**Memoirs of the American Mathematical Society**

2012;
69 pp;
Softcover

MSC: Primary 34; 76; 83; 85;

Print ISBN: 978-0-8218-5358-0

Product Code: MEMO/218/1025

List Price: $58.00

AMS Member Price: $34.80

MAA Member Price: $52.20

**Electronic ISBN: 978-0-8218-9012-7
Product Code: MEMO/218/1025.E**

List Price: $58.00

AMS Member Price: $34.80

MAA Member Price: $52.20

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

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*Joel Smoller; John Blake Temple*

The authors 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, they 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.

The authors prove that all of the self-similar
spacetimes in the family are distinct from the non-critical \(k\neq0\)
Friedmann spacetimes, thereby characterizing the critical
\(k=0\) Friedmann universe as the unique spacetime lying at the
intersection of these two one-parameter families. They 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, they 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.

#### Table of Contents

# Table of Contents

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

- Chapter 1. Introduction 18 free
- Chapter 2. Self-Similar Coordinates for the k=0 FRW Spacetime 714 free
- Chapter 3. The Expanding Wave Equations 1118
- Chapter 4. Canonical Co-moving Coordinates and Comparison with the k=0 FRW Spacetimes 2128
- Chapter 5. Leading Order Corrections to the Standard Model Induced by the Expanding Waves 2532
- Chapter 6. A Foliation of the Expanding Wave Spacetimes into Flat Spacelike Hypersurfaces with Modified Scale Factor R(t)=ta. 4956
- Chapter 7. Expanding Wave Corrections to the Standard Model in Approximate Co-moving Coordinates 5158
- Chapter 8. Redshift vs Luminosity Relations and the Anomalous Acceleration 5562
- Chapter 9. Appendix: The Mirror Problem 6370
- Chapter 10. Concluding Remarks 6572
- Bibliography 6976