For a spatially flat ( ) FRW universe with no separate term (a cosmological constant

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

For a spatially flat (K=0K=0) FRW universe with no separate Λ\Lambda term (a cosmological constant, if present, enters as a fluid with w=−1w=-1), take as given the two Friedmann equations H2=8πGρ3c2H^2=\dfrac{8\pi G\rho}{3c^2} and a¨a=−4πG3c2(ρ+3p)\dfrac{\ddot a}{a}=-\dfrac{4\pi G}{3c^2}(\rho+3p), and the fluid-equation result ρ∝a−3(1+w)\rho\propto a^{-3(1+w)} for a component of constant w≡p/ρw\equiv p/\rho. ((a)) For radiation (w=1/3w=1/3), write ρ=ρr,0a−4\rho=\rho_{r,0}a^{-4} (with ρr,0\rho_{r,0} a constant) in the first Friedmann equation, separate variables in aa and tt, and integrate with a(0)=0a(0)=0 to show a(t)=(32πGρr,03c2)1/4t1/2a(t)=\left(\dfrac{32\pi G\rho_{r,0}}{3c^2}\right)^{1/4}t^{1/2}. ((b)) For a cosmological-constant-only fluid (w=−1w=-1, constant ρΛ\rho_\Lambda), substitute into the same equation to show a(t)=a(t1) eHΛ(t−t1)a(t)=a(t_1)\,e^{H_\Lambda(t-t_1)} for any reference time t1t_1, with HΛ=8πGρΛ/3c2H_\Lambda=\sqrt{8\pi G\rho_\Lambda/3c^2} constant. ((c)) Substitute each solution into the acceleration equation and confirm that both solve it too, using ρ+3p=2ρ\rho+3p=2\rho for radiation and ρ+3p=−2ρ\rho+3p=-2\rho for the Λ\Lambda-fluid.

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