# Fate of the Universe

1.We have heard of simple toy models of the Universe that were studied by Einstein, de Sitter, and Friedman. If our Universe was 100,000 times denser than it actually is, what fate would the toy-model predict, a Big Crunch, Eternal Expansion, or a Big Rip? Why?

2. In the question above, would the universe have a finite size or would it be indefinitely large?

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#### Solution Summary

Fate of the universe is discussed.

Expansion of the Universe by Friedmann's Equation

The expansion of the Universe is described by Friedmann's equation,

(R/R)^2 = 8piGp/3 - (kc^2)/(R^2) + lambda/3,

where R is the characteristic scale of the Universe, R is its derivative with respect to time, p is the mass density of the Universe, k is a constant which can have values of -1, -, + 1 and lambda is Einstein's Cosmological Constant. By expressing the left-hand side of Friedmann's equation in terms of Hubble's constant H = R/R obtain an expression for the critical mass density of the Universe p_c at which k = 0, in the case where lambda = 0 also. Using the value for Hubble's constant given below, obtain the value of p_c and express your answer in units of atomic mass units per cubic meter.

Rewrite Friedmann's equation in the form

1 = Ohm_M - (kc^2)/((H^2((R^2)) + Ohm_lambda

in the case where H>0 and obtain expressions for Ohm_M and Ohm_lambda involving p and lambda respectively. Obtain an expression for R in terms of k, Ohm_M and Ohm_lambda and hence five the conditions required to have k = -1, k = 0 or k == +1.

Show that, when k = +1, lambda = 0 and p is dominated by matter with zero pressure, the value of R at which the Universe stops expanding is

(R_0) (Ohm_1)/(Ohm_0 - 1)

where R_0 is the present-day value of R and Ohm_0 is the present day value of Ohm_M. What is the fate of the Universe if p > p_c and if p < p_c?

By considering the relative importance of the three terms on the right hand side of Friedmann's equation, determine the behavior of R(t) as t --> infinity when lambda is constant and greater than 0, and hence discuss the fate fo such a Universe. Calculate the value of Hubble's constant at late times in such a Universe, if the present value for Ohm_lambda is 0.7.

[Assume a present-day value for Hubble's constant of H_0 = 72kms^-1 Mpc^-1.]

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