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.
Highlights
Unitarity is a central pillar of quantum mechanics
We derive Cosmological Cutting Rules for the wavefunction coefficients ψn for any number of external legs and to any loop order
– Take the discontinuity (defined in (3.3)) of all possible subdiagrams by analytically continuing all external legs except those arising from the cutting of an internal line
Summary
Unitarity is a central pillar of quantum mechanics. On the one hand, the positive norm of states in the Hilbert space is essential for ensuring that probabilities are positive. We discuss the “bootstrap” approach, namely the prospect of using the powerful constraints of unitarity, combined with other basic principles such as locality, the choice of vacuum and symmetries as a computational tool to derive observables, and potentially bypass the traditional bulk in-in calculation This approach has a demonstrated track record for the calculation of amplitudes [11,12,13], and has gained much traction in the cosmological context. Very promising results in this direction have already been derived using constraints from factorization [36], the formulation of very general boostless Bootstrap Rules [45], and the recently derived Manifestly Local Test and partial-energy recursion relations [46] From this perspective our Cosmological Cutting Rules add a powerful tool to bootstrap in full generality higher order correlators form lower order ones, and in particular exchange and loop diagrams
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