Multiplescales approach to the averaging problem in cosmology
Abstract
The Universe is homogeneous and isotropic on large scales, so on those scales it is usually modelled as a FriedmannLemaítreRobertsonWalker (FLRW) spacetime. The nonlinearity of the Einstein field equations raises concern over averaging over smallscale deviations form homogeneity and isotropy, with possible implications on the applicability of the FLRW metric to the Universe, even on large scales. Here I present a technique, based on the multiplescales method of singular perturbation theory, to handle the smallscale inhomogeneities consistently. I obtain a leading order effective Einstein equation for the largescale spacetime metric, which contains a backreaction term. The derivation relies on a series of consistency conditions, that ensure that the growth of deviations from the largescale spacetime metric do not grow unboundedly; criteria for their satisfiability are discussed, and it is shown that they are indeed satisfied if matter is nonrelativistic on small scales. The analysis is performed in harmonic gauge, and conversion to other gauges is discussed. I estimate the magnitude of the backreaction term relative to the critical density of the Universe in the example of an NFW halo, and find it to be of the order of a few percent. In this example, the backreaction term is interpreted as a contribution of the energydensity of gravitational potential energy, averaged over the smallscale, to the total energymomentum tensor.
 Publication:

Journal of Cosmology and Astroparticle Physics
 Pub Date:
 February 2021
 DOI:
 10.1088/14757516/2021/02/049
 arXiv:
 arXiv:2005.03026
 Bibcode:
 2021JCAP...02..049G
 Keywords:

 General Relativity and Quantum Cosmology;
 Astrophysics  Cosmology and Nongalactic Astrophysics
 EPrint:
 (V1) Submitted for publication. Comment welcome. (V4) Version accepted for publication in JCAP. Major update relative to earlier versions