# r dt at distance traveled by light beam a is ct

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Unformatted text preview: r ,0 = H0 % + + ΩΛ ,0 % dt \$ & ( ( ' •  Given H0, Ωm,0, Ωr,0 & ΩΛ , this can be integrated backwards w.r.t. 8me to ﬁnd t(a). 5/23/13 UW, ASTR 323, SP13, Eric Agol ©2013 14 Expansion rate •  In a ﬂat Universe where one component dominates, da 1/2 − (1+3w )/2 dt = H0Ω0 a •  Integra8ng this (for w≠- 1) gives: a da!a!(1+3w )/2 = ∫ 0 2a3( 3( )/2 ) = H0Ω1/2t 0 •  For radia8on, w=1/3, so a∝ •  For cold maZer, w=0, so a∝ 5/23/13 UW, ASTR 323, SP13, Eric Agol ©2013 15 Expansion History 102 ΩΛ=0.685 101 a∝eHt 100 Ωm,0=0.315 a(t) 10-1 a(t)= 10 (1+z)- 1 10 a∝t2/3 -2 -3 a∝t1/2 10-4 Ωr,0=4.6e-5 10-5 10-8 10-6 10-4 10-2 100 102 H 0t 5/23/13 UW, ASTR 323, SP13, Eric Agol ©2013 16 Time versus redshiS •  The lookback 8me is 8me that has passed since light was emiZed at a par8cular redshiS •  Compute from the metric: z = 1 / a − 1,\$so dz / dt = −H / a = −H(1 + z),\$so dz dt = − H(1 + z) •  Thus the lookback 8me is: t0 ∫ dt = te 5/23/13 z ∫ 0 dz" 1 ≈ H(z")(1 + z&...
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## This document was uploaded on 04/05/2014.

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