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 Incoming links: Frozen contribution to a de Sitter scalar spectrum

For an exactly massless minimally coupled scalar field on flat-slicing de Sitter spacetime with constant

H

, use natural units and neglect its metric backreaction. With

a = - 1/ (H τ)

and the Bunch-Davies vacuum, its canonical mode is

u_{k} (τ) = \frac{e ^{- ik τ}}{2 k} (1 - \frac{i}{k τ}), δ ϕ_{k} = u_{k} / a .

With power spectrum convention

⟨ δ ϕ_{k} δ ϕ_{k^{'}} ⟩ = (2 π)^{3} δ^{3} (k + k^{'}) P_{δ ϕ} (k, τ)

, taking the squared mode amplitude gives

P_{δ ϕ} (k, τ) = \frac{H ^{2}}{2 k ^{3}} (1 + k^{2} τ^{2}), Δ_{δ ϕ}^{2} = \frac{H ^{2}}{4 π ^{2}} (1 + k^{2} τ^{2}) .

The constant part is the frozen contribution to a de Sitter scalar spectrum; the remaining part is the subhorizon vacuum contribution to a de Sitter scalar spectrum. A merely light but nonzero mass changes the exact mode equation, so the displayed massless solution is then an approximation rather than an exact identity. Nonminimal curvature coupling also changes it.

The massless infinite-volume two-point correlation function has an infrared divergence. A finite-volume or finite-duration infrared prescription is needed, and coincident products require ultraviolet regulation or smearing. A finite spectrum at each fixed nonzero

k

does not by itself make the all-mode equal-position integral finite.