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probset3

# probset3 - COSMOLOGY Problem Set 3 Problem 1 Matter...

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COSMOLOGY Problem Set 3 April 12, 2010 Problem 1 : Matter Dominated Evolution Consider a “Matter Dominated” (MD: p = 0, Λ = 0) Robertson-Walker-Friedman cosmological model. a) Find the density parameter as a function of the redshift, Ω( z ), and its present value Ω 0 . In the limit of high redshift (e.g., Ω 0 z > 1) find Ω( z ). [10] b) In terms of H 0 and Ω 0 , find the Hubble parameter as a function of redshift, H ( z ). In the limit of high redshift (e.g., Ω 0 z > 1) find H ( z ). [10] c) For Ω 0 < 1, find an analytic expression for the ratio of the present age to the Hubble age, H 0 t 0 , as a function of Ω 0 . How old is this Universe at present if Ω 0 = 0 . 3 (you may assume H 0 = 70 km/s/Mpc)? [10] d) How old would this Universe be at present if Ω 0 = 1? [5] Problem 2 : MD, High Density For a MD, RWF model with Ω 0 = 2 find: [Note: You may assume here that a 0 c/H 0 .] a) The present comoving radial coordinate of the horizon, Θ H ( t 0 ). What fraction of the entire Universe is observable (in principle) at t 0 ? [Note that this is a closed, finite universe.] [10] b) The comoving radial coordinate of the horizon at the time of maximum expansion, Θ H ( t max ) (where a max = a (

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