## An introduction to mathematical cosmology by Jamal Nazrul Islam

By Jamal Nazrul Islam

This e-book is a concise creation to the mathematical points of the starting place, constitution and evolution of the universe. The publication starts with a short evaluate of observational cosmology and basic relativity, and is going directly to speak 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 publication 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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If we put ␦Iϭ0, we will obtain Einstein’s vacuum ﬁeld equations: R ϭ 0. 86a) where R* ϭ (⌫ , Ϫ ⌫ ,), Qϭ (⌫ ⌫ Ϫ ⌫ ⌫ ). We ﬁrst remove the second derivatives of the these occur in the expression R*. 86b) ) ⌫ . 87) TLFeBOOK 29 Some special topics in general relativity We can use the divergence theorem to convert the ﬁrst two integrals into surface integrals over Ѩ⍀, and so they will not contribute to the variation ␦I since the ␦ vanish on Ѩ⍀. 7) one can show that ( ) , ϭ(Ϫ ⌫ Ϫ  ␣⌫ ϩ ␣ ⌫ ) .

4) one can readily show that ␦ ϭϪ ␣ ␦ ␣. 68) one can obtain the following relation: ␦ ϭ 21 ␣␦ ␣, so that ␦( ) ϭϪ( ␣  ) ␦ 1 ␣ Ϫ2 ␣. 22)). 26) can also be obtained by a variation principle, if it is not a null geodesic. 99) A which represents the ‘length’ (in the case of time-like curves this is the proper time) from the point A to the point B of the curve. Let each point of the curve with coordinate x be moved to x ϩdx. If dx is an element along the curve, we have ds2 ϭ dx dx.

This set of curves is referred to as the congruence of curves generated by the given vector ﬁeld. In general there is a unique member of this congruence passing through any given point. A particular member of the congruence is sometimes referred to as an orbit. Consider now the vector ﬁeld given by ( 0, 1, 2, 3)ϭ(1,0,0,0). 30) we see that the congruence of this vector ﬁeld is the set of curves given by (x0 ϭ ,x1 ϭconstant, x2 ϭconstant, x3 ϭconstant). 31) This vector ﬁeld is also referred to as the vector ﬁeld Ѩ/Ѩx0.