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# How do you determine the characteristic scale of the universe and Hubbles constant in late times?

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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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#### Solution Summary

The expansion of the universe by Friedmann's equation is analyzed.

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