What is the distance to the CMB?

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 shift follows 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 1%. We argue that neglecting this shift could lead to a significant bias in the background cosmological parameters, because it alters the meaning of the background model. A naive adjustment of CMB parameter estimation if this shift is really a correction to the background would raise the H0 value inferred from the CMB by 5%, potentially removing the tension with local measurements of H0. Other CMB parameters which depend on the distance would also be shifted by ~1sigma when combined with local H0 data. While our estimations rely on a simplistic analysis, they nevertheless illustrate that accurately defining the background model in terms of the expectation values of observables is critical when we aim to determine the model parameters at the sub-percent level.
Comment: 15 pages, 3 figures. v3 is the version to appear in JCAP