An Introduction to Mathematical Cosmology by J. N. Islam

By J. N. Islam

This e-book presents a concise advent to the mathematical features of the starting place, constitution and evolution of the universe. The publication starts with a short evaluate of observational and theoretical cosmology, besides a quick creation of basic relativity. It then is going directly to speak about Friedmann versions, the Hubble consistent and deceleration parameter, singularities, the early universe, inflation, quantum cosmology and the far away way forward for the universe. This re-creation includes a rigorous derivation of the Robertson-Walker metric. It additionally discusses the bounds to the parameter area via numerous theoretical and observational constraints, and provides a brand new inflationary resolution for a 6th measure capability. This booklet is appropriate as a textbook for complicated undergraduates and starting graduate scholars. it's going to even be of curiosity to cosmologists, astrophysicists, utilized mathematicians and mathematical physicists.

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The parameter t can be chosen to measure the proper time along a geodesic. Now introduce spatial coordinates (x1, x2, x3) which are constant along any geodesic. Thus, for each galaxy the coordinates (x1, x2, x3) are constant. 1) where the hij are functions of (t, x1, x2, x3) and as usual repeated indices are to be summed over (Latin indices take values 1, 2, 3). 1) incorporates the properties described above can be seen as follows. Let the worldline of a galaxy be given by x␮(␶), where ␶ is A simple derivation 39 the proper time along the galaxy.

Put ␮ ϭ1, ␯ ϭ2 to get A1;2 ϪA2;1 ϭA1,2 ϪA2,1. 81) 28 Introduction to general relativity where the integral at the end is over the perimeter ѨS of the area S. We will express this in an invariant manner. An element of surface dS␮␯ given by two infinitesimal contravariant vectors ␰ ␮ and ␨ ␮ is given by dS␮␯ ϭ ␰ ␮␨ ␯ Ϫ ␰ ␯␨ ␮. 82) For example, if ␰ ␮ ϭ(0,dx1,0,0), ␨ ␮ ϭ(0,0,dx2,0), then dS12 ϭdx1dx2, dS21 ϭϪdx1dx2, the other components being zero. 81) becomes 1 2 ΎΎ (A␮;␯ ϪA␯;␮)dS␮␯ ϭ surface Ύ A␮dx␮.

These properties are reflected in the tensor (this discussion is taken from Dirac 1975, 1996, p. 113) 34 Introduction to general relativity with T␮␯ giving the density and flux of energy and momentum. The symmetric tensor T␮␯ is the material energy–momentum tensor. 113) we get T␮␯;␯ ϭ(␳u␮u␯);␯ ϭ(␳u␯);␯u␮ ϩ ␳u␯u␮;␯ ϭ ␳u␯u␮;␯. 104), (u␮,␯u␯ ϩ ⌫␮␯␴u␯u␴)ϭ(u␮,␯ ϩ ⌫␮␯␴u␴)u␯ ϭu␮;␯u␯ ϭ0. 22). 23), being obtained from the latter by setting pϭ0. 116). This zero-pressure form of matter is usually referred to as ‘dust’, and arises in various situations including cosmological ones, as we shall see later.

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