No-go Theorem for Cosmological Parity Violation

A no-go theorem for parity-violation in even $D$-dimensional spacetimes invariant under $ISO(d)$ and dilatations (as well as the implications for odd $D$) is derived. For the case of real massless scalar and gravitons (as well as any massless even integer spin-$s$ field) at $\mathcal{I}^+$, the reality of wavefunction coefficients in Fourier space to all orders in perturbation theory (any order in loops) coming from a local, unitary, IR- and UV-finite theory, which start from the initial \CRT-invariant Bunch-Davies state in the infinite past, is proven. From this it is inferred that a parity-odd correlator with any massless scalar fields and even integer spin-$s$ fields vanishes in the presence of any number of interactions of massless fields. The same is true for correlators with an even number of conformally-coupled and massless odd integer spin-$s$ external fields, which is used to derive the cosmological analogue of Furry's theorem. The fundamental implications of \CRT symmetry for theories with chemical potentials, such as Chern-Simons and Axion inflation, is also discussed. Given the recent interest in parity-violation coming from observational claims of parity-violation detection, these results provide clear constraints on parity-violating models of inflation and establish the measurement of any parity-odd correlator as an exceptionally sensitive probe of new physics beyond vanilla inflation.