A spatially flat ( ) FRW universe contains dust only

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Differential geometry of curved spacetime · Einstein's field equations

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

Problem

A spatially flat (

K=0

) FRW universe contains dust only, with

\Omega_{\rm m}=1

and present Hubble parameter

H_0 = 70

km s

^{-1}

Mpc

^{-1}

. Recall that the covariant conservation of the perfect-fluid stress-energy tensor,

\nabla_\mu T^{\mu\nu}=0

, gives

\rho\propto a^{-3}

for dust. ((a)) Find the scale factor

a_{\rm em}

at which the dust density was

8

times today's critical density. ((b)) Use the first Friedmann equation to find

H

at that scale factor, in km s

^{-1}

Mpc

^{-1}

. ((c)) A photon travels along the

x

-axis of comoving Cartesian coordinates from that time to today. Explain why

\partial_t

is not a Killing vector of this spacetime (unlike the static Schwarzschild spacetime of Chapter 15, whose timelike Killing vector gives a conserved photon energy for a static observer), identify which Killing vector instead supplies a conserved quantity for this photon, and use it to give the redshift

z

of the photon.

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