Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 1 / 15B / a

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- Solution a

Solution

Write

A = 8 π G / (3 c^{2})

and

H = \overset{a}{˙} / a

. Differentiating the Friedmann equation gives

2 H \dot{H} - 2 \frac{K c ^{2}}{a ^{2}} H = A \overset{ρ}{˙} .

The cosmological perfect-fluid continuity equation says

\overset{ρ}{˙} = - 3 H (ρ + P)

, so, first where

H \neq = 0

and then everywhere by continuity,

\dot{H} - \frac{K c ^{2}}{a ^{2}} = - \frac{4 π G}{c ^{2}} (ρ + P) .

Using

K c^{2} / a^{2} = A ρ - H^{2}

from the original equation gives

\dot{H} + H^{2} = - \frac{4 π G}{3 c ^{2}} (ρ + 3 P) .

Since

\dot{H} + H^{2} = \overset{a}{¨} / a

for the Hubble parameter, this is the Friedmann acceleration equation.

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