CMB Anisotropies: Total Angular Momentum Method

TL;DR

The paper presents a total angular momentum representation to unify the treatment of CMB temperature and polarization anisotropies across scalar, vector, and tensor perturbations in a flat geometry. It derives angular and radial normal modes, reformulates the Boltzmann transport and Einstein equations in a compact, separable form, and provides integral solutions that map source terms to observable sky patterns. A key result is that polarization parity yields robust, model-independent signatures: scalars produce mainly electric polarization with no magnetic component, vectors yield dominant magnetic polarization, and tensors yield comparable E- and B-mode power with distinct cross-correlations, enabling discrimination among perturbation types and seeded models. The framework illuminates how causal scaling seeds (e.g., defects) imprint characteristic large-angle polarization and cross-spectra, offering a powerful diagnostic for isocurvature versus adiabatic scenarios and a versatile method extendable to open geometries and non-standard seeds.

Abstract

A total angular momentum representation simplifies the radiation transport problem for temperature and polarization anisotropy in the CMB. Scattering terms couple only the quadrupole moments of the distributions and each moment corresponds directly to the observable angular pattern on the sky. We develop and employ these techniques to study the general properties of anisotropy generation from scalar, vector and tensor perturbations to the metric and the matter, both in the cosmological fluids and from any seed perturbations (e.g. defects) that may be present. The simpler, more transparent form and derivation of the Boltzmann equations brings out the geometric and model-independent aspects of temperature and polarization anisotropy formation. Large angle scalar polarization provides a robust means to distinguish between isocurvature and adiabatic models for structure formation in principle. Vector modes have the unique property that the CMB polarization is dominated by magnetic type parity at small angles (a factor of 6 in power compared with 0 for the scalars and 8/13 for the tensors) and hence potentially distinguishable independent of the model for the seed. The tensor modes produce a different sign from the scalars and vectors for the temperature-polarization correlations at large angles. We explore conditions under which one perturbation type may dominate over the others including a detailed treatment of the photon-baryon fluid before recombination.

Figures (10)

• Figure 1: Addition theorem and scattering geometry. The addition theorem for spin-$s$ harmonics Eqn. (\ref{['eqn:composition']}) is established by their relation to rotations Eqn. (\ref{['eqn:spinbasis']}) and by noting that a rotation from $(\theta',\phi')$ through the origin (pole) to $(\theta,\phi)$ is equivalent to a direct rotation by the Euler angles $(\alpha,\beta,\gamma)$. For the scattering problem of Eqn. (\ref{['eqn:RSR']}), these angles represent the rotation by $\alpha$ from the $\hat{k} = \hat{e}_3$ frame to the scattering frame, by the scattering angle $\beta$, and by $\gamma$ back into the $\hat{k}$ frame.
• Figure 2: Projection effects. A plane wave $\exp(i \vec{k} \cdot \vec{x})$ can be decomposed into $j_\ell(kr) Y_\ell^0$ and hence carries an "orbital" angular dependence. A plane wave source at distance $r$ thus contributes angular power to $\ell \approx kr$ at $\theta=\pi/2$ but also to larger angles $\ell \ll kr$ at $\theta=0$ which is encapsulated into the structure of $j_\ell$ (see Fig. \ref{['fig:radialtemp']}). If the source has an intrinsic angular dependence, the distribution of power is altered. For an aligned dipole $Y_1^0 \propto \cos\theta$ ('figure 8's) power at $\theta=\pi/2$ or $\ell \approx kr$ is suppressed. These arguments are generalized for other intrinsic angular dependences in the text.
• Figure 3: Radial spin-0 (temperature) modes. The angular power in a plane wave (left panel, top) is modified due to the intrinsic angular structure of the source as discussed in the text. The left panel corresponds to the power in scalar ($m=0$) monopole $G_0^0$, dipole $G_1^0$, and quadrupole $G_2^0$ sources (top to bottom); the right panel to that in vector ($m=1$) dipole $G_1^{\pm 1}$ and quadrupole $G_2^{\pm 1}$ sources and a tensor ($m=2$) quadrupole $G_2^{\pm 2}$ source (top to bottom). Note the differences in how sharply peaked the power is at $\ell \approx kr$ and how fast power falls as $\ell \ll kr$. The argument of the radial functions $kr=100$ here.
• Figure 4: Radial spin-2 (polarization) modes. Displayed is the angular power in a plane-wave spin-2 source. The top panel shows that vector ($m=1$, upper panel) sources are dominated by $B$-parity contributions, whereas tensor ($m=2$, lower panel) sources have comparable but less power in the $B$-parity. Note that the power is strongly peaked at $\ell = kr$ for the $B$-parity vectors and $E$-parity tensors. The argument of the radial functions $kr=100$ here.
• Figure 5: Spin-0 $\times$ Spin-2 (temperature $\times$ polarization) modes. Displayed is the cross angular power in plane wave spin-0 and spin-2 sources. The top panel shows that a scalar monopole ($m=0$) source correlates with a scalar spin-2 (polarization quadrupole) source whereas the tensor quadrupole ($m=2$) anticorrelates with a tensor spin-2 source. Vector dipole ($m=1$) sources oscillate in their correlation with vector spin-2 sources and contribute negligible once modes are superimposed. One must go to vector quadrupole sources (lower panel) for a strong correlation. The argument of the radial functions $kr=100$ here.