Matter growth equation as a function of scale factor

In a matter-plus-cosmological-constant universe, changing the independent variable from cosmic time to

a

gives

\frac{d ^{2} δ _{m}}{d a ^{2}} + (\frac{d l o g H}{d a} + \frac{3}{a}) \frac{d δ _{m}}{d a} - \frac{3 Ω _{m, 0} H _{0}^{2}}{2 a ^{5} H ^{2}} δ_{m} = 0.

Table of contents
- Integral linear growth factor in a matter-Lambda universe Matter growth equation as a function of scale factor
  - Galaxy-formation bound on the cosmological constant Integral linear growth factor in a matter-Lambda universe

Integral linear growth factor in a matter-Lambda universe ( $D_{+} (a)$ )

The growing solution of the matter perturbation equation in a spatially flat matter-plus-cosmological-constant universe is

D_{+} (a) = \frac{5}{2} Ω_{m, 0} H_{0}^{2} H (a) \int_{0}^{a} \frac{d a ^{'}}{a ^{'3} H ( a ^{'} ) ^{3}},

normalized so that

D_{+} (a) \sim a

during early matter domination.

Galaxy-formation bound on the cosmological constant

If a perturbation has amplitude

A ≪ 1

at decoupling, it needs a growth factor of at least

A^{- 1}

before cosmological-constant domination freezes its growth. The order-of-magnitude requirement is

ρ_{Λ} ≲ A^{3} ρ_{m} (t_{dec}),

up to the collapse threshold and the gradual transition encoded by the exact growth factor.

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Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 310 / 3 / c / Solution

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