ps2 - Phys/Astro 228 Extragalactic Astronomy and Cosmology...

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Unformatted text preview: Phys/Astro 228: Extragalactic Astronomy and Cosmology September 15 2006 C.–P. Ma Problem Set 2 Due in class Friday September 22 Readings: Three supernova papers: Riess et al (1998), Perlmutter et al (1999), and Riess et al (2004); see links at course website. 1. Hubble Diagram I: Distance vs. Redshift In class, we derived an analytic expression for the comoving distance r vs. redshift z for a matter-only ( w = 0) universe. (a) Generalize this derivation, starting with the Robertson-Walker metric. Show that for arbitrary Ω ,m and Ω , Λ , the relation is r = | k | − 1 / 2 sinn c | k | 1 / 2 H integraldisplay z dz ′ radicalBig Ω ,m (1 + z ′ ) 3 + Ω , Λ + (1- Ω ,m- Ω , Λ )(1 + z ′ ) 2 , (1) where sinn = sin for k > 0, sinn = sinh for k < 0, and sinn is absent for k = 0. (b) For small redshift, expand the expression above and show that for all three cases of k , r = c H bracketleftbigg z- 1 2 (1 + q ) z 2 bracketrightbigg + O ( z 3 ) , (2) where q is the deceleration parameter. As the next problem will show, this is the key equation usedis the deceleration parameter....
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