Baumann's Cosmology (7)

Observations of CMB have played a pivotal role in the standard cosmological model. They provided the first evidence that the primordial fluctuations originated from quantum fluctuations during period of inflation. The fluctuations were captured when they still small and accurately described by linear perturbation theory. The physics of the CMB can be understood from first principle unlike the nonlinear structure formation in Chapter 5. Small fluctuations in the primordial plasma evolve under a well-defined set of equations, allowing very accurate predictions of the expected CMB temperature anisotropies.
   Fig. 7.1 is the stunning image of the universe in only 370,000 years old. It shows variations of intensity of the CMB photons at time of photon decoupling. The CMB temperature fluctuations are analyzed stastically as a function of their angular separation. The result is the angular power spectrum shown in Fig. 7.2 which shows a fit of the theoretical prediction for the CMB spectrum to data remarkably. It's usefule to separate them into three regimes as I, II and III:

  • Region I: Large angle correlations are sourced by fluctuations that were still outside the horizon at recombination. These are a direct probe of the initial conditions.
  • Region II: Perturbations with shorter wavelengths entered the horizon before recombination. Inside the horizon the perturbations in the tightly-coupled photon-baryon fluid  propagate as sound waves supported by the large photon pressure. The oscillation frequency is a function of their wavelength, so that different modes are captured at different moments in their evolution at photon decoupling. This is the origin of the oscillatory pattern seen in the angular power spectrum.
  • Region III: On small scales, the random diffusion of the photon has led to a damping of the eave amplitude. This suppresses the amplitude of the CMB power spectrum on small angular scales (large multipole moments).

The goal is to understand how the cosmological parameters determine the shape of the power spectrum, so that we can understand how the CMB measurements constrain these parameters. The evolution of fluctuations in a hydrodynamic approximation will be presented. This approximation works well, but breaks down below the mean free path of the photons. A fully self-consistent treatment of the CMB anisotropies by Boltzmann equation will be given in Appendix B.

          7.1 Anisotropies in the First Light
The largest anisotropy in the CMB is a temperature dipole of magnitude 3.36 mK, which is due to the motion of the Solar system with respect to the CMB rest frame. As Problem 7.1 the motion creates a Doppler shift and hence a temperature anisotropy
    (7.1)   𝛿𝛵(𝐧̂)/𝛵 = 𝐧̂ ⋅ 𝐯 = 𝑣 cos(𝜃),
where 𝐧̂ is a unit vector pointing in the observer's line of sight,which is the opposite to the momentum of the incoming radiation 𝐩̂ and 𝐯 is the velocity of the observer (see Fig. 7.3). The temperature is higher if we move toward the radiation (𝐧̂ ⋅ 𝐯 = 𝑣) and smaller if we move away from it (𝐧̂ ⋅ 𝐯 = -𝑣). Fitting this dipolar anisotropy to the data, we find 𝑣 = 368 km/s.². After subtracting the dipole, we are left with the cosmological signal in Fig. 7.1.

          7.1.1 Angular Power Spectrum
Let 𝛵(𝐧̂) ≡ 𝛵̄₀[1 + 𝛩(𝐧̂)] be the measured CMB temperature in the direction 𝐧̂ in the sky, where 𝛵̄₀ is the average CMB temperature today and we introduced the fractional temerature fluctuation 𝛩(𝐧̂) ≡ 𝛿𝛵(𝐧̂)/𝛵̄₀. Comparing the temperature at two distinct 𝐧̂ and 𝐧̂ʹ (see Fig. 7.4) gives the two point correlation function
    (7.2)   𝐶(𝜃) ≡ ⟨𝛩(𝐧̂)𝛩(𝐧̂ʹ)⟩,
where cos(𝜃) = 𝐧̂ ⋅ 𝐧̂ʹ. The angle bracket denote an average over an ensemble universe. Our universe is only one member of this ensemble, but we can estimate the ensemble average by dividing the CMB into independent patches and averaging over the correlations in each patch. For large-angle correlations, the number of independent patches is small and this estimate will have a large variance.
   Given that we observe fluctuations on the spherical surface of last scattering, it is convenient to expand the temperature field in spherical harmonics
    (7.3)   𝛩(𝐧̂) = ∑^∞_𝑙=2∑^𝑙_𝑚=-𝑙 𝑎_𝑙𝑚𝑌_𝑙𝑚(𝐧̂),
where the expansion coefficients 𝑎_𝑙𝑚 are called multipole moments. A few relevant mathematical properties of the spherical harmonics are reviewed in Appendix D. There are various phase conventions for the 𝑌_𝑙𝑚; We will adopt 𝑌^*_𝑙𝑚 = (-1)^𝑚, so that 𝑎^*_𝑙𝑚 = (-1)^𝑚𝑎^*_𝑙,-𝑚 for a real field 𝛩(𝐧̂). The two-point function of the multipole moments is defined as
    (7.4)   ⟨𝑎^*_𝑙𝑚𝑎^*_𝑙ʹ𝑚ʹ⟩ = 𝐶_𝑙𝛿_{𝑙𝑙ʹ𝑚𝑚ʹ},
where 𝐶_𝑙 is the angular power spectrum and the Kronecker deltas are a consequence of statistical isotropy. The angular power spectrum is the harmonic space equivalent of the two-point correlation function in real space. Substituting (7.3)  into (7.2), we get
    (7.5)   𝐶(𝜃) = ⟨𝛩(𝐧̂)𝛩(𝐧̂ʹ)⟩ = ∑_𝑙𝑚∑_𝑙ʹ𝑚ʹ⟨𝑎^*_𝑙𝑚𝑎^*_{𝑙ʹ,𝑚ʹ}⟩𝑌_𝑙𝑚(𝐧̂)𝑌^*_𝑙ʹ𝑚ʹ(𝐧̂ʹ) = ∑_𝑙𝐶_𝑙∑_𝑚𝑌_𝑙𝑚(𝐧̂)𝑌^*_𝑙ʹ𝑚ʹ(𝐧̂ʹ) = ∑_𝑙(2𝑙 + 1)/4π 𝐶_𝑙𝑃_𝑙(cos(𝜃)),
where 𝑃_𝑙(cos(𝜃)) are Legendre polynomials (see Appendix D). Wee see that the moments of the angular power spectrum appear as coefficients in an expansion of 𝐶(𝜃) in terms of Legendre polynomials. Using the orthogonality of the Legendre polynomials, we can also write
    (7.6)   𝐶_𝑙 = 2π ∫¹_-1d cos(𝜃) 𝐶(𝜃)𝑃_𝑙(cos(𝜃)).
   The variance of the temperature anisotropy field is
    (7.7)   𝐶₀ = ∑_𝑙(2𝑙 + 1)/4π 𝐶_𝑙 ≈ ∫ d ln 𝑙 (2𝑙 + 1) 𝑙𝐶_𝑙/4π ≈ ∫ d ln 𝑙 𝑙(𝑙 + 1)𝐶_𝑙/2π,
where the final equality holds for 𝑙 ≫ 1. The power per logarithmic interval in 𝑙 is
    (7.8)   ∆²_𝛵 ≡ 𝑙(𝑙 + 1)/2π 𝐶_𝑙𝛵̄₀².
We usually plot the CMB power spectrum as ∆²_𝛵, which on large scales (small multiples) will be independent of 𝑙 if the primordial fluctuations are scale invariant (See Section 7.3.2). Note that our ∆²_𝛵 is the same as 𝓓_𝑙 in the Plank papers.
   Finally we return to the cosmic variance. For fixed 𝑙, we have 2𝑙 + 1 different 𝑎_𝑙𝑚's, allowing for 2𝑙 + 1 independent of 𝑙 independent estimates of the true 𝐶_𝑙's. Imagine that we made a full-sky noise-free observation of the temperature field 𝛩(𝐧̂) and extracted its multipole moments 𝑎_𝑙𝑚. An estimator for 𝐶_𝑙 is
    (7.9)   𝐶̂_𝑙 ≡ 1/(2𝑙 + 1) ∑_𝑚∣𝑎_𝑙𝑚∣².
This estimator is unbiased in the sense that ⟨𝐶̂_𝑙⟩ = 𝐶_𝑙. However, the estimator has a nonzero variance corresponding to an irreducible error in our determination of the true power spectrum
    (7.10)   ∆𝐶_𝑙/𝐶_𝑙 ≡ √⟨(𝐶_𝑙 - 𝐶̂_𝑙)⟩/𝐶_𝑙 = √[2/(2𝑙 + 1)].
This cosmic variance is largest for small 𝑙, corresponding to large scales. this explain why the error bars in Fig. 7.2, which contain both cosmic variance and instrumental noise, are largest for small multipoles. Curren measurements from Plank are dominated by cosmic variance up to 𝑙 ~ 2000.

Exercise 7.1   Use Wick's theorem, to derive (7.10)
    (7.11)   ⟨𝑎_𝑙𝑚𝑎^*_𝑙𝑚𝑎_𝑙𝑚ʹ𝑎^*_𝑙𝑚ʹ ⟩ = ⟨𝑎_𝑙𝑚𝑎^*_𝑙𝑚⟩⟨𝑎_𝑙𝑚ʹ𝑎^*_𝑙𝑚ʹ⟩ + ⟨𝑎_𝑙𝑚𝑎^*_𝑙𝑚ʹ⟩ + ⟨𝑎_𝑙𝑚ʹ𝑎^*_𝑙𝑚ʹ⟩⟨𝑎^*_𝑙𝑚𝑎_𝑙𝑚ʹ⟩. [Solution]   Using ⟨𝐶̂_𝑙⟩ = 𝐶_𝑙 and Wick's theorem  
   (a)   ∆𝐶_𝑙² = ⟨(𝐶̂_𝑙 - 𝐶_𝑙)²⟩ = ⟨𝐶̂_𝑙𝐶̂_𝑙 - 2𝐶_𝑙⟨𝐶̂_𝑙⟩ + 𝐶_𝑙² = 1/(2𝑙 + 1)² ∑_𝑚𝑚ʹ⟨𝑎_𝑙𝑚𝑎^*_𝑙𝑚𝑎_𝑙𝑚ʹ𝑎^*_𝑙𝑚ʹ⟩ - 𝐶_𝑙².
   (b)   ∆𝐶_𝑙² = 1/(2𝑙 + 1)² ∑_𝑚𝑚ʹ(⟨𝑎_𝑙𝑚𝑎_𝑙𝑚ʹ⟩⟨𝑎^*_𝑙𝑚𝑎^*_𝑙𝑚ʹ⟩ + ⟨𝑎_𝑙𝑚𝑎^*_𝑙𝑚ʹ⟩⟨𝑎^*_𝑙𝑚𝑎_𝑙𝑚ʹ⟩)
   (c)   ⟨𝑎_𝑙𝑚𝑎_𝑙𝑚ʹ⟩ = (-1)^𝑚ʹ⟨𝑎_𝑙𝑚𝑎^*_𝑙,-𝑚ʹ⟩ = (-1)^𝑚ʹ𝐶_𝑙𝛿_𝑚,-𝑚ʹ,     ⟨𝑎^*_𝑙𝑚𝑎^*_𝑙𝑚ʹ⟩ = (-1)^𝑚ʹ⟨𝑎^*_𝑙𝑚𝑎_𝑙,-𝑚ʹ⟩ = (-1)^𝑚ʹ𝐶_𝑙𝛿_𝑚,-𝑚ʹ,    
        ⟨𝑎_𝑙𝑚𝑎^*_𝑙𝑚ʹ⟩ = 𝐶_𝑙𝛿_𝑚,𝑚ʹ,     ⟨𝑎^*_𝑙𝑚𝑎_𝑙𝑚ʹ⟩ = 𝐶_𝑙𝛿_𝑚,𝑚ʹ.     ⇒    (c)   ∆𝐶_𝑙² = 2/(2𝑙 + 1)² 𝐶_𝑙².     ⇒     ∆𝐶_𝑙/𝐶_𝑙 = √[2/(2𝑙 + 1)].  ▮

          7.1.2 A Road Map
Our main goal is now to understand how the observed power spectrum of CMB anisotropies is related to the spectrum of initial curvature perturbation:
             ∆_𝓡²(𝑘) ≡ 𝑘³/2π² 𝒫_𝓡(𝑘)   →   𝐶_𝑙 = 4π ∫ d ln 𝑘 𝛩_𝑙²(𝑘)∆_𝓡²(𝑘)
The transfer function 𝛩_𝑙(𝑘) that maps ∆_𝓡²(𝑘) to 𝐶₁ captures the evolution of the fluctuation in the primordial plasma, the free streaming of the photon after decoupling and the projection of the anisotropies onto the sky.
             𝓡(0,𝑘)  ^{evolution (§7.4)}→  [𝛿_𝛾,  𝛹,  𝑣_𝑏]_𝜂*  ^{free streaming (§7.2); projection (§7.3)}→  𝛿𝛵(𝐧̂)
where 𝓡 is the initial curvature perturbations, which requires us to follow the evolution of coupled perturbations from early times until decoupling and 𝛿𝛵(𝐧̂) is fluctuations on the surface of last-scattering. The CMB anisotropies depends on the fluctuation in the photon density (𝛿_𝛾), the gravitational potential (𝛹) and the baryon density (𝑣_𝑏) at the time of decoupling (𝜂_*). We will explain that the projection of these inhomogeneities onto the observer's sky leads to nontrivial angular variation. Since the evolution is linear, we can study each Fourier mode separately, and the final anisotropy spectrum is obtained by summing over many Fourier modes weighted by the spectrum of the initial conditions.

          7.2 Photons in a Clumpy Universe
We begin with the free streaming of the photon after decoupling. As the photons travel through the inhomogeneous universe they gain and lose their energy, which will affect the observed temperature anisotropies.

          7.2.1 Gravitational redshift
The evolution of the photons after decoupling is governed by the geodesic equation (see Chapter 2 and Appendix A)
    (7.12)   d𝑃^𝜇/d𝜆 = -𝛤^𝜇_𝜈𝜌𝑃^𝜈𝑃^𝜌,
where 𝑃^𝜇 = d𝑥^𝜇/d𝜆. We will work in Newtonian gauge as
    (7.13)   d𝑠² = 𝑎²(𝜂)[-(1 + 2𝛹)d𝜂² + (1 - 2𝛷)𝛿_𝑖𝑗d𝑥^𝑖d𝑦^𝑗].
The case of tensor perturbations will be explored in Problem 7.6.
    (7.14)   d𝑃^𝜇/d𝜆 = d𝜂/d𝜆 d𝑃^𝜇/d𝜂 = 𝑃⁰ d𝑃^𝜇/d𝜂.
To determine the evolution of the photon energy, we consider the time component of (7.12):
    (7.15)   d𝑃^𝜇/d𝜂 = -𝛤⁰_𝜈𝜌 𝑃^𝜈𝑃^𝜌/𝑃⁰ = -𝛤⁰₀₀𝑃⁰ - 2𝛤⁰_0𝑖𝑃^𝑖 - 𝛤⁰_𝑖𝑗 𝑃^𝑖𝑃^𝑗/𝑃⁰
                    = -(𝓗 + 𝛹ʹ)𝑃⁰ - 2∂_𝑖𝛹𝑃^𝑖 - [𝓗 - 𝛷ʹ - 2𝓗(𝛷 + 𝛹)]𝛿_𝑖𝑗 𝑃^𝑖𝑃^𝑗/𝑃⁰.
The 4-momentum components 𝑃^𝜇 = (𝑃⁰, 𝑃^𝑖) are defined in the coordinate frame, while what an observer actually measures as the photon energy and momentum are components 𝑃^𝜇̂ = (𝑃⁰, 𝑃^𝑖̂) in their local inertial frame. to relate these we use that
    (7.16)   𝜂_𝜇̂𝜈̂𝑃^𝜇̂𝑃^𝜈̂ = 𝑔_𝜇𝜈𝑃^𝜇𝑃^𝜈,     -𝛦² + 𝛿_𝑖𝑗𝑃^𝑖𝑃^𝑗 = 𝑔₀₀(𝑃⁰)² + 𝑔_𝑖𝑗𝑃^𝑖𝑃^𝑗.
The energy and momentum in the local inertial frame can therefore be written as³ [RE 2.55-6]
    (7.17)   𝐸 = √-𝑔₀₀ 𝑃⁰,     𝑝² ≡ 𝑔_𝑖𝑗𝑃^𝑖𝑃^𝑗 = 𝛿_𝑖𝑗𝑃^𝑖̂𝑃^𝑗̂,     and hence we have
    (7.18)   𝑃⁰ = 𝐸/√-𝑔₀₀ = 𝐸/√[𝑎²(1 + 2𝛹)] = 𝐸/𝑎 (1 - 𝛹),      𝑃^𝑖 = 𝐸/√[𝑎²(1 - 2𝛷)𝑝̂^𝑖] = 𝐸/𝑎 (1 + 𝛷)𝑝̂^𝑖,
where 𝑝̂^𝑖 is the unit vector in the photon's direction of propagation. Substituting (7.18) into (7.15), we get
    (7.19)   d/d𝜂 [𝐸/𝑎 (1 - 𝛹)] = -(𝓗 + 𝛹ʹ)𝐸/𝑎 (1 - 𝛹) - 2∂_𝑖𝛹 𝐸/𝑎 (1 + 𝛷)𝑝̂^𝑖 - [𝓗 - 𝛷ʹ - 2𝓗(𝛷 + 𝛹)] 𝐸/𝑎 (1 + 𝛷)²/(1 - 𝛹),    
which, at first order cleans up
    (7.20)   1/𝐸 d𝐸/d𝜂 = -𝓗 + 𝛷ʹ - 𝑝̂^𝑖∂_𝑖𝛹,
where we used that
    (7.21)   d𝛹/d𝜂 = ∂𝛹/∂𝜂 + d𝑥^𝑖/d𝜂 ∂_𝑖𝛹 ≡ 𝛹ʹ + 𝑝̂^𝑖∂_𝑖𝛹.
The right-hand side of (7.20) has a clear physical interpretation: The first term describes the redshifting of photo energy due to the expansion of the universe, 𝐸 ∝ 𝑎^-1. The metric potential 𝛷 can be viewed as a local perturbation of the scale factor, 𝑎˜(𝜂, 𝐱) = 𝑎(𝜂)(1 - 𝛷). The 𝛷ʹ term in (7.20) simply captures the fact that the photon energy now decrease as 𝐸 ∝ 𝑎˜^-1. The third term proportional to  ∂_𝑖𝛹 describes gravitational redshift (or blueshift) as the photon travels out of (or falls into) a gravitational well. Using (7.21) we cab write
    (7.22)   𝑝̂^𝑖∂_𝑖𝛹 = d𝛹/d𝜂 - ∂𝛹/∂𝜂 ≡ d𝛹/d𝜂 - 𝛹ʹ,     so that (7.20) becomes
    (7.23)   d ln(𝑎𝐸)/d𝜂 = -d𝛹/d𝜂 + 𝛷ʹ + 𝛹ʹ.
This equation determines how he inhomogeneities in he spacetime affect the photon energy, beyond the usual redshifting due to the expansion of the universe.

          7.2.2 Line-of-Sight Solution
By integrating the geodesic equation (7.23) we can relate the energy of CMB photon at decoupling to its energy today, when it enters our detectors. We will assume that recombination was nearly instanteneous so that all photons were released at the same time 𝜂*. This assumption is justified by the  visibility function (see Section 3.2.5), which we define in terms of conformal time:
    (7.24)   𝑔(𝜂) ≡ d/d𝜂 𝑒-𝜏 = -𝜏ʹ𝑒-𝜏,
where 𝜏 is the optical depth. The visibility function describes the probability that a photon last scattered in the interval [𝜂, 𝜂 + d𝜂]. Fig. 7.5 shows that the visibility function is sharply peaked at 𝜂* ≈ 373 000 yrs (or 𝑧* ≈ 1090), the moment of last scattering.
   Letting our location be the origin of coordinates, 𝐱0 ≡ 0, the photons in a direction 𝐧̂ were emitted at the point 𝐱* ≡ 𝜒*𝐧̂, where 𝜒* = 𝜂0 - 𝜂* is the distance to the last-scattering surface (in a flat universe). Integrating (7.23) from the time of emission 𝜂* to the time of observation 𝜂0, we get
    (7.25)   ln(𝑎𝐸)0 = ln(𝑎𝐸)* - (𝛹0 - 𝛹*) + ∫𝜂0𝜂*d𝜂 (𝛷ʹ + 𝛹ʹ).
After decoupling, the photon distribution function maintains the same shape, because all photons simply move along geodesics. Since the Bose-Einstein distribution is a function of 𝛦/𝛵, this means 𝛵 ∝ 𝛦 (see Chapter 3). So we can relate
the perturbed photon energy to the temperature anisotropy:
    (7.26)   𝑎𝐸 ∝𝑎𝛵̄ (1 + 𝛿𝛵/𝛵̄),
where 𝛵̄ is the mean temperature. Taylor expanding the logaritims in (7.25) to the first order [ln(1 + 𝑥)] ≈ 𝑥, (𝑥 = 0).] and since 𝑎0𝛵̄0 = 𝑎*𝛵̄*, we find
    (7.27)   𝛿𝛵/𝛵̄∣0 = 𝛿𝛵/𝛵̄∣* + 𝛹* + ∫𝜂0𝜂*d𝜂 (𝛷ʹ + 𝛹ʹ),
where we drop 𝛹0 since it only contribute to the 𝑙 = 0 monopole and is hence unobservable.   [verification needed]

          7.2.3 Fluctuations at Last-Scattering
We see from (7.27) that the observed temperature fluctuation today, 𝛿𝛵₀, depend on the intrinsic temperature fluctuations at the moment of the last-scattering, 𝛿𝛵_*. The latter have two distinct sources:

•  First, there are fluctuations due to photon density at recombination; since 𝜌_𝛾 ∝𝛵⁴ [RE (3.35)], 𝛿𝛵/𝛵 ⊂ 𝛿_𝛾/4. So the regions of enhanced photon density regions are slightly hotter and vice versa.

•  Second, there are there are fluctuations due to bulk velocity of the electrons at recombination. When the photons scatter off these electrons receive an additional Doppler shift of the form 𝛿𝛵/𝛵 ⊂ 𝐩̂ ⋅ 𝐯_𝑒, where 𝐩̂ is a unit vector associated with the momentum of the scattered photon (see Fig. 7.6).⁴ A formal derivation using Boltzmann equation will be in Appendix B. Similarly with (7.1) the Doppler-induced temperature anisotropy due to our motion relative to the CMB frame.

   Since 𝐩̂ = -𝐧̂ and the electrons are strongly interacting with the baryons, we can use 𝐯𝑒 = 𝐯𝑏 and write 𝛿𝛵/𝛵 ⊂ -𝐧̂ ⋅ 𝐯𝑏. Combing two sources we get
    (7.28)   𝛿𝛵/𝛵̄̄∣* = (1/4 𝛿𝛾 - 𝐧̂ ⋅ 𝐯𝑏)*,
where * indicates that a quantity is evaluated at 𝜂* and the position 𝐱* on the surface of last-scattering.
    (7.29)   𝛿𝛵/𝛵̄̄ (𝐧̂) = (1/4 𝛿𝛾 + 𝛹)*[← SW] - (𝐧̂ ⋅ 𝐯𝑏)*[← Doppler] + ∫𝜂0𝜂* d𝜂 (𝛷ʹ + 𝛹ʹ)[← ISW].
where we dropped the subscript 0 on the observed 𝛿𝛵/𝛵̄̄. The answer in (7.29) can be separated into three contributions:

• SW  The first term is the so-called Sachs-Wolfe term [8]. It contains the intrinsic temperature fluctuations associated to the photon density fluctuations at last-scattering, 𝛿_𝛾/4 with the induced temperature perturbation 𝛹_* arising from the the gravitational redshift of the photons. In our conventions the negative 𝛹_* leads to a temperature decrement, as expected since a photon loses energy when climbing out of a potential well. • Doppler  The Doppler term that we discussed before. Since the baryon velocity vanishes on superhorizon scales. We will find that the oscillations in the Doppler term are out of phase with the oscillations in the Sachs-Wolfe term, so that adding the Doppler contribution reduces the contrast between the peaks and troughs in the spectrum (see the left panel of Fig. 7.7. • ISW  The last term describes the additional gravitational redshift due to the evolution of the metric potentials along the line-of-sight. We call this the integrated Sachs-Wolfe (ISW) effect. A nonzero ISW effect requires that potential changes with time. At early times the residual amount of radiation gives 𝛷ʹ ≠ 0, which results in a nonzero early ISW effect. Adding this contribution raises the height of the first peak of the spectrum (see the right panel of Fig. 7.7). Similarly, at late times, dark energy becomes relevant, leading to a late ISW effect, which adds power for low 𝑙.

          7.3 Anisotropies from Inhomogeneities
The line-sight-solution (7.29) shows how the observed CMB temperature fluctuations in a given direction are related to fluctuations at the surface of lass-scattering. In this section how the inhomogeneities in the primordial plasma give rise to correlations between the temperatures in different directions.

          7.3.1 Spatial-to-Angular Projection
The evolution of the primordial fluctuations is best studied in Fourier space because of its linearity. Fig. 7.8 shows how a single Fourier mode cause angular variations in the CMB temperature. Let us for a moment ignore the ISW contribution.The temperature fluctuation in 𝐧̂ is then directly related to those at the point 𝐱_* = 𝜒_*𝐧̂ on the surface of last scattering. For scalar fluctuation, we have 𝐯_𝑏 = 𝑖𝐤̂𝑣_𝑏 and hence the (7.29) can be written as
    (7.30)   𝛩(𝐧̂) = 𝛿𝛵/𝛵̄̄ (𝐧̂) = { d³𝑘/(2π)³ 𝑒𝑖𝐤⋅(𝜒_*𝐧̂) [𝐹(𝜂_*, 𝐤) - 𝑖(𝐤̂ ⋅ 𝐧̂)𝐺(𝜂_*, 𝐤)],
where 𝐹 ≡ 1/4 𝛿_𝛾 + 𝛹 and 𝐺 ≡ 𝑣_𝑏. Since the evolution is linear, it's convenient to factor out the initial curvature perturbation 𝓡_𝑖(𝐤) ≡ 𝓡(0, 𝐤) and write  
    (7.31)   𝛩(𝐧̂) = 𝛿𝛵/𝛵̄̄ (𝐧̂) = { d³𝑘/(2π)³ 𝑒𝑖𝐤⋅(𝜒_*𝐧̂) [𝐹_*(𝑘) - 𝑖(𝐤̂ ⋅ 𝐧̂)𝐺_*(𝑘)]𝓡_𝑖(𝐤),
where 𝐹_*(𝑘) ≡ 𝐹(𝜂_*, 𝐤) /𝓡_𝑖(𝐤) and 𝐺_*(𝑘) ≡ 𝐺(𝜂_*, 𝐤)/𝓡_𝑖(𝐤) are the transfer functions for the SW and Doppler terms. Note that these transfer functions only depend on 𝑘 ≡ ∣𝐤∣, while the initial perturbations are a function of its direction. We then use the Legendre expansion of the exponential
    (7.32)   𝑒𝑖𝐤⋅(𝜒_*𝐧̂) = ∑_𝑙𝑖^𝑙(2𝑙 + 1)𝑗_𝑙𝑃_𝑙(𝐤̂ ⋅ 𝐧̂),
where 𝑗_𝑙 are the spherical Bessel functions (see Appendix D). The factor of 𝑖(𝐤̂ ⋅ 𝐧̂) of Doppler term leads to
    (7.33)   𝑖(𝐤̂ ⋅ 𝐧̂)𝑒^{𝑖𝜒_*𝐤⋅𝐧̂} = d/d(𝑘𝜒_*) 𝑒^{𝑖(𝑘𝜒_*)𝐤̂⋅𝐧̂} = ∑_𝑙𝑖^𝑙(2𝑙 + 1)𝑗_𝑙ʹ(𝑘𝜒_*)𝑃_𝑙(𝐤̂ ⋅ 𝐧̂),
where the prime denote a derivative with respect to the argument of the spherical Bessel function. Substituting (7.32) and (7.33) into (7.31) we get
    (7.34)   𝛩(𝐧̂) = ∑_𝑙𝑖^𝑙(2𝑙 + 1) ∫ d³𝑘/(2π)³ 𝛩_𝑙(𝑘)𝓡_𝑖(𝐤)𝑃_𝑙(𝐤̂ ⋅ 𝐧̂),
where we defined the transfer function
    (7.35)  𝛩_𝑙(𝑘) ≡ 𝐹_*(𝑘)𝑗_𝑙(𝑘𝜒_*) - 𝐺_*(𝑘)𝑗_𝑙ʹ(𝑘𝜒_*).
Inserting (7.34) into the two-point function (7.2) and using
    (7.36-8)  ⟨𝓡_𝑖(𝐤)𝓡_𝑖(𝐤ʹ)⟩ = 2π²/𝑘³ ∆_𝓡²(𝑘)(2π)³𝛿_𝐷(𝐤 + 𝐤ʹ),      𝑃_𝑙(𝐤̂ ⋅ 𝐧̂) = (-1)_𝑙𝑃_𝑙(𝐤̂ ⋅ 𝐧̂),    ∫ d𝐤̂ 𝑃_𝑙(𝐤̂ ⋅ 𝐧̂)𝑃_𝑙ʹ(𝐤̂ ⋅ 𝐧̂ʹ) = 4π/(2𝑙 + 1) 𝑃_𝑙(𝐧̂ ⋅ 𝐧̂ʹ)𝛿_𝑙𝑙ʹ,
we find
    (7.39)   ⟨𝛩(𝐧̂)𝛩(𝐧̂ʹ)⟩ = ∑_𝑙 (2𝑙 + 1)/4π [4π ∫ d ln 𝑘 𝛩_𝑙²(𝑘)∆_𝓡²(𝑘)]𝑃_𝑙(𝐧̂ ⋅ 𝐧̂ʹ).
Comparing this to (7.5), we identify the angular power spectrum as
    (7.40)   𝐶_𝑙 = 4π ∫ d ln 𝑘 𝛩_𝑙²(𝑘)∆_𝓡²(𝑘). We see that the CMB power spectrum is determined by the power spectrum of the primordial curvature perturbations, ∆_𝓡²(𝑘), and the transfer function 𝛩_𝑙(𝑘). The latter describes both the evolution until decoupling and the projection onto the surface of last-scattering.

Exercise 7.2 Using the addition theorem
    (7.41)   𝑃_𝑙(𝐤̂ ⋅ 𝐧̂) = ∑_{∣𝑚∣≤𝑙} 4π/(2𝑙 + 1) 𝑌^*_𝑙𝑚(𝐤̂)𝑌_𝑙𝑚(𝐧̂),
   show that the line-of-sight solution (7.34) can be written in the form (7.3) with
    (7.42)   𝑎_𝑙𝑚 = 4π𝑖^𝑙 ∫ d³𝑘/(2π)³ 𝛩_𝑙(𝑘)𝓡_𝑖(𝐤) 𝑌_𝑙𝑚^*(𝐤̂).    Show that substituting this into (7.4) gives the result in (7.40).
[Solution]   Substituting (7.41) into (7.34) gives
   (a)   𝛩(𝐧̂) = ∑_𝑙 𝑖^𝑙(2𝑙 + 1) ∫ d³𝑘/(2π)³ 𝛩_𝑙(𝑘)𝓡_𝑖(𝐤)∑_{∣𝑚∣≤𝑙} 4π/(2𝑙 + 1) 𝑌^*_𝑙𝑚(𝐤̂)𝑌_𝑙𝑚(𝐧̂) = ∑_𝑙𝑚[4π𝑖^𝑙 ∫ d³𝑘/(2π)³ 𝛩_𝑙(𝑘)𝓡_𝑖𝑌^*_𝑙𝑚(𝐤̂)]𝑌_𝑙𝑚(𝐧̂),
Since 𝛩(𝐧̂) = ∑^∞_𝑙=2∑^𝑙_𝑚=-𝑙 𝑎_𝑙𝑚𝑌_𝑙𝑚(𝐧̂) in (7.3) we know
   (b)   𝑎_𝑙𝑚 = 4π𝑖^𝑙 ∫ d³𝑘/(2π)³ 𝛩_𝑙(𝑘)𝓡_𝑖(𝐤)𝑌^*_𝑙𝑚(𝐤̂).          
Since ⟨𝑎^*_𝑙𝑚𝑎^*_𝑙ʹ𝑚ʹ⟩ = 𝐶_𝑙𝛿_{𝑙𝑙ʹ𝑚𝑚ʹ} in (7.4)
   (c)  ⟨𝑎^*_𝑙𝑚𝑎^*_𝑙ʹ𝑚ʹ⟩ = (4π)²𝑖^𝑙-𝑙ʹ ∫ d³𝑘/(2π)³ ∫ d³𝑘ʹ/(2π)³ 𝛩_𝑙(𝑘)𝛩^*_𝑙(𝑘ʹ)⟨𝓡_𝑖(𝐤)𝓡^*_𝑖(𝐤)⟩𝑌^*_𝑙𝑚(𝐤̂)𝑌_𝑙ʹ𝑚ʹ(𝐤̂ʹ)
                  = (4π)²𝑖^𝑙-𝑙ʹ ∫ d³𝑘/(2π)³ 𝛩_𝑙² 2π³/𝑘³ ∆_𝓡(𝑘) 𝑌^*_𝑙𝑚(𝐤̂)𝑌_𝑙ʹ𝑚ʹ(𝐤̂ʹ) = 4π ∫ d ln 𝑘 𝛩_𝑙²(𝑘)∆_𝓡²(𝑘) 𝛿_𝑙𝑙ʹ𝛿_𝑚𝑚ʹ     ⇒
   (d)  𝐶_𝑙 = 4π ∫ d ln 𝑘 𝛩_𝑙²(𝑘)∆_𝓡²(𝑘).  ▮ 

   Including the ISW contribution, and repeating the same step, we find that the complete transfer function is
    (7.43)   𝛩𝑙(𝑘) = 𝐹*(𝑘)𝑗𝑙(𝑘𝜒*) - 𝐺*(𝑘)𝑗𝑙ʹ(𝑘𝜒*) + ∫𝜂0𝜂*d𝜂 (𝛷ʹ + 𝛹ʹ)𝑗𝑙(𝑘𝜒),
where 𝜒(𝜂) = 𝜂0 - 𝜂. Substituting (7.43) into (7.40), in principle leads to six terms: the power spectrum, 𝐶𝑙SW, that of Doppler term, 𝐶𝑙D, that of the ISW term, 𝐶𝑙ISW, and three cross spectra. In practice, the cross spectra are small, although the cross relation between the SW and ISW terms plays a non-negligible role in the height of the first peak.

          7.3.2 Large Scales: Sachs-Wolfe Effect
The low multipole of the CMB spectrum (𝑙 < 100) are created by superhorizon fluctuations at recombination. From Fig. 7.7 we see that this regime is dominated by the Sachs-Wolfe term. Let us evaluate it for adiabatic initial conditions.
   Since decoupling ocuurs during the matter era, the superhorizon limit of the fluctuation  implies -2𝛹 ≈ -2𝛷 ≈ 𝛿 = 𝛿_𝑚 = 3/4 𝛿_𝛾 and the observed MB temperature fluctuation on large scale are
    (7.44)   𝛩(𝐧̂) ≈ (1/4 𝛿_𝛾 + 𝛹)_* = 1/3 𝛹_* = 1/5 𝓡_𝑖,
where te final equality follows from (6.155). Two points are worth noting: First since there has been no evolution on large scales, this limits of the CMB spectrum directly probes the initial conditions. Second, the gravitational redshift has won over the intrinsic temperature fluctuations. This means that an overdensity at last-sccattering (𝛿_𝛾,* > 0), corresponding to a potential well (𝛹_* < 0), leads to a cold spot in the CMB map (𝛩 < 0). Conversely a hot spot corresponds to an underdensity at last scattering.
   The transfer function (7.43) for large-scale Sachs-Wolfe term then simply is
    (7.45)   𝛩_𝑙^SW(𝑘) = 1/5 𝑗_𝑙(𝑘𝜒_*).
and the power spectrum (7.40) becomes
    (7.46)   𝐶_𝑙^SW = 4π/25 ∫^∞₀ d ln 𝑘 ∆_𝓡²(𝑘)𝑗_𝑙²(𝑘𝜒_*).
For a primordial spectrum of a power law form ∆_𝓡²(𝑘) = 𝛢_𝑠(𝑘/𝑘₀)^{𝑛_𝑠 - 1}, the integral can be evaluated analytically  and we find
    (7.47)   𝐶_𝑙^SW = 4π/25 𝛢_𝑠(𝑘₀𝜒_*)^{1 - 𝑛_𝑠}2^{𝑛_𝑠 - 4}π 𝛤(3 - 𝑛_𝑠)/𝛤²(4 - 𝑛_𝑠/2) 𝛤[𝑙 + (𝑛_𝑠 - 1)/2]/𝛤[𝑙 + 2 - (𝑛_𝑠 - 1)/2].
For 𝑛_𝑠 = 1 the first ratio of gamma functions becomes 𝛤(2)/𝛤²(3/2) = 4/π,^{[verification needed]} while the second ratio is [𝑙(𝑙 + 1)]^-1.[verification needed] Moreover the scale dependence from (𝑘₀𝜒_*)^{1 - 𝑛_𝑠} disappears, as expected for a spectrum with no intrinsic scale, and the power spectrum becomes
    (7.48)   𝑙(𝑙 + 1)/2π 𝐶_𝑙^SW = 𝛢_𝑠/25.
We see that a scale-invariant primordial spectrum 𝑘³𝒫_𝓡(𝑘) = const, corresponds to the combination 𝑙(𝑙 + 1)𝐶_𝑙 being a constant, independent of the multipole moment 𝑙. The amplitude of large-scale CMB spectrum is a direct measure of the amplitude 𝛢_𝑠.

          7.3.3 Small Scales: Sound waves
The larger multipoles (𝑙 > 100) are sourced by small-scale fluctuations on sub-Hubble scales. Here we just briefly comment on the projection effect for these scales.
   For large values of 𝑙, the spherical Bessel function 𝑗_𝑙(𝑥) is peaked near 𝑥 ≈ 𝑙 (cf. Fig. 7.9) and acts like a delta function in the integral (7.40).  The derivative 𝑗_𝑙ʹ is not as sharply peaked at 𝑥 ≈ 𝑙, so the projection from wavenumber 𝑘 to multipole 𝑙 is less sharp for Doppler term.⁵ We will indicate this by writing 𝑥 ~ 𝑙 rather than 𝑥 ≈ 𝑙. Therefore we have
    (7.49)   𝑙(𝑙 + 1)/2π 𝐶_𝑙^SW ~ 𝐹_*²(𝑘)∆_𝓡²(𝑘)∣_{𝑘≈𝑙/𝜒*},
    (7.50)   𝑙(𝑙 + 1)/2π 𝐶_𝑙^D ~ 𝐺_*²(𝑘)∆_𝓡²(𝑘)∣_{𝑘~𝑙/𝜒*},
where 𝐹_*(𝑘) and 𝐺_*(𝑘) are the transfer functions defined in (7.31), which is derived more rigorously in Exercise 7.3. For a scale-invariant initial conditions, ∆_𝓡²(𝑘) = const, 𝐶_𝑙^SW and 𝐶_𝑙^D are determined simply by the squares of the transfer functions at 𝑘 = 𝑙/𝜒_*. Oscillations in 𝑘become oscillations in 𝑙.

Exercise 7.3   For large 𝑙, the sperical Bessel functions can be approximated as
    (7.51)    𝑗_𝑙(𝑥) → { cos(𝑏)cos[𝜈(tan(𝑏) - 𝑏) - π/4]/𝜈√(sin(𝑏)     𝑥 > 𝜈; 0      𝑥 < 𝜈,
   where 𝜈 ≡ 𝑙 + 1/2 and cos(𝑏) ≡ 𝜈/𝑥, with 0 ≤ 𝑏 ≤ π/2. Use this to show the following equations can be written
    (7.52)    𝑙(𝑙 + 1)/2π 𝐶_𝑙^SW = ∫^∞₁ d𝛽/𝛽²√(𝛽² - 1) 𝐹_*²(𝑙𝛽/𝜒_*)∆_𝓡²(𝑙𝛽/𝜒_*),
    (7.53)    𝑙(𝑙 + 1)/2π 𝐶_𝑙^D = ∫^∞₁ d𝛽/𝛽²√(𝛽² - 1) 𝐺_*²(𝑙𝛽/𝜒_*)∆_𝓡²(𝑙𝛽/𝜒_*),
where we see explicitly that the intergrand of the Doppler contribution vanishes for 𝛽 = 1. Given the transfer functions, these integrals are easy to evaluate numerically. Most of the contribution comes from 𝛽 ~ 1, so the intergrals can be performed with a finite cutoff, say 𝛽 = 5, without losing much accuracy [Weinberg].^{[verification needed]}

² The inferred speed has three components: (1) The orbital velocity of the Solar system with respect to the center of our Galaxy; (2)the velocity of our Galaxy with respect to the center-of-mass of the Local Group galaxies; and (3) the velocity of the Local Group with respect to the CMB rest frame. (1) and (2) leads to about 307 km/s, but (3) almost exactly in opposite direction is 626 ± 30 km/s [6]. Part of this (about 220 km/s) can be accounted for by the pull of the nearby Virgo cluster of galaxies. Explaining the rest of the bulk motion of the Local Group remains an open problem (but see [7]).
³ The condition in (7.16) only fixes the relationship between the components in the two frames up to a Lorentz transformation. We assume that the observer is at rest and that their spatial vectors are aligned with the spatial coordinate directions.
⁴ This temperature fluctuation as arising because the photon are emitted with different peculiar velocities at each point on the surface. The projection of these velocities onto the line-of sight describes the Doppler shift of the photon energy.
⁵ In fact he contribution from 𝑥 ≈ 𝑙 actually vanishes in The Doppler integral; see (7.53). For a plane wave travelling in the 𝑧-direction (see Fig. 7.8), the condition 𝑥 ≈ 𝑙 corresponds to lines-of-sight in the 𝑥𝑦-plane. The Doppler effect for such light-of sigh is zero because they are perpendicular to the velocity vectors at the last-scattering surface, 𝐯_𝑏 = 𝑖𝑧̂𝑣_𝑏, so that 𝐯_𝑏 ⋅ 𝐧̂ = 0. Moving away from 𝑥𝑦-plane we start  to picking up nonzer Dopller effects, but 𝑙 < 𝑥.

          7.4 Primordial Sound Waves
To complete the derivation of the CMB power spectrum, we have to compute the transfer functions describing the evolution before recombination:
             𝓡(0,𝑘)  ^{transfer function} →  [𝛿_𝛾,  𝛹,  𝑣_𝑏]_𝜂*
A numerical solution of the transfer function of the photon density contrast 𝛿_𝛾 is shown in Fig. 7.10.

•Sound horizon  Sound waves can only propagate if their wavelengths are smaller than the sound horizon
    (7.54)   𝑟_𝑠(𝜂) ≡ ∫^𝜂₀ d𝜂̄ 𝑐_𝑠(𝜂̄),
where 𝑐_𝑠 is the sound speed. [RE (5.11)] In Problem 7.2 we will show the comoving sound horizon at decoupling is  𝑟_𝑠,* ≈ 135 Mpc. The effect of sound waves will only important for modes with 𝑘𝑟_𝑠,* > 1. Since 𝑙 ~ 𝑘𝜒_*, with 𝜒_* ≈ 14 Gpc, this implies that sound waves should affect the CMB spectrum for 𝑙 > 100. This is indeed what is seen in the left panel of Fig. 7.7.

•Damping scale  On small scales photon diffusion  leads to a damping of the waves. High-energy photons in hot regions can diffuse into cold regions, thereby erasing the temperature difference. This diffusion can be modeled as a random walk, with the step size being photon's mean free path 𝓁_𝛾 = 1/(𝑎𝑛_𝑒𝜎_𝛵). In a time interval ∆𝜂, the number of scattering is 𝛮 = ∆𝜂/𝓁_𝛾 and the mean-squared distance explored by the random walk is ⟨∆𝑟²⟩ = 𝛮𝓁_𝛾² = 𝓁_𝛾∆𝜂. Integrating this from 𝜂 = 0 to a time 𝜂, we get the squared diffusion length
    (7.55)   𝑟_𝐷²(𝜂) ~ ∫^𝜂₀ d𝜂̄ 𝓁_𝛾(𝜂̄).
We see that the diffusion length is roughly the geometric mean between the age of the universe in conformal time and the mean free path of the photons. In Problem 7.3 we will show that the diffusion length at decoupling is 𝑟_𝐷,* ≈ 7 Mpc. We will see in Section 7.4.4 that on scale smaller than the diffusion length, 𝑘𝑟_𝐷,* > 1, the fluctuation are suppressed by a factor of 𝑒_{-2(𝑘𝑟𝐷,*)²}. In harmonic space, the damping scale becomes 𝑙_𝐷 ~ 𝜒_*/√2𝑟_𝐷,* ~ 1400, so we expect the CMB fluctuations to be exponentially damped for 𝑙 > 1400. In Section 7.4.4 we will improve this estimate.

          7.4.1 Photon-Baryon Dynamics
Here we will re-derive the equations Thomson scattering does not exchange energy, so the the continuity equations for the photons and baryons still apply in their original forms [RE (5.14) (6.88) (6.85)]

          Hydrodynamic equations
At linear order, the Thomson scattering does not exchange energy, so the continuity equations for the photons and baryons still apply in their original forms
    (7.56)   𝛿_𝛾ʹ = 4/3 𝑘𝑣_𝛾 + 4𝛷ʹ,
    (7.57)   𝛿_𝑏ʹ = 𝑘𝑣_𝑏 + 3𝛷ʹ.
Momentum on the other hand is exchanged by the scattering, so the Euler equations are modified. The Euler equation for photons becomes
    (7.58)   𝑣_𝛾ʹ + 1/4 𝑘𝛿_𝛾 - 2/3 𝑘𝛱_𝛾 + 𝑘𝛹 = -𝛤(𝑣_𝛾 -𝑣_𝑏),
where the source term on the right-hand side-called drag term-describes the momentum transfer between the photons and baryons. A formal derivation requires the Boltzmann equation and is given in Appendix B. The momentum transfer is proportional to the scattering rate, 𝛤, and to the difference between the photon and baryon velocities, 𝑣_𝛾-𝑏 ≡ 𝑣_𝛾 - 𝑣_𝑏 (also called the slip velocity). Since the scattering tries to make the photons move together with baryons, so that 𝑣_𝛾 → 𝑣_𝑏 for large 𝛤. And the Euler equation for baryons is
    (7.59)   𝑣_𝑏ʹ + 𝓗𝑣_𝑏 + 𝑘𝛹 = 𝛤/𝑅((𝑣_𝛾 - 𝑣_𝑏),
where the right-hand side is fixed by the requirement that the combined density must be conserved. The factor of 1/𝑅 simply arise because the baryons' momentum density is smaller than the photons' by a factor of 𝑅 ≡ 3/4 𝜌̄_𝑏/𝜌̄_𝛾. [RE (6.191)]

          Tight-coupling limit
We first study these equations in the limit of large scattering rate. Let us begin by rearranging (7.59) as
    (7.60)   𝑣_𝛾-𝑏 = 𝑅/𝛤 [𝑣_𝑏ʹ + 𝓗𝑣_𝑏 + 𝑘𝛹].
To lowest order in 𝛤^-1 the slip velocity is small, 𝑣_𝛾-𝑏 ≈ 0. However, since 𝑣_𝛾-𝑏 is multiplied by 𝛤 in the Euler equations, we need the next-to-leading order solution for it. We obtain this by substituting 𝑣_𝑏 ≈ 𝑣_𝛾, into the (7.60)
    (7.61)   𝑣_𝛾-𝑏 = 𝑅/𝛤 [𝑣_𝛾ʹ + 𝓗𝑣_𝛾 + 𝑘𝛹].
Using this in (7.58), we get
    (7.62)   𝑣_𝛾ʹ + 𝑅𝓗/(1 + 𝑅) 𝑣_𝛾 + 1/4(1 + 𝑅) 𝑘𝛿_𝛾 + 𝑘𝛹 = 0,
where we dropped the photon anisotropic stress 𝛱_𝛾 which is suppressed for large 𝛤 (see below). Combining this with the continuity equation (7.56), we find a second-order equation for the photon density contrast
    (7.63)   𝛿_𝛾ʺ + 𝑅ʹ/(1+ 𝑅) 𝛿_𝛾ʹ + 𝑘²𝑐_𝑠²𝛿_𝛾 = -3/4 𝑘²𝛹 + 4𝛷ʺ + 4𝑅ʹ/(1+ 𝑅) 𝛷ʹ,
where  we used 𝑅𝓗 = 𝑅ʹ (since 𝑅 ∝𝑎) and introduce the sound speed 𝑐_𝑠² = [3(1 + 𝑅)]^-1. [RE (6.198)] This is the same as (6.197) obtained in Section 6.4.

          7.4.2 High-Frequency Solution
We will study to the solutions to master equation (7.63). We can make progress by studying special limits. We first consider the high-frequency limits, 𝑘 ≫ 𝓗. Following Weinberg [Weinberg], we split the solution into "fast" and "slow" modes
    (7.64)   𝑓 = 𝑓^fast + 𝑓^slow,
where 𝑓 ≡ {𝛿_𝛾, 𝛹, 𝛷). The fast modes evolve on a scale 𝑘^-1, so that ∂_𝜂𝑓^fast ~ 𝑘𝑓^fast, while the slow modes vary on the Hubble time scale 𝓗^-1, so that ∂_𝜂𝑓^slow ~ 𝓗𝑓^slow. Dark matter fluctuations do not have any fast modes (because the dark matter is pressureless). The fast modes of the gravitational potential is sourced by the radiation fluctuations. From the Poisson equation, we have
    (7.65)   -𝑘²𝛷^fast = 4π𝐺𝑎²𝜌̄_𝛾𝛿_𝛾^fast ≲ 𝓗²𝛿_𝛾^fast,
so that the gravitational potential 𝛷^fast is suppressed by a factor of 𝓗²/𝑘² ≪ 1 in the high-frequency limits. The fast mode of the photon density contrast obeys the homogeneous equation
    (7.66)   𝛿_𝛾ʺ + 𝑅ʹ/(1+ 𝑅) 𝛿_𝛾ʹ + 𝑘²𝑐_𝑠²𝛿_𝛾 = 0     (fast mode).
This equation can be solved in a WKB approximation (see Problem 7.4). Consider the ansatz
    (7.67)   𝛿_𝛾 = 𝛢(𝜂)exp[±𝑖𝑘 ∫^𝜂₀ 𝑐_𝑠(𝜂̄) d𝜂̄] ≡ 𝛢(𝜂)𝑒^{±𝑖𝜑(𝜂)},
where the function 𝛢(𝜂) is slowly varying on the timescale of the oscillations. Taking time derivatives of (7.67). we get
    (7.68)   𝛿_𝛾ʹ = 𝛢ʹ𝑒^±𝑖𝜑 ± 𝑖𝑘𝑐_𝑠𝛢𝑒^±𝑖𝜑 ≈ 𝑖𝑘𝑐_𝑠𝛿_𝛾,    
    (7.69)   𝛿_𝛾ʺ = 𝛢ʹ𝑒^±𝑖𝜑 ± 2𝑖𝑘𝑐_𝑠𝛢ʹ𝑒^±𝑖𝜑 ± 𝑖𝑘𝑐_𝑠ʹ𝛢𝑒^𝑖𝜑 - 𝑘²𝑐_𝑠²𝛢𝑒^𝑖𝜑 ≈ [± 2𝑖𝑘𝑐_𝑠𝛢ʹ/𝛢 ∓ 𝑖𝑘𝑐_𝑠/2 𝑅ʹ/(1+ 𝑅) - 𝑘²𝑐_𝑠²]𝛿_𝛾,
where we used 𝛢ʹ ≪ 𝑖𝑘𝑐_𝑠𝛢, 𝛢ʺ ≪ 𝑖𝑘𝑐_𝑠𝛢ʹ and
    (7.70)   𝑐_𝑠ʹ = d/d𝜂 [1/√3(1 + 𝑅)] = -1/2 𝑐_𝑠 𝑅ʹ/(1+ 𝑅).
Substituting (7.68) and (7.69) into (7.66) then gives
    (7.71)   𝛢ʹ/𝛢 = -1/4 𝑅ʹ/(1+ 𝑅)     ⇒     𝛢(𝜂) ∝(1 + 𝑅)^-1/4,     [taking the derivative of logarithm of both sides]
and the solution of (7.66) takes the form
    (7.72)   𝛿_𝛾^fast = (1 + 𝑅)^-1/4[𝐶 cos(𝜑) + 𝐷 sin(𝜑)].
The integration constants 𝐶 and 𝐷 are fixed by the superhorizon initial conditions. For adiabatic initial conditions, we expect 𝐶 ≫ 𝐷 and oscillating part of the solution is a pure cosine.
   To complete the solution we must add the slow mode of the photon density contrast. In this case we can drop all time derivative terms in (7.63) and the solution becomes
    (7.73)   𝛿_𝛾^slow = -4(1 + 𝑅)𝜓^slow,
where 𝜓^slow is now also sourced by the dark matter perturbations. The full WKB solution is
    (7.74)   𝛿_𝛾 = (1 + 𝑅)^-1/4[𝐶 cos(𝜑) + 𝐷 sin(𝜑)] - 4(1 + 𝑅)𝜓.
We see that the gravitational force term has shifted the zero-point of the oscillations. Since the gravitational potential decays on sub-Hubble scales during the radiation era, the size of the zero-point shift will depend on the time that the mode has spent inside the horizon during the era (which is dedicated by the wavelength of the fluctuation).
   The transfer function for the Sachs-Wolfe term is
    (7.75)   𝐹_*(𝑘) = 1/4 𝛿_𝛾(𝜂_*, 𝐤) + 𝛹(𝜂_*, 𝐤) = 1/4 (1 + 𝑅)^-1/4[𝐶 cos(𝜑_*) + 𝐷 sin(𝜑_*)]/𝓡_𝑖(𝐤) - 𝑅 𝛹(𝜂_*, 𝐤)/𝓡_𝑖(𝐤), [RE (6.55) (6.150) (6.155) (6.165) (7.31) (7.44)]
with 𝜑_* = 𝑘𝑟_𝑠,*. In Section 7.3.3 we saw that the angular power spectrum is given roughly by ∣𝐹_*(𝑘)∣², with 𝑘 ≈ 𝑙/𝜒_*. Since 𝐶 ≫ 𝐷, we expect the peaks of the spectrum to be at [RE (7.3), p 267, (7.49)]
    (7.76)   𝑙_𝑛 = 𝑛π 𝜒_*/𝑟_𝑠,* ≈ 𝑛π (14 Gpc)/(145 Mpc) ≈ 𝑛 × 302,
where 𝑛 ∊ 𝐙(interger). However the prediction does not match the measured locations of the first few peaks, 𝑙₁ = 220.0±0.5, 𝑙₂ = 537.5±0.7 and 𝑙₂ = 810.8±0.7 [11]. A graphical illustration of this mismatch is given in Fig. 7.11. While the prediction (7.76) works well for the peak spacing, the predicted peak locations is shifted by about one fourth of the oscillation period here.

          7.4.3 Semi-Analytic Solution^*
There is no analytic solution for the  dynamics of the photon-baryon fluid valid at all times and for all wavelengths. However Weinberg found an good semi-analytic solution using interpolation between two analytic solution for 𝑘 ≪ 𝑘_eq and 𝑘 ≫ 𝑘_eq.  The following is a summary [Weinberg]. An alternative solution by Hu and Sugiyama [12] is discussed in Problem 7.5.

•  Modes with 𝑘 < 𝑘_eq are outside the horizon during the radiation era and enter the horizon during the matter era. The superhorizon evolution of the fluctuations is known analytically. In the matter era an analytic solution is possible on all scales. By matching this to the superhorizon solution we obtain an analytic solution for 𝑘 < 𝑘_eq (see Fig. 7.12). For realistic parameters, the time of decoupling 𝜂_* relatively close to 𝜂_eq, assuming 𝜂_* ≫ 𝜂_eq is still useful because it motivates an ansaz for the interpolation function, which will apply at 𝜂_*.
•  Modes with 𝑘 > 𝑘_eq instead enter the horizon during the radiation era. In the radiation era an analytic solution exists for all 𝑘. Once the mode is deep inside the horizon, an analytic solution is possible  that remains valid when matter becomes important. By matching the two solutions in the region of overlap-i.e. for subhorizon modes in the radiation era-we obtain an analytic solution for 𝑘 > 𝑘_eq with yhe correct initial conditions  (see Fig. 7.12).
The two analytic solutions will first be derived and then Weinberg's interpolation will be described.

          Long-wavelength solution We first consider modes with 𝑘 < 𝑘_eq. They are derived in Section 6.2 and Problem 6.1. There we found that the superhorizon limit of the photon density contrast in the matter era is
    (7.77)  𝛿_𝛾  = 4/3 𝛿_𝑐 = -8/3 𝛷     (𝑘 < 𝓗 < 𝑘_eq),
where we used that 𝛿_𝑐 = -2𝛷. In terms of the curvature perturbation 𝓡_𝑖(𝐤), we have [RE (6.150) (6.155)]
    (7.78)  𝛹 ≈ 𝛷 = 3/5 𝓡_𝑖,     𝛿_𝛾 = -8/5 𝓡_𝑖     (𝑘 < 𝓗 < 𝑘_eq).
We will use this as an initial condition for the modes entering the horizon during the matter era.
   Initially the effect of baryon is still negligible, 𝑅 ≪ 1, [RE (6.191)] and the (7.63) becomes
    (7.79)  𝛿_𝛾ʺ + 1/3 𝑘²𝛿_𝛾 = -4/3 𝑘²𝛹,
where we used that 𝛷 is a constant during the matter era. This equation is valid on all scales. Its solution is
    (7.80)  𝛿_𝛾 = 𝐶 cos(𝜑) + 𝐷 sin(𝜑) - 4𝛹,
where 𝜑 = 𝑘𝜂/√3. Matching the 𝜑 ≪ 1 limit of (7.80) to the superhorizon limit (7.78), we get  
    (7.81)  𝛿_𝛾 = 4/5 𝓡_𝑖[cos(𝜑) - 3].
Eventually 𝑅 becomes large and with 𝑘 > 𝓗 this happens when they are already deep inside the horizon. Then the WKB solution (7.74) applies and we have
    (7.82)  𝛿_𝛾 = 4/5 𝓡_𝑖[(1 + 𝑅)^-1/4 cos(𝜑) - 3(1 + 𝑅)]     (𝑘 < 𝑘_eq),
where 𝜑 = 𝑘 ∫ d𝜂/√3(1 + 𝑅). This solution is smoothly connected to the solution in (7.81).

          Short-wavelength solution
Next, we consider modes with 𝑘 > 𝑘_eq. which entered the Hubble radius during the radiation era. We first derive an analytic solution in the radiation era and then mach its the high-frequency limit to WKB solution (7.74).
   During the radiation era with 𝑅 ≪ 1 and the effects of the baryons can be ignored. This simplifies the (7.63) to
    (7.83)  𝛿_𝛾ʺ + 1/3 𝑘²𝛿_𝛾 = -4/3 𝑘²𝛹 + 4𝛷ʺ,
where the time dependence of metric potentials must now be account for. Defining 𝑑_𝛾 ≡ 𝛿_𝛾 - 4𝛷 [RE (6.127)] and 𝜑 = 𝑘𝜂/√3, we get
    (7.84)  𝑑_𝛾ʺ + 𝑑_𝛾 = -4(𝛷 + 𝛹),
where the prime is for a derivative respect to 𝜑. The solution is
    (7.85)  𝑑_𝛾 = 𝐶 cos(𝜑) + 𝐷 sin(𝜑) - 4∫^𝜑₀ d𝜑̂ sin(𝜑 - 𝜑̂) (𝛷 + 𝛹)(𝜑̂),
where 𝐶 = -4𝓡_𝑖 and 𝐷 = 0 for adiabatic initial conditions.

Exercise 7.4   Let 𝑆₁ = cos(𝜑) and 𝑆₂ = sin(𝜑) be the two homogeneous solutions of (7.84). Using
    (7.86)   𝐺(𝜑, 𝜑̂) = [𝑆₁(𝜑̂)𝑆₂(𝜑) - 𝑆₁(𝜑)𝑆₂(𝜑̂)]/[𝑆₁(𝜑)𝑆₂ʹ(𝜑) - 𝑆₁ʹ(𝜑)𝑆₂(𝜑)],
   confirm the form of the Green's function in (7.85).
[Solution]     𝐺(𝜑, 𝜑̂) = [cos(𝜑̂)sin(𝜑) - cos(𝜑)sin(𝜑̂)]/[cos²(𝜑) + sin²(𝜑)] = cos(𝜑̂)sin(𝜑) - cos(𝜑)sin(𝜑̂) = sin(𝜑 - 𝜑̂).  ▮

   Using the relation sin(𝜑 - 𝜑̂) = sin(𝜑)cos(𝜑̂) - cos(𝜑)sin(𝜑̂), (7.85) can be written as
    (7.87)  𝑑_𝛾 = (𝐶 + ∆𝐶) cos(𝜑) + ∆𝐷 sin(𝜑),
    (7.88-9)  ∆𝐶 = 4∫𝜑₀ d𝜑̂ sin(𝜑̂) (𝛷 + 𝛹)(𝜑̂),     ∆𝐷 = -4∫𝜑₀ d𝜑̂ cos(𝜑̂) (𝛷 + 𝛹)(𝜑̂).
To evaluate this we need the solutions for 𝛷(𝜂) and 𝛹(𝜂), which were found in Section 6.3.1. Ignoring the anisotropic stress due to neutrinos, we obtained [RE (6.165)]
    (7.90)  𝛷 = 𝛹 = 2𝓡_𝑖 [sin(𝜑) - 𝜑 cos(𝜑)]/𝜑³.
Substituting this into (7.88) and (7.89), we get
    (7.91-2)  ∆𝐶 = 8𝓡_𝑖 [1- sin²(𝜑)/𝜑²],     ∆𝐷 = 8𝓡_𝑖 [cos(𝜑) sin(𝜑) - 𝜑]/𝜑²,
and (7.87) becomes
    (7.93)  𝑑_𝛾 = 4𝓡_𝑖 [cos(𝜑) - 2sin(𝜑)/𝜑].
This is a remarkably simple solution for the fluctuations in the radiation era, valid on all scales. The solution is only a pure cosine curve in the high-frequency limits, 𝜑 ≫ 1. For intermediate wavelengths the gravitational evolution has led to an admixture.
   To extend this into the matter era, we match the 𝜑 ≫ 1 limit of (7.93), 𝑑_𝛾 → 𝛿_𝛾 ≈ 4𝓡_𝑖cos(𝜑) to the WKB solution (7.74). The resulting solution is
    (7.94)  𝛿_𝛾 = 4𝓡_𝑖(1 + 𝑅)^-1/4 cos(𝜑) - 4(1 + 𝑅)𝛹.
To complete the analysis we need to relate 𝛹 to 𝓡_𝑖. in the matter era, we have [RE (6.53) (6.134)]
    (7.95)  𝛹 ≈ 3/2 𝓗²/𝑘² 𝛿_𝑐,
so 𝛹 can be obtained from the solution for the dark matter density contrast 𝛿_𝑐. It will be shown below (7.98)
    (7.96-7)  𝛿_𝑐 ≈ -1.54𝓡_𝑖 ln (0.15 𝑘/𝑘_eq)(𝑘_eq 𝜂)²     (𝑘 > 𝑘_eq);     𝛹 ≈ 3/5 𝓡_𝑖 ln(0.15 𝑘/𝑘_eq)/(0.31 𝑘/𝑘_eq)².
Feeding this into (7.94), we obtain the solution for the photon density contrast in the matter era:
    (7.98)  𝛿_𝛾 = 4/5 𝓡_𝑖[5(1 + 𝑅)^-1/4 cos(𝜑) - 3(1 + 𝑅) ln(0.15 𝑘/𝑘_eq)/(0.31 𝑘/𝑘_eq)²]     (𝑘 > 𝑘_eq). Compared to the long-wavelength solution (7.82), he amplitude is enhanced by a factor of 5 and the zero-point is shfted by a nontrivial transfer function.
Derivation   In the following we derive the dark matter transfer function for 𝑘 > 𝑘_eq. The continiuty and Euler equations for dark matter are [RE (6.85-6)]
    (7.99-100)   𝛿_𝑐ʹ = 𝑘𝑣_𝑐 + 3𝛷ʹ,     𝑣_𝑐ʹ + 𝓗𝑣_𝑐 = -𝑘𝛹.
Substituting (7.99) into (7.100) and defining 𝑑_𝑐 ≡ 𝛿_𝑐 - 3𝛹,
    (7.101)   (𝑎𝑑_𝑐ʹ)ʹ = -3𝑎𝛹,   ^{[verification needed]}
where primes denote derivatives with respect to 𝜑 ≡ 𝑘𝜂/√3, Integrating (7.101), we get
    (7.102)   𝑑_𝑐(𝜑) = -3∫^𝜑₀ d𝜑̂/𝑎 ∫^𝜑̂₀ 𝑎𝛹(𝜑̄) d𝜑̄.
This result is exact and valid for all 𝑘. During the radiation era the gravitational potential is determined by the radiation and 𝛹 can be treated as an external source as far as the evolution of dark matter is concerned. We can use (7.90) and 𝑎 ∝ 𝜂 ∝ 𝜑 to evaluate the integral
    (7.103)   𝑑_𝑐(𝜑) = -3∫^𝜑₀ d𝜑̂/𝜑̂ ∫^𝜑̂₀ 𝑎𝛹(𝜑̄) d𝜑̄ = -6𝓡_𝑖∫^𝜑₀ d𝜑̂/𝜑̂ ∫^𝜑̂₀ [sin(𝜑̄) - 𝜑̄ cos(𝜑̄)]/𝜑̄² d𝜑̄ = -6𝓡_𝑖∫^𝜑₀ d𝜑̂/𝜑̂ [1 - sin(𝜑̂)/𝜑̂] + 𝑑_𝑐(0),
where the integration constant is fixed by the superhorizon limit, 𝑑_𝑐(0) = -3𝓡_𝑖. ^{[verification needed]} Solving the integration, we find
    (7.104)   𝑑_𝑐(𝜑) = 6𝓡_𝑖[Ci(𝜑) - sin(𝜑)/𝜑 - ln 𝜑]∣^𝜑₀ - 3𝓡_𝑖,
where Ci(𝜑) is the cosine integral
    (7.105)   Ci(𝜑) ≡ -∫^∞_𝜑 cos(𝜑̂)/𝜑̂ d𝜑̂. [added '-' unlike the text; RE Wikipedia Trigonometric integral]
The superhorizon limit of the cosine integral is
    (7.106)   lim_𝜑→0 Ci(𝜑) = 𝛾_𝐸 + ln 𝜑 - 𝑂(𝜑²),  [RE 𝑂(𝜑²) = 𝜑²/4]
where 𝛾_𝐸 = 0.5772 ... is the Euler-Mascheroni constant, while the high-frequency limit, 𝜑 → ∞, vanishies by definition.
    (7.107)   𝑑_𝑐(𝜑)  →^𝜑≫1  𝑑_𝑐(𝜑) = 6𝓡_𝑖[1/2 - 𝛾_𝐸 - ln 𝜑 + 𝑂(𝜑^-1)],    
which describes the evolution after the mode entered the horizon but before matter-radiation equality.
We would like to compare this solution to the solution (6.181) for matter fluctuation on sub-Hubble scales (but not restricted to the radiation era):
    (7.108)   𝛿_𝑐 = 𝐶₁(1 + 3/2 𝑦) + 𝐶₂[(1 + 3/2 𝑦) ln[{√(1 + 𝑦) + 1}/{√(1 + 𝑦) - 1}] - 3√(1 + 𝑦)]. where 𝑦 ≡ 𝑎/𝑎_eq. In the limit 𝑦 ≪ 1, this solution gives
    (7.109)   𝛿_𝑐  →^𝑦≪1  (𝐶₁ -3𝐶₂) - 𝐶₂ ln(𝑦/4) + 𝑂(𝑦) = [𝐶₁ -3𝐶₂) - 𝐶₂ ln[2/√3(√2 - 1) 𝑘/𝑘_eq]] - 𝐶₂ ln 𝜑,  
where we used (2.162) to relate 𝑦 ∝ 𝜂 to 𝜑 [= 𝑘𝜂/√3] and introduced 𝑘_eq = 𝜂_eq^-1. This is consistent with (7.107) if
    (7.110)   𝐶₁ = 6𝓡_𝑖 [7/2 - 𝛾_𝐸 - ln[2/√3(√2 - 1) 𝑘/𝑘_eq]] ≈ -6𝓡_𝑖 ln(0.15 𝑘/𝑘_eq).  
    (7.111)   𝐶₂ = 6𝓡_𝑖.
During the matter-dominate era (𝑦 ≫ 1), the second term in (7.108) is a decaying mode and can be dropped. We then find
    (7.112)   𝛿_𝑐  →^𝑦≫1 3/2  𝐶₁𝑦 = -9𝓡_𝑖 ln(0.15 𝑘/𝑘_eq) (𝑘_eq𝜂)²(√2 - 1)² ≈ -1.54𝓡_𝑖 ln(0.15 𝑘/𝑘_eq) (𝑘_eq𝜂)²,
where we used (2.162) for 𝑦 ∝ 𝜂² in the matter era. This solution has the ln 𝑘 scaling that we found in Section 6.3.2, but now includes the precise numerical factors that are required to relate 𝛿_𝑐 to 𝓡_𝑖.

          Solution at last-scattering The analytic solutions (7.82) and (7.98) imply the following limits of Sachs-Wolfe transfer function at last-scattering:
    (7.113)   𝐹_*(𝑘) ≡ (1/4 𝛿_𝛾 + 𝛹)_*/𝓡_𝑖   [RE 7.31)]
              = { 1/5 [1/(1 + 𝑅_*)^1/4 cos(𝑘𝑟_𝑠,*) - 3𝑅_*]  𝑘 < 𝑘_eq; 1/5 [5/(1 + 𝑅_*)^1/4 cos(𝑘𝑟_𝑠,*) - 3𝑅_* ln(0.15𝑘/𝑘_eq)/(0.31𝑘/𝑘_eq)²]  𝑘 > 𝑘_eq
We would like to find a solution that interpolates between these two limits valid with 𝑘 ~ 𝑘_eq. Motivated by the solutions in (7.113), Weinberg proposed the following ansatz
    (7.114)   𝐹_*(𝑘) = 1/5 [𝑆(𝑘)/(1 + 𝑅_*)^1/4 cos[𝑘𝑟_𝑠,* + 𝜃(𝑘)] - 3𝑅_*𝛵(𝑘)],
with 𝛵(𝑘), 𝑆(𝑘) and 𝜃(𝑘) are momentum-dependent interpolation functions, with limits
    (7.115)   𝑘 ≪ 𝑘_eq:     𝑆 → 1,     𝛵 → 1,     𝜃 → 1,
              𝑘 ≫ 𝑘_eq:     𝑆 → 5,     𝛵 → ln(0.15𝑘/𝑘_eq)/(0.31𝑘/𝑘_eq)²,     𝜃 → 0,    
The interpolation between 𝑘 ≪ 𝑘_eq and 𝑘 ≫ 𝑘_eq can almost be done "by hand," with reasonable results for the CMB spectrum. The result is summarized by the following fitting functions [Weinberg]
    (7.116)   𝑆(𝜅) ≡ [[1 + (1.209𝜅)² + (0.5116𝜅)⁴ + √5(0.1657𝜅)⁶]/[1 + (0.9459𝜅)² + (0.249𝜅)⁴ + √(0.167𝜅)⁶]]²,
              𝛵(𝜅) ≡ [ln[1 + (0.124𝜅)²]/(0.124𝜅)² [[1 + (1.257𝜅)² + (0.4452𝜅)⁴ + (0.2197𝜅)⁶]/[1 + (1.606𝜅)² + (0.8568𝜅)⁴ + (0.3927𝜅)⁶]]^1/2,
              𝜃(𝜅) ≡ [[(1.1547𝜅)² + (0.5986𝜅)⁴ + (0.2578𝜅)⁶]/[1 + (1.723𝜅)² + (0.8707𝜅)⁴ + (0.458𝜅)⁶ + (0.2204𝜅)⁸]]^1/2,
where 𝜅 ≡ √2𝑘/𝑘_eq. Plots of these fitting functions are given in Fig. 7.13. Note that it takes quite long for these functions to reach the high-frequency limit, especially 𝑆(𝜅).
   The function 𝜃(𝜅) leads to shifts in the CMB peak positions relative to the high-frequency solution in Section 7.4.1.⁷ As Weinberg showed, the agreement with the observed peak positions is now "almost embarrassingly good" [Weinberg]. Indeed Weinberg's SW transfer function (7.114) shows perfect agreement of the oscillatoty part with the exact numerical result in Fig. 7.14. The disagreement at small 𝑘 is expected since Weinberg's solution only applies to subhorizon modes at last-scattering. In this low-frequency limit, we should take 𝑅_* → 0 in (7.114) since sound waves are not important on these large scales. This reproduce the correct SW limit of Section 7.3.2.
   The solution above does not contain the effects of neutrinos. The neutrinos's anisotropic stress on the superhorizon initial conditions for metric potential is small as we see in Problem 6.2. The free-straming neutrinos leads to a small phase shift  in the high -frequency limit of the solution [13, 14], 𝜃(𝑘 ≫ 𝑘_eq) → 0.063π. These effects have to be considered to match the precision of current experiments, but are't essential for understanding the main feature of the CMB spectrum.
   From the solution for photon density contrast, we can infer the baryon velocity and hence the transfer functon for the Doppler contribution. Using the continuity equation for photon during the matter era reads 𝛿_𝛾ʹ 4/3 𝑘𝑣_𝛾, we obtain 𝐺_*(𝑘) = (𝑣_𝑏)_*/𝓡_𝑖 ≈ (𝑣_𝛾)_*/𝓡_𝑖 = 3(𝛿_𝛾ʹ)_*/(4𝓡_𝑖𝑘). Taking the derivative of the solution for 𝛿_𝛾 with respect to conformal tme, we find
    (7.117)   𝐺_*(𝑘) = -√3/5 𝑆(𝑘)/(1 + 𝑅_*)^3/4 sin[𝑘𝑟_𝑠,* + 𝜃(𝑘)].
We see that the oscillations in the Doppler term are exactly out of phase with those in the photon density.

          7.4.4 Small-Scaling Damping
We saw in the plots of the CMB spectrum that the ower is suppressed for multipoles (𝑙 > 1000). This damping is nnotyet included. There are two sources of damping: First, photon deiffusion between hot and cold regions erases tempurature differences on small scales. Second, the surface of last-scattering has a finite thickness. Averaging over this finite width suppresses short-wavelength fluctuations. We will derive these two effects.

          Silk damping since the effect of photon diffusion is only important deep inside the Hubble radius, we can ignore the effects of expansion: e.g. 𝓗𝑣_𝑏 ≪ 𝑣_𝑏ʹ in the Euler equation (7.59). We can also drop the gravitational forcing terms, since -𝑘²𝛷 ~ 𝓗²𝛿, so that 𝑘𝛹 is small relative to the photon pressure term, 𝑘𝛿_𝛾, and the change in the baryon momentum density, 𝑅𝑣_𝑏ʹ. The Euler equations are
    (7.118-9)   𝑣_𝛾ʹ + 1/4 𝑘𝛿_𝛾 - 2/3 𝑘𝛱_𝛾 = -𝛤𝑣_𝛾-𝑏,     𝑅𝑣_𝑏ʹ = +𝛤𝑣_𝛾-𝑏,
we restored the photon anisotropic stress 𝛱_𝛾. Above we studied these equations at leading order in an expansion in 𝑘/𝛤 = 𝑘𝓁_𝛾 < 1. ^{[verification needed]} We will extend the treatment to include the next-to-leading order correction capturing the damping fluctuations. Combining the two Euler equations, we get the slip velocity equation
    (7.120)   𝑣ʹ_𝛾-𝑏 = - (1 + 𝑅)/𝑅 𝛤_{𝑣𝛾-𝑏} - 1/4 𝑘𝛿_𝛾 + 2/3 𝑘𝛱_𝛾.
In the tight-coupling limit we can drop the photon anisotropic stress and the time variation of the slip velocity. We then have
    (7.121)   𝑣_𝛾-𝑏 ≈ - 𝑅/(1 + 𝑅) 𝑘/𝛤 𝛿_𝛾/4.
his equation has an interesting physical interpretation: the slip velocity characterizes the bulk velocity of the photons in the rest frame of the baryons and is therefore associated with an energy flux in that frame. Since 𝜌_𝛾 ∝ 𝛵_𝛾⁴, the right-hand side of (7.121) can be written  as -𝓁_𝛾𝛁𝛵_𝛾/𝛵̄, which means That it describes the energy flux  coming from a gradient in the photon temperature. A finite slip velocity is therefore associated with thermal conduction.
   Adding the two Euler equations, we find
    (7.122)   𝑅𝑣_𝑏ʹ + 𝑣_𝛾ʹ + 1/4 𝑘𝛿_𝛾 - 2/3 𝑘𝛱_𝛾 = 0.
Using the conduction equation (7.121), the time derivative of the baryon velocity can be written as
    (7.123)   𝑣_𝑏ʹ = 𝑣_𝛾ʹ - 𝑣ʹ_𝛾-𝑏 ≈ 𝑣_𝛾ʹ + 𝑅/(1 + 𝑅) 𝑘/𝛤 𝛿_𝛾ʹ/4,
where we ignored time derivatives of 𝑅 and 𝛤 on sub-Hubble scales. Equation (7.122) becomes
    (7.124)   (1 + 𝑅)𝑣_𝛾ʹ = -1/4 𝑘𝛿_𝛾 - 𝑅²/4(1 + 𝑅) 𝑘/𝛤 𝛿_𝛾ʹ + 2/3 𝑘𝛱_𝛾.
In Appendix B, we show that the photon anisotropic stress is related to the gradient of the bulk velocity (see section B.3.1)
    (7.125)  𝛱_𝛾 ≈ -4/9 𝑘/𝛤 𝑣_𝛾,
which leads t an effective photon viscosity. Equation (7.124) reads
    (7.126)  (1 + 𝑅)𝑣_𝛾ʹ = -1/4 𝑘𝛿_𝛾 - 1/4 [8/9 + 𝑅²/(1 + 𝑅)] 𝑘/𝛤 𝛿_𝛾ʹ,
where we used the continuity equation on sub-Hubble scales, 𝛿_𝛾ʹ = 4/3 𝑘𝑣_𝛾. [RE (6.88)] Taking a time derivative of the continuity equation, 𝛿_𝛾ʺ = 4/3 𝑘𝑣_𝛾ʹ, and substituting (7.126) hen gives
    (7.127)  𝛿_𝛾ʺ + 𝑘²/3(1 + 𝑅) 1/𝛤 [8/9 + 𝑅²/(1 + 𝑅)]𝛿_𝛾ʹ + 𝑘²𝑐_𝑠² 𝛿_𝛾 = 0. We see that going beyond leading order in the tight-coupling approximation has introduced a friction term to 𝛿_𝛾ʹ. This friction is a combination of thermal conduction and photon viscosity. The size of the friction term depends on the wavenumber 𝑘. The associated damping of the fluctuations is called Silk damping [15].
   We can solve (7.127) in a WKB approximation. Consider the ansatz
    (7.128)  𝛿_𝛾 ∝ exp[𝑖∫^𝜂₀ 𝜔(𝜂') d𝜂'],     with     𝜔ʹ ≪ 𝜔².
Taking time derivatives of it, we get
    (7.129)  𝛿_𝛾ʹ = 𝑖𝜔𝛿_𝛾
    (7.130)  𝛿_𝛾ʺ = (-𝜔² + 𝑖𝜔ʹ)𝛿_𝛾 ≈ -𝜔²𝛿_𝛾,    
and using this in (7.127), we find
    (7.131)  (𝑘²𝑐_𝑠² - 𝜔²) + 𝑘²/3(1 + 𝑅) 1/𝛤 [8/9 + 𝑅²/(1 + 𝑅)]𝑖𝜔 = 0.
Substituting 𝜔 = 𝑘𝑐_𝑠 + 𝛿𝜔, and expanding in small 𝛿𝜔, gives
    (7.132)   𝛿𝜔 = 𝑖𝑘²/6(1 + 𝑅) 1/𝛤 [8/9 + 𝑅²/(1 + 𝑅)] [Derivation   𝑘²𝑐_𝑠² - 𝜔² + 2𝛿𝜔 = 0.  ⇒  𝜔 = 1/2 ±√(4𝑘𝑐_𝑠² - 2𝛿²) + ≈ 𝑘𝑐_𝑠 + 𝛿𝜔.  ▮]
Since the correction is imaginary, the solution (7.128) receives an exponential suppression
    (7.133)   𝛿_𝛾 ∝ 𝑒^{-𝑘2/𝑘_𝑆2} exp[𝑖𝑘 ∫ 𝑐_𝑠(𝜂') d𝜂'],
where the damping wavenumber is
    (7.134)   𝑘_𝑆^-2(𝜂) ≡ ∫^𝜂₀ d𝜂'/[6(1 + 𝑅)𝛤(𝜂') [8/9 + 𝑅²/(1 + 𝑅)].
Including polarization corrections to the scattering would give the same result with 8/9 → 16/15 [16]. We see that diffusion damping can be modeled by an 𝑒^{-𝑘2/𝑘_𝑆2} envelope to the oscillatory part in the previous section. Therefore this replaces the transfer function 𝑆(𝑘) in Weinberg's solutions (7.114) and (7.117) with 𝑒^{-𝑘2/𝑘_𝑆,*2}𝑆(𝑘) where 𝑘_𝑆,* is given by (7.134) evaluated at last-scattering. For standard cosmological parameters, a good estimate of the Silk damping scale is (see Problem 7.3)
    (7.135)   𝑘_𝑆,*^-1 ≈ 7.2 Mpc.
This corresponds to damping in the CMB power spectrum for multipoles larger than 𝑙_𝑆 ~ 𝑘_𝑆,*𝜒_*/√2 ~ 1370.

          Landau damping
So far we have assumed that recombination was an instantaneous process,, corresponding to the visibility function being a perfect delta function. But as we see from Fig. 7.5 , in reality it has a finite width. For short-wavelength fluctuations this has to be taken into account.
   Near its maximum, the visiblity function can be approximated by a Gaussian
    (7.136)   𝑔(𝜂) = exp[-(𝜂 - 𝜂*)2/3𝜎𝑔2]/√(2π)𝜎𝑔,
where 𝜎𝑔 ≈ 15.5 Mpc. The fluctuations must now be averaged over this finite width of the function. This averaging makes little difference for slowly evolving fluctuations which can be enhanced at the mean time of decoupling 𝜂*. The amplitude of the rapidly oscillating part can be significantly reduced by averaging. This effect is called Landau damping because it is similar to the damping that occurs for oscillations with a spread frequencies.
   In Appendix B we derive the line-of-sight solution in the visibility function; see Section B.2. For Sachs-Wolfe term, we find
    (7.137)  𝛳SW𝑙(𝑘) = ∫𝜂00 d𝜂 𝑔(𝜂) (1/4 𝛿𝛾 + 𝛹) 𝑗𝑖(𝑘𝜒),
where 𝜒(𝜂) = 𝜂 - 𝜂*. Over the small range of support of the visibility function, the Bessel function can be time independent and be pulled out of the integral. The remaining integral is
    (7.138)  ∫𝜂00 d𝜂 𝑔(𝜂) cos[∫𝜂0 𝜔(𝜂') d𝜂'] ≈ ∫𝜂00 d𝜂 𝑔(𝜂) cos[∫𝜂*0 𝜔(𝜂') d𝜂' + 𝜔*(𝜂 - 𝜂*)] ≈ exp(-𝜔*2𝜎𝑔2/2) cos[∫𝜂*0 𝜔(𝜂') d𝜂'],
where 𝜔* = 𝜔(𝜂*) = 𝑘𝑐𝑠,*. We see that the averaging induced an exponential damping factor exp(-𝑘2/𝑘𝐿,*2), with a scale
    (7.139)  𝑘𝐿,*-1 = 𝑐𝑠,*𝜎𝑔/√2 = 𝜎𝑔/√(6(1 + 𝑅*) ≈ 5.0 Mpc.
Since the functional dependence of this effect is the same as for Silk damping, we can combine the two. To obtain the effective damping scale, the two damping scales must be added in quadrature
    (7.140)  𝑘𝐷,*-1 = √(𝑘𝑆,*-2 + 𝑘𝐿,*-2) ≈ 8.8 Mpc.
The multiple mement corresponding to this damping scale is 𝑙𝐷 ~ 𝑘𝐷,*𝜒*/√2 ~ 1125, which agrees with the observed CMB spectrum in Fig. 7.2.

⁷ The phase shift analytically is possible but quite nontrivial. The phase shift 𝜃(𝜅) has multiple origins, such as the breakdown of tight coupling, neutrino free-streaming and a transient of gravitational potential. Moreover the significant part of mismatch in the predicted peak locations (7.76) comes from that the 𝑙 ~ 𝑘𝜒_* mapping is not very accurate [11]and this account for about half of the shifts seen in Fig. 7.11 and the rest is captured by the nontrivial phase shift 𝜃(𝜅).

          7.4.5 Summary of Results Weinberg's semi-analytic solutions for Sachs-Wolfe and Doppler transfer functions are
    (7.141)   𝐹_*(𝑘) = 1/5 [𝑒^{-𝑘2/𝑘_𝐷,*2}𝑆(𝑘)/(1 + 𝑅_*)^1/4 cos[𝑘𝑟_𝑠,* + 𝜃(𝑘)] - 3𝑅_*𝛵(𝑘)],
    (7.142)   𝐺_*(𝑘) = -√3/5 𝑒^{-𝑘2/𝑘_𝐷,*2}𝑆(𝑘)/(1 + 𝑅_*)^3/4 sin[𝑘𝑟_𝑠,* + 𝜃(𝑘)]
here the interpolation functions 𝑆(𝑘), 𝛵(𝑘) and 𝜃(𝑘) are defined in (7.116) and plotted in Fig. 7.13. Using the result of Exercise 7.3, the power spectrum can be written as
    (7.143)   𝑙(𝑙 + 1)/2π 𝐶_𝑙 = ∫^∞₁ d𝛽/𝛽²√(𝛽² - 1) [𝐹_*²(𝑙𝛽/𝜒_*) + (𝛽² - 1)/𝛽² 𝐺_*²(𝑙𝛽/𝜒_*)]∆_𝓡²(𝑙𝛽/𝜒_*),
The integral converges rapidly and accurate results are obtained with a relatively low cutoff at 𝛽 = 5 [Weinberg]. Since most of support is near 𝛽 = 1, the power spectrum is the sum of the two transfer functions, evaluated at 𝑘 ≈ 𝑙/𝜒_* . The full spectrum must also include the ISW contribution and the cross correlations between the (I)SW and Doppler contributions.
   Fig. 7.15 shows the different contributions to the CMB power spectrum. We see that most of the shape of the spectrum comes from SW term, with the Doppler term slightly reducing the contrast between the peaks and troughs. The ISW effect has a significant effect on the height of the first peak.We will discuss how the shape of the spectrum depends on the cosmological parameters.

          7.5 Understanding the Power Spectrum
The 𝛬CDM model is defined by six parameters, which are
            {𝛢_𝑠, 𝑛_𝑠, 𝜔_𝑏, 𝜔_𝑚, 𝛺_𝛬, 𝜏}
𝛢_𝑠 and 𝑛_𝑠 describe the spectrum of the initial curvature perturbations. The remaining four parameters determine the evolution of the fluctuations as well as their projection onto the sky. The physical baryon and mater densities, 𝜔_𝑏 ≡ 𝛺_𝑏𝘩² and 𝜔_𝑚 ≡ 𝛺_𝑚𝘩², where 𝘩 is the dimensionless Hubble constant. The fractional density in the cosmological constant is 𝛺_𝛬. The photon density is fixed by the measured CMB temperature 𝛵₀ = 2.725 K. Neutrinos are treated as massless, so that their density is fixed  in terms of the photon density. A nonzero neutrino mass is easy to incorporate and can be constrained by its effect on the gravitationally lensed spectrum. The Hubble constant 𝐻₀ (or 𝘩) is not a free parameter because we are assuming a flat universe, so that 𝘩 = √[𝜔_𝑚/(1 - 𝛺_𝛬)]. The final parameter 𝜏 is the integrated optical depth to recombination, [RE (3.87)] which is nonzero because the light of the first stars reionized the universe. The measured values of the base parameters of the 𝛬CDM cosmology are given in Table 7.1.

          7.5.1 Peak Locations
The key characteristics of the CMB spectrum are the positions and heights of the acoustic peaks. The angular scale of the acoustic  peaks is given by 𝜃_𝑠 ≡ 𝑟_𝑠,*/𝑑_𝛢,*, where 𝑟_𝑠,* is the sound horizon at decoupling and 𝑑_𝛢,* is the comoving angular diameter distance to the last-scattering surface (see Section 2.2.3). In this chapter we assumed a flat universe where 𝑑_𝛢,* = 𝜒_*. The sound horizon depends on pre-recombination physics, so in 𝛬CDM model it is determined by 𝜔_𝑏 and 𝜔_𝑚 alone (for fixed radiation density). On the other hand the distance to last-scattering depends on the geometry (𝛺_𝑘, 𝐻₀ and the energy content (𝛺_𝛬, 𝛺_𝑚) after recombination. Breaking the degeneracy between pre- and post-recombination physics requires additional data set(e.g. BAO data or supernovae).

          7.5.2 Peak Heights
The peak heights are determined by four distinct effects:

1. Diffusion damping  Recall that the diffusion length is roughly the geometric mean of the photon mean free path and the age of the universe at recombination. The former depends on the baryon density (𝜔_𝑏), while the latter is a function of the total matter density (𝜔_𝑚).
2. Early ISW effect  As in Section 7.2.3 the residual radiation density at recombination lead to a time dependence in the potential, 𝛷ʹ ≠ 0, which affects the gravitational redshift of the CMB photons. Since 𝛷 is small on small scales, the main effect is on the first peak of the spectrum.⁸
3. Baryon loading  Baryon add extra weight to the fluid which displaces the zeropoint of the acoustic oscillations. This "baryon loading" changes the relative heights of the odd and even peaks. The heights of alternating peaks are therefore a measure of the baryon density.
4. Radiation driving  Changes in the relative amount of matter and radiation affect the peak heights through an effect called "radiation driving." Reducing the amount of matter increases the height of the first few peaks. Since the radiation density is fixed by the CMB temperature, this provides a way to measure the total matter density of the universe. The last two effects deserve further explanation.

          Baryon loading Baryons contribute to both the initial and gravitational mass of the photon-baryon fluid, 𝑚_eff ∝1 + 𝑅, but not to the pressure. Increasing the baryon density change the pressure and gravity in the evolution of the photon-baryon fluid.
   One characteristic effect of this "baryon loading" is the shift of the zero-point of the oscillations, described by the -3𝑅𝛵(𝑘) term in (7.114). Since the CMB power spectrum depends on the square of the the transfer function, this shift of the zero-point leads to dfferences in the heights of alternating peaks (see Fig. 7.16). The effect is clearly seen in Weinberg's solution for SW transfer function, although the impact on the heights of the third and fifth peaks is masked by the damping of fluctuations.
   The time dependence of the baryon-to-photon ratio 𝑅 affects the amplitude of the acoustic oscillation through the WKB factor (1 + 𝑅)^-1/4 in (7.11). Consider the analogy with, in classic mechanics, the ratio of the energy 𝐸 = 1/2 𝑚_eff𝜔²𝛢² to the frequency 𝜔 of an oscillator is an adiabatic invariant. In our case, the evolution of 𝑅 induces a slow evolution of the sound speed and hence the oscillation frequency 𝜔 ∝ (1 + 𝑅^-1/2. Combining this with 𝑚_eff ∝ (1 + 𝑅), the constraint 𝑚_eff𝜔𝛢² = const leads to 𝛢 ∝ (1 + 𝑅)^-1/4.

          Radiation driving
As in Section 6.3.1, the gravitational potential decays when a mode enters the horizon during the radiation era. This leads to a boost the amplitude of the fluctuation captured by the transfer function 𝑆(𝑘) in (7.11.1). Intuitively, this can be understand as follows: the moment at which the potential starts to decay coincides with the compression of the photon-baryon fluid, since both begin at horizon crossing.  The potential assists the first compression of the fluid through a  near-resonant driving force, but doesn't resist the subsequent rarefaction of the fluid (see Fig. 7.17). That amplitude of the acoustic oscillations therefore increase. Additionally the time-dependent potential 𝛷(𝜂̄) induces a time dilation effect in the force term cf. (7.63), which increase the amplitude of the photon fluctuations. Heuristically an overdensity (with negative a 𝛷) induces a "stretching: of the space as in (7.13). As the potential decays the space "re-contracts" and the photons are blueshifted. The combination of the two effects boosts by a factor of about 5. The effect is clearly seen in Weinberg's solution for the SW transfer function, cf. Fig. 7.18.
   Only modes that enters the sound horizon during radiation-dominated epoch experience this "radiation driving." We expect the amplitude of the peaks in the CMB to increase as we go from low multipoles to high ones, with the point of the transition depending of the relative amount of matter and radiation in the universe.

          7.5.3 𝛬CDM Cosmology
WE are now ready to explain how the CMB spectrum depends on the base parameters of the 𝛬CDM model, {𝛢_𝑠, 𝑛_𝑠, 𝜔_𝑏, 𝜔_𝑚, 𝛺_𝛬, 𝜏}.

          Initial conditions
Our computation of the CMB spectrum assume a power law ansatz for the power spectrum of primordial curvature perturbations
    (7.144)   ∆_𝑅²(𝑘) = 𝛢_𝑠(𝑘/𝑘₀₎)^{𝑛_𝑠 - 1},
where 𝑘₀ = 0.05 Mpc^-1 is an arbitrary chosen pivot scale. Since the evolution is linear, it is easy to understand how changes in the amplitude 𝛢_𝑠 and the tilt 𝑛_𝑠 affect the observed spectrum. 𝛢_𝑠s simply rescale the overall amplitude, while 𝑛_𝑠s changes the relative amount of power on large and small scales (see Fig. 7.19).
   One of the most remarkable discoveries from CMB is that the primordial spectrum is close to scale invariant, but not exactly: 𝑛_𝑠 = 0.965 ± 0.004 [17]. In Chapter 8 we will see this deviation is a prediction of inflation, where it arises from slow decay of the inflationary energy density. The measurement of the parameter 𝑛_𝑠 is our first measurement of the dynamics during inflation. We will study more about this in the next chapter.

          Reionization
There is an effect, not yet mentioned, that also affects the overall amplitude. The ultraviolet light from the first stars reionized the universe, so that the CMB photons have a small probability to scatter off the free electrons in the late universe. To quantify this, consider a photon  on a direction 𝐧̂ , with temperature 𝛵(𝐧̂) = 𝛵̄[1 + 𝛩(𝐧̂)]. The probability of no scattering  to occur is 𝑒^-𝑟, while the probability of rescattering is 1 - 𝑒^-𝑟. The temperature of the rescattered photons will be the temperature 𝛵̄ of the equilibrated ionized regions. The observed temperature today is
    (7.145)   𝛵₀(𝐧̂) = 𝛵₀̄[1 + 𝛩(𝐧̂)]𝑒^-𝑟 + 𝛵₀̄(1 - 𝑒^-𝑟) = 𝛵₀̄[1 + 𝛩(𝐧̂)𝑒^-𝑟].
We see that the observed anisotropies are suppressed by a factor of 𝑒^-𝑟 and the power spectrum becomes
    (7.146)   𝐶_𝑙 → 𝐶_𝑙𝑒^-2𝑟.
The effect only occurs on scales that are smaller than the horizon at reionization corresponding to multipoles 𝑙 > 0. Since the power spectrum is uniformly reduced by a factor of 𝑒^-2𝑟, the impact of reionization is degenerate with a change of amplitude 𝛢_𝑠; observation only constrain the combination 𝛢_𝑠𝑒^-2𝑟 (see Fig. 7.20). To break this degeneracy requires an independent measurement of the optical depth 𝜏. This is provided by CMB polarization (see Section 7.6).

          Baryons
A change in the baryon density 𝜔_𝑏 has three main effects:

1. It changes the sound  speed and hence lead to a different sound horizon at recombination,
    (7.147)   𝑟𝑠,* = ∫𝑎*0 (d ln 𝑎)/𝑎𝐻 𝑐𝑠(𝑎).
   Associated with the change in 𝑟𝑠,* is a change in angular scale of the sound horizon 𝜃𝑠 = 𝑟𝑠,*/𝜒*, where s the comoving distance to the last-scattering surface
    (7.148)   𝜒* = ∫1𝑎* (d ln 𝑎)/𝑎𝐻.
   The value of 𝜃𝑠 determines the peak locations of the CMB spectrum.
2. It changes the zero-point of the acoustic oscillations, enhancing the difference in the heights of odd and even peaks in the spectrum (see Section 7.5.2).
3. It affects the damping of the fluctuations. Reducing the baryon density reduces the tight coupling between the photons and baryons and hence increases the damping effect.

The effect on the peak locations is degenerate with other parameters (such as the dark energy density 𝛺_𝛬) that change the distance to last-scattering 𝜒_* and hence the observed angular scale 𝜃_𝑠. In the left panel of Fig. 21. the spectrum for some 𝜔_𝑏 but fixed 𝜃_𝑠 was therefore plotted. This removes the effect on the peak locations and highlights the effect on the peak amplitudes. We see that increasing 𝜔_𝑏 raises the first peak and suppresses the second peak, as expected the discussion on baryon loading. Fr the higher peaks the Silk damping and the WKB factor (1 + R)^-1/4 mask the effect. Looking closely at the spectrum for 𝑙 > 1000, we also see the enhanced damping of the fluctuations for reduced baryon density.
   The observation of the CMB lead to 𝜔_𝑏 = 0.02237 ± 0.00015 or 𝛺_𝑏 = 0.0493 ± 0.0006 [17]. As in Section 3.2.4, this value is consistent with the value required by BBN.

          Dark Matter
A change in 𝜔_𝑚 has four main effects:

1. It changes the Sound horizon t recombination through a change in the evolution of the Hubble rate cf (7.147). It also affects 𝜒_* The combination of the two effects determines the angular scale of 𝜃_𝑠 and hence the peak locations.
2. It shifts the time of matter-radiation equality. Reducing the matter density shifts the moment of equality to a later time which enhances the radiation driving effect described in Section 7.5.2 and hence increases the peak heights.
3. It determines the early ISW effect by changing the relative amount matter and radiation at the time of recombination. Reducing the matter density increases the early ISW effect, which leads to a boost in the height of the first peak.
4. It affects the diffusion scale by changing the time to recombination.

In the right panel of Fig. 7.21, The CMB spectrum for different value of 𝜔_𝑚. As expected, reducing the matter density leads to a boost in the amplitudes of the few peaks. The effect saturates for lower^* peaks
which aren't t affected by changes in the matter density. [correction^*: higher in original text]    The observations give 𝜔_𝑚 = 0.1430 ± 0.0011 or 𝛺_𝑚 = 0.3153 ± 0.0073 [17]. Comparing this to the measurement of the baryon density, we conclude that the universe must have non-baryonic dark matter. The measured amount of the dark matter is consistent with the requirement from formation discussed in Chapter 5 and 6.

          Dark Energy
Changing the dark energy density 𝛺_𝛬 affects the CMB spectrum mainly through its effect on the distance to the last-scattering surface. The latter depends on the Hubble rate after recombination, which in a flat universe is
    (7.149)   𝛨²(𝑎) = 𝛨₀² (𝛺_𝑚𝑎^-3 + 𝛺_𝛬).
Since the matter density is constrained by its effect o the peak heights, this allows us to measure the dark energy density. Increasing 𝛺_𝛬 at fixed 𝛺_𝑚𝘩² increases the Hubble rate and decreases the distance to last-scattering, so that the peaks move to larger angular scale (lower 𝑙). This is what is seen in Fig. 7.22. Since dark energy affects the CMB mostly through its effect on the distance to last-scattering, it is degenerate with any other parameter that also changes its distance, such as spatial curvature. Breaking this geometric degeneracy requires either external data data set or a measurement of CMB lensing. Combing measurements of the CMB and BAO gives 𝛺_𝛬 = 0.685 ± 0.007 [17].
   As we can see from Fig. 7.22, increasing the amount of dark energy also leads to an increase in the power on large scales (low 𝑙). This arises from late ISW effect, The onset of dark energy induces a decay of the gravitational potential, which reduces the net gravitational redshift in traversing the large-scale time-varying gravitational potentials and hence increases the large-scale CMB temperature fluctuations.

          Curvature
We assumed a spatially flat universe until now. We may ask how the CMB can test if the universe is really flat. Physical scales on the last-scattering surface are related to the observed angular scales by the (comoving) angular diameter distance 𝑑_𝛢,*. For small deviation from a flat universe, we have
    (7.150)   𝑑_𝛢,* = { 𝑅₀ sin(𝜒_*/𝑅₀) ≈ 𝜒_*(1 - 𝜒_*²/6𝑅₀²)   (positively curved); 𝑅₀ sinh(𝜒_*/𝑅₀) ≈ 𝜒_*(1 + 𝜒_*²/6𝑅₀²)   (negatively curved).
We see that the angular diameter distance increases for a negatively curved universe. The peaks in the CMB spectrum move to smaller scale (larger multipoles) and vice versa. This is what is seen in Fig. 7.22. (Recall 𝛺_𝑘 < 0 for a positvely curved universe.) The figure also illustrates on the low-𝑙 spectrum the significant effect through its impact on the late ISW effect.

          Hubble constant
Our parameterization of 𝛬CDM model in (7.144) did not include Hubble constant. Assuming a flat universe the Friedmann equation relates the Hubble constant to 𝛺_𝑚 and 𝛺_𝛬. We could also have used Hubble constant  as a free parameter and made the dark energy density a derived parameter. In that case, the Hubble rate in (7.149) would be written as
    (7.151)   𝐻²(𝑎) ∝ (𝜔_𝑚𝑎^-3 + 𝘩² - 𝜔_𝑚).  
Constraining 𝜔_𝑚 through its effect on the peak heights (and/or external data sets) then allows the CMB to measure the Hubble constant 𝘩 (𝐻₀). These observation finds 𝘩 = 0.674 ± 0.005 [17]. Interestingly, there is somewhat in tension with local measurements using supernovae, which gives 𝘩 = 0.730 ± 0.010 [18]. This so called "Hubble tension" remain an important open problem of the 𝛬CDM model.

          7.5.4 Beyond 𝛬CDM
As extensions beyond the 𝛬CDM cosmology, we will discuss the two well-motivated examples: extra relativistic species and tensor modes.

          Extra relativistic species
In the 𝛬CDM model, the radiation density is fixed in therms of the CMB temperature. However extra ligh species or non-standard neutrino properties can lead to extra radiation density.
   The total energy density in relativistic species is often defined as [RE (3.66)]
    (7.152)   𝜌_𝑟 = [1 + 7/8 (4/11)^4/3 𝑁_eff]𝜌_𝛾,
where 𝜌_𝛾 is the energy density of photons. 𝑁_eff is referred to as "effective number of neutrinos," although there may be contributions to it that have nothing to do with neutrinos. The Standard Model predicts 𝑁_eff = 3.046 and the current constraint from the Planck satellite is 𝑁_eff = 2.99±0.17 [17]. Deviations from the standard value may arise if the neutrinos have non-standard properties or if new physics at high energies leads to additional weakly coupled light species (see Problem 3.5).
   The main effect of adding extra radiation to the early universe is to increase the damping of the CMB spectrum (see Fig. 7.23. Increasing 𝑁_eff increases 𝐻_*, the expansion rate at recombination. This would change both the damping scale 𝜃_𝐷 and the acoustic scale 𝜃_𝑠, with the ratio scaling as [RE (7.54-55) (7.147)]
    (7.153)   𝜃_𝐷/𝜃_𝑠 = 1/𝑟_𝑠,*𝑘_𝐷 ∝ 1/𝐻_*^-1𝐻_*^1/2 = 𝐻_*^1/2.  
Since 𝜃_𝑠 is measured very accurately by the peak locations, we need to keep it fixed. His can be done, for example, by simultaneously increasing  the Hubble constant 𝐻₀. Increasing 𝑁_eff (and hence 𝐻_*) at fixed 𝜃_𝑠 then implies larger 𝜃_𝐷, so that the damping kicks in at larger scales, reducing the power in the damping tail (see Fig. &.23). By accurately measuring the small-scale CMB anisotropies observations can put a constraint on the number of relativistic species at recombination.
   The main limiting factor in these measurements is a degeneracy with the primordial helium fraction 𝑌_𝑃 ≡ 4𝑛_He/𝑛_𝑏. [RE (3.115)(3.141)] At fixed 𝜔_𝑏, increasing 𝑌_𝑃 decreases the number density of free electrons, which increases the diffusion length and hence reduces the power in the damping tail. 𝑌_𝑃 and 𝑁_eff are anti-correlated.^{[verification needed]} This degeneracy is broken by the phase shift induced by free-streaming relativistic species [13, 14].    As shown in Problem 3.5 light particles that decoupled before the QCD phase transition contribute at the percent level to the radiation density of the universe. Although such a signal is an order of magnitude smaller than the sensitivity of current CMB observations, it is within reach of the next generation of CMB experiments [19].

          Gravitational waves
In Problem 7.6 we will explore the impact of tensor metric fluctuations, 𝛿𝑔𝑖𝑗 = 𝑎2𝘩𝑖𝑗, on the CMB temperature fluctuations. [RE (6.208)] The presence of these gravitational waves creates temperature anisotropies at last-scattering, as can be determined from geodesic equation for photons. In Problem 7.6 we will find the line-of -sight solution is
    (7.154)  𝛳(𝑡)(𝐧̂) = -1/2 ∫𝜂0𝜂* d𝜂 𝘩𝑖𝑗ʹ𝜂̂𝑖𝜂̂𝑗.
Moreover we will show that the tensor-induced power spectrum is
    (7.155)  𝐶𝑙(𝑡) = 4π ∫ d ln 𝑘 ∣𝛳𝑙(𝑡)(𝑘)∣2 ∆𝘩2(𝑘),
where ∆𝘩2(𝑘) is the primordial power spectrum of the tensor mode (as defined in Section 6.5). In problem 7.6, we will derive an explicit form for tensor function 𝛳𝑙(𝑡)(𝑘). A scale-invariant tensor spectrum ∆𝘩2(𝑘) ≈ const, lead to 𝑙(𝑙 + 1)𝐶𝑙(𝑡) ≈ const on large scales. On small scales we expect the signal to be suppressed since the amplitude of the gravitational waves decays inside the horizon. This is confirmed by the numerical result in Fig. 7.24.    The measurement of the large-scale CMB temperature anisotropy spectrum constrains the tensor amplitude to 𝑟 < 0.1 [170, where 𝑟 ≡ ∆𝘩2(𝑘0)/∆𝑅2(𝑘0) is the tensor-to-scalar ratio. Stronger constrains on 𝑟 require the measurement of CMB polarization, especially its B-mode type.

          7.6 A Glimpse at CMB Polarization^*
Measurement of the CMB polarlization are at the forefront of observational cosmology. In this section an introduction to the physics of CMB polarization will be given. The goal is to provide an intuitive understanding of E/B decomposition of the polarization signal and to explain why B-modes are a unique signature of primordial tensor modes.

          7.6.1 Polarization from Scattering
Consider the scattering of radiation by a free electron as illustrated in Fog. 7.25. Imagine that the angle between the incoming and outgoing rays is 90^∘; e.g. let the incoming and outgoing rays be in the directions -𝐱̂ and +𝐳̂, respectively. Even if the incoming radiation is unpolarized, the outgoing radiation will become polarized, This is because the polarization direction must be transverse to the direction of propagation. For the scattering shown in the left panel of Fig. 7.25 all of the intensity along the 𝑧-axis blocked and only that along the 𝑦-axis is transmitted. The net result is polarization in the direction 𝐲̂.
   In general, radiation will come from all directions and scatter into all directions. If the incident radiation is isotropic, then the net result after scattering will still be unpolaarized radiation. To see this, consider adding a ray of ulpolarized radiation in the direction -𝐲̂ as the right panel in Fig. 7.25. The sum of the polarized radiation created by the two incoming rays will be unpolarized radiation. To create a net polarization, we require anisotropy in the incoming radiation.

Exercise 7.5   Show with a sketch that the scattering of dipolar anisotropy also does not lead to a net polarization.
[Solution]   If we make the modified Fig. 7.26 where the intensity of the radiation coming from ±𝐱̂ is the same as that coming from ±𝐲̂ and 𝐱̂ is for cold and hot region while 𝐲̂ is for dipolar anisotropy, it becomes a sketch that the scattering of dipolar anisotropy also does not lead to a net polarization.  ▮

Fig. 7.26 shows that polarization is generated if the incident radiation has a quadrupolar anisotropy. The intensity of the radiation coming from the directions ±𝐲̂ is now larger than that coming from ±𝐱̂. Then the outgoing radiation will be polarized in the 𝐱̂-direction. It is useful to remember that the polarization direction of the outgoing radiation is aligned with the hot regions.
   The polarization pattern observed on the sky will be the result of the scattering of the temperature inhomogeneities in the primordial plasma. These inhomogeneities are best described in a Fourier decomposition, with each mode describing a plain wave perturbation. It is useful to first determine the polarization generated by a single plane wave. Consider a plane wave photon density perturbation propagating in a direction transverse to the line-of-sight (see Fig. 7.27). The plane wave corresponds to hot and cold modulations of the photon temperature. Using our earlier assertion that the polarization is aligned with the hot regions, we predict the polarization pattern shown in Fig. 7.27. The polarization  directions are ether parallel or perpendicular to the wavevector 𝐤. We call such a polarization pattern an E-mode.
   Rotating all polarization directions by 45 degrees would give so-called B-mode polarization (see Fig. 7.28). By symmetry, this type of polarization pattern cannot be generated by scalar (density) fluctuations. The B-mode patterns is a key signature of gravitational waves in the early universe. We will discuss this in detail in Section 7.6.5.  

          7.6.2 Statistics of CMB Polarization

          Stokes parameters
Consider a monochromatic electromagnetic plane wave propagating in the 𝑧-direction. The electric field of the wave i
s     (7.156)   𝑬(𝑧, 𝑡) = Re[(𝐸_𝑥𝐱̂ + 𝐸_𝑦𝐲̂) 𝑒^{𝑖(𝑘𝑧 - 𝜔𝑡)}],    [Re(𝑥): 'real part' of 𝑥]
where 𝜔 = 𝑘𝑐. We can define the complex amplitudes as 𝐸_𝑥 = ∣𝐸_𝑥∣𝑒^{𝑖𝜑_𝑥} and 𝐸_𝑦 = ∣𝐸_𝑦∣𝑒^{𝑖𝜑_𝑦}. At a given location, taking 𝑧 = 0, we have
    (7.157)   𝑬(𝑡) = ∣𝐸_𝑥∣ cos(𝜔𝑡) 𝐱̂ + ∣𝐸_𝑦∣ cos(𝜔𝑡 - 𝜑) 𝐲̂,
where we defined 𝜑 ≡ 𝜑_𝑦 - 𝜑_𝑥 and chose the origin of time so that 𝜑_𝑥 ≡ 0. In general, the field vector in (7.157) traces out an ellipse in the 𝑥𝑦-plane. For 𝜑 = 0 (or π), the 𝑥- and 𝑦- components of the field oscillate in phase (anti-phase) and the ellipse degenerates into a line. We say that the wave is linearly polarized. For 𝜑 = ±π/2, the wave is circularly polarized.
   The polarization of electromagnetic radiation is then defined by the Stokes parameters
    (7.158)   𝐼 ≡ ∣𝐸_𝑥∣² + ∣𝐸_𝑦∣²,     𝑄 ≡ ∣𝐸_𝑥∣² - ∣𝐸_𝑦∣²,     𝑈 ≡ 2∣𝐸_𝑥∣∣𝐸_𝑦∣ cos(𝜑),     𝑉 ≡ 2∣𝐸_𝑥∣∣𝐸_𝑦∣ sin(𝜑),
where 𝐼 measures the intensity of the radiation, while 𝑄, 𝑈, 𝑉 describes its polarization state. The expression in (7.158) describes fully polarized radiation, with 𝐼² = 𝑄² + 𝑈² + 𝑉², which the CMB is certainly not; the polarization amplitude of the CMB is less than 10% of the temperature anisotropy. Recall that, in SI units, the energy density of an elevtromagnetic wave is 𝜌 = 𝜀₀∣𝐸∣² and its intensity is 𝓘 = 𝜌𝑐 = 𝑐𝜀₀∣𝐸∣². Averaged over many oscillation periods, we have
    (7.159)   ⟨𝓘⟩ = 𝑐𝜀₀⟨∣𝐸∣²⟩ = 1/2 𝑐𝜀₀(∣𝐸_𝑥∣² + ∣𝐸_𝑦∣²) = 1/2 𝑐𝜀₀𝐼. In units with 𝑐 = 𝜀₀ ≡ 1, we get 𝐼 = 2⟨𝓘⟩, so that the Stokes parameter 𝐼 indeed is a measure of the time-averaged intensity.
   The Stokes parameter 𝑉 vanishes for linear polarization (with 𝜑 = 0, π). Since circular polarization usually isn't produced in the early universe, we will set 𝑉 ≡ 0 in the following. Then 𝑄 and 𝑈 describe the linear polarization of the radiation. 𝑄 is the difference between the intensity of the radiation along 𝑥-axis, ∣𝐸_𝑥∣² and that along 𝑦-axis, ∣𝐸_𝑦∣². Radiation polarized along 𝑥-axis has 𝑄 > 0, while Radiation polarized along 𝑦-axis has 𝑄 < 0. Similarly, the parameter 𝑈 can be thought of as the difference in the intensity along two contribute axes 𝑎 and 𝑏 that are rotated counterclockwise by 45^∘ (see Fig. 7.20):
    (7.160)   𝑈 = ∣𝐸_𝑎∣² - ∣𝐸_𝑏∣² = 2Re[𝐸_𝑥𝐸_𝑦^*],
where we used that 𝐸_𝑥,𝑦 = (𝐸_𝑎 ± 𝑖𝐸_𝑏)/√2. Radiation polarized along the 𝑎-axis has 𝑈 > 0, while radiation polarized along 𝑏-axis has 𝑈 < 0.
   Rotating the 𝑥 and 𝑦 axes counterclockwise by an angle 𝜙 transforms the Stokes parameters
    (7.161)  ⌈𝐸_𝑥ʹ⌉ = ⌈ cos(𝜙)   sin(𝜙)⌉   ⌈𝐸_𝑥⌉  
                  ⌊𝐸_𝑦ʹ⌋    ⌊-sin(𝜙)   cos(𝜙)⌋   ⌊𝐸_𝑦⌋
                  ⌈𝑄ʹ⌉ = ⌈ cos(2𝜙)  sin(2𝜙)⌉   ⌈𝑄⌉
                  ⌊𝑈ʹ⌋    ⌊-sin(2𝜙)  cos(2𝜙)⌋   ⌊𝑈⌋.
  The transformation of the Stokes parameters can also be written as
    (7.162)  𝑄ʹ ± 𝑖𝑈ʹ = 𝑒^∓2𝑖𝜙(𝑄 ± 𝑖𝑈),
which shows that polarization transform like a spin-2 field. To avoid this ambiguity, we will introduce the coordinate-independent E- and B-modes of the polarization field.

          Decomposition into E- and B-modes On scales smaller than a few degrees the sky cam be approximated as flat. This flat-sky limit greatly simplifies the mathematical treatment of CMB polarization, so we will use the following. Let 𝜽 = (𝜃_𝑥, 𝜃_𝑦) be the position on a flat region of the sky and definethe symmetric and trace-free polarization tensor as     (7.163)  𝑃_𝑎𝑏(𝜽) ≡ ⌈𝑄(𝜽)   𝑈(𝜽)⌉
                                 ⌊𝑈(𝜽)  -𝑄(𝜽)⌋
Being made out of the Stokes parameters, this polarization tensor also transforms under a rotation  of the coordinate. However by taking spatial derivatives of 𝑃_𝑎𝑏 we can define two quantities invariant under rotations
    (7.164)  ∇²𝐸 ≡ ∂_𝑎∂_𝑏𝑃_𝑎𝑏,     ∇²𝛣 ≡ 𝜖_𝑎𝑐∂_𝑏∂_𝑐𝑃_𝑎𝑏,
where 𝜖_𝑎𝑐 is the antisymmetric tensor. In fact, these definitions of E- and B-modes even hold beyond the flat-sky limit if we promote the derivatives to covariant derivatives on the sphere. Writing (7.164) out in terms of the Stokes parameters, we get
    (7.165)  ∇²𝐸 ≡ (∂_𝑥² - ∂_𝑦²)𝑄 + 2∂_𝑥∂_𝑦𝑈,     ∇²𝛣 ≡ (∂_𝑥² - ∂_𝑦²)𝑈 - 2∂_𝑥∂_𝑦𝑄.
We note that the relation between the E- and B-modes and the Stokes parameters is non-local.
    The E/B decomposition of the polarization is similar to the decomposition of a vector field  into a gradient of a scalar function and the curl of a vector
    (7.166)  𝐕 = 𝛁𝐺 - 𝛁 × 𝐂,
where 𝛁 ⋅ 𝐂 = 0. Taking the divergence and the curl of the vector, we can isolate the two components
    (7.167)  ∇²𝐺 = 𝛁 ⋅ 𝐕,     ∇²𝐂 = 𝛁 × 𝐕,    
which indeed looks very similar to (7.164). This decomposition of a vector field into a curl-free gradient part and a divergence-free curl part also explains the names E-mode and B-mode: in electrostatics, the electric field has vanishing curl, 𝛁 × 𝐄 = 0, [RE 𝛁 × 𝐄 = -∂𝐁/∂𝑡 in Maxwell's equation] while a magnetic field has zero divergence, 𝛁 ⋅ 𝐁 = 0.

          Polarization power spectra
The CMB map is now described by the temperature 𝛵 and the polarization modes 𝐸 and 𝛣, which collectively call 𝑋 ≡ {𝛵, 𝐸, 𝛣}. The main benefit of working in the flat-sky limit is that we can use ordinary Fourier transform instead of spherical harmonics
    (7.168)  𝑋(𝒍) = ∫ d²𝜽 𝑋(𝜽) 𝑒^{-𝑖𝒍⋅𝜽},
where 𝒍 is the 2-dimensional wavevector conjugate to the position 𝜽. The relations in (7.165) imply that the Fourier components of the E- and B-modes satisfy
    (7.169)  𝐸(𝒍) = [(𝑙_𝑥² - 𝑙_𝑦²)𝑄(𝒍) + 2𝑙_𝑥𝑙_𝑦𝑈(𝒍)]/(𝑙_𝑥² + 𝑙_𝑦²),     𝛣(𝒍) = [(𝑙_𝑥² - 𝑙_𝑦²)𝑈(𝒍) - 2𝑙_𝑥𝑙_𝑦𝑄(𝒍)]/(𝑙_𝑥² + 𝑙_𝑦²).
We should confirm that although 𝑄(𝒍) and 𝑈(𝒍) transform under a rotation of the 𝜽_𝑥- and 𝜽_𝑦-axes, 𝐸(𝒍) and 𝛣(𝒍) are invariant. The power spectra of the fluctuations are defined as
    (7.170)  ⟨𝑋(𝒍)𝑌(𝒍ʹ)⟩ = (2π)²𝛿_𝐷(𝒍 + 𝒍ʹ)𝐶_𝑙^𝑋𝑌,
where 𝑋, 𝑌 = {𝛵, 𝐸, 𝛣}. In principle, this corresponds to six different spectra. But in practice only fiur spectra of them are expected to be nonzero. To see this, consider a reflection about 𝑥-axis (called a parity inversion).
    (7.171)  𝜃_𝑦 → -𝜃_𝑦,    𝑄 →  𝑄,     𝑈 →  -𝑈,     𝑙_𝑥 →  𝑙_𝑥,     𝑙_𝑦 → -𝑙_𝑦.
From (7.169) we see that E-mode is invariant under this reflection, while the B-mode change sign:
    (7.172)  𝐸(𝒍)  →  𝐸(𝒍),     𝛣(𝒍)  →    -𝛣(𝒍).
The field 𝐸 is therefore a parity-even scalar field (like 𝛵), while 𝛣 is a parity-odd pseudoscalar field. IF the physics of the primordial universe is parity conserving, then we expect 𝐶_𝑙^TB = 𝐶_𝑙^EB = 0.
   On the full sky, we have to use spherical harmonics to describe the polarization field and generalize our definitions of E- and B-modes. We will just sketch the logic and leave the complicate mathematical details to [20. 21].
   We saw that polarization is a spin-2 field, cf. (7.162). So we can't use ordinary scalar spherical harmonics to describe its decomposition into multipole moments. Instead we must use so-called spin-weighted sperical harmonics [20, 21]. Then the polarization field can be written as
    (7.173)  𝑄(𝐧̂) ± 𝑖𝑈(𝐧̂) = ∑_𝑙𝑚 𝑎_{±2,𝑙𝑚 ±}𝑌_𝑙𝑚(𝐧̂),
where the explicit forms of the spin-2 spherical harmonics _±𝑌_𝑙𝑚(𝐧̂) can be found in [20, 21]. The multipole coefficient 𝑎_2,𝑙𝑚 and 𝑎_-2,𝑙𝑚 can be combined into parity even and parity-odd combinations:
    (7.174)  𝑎_𝑙𝑚^𝐸 ≡ -(𝑎_2,𝑙𝑚 + 𝑎_-2,𝑙𝑚)/2,     𝑎_𝑙𝑚^𝛣 ≡ -(𝑎_2,𝑙𝑚 + 𝑎_-2,𝑙𝑚)/2𝑖,
which are multipole coefficients of the E- and B-modes:
    (7.175)  𝐸(𝐧̂) = ∑_𝑙𝑚 𝑎_{𝐸,𝑙𝑚} 𝑌_𝑙𝑚(𝐧̂),     𝛣(𝐧̂) = ∑_𝑙𝑚 𝑎_{𝛣,𝑙𝑚} 𝑌_𝑙𝑚(𝐧̂).
The polarization power spectra are then defined in the usual way:
    (7.176)  ⟨𝑎_𝑙𝑚𝑎_𝑙ʹ𝑚ʹ^𝐸*⟩ = 𝐶_𝑙^TE𝛿_𝑙𝑙ʹ𝛿_𝑚𝑚ʹ,      ⟨𝑎_𝑙𝑚^𝐸𝑎_𝑙ʹ𝑚ʹ^𝐸*⟩ = 𝐶_𝑙^EE𝛿_𝑙𝑙ʹ𝛿_𝑚𝑚ʹ,     ⟨𝑎_𝑙𝑚^𝛣𝑎_𝑙ʹ𝑚ʹ^𝛣*⟩ = 𝐶_𝑙^BB𝛿_𝑙𝑙ʹ𝛿_𝑚𝑚ʹ.
CMB polarization was first detected in 2002 by the Degree Angular Scale Interferomer (DASI) [22]. It has sice been measured with increasing precision by the WMAP and Planck satellites, as well as a number of ground-based experiments, Fig. 7.30 shows the TE and EE spectra measured by Planck [2] and ACT [23].

          7.6.3 Visualizing E- and B-modes
We will relate the formal definitions of E- and B-modes to the pictures in Section 7.6.1. Wee return to the flat-sky approximation. For simplicity we align the 𝑥-axis of our coordinates with the position vector 𝜽, and let 𝜙_𝒍 be the angle of the wavevector 𝒍 = 𝑙[cos(𝜙_𝒍), sin(𝜙_𝒍)]. The Fourier modes in (7.169) then are
    (7.177)   𝐸(𝒍) =  +𝑄(𝒍) cos(2𝜙_𝒍) + 𝑈(𝒍) sin(2𝜙_𝒍),     𝛣(𝒍) =  -𝑄(𝒍) sin(2𝜙_𝒍) + 𝑈(𝒍) cos(2𝜙_𝒍).
Inverting these expressions leads to write the Stokes parameters in terms of 𝐸 and 𝛣:
    (7.178)   𝑄(𝒍) = 𝐸(𝒍) cos(2𝜙_𝒍) - 𝛣(𝒍) sin(2𝜙_𝒍),     𝑈(𝒍) = 𝐸(𝒍) sin(2𝜙_𝒍) - 𝛣(𝒍) cos(2𝜙_𝒍),
The polarization generated by a single plane wave then is
    (7.179)   𝑄(𝜽) = Re[𝐸(𝒍) cos(2𝜙_𝒍) - 𝛣(𝒍) sin(2𝜙_𝒍) 𝑒^{𝑖𝒍⋅𝜽}],     𝑈(𝜽) = Re[𝐸(𝒍) cos(2𝜙_𝒍) + 𝛣(𝒍) sin(2𝜙_𝒍) 𝑒^{𝑖𝒍⋅𝜽}],
where the factor of 𝑒^{𝑖𝒍⋅𝜽} describes the modulation of the polarization amplitude induced by the plane wave perturbation. From this expression we can infer the polarization patterns corresponding to pure E- and B-mode created by a plane wave perturbation.
   For a pure E-mode, we have
    (7.180)   𝑄(𝜽) = Re[𝐸(𝒍) cos(2𝜙_𝒍) 𝑒^{𝑖𝒍⋅𝜽}],     𝑈(𝜽) = Re[𝐸(𝒍) sin(2𝜙_𝒍) 𝑒^{𝑖𝒍⋅𝜽}].
In the top of Fig. 7.31 The corresponding polarization pattern for two representative orientations of the wavevector 𝒍 was showed. In both cases, the polarization directions are parallel or perpendicular to the wavevector. This defining feature of E-mode polarization holds for its any orientation. Another way diagnose the presence of the E-mode is to rotate the coordinate system so that the 𝑥ʹ-axis is aligned with direction 𝒍. The Stokes 𝑈 parameter will then vanish and the polarization will be pure 𝑄. This is another coordinate-independent way define the E-mode.
   For a pure B-mode, we have
    (7.181)   𝑄(𝜽) = -Re[𝛣(𝒍) sin(2𝜙_𝒍) 𝑒^{𝑖𝒍⋅𝜽}],     𝑈(𝜽) = +Re[𝛣(𝒍) cos(2𝜙_𝒍) 𝑒^{𝑖𝒍⋅𝜽}].
In the bottom of Fig. 7.31, as before the corresponding polarization pattern for the same two representative orientations of the wavevector 𝒍 was showed. The polarization directions are now tilted by 45^∘ with the respect to the wavevector, which is the defining characteristic of B-mode polarization. Rotating the coordinates so that the 𝑥ʹ-axis is aligned with direction 𝒍. The Stokes 𝑄 parameter would vanish and the polarization is pure 𝑈.
   More complex polarization patterns arise from the superposition of the polarization patterns created by plane wave perturbations. In Fig. 7.32 we consider the polarization generated by a radial wave in the 𝑥𝑦-plane. The wave can be thought of as a superposition of plane waves with equal phases and amplitude, but different azimuthal angle 𝜙_𝒍. We can convince ourselves that a superposition of the polarization patterns shown in Fig. 7.31 (for varying 𝜙_𝒍) produces the polarization in Fig. 7.32. We see that the E-mode pattern alternates between being radial and tangential, while the B-mode polarization has a characteristic "swirl" pattern.

          7.6.4 E-modes from Scalars
We will now discuss the contents with more equations and a larger degree of generality and follow the excellent treatment in Dodelson's book [Dodelson]. In this section we will assume that the primordial fluctuations are sourced by adiabatic scalar fluctuations and show that these do not produce B-modes.

          Thomson scattering revisited
In Section 7.6.1 we saw graphically how unpolarized radiation becomes polarized by scattering. Mathematically this is described by the dependence of the scattering cross section on the polarization of incoming and the outgoing radiation. If the polarization direction of the incoming radiation is 𝛜̂ʹ and that of the outgoing radiation is 𝛜̂, then the differential cross section is
    (7.182)   d𝜎/d𝛺 = 3𝜎_𝛵/8π ∣𝛜̂ ⋅ 𝛜̂ʹ∣².
The dependence on 𝛜̂ ⋅ 𝛜̂ʹ enforces that only the component of the incoming polarization that is orthogonal to the direction of the outgoing radiation is transmitted.
   Without loss of generality, we can take the direction of the outgoing radiation to be 𝐳̂ and 𝛜̂₁ ≡ 𝐱̂ and 𝛜̂₂ ≡ 𝐲̂ to be an orthogonal basis of polarizational vectots (see Fig. 7.33). The 𝑄-type polarization of the outgoing radiation is
    (7.183)   𝑄(𝐳̂) = 𝛢 ∫ d𝛺ʹ 𝑓(𝐧̂ʹ) ∑²_𝑗=1 (∣𝐱̂ ⋅ 𝛜̂ʹ_𝑗∣² - ∣𝐲̂ ⋅ 𝛜̂ʹ_𝑗∣²),
where 𝑓(𝐧̂ʹ) is the distribution function of the incoming radiation and 𝛢 is the overall normalization (which would be fixed by the intensity of the incoming radiation). We assumed that incident radiation is unpolarized, so 𝑓(𝐧̂ʹ) doesn't depend on 𝛜̂ʹ_𝑗. To evaluate the inner product in (7.183), we write the polarization basis of the incoming radiation in Cartesian coordinates
    (7.184)   𝛜̂ʹ₁ ≡ -𝜽̂ʹ = [-cos(𝜃ʹ) cos(𝜙ʹ), -cos(𝜃ʹ) sin(𝜙ʹ), sin(𝜃ʹ)],     𝛜̂ʹ₂ ≡ -𝝓̂ʹ = [-sin(𝜙ʹ), cos(𝜙ʹ), 0]
We then get  
    (7.185)   𝑄(𝐳̂) = 𝛢 ∫ d𝛺ʹ 𝑓(𝐧̂ʹ) [cos²(𝜃ʹ) cos²(𝜙ʹ) + sin²(𝜙ʹ) - cos²(𝜃ʹ) sin²(𝜙ʹ) - cos²(𝜙ʹ)] = -𝛢 ∫ d𝛺ʹ 𝑓(𝐧̂ʹ) sin²(𝜃ʹ) cos(2𝜙ʹ)
The combination of angles in this integral is proportional to a sum of 𝑙 = 2 spherical harmonics, sin²(𝜃ʹ) cos(2𝜙ʹ) ∝[𝑌₂₂ + 𝑌_2,-2](𝐧̂ʹ); see (D.6) in Appendix D. By orthogonality of the spherical harmonics, the integral will pick out the 𝑙 = 2, 𝑚 = ±2 components of 𝑓(𝐧̂ʹ). We derived the crucial result that a nonzero 𝑄 will only be produced if the incident radiation has a nonzero quadrupole moment. We saw this in pictures before, but it is nice to have it drop out of the equations.
   The 𝑈-type polarization of the outgoing radiation is defined with respect to the polarization basis 𝛜̂_𝑎 = (𝐱̂ + 𝐲̂)/√2 and 𝛜̂_𝑏 = (𝐱̂ - 𝐲̂)/√2, We then find  
    (7.186)   𝑈(𝐳̂) = 𝛢 ∫ d𝛺ʹ 𝑓(𝐧̂ʹ) ∑²_𝑗=1 [∣(𝐱̂ + 𝐲̂) ⋅ 𝛜̂ʹ_𝑗∣² - ∣(𝐱̂ - 𝐲̂) ⋅ 𝛜̂ʹ_𝑗∣²] = -𝛢 ∫ d𝛺ʹ 𝑓(𝐧̂ʹ) sin²(𝜃ʹ) sin(2𝜙ʹ)
where we substituted (7.184) to obtain the result in the second line. The angular dependence of the integrand is now proportional to the difference of 𝑙 = 2 spherical harmonics, sin²(𝜃ʹ) sin(2𝜙ʹ) ∝ [𝑌₂₂ - 𝑌_2,-2](𝐧̂ʹ), so that again only the quadruople moment of 𝑓(𝐧̂ʹ) produces polarization.

          Polarization from a single plane wave
WE would like to predict the polarization signals generated by the scattering of an inhomogeneous distributions of photons. We first write the perturbed photon distribution function as
    (7.187)   𝑓(𝐧̂ʹ, 𝐱) = exp(𝐸/𝛵̄[1 + 𝛩(𝐧̂ʹ, 𝐱)] -1)^-1 ≈ 𝑓̄ - d𝑓̄/(d ln 𝐸) 𝛩(𝐧̂ʹ, 𝐱)
It's convenient to describe the spatial dependence of the fluctuation s in Fourier space, 𝜭(𝐧̂ʹ, 𝐱) → 𝜭(𝐧̂ʹ, 𝐱). The fluctuations created by scalar density perturbations have an axial symmetry. The perturbations in the direction 𝐧̂ʹ only depend on magnitude of the wavevector 𝑘 and the angle defined by 𝜇ʹ = 𝐤̂ ⋅ 𝐧̂ʹ. We then write the temperature perturbation as an expansion in Legendre polynomials
    (7.188)   𝛩(𝑘, 𝜇ʹ) = ∑_𝑙 𝑖^𝑙(2𝑙 + 1) 𝛩_𝑙(𝑘) 𝑃_𝑙(𝜇ʹ).
Only the 𝑙 = 2 quadrupole moment of this expansion leads to polarization. Substituting this quadrupole into (7.185) and (7.186), we get
    (7.189)   𝑄(𝐳̂) ∝𝛩₂(𝑘) ∫^π₀ d𝜃ʹ sin(𝜃ʹ) ∫^2π₀ d𝜙ʹ 𝛲₂(𝜇ʹ) sin²(𝜃ʹ) cos(2𝜙ʹ),     𝑈(𝐳̂) ∝𝛩₂(𝑘) ∫^π₀ d𝜃ʹ sin(𝜃ʹ) ∫^2π₀ d𝜙ʹ 𝛲₂(𝜇ʹ) sin²(𝜃ʹ) sin(2𝜙ʹ).
To evaluate the angular integrals, we must first write 𝜇ʹ in terms of 𝜃ʹ and 𝜙ʹ. Following Dodelson [Dodelson], we will do this in three steps:

1.  First, we take the wavevector to be parallel to 𝑥-axis, 𝐤∥𝐱̂. This gives
    (7.190)   𝜇ʹ = 𝐤 ⋅ 𝐧̂ʹ = 𝐱̂ ⋅ 𝐧̂ʹ = 𝑛̂_𝑥ʹ = sin(𝜃ʹ) cos(𝜙ʹ),
   and hence [RE Appendix D.3 (D.13): 𝑃̄₂ = 1/2 (3𝜇₂ - 1)]
    (7.191)   𝑃̄₂(𝜇ʹ) = 3/2(𝜇ʹ)² - 1/2 = 3/2[sin(𝜃ʹ) cos(𝜙ʹ)]² - 1/2.
   Only the first term of the Legendre polynomial survives the angular integration in (7.180) and we get
    (7.192)   𝑄(𝐳̂, 𝐤∥𝐱̂) ∝ 4π/5 𝛩₂(𝑘),     𝑈(𝐳̂, 𝐤∥𝐱̂) ∝0.
   This is consistent with what we found before: the polarization created by density perturbations has vanishing Stokes parameter 𝑈 when the 𝑥-axis is aligned with the wavevector of the plane wave.
2.  Next, we take the wavevector to lie in the 𝑥𝑦-plane, 𝐤⊥𝐲̂. Writing
    (7.193)   𝐤̂ = [sin(𝜃_𝑘), 0, cos(𝜃_𝑘)].
   we get
    (7.194)   𝜇ʹ = 𝐤 ⋅ 𝐧̂ʹ = sin(𝜃_𝑘) 𝑛̂_𝑥ʹ + cos(𝜃_𝑘) 𝑛̂_𝑥ʹ = sin(𝜃_𝑘) sin(𝜃ʹ) cos(𝜙ʹ) + cos(𝜃_𝑘) cos(𝜃ʹ),
   and hence
    (7.195)   𝑃̄₂(𝜇ʹ) = 3/2[sin(𝜃ʹ) cos(𝜙ʹ)]² + ⋅ ⋅ ⋅ ,
   where the ellipse stand for terms that lead to vanishing contributions when integrated against cos(2𝜙ʹ) or sin(2𝜙ʹ). The only difference with the relevant term in (7.191) is the extra factor of sin²(𝜃_𝑘). WE therefore get
    (7.196)   𝑄(𝐳̂, 𝐤⊥𝐱̂) ∝4π/5 𝛩₂(𝑘),     𝑈(𝐳̂, 𝐤⊥𝐱̂) ∝0.
   Projected onto the 𝑥𝑦-plane the direction of the wavevector is still aligned with the 𝑥-axis, so the Stokes parameter 𝑈 still vanishes.
3.  Finally, we consider a wavevector pointing in an arbitrary direction. Its components are
    (7.197)   𝐤̂ = [sin(𝜃_𝑘) cos(𝜙_𝑘), sin(𝜃_𝑘) sin(𝜙_𝑘), cos(𝜃_𝑘)].
    (7.198)   𝑄(𝐳̂, 𝐤) ∝ 4π/5 sin²(𝜃_𝑘) cos(2𝜙_𝑘) 𝛩₂(𝑘),     𝑈(𝐳̂, 𝐤) ∝ 4π/5 sin²(𝜃_𝑘) sin(2𝜙_𝑘) 𝛩₂(𝑘).

   Comparison with (7.178) reveals that this is indeed a pure 𝐸-mode:
    (7.199)   𝐸(𝐳̂, 𝐤) ∝ 4π/5 sin²(𝜃_𝑘) 𝛩₂(𝑘).   𝐸(𝐳̂, 𝐤) ∝ 0,
   where we have implicitly used that 𝒍 is the projection of 𝐤 onto the plane of the sky to identify 𝜙_𝑘 with 𝜙_𝒍.

Exercise 7.6   Derive the result in (7.198)
[Solution]   For a fluctuationh general wavevector 𝐤, using
   (a)   𝐤̂ = [sin(𝜃_𝑘) cos(𝜙_𝑘), sin(𝜃_𝑘) sin(𝜙_𝑘), cos(𝜃_𝑘)],    𝐧̂ = [sin(𝜃ʹ) cos(𝜙ʹ), sin(𝜃ʹ) sin(𝜙ʹ), cos(𝜃ʹ)].
   (b)   𝜇ʹ ≡ 𝐤̂ ⋅ 𝐧̂ = sin(𝜃_𝑘) cos(𝜙_𝑘 sin(𝜃ʹ) cos(𝜙ʹ) + sin(𝜃_𝑘) sin(𝜙_𝑘 sin(𝜃ʹ) sin(𝜙ʹ) + cos(𝜃_𝑘) cos(𝜃ʹ).
   (c)   𝑃̄₂(𝜇ʹ) = 3/2(𝜇ʹ)² - 1/2 = 3/2 (sin²(𝜃_𝑘) cos²(𝜙_𝑘) [sin(𝜃ʹ) cos(𝜙ʹ)]²
                               + sin²(𝜃_𝑘) sin²(𝜙_𝑘) [sin(𝜃ʹ) sin(𝜙ʹ)]² + 2 sin²(𝜃_𝑘) cos(𝜙_𝑘) sin(𝜙_𝑘) sin²(𝜃ʹ) cos(𝜙ʹ) sin(𝜙ʹ)) + ⋅ ⋅ ⋅ ,
Substituting this into (7.189), we get
   (d)   𝑄(𝐳̂, 𝐤) ∝ 𝛩₂(𝑘) ∫^π₀ d𝜃ʹ sin(𝜃ʹ) ∫^2π₀ d𝜙ʹ 𝛲₂(𝜇ʹ) sin²(𝜃ʹ) cos(2𝜙ʹ)
             ∝ 4π/5 𝛩₂(𝑘) sin²(𝜃_𝑘) [cos²(𝜙_𝑘) - sin²(𝜙_𝑘)] = 4π/5 sin²(𝜃_𝑘) cos(2𝜙_𝑘) 𝛩₂(𝑘).
   (e)   𝑈(𝐳̂, 𝐤) ∝ 𝛩₂(𝑘) ∫^π₀ d𝜃ʹ sin(𝜃ʹ) ∫^2π₀ d𝜙ʹ 𝛲₂(𝜇ʹ) sin²(𝜃ʹ) sin(2𝜙ʹ)
             ∝ 4π/5 𝛩₂(𝑘) sin²(𝜃_𝑘) × 2 sin(𝜃_𝑘) cos(𝜙_𝑘) = 4π/5 sin²(𝜃_𝑘) sin(2𝜙_𝑘) 𝛩₂(𝑘).  ▮

   Ultimately we need the result for a general line-of-sight 𝐧̂. To obtain this we perform 𝐳̂ → -𝐧̂ and 𝐧̂ = -𝐩. Since the E-mode is invariant under this rotation, we must simply replace cos(𝜃_𝑘) with -𝐧̂ ⋅ 𝐤̂ and (7.199) becomes
    (7.200)   𝐸(𝐧̂, 𝐤̂) ∝ 4π/5 [1 - (𝐧̂ ⋅ 𝐤̂)²] 𝛩₂(𝑘),     𝛣(𝐧̂, 𝐤̂) ∝ 0.
This result assumes the flat-sky approximation and hence only applies to directions 𝐧̂ near the 𝑧-axis. The general result is much more complicated, but the same conclusion holds: no B-modes are generated by scalar density fluctuations.

          Quadruple from velocity gradients
We saw that polarization is generated by the scattering of a quadrupolar anisotropy. But in the tight-coupling regim, this anosotropy is erased by the frequent scattering of the photons. Polarization is therefore only generated near recombination when anisotropies grow by free streaming. Since the electron density drops sharply at recombination there is only a short time window of the significant polarization of the CMB.
   In density perturbations the quadrupole moment is produced by velocity gradients in the photon-baryon fluid. Refer to Appendix B for the explicit derivation using Boltzmann equation. Here a more heuristic explanation will be given [27]. Consider scattering at 𝐱₀. The scattered photon came from a position 𝐱 = 𝐱₀ + 𝓁_𝛾𝐧̂ʹ, where 𝓁_𝛾 = 𝛤^-1 is the mean free path. The velocity of photon of the photon-baryon fluid at that position was
    (7.201)   𝐯_𝑏(𝐱) ≈ 𝐯_𝑏(𝐱₀) + 𝓁_𝛾𝐧̂ʹ ⋅ 𝛁 𝐯_𝑏(𝐱₀).
The temperature seen by the scatterer at 𝐱₀ is Doppler shifted and hence given by
    (7.202)   𝛩(𝐱₀, 𝐧̂ʹ) ≡ 𝛿𝛵(𝐱₀, 𝐧̂ʹ)/𝛵̄ = 𝐧̂ʹ ⋅ [𝐯_𝑏(𝐱) -  𝐯_𝑏(𝐱₀)] ≈ 𝓁_𝛾𝑛̂_𝑖ʹ𝑛̂_𝑗ʹ∂_𝑖(𝐯_𝑏)_𝑗.
Being quadradtic in 𝐧̂ʹ, the induced temperature anisotropy has the required quadrupole component 𝛩₂. The Fourier transform of (7.202) is
    (7.203)   𝛩(𝐤, 𝐧̂ʹ) = ∫ d³ 𝑥 𝑒^{-𝑖𝐤, 𝐱} 𝛩(𝐱, 𝐧̂ʹ) = 𝓁_𝛾𝑛̂_𝑖ʹ𝑛̂_𝑗(𝑖𝑘^𝑖)(𝑗𝑘^𝑗 𝑣_𝑏) = -𝑘/𝛤 𝑣_𝑏 (𝐤̂ ⋅ 𝐧̂ʹ)² ⊂ -2/3 𝑘/𝛤 𝑣_𝑏 𝑃̄₂(𝜇ʹ),  
from which we can read off 𝛩₂(𝑘).

          E-mode spectrum
Let 𝐱₀ now be a point on the surface of last-scattering. Using (7.200) and (7.203), the E-mode polarization created by a plane wave perturbation is
    (7.204)   𝐸(𝐧̂, 𝐤) ∝ [1 - (𝐧̂ ⋅ 𝐤̂)²] 𝛩₂(𝑘) 𝑒^{𝑖𝐤 ⋅ 𝐱₀} ∝ 𝑘/𝛤 𝑣_𝑏(𝜂_*, 𝐤)(1 - 𝜇²) 𝑒^{𝑖𝑘𝜂₀𝜇},
where we include the factor of 𝑒^{𝑖𝐤 ⋅ 𝐱₀} to describe the modulation of the amplitude as a function of the position 𝐱₀ = (𝜂₀ - 𝜂_*)𝐧̂ ≈ 𝜂₀𝐧̂. Next we expand the exponential in spherical Bessel functions and extract the multipole moment 𝐸_𝑙(𝐤). The details are given in Appendix B, where we find that
    (7.205)   𝐸_𝑙(𝐤) ∝ 𝑘/𝛤 𝑣_𝑏(𝜂_*, 𝐤) √(1 + 2)𝑙!/(1 - 2)𝑙! 𝑗_𝑙(𝑘𝜂₀)/(𝑘𝜂₀)².
For large 𝑙, the square root becomes 𝑙² and the polarization transfer function is
    (7.206)   𝛩_𝐸,𝑙(𝑘) ≡ 𝐸_𝑙(𝐤)/𝓡_𝑖(𝐤) ∝ 𝑘/𝛤 𝐺_*(𝑘) 𝑙²/(𝑘𝜂₀)² 𝑗_𝑙(𝑘𝜂₀),
where 𝐺_*(𝑘) ≡ 𝑣_𝑏(𝜂_*, 𝐤)/𝓡_𝑖(𝐤) is the transfer function of the photon-baryon velocity. The E-mode power spectrum then is
    (7.207)   𝐶^EE_𝑙 = 4π ∫^∞₀ d ln 𝑘 𝛩_𝐸,𝑙²(𝑘) ∆²_𝓡(𝑘).
A more exact treatment would derive 𝛩_𝐸,𝑙(𝑘) from the Boltzmann equation (see Appendix B)
   Fig. 7.34 shows the predicted E-mode spectrum and compares it to the temperature power spectrum. Since the E-mode signal is created by the gradient of the fluid velocity, its oscillations are expected to be out of phase with oscillations in the temperature spectrum; recall 𝛿_𝛾ʹ = -4/3 𝛁 ⋅ 𝐯_𝑏. This is indeed seen in  Fig. 7.34: the peak of E-mode spectrum coincide with the troughs of the temperature power spectrum. The polarization spectrum also has a pronounced peak at 𝑙 < 10, which is due to the rescattering of the CMB photons after reionization. The size of this reionization peak depends on the optical depth 𝜏, so that measurements of the low-𝑙 polarization help to break the degeneracy between 𝜏 and 𝛢_𝑠  that we found for the temperature fluctuations (see Section 7.5.3).

          7.6.4 B-modes from Tensors
We saw the primordial density fluctuation cannot produce a B-mode polarization signal. Such a signal is therefore a unique discovery channel for primordial gravitational waves. We found that the scattering of incident radiation with temperature anisotropies 𝛩(𝐧̂ʹ) creates the creates the following polarization for outgoing radiation near the 𝑧-axis:
    (7.208)   𝑄(𝐳) ∝∫ d𝛺ʹ 𝛩(𝐧̂ʹ) sin²(𝜃ʹ) cos(2𝜙ʹ),      𝑈(𝐳) ∝∫ d𝛺ʹ 𝛩(𝐧̂ʹ) sin²(𝜃ʹ) sin(2𝜙ʹ).
We will show how a quadrupole anisotropy is generated by gravitational waves.

          Quadrupole from gravitational waves
Consider a gravitational wave propagating in the 𝑧-direction. The perturbation of the spatial metric is
    (7.209)   𝛿𝑔_𝑖𝑗 = 𝑎²𝘩_𝑖𝑗 = 𝑎² ⌈ 𝘩₊  𝘩_×  0 ⌉
                                              ┃ 𝘩_× -𝘩₊  0 ┃
                                               ⌊ 0    0    0  ⌋
The presence of this perturbation affects the free streaming of the photons. Using (7.209) in the geodesic equation gives (see Problem 7.6)
    (7.210)   d𝛩^(𝑡)/d𝜂 = d ln(𝑎𝐸)/d𝜂 = -1/2 ∂𝘩_𝑖𝑗/∂𝜂 (𝐧̂ʹ)^𝑖(𝐧̂ʹ)^𝑗 = -1/2 ∂𝘩₊/∂𝜂 [(𝑛̂_𝑥ʹ)² - (𝑛̂_𝑦ʹ)²] - ∂𝘩_×/∂𝜂 𝑛̂_𝑥ʹ𝑛̂_𝑦ʹ
                                  = -1/2 sin²(𝜃ʹ) [∂𝘩₊/∂𝜂 cos(2𝜙ʹ) + ∂𝘩_×/∂𝜂 sin(2𝜙ʹ)],
where we wrote the direction of the incoming photon as
    (7.211)   𝐧̂ʹ = [sin(𝜃ʹ) cos(𝜙ʹ), sin(𝜃ʹ) sin(𝜙ʹ), cos(𝜃ʹ)].
The angular dependence in (7.210) induces a quadrupole moment in the radiation field and the scattering of this tensor-induced temperature anisotropy will then lead to polarization.
   To understand this intuitively, consider how a gravitational wave propagating in the 𝑧-direction distorts space in the 𝑥𝑦-plane (see Fig. 7.35. The key characteristic of gravitational waves is that these distortions are anisotropic, streching space in one direction and compressing it in an orthogonal direction. Along the expanding direction, the photons are redshifted and become colder, while in the contracting direction, the photons are blueshifted and become hotter. This produces precisely the temperature quadrupole required to create CMB polarization. Since the gravitational wave has the two polarizaion modes rotated by 45^∘, we get two polarization patterns with a relative orientation of 45^∘. This is the fundamental reason why gravitational waves produce both 𝐸- and 𝛣-modes. We will prove more rigorously now.

          B-modes from gravitational waves
Let us first focus on the × polarization of the gravitational wave. The induced temperature fluctuations then have the following angular dependence:
    (7.212)   𝛩_×^(𝑡)(𝐧̂ʹ, 𝐤∥𝐳̂) ∝ 1/2 sin²(𝜃ʹ) sin(2𝜙ʹ) = 𝑛̂ʹ_𝑥𝑛̂ʹ_𝑦.
This is the result for a wavevector pointing in the 𝑧-direction, but we need more general case. To show that the gravitational waves produce B-modes, it suffices to consider the special case
    (7.213)   𝐤̂ = [sin(𝜃_𝑘), 0, cos(𝜃_𝑘) ⊥ 𝐲̂,
where 𝜃_𝑘 is the angle between 𝐤̂ and the inverse of the line-of-sight, -𝐧̂. The projection of this wavevector onto the sky is parallel to 𝑥-axis (𝜙_𝒍 = 0), so it should lead to vanishing 𝑈 if the polarization is a pure E-mode; cf. (7.180). Conversely, finding 𝑈 ≠ 0 for this wavevectore would prove that a nonzero B-mode was generated.
   We first need to transform the result in (7.212), so that it gives the temperature anisotropy induced by the wavevectore in (7.213). We achieve this by rotating the coordinate system by an angle -𝜃_𝑘 around the 𝑦-axis. This gives
    (7.214)   𝑛̂ʹ_𝑥 → cos(𝜃_𝑘)𝑛̂ʹ_𝑥 - sin(𝜃_𝑘)𝑛̂ʹ_𝑧,     𝑛̂ʹ_𝑦 → 𝑛̂ʹ_𝑦,
and hence
    (7.215)   𝛩_×^(𝑡)(𝐧̂ʹ, 𝐤⊥𝐲̂) ∝cos(𝜃_𝑘)𝑛̂ʹ_𝑥𝑛̂ʹ_𝑦 - sin(𝜃_𝑘)𝑛̂ʹ_𝑦𝑛̂ʹ_𝑧 ∝ cos(𝜃_𝑘)[sin²(𝜃ʹ) sin(2𝜙ʹ)] - sin(𝜃_𝑘) [sin(𝜃ʹ) cos(𝜃ʹ) sin(𝜙ʹ)].
The second term leads to a vanishing contribution in (7.208), while the first term gives
    (7.216-7)   𝑄_×(𝐳̂ʹ, 𝐤⊥𝐲̂) = 0,    𝑈_×(𝐳̂ʹ, 𝐤⊥𝐲̂) ∝cos(𝜃_𝑘) ∫¹_-1 d cos(𝜃ʹ) sin⁴(𝜃ʹ) ∫^2π₀ d𝜙ʹ sin²(2𝜙ʹ) = 16π/15 cos(𝜃_𝑘).
Except for a plane wave that is oriented precisely orthogonal to the line-of sight (𝜃_𝑘 =90^∘), we found a nonzero Stokes parameter 𝑈. Gravitational waves indeed produce B-mode polarization.

Exercise 7.7   Repeating the above analysis for the + polarization of the gravitational wave, show that
    (7.218-9)   𝑄₊(𝐳̂ʹ, 𝐤⊥𝐲̂) ∝4π/15 [cos²(𝜃_𝑘) + 1],    𝑈₊(𝐳̂ʹ, 𝐤⊥𝐲̂) = 0.
We see that this polarization signal has vanishing 𝑈 parameter and is hence an E-mode.
[Solution]   We can start equations, the temperature anisotropies induced by a gravitational waves propagating in 𝑧-direction (7.210). For the + polarization, this implies as follows
   (a)   𝐧̂ʹ = [sin(𝜃ʹ) cos(𝜙ʹ), sin(𝜃ʹ) sin(𝜙ʹ), cos(𝜃ʹ)]
   (b)   𝛩₊^(𝑡)(𝐧̂ʹ, 𝐤∥𝐳̂) ∝ (𝑛̂_𝑥ʹ)² - (𝑛̂_𝑦ʹ)² = sin²(𝜃ʹ) cos(2𝜙ʹ).
We would like to the result for a more general wavevector 𝐤⊥𝐲̂:
   (c)   𝐤̂ = [sin(𝜃_𝑘), 0, cos(𝜃_𝑘).
We achieve this by rotating the coordinate system by an angle -𝜃_𝑘 around the 𝑦-axis. This gives
   (d)   𝑛̂ʹ_𝑥 → cos(𝜃_𝑘) 𝑛̂ʹ_𝑥 - sin(𝜃_𝑘) 𝑛̂ʹ_𝑧,     𝑛̂ʹ_𝑦 → 𝑛̂ʹ_𝑦,    and hence
   (e)   𝛩₊^(𝑡)(𝐧̂ʹ, 𝐤⊥𝐲̂) ∝[cos(𝜃_𝑘) 𝑛̂ʹ_𝑥 - sin(𝜃_𝑘) 𝑛̂ʹ_𝑦]² ∝ [cos(𝜃_𝑘) sin(𝜃ʹ) cos(𝜙ʹ) - sin(𝜃_𝑘) cos(𝜃ʹ)]² - [sin(𝜃ʹ) sin(𝜙ʹ)]²
Substituting this into
   (f)   𝑄₊(𝐳) ∝ ∫ d𝛺ʹ 𝛩₊^(𝑡)(𝐧̂ʹ) sin²(𝜃ʹ) cos(2𝜙ʹ),      𝑈₊(𝐳) ∝ ∫ d𝛺ʹ 𝛩₊^(𝑡)(𝐧̂ʹ) sin²(𝜃ʹ) sin(2𝜙ʹ).
We get
   (g)   𝑄₊(𝐳, 𝐤⊥𝐲̂) ∝ 4π/15 [cos²(𝜃_𝑘) + 1],   𝑈₊(𝐳, 𝐤⊥𝐲̂) = 0.  ▮ 

In Summary, we have found that gravitational waves produce both E- and B-modes, while density fluctuations only create E-modes.

          B-mode spectrum
The solid line in Fig. 7.36 shows the B-mode spectrum expected for a scale-invariant spectrum of primordial gravitational waves. The damping of the signal for 𝑙 > 100 is caused by the finite thickness of the last-scattering surface [31], while the peak at 𝑙 < 10 is again due to the rescattering of the CMB photons after reionization. THe oscillations in the spectrum are analogous to the acoustic oscillation in the angular power spectrum from scalar perturbations, but the frequency is set by the speed of light and not the sound speed of photon-baryon fluid. The peaks of the spectrum are at 𝑙_𝑛 ≈ 𝑛π𝜒_*/𝜂_*. Note that the amplitude of the signal not predicted, but depends on unknown tensor-to scalar ratio 𝑟. Recent measurement of CMB polarization imply the upper bound 𝑟 < 0.035 [32].
   Although scalar fluctuations don't produce B-mode polarization at the moment of last-scattering, they do create B-modes at later times via gravitational lensing. As the CMB photons travel through the large-scale structure of the universe they get deflected and E-mode polarization gets converted into B-modes. The resulting lensing-induced B-mode power spectrum is shown as the dashed line in Fig. 7.36. The signal has recently been detected by BICEP, SPTpol and Polarbear. While the lensing B-modes provide an interesting consistency check, they are a foreground in the search for primordial B-modes.

          7.7 Summary
In yhis chapter we have studied the physics of the CMB anisotropies. We related observed temperature fluctuation 𝛿𝛵(𝐧̂) to the primordial curvature perturbations 𝓡𝑖(𝐤) ≡ 𝓡(0, 𝐤) and computed the two-point correlation function, ⟨𝛿𝛵(𝐧̂)𝛿𝛵(𝐧̂ʹ)⟩ given the power spectrum ⟨𝓡𝑖(𝐤)𝓡𝑖(𝐤ʹ)⟩. The key steps involved in the calculation are summarized by the following schematic:
             𝓡(0,𝑘)  → evolution (§7.4) →  [𝛿𝛾,  𝛹,  𝑣𝑏]𝜂*  → free streaming (§7.2); projection (§7.3) →  𝛿𝛵(𝐧̂).
we worked through this in reverse order:
•  Free streaming  By intergrating the geodesic equation along the line-of sight, we obtained a relation between the observed temperature fluctuations in a particular direction and corresponding fluctuations on the surface of last-scattering
    (7.220)   𝛿𝛵/𝛵̄̄ (𝐧̂) = (1/4 𝛿𝛾 + 𝛹)* - (𝐧̂ ⋅ 𝐯𝑏)* + ∫𝜂0𝜂*d𝜂 (𝛷ʹ + 𝛹ʹ),
   where the subscript * indicates quantities evaluated at last-sccattering. The three terms on he right-hand side of (7.220) are the Sachs-Wolfe (SW) term, the Doppler term and the integrated Scachs-Wolfe (ISW) term, respectively.

•  Projection  We then wrote the temperature anisotropies as a sum over spherical harmonics
    (7.221)   𝛿𝛵/𝛵̄̄ (𝐧̂) = ∑_𝑙𝑚 𝑎_𝑙𝑚𝑌_𝑙𝑚(𝐧̂),
   Ignoring the ISW contribution, the multipole moments arising from the line-of-sight solution (7.220) are
    (7.222)   𝑎_𝑙𝑚 = 4π𝑖^𝑙∫ d³𝑘/(2π)³ [𝐹_*(𝑘)𝑗_𝑙(𝑘𝜒_*) - 𝐺_*(𝑘)𝑗_𝑙ʹ(𝑘𝜒_*)]𝓡_𝑖(𝐤) 𝑌_𝑙𝑚^*(𝐤̂).
   where 𝐹_*(𝑘)𝑗_𝑙(𝑘𝜒_*) - 𝐺_*(𝑘)𝑗_𝑙ʹ(𝑘𝜒_*) ≡ 𝛩_𝑙(𝑘) as well as 𝐹_*(𝑘) ≡ 𝐹(𝜂_*, 𝐤) /𝓡_𝑖(𝐤) and 𝐺_*(𝑘) ≡ 𝐺(𝜂_*, 𝐤)/𝓡_𝑖(𝐤) are transfer functions for Sachs-Wolfe and Doppler terms, respectively. The two-point function of the multipole moments, ⟨𝑎^*_𝑙𝑚𝑎^*_𝑙ʹ𝑚ʹ⟩ = 𝐶_𝑙𝛿_{𝑙𝑙ʹ𝑚𝑚ʹ}, defines the angular power spectrum 𝐶_𝑙. For the solution (7.222), The power spectrum is given by  
    (7.223)   𝐶_𝑙 = 4π ∫ d ln 𝑘 𝛩_𝑙²(𝑘)∆_𝓡²(𝑘).
   We see that the power spectrum is determined by the power spectrum of the primordial curvature perturbations, ∆_𝓡²(𝑘), and the transfer function, 𝛩_𝑙(𝑘). The latter describes both the evolution until decoupling and the projection onto the surface of last-scattering.

•  Evolution  To determine the transfer functions 𝐹_*(𝑘) and 𝐺_*(𝑘), we studied the evolution of fluctuations in the primordial plasma. As long as photons and baryons are tightly coupled, they can be treated as  single fluid. Fluctuations in the photon density the equation of a forced harmonic oscillator
    (7.224)   𝛿_𝛾ʺ + 𝑅ʹ/(1+ 𝑅) 𝛿_𝛾ʹ + 𝑘²𝑐_𝑠²𝛿_𝛾 = -3/4 𝑘²𝛹 + 4𝛷ʺ + 4𝑅ʹ/(1+ 𝑅) 𝛷ʹ,
   where 𝑅 ≡ 3/4 𝜌̄_𝑏/𝜌̄_𝛾 and 𝑐_𝑠² = [3(1 + 𝑅)]^-1. This describes the propagation of sound waves in the plasma. On small scales, the time dependence of the metric potentials can be ignored and that of the baryon-to-photon ratio 𝑅(𝑡) can be accounted for in a WKB approximation. The high-frequency solution of (7.224) then is
    (7.225)   𝛿_𝛾 = (1 + 𝑅)^-1/4[𝐶 cos(𝑘𝑟_𝑠) + 𝐷 sin(𝑘𝑟_𝑠)] - 4(1 + 𝑅)𝜓,
   where 𝑟_𝑠 = ∫ d𝜂/√[3(1 + 𝑅)] is the sound horizon. The constant 𝐶 and 𝐷 must be determined by matching to the superhorizon limit, with 𝐶 ≫ 𝐷 for adiabatic initial conditions. We als discussed a more accurate semi-analytic solution to (7.224) due to Weinberg [Weinberg]:  
    (7.226)   𝛿_𝛾 = 4𝓡_𝑖/5 [𝑆(𝑘)/(1 + 𝑅)^1/4 cos[𝑘𝑟_𝑠 + 𝜃(𝑘)] - (1 + 3𝑅) 𝛵(𝑘)],  
   where 𝑆(𝑘), 𝛵(𝑘) and 𝜃(𝑘) are defined as interpolation functions in (7.116). Finally we included photon viscosity which becomes important on scales less than the diffusion length of the photons and leads to an exponential damping of the fluctuations.

   We explained how the shape of the CMB power spectrum depends on the 𝛬CDM parameters {𝛢_𝑠, 𝑛_𝑠, 𝜔_𝑏, 𝜔_𝑚, 𝛺_𝛬, 𝜏}. The main effects are summarized in Table 7.2. Three length scales are imprinted in the spectrum:

• 𝑟_𝑠,*:  The peak location depend on the sound horizon at last-scattering.
• 𝑘_𝐷^-1:  The damping of the spectrum is determined by the diffusion scale.
• 𝑑_𝛢,*:  The map to angular scales depends on the distance to last-scattering.

The sound horizon and the diffusion scale are fixed by pre-combination physics ad hence in the 𝛬CDM model depend only on 𝜔_𝑏 and 𝜔_𝑚 (for fixed radiation density). These parameters also determine the peak heights through baryon loading (𝜔_𝑏) and radiation driving (𝜔_𝑚). The angular diameter distance to last-scattering is sensitive to the geometry (𝛺_𝛬, 𝐻₀) and the energy content (𝛺_𝑚, 𝛺_𝛬) of the late universe.
   Finally, a brief introduction to CMB polarization was given. It was shown that polarization is generated by scattering if the temperature anisotropies seen by the electrons have a quadrupole moment. The signal can be seperated into a curl-free E-mode and a divergenceless B-mode. We prove that density fluctuations only produce E-modes, making the B-mode signal a unique signature of primordial gravitational waves (see Fig. 7.37). As we will discuss in the next chapter, these gravitational waves carry critical information about the physics of inflation.