By J. N. Islam
This e-book offers a concise creation to the mathematical points of the foundation, constitution and evolution of the universe. The e-book starts off with a quick review of observational and theoretical cosmology, in addition to a quick advent of normal 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 away way forward for the universe. This new version encompasses a rigorous derivation of the Robertson-Walker metric. It additionally discusses the boundaries to the parameter area via numerous theoretical and observational constraints, and provides a brand new inflationary answer for a 6th measure strength. This ebook is appropriate as a textbook for complex 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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Additional resources for An Introduction to Mathematical Cosmology
2 Tensor densities Tensor densities are needed in some contexts, such as volume and surface integrals. The latter are used in formulating an action principle from which ﬁeld equations can be derived in a convenient manner. We shall use this principle to obtain the ﬁeld equations with a scalar (Higgs) ﬁeld in connection with inﬂationary cosmologies. Consider a transformation from coordinates x to xЈ. 52) 22 Introduction to general relativity where J is the Jacobian of the transformation given by Έ Έ ѨxЈ0 Ѩx0 ѨxЈ1 0 1 2 3 Ѩx0 Ѩ(xЈ ,xЈ ,xЈ ,xЈ ) Jϭ ϵ ѨxЈ2 Ѩ(x0,x1,x2,x3 ) Ѩx0 ѨxЈ3 Ѩx0 ѨxЈ0 Ѩx1 ѨxЈ1 Ѩx1 ѨxЈ2 Ѩx1 ѨxЈ3 Ѩx1 ѨxЈ0 Ѩx2 ѨxЈ1 Ѩx2 ѨxЈ2 Ѩx2 ѨxЈ3 Ѩx2 ѨxЈ0 Ѩx3 ѨxЈ1 Ѩx3 .
There is one and only one such geodesic passing through each regular (that is, a point which is not a singularity) space-time point. This assumption is satisﬁed to a high degree of accuracy in the actual universe. The deviation from the general motion postulated here is observed to be random and small. The concept of a singular point introduced here will be elucidated in the next chapter and in Chapter 7. We assume that the bundle of geodesics satisfying Weyl’s postulate possesses a set of space-like hypersurfaces orthogonal to them.
We will deal with these in detail later. We will now give a brief discussion of the manner in which the Robertson–Walker metric is derived more rigorously with the help of Killing vectors. A space is said to be homogeneous if there exists an inﬁnitesimal isometry of the metric which can carry any point into any other point in its neighbourhood. From the discussion of Killing vectors it follows that this implies the existence of Killing vectors of the metric which at any point can take all possible values.
An Introduction to Mathematical Cosmology by J. N. Islam