Renormalisation of IR divergences and holography in de Sitter

TL;DR

This paper develops a renormalisation procedure for IR divergences in de Sitter space correlators, linking them to holographic dualities and establishing formulas relating de Sitter correlators to conformal field theory correlators.

Contribution

It introduces a novel renormalisation method for IR divergences in de Sitter correlators using boundary counterterms, connecting holography in de Sitter to AdS via analytic continuation.

Findings

IR divergences in de Sitter are renormalised by boundary counterterms.

Holographic formulas relate de Sitter correlators to CFT correlators up to four points.

Analytic continuation links de Sitter renormalisation to AdS conformal anomalies.

Abstract

We formulate a renormalisation procedure for IR divergences of tree-level in-in late-time de Sitter correlators. These divergences are due to the infinite volume of spacetime and are analogous to the divergences that appear in AdS dealt with by holographic renormalisation. Regulating the theory using dimensional regularisation, we show that one can remove all infinities by adding local counterterms at the future boundary of dS in the Schwinger-Keldysh path integral. The counterterms amount to renormalising the late-time bulk field. We frame the discussion in terms of bulk scalar fields in dS, using tree-level correlators of massless and conformal scalars for illustration. The relation to AdS via analytic continuation is discussed, and we show that different versions of the analytic continuation appearing in the literature are equivalent to each other. In AdS, one needs to add counterterms that are related to conformal anomalies, and also to renormalise the source part of the bulk field. The analytic continuation to dS projects out the traditional AdS counterterms, and links the renormalisation of the sources to the renormalisation of the late-time bulk field. We use these results to establish holographic formulae that relate tree-level dS in-in correlators to CFT correlators at up to four points, and we provide two proofs: one using the connection between the dS wavefunction and the partition function of the dual CFT, and a second by direct evaluation of the in-in correlators using the Schwinger-Keldysh formalism. The renormalisation of the bulk IR divergences is mapped by these formulae to UV renormalisation of the dual CFT via local counterterms, providing structural support for a possible duality. We also recast the regulated holographic formulae in terms of the AdS amplitudes of shadow fields, but show that this relation breaks down when renormalisation is required.