An Introduction to Mathematical Cosmology

by J. N. Islam

• ISBN13: 9780521499736
• ISBN10: 0521499739
• eBook ISBN(s): 9780511037931
• Edition: 2nd
• Format: Paperback
• Copyright: 2001-12-17
• Publisher: Cambridge University Press

Summary

This book provides a concise introduction to the mathematical aspects of the origin, structure and evolution of the universe. The book begins with a brief overview of observational and theoretical cosmology, along with a short introduction to general relativity. It then goes on to discuss Friedmann models, the Hubble constant and deceleration parameter, singularities, the early universe, inflation, quantum cosmology and the distant future of the universe. This new edition contains a rigorous derivation of the Robertson-Walker metric. It also discusses the limits to the parameter space through various theoretical and observational constraints, and presents a new inflationary solution for a sixth degree potential. This book is suitable as a textbook for advanced undergraduates and beginning graduate students. It will also be of interest to cosmologists, astrophysicists, applied mathematicians and mathematical physicists.

Table of Contents

Preface to the first edition	ix
Preface to the second edition	xi
Some basic concepts and an overview of cosmology	1	(11)
Introduction to general relativity	12	(25)
Summary of general relativity	12	(6)
Some special topics in general relativity	18	(19)
Killing vectors	18	(3)
Tensor densities	21	(3)
Gauss and Stokes theorems	24	(4)
The action principle for gravitation	28	(4)
Some further topics	32	(5)
The Robertson-Walker metric	37	(23)
A simple derivation of the Robertson-Walker metric	37	(5)
Some geometric properties of the Robertson-Walker metric	42	(3)
Some kinematic properties of the Robertson-Walker metric	45	(6)
The Einstein equations for the Robertson-Walker metric	51	(2)
Rigorous derivation of the Robertson-Walker metric	53	(7)
The Friedmann models	60	(16)
Introduction	60	(4)
Exact solution for zero pressure	64	(3)
Solution for pure radiation	67	(1)
Behaviour near t = 0	68	(1)
Exact solution connecting radiation and matter eras	68	(3)
The red-shift versus distance relation	71	(2)
Particle and event horizons	73	(3)
The Hubble constant and the deceleration parameter	76	(18)
Introduction	76	(1)
Measurement of H0	77	(3)
Measurement of q0	80	(5)
Further remarks about observational cosmology	85	(9)
Appendix to Chapter 5	90	(4)
Models with a cosmological constant	94	(18)
Introduction	94	(4)
Further remarks about the cosmological constant	98	(2)
Limits on the cosmological constant	100	(2)
Some recent developments regarding the cosmological constant and related matters	102	(10)
Introduction	102	(2)
An exact solution with cosmological constant	104	(3)
Restriction of parameter space	107	(5)
Singularities in cosmology	112	(16)
Introduction	112	(1)
Homogeneous cosmologies	113	(2)
Some results of general relativistic hydrodynamics	115	(3)
Definition of singularities	118	(2)
An example of a singularity theorem	120	(1)
An anisotropic model	121	(1)
The oscillatory approach to singularities	122	(4)
A singularity-free universe?	126	(2)
The early universe	128	(38)
Introduction	128	(7)
The very early universe	135	(7)
Equations in the early universe	142	(1)
Black-body radiation and the temperature of the early universe	143	(5)
Evolution of the mass-energy density	148	(5)
Nucleosynthesis in the early universe	153	(6)
Further remarks about helium and deuterium	159	(5)
Neutrino types and masses	164	(2)
The very early universe and inflation	166	(23)
Introduction	166	(1)
Inflationary models - qualitative discussion	167	(7)
Inflationary models - quantitative description	174	(4)
An exact inflationary solution	178	(2)
Further remarks on inflation	180	(3)
More inflationary solutions	183	(6)
Appendix to Chapter 9	186	(3)
Quantum cosmology	189	(22)
Introduction	189	(2)
Hamiltonian formalism	191	(4)
The Schrodinger functional equation for a scalar field	195	(2)
A functional differential equation	197	(2)
Solution for a scalar field	199	(1)
The free electromagnetic field	199	(2)
The Wheeler-De Witt equation	201	(1)
Path integrals	202	(4)
Conformal fluctuations	206	(3)
Further remarks about quantum cosmology	209	(2)
The distant future of the universe	211	(9)
Introduction	211	(1)
Three ways for a star to die	211	(2)
Galactic and supergalactic black holes	213	(2)
Black-hole evaporation	215	(1)
Slow and subtle changes	216	(2)
A collapsing universe	218	(2)
Appendix	220	(18)
Bibliography	238	(9)
Index	247