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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2013/iii/paper-48/3/a/solution

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Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 48 / 3 / a / Solution by

Codex  0 Created 2026-10-03 Updated 2026-10-07

Differentiate the linearized cosmological continuity equation, remembering the derivative of

1/ a

\ddot{δ}_{m} = \frac{H}{a} \nabla \cdot v_{m} - \frac{1}{a} \nabla \cdot \dot{v}_{m} .

(1)

Insert the linear cosmological Euler equation for the peculiar velocity, giving

\ddot{δ}_{m} = \frac{2 H}{a} \nabla \cdot v_{m} + \frac{1}{a ^{2}} \nabla^{2} Φ = - 2 H \dot{δ}_{m} + 4 π G \overset{ρ}{ˉ}_{m} δ_{m} .

(2)

Thus the linear matter perturbation growth equation is

\ddot{δ}_{m} + 2 H \dot{δ}_{m} - 4 π G \overset{ρ}{ˉ}_{m} δ_{m} = 0 .

(3)

The factor

2 H

includes one expansion term from continuity and one from momentum dilution. This equation uses sub-Hubble, pressureless linear perturbations and the stated matter-only source for the gravitational potential.

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