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

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Unformatted text preview: rse in radial direc8on: d2 = (cdt)2 − a(t)2 dr02 = cdt •  So dr0=- ________ ∫ dr! = − ∫ a(t) •  Trick: da/dt = H(t)a(t), so dt = da/(Ha) r0 0 r0 1/(1+z ) r0 = ∫ dr0! = − 0 5/28/13 ∫ 1 t 0 t0 cda c 1 da = ∫ Ha H0 1/(1+z ) Ω a−2 + Ω a−1 + Ω a2 r ,0 m ,0 Λ ,0 UW, ASTR 323, SP13, Eric Agol ©2013 7 Time Dila8on •  The 8me between events at redshig 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( ) 5/28/13 0 UW, ASTR 323, SP13, Eric Agol ©2013 0 0 e e 8 Luminosity & angular diameter distances # L &1 /2 dL = % ( !! \$ 4πF ' #R...
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