The Damping Tail of CMB Anisotropies

TL;DR

The paper develops a physics-driven decomposition of the CMB damping tail into transfer-function components for diffusion damping, reionization damping, and gravitational driving, enabling model-independent reconstruction of the background cosmology from small-scale data. It combines Boltzmann formalism, analytic estimates, and numerical calibration to produce compact envelopes ${\\cal D}_\\ell$ and ${\\cal R}_\\ell$ that describe damping, plus a potential envelope ${\\cal P}_\\ell$ and baryon-drag signatures that reveal structure formation details. By removing the model-independent damping effects, the authors expose model-dependent signatures such as the baryon drag modulation of acoustic peaks and the evolution of metric potentials, which can distinguish between inflationary and defect scenarios and constrain curvature. The framework provides practical diagnostics for curvature, reionization epoch, and baryon content, offering a pathway to extract cosmological parameters from high-precision small-scale CMB measurements with future missions.

Abstract

By decomposing the damping tail of CMB anisotropies into a series of transfer functions representing individual physical effects, we provide ingredients that will aid in the reconstruction of the cosmological model from small-scale CMB anisotropy data. We accurately calibrate the model-independent effects of diffusion and reionization damping which provide potentially the most robust information on the background cosmology. Removing these effects, we uncover model-dependent processes such as the acoustic peak modulation and gravitational enhancement that can help distinguish between alternate models of structure formation and provide windows into the evolution of fluctuations at various stages in their growth.

Figures (10)

• Figure 1: Diffusion damping calibration. In the absence of both diffusion damping and gravitational sources, the rms temperature fluctuation at recombination (short-dashed line) exhibits simple acoustic oscillations. These are mapped onto anisotropies on the sky in a near one to one fashion (solid line). The inclusion of diffusion terms in the Boltzmann equation allows for a simple numerical calibration of its effects.
• Figure 2: Reionization damping calibration. By removing the relative Doppler effect from a reionized Boltzmann calculation and comparing the result to the same model (here standard CDM $\Omega_0=1$, $h=0.5$, $\Omega_b h^2=0.0125$) with no reionization, the effects of rescattering damping are isolated. The reionization damping envelope is fit by two parameters, the optical depth during reionization and the horizon scale at last scattering (see Eq. 24).
• Figure 3: Diffusion scale calibration. Analytic estimates of $k_D(\Omega_0 h^2, \Omega_b h^2)$ based on the tight-coupling approximation trace the results to reasonable accuracy and explains their general behavior (HSc, Eq. E4). The fitting function of Eq. (17) tracks the numerical calibration to better than the $1\%$ level.
• Figure 4: Baryon drag and its potential dependence. Baryon inertia in the fluid displaces the zero point of the temperature oscillations leading to alternating peak heights as a function of scale at last scattering. The magnitude of the displacement is $R_*\Psi(\eta_*)$, and by removing it the monotonic variation of heights due to the potential envelope is uncovered (upper panel). The fractional effect is of order $R_*\Psi(\eta_*,k)/\Psi(0,k)$ and can be adequately described by the matter transfer function $T(k)$ (lower panel). The model here is CDM with $\Omega_0=1$, $h=1$ and $\Omega_bh^2 = 0.025$.
• Figure 5: Uncovering Baryon Drag in a low baryon universe. Diffusion damping obscures the baryon drag signal especially in a low baryon universe (here $\Omega_b h^2=0.075$ in an otherwise standard CDM model). Employing the numerical calibration of the damping tail, we recover the alternations.