Equal-time two-point function of a de Sitter scalar ($P_{\delta \varphi }(k,\tau )=H^2(1+k^2{\tau }^2)/(2k^3)$)

For a light scalar in the Bunch-Davies vacuum with metric perturbations neglected, the mode $e^{-ik\tau }(1-i/(k\tau ))/\sqrt{2k}$ gives the displayed equal-time spectrum. The dimensionless power tends to $H^2/(4{\pi }^2)$ 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.

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.

The equal-time two-point function of a de Sitter scalar has a contribution $P_{\mathrm{vac}}=1/(2k{a}^2)$, 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.