Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 312 / 1

Table of contents
- i 1
  - Solution i
- ii 1
  - Solution ii

i

Solution

Write

ϕ = \overset{ˉ}{ϕ} + φ

and evaluate derivatives of

P

on the homogeneous background

\overset{ˉ}{X} = \frac{ϕ ˉ ^{'2}}{2 a ^{2}} .

The first- and second-order changes in

X

are

δ_{1} X = \frac{ϕ ˉ ^{'} φ ^{'}}{a ^{2}}, δ_{2} X = \frac{φ ^{'2} - ( \nabla φ ) ^{2}}{2 a ^{2}} .

After using the background equation to remove the linear action, the quadratic action of the P(X, phi) scalar field theory is

S_{2} = \frac{1}{2} \int d τ d^{3} x {a^{2} A φ^{'2} - a^{2} P_{X} (\nabla φ)^{2} + 2 a^{2} P_{Xϕ} \overset{ˉ}{ϕ}^{'} φ φ^{'} + a^{4} P_{ϕϕ} φ^{2}},

where

A = P_{X} + 2 \overset{ˉ}{X} P_{XX} .

Varying gives

(a^{2} A φ^{'})^{'} - a^{2} P_{X} \nabla^{2} φ + [(a^{2} P_{Xϕ} \overset{ˉ}{ϕ}^{'})^{'} - a^{4} P_{ϕϕ}] φ = 0.

The Sound speed of a P(X, phi) scalar perturbation is

c_{s}^{2} = \frac{P _{X}}{P _{X} + 2 X ˉ P _{XX}} = \frac{P _{X}}{A} .

At leading slow variation, take

H

,

c_{s}

, and the kinetic coefficients as nearly constant and neglect the effective mass and their logarithmic derivatives. The Fourier equation then becomes

φ_{k}^{''} + 2 \frac{a ^{'}}{a} φ_{k}^{'} + c_{s}^{2} k^{2} φ_{k} = 0.

For

u_{k} = a φ_{k}

and de Sitter spacetime

a^{''} / a = 2/ τ^{2}

, this is

u_{k}^{''} + (c_{s}^{2} k^{2} - \frac{2}{τ ^{2}}) u_{k} = 0.

Two independent solutions are

u_{k}^{\pm} = (1 \pm \frac{i}{c _{s} k τ}) e^{\pm i c_{s} k τ} .

The Bunch-Davies vacuum selects the positive-frequency behavior

e^{- i c_{s} k τ}

- c_{s} k τ \to \infty

. Canonical normalization of

v = a A φ

gives the general amplitude; with the field normalization

P_{X} ≃ 1

, so

A ≃ c_{s}^{- 2}

, it reduces, up to an overall phase, to

φ_{k} (τ) = \frac{H}{2 c _{s} k ^{3}} (1 + i c_{s} k τ) e^{- i c_{s} k τ} .

Solved by gpt-5.6-sol high.

ii

Solution

At cubic order, the expansion of

P (X, ϕ)

produces the schematic operators

φ^{'3}, q u a d φ^{'} (\nabla φ)^{2}, q u a d φ φ^{'2}, q u a d φ (\nabla φ)^{2}, q u a d φ^{2} φ^{'}, q u a d φ^{3} .

The derivative self-interactions involving

P_{XX}

and

P_{XXX}

are generally largest when

c_{s} ≪ 1

, while coefficients containing explicit

ϕ

derivatives of a slowly varying

P

are commonly slow-variation suppressed. The former therefore tend to dominate primordial non-Gaussianity.

The terms contributing to

φ φ^{'2}

are

a^{4} P_{Xϕ} φ δ_{2} X + \frac{a ^{4}}{2} P_{XXϕ} φ (δ_{1} X)^{2} .

Thus

S_{3} \supset \int d τ d^{3} x a^{2} λ_{0} φ φ^{'2}, λ_{0} = \frac{1}{2} (P_{Xϕ} + 2 \overset{ˉ}{X} P_{XXϕ}) .

Equivalently, if the complete time-dependent coefficient is called

λ

, then

λ (τ) = a^{2} λ_{0}

.

Treat

λ_{0}

as constant at leading slow variation. To cubic order the corresponding interaction Hamiltonian is

H_{I} = - a^{2} λ_{0} \int_{p_{1} p_{2} p_{3}} (2 π)^{3} δ^{(3)} (p_{1} + p_{2} + p_{3}) φ_{p_{1}} φ_{p_{2}}^{'} φ_{p_{3}}^{'} .

For

q_{i} = c_{s} k_{i}

, the mode function above satisfies

φ_{k} (0) = A_{k}, φ_{k}^{*'} (τ) = A_{k} q_{k}^{2} τ e^{i q_{k} τ}, A_{k} = \frac{H}{2 c _{s} k ^{3}} .

K = k_{1} + k_{2} + k_{3}

, the needed regulated integral is

\int_{- \infty (1 - i ϵ)}^{0} (1 - i q_{i} τ) e^{i c_{s} K τ} d τ = - \frac{i}{c _{s}} (\frac{1}{K} + \frac{k _{i}}{K ^{2}}) .

There are three choices for the undifferentiated leg and two contractions interchanging the differentiated legs. The stated in-in formalism formula therefore gives

⟨ φ_{k_{1}} φ_{k_{2}} φ_{k_{3}} ⟩ = (2 π)^{3} δ^{(3)} (k_{1} + k_{2} + k_{3}) B_{λ} (k_{1}, k_{2}, k_{3}),

with

B_{λ} = \frac{λ _{0} H ^{4}}{2 ( k _{1} k _{2} k _{3} ) ^{3}} cyc \sum k_{j}^{2} k_{l}^{2} (\frac{1}{K} + \frac{k _{i}}{K ^{2}}) .

The powers of

c_{s}

cancel for this vertex with the normalization specified in part (i). Reversing the convention for the sign of

H_{I}

reverses the displayed overall sign but not the momentum shape of the primordial bispectrum.

Solved by gpt-5.6-sol high.

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