Can the CMB be Odd? Effect of a Parity-Violating Matter 4-Point Function on the Low-$\ell$ CMB Trispectrum

TL;DR

This paper analyzes whether parity-violating signals in the large-scale structure 4PCF can imprint on the primary CMB through the low-$\ell$ Sachs-Wolfe trispectrum. By linking the CMB trispectrum to the primordial matter trispectrum and then to the linear matter 4PCF, the authors derive a tractable analytic framework using an approximate matter transfer function and angular-spectral expansions. They find a robust geometric suppression of the odd-parity PV signal in the CMB trispectrum by a factor of order $R/\chi_* \simeq 1\%$ for $R$ around a few hundred Mpc/$h$ and $\chi_* \approx 2\times 10^4$ Mpc/$h$, implying that non-detection in the CMB does not necessarily contradict PV in LSS. The work further clarifies when odd-parity inputs can evade suppression (e.g., in even-parity inputs) and develops asymptotic arguments showing the odd CMB trispectrum is generally suppressed in the SW-dominated regime, with a reduced trispectrum formalism and asymptotic insights offered for broader applicability.

Abstract

Here we explore from a theoretical perspective the sensitivity of the primary CMB anisotropy trispectrum to parity violation (PV) in large-scale structure (LSS). We focus on the Sachs-Wolfe term, which dominates at $\ell < 40$, after which the Doppler term takes over. We consider a model where the PV is only present out to some maximal scale $R$ of order a few hundred Mpc/$h$, consistent with what recent LSS 4PCF measurements seem to indicate. We find that the odd CMB trispectrum must be suppressed by at least one factor of $R/χ_*$$\simeq$1\%, with $χ_*$ the distance to last scattering, relative to the input matter 4PCF. Thus, a non-detection of PV in the CMB trispectrum is \textit{not necessarily inconsistent} with a genuine detection of PV in the LSS 4PCF.