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 of 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.

activity in the universe at red-shifts of about 2–4 than there is now. This does indicate evolution of the universe and is consistent with the existence of the cosmic background radiation. Radio astronomy has provided a valuable additional approach to observational cosmology. One of the reasons for its importance is that numerous faint radio sources have been detected, many of which lie presumably at great distances, which have not been optically identiﬁed and probably cannot be so identiﬁed, at

aspects. In this section we consider some more recent developments which have both theoretical and observational aspects; the latter can be considered as extensions of observational cosmology discussed in the last chapter. For convenience we may repeat some earlier remarks. There are some uncertainties, as usual, both theoretically and observationally, but we will attempt to present a balanced picture and try to convey the ‘ﬂavour’ of the current research. We will rely largely on the reviews by

ϩconstantϭ1.09/TЈ2 (s)ϩconstant, (8.41) where TЈ is the temperature measured in units of 1010 K. Thus the temperature takes 0.0108 s to drop from TЈϭ102 (that is, Tϭ1012 K) to TЈϭ10 K (Tϭ1011 K) and another 1.079 s to drop to TЈӍ1, (Tϭ1010 K). These values are roughly consistent with the ‘ﬁrst frame’ time and temperature tϭ0.01 s, Tϭ1011 K, and ‘third frame’ tϭ1.1 s, Tϭ1010 K. (ii) 5.5ϫ109 KϾT Ͼ 109 K We have mc ϭ0.51 MeV, so that the rest mass of an electron–positron pair is about 1.02 MeV.

(nϩ →pϩeϪ) (n→p)ϭA Ύ ΄1 Ϫ (q ϩ Q) ΅ m2e 1/2 2 (qϩQ)2q2 dq[1ϩexp(q/kT)]Ϫ1 (n→p)ϭ (n→ϫ{1ϩexp[Ϫ(qϩQ)/kT]}Ϫ1, (8.58a) (p→n)ϭ (pϩeϪ ϩ ¯ →n)ϩ (pϩ ¯ →nϩeϩ)ϩ (pϩeϪ →nϩ ) (n→p)ϭA Ύ΄ 1Ϫ m2e (q ϩ Q) 2 ΅ 1/2 (qϩQ)2q2 dq[1ϩexp(Ϫq/kT)]Ϫ1 (n→p)ϭ (n→ϫ{1ϩexp[(qϩQ)/kT]}Ϫ1. (8.58b) Here T is the temperature of the electrons, photons and nucleons and T is the neutrino temperature; below about 1010 K, T and T are diﬀerent and are given by (8.19). The integration in (8.58a) and

and Bertschinger, 1989) in which a special phase transition is not needed, that is, V() can have a signiﬁcant barrier between the true and false vacuum phases. Steinhardt (1990) shows that this model accommodates initial conditions leading to ⍀ р0.5. In ‘extended inﬂation’ the defects of ‘old inﬂation’ are avoided if the eﬀective gravitational constant, G, varies with time during inﬂation. 9.6 More inﬂationary solutions Ellis and Madsen (1991) ﬁnd a number of exact cosmological solutions with a