Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 310 / 1

Table of contents
- a 1
  - Solution a
- b 1
  - Solution b
- c 1
  - Solution c
- d 1
  - Solution d

a

Solution

For a component with constant barotropic equation of state

P = wρ

, the cosmological perfect-fluid continuity equation gives

\frac{d ρ}{ρ} = - 3 (1 + w) \frac{d a}{a} .

Hence

ρ_{i} = ρ_{i, 0} a^{- 3 (1 + w_{i})}

. Choose the present scale factor to be

a_{0} = 1

, so the cosmological redshift obeys

1 + z = a^{- 1}

, and define the present cosmological density parameter by

Ω_{i, 0} = \frac{ρ _{i, 0}}{ρ _{crit, 0}}, ρ_{crit, 0} = \frac{3 H _{0}^{2}}{8 π G} .

Substitution in the spatially flat Friedmann equation then yields

H (z) = H_{0} [i \sum Ω_{i, 0} (1 + z)^{3 (1 + w_{i})}]^{1/2} .

b

Solution

The Friedmann acceleration equation and

H = \overset{a}{˙} / a

give the deceleration parameter

q = - \frac{a ¨}{a H ^{2}} = \frac{4 π G}{3 H ^{2}} \sum_{i} ρ_{i} (1 + 3 w_{i}) = \frac{1}{2} i \sum Ω_{i} (z) (1 + 3 w_{i}) .

An Einstein-de Sitter universe contains only pressureless matter, so

q_{0} = 1/2

and cannot explain the observed negative value. For the stated Lambda-CDM model,

q_{0} = \frac{1}{2} [0.3 + 0.7 (1 - 3)] = - 0.55,

so its cosmological constant produces the required accelerating universe.

c

Solution

The age of an FLRW universe follows from

d t = d a / (a H)

. Neglecting radiation and spatial curvature gives

t_{0} = \frac{1}{H _{0}} \int_{0}^{1} \frac{d a}{a Ω _{m, 0} a ^{- 3} + Ω _{Λ, 0}} = \frac{1}{H _{0}} \int_{0}^{1} \frac{a ^{1/2} d a}{Ω _{m, 0} + Ω _{Λ, 0} a ^{3}} .

For an Einstein-de Sitter universe,

Ω_{m, 0} = 1

and

Ω_{Λ, 0} = 0

, so

t_{0} = H_{0}^{- 1} \int_{0}^{1} a^{1/2} d a = \frac{2}{3} H_{0}^{- 1} .

d

Solution

The measured Hubble time would give the Einstein-de Sitter age

t_{0} = \frac{2}{3} (14 Gyr) ≃ 9.3 Gyr,

which is less than the measured

12 Gyr

age of stars that must themselves be younger than the universe. The measurements are therefore incompatible with an Einstein-de Sitter universe. For the stated spatially flat Lambda-CDM model, direct evaluation of the preceding integral gives

H_{0} t_{0} = \frac{2}{3 0.7} arsinh \frac{0.7}{0.3} ≃ 0.964,

and hence

t_{0} ≃ 13.5 Gyr

. The period of accelerating expansion caused by the cosmological constant therefore allows an age consistent with the stellar lower bound.

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