Gravitational Time Delay Effects on CMB Anisotropies

TL;DR

This paper quantifies gravitational time delay as a second-order effect on CMB temperature and polarization. By developing a perturbative formalism that treats time delay and lensing as correlated radial and angular modulations of the last-scattering surface, the authors compute the resulting changes to power spectra and bispectra. They find that time delay alters the temperature and polarization spectra at about the 10^{-4}–10^{-3} level, with the most noticeable impact on the temperature–polarization cross-spectrum near $\ell\sim1000$, where the effect is comparable to cosmic variance. The delay-induced bispectrum is several orders of magnitude smaller and effectively undetectable, indicating that time delay is a small but non-negligible systematic for future high-precision CMB experiments.

Abstract

We study the effect of gravitational time delay on the power spectra and bispectra of the cosmic microwave background (CMB) temperature and polarization anisotropies. The time delay effect modulates the spatial surface at recombination on which temperature anisotropies are observed, typically by ~1 Mpc. While this is a relatively large shift, its observable effects in the temperature and polarization fields are suppressed by geometric considerations. The leading order effect is from its correlation with the closely related gravitational lensing effect. The change to the temperature-polarization cross power spectrum is of order 0.1% and is hence comparable to the cosmic variance for the power in the multipoles around l~1000. While unlikely to be extracted from the data in its own right, its omission in modeling would produce a systematic error comparable to this limiting statistical error and, in principle, is relevant for future high precision experiments. Contributions to the bispectra result mainly from correlations with the Sachs-Wolfe effect and may safely be neglected in a low density universe.

Figures (8)

• Figure 1: Gravitational Lensing vs. Time Delay. Lensing introduces an angular perturbation in the mapping of a plane-wave source field at recombination onto anisotropies today. Time delay introduces a radial modulation. When the wavevector is perpendicular to the line-of-sight, features in the angular spectrum -- such as the acoustic peaks -- are created, geometrically distinguishing the otherwise similar lensing and delay effects.
• Figure 2: Power spectra for the lensing deflection angles (${\phi}{\phi}$), time-delay (${d}{d}$) and deflection-delay cross correlation (${\phi}{d}$). The underlying lensing potential spectrum $C_\ell^{{\phi}{\phi}}$ and cross spectrum $C_\ell^{{\phi}{d}}$ are weighted by $\ell(\ell+1)$ and $[\ell(\ell+1)]^{1/2}$ respectively to reflect the angular gradients in the deflection angles.
• Figure 3: Projection functions for the lensing and delay effects. Lensing involves $j_\ell(kr)$ whose strong peak in $\ell$ can retain source features; the $j_\ell'(kr)$ of the time-delay cannot. The product of these functions reflects the cross correlation and is suppressed due to the phase difference.
• Figure 4: Power spectra for the angular ($\ell(\ell+1) T_\ell^{00}$), radial ($T_\ell^{11}$) and angular-radial cross gradients $(\sqrt{\ell(\ell+1)} T_\ell^{01})$ of the primary anisotropies. The cross gradient spectrum has been weighted to reflect the $\ell$ contributions from the angular gradient.
• Figure 5: Delay, lensing and delay-lensing (cross) perturbations to the CMB temperature power spectrum for the fiducial $\Lambda$CDM model. The cross spectrum dominates the delay spectrum but both produce negligible changes to the primary anisotropies unlike lensing.