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 Incoming links: Conformal-time quadratic action for an inflaton perturbation

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 310 / 4 / b / Solution Created 2026-10-03 Updated 2026-10-05

Subtract the homogeneous background equation and linearize the inflaton equation of motion. The field perturbation satisfies

δ ϕ^{''} + 2 \frac{a ^{'}}{a} δ ϕ^{'} - \nabla^{2} δ ϕ + a^{2} V_{, ϕϕ} (\overset{ˉ}{ϕ}) δ ϕ = 0.

With

f = a δ ϕ

, direct differentiation cancels the first-derivative term and gives

f^{''} - \nabla^{2} f + (a^{2} V_{, ϕϕ} - \frac{a ^{''}}{a}) f = 0.

The conformal-time quadratic action for an inflaton perturbation, after a boundary term is discarded, is

S^{(2)} = \frac{1}{2} \int d τ d^{3} x {(f^{'})^{2} - (\nabla f)^{2} + (\frac{a ^{''}}{a} - a^{2} V_{, ϕϕ}) f^{2}} .

For the specified de Sitter spacetime background,

a^{''} / a = 2/ τ^{2} = 2 a^{2} H^{2}

. The light-field approximation

∣ V_{, ϕϕ} ∣ ≪ 2 H^{2}

therefore permits dropping the mass term. Taking a spatial Fourier transform yields the Canonically rescaled de Sitter scalar mode equation

f_{k}^{''} + (k^{2} - \frac{a ^{''}}{a}) f_{k} = 0, \frac{a ^{''}}{a} = \frac{2}{τ ^{2}} .

Its kinetic term is canonical, with momentum

π_{f} = f^{'}

. Each independent real Fourier component consequently has the canonical coordinate/momentum algebra of a quantum harmonic oscillator, with time-dependent squared frequency

ω_{k}^{2} = k^{2} - a^{''} / a

. This is why canonical quantization of a real scalar field proceeds by oscillator creation and annihilation operators. On superhorizon scales the squared frequency is negative: the system is then an inverted time-dependent oscillator, not a stationary oscillator with a positive frequency. The canonical commutation relations remain valid.

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