CMB Anisotropies from a Gradient Mode

TL;DR

This paper demonstrates that a linear gradient mode does not produce observable effects on CMB anisotropies if Maldacena's consistency condition holds, and explores second-order effects including an induced quadrupole moment.

Contribution

The paper extends the Sachs-Wolfe formula to include gradient modes and analyzes their second-order effects on CMB anisotropies, confirming the non-observability of linear gradient modes.

Findings

Gradient modes do not cause hemispherical asymmetry in CMB power.

A gradient mode induces an observable quadrupole moment.

Superposition of quadratic perturbations can cancel the induced quadrupole.

Abstract

A linear gradient mode must have no observable dynamical effect on short distance physics. We confirm this by showing that if there was such a gradient mode extending across the whole observable Universe, it would not cause any hemispherical asymmetry in the power of CMB anisotropies, as long as Maldacena's consistency condition is satisfied. To study the effect of the long wavelength mode on short wavelength modes, we generalize the existing second order Sachs-Wolfe formula in the squeezed limit to include a gradient in the long mode and to account for the change in the location of the last scattering surface induced by this mode. Next, we consider effects that are of second order in the long mode. A gradient mode $\Phi = \boldsymbol q\cdot \boldsymbol x$ generated in Single-field inflation is shown to induce an observable quadrupole moment. For instance, in a matter-dominated model it is equal to $Q=5 (\boldsymbol q\cdot \boldsymbol x)^2 /18$. This quadrupole can be canceled by superposition of a quadratic perturbation. The result is shown to be a nonlinear extension of Weinberg's adiabatic modes: a long-wavelength physical mode which looks locally like a coordinate transformation.