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General relativistic self-similar waves that induce an anomalous acceleration into the standard model of cosmology

About this Title

Joel Smoller, Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109 and Blake Temple, Department of Mathematics, University of California, Davis, Davis, California 95616

Publication: Memoirs of the American Mathematical Society
Publication Year: 2012; Volume 218, Number 1025
ISBNs: 978-0-8218-5358-0 (print); 978-0-8218-9012-7 (online)
DOI: https://doi.org/10.1090/S0065-9266-2011-00641-6
Published electronically: November 3, 2011
MSC: Primary 34A05, 76L05, 83F05, 85A40

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Table of Contents

Chapters

  • 1. Introduction
  • 2. Self-Similar Coordinates for the $k=0$ FRW Spacetime
  • 3. The Expanding Wave Equations
  • 4. Canonical Co-moving Coordinates and Comparison with the $k\neq 0$ FRW Spacetimes
  • 5. Leading Order Corrections to the Standard Model Induced by the Expanding Waves
  • 6. A Foliation of the Expanding Wave Spacetimes into Flat Spacelike Hypersurfaces with Modified Scale Factor $R(t)=t^{a}$.
  • 7. Expanding Wave Corrections to the Standard Model in Approximate Co-moving Coordinates
  • 8. Redshift vs Luminosity Relations and the Anomalous Acceleration
  • 9. Appendix: The Mirror Problem
  • 10. Concluding Remarks

Abstract

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 $k\neq 0$ 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 self-similar 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.

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References
  • H. Alnes, M. Amarzguioui, Ø. Grøn, An inhomogeneous alternative to dark energy?, arXiv:astro-ph/0512006v2 18 Apr 2006.
  • C.Copi, D. Huterer, D.J. Schwarz, G.D. Starkman, On the large-angle anomalies of the microwave sky, Mon. Not. R. Astron. Soc., (2005) pp. 1-26.
  • T. Clifton, P.G. Ferreira, K. Land, Living in a void: testing the Copernican principle with distant supernovae, Phys. Rev. Lett., 101, (2008), 131302 (arXiv:0807.1443v2 [astro-ph] 29 Sep 2008)
  • T. Clifton and P.G. Ferreira, Does dark energy really exist?, Sci. Am., 2009, April (2009), 48-55
  • Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1955. MR 0069338
  • Douglas M. Eardley, Self-similar spacetimes: geometry and dynamics, Comm. Math. Phys. 37 (1974), 287–309. MR 351368
  • I. M. H. Etherington, Republication of: “LX. On the definition of distance in general relativity”, Gen. Relativity Gravitation 39 (2007), no. 7, 1055–1067. Republished from Philos. Mag. (7) 15 (1933), 761–773. MR 2322617, DOI 10.1007/s10714-007-0447-x
  • James Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715. MR 194770, DOI 10.1002/cpa.3160180408
  • James Glimm and Peter D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoirs of the American Mathematical Society, No. 101, American Mathematical Society, Providence, R.I., 1970. MR 0265767
  • Jeff Groah and Blake Temple, Shock-wave solutions of the Einstein equations with perfect fluid sources: existence and consistency by a locally inertial Glimm scheme, Mem. Amer. Math. Soc. 172 (2004), no. 813, vi+84. MR 2097533, DOI 10.1090/memo/0813
  • Øyvind Grøn and Sigbjørn Hervik, Einstein’s general theory of relativity, Springer, New York, 2007. With modern applications in cosmology. MR 2351140
  • Philip Hartman, A lemma in the theory of structural stability of differential equations, Proc. Amer. Math. Soc. 11 (1960), 610–620. MR 121542, DOI 10.1090/S0002-9939-1960-0121542-7
  • P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
  • Jerzy Plebański and Andrzej Krasiński, An introduction to general relativity and cosmology, Cambridge University Press, Cambridge, 2006. MR 2263604
  • Joel Smoller and Blake Temple, Global solutions of the relativistic Euler equations, Comm. Math. Phys. 156 (1993), no. 1, 67–99. MR 1234105
  • Joel Smoller and Blake Temple, Cosmology, black holes and shock waves beyond the Hubble length, Methods Appl. Anal. 11 (2004), no. 1, 77–132. MR 2128353, DOI 10.4310/MAA.2004.v11.n1.a7
  • Joel Smoller and Blake Temple, Shock-wave cosmology inside a black hole, Proc. Natl. Acad. Sci. USA 100 (2003), no. 20, 11216–11218. MR 2007847, DOI 10.1073/pnas.1833875100
  • Blake Temple and Joel Smoller, Expanding wave solutions of the Einstein equations that induce an anomalous acceleration into the Standard Model of Cosmology, Proc. Natl. Acad. Sci. USA 106 (2009), no. 34, 14213–14218. MR 2539730, DOI 10.1073/pnas.0901627106
  • B. Temple and J. Smoller, Answers to Questions Posed by Reporters: Temple-Smoller GR expanding waves, August 19, 2009. $http://www.math.ucdavis.edu/~temple/$.
  • B. Temple, Numerical Refinement of a Finite Mass Shock-Wave Cosmology, AMS National Meeting, Special Session Numerical Relativity, New Orleans (2007): $http://www.math.ucdavis.edu/\sim temple/talks/NumericalShockWaveCosTalk.pdf$.
  • S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity, John Wiley & Sons, New York, 1972.