Bunch-Davies vacuum

The Bunch-Davies vacuum selects the mode that approaches the positive-frequency ground-state mode of angular frequency

c k

in the far past

τ \to - \infty

Table of contents

Bunch-Davies mode normalization in a finite comoving volume ( $f_{k} = H (- τ + i / k) e^{- ik τ} / 2 k V, a^{2} V (f_{k} f_{k}^{'} - f_{k}^{} f_{k}^{'}) = i$ )

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For a minimally coupled massless scalar field in flat de Sitter spacetime,

a = - 1/ (H τ)

and the Canonically rescaled de Sitter scalar mode is

u_{k} = a f_{k}

. The equation

u_{k}^{''} + (k^{2} - 2/ τ^{2}) u_{k} = 0

and the positive-frequency condition

u_{k} \sim e^{- ik τ} / 2 k V

select the displayed mode. Direct differentiation verifies its Wronskian normalization, which ensures the equal-time canonical commutation relations with box-normalized creation and annihilation operators. A conformally coupled massless scalar has a different mode equation.

Cosmological horizon exit

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A Fourier mode exits the Hubble horizon during inflation when

k = a H

. On superhorizon scales

k ≪ a H

, the growing mode of a sufficiently light scalar perturbation approaches a constant.

Superhorizon scale

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A cosmological mode is on a superhorizon scale when its physical wavenumber is smaller than the Hubble expansion rate,

k / a ≪ H

, equivalently

k ≪ a H

Scale-invariant inflationary power spectrum

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A Fourier-space variance

P_{k}

in three dimensions is scale invariant when the dimensionless power per logarithmic wavenumber interval,

Δ^{2} (k) = \frac{k ^{3} P _{k}}{2 π ^{2}},

(1)

is independent of

k

. For a massless inflaton mode at late times,

P_{k} ⟶ \frac{ℏ H _{inf}^{2}}{2 c ^{3} k ^{3}},

(2)

Δ^{2} (k)

is constant.

Infrared qualification of a scale-invariant covariance ( $C (r) - C (r_{0}) = - \frac{A}{2 π ^{2}} lo g (r / r_{0})$ )

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An exact

P (k) = A / k^{3}

has divergent unsubtracted point variance. Its finite isotropic covariance difference is the displayed logarithm, unchanged by simultaneous dilation of

r, r_{0}

. Thus the nonzero-mode scaling statement has a natural subtraction interpretation. Exact dilation invariance of a continuous finite-variance isotropic covariance itself would instead force a constant covariance, corresponding only to a zero-mode spectral measure. Cutoffs or a zero-mode subtraction must accompany the nontrivial spectrum.

Equal-time two-point function of a de Sitter scalar ( $P_{δ ϕ} (k, τ) = H^{2} (1 + k^{2} τ^{2}) / (2 k^{3})$ )

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For a light scalar in the Bunch-Davies vacuum with metric perturbations neglected, the mode

e^{- ik τ} (1 - i / (k τ)) / 2 k

gives the displayed equal-time spectrum. The dimensionless power tends to

H^{2} / (4 π^{2})

on superhorizon scales. The exact massless infinite-volume correlation has an infrared divergence; a finite-volume or finite-duration infrared regulator, and smearing for coincident products, makes the correlators well defined.

Frozen contribution to a de Sitter scalar spectrum ( $P_{frozen} = H^{2} / (2 k^{3}), Δ_{frozen}^{2} = H^{2} / (4 π^{2})$ )

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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.

Subhorizon vacuum contribution to a de Sitter scalar spectrum ( $Δ_{vac}^{2} (k, τ) = k^{2} / (4 π^{2} a^{2})$ )

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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.

Primordial scalar amplitude ( $A_{s}$ )

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At leading slow-roll order, the dimensionless primordial curvature-perturbation amplitude at horizon exit is

A_{s} = \frac{H ^{2}}{8 π ^{2} ϵ _{V} M _{Pl}^{2}} = \frac{V}{24 π ^{2} M _{Pl}^{4} ϵ _{V}} .

(1)

Scalar spectral index ( $n_{s}$ )

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The scalar spectral index describes the departure of the primordial scalar spectrum from scale invariance. To first order in potential slow-roll parameters,

n_{s} - 1 ≃ - 6 ϵ_{V} + 2 η_{V} .

(1)

Running of the scalar spectral index ( $α_{s} = d n_{s} / d lo g k ≃ 16 ϵ_{V} η_{V} - 24 ϵ_{V}^{2} - 2 ξ_{V}^{2}$ )

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The running describes the scale dependence of the scalar spectral index. Differentiating

n_{s} - 1 ≃ - 6 ϵ_{V} + 2 η_{V}

with the potential slow-roll flow equations gives the displayed second-order slow-roll result. It includes the third potential slow-roll parameter and is generally smaller than the leading spectral tilt.