An introduction to mathematical cosmology by Jamal Nazrul Islam

By Jamal Nazrul Islam

This booklet is a concise advent to the mathematical features of the beginning, constitution and evolution of the universe. The ebook starts with a short assessment of observational cosmology and common relativity, and is going directly to talk about Friedmann types, the Hubble consistent, types with a cosmological consistent, singularities, the early universe, inflation and quantum cosmology. This ebook is rounded off with a bankruptcy at the far-off way forward for the universe. The ebook is written as a textbook for complicated undergraduates and starting graduate scholars. it is going to even be of curiosity to cosmologists, astrophysicists, astronomers, utilized mathematicians and mathematical physicists.

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This is an intuitive statement of the Cosmological Principle which needs to be made more precise. For example, what does one mean by ‘a particular time’? In Newtonian physics this concept is unambiguous. In special relativity the concept becomes well-defined if one chooses a particular inertial frame. In general relativity, however, there are no global inertial frames. To define ‘a moment of time’ in general relativity which is valid globally, a particular set of circumstances are necessary, which, in fact, are satisfied by a homogeneous and isotropic universe.

In fact ⌫␮00 ϭ 12 ␮␯(2 ␯0,0 Ϫ 00,␯). 2) are indeed geodesics. 1) does not incorporate the property that space is homogeneous and isotropic. This form of the metric can be used, with the help of a special coordinate system obtained by singling out a particular typical galaxy, to derive some general properties of the universe without the assumptions of homogeneity and isotropy (see, for example, Raychaudhuri (1955)). 1) when space is homogeneous and isotropic. The spatial separation on the same hypersurface t ϭconstant of two nearby galaxies at coordinates (x1, x2, x3) and (x1 ϩ⌬x1, x2 ϩ⌬x2, x3 ϩ⌬x3) is d␴2 ϭhij ⌬xi⌬xj.

Consider the three-space given by 1 2 3 d␴Ј2 ϭ␥ijdx i dx j. 9) We assume this three-space to be homogeneous and isotropic. According to a theorem of differential geometry, this must be a space of constant curvature (see, for example, Eisenhart (1926) or Weinberg (1972)). In such a space the Riemann tensor can be constructed from the metric (and not its derivatives) and constant tensors only. 10) where k is a constant. 10) if the ␥ij are chosen to be given by the following metric (Weinberg 1972, Chapter 13): d␴Ј2 ϭ(1ϩ 14krЈ2)Ϫ2[(dx1)2 ϩ(dx2)2 ϩ(dx3)2], rЈ2 ϭ(x1)2 ϩ(x2)2 ϩ(x3)2.

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