Cosmic Microwave Background Fluctuations from Gravitational Waves: An Analytic Approach

TL;DR

This paper develops an analytic framework to compute CMB temperature and polarization power spectra $C_l^{TT}$, $C_l^{EE}$, and $C_l^{BB}$ sourced by inflationary gravitational waves, complementing precise numerical results with physical insight into the tensor spectra. By combining exact tensor Boltzmann equations with practical approximations—most notably a Gaussian visibility function, projection-factor envelopes, and a WKB treatment of the radiation–matter transition—the authors derive scaling relations and explain the origin of peaks and phase-damping, linking spectrum features to three key scales: the horizon at recombination, the horizon at matter–radiation equality, and the last-scattering surface width. They show how the tensor amplitude and peak locations depend on the tensor power spectrum, recombination details, and neutrino anisotropic stress, highlighting that polarization features offer clean information about the early expansion history and the inflationary energy scale. The analytic approach provides intuitive, fast estimates that illuminate data interpretation and guide future analyses, while remaining consistent with full numerical results when all effects are included.

Abstract

We develop an analytic approach to calculation of the temperature and polarisation power spectra of the cosmic microwave background due to inflationary gravitational waves. This approach complements the more precise numerical results by providing insight into the physical origins of the features in the power spectra. We explore the use of analytic approximations for the gravitational-wave evolution, making use of the WKB approach to handle the radiation-matter transition. In the process, we describe scaling relations for the temperature and polarisation power spectra. We illustrate the dependence of the amplitude, shape, and peak locations on the details of recombination, the gravitational-wave power spectrum, and the cosmological parameters, and explain the origin of the peak locations in the temperature and polarisation power spectra. The decline in power on small scales in the polarisation power spectra is discussed in terms of phase-damping. In an appendix we detail numerical techniques for integrating the gravitational-wave evolution in the presence of anisotropic stress from free-streaming neutrinos.

Figures (15)

• Figure 1: Comparison of power generated by the two source terms for temperature anisotropy. Plotted are the total power (solid line), ISW term only (dotted line), and $g\Psi$ term only (dashed line). The $g\Psi$ term is essentially negligible at all $l$. The normalisation here, and in all plots, is specified by setting $A_T=1/8(4\pi)^2$ and $n_T=0$.
• Figure 2: Evolution of a gravitational-wave. Wavenumber $k$ satisfies $k\tau_{\rm{eq}}=10$. Shown solutions are numerical without anisotropic stress (solid curve), numerical with anisotropic stress (dotted curve), radiation (long dashed curve), matter (short dashed curve), and WKB (dot-dashed curve). The two vertical lines denote $\tau=1/k$ and $\tau_{\rm{eq}}$.
• Figure 3: Tensor power spectra. Curves from top to bottom are $C^{TT}_l$, $C^{EE}_l$, and $C^{BB}_l$. Vertical lines indicate important angular scales, from left to right: horizon at recombination, $\tau_R$, horizon at matter-radiation equality, $\tau_{eq}$, and the width of the last-scattering surface, $\Delta\tau_R$.
• Figure 4: T and B power spectra calculated using approximate forms for the gravitational-wave amplitude $h$. Plotted are the results using $h$ from the full numerical calculation (solid curve) and from the radiation-dominated (long dashed curve), matter-dominated (dot-short dashed curve), instantaneous-transition (dotted curve), and the WKB (dot-long dashed curve) approximations.
• Figure 5: Recombination history. Plotted are the visibility function $g(\tau)$ and the optical depth $\kappa$ calculated numerically (solid curves) and the approximations described in the text (dashed curves) using $\Delta\tau_R=15.7$.