General Relativistic Self-similar Waves that Induce an Anomalous Acceleration into the Standard Model of Cosmology

We prove that the Einstein equations for a spherically symmetric spacetime in Standard Schwarzschild Coordinates (SSC) close to form a system of three ordinary differential equations for a family of self-similar expansion waves, and the critical (k = 0) Friedmann universe associated with the pure radiation phase of the Standard Model of Cosmology is embedded as a single point in this family. Removing a scaling law and imposing regularity at the center, we prove that the family reduces to an implicitly defined one-parameter family of distinct spacetimes determined by the value of a new acceleration parameter a, such that a = 1 corresponds to the Standard Model. We prove that all of the self-similar spacetimes in the family are distinct from the non-critical k != 0 Friedmann spacetimes, thereby characterizing the critical k = 0 Friedmann universe as the unique spacetime lying at the intersection of these two one-parameter families. We then present a mathematically rigorous analysis of solutions near the singular point at the center, deriving the expansion of solutions up to fourth order in the fractional distance to the Hubble Length. Finally, we use these rigorous estimates to calculate the exact leading order quadratic and cubic corrections to the redshift vs luminosity relation for an observer at the center. It follows by continuity that corrections 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 the free parameter a. The third order correction is then a prediction. Since selfsimilar expanding waves represent possible time-asymptotic wave patterns for the conservation laws associated with the highly nonlinear radiation phase, we propose to further investigate the possibility that these corrections to the Standard Model might be the source of the anomalous acceleration of the galaxies, an explanation wholly within Einstein’s equations with classical sources, and not requiring Dark Energy or the cosmological constant.1 Received by the editor May 13, 2010 and, in revised form, December 10, 2010. Article electronically published on November 3, 2011; S 0065-9266(2011)00641-6. 2010 Mathematics Subject Classification. Primary 34A05, 76L05, 83F05, 85A40. The first author was supported by NSF Applied Mathematics Grant Number DMS-060-3754. The second author was supported by NSF Applied Mathematics Grant Number DMS-0707532. The second author originally proposed the idea that a secondary expansion wave reflected backwards from the cosmic shock wave constructed in [17] might account for the anomalous acceleration of the galaxies in the talk [20] and NSF proposal DMS-060-3754, cf. [18]. c ©2011 American Mathematical Society 1This paper fills in the proofs and extends the results quoted in the authors’ PNAS article [18].