Cosmological Cutting Rules

Abstract

Primordial perturbations in our universe are believed to have a quantum origin, and can be described by the wavefunction of the universe (or equivalently, cosmological correlators). It follows that these observables must carry the imprint of the founding principle of quantum mechanics: unitary time evolution. Indeed, it was recently discovered that unitarity implies an infinite set of relations among tree-level wavefunction coefficients, dubbed the Cosmological Optical Theorem. Here, we show that unitarity leads to a systematic set of “Cosmological Cutting Rules” which constrain wavefunction coefficients for any number of fields and to any loop order. These rules fix the discontinuity of an n-loop diagram in terms of lower-loop diagrams and the discontinuity of tree-level diagrams in terms of tree-level diagrams with fewer external fields. Our results apply with remarkable generality, namely for arbitrary interactions of fields of any mass and any spin with a Bunch-Davies vacuum around a very general class of FLRW spacetimes. As an application, we show how one-loop corrections in the Effective Field Theory of inflation are fixed by tree-level calculations and discuss related perturbative unitarity bounds. These findings greatly extend the potential of using unitarity to bootstrap cosmological observables and to restrict the space of consistent effective field theories on curved spacetimes.

From locality and unitarity to cosmological correlators

“…To this end, we perform complex shifts of the partial energies of a given diagram, which are the sums of the energies flowing into any given sub-diagram, and are the only allowed singularities at tree-level together with the total energy. The residues of all the poles in the complex shift are fully fixed by the Cosmological Optical Theorem [32] and its perturbative manifestation in the recently derived Cosmological Cutting Rules [36,37]. This is in contrast to the factorization limits discussed for example in [26], which fix only the leading order singularities.…”

“…We choose to work with the wavefunction as opposed to correlators for several reasons. At the technical level, the constraints from unitarity and the choice of vacuum derived in [32,[35][36][37] can be expressed very straightforwardly in terms of the wavefunction coefficients, but become very cumbersome in terms of the correlators, since in general one needs to include correlators of products of both the fields and their conjugate momenta. At a more abstract level, in a conjectural holographic approach to quantum field theory and quantum gravity in de Sitter, it is the wavefunction that is naturally computed by a conformal field theory, in analogy with the AdS/CFT correspondence and along the lines of the proposal in [30] (see also [38,39] for a different perspective).…”

“…This relation can be thought of as a Cosmological Optical Theorem (COT) providing a nonlinear relation for the discontinuities of correlators. The COT is also remarkably general: it is valid for fields of any mass and spin on a very general class of FLRW spacetimes, including de Sitter, inflation and ΛCDM [37], as well as to any loop order in perturbation theory [36] (where it takes the form of Cosmological Cutting Rules). In this paper we show how these relations can be used to bootstrap exchange n-point functions arising from boost-breaking interactions.…”

“…In section 5 we then turned to bootstrapping exchange diagrams and introduced partial energy recursion relations, which allow for efficient computation of 4-point exchange diagrams given a pair of 3-point sub-diagrams. When used in conjunction with the Cosmological Optical Theorem (COT) [32] (see also upcoming work [36,37]), these recursion relations fix the residues of all leading and sub-leading partial energy poles and therefore fix the 4-point exchange diagram up to the presence of a boundary term with only total energy poles. The boundary term is then fixed by the MLT and the COT up to contact contributions from quartic interactions, which can always be chosen at will.…”

From amplitudes to contact cosmological correlators

“…[24][25][26], however we will not discuss this in the present work. As a technical note, we should mention that we work at the level of the IR finite part of the wavefunction coefficients, which are forced to be real by unitarity in the form of the cosmological optical theorem [11,16,[27][28][29] -see also refs. [30][31][32] for the anti de Sitter (AdS) side of this story and refs.…”