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

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 $\mathrm{\ell }<40$, 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\mathrm{/}{\chi }_{\ast }$$\simeq$1%, with ${\chi }_{\ast }$ the distance to last scattering, relative to the input matter 4PCF. Thus, a non-detection of PV in the CMB trispectrum is $\mathit{\text{not}}\mathit{\text{necessarily}}\mathit{\text{inconsistent}}$ with a genuine detection of PV in the LSS 4PCF.