In a spatially flat FRW spacetime,

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Astrophysics & Cosmology · Einstein's field equations

The Geometry of Gravity, Ch FRW cosmology as an exact solution

Problem

In a spatially flat FRW spacetime, ds2=−c2dt2+a(t)2[dχ2+χ2dΩ2]ds^2=-c^2dt^2+a(t)^2\left[d\chi^2+\chi^2d\Omega^2\right], a comoving observer at χ=0\chi=0 receives a radial light ray from a comoving emitter at χ=χem\chi=\chi_{\rm em}. ((a)) Starting from the radial null condition c dt/a(t)=dχc\,dt/a(t)=d\chi, show that two wave crests emitted a coordinate-time interval Δtem\Delta t_{\rm em} apart, both travelling the same χem\chi_{\rm em}, are received an interval Δt0\Delta t_0 apart with Δt0/Δtem=a(t0)/a(tem)\Delta t_0/\Delta t_{\rm em}=a(t_0)/a(t_{\rm em}), and hence 1+z=1/a(tem)1+z=1/a(t_{\rm em}) with the normalisation a(t0)=1a(t_0)=1. ((b)) A flat, dust-only model of this universe has a(t)=(t/t0)2/3a(t)=(t/t_0)^{2/3}, with present age t0=9.6t_0=9.6 Gyr. Find the redshift of light emitted at tem=1.2t_{\rm em}=1.2 Gyr. ((c)) That light is the hydrogen Balmer-alpha (Hα\alpha) line, of rest-frame wavelength λem=656.3\lambda_{\rm em}=656.3 nm. Find the wavelength at which it is observed today.

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