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 Incoming links: Equal-time two-point function of a de Sitter scalar

Frozen contribution to a de Sitter scalar spectrum 2026-10-14

The second term in the equal-time two-point function of a de Sitter scalar is independent of time for constant

H

. It dominates on superhorizon scales and has constant dimensionless power spectrum, giving a scale-invariant inflationary power spectrum. The infrared qualification of a scale-invariant covariance still applies to the formally infinite-volume coincident variance.

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

Use the Fourier normalization of the PDF. The creation and annihilation operators obey

[a_{k}, a_{k^{'}}^{†}] = δ^{3} (k - k^{'}), [a_{k}, a_{k^{'}}] = [a_{k}^{†}, a_{k^{'}}^{†}] = 0.

There is no

(2 π)^{3}

in this commutator because the expansion uses

(2 π)^{- 3/2}

. The Bunch-Davies vacuum satisfies

a_{k} ∣0 ⟩ = 0

. Write

u_{k} = f_{k}^{*}

for the coefficient of the annihilation operator. Only the contraction

⟨ 0∣ a_{k} a_{k^{'}}^{†} ∣0 ⟩ = δ^{3} (k - k^{'})

contributes, so

⟨ 0∣ δ ϕ (τ, x) δ ϕ (τ, x + r) ∣0 ⟩ = \int \frac{d ^{3} k}{( 2 π ) ^{3}} \frac{∣ u _{k} ( τ ) ∣ ^{2}}{a ^{2}} e^{- i k \cdot r} .

The given mode has

∣ u_{k} ∣^{2} = \frac{1}{2 k} (1 + \frac{1}{k ^{2} τ ^{2}}) .

Since

a^{- 2} = H^{2} τ^{2}

, the equal-time two-point function of a de Sitter scalar therefore has dimensional power spectrum

P_{δ ϕ} (k, τ) = \frac{H ^{2}}{2 k ^{3}} (1 + k^{2} τ^{2}) .

Comparing with the dimensionless power convention in the hint gives

Δ_{δ ϕ}^{2} (k, τ) = \frac{k ^{3}}{2 π ^{2}} P_{δ ϕ} (k, τ) = (\frac{H}{2 π})^{2} (1 + k^{2} τ^{2}) .

Equivalently, angular integration expresses the correlation as

\int_{0}^{\infty} (d k / k) Δ_{δ ϕ}^{2} sin (k r) / (k r)

, where

r = ∣ r ∣

.

On superhorizon scales,

k ≪ a H

is

∣ k τ ∣ ≪ 1

, and

Δ_{δ ϕ}^{2} ⟶ (\frac{H}{2 π})^{2} .

This scale-invariant inflationary power spectrum gives equal variance per logarithmic wavenumber interval. It is exactly scale invariant in the massless constant-

H

approximation here; slow evolution of

H

and small field mass produce a small spectral tilt. Such quantum primordial perturbations supply the initial fluctuations that subsequently seed cosmological structure after conversion to curvature and density perturbations. The curvature conversion itself requires the metric perturbations excluded from this calculation.

The continuum correlation is understood with the usual infrared regulator, such as finite volume or finite inflationary duration. The strictly massless infinite-volume model has a logarithmic infrared divergence. Coincident products also require ultraviolet regularization or smearing; the finite mode spectrum is unaffected by these qualifications.

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Subhorizon vacuum contribution to a de Sitter scalar spectrum 2026-10-14

The equal-time two-point function of a de Sitter scalar has a contribution

P_{vac} = 1/ (2 k a^{2})

, giving the displayed dimensionless power spectrum per logarithmic wavenumber. At fixed comoving wavenumber it decreases as

a^{- 2}

; on short physical scales it reproduces the vacuum fluctuations of a massless scalar field in Minkowski spacetime. Its coincident integral is ultraviolet divergent, so the fixed-mode decrease cannot be interchanged with an unregulated integral over all modes.

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