Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 49 / 4 / i

Solution

Take regular adiabatic initial conditions and a common scale-invariant primordial amplitude,

k^{3/2} Φ_{prim} (k) = constant

. Let

k_{eq} = H_{eq}

. The three histories are distinguished by their time of cosmological horizon crossing:

For $k ≪ k_{eq}$ , the mode remains outside the Hubble radius throughout radiation domination. Its potential remains nearly constant through equality, changing to $9/10$ of the primordial radiation value for a purely adiabatic transition. It stays constant after later horizon entry in matter domination.
For $k ≫ k_{eq}$ , horizon entry occurs during radiation domination. Radiation pressure produces oscillations and a rapidly decreasing potential. The small cold dark matter component eventually supports a much smaller constant matter-era potential; the pure-radiation oscillations are not continued forever after equality.
For $k \sim k_{eq}$ , the transition and horizon entry overlap, giving an intermediate suppression before the potential approaches a constant.

The

9/10

factor follows from conserving the large-scale adiabatic curvature: the constant potential is proportional to

3 (1 + w) / (5 + 3 w)

(3/5) / (2/3) = 9/10

Φ_{late} = (9/10) T (k) Φ_{prim}

. Then

T \to 1

on large scales, while

T \sim (k_{eq} / k)^{2} ln (k / k_{eq})

far below the equality length, as explained by the density growth below.

Late amplitudes are $Φ_{late} = (9/10) T (k) Φ_{prim}$ , with $T ≃ 1$ for $k ≪ k_{eq}$ and $T ≪ 1$ for $k ≫ k_{eq}$ .

The upper panel sketches

k^{3/2} Φ

for the three modes. The lower panel provides the corresponding density contrast histories used in the next part. A negative common primordial potential was chosen so the growing density is positive; this arbitrary phase has no effect on a cosmological density power spectrum.