# An introduction to mathematical cosmology by J. N. Islam

By J. N. Islam

This booklet presents a concise creation to the mathematical elements of the starting place, constitution and evolution of the universe. The e-book starts with a quick assessment of observational and theoretical cosmology, in addition to a quick advent of basic relativity. It then is going directly to talk about Friedmann versions, the Hubble consistent and deceleration parameter, singularities, the early universe, inflation, quantum cosmology and the far-off way forward for the universe. This re-creation includes a rigorous derivation of the Robertson-Walker metric. It additionally discusses the bounds to the parameter area via quite a few theoretical and observational constraints, and provides a brand new inflationary answer for a 6th measure capability. This booklet is acceptable as a textbook for complicated undergraduates and starting graduate scholars. it is going to even be of curiosity to cosmologists, astrophysicists, utilized mathematicians and mathematical physicists.

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An equivalent description, known as Weyl’s postulate (Weyl, 1923) is to assume that the worldlines of galaxies are a bundle or congruence of geodesics in space-time diverging from a point in the (ﬁnite or inﬁnitely distant) past, or converging to such a point in the future, or both. These geodesics are non-intersecting, except possibly at a singular point in the past or future or both. There is one and only one such geodesic passing through each regular (that is, a point which is not a singularity) space-time point.

32), the ‘radius’ of this sphere, that is, the distance from the ‘centre’ given by ϭ0 to the surface given by ϭ 0 along ϭ constant and ϭconstant, is R0. Thus the surface area is larger than that of a sphere of radius R0 in Euclidean space. In this case the range of the coordinates (, , ) is: 0ഛ ഛϱ,0ഛ ഛ , 0ഛ Ͻ 2. The spatial volume is inﬁnite. 3 Some kinematic properties of the Robertson–Walker metric We have seen that galaxies have ﬁxed spatial coordinates, that is, they are at rest in the coordinate system deﬁned above.

The circumference of the circle on the sphere given by Јϭconstantϭ Ј0 is 2RЈ sinЈ0, while the distance of this circle from the north pole O is RЈЈ0. The circumference of this circle reaches a maximum at 0Ј ϭ /2, after which it decreases until it reaches zero at Ј0 ϭ , when the distance from the north pole along the surface is RЈ, analogously to the previous case. 24), the entire surface is swept by the coordinate range 0ഛ ഛ , 0 ഛ ഛ , 0 ഛ ഛ2. 20) is 44 The Robertson–Walker metric RЈsincЈ0 RЈ cЈ0 cЈϭp/2 Fig.