CMB spectroscopy at third-order in cosmological perturbations

Early energy injection to the Cosmic Microwave Background (CMB) from dissipation of acoustic waves generates deviations from the blackbody spectrum not only at second-order but also at third-order in cosmological perturbations. We compute this new spectral distortion $\mathcal{\kappa }$ based on third-order cosmological perturbation theory and show that $\kappa$ arises as a result of mode coupling between spectral distortions and temperature perturbations. The ensemble average of $\kappa$ can be directly sourced by (integrated) primordial non-Gaussianity. In particular, we roughly estimate the signal as $\kappa =f_{\mathrm{NL}}^{\mathrm{loc}}\times \mathcal{O}({10}^{-18})$ for local type scale-independent non-Gaussianity. The signal is incredibly tiny; however, we argue that it carries a specific frequency dependence different from other types of CMB spectral distortions. Also, it should be noticed that $\kappa$ is sensitive to extremely squeezed shapes of primordial bispectra that cannot be constrained by the CMB anisotropies. Finally, we comment on other possible applications of our results.

Figure 1

Hierarchy of the scales. The dashed arrow corresponds to the Fourier momenta in the convolutions of ${\mathrm{\Theta }}_2^2$ or ${\mathrm{\Theta }}_{1g}^2$. The solid and dotted arrows are those of the $y$ and $\kappa$, respectively. For the left squeezed shape, the superhorizon $y$ is produced from ${\mathbf{k}}_{1(2)}$ modes in the earlier stage. Then, $y$ enters the horizon and produces $\kappa$ of $\mathbf{k}+{\mathbf{k}}^{\prime }$ modes. For the equilateral shape, $y$ and $\kappa$ are produced from ${\mathbf{k}}_1$, ${\mathbf{k}}_2$ and ${\mathbf{k}}^{\prime }$ modes simultaneously. In this case, our assumption behind Eq. (42) is no more available and we need to account thoroughly for the nonlinear evolution of $y$. In any case, we consider $|\mathbf{k}+{\mathbf{k}}^{\prime }|\to 0$ limit when we calculate the ensemble average of $\kappa$.

Figure 2

Transfer functions for ${\mathrm{\Theta }}_1-V_1$ (top left), ${\mathrm{\Theta }}_2$ (top right), $y_1$ (bottom left) and $y_2$ (bottom right). The horizontal axis is the redshift. In contrast to the temperature multipoles, $y$ multipoles do not oscillate in the earlier epoch because $y$ does not contribute to the velocity of photons because of the frequency dependence of $\mathcal{Y}$.

Figure 3

The Fourier space window functions for the spectral distortions in units of $f_{\mathrm{NL}}^{\mathrm{loc}}⟪y⟫k^3{P}_{\zeta }(k)/2{\pi }^3$ (solid line) and $k^3{P}_{\zeta }(k)/2{\pi }^3$ (dashed line).