By Jamal Nazrul Islam

This e-book is a concise creation to the mathematical features of the foundation, constitution and evolution of the universe. The ebook starts off with a quick evaluation of observational cosmology and normal relativity, and is going directly to talk about Friedmann types, the Hubble consistent, versions with a cosmological consistent, singularities, the early universe, inflation and quantum cosmology. This e-book is rounded off with a bankruptcy at the far away way forward for the universe. The publication is written as a textbook for complicated undergraduates and starting graduate scholars. it's going to even be of curiosity to cosmologists, astrophysicists, astronomers, utilized mathematicians and mathematical physicists.

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91) Although the integrand is not a scalar quantity, it is convenient for the present purpose, since it contains only ␮␯ and their first derivatives, being homogeneous of the second degree in the derivatives. In dynamical problems the action I is in fact the time integral of the Lagrangian L, so that the latter is given by Ύ L ϭ LdxЈdx2dx3, where L is the Lagrangian density, and the action I can be taken as a time integral of L, and a space-time integral of L, as follows: Ύ Ύ Iϭ Ldx0 ϭ Ld4x; Lϭ R.

33) TLFeBOOK 18 Introduction to general relativity As is well known, differential geometry and, correspondingly, general relativity can be developed independently of coordinates and components. We shall not be concerned with this approach except incidentally (see, for example, Hawking and Ellis, 1973). We will now consider some special topics in general relativity which may not all be used directly in the following chapters, but which may be useful in some contexts in cosmological studies. 22) by setting T␮␯ ϭ0) are a set of coupled non-linear partial differential equations for the ten unknown functions ␮␯.

24), the entire surface is swept by the coordinate range 0ഛ ␺ ഛ ␲, 0 ഛ ␪ ഛ ␲, 0 ഛ ␾ ഛ2␲. 20) is TLFeBOOK 44 The Robertson–Walker metric RЈsincЈ0 RЈ cЈ0 cЈϭp/2 Fig. 2. Diagram to illustrate the analogy between the surface of a twosphere and three-space of positive curvature. 26) which is finite. Here (3) is the determinant of the three-dimensional metric. 27) which is the ordinary three-dimensional Euclidean space. 28) gives d␴21 ϭdx2 ϩdy2 ϩdz2. 29) The range of (␺, ␪, ␾) is 0ഛ ␺ Ͻϱ, 0 ഛ ␪ ഛ ␲, 0 ഛ ␾ ഛ2␲, and the spatial volume is infinite.

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