 Solution

ID: past-exam-of-the-relativity-course-of-the-university-of-cambridge/2013/paper-1/4/solution

redshift. An incoming radial null ray satisfies

d χ = - c d t / R (t)

. Consecutive wave crests follow the same comoving path, so the endpoint changes in the fixed path integral obey

d t_{0} / R_{0} = d t_{e} / R_{e}

. Comoving clocks read cosmic time; hence their measured periods and frequencies satisfy

1 + z = \frac{λ _{0}}{λ _{e}} = \frac{d t _{0}}{d t _{e}} = \frac{R _{0}}{R _{e}} .

The physical wavelength stretches with the scale factor during propagation. It is not necessary to assign a unique global inertial-frame recession velocity to a distant source.

Radial coordinate and cosmological look-back time. Differentiating

1 + z = R_{0} / R (t)

gives

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

. Combining this with the null-ray integral yields

χ (z) = \frac{c}{R _{0}} \int_{0}^{z} \frac{d z ˉ}{H ( z ˉ )}, t_{0} - t_{e} = \int_{0}^{z} \frac{d z ˉ}{( 1 + z ˉ ) H ( z ˉ )} .

The extra factor

1 + z

in the second denominator records the conversion from scale-factor change to elapsed cosmic time.

proper-motion distance. A small transverse displacement at emission has proper length

d ℓ = R_{e} S (χ) d θ

. Define its source-frame transverse speed as

v = d ℓ / d t_{e}

d t_{0} = (1 + z) d t_{e}

. Therefore

\dot{θ}_{0} = \frac{v}{( 1 + z ) R _{e} S ( χ )}, d_{M} = \frac{v}{θ ˙ _{0}} = R_{0} S (χ (z)) .

Equivalently,

d_{M} = (1 + z) d_{A}

d_{A} = R_{e} S (χ)

. The observer-time versus source-time distinction is essential. This calculation uses the stated transverse speed convention and the homogeneous background, without additional peculiar radial relativistic Doppler effect or gravitational lensing.

redshift shell volume at the present epoch. The induced spatial metric tensor on the present slice gives

d V_{0} = R_{0}^{3} S^{2} (χ) d χ d Ω

. Since

d χ / d z = c / (R_{0} H)

d V_{0} = \frac{c R _{0}^{2} S ^{2} ( χ ( z ))}{H ( z )} d z d Ω = \frac{c d _{M}^{2} ( z )}{H ( z )} d z d Ω .

This is the volume at the present epoch of the comoving shell intersected by the past light cone. Its proper volume at emission is smaller by

(1 + z)^{3}

; a conserved population's proper number density is larger by the same factor, preserving its count.

Einstein-de Sitter cumulative number counts. In flat geometry

S (χ) = χ

. Inserting the stated Hubble parameter into the radial integral gives

d_{M} (z) = \frac{2 c}{H _{0}} [1 - (1 + z)^{- 1/2}] .

d N = n_{0} d V_{0}

. With no members of the population before its formation epoch,

N = \frac{4 c ^{3} n _{0}}{H _{0}^{3}} \int_{0}^{z_{f}} \frac{[ 1 - ( 1 + z ) ^{- 1/2} ] ^{2}}{( 1 + z ) ^{3/2}} d z .

As an independent check, substitution

u = (1 + z)^{- 1/2}

evaluates this to

N = \frac{8 c ^{3} n _{0}}{3 H _{0}^{3}} [1 - (1 + z_{f})^{- 1/2}]^{3} = \frac{n _{0} d _{M}^{3} ( z _{f} )}{3} .

This is exactly the Euclidean comoving volume per unit solid angle in the flat model. It assumes a complete count with the specified fixed population; luminosity selection, mergers or subsequent number-density evolution would require additional factors.

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