Cosmological ensemble and directional averages of observables

We show that at second order, ensemble averages of observables and directional averages do not commute due to gravitational lensing - observing the same thing in many directions over the sky is not the same as taking an ensemble average. In principle this non-commutativity is significant for a variety of quantities that we often use as observables and can lead to a bias in parameter estimation. We derive the relation between the ensemble average and the directional average of an observable, at second order in perturbation theory. We discuss the relevance of these two types of averages for making predictions of cosmological observables, focusing on observables related to distances and magnitudes. In particular, we show that the ensemble average of the distance in a given observed direction is increased by gravitational lensing, whereas the directional average of the distance is decreased. For a generic observable, there exists a particular function of the observable that is not affected by second-order lensing perturbations. We also show that standard areas have an advantage over standard rulers, and we discuss the subtleties involved in averaging in the case of supernova observations.