The success of precision cosmology depends not only on accurate observations, but also on the theoretical model -- which must be understood to at least the same level of precision. Subtle relativistic effects can lead to biased measurements if they are neglected.

One such effect gives a systematic shift in the distance-redshift relation away from its background value, due to the non-linear relativistic conservation of total photon flux. We also show directly how this shiftfollows from a fully relativistic analysis of the geodesic deviation equation.

We derive the expectation value of the shift using second-order perturbations about a concordance background, and show that

the distance to last scattering is increased by \%. Neglecting this shift leads to significant bias in the background cosmological parameters.

Correcting the CMB analysis for this shift removes the tension with local measurements of $H_0$, by raising the $H_0$ value inferred from the CMB by 5%. Other CMB parameters which depend on the distance are shifted

by ~1sigma when combined with local $H_0$ data.Give a non-specialist summary of your paper.