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probset2

# probset2 - COSMOLOGY Problem Set 2 April 7 2010 Problem 1...

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COSMOLOGY Problem Set 2 April 7, 2010 Problem 1 : Collapse of a Uniform Sphere This is really a problem in Newtonian mechanics. Consider a uniform (homoge- neous) sphere of mass M whose initial radius (at t = 0) is R 0 . The sphere is released from rest at t = 0, collapsing under its own gravity. a) In terms of M and R 0 , find the time, t coll , for the sphere to collapse to R = 0. [10] b) Suppose the same uniform sphere of mass M were released from R 0 = αR 0 . Find the ratio of the new collapse time t coll to t coll (from part a). [5] c) In the limit R R 0 , where R = R ( t ) is the radius of the collapsing sphere ( t > t 0 ), find how rapidly the collapse is accelerating; i.e., find the de celeration parameter q . (Recall that q 1 H 2 [ 1 R ( d 2 R dt 2 )].) [5] Problem 2 : Power-Law Expanding Universe There is an interesting class of cosmological models for which the 3-space curvature may be neglected ( k = 0) and the time-evolution of the scale factor is a power law ( a t α , where 0 < α < 1). For this class of models find (in terms of α ):

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probset2 - COSMOLOGY Problem Set 2 April 7 2010 Problem 1...

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