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.