Past exam of the mathematics course of the University of Cambridge / 2022 / 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

The cosmological perfect-fluid continuity equation is

\overset{ρ}{˙} + 3 H (ρ + p) = 0.

For constant equation-of-state parameter

w = p / ρ

, integration gives

ρ_{i} (a) = ρ_{i, 0} a^{- 3 (1 + w_{i})} = ρ_{i, 0} (1 + z)^{3 (1 + w_{i})},

where

a_{0} = 1

. Define the present cosmological density parameter

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

The Friedmann equation then gives

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

Spatial curvature may be included as an effective

w = - 1/3

component. Once the parameters are specified, the scale factor follows from the first-order equation

\overset{a}{˙} = a H (a)

, or equivalently

t (a) - t (a_{i}) = \int_{a_{i}}^{a} \frac{d a ^{'}}{a ^{'} H ( a ^{'} )} .

b

Solution

For a variable equation of state, continuity gives

ρ_{DE} (z) = ρ_{DE, 0} exp [3 \int_{0}^{z} \frac{1 + w ( z ^{'} )}{1 + z ^{'}}, d z^{'}] .

For the Chevallier-Polarski-Linder parametrization

w (z) = w_{0} + w_{a} \frac{z}{1 + z},

the integral is

(1 + w_{0} + w_{a}) lo g (1 + z) - w_{a} \frac{z}{1 + z} .

Consequently

X (Ω_{DE, 0}, z, w_{0}, w_{a}) = Ω_{DE, 0} (1 + z)^{3 (1 + w_{0} + w_{a})} e^{- 3 w_{a} z / (1 + z)}

and

H (z) = H_{0} [Ω_{m, 0} (1 + z)^{3} + X]^{1/2} .

c

Solution

For successive wavecrests from the same comoving source,

1 + z = a (t_{0}) / a (t_{s})

. Therefore

Δ z = \frac{a ( t _{0} + Δ t _{0} )}{a ( t _{s} + Δ t _{s} )} - \frac{a ( t _{0} )}{a ( t _{s} )} .

The same radial null path implies equality of the elapsed conformal time,

\frac{Δ t _{s}}{a ( t _{s} )} = \frac{Δ t _{0}}{a ( t _{0} )}, Δ t_{s} = \frac{Δ t _{0}}{1 + z _{1}} .

Expanding both scale factors to first order gives the redshift drift

Δ z = [(1 + z_{1}) H_{0} - H (z_{1})] Δ t_{0} = H_{0} Δ t_{0} [1 + z_{1} - E (z_{1})] .

d

Solution

Measurements at many source redshifts directly sample the function

H_{0} [1 + z - E (z)]

. In principle, sufficiently many precise and well-spaced measurements can fit

H_{0}, Ω_{m, 0}, Ω_{DE, 0}, w_{0}, w_{a}

simultaneously; a single redshift supplies only one combination and cannot. In a spatially flat two-component model,

Ω_{m, 0} + Ω_{DE, 0} = 1

, reducing the number of independent parameters by one. In practice the signal accumulated over ten years is extremely small and parameter degeneracies require broad redshift coverage and complementary data.

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