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Unformatted text preview: Physics 523, Introduction to Relativity Homework 8 Due Tuesday, 10 th January 2012 Hans Bantilan Angular Size in an FRW Spacetime Consider a matter-dominated, flat FRW spacetime so that the metric in comoving coordinates can be written as g ij dx i dx j =- dt 2 + a 2 ( t ) R 2 ( d 2 + S k 2 ( ) d 2 ) where a ( t ) is the scale factor, R is the characteristic length defined such that the curvature parameter in n dimensions 1 is = Ricci/ ( n ( n- 1)) = k/R 2 , and S k ( ) = sin( ) k = 1 (closed) k = 0 (flat) sinh( ) k =- 1 (open) . We are to determine the angular size ( z ) of an object of physical size L and at redshift z in this spacetime. Our point of entry is the relation between luminosity distance to the object d L ( z ) = (1 + z ) R S k R- 1 H- 1 Z z dz 1 E ( z ) which is inferred from E ( z ) = H ( z ) /H , where H ( z ) is the Hubble parameter 2 , and the angular distance to the object d A ( z ) = L ( z ) inferred from the proper size L and angular size ( z ) of the object. This relation is given by d L = (1 + z ) 2 d A . The flatness condition k = 0 determines the form of the S k function to be S k R- 1 H- 1 Z z dz 1 E ( z ) = R- 1 H- 1 Z z dz 1 E ( z ) ....
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This note was uploaded on 02/02/2012 for the course PHY 523 at Princeton.