The consequences of parity symmetry for higher-order statistics of cosmic shear and other polar fields

We consider the parity transformation and the consequences of parity invariance on the n-point correlation function of the cosmic shear caused by gravitational lensing by the large-scale structure, or any other polar ( or spin-2) field. The decomposition of the shear field into E- and B-modes then yields the result that any correlation function which contains an odd number of B-mode shear components vanishes for parity-invariant random shear fields. In particular, this result implies that the expectation value of the third-order cross aperture statistics, [M-x(3)], vanishes for parity-invariant shear fields. Therefore, a significant detection of a non-zero value of [M-x(3)] in a cosmic shear survey does not indicate the presence of B-modes, but of an underestimate of the statistical uncertainty or cosmic variance, or a remaining systematic effect in the shear measurement. We argue that the parity invariance provides a very specific diagnostic of systematic effects in shear data. Our results apply as well to the linear polarization of the cosmic microwave background.