Cosmological ensemble and directional averages of observables

Understand

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.