Line-of-Sight CMB Anisotropies

The CMB sky is a projection of local photon sources distributed along each past-directed ray. The line-of-sight solution makes that statement precise: a source at conformal time $\eta$ is attenuated by the optical depth to today and projected onto angular multipole $\ell$ by a spherical Bessel kernel. This separates intrinsic temperature, gravitational redshift, Doppler, time-dependent-potential, and polarization contributions without counting any of them twice.

Required background. Probability and Inference for Gravity supplies Probability and Statistical Inference for random-field and angular-power-spectrum normalization.

Helpful background. The Einstein Equation supplies Einstein Equations and metric perturbations; Relativistic Matter and Compact Objects supplies Relativistic Matter and Compact Objects stress–energy variables; Relativistic Cosmology supplies the background, recombination, and perturbation transfer; and The Wave Equation and Characteristics supplies Waves, Green Functions, and Geometric Optics for retarded propagation and angular projection.

Photon transport through the visibility window

Work in spatially flat Newtonian gauge with the chapter conventions

$$ds^2=a^2\left[-(1+2\Phi)d\eta^2 +(1-2\Psi)\delta_{ij}dx^i dx^j\right], \qquad e^{i\mathbf k\cdot\mathbf x}.$$

Here $\eta$ is conformal time and a prime denotes $d/d\eta$. Let $\Theta(\eta,\mathbf k,\mu)=\delta T/T$ and $\mu=\hat{\mathbf k}\cdot\mathbf n$, where $\mathbf n$ is the photon propagation direction toward the observer; the viewing direction on the sky is $-\mathbf n$. Define the scalar baryon velocity by $\mathbf n\cdot\mathbf v_b=-i\mu v_b$, so $v_b=\theta_b/k$ in the velocity-divergence convention of the preceding chapter. With the positive conformal scattering rate $\dot\kappa=an_e\sigma_T$, the optical depth from $\eta$ to the observer and the visibility density are

$$\kappa(\eta)=\int_\eta^{\eta_0}\dot\kappa(\tilde\eta)\,d\tilde\eta, \qquad g(\eta)=\dot\kappa e^{-\kappa}.$$

Thus $g(\eta)d\eta$ is the probability, in this idealized Thomson history, that a photon last scattered in $d\eta$. Its integral is $1-e^{-\kappa(0)}$, approximately unity when the early plasma is opaque; reionization adds a later visibility feature rather than a second independent probability.

For the polarization brightness combination

$$\mathcal P=\frac14 \left(F_{\gamma2}+G_{\gamma0}+G_{\gamma2}\right),$$

the scalar temperature equation can be written

$$\Theta'+ik\mu\Theta =\Psi'-ik\mu\Phi +\dot\kappa\left[-\Theta+\Theta_0-i\mu v_b -\frac12P_2(\mu)\mathcal P\right].$$

The collision normalization follows from the Thomson intensity and polarization operators in Ma and Bertschinger 1995, §5.5, printed pp. 21–22, eqs. (59)–(64) (Open PDF).

Integrating the transport equation

Set $X=\Theta+\Phi$. Multiplying its first-order equation by the integrating factor $e^{ik\mu\eta-\kappa}$ and taking the opaque early boundary gives

$$\begin{aligned} X(\eta_0,\mu,k)=\int_0^{\eta_0}d\eta\, e^{-\kappa}e^{-ik\mu(\eta_0-\eta)} \bigg\{&\Phi'+\Psi'\\ &+\dot\kappa\left[\Theta_0+\Phi-i\mu v_b -\frac12P_2(\mu)\mathcal P\right]\bigg\}. \end{aligned}$$

This is already the answer in direction space. It also shows why the ordinary and integrated Sachs–Wolfe terms are different: $\Theta_0+\Phi$ is sampled where $g$ is nonzero, whereas $\Phi'+\Psi'$ accumulates wherever the photon is free streaming. Moving the latter into the visibility term would double count potential evolution near last scattering.

Expand

$$\Theta(\mu)=\sum_{\ell=0}^{\infty} (-i)^\ell(2\ell+1)\Theta_\ell P_\ell(\mu), \qquad D(\eta)=\eta_0-\eta.$$

The plane-wave identity and derivatives of $j_\ell(kD)$ project the four local angular structures separately. For multipoles $\ell\ge2$, with the observer monopole and kinematic dipole removed,

$$\begin{aligned} \Theta_\ell(k,\eta_0)=\int_0^{\eta_0}d\eta\,\bigg[& g(\Theta_0+\Phi)j_\ell(kD) +g v_b j_\ell'(kD)\\ &+e^{-\kappa}(\Phi'+\Psi')j_\ell(kD)\\ &+\frac{g\mathcal P}{4} \left\{j_\ell(kD)+3j_\ell''(kD)\right\}\bigg]. \end{aligned}$$

Primes on the potentials denote $d/d\eta$; primes on $j_\ell$ denote derivatives with respect to its dimensionless argument. The four rows are, respectively, effective temperature at scattering, Doppler projection, integrated Sachs–Wolfe evolution, and the temperature projection of the polarization quadrupole. Hu and Dodelson give the instantaneous version and its integral generalization in Hu and Dodelson 2002, §3.8, preprint printed pp. 22–25, eqs. (21)–(23) (Open PDF); their two metric potentials must be translated together to the site convention used above.

The projection is broad rather than a rule $\ell=kD$. For large $\ell$, $j_\ell(kD)$ is concentrated near $kD\sim\ell$, but it has finite width and oscillatory tails. The Doppler kernel $j_\ell'$ has a different shape, while a finite visibility width averages sources evaluated at different acoustic phases. Consequently a feature in wavenumber space is never copied point by point onto the sky, and an angular peak cannot be assigned to one local source without the full projection.

The instantaneous Sachs–Wolfe check

Let $g(\eta)\to\delta(\eta-\eta_*)$, neglect velocity, polarization, and late potential evolution, and take a matter-dominated adiabatic superhorizon mode with $\Phi=\Psi$ constant. The Einstein constraint gives $\delta_m=-2\Phi$, adiabaticity gives $\delta_\gamma=4\delta_m/3$, and hence

$$\Theta_0+\Phi=\frac14\delta_\gamma+\Phi=\frac13\Phi.$$

The line-of-sight solution therefore reduces to

$$\Theta_\ell(k,\eta_0)=\frac13\Phi(k,\eta_*) j_\ell[k(\eta_0-\eta_*)],$$

the ordinary Sachs–Wolfe limit. Ma and Bertschinger obtain the same transfer and its angular spectrum in Ma and Bertschinger 1995, §6, printed pp. 26–28, eqs. (82)–(91) (Open PDF). This check fails deliberately when radiation driving, anisotropic stress, finite visibility width, reionization, or evolving late-time potentials matter.

The signed relation to primordial curvature is also useful. In this matter-era limit, the curvature constraint gives $\Phi=-3\mathcal R/5$, hence $T_\ell^T=-j_\ell[k(\eta_0-\eta_*)]/5$ when transfer is normalized to $\mathcal R$. The minus sign matters for cross-correlations even though it disappears from the temperature auto-spectrum.

From transfer functions to angular power

Normalize the primordial curvature perturbation by

$$\langle\mathcal R(\mathbf k)\mathcal R^*(\mathbf k')\rangle =(2\pi)^3\delta^3(\mathbf k-\mathbf k')P_{\mathcal R}(k), \qquad \mathcal P_{\mathcal R}(k)=\frac{k^3}{2\pi^2}P_{\mathcal R}(k).$$

If $\Theta_\ell=T_\ell^T\mathcal R$, statistical isotropy gives

$$C_\ell^{TT}=4\pi\int d\ln k\, \mathcal P_{\mathcal R}(k)|T_\ell^T(k)|^2.$$

This ensemble prediction is not a frequency map, a beam-convolved spectrum, or a likelihood. The monopole is unobservable as an anisotropy and the measured dipole is dominated by observer motion. Curvature replaces the flat $j_\ell$ kernels by the appropriate hyperspherical functions; spectral distortions, nonlinear secondaries, and frequency-dependent foregrounds require a larger transport and measurement model. Tong develops the large-angle redshift and power-spectrum logic in Tong 2019, §§3.4.1–3.4.2, printed pp. 171–175 (PDF).

Exercises

Starting from the direction-space solution, derive the Doppler kernel and the polarization kernel in the multipole equation.

Solution

Write $x=kD$. Because $\partial_xe^{-ix\mu}=-i\mu e^{-ix\mu}$, the source $-i\mu v_b$ projects to $v_bj_\ell'(x)$. Also

$$P_2(\mu)e^{-ix\mu} =-\frac12\left(3\partial_x^2+1\right)e^{-ix\mu}.$$

Multiplying this by the collision coefficient $-\mathcal P/2$ gives $\mathcal P(3\partial_x^2+1)/4$. It therefore projects to $\mathcal P[j_\ell(x)+3j_\ell''(x)]/4$, with the common visibility factor $g$. The two kernels differ because a velocity is a dipolar source whereas Thomson polarization feeds the temperature equation through a quadrupole.

References

• Hu, W., and S. Dodelson, “Cosmic Microwave Background Anisotropies,” Annual Review of Astronomy and Astrophysics 40, 171–216 (2002), DOI, Open PDF.
• Ma, C.-P., and E. Bertschinger, “Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges,” The Astrophysical Journal 455, 7–25 (1995), DOI, Open PDF.
• Tong, D., Cosmology, lecture notes (2019), Official PDF.