Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 57 / 4 / ii

Solution

Factor the momentum dependence into

G = (2 π)^{3} δ^{3} (\sum k_{i}) \frac{S}{k _{1}^{3} k _{2}^{3} k _{3}^{3}} (- K + \frac{s _{2}}{K} + \frac{s _{3}}{K ^{2}}) .

Write

s = + 1

for the intended target, obtained with the Hamiltonian sign repair in part (i), and

s = - 1

for the literal stated Hamiltonian. The same sign choice applies to both limits and cancels in their amplitude ratio. The primordial bispectrum contribution is

s H^{4} (ϵ + 1 - c_{s}^{2}) G / (16 ϵ^{2} c_{s}^{4})

. Hence

c_{s}^{2} \to 1 : ⟨ ζ^{3} ⟩_{c} ≃ s \frac{H ^{4}}{16 ϵ} G, c_{s}^{2} ≪ 1 : ⟨ ζ^{3} ⟩_{c} ≃ s \frac{H ^{4} ( 1 + ϵ )}{16 ϵ ^{2} c _{s}^{4}} G .

At fixed

H, ϵ

\frac{∣ B ( c _{s} ) ∣}{∣ B ( 1 ) ∣} = \frac{ϵ + 1 - c _{s}^{2}}{ϵ c _{s}^{4}} ≃ \frac{101}{c _{s}^{4}} (ϵ = 0.01, c_{s}^{2} ≪ 1) .

Keeping only the leading slow-roll order makes this

100/ c_{s}^{4}

. To compare the two pieces of this same interaction at a fixed

c_{s}

, their ratio is instead

(1 - c_{s}^{2}) / ϵ ≃ 100 (1 - c_{s}^{2})

. Thus the noncanonical piece overtakes the slow-roll piece once

1 - c_{s}^{2} ≫ 0.01

, while both have the common raw

c_{s}^{- 4}

enhancement from the interaction and its external mode normalization. For instance

c_{s}^{2} = 0.1

gives an exact ratio

9100

to the canonical limit with these fixed parameters.

P_{ζ} (k) = H^{2} / (4 ϵ c_{s} k^{3})

, so the ratio

B / P_{ζ}^{2}

scales as

(ϵ + 1 - c_{s}^{2}) / c_{s}^{2}

at fixed shape. The small-sound-speed enhancement in normalized non-Gaussianity is therefore

c_{s}^{- 2}

. These are tree-level weak-interaction predictions; approaching arbitrarily small

c_{s}

eventually requires checking the perturbative regime.

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