Past exam of the mathematics course of the University of Cambridge / 2024 / 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

P = wρ

and constant

w

, the cosmological perfect-fluid continuity equation becomes

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

Integration gives the constant-equation-of-state density scaling

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

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 into the spatially flat Friedmann equation gives the Hubble parameter for constant-equation-of-state components

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

Spatial flatness is what allows the expression to contain only the listed density components, with

\sum_{i} Ω_{i, 0} = 1

.

b

Solution

For dynamical dark energy, use

d z / d t = - (1 + z) H

in the continuity equation to obtain

\frac{d l o g ρ _{DE}}{d z} = \frac{3 [ 1 + w ( z )]}{1 + z} .

Therefore the variable dark-energy equation of state gives

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

Combining this with

ρ_{m} = ρ_{m, 0} (1 + z)^{3}

in the Friedmann equation yields

H (z) = H_{0} {Ω_{m, 0} (1 + z)^{3} + Ω_{DE, 0} (1 + z)^{3} exp [\int_{0}^{z} \frac{3 w ( z ^{'} )}{X ( z ^{'} )} d z^{'}]}^{1/2},

where

X (z) = 1 + z .

This is the Hubble parameter for matter and dynamical dark energy.

c

Solution

The flux relation is

F = L / (4 π d_{L}^{2})

with luminosity distance

d_{L} = (1 + z) χ (z)

. In a spatially flat universe,

χ (z) = c \int_{0}^{z} \frac{d z ^{'}}{H ( z ^{'} )}, d_{L} (z) = (1 + z) c \int_{0}^{z} \frac{d z ^{'}}{H ( z ^{'} )} .

Thus Type Ia supernova cosmology measures the distance-redshift curve. Equation (2) makes

H (z)

depend on an integral of

w (z)

, while

d_{L}

introduces a second integral. Fitting predicted distances to many supernova fluxes over a range of redshifts therefore constrains parameters or bins describing

w (z)

, although these integrations smooth fine redshift structure.

The common luminosity

L

need not be known to constrain the shape of

w (z)

. An unknown

L

multiplies every inferred distance by the same factor and is degenerate with the overall scale

H_{0}^{- 1}

, or equivalently with the supernova absolute magnitude. Relative distances at different redshifts still determine the shape of the expansion history. An external calibration is needed to determine the absolute distance scale.

d

Solution

All galaxies in the population share the same formation time

t_{f}

, so the difference between their stellar ages equals the difference between their cosmic emission times. The redshift-time relation gives

\frac{d z}{d t} = - (1 + z) H (z) .

For a close pair with

∣Δ z ∣ ≪ 1

,

Δ t ≃ - \frac{Δ z}{( 1 + z ) H ( z )}, H (z) ≃ - \frac{1}{1 + z} \frac{Δ z}{Δ t} .

A cosmic chronometer measurement therefore reconstructs

H (z)

directly from differential galaxy ages. Inserting those values into the matter-plus-dark-energy Friedmann expression constrains

w (z)

.

Supernova distances integrate

1/ H (z)

, while

H (z)

already contains an integral of

w (z)

. Cosmic chronometers avoid the distance integral, so rapid oscillations in

w (z)

suffer one fewer smoothing operation and can leave a more visible signal.

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