# Rt 8me m 0 d 2a 2 r 0 h0 3 2 0a 2 a dt 2a

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Unformatted text preview: quot;) H0 z ∫ 0 dz" (1 + z") Ωm ,0 (1 + z")3 + ΩΛ ,0 UW, ASTR 323, SP13, Eric Agol ©2013 17 Time Dila8on •  The 8me between events at redshiS z: Light beam Sent received A te t0 B te+Δte t0+Δt0 •  Distance traveled by light beam A is: c dr ! = r = ∫ dt ∫ a(t ) •  Distance traveled by light beam A is: cΔt cΔt c c dr ! = r = ∫ dt ≈ ∫ dt + − ∫ a(t ) a(t ) a(t ) a(t ) r0 0 r0 0 t0 0 te t0 +Δt0 0 0 t0 te +Δte te •  So Δt0=Δte( ) 5/23/13 0 UW, ASTR 323, SP13, Eric Agol ©2013 0 0 e e 18 Proper distance •  r(t)=a(t)r0 - instantaneous separa8on – _________ to measure! •  What we can measure is the trajectory of light for which dl =___. In a ﬂat Universe in radial d2 = (cdt )2 − a(t )2 dr02 = direc8on: cdt •  So dr0=- ________ ∫ dr! = − ∫ a(t) •  Trick: da/dt = H(t)a(t),...
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