A spatially flat ( ) FRW universe contains radiation 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 radiation only, with

\Omega_{\rm r}=1

and present Hubble parameter

H_0=70

km s

^{-1}

Mpc

^{-1}

. ((a)) Using the perfect-fluid stress-energy tensor of Chapter 13,

T^{\mu\nu}=(\rho+p)u^\mu u^\nu/c^2+pg^{\mu\nu}

, with the radiation equation of state

p=\rho/3

, and the first Friedmann equation, find

H

at scale factor

a=0.01

, in km s

^{-1}

Mpc

^{-1}

. ((b)) Find the redshift

z

of a photon emitted at

a=0.01

and received today. ((c)) Chapter 15 derives the gravitational redshift of a 1.4 solar-mass neutron star of radius 12 km (

z=0.235

) as the ratio of

-u^\mu k_\mu

for static observers at two radii, built from the timelike Killing vector of the static Schwarzschild spacetime. State the one respect in which this cosmological redshift is built from the same construction, and the one respect in which the observers and Killing vectors involved differ.

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