By Jamal Nazrul Islam

This booklet is a concise creation to the mathematical points of the beginning, constitution and evolution of the universe. The e-book starts off with a quick review of observational cosmology and basic relativity, and is going directly to talk about Friedmann versions, the Hubble consistent, types with a cosmological consistent, singularities, the early universe, inflation and quantum cosmology. This booklet is rounded off with a bankruptcy at the far-off way forward for the universe. The booklet 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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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 Adx. 83) perimeter In this form Stokes’s theorem can be used for curved (Riemannian) spaces. 84) ⍀ where ⍀ is a given four-dimensional region. 60) we see that I is a scalar (invariant) quantity. We will consider the variation of the quantity ␦I, when the are varied by an inﬁnitesimal amount: → ϩ ␦ , such that the variations vanish on the boundary Ѩ⍀ of ⍀.

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. 19). 19) describes the intrinsic properties of the surface and does not depend on its embedding in a higherdimensional space, although in this simple case it is natural to think in terms of the surface of an ordinary sphere in three dimensions.

119a,b) imply ⌫ i0j ϭ0, i,jϭ 1,2,3. 106) as follows: 00 (u0)2 ϩ ij uiu j ϭ1. 121) For particles moving slowly with respect to the speed of light, the second set of terms on the left hand side is small compared to the ﬁrst term (since the ui are of the order of /c, where is a typical velocity), so we get 00 (u0)2 ϭ1. ), i dui/dsϭϪ ⌫ 00 (u0)2 ϭ(1/2) ij 00,j (u0)2. 123a) To ﬁrst order we also have dui/dsϭ (Ѩui/Ѩx)(dx/ds)ϭ(Ѩui/Ѩx0)u0. 122) u0 ϭ( 00)Ϫ2) (Ѩui/Ѩx0)ϭ(1/2) u0 ϭ ij 00,j ij ( 1 2 ) .