A spatially flat ( ) universe contains only dust ( ) and 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

A spatially flat (K=0K=0) universe contains only dust (w=0w=0) and a cosmological constant, with Ωm+ΩΛ=1\Omega_{\rm m} + \Omega_\Lambda = 1 today. ((a)) Using the acceleration equation a¨/a=−(4πG/3c2)(ρ+3p)\ddot a/a = -(4\pi G/3c^2)(\rho+3p) applied to the sum of the two components, derive an expression for the scale factor aacca_{\rm acc} at which the expansion switches from decelerating to accelerating, in terms of Ωm\Omega_{\rm m} and ΩΛ\Omega_\Lambda. ((b)) Evaluate aacca_{\rm acc} and the corresponding redshift zacc=1/aacc−1z_{\rm acc} = 1/a_{\rm acc}-1 for (i) the measured values Ωm=0.311\Omega_{\rm m}=0.311, ΩΛ=0.689\Omega_\Lambda=0.689, and (ii) a hypothetical mixture Ωm=0.5\Omega_{\rm m}=0.5, ΩΛ=0.5\Omega_\Lambda=0.5. State which mixture makes the transition to acceleration happen more recently.

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