By J. N. Islam

ISBN-10: 0511018495

ISBN-13: 9780511018497

This booklet presents a concise advent to the mathematical features of the starting place, constitution and evolution of the universe. The e-book starts with a quick assessment of observational and theoretical cosmology, in addition to a brief advent of common relativity. It then is going directly to talk about Friedmann types, the Hubble consistent and deceleration parameter, singularities, the early universe, inflation, quantum cosmology and the far-off way forward for the universe. This new version features a rigorous derivation of the Robertson-Walker metric. It additionally discusses the bounds to the parameter house via a variety of theoretical and observational constraints, and provides a brand new inflationary answer for a 6th measure capability. This publication is appropriate as a textbook for complicated undergraduates and starting graduate scholars. it is going to even be of curiosity to cosmologists, astrophysicists, utilized mathematicians and mathematical physicists.

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**Example text**

14) becomes ds2 ϭc2 dt2 ϪR2(t)[d2 ϩsin2(d2 ϩsin2 d2)]. 18) Some insight may be gained by embedding the spatial part of this metric in a four-dimensional Euclidean space. 18) can, in fact, be so embedded. Before proceeding to do this, we consider a simple example of embedding, namely, that of the space given by the twodimensional metric dЈ2 ϭa2(d2 ϩsin2 d2). 19) This, of course, is just the surface of a two-sphere and is represented by the equation x2 ϩy2 ϩz2 ϭa2 in ordinary three-dimensional Euclidean space.

14) becomes ds2 ϭc2 dt2 ϪR2(t)[d2 ϩsin2(d2 ϩsin2 d2)]. 18) Some insight may be gained by embedding the spatial part of this metric in a four-dimensional Euclidean space. 18) can, in fact, be so embedded. Before proceeding to do this, we consider a simple example of embedding, namely, that of the space given by the twodimensional metric dЈ2 ϭa2(d2 ϩsin2 d2). 19) This, of course, is just the surface of a two-sphere and is represented by the equation x2 ϩy2 ϩz2 ϭa2 in ordinary three-dimensional Euclidean space.

108) However, having set up this coordinate system for the worldlines, or the ﬂow of matter, we will simply assume, as mentioned, that the four-velocity vector u is a function of the position (or ‘event’) coordinates x. 106), which is the same as the ordinary derivative for a scalar, we get 0ϭ( u) ϭ( , u u ); ϭ u (u ;u ϩu u ;), where we have used the fact that the covariant derivative of and the symmetry of . 109) we get uu; ϭ0. 111) in electromagnetic theory, so we can deﬁne a scalar ﬁeld and the corresponding vector ﬁeld u which determine the density and ﬂow of matter.

### An introduction to mathematical cosmology by J. N. Islam

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