A One Parameter Family of Expanding Wave Solutions of the Einstein Equations That Induces an Anomalous Acceleration Into the Standard Model of Cosmology

Smoller, Joel
Temple, Blake
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We derive a new set of equations which describe a continuous one parameter family of expanding wave solutions of the Einstein equations such that the Friedmann universe associated with the pure radiation phase of the Standard Model of Cosmology, is embedded as a single point in this family. All of the spacetime metrics associated with this family satisfy the equation of state $p=\rho c^2/3$, correct for the pure radiation phase after inflation in the Standard Model of the Big Bang. By expanding solutions about the center to leading order in the Hubble length, the family reduces to a one-parameter family of expanding spacetimes that represent a perturbation of the Standard Model. We then derive a co-moving coordinate system in which the perturbed spacetimes can be compared with the Standard Model. In this coordinate system we calculate the correction to the Hubble constant, as well as the exact leading order quadratic correction to the redshift vs luminosity relation for an observer at the center of the expanding FRW spacetime. The leading order correction to the redshift vs luminosity relation entails an adjustable free parameter that introduces an anomalous acceleration. We conclude that any correction to the redshift vs luminosity relation observed after the radiation phase of the Big Bang can be accounted for, at the leading order quadratic level, by adjustment of this free parameter. Since exact non-interacting expanding waves represent possible time-asymptotic wave patterns for conservation laws, we propose to further investigate the possibility that these corrections to the Standard Model might account for the anomalous acceleration of the galaxies, without the introduction of the cosmological constant.
Comment: 16 pages
Astrophysics - Cosmology and Nongalactic Astrophysics, General Relativity and Quantum Cosmology, Mathematics - Analysis of PDEs