Past exam of the astrophysics course of the University of Oxford / 2022 / 6 / b

Solution

Within the equal-expansion construction of a, subtract the flat-background Friedmann equation to obtain

δ ρ = \frac{3 K}{8 π G a ^{2}}, δ = \frac{3 K}{8 π G a ^{2} ρ _{b}} .

The background density scales as

ρ_{b} \propto a^{- 4}

ρ_{b} \propto a^{- 3}

during matter domination. Therefore

δ \propto a^{2} (a ≪ a_{eq}), δ \propto a (a ≫ a_{eq}) .

These are the regular adiabatic growing-mode scalings on scales outside the Hubble radius, with a specified time slicing. In a consistent comoving description the numerical relation between curvature and density contrast differs from the naive subtraction, but the powers agree. The result does not imply that every gauge's density contrast, every decaying mode, or subhorizon dark matter during radiation domination has these powers.

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