed by i, dt is the data in a time ordered stream indexed by t. Number of time ordered data will be of the order 1010 for a satellite! number of pixels 106 − 107.

• The noise nt is drawn from a distribution with a known power spectrum

hntnt0 i = Cd,tt0 Pointing Matrix • The pointing matrix P is the mapping between pixel space and the time ordered data • Simplest incarnation: row with all zeros except one column which just says what point in the sky the telescope is pointing at that time   0 0 1 ... 0    1 0 0 ... 0      P =  0 1 0 ... 0       ......    0 0 1 ... 0

• More generally encorporates differencing, beam, rotation (for polarization) Maximum Likelihood Mapmaking

• What is the best estimator of the underlying map Θi • Likelihood function: the probability of getting the data given the theory L ≡ P [data|theory]. In this case, the theory is the set of

parameters Θi. 1 1 −1 L (d ) = √ exp − (d − P Θ ) C 0 (d 0 − P 0 Θ ) . Θ t N /2 t ti i d,tt t t j j (2π) t det Cd 2

• Bayes theorem says that P [Θi|dt], the probability that the

temperatures are equal to Θi given the data, is proportional to the

likelihood function times a prior P (Θi), taken to be uniform

P [Θi|dt] ∝ P [dt|Θi] ≡ LΘ(dt) Maximum Likelihood Mapmaking

• Maximizing the likelihood of Θi is simple since the log-likelihood is quadratic.

• Differentiating the argument of the exponential with respect to Θi and setting to zero leads immediately to the estimator

ˆ −1 Θi = CN,ijPjtCd,tt0 dt0 ,

tr −1 −1 where CN ≡ (P Cd P) is the covariance of the estimator • Given the large dimension of the time ordered data, direct matrix manipulation is unfeasible. A key simplifying assumption is the 0 stationarity of the noise, that Cd,tt0 depends only on t − t (temporal statistical homogeneity) Power Spectrum • The next step in the chain of inference is the power spectrum extraction. Here the correlation between pixels is modelled through the power spectrum

X 2 CS,ij ≡ hΘiΘji = ∆T,`W`,ij `

• Here W`, the window function, is derived by writing down the expansion of Θ(nˆ) in harmonic space, including smoothing by the beam and pixelization • For example in the simple case of a gaussian beam of width σ it is

proportional to the Legendre polynomial P`(nˆi · nˆj) for the pixel 2 −`(`+1)σ2 separation multiplied by b` ∝ e Band Powers • In principle the underlying theory to extract from maximum likelihood is the power spectrum at every ` • However with a ﬁnite patch of sky, it is not possible to extract multipoles separated by ∆` < 2π/L where L is the dimension of the survey 2 • So consider instead a theory parameterization of ∆T,` constant in bands of ∆` chosen to match the survey forming a set of band

powers Ba • The likelihood of the bandpowers given the pixelized data is

1 1 L (Θ ) = √ exp − Θ C−1 Θ B i N /2 i Θ,ij j (2π) p det CΘ 2

where CΘ = CS + CN and Np is the number of pixels in the map. Band Power Esitmation

• As before, LB is Gaussian in the anisotropies Θi, but in this case

Θi are not the parameters to be determined; the theoretical

parameters are the Ba, upon which the covariance matrix depends. • The likelihood function is not Gaussian in the parameters, and there is no simple, analytic way to ﬁnd the maximum likelihood bandpowers • Iterative approach to maximizing the likelihood: take a trial point (0) Ba and improve estimate based a Newton-Rhapson approach to ﬁnding zeros

ˆ ˆ(0) ∂ ln LB Ba = Ba + FB,ab ∂Bb ˆ(0) 1 −1 −1 ∂CΘ,jk −1 −1 ∂CΘ,ji = Ba + FB,ab ΘiCΘ,ij CΘ,klΘl − CΘ,ij , 2 ∂Bb ∂Bb Fisher Matrix • The expectation value of the local curvature is the Fisher matrix

2 ∂ ln LΘ FB,ab ≡ − ∂Ba∂Bb

1 −1 ∂CΘ,jk −1 ∂CΘ,li = CΘ,ij CΘ,kl . 2 ∂Ba ∂Bb • This is a general statement: for a gaussian distribution the Fisher matrix 1 F = Tr[C−1C C−1C ] ab 2 ,a ,b • Kramer-Rao identity says that the best possible covariance matrix on a set of parameters is C = F−1 • Thus, the iteration returns an estimate of the covariance matrix of

the estimators CB Cosmological Parameters • The probability distribution of the bandpowers given the

cosmological parameters ci is not Gaussian but it is often an adequate approximation

1 1 L (Bˆ ) ≈ √ exp − (Bˆ − B )C−1 (Bˆ − B ) c a N /2 a a B,ab b b (2π) c detCB 2 • Grid based approaches evaluate the likelihood in cosmological parameter space and maximize • Faster approaches monte carlo the exploration of the likelihood space intelligently (“Monte Carlo Markov Chains”) • Since the number of cosmological parameters in the working

model is Nc ∼ 10 this represents a ﬁnal radical compression of information in the original timestream which recall has up to 10 Nt ∼ 10 data points. Parameter Forecasts • The Fisher matrix of the cosmological parameters becomes

∂Ba −1 ∂Bb Fc,ij = CB,ab . ∂ci ∂cj which is the error propagation formula discussed above • The Fisher matrix can be more accurately deﬁned for an experiment by taking the pixel covariance and using the general formula for the Fisher matrix of gaussian data • Corrects for edge effects with the approximate effect of

ΘΘ ΘΘ X (2` + 1)fsky ∂C ∂C F = ` ` µν ΘΘ NN 2 2(C + C ) ∂pµ ∂pν ` ` `

where the sky fraction fsky quantiﬁes the loss of independent modes due to the sky cut