Search Results for « Cosmological correlators »

Abstract Much of the structure of cosmological correlators is controlled by their singularities, which in turn are fixed in terms of flat-space scattering amplitudes. An important challenge is to interpolate between the singular limits to determine the full correlators at arbitrary kinematics. This is particularly relevant because the singularities of correlators are not directly observable, but can only be accessed by analytic continuation. In this paper, we study rational correlators — including those of gauge fields, gravitons, and the inflaton — whose only singularities at tree level are poles and whose behavior away from these poles is strongly constrained by unitarity and locality. We describe how unitarity translates into a set of cutting rules that consistent correlators must satisfy, and explain how this can be used to bootstrap correlators given information about their singularities. We also derive recursion relations that allow the iterative construction of more complicated correlators from simpler building blocks. In flat space, all energy singularities are simple poles, so that the combination of unitarity constraints and recursion relations provides an efficient way to bootstrap the full correlators. In many cases, these flat-space correlators can then be transformed into their more complex de Sitter counterparts. As an example of this procedure, we derive the correlator associated to graviton Compton scattering in de Sitter space, though the methods are much more widely applicable.

Abstract The initial conditions of our universe appear to us in the form of a classical probability distribution that we probe with cosmological observations. In the current leading paradigm, this probability distribution arises from a quantum mechanical wavefunction of the universe. Here we ask what the imprint of quantum mechanics is on the late time observables. We show that the requirement of unitary time evolution, colloquially the conservation of probabilities, fixes the analytic structure of the wavefunction and of all the cosmological correlators it encodes. In particular, we derive in perturbation theory an infinite set of single-cut rules that generalize the Cosmological Optical Theorem and relate a certain discontinuity of any tree-level n-point function to that of lower-point functions. Our rules are closely related to, but distinct from the recently derived Cosmological Cutting Rules. They follow from the choice of the Bunch-Davies vacuum and a simple property of the (bulk-to-bulk) propagator and are astoundingly general: we prove that they are valid for fields with a linear dispersion relation and any mass, any integer spin and arbitrary local interactions with any number of derivatives. They also apply to general FLRW spacetimes admitting a Bunch-Davies vacuum, including de Sitter, slow-roll inflation, power-law cosmologies and even resonant oscillations in axion monodromy. We verify the single-cut rules in a number of non-trivial examples, including four massless scalars exchanging a massive scalar, as relevant for cosmological collider physics, four gravitons exchanging a graviton, and a scalar five-point function.

Abstract Developing our understanding of how correlations evolve during inflation is crucial if we are to extract information about the early Universe from our late-time observables. To that end, we revisit the time evolution of scalar field correlators on de Sitter spacetime in the Schrödinger picture. By direct manipulation of the Schrödinger equation, we write down simple “equations of motion” for the coefficients which determine the wavefunction. Rather than specify a particular interaction Hamiltonian, we assume only very basic properties (unitarity, de Sitter invariance and locality) to derive general consequences for the wavefunction’s evolution. In particular, we identify a number of “constants of motion” — properties of the initial state which are conserved by any unitary dynamics — and show how this can be used to partially fix the cubic and quartic wavefunction coefficients at weak coupling. We further constrain the time evolution by deriving constraints from the de Sitter isometries and show that these reduce to the familiar conformal Ward identities at late times. Finally, we show how the evolution of a state from the conformal boundary into the bulk can be described via a number of “transfer functions” which are analytic outside the horizon for any local interaction. These objects exhibit divergences for particular values of the scalar mass, and we show how such divergences can be removed by a renormalisation of the boundary wavefunction — this is equivalent to performing a “Boundary Operator Expansion” which expresses the bulk operators in terms of regulated boundary operators. Altogether, this improved understanding of the wavefunction in the bulk of de Sitter complements recent advances from a purely boundary perspective, and reveals new structure in cosmological correlators.

Abstract In the standard approach to deriving inflationary predictions, we evolve a vacuum state in time according to the rules of a given model. Since the only observables are the future values of correlators and not their time evolution, this brings about a large degeneracy: a vast number of different models are mapped to the same minute number of observables. Furthermore, due to the lack of time-translation invariance, even tree-level calculations require an increasing number of nested integrals that quickly become intractable. Here we ask how much of the final observables can be “bootstrapped” directly from locality, unitarity and symmetries. To this end, we introduce two new “boostless” bootstrap tools to efficiently compute tree-level cosmological correlators/wavefunctions without any assumption about de Sitter boosts. The first is a Manifestly Local Test (MLT) that any n-point (wave)function of massless scalars or gravitons must satisfy if it is to arise from a manifestly local theory. When combined with a sub-set of the recently proposed Bootstrap Rules, this allows us to compute explicitly all bispectra to all orders in derivatives for a single scalar. Since we don’t invoke soft theorems, this can also be extended to multi-field inflation. The second is a partial energy recursion relation that allows us to compute exchange correlators. Combining a bespoke complex shift of the partial energies with Cauchy’s integral theorem and the Cosmological Optical Theorem, we fix exchange correlators up to a boundary term. The latter can be determined up to contact interactions using unitarity and manifest locality. As an illustration, we use these tools to bootstrap scalar inflationary trispectra due to graviton exchange and inflaton self-interactions.

Abstract: Developing our understanding of how correlations evolve during inflation is crucial if we are to extract information about the early Universe from our late-time observables. To that end, we revisit the time evolution of scalar field correlators on de Sitter spacetime in the Schrödinger picture. By direct manipulation of the Schrödinger equation, we write down simple “equations of motion” for the coefficients which determine the wavefunction. Rather than specify a particular interaction Hamiltonian, we assume only very basic properties (unitarity, de Sitter invariance and locality) to derive general consequences for the wavefunction’s evolution. In particular, we identify a number of “constants of motion” — properties of the initial state which are conserved by any unitary dynamics — and show how this can be used to partially fix the cubic and quartic wavefunction coefficients at weak coupling. We further constrain the time evolution by deriving constraints from the de Sitter isometries and show that these reduce to the familiar conformal Ward identities at late times. Finally, we show how the evolution of a state from the conformal boundary into the bulk can be described via a number of “transfer functions” which are analytic outside the horizon for any local interaction. These objects exhibit divergences for particular values of the scalar mass, and we show how such divergences can be removed by a renormalisation of the boundary wavefunction — this is equivalent to performing a “Boundary Operator Expansion” which expresses the bulk operators in terms of regulated boundary operators. Altogether, this improved understanding of the wavefunction in the bulk of de Sitter complements recent advances from a purely boundary perspective, and reveals new structure in cosmological correlators.

Abstract We propose a Mellin space approach to the evaluation of late-time momentum-space correlation functions of quantum fields in (d + 1)-dimensional de Sitter space. The Mellin-Barnes representation makes manifest the analytic structure of late-time correlators and, more generally, provides a convenient general d framework for the study of conformal correlators in momentum space. In this work we focus on tree-level correlation functions of general scalars as a prototype, including n-point contact diagrams and 4-point exchanges. For generic scalars, both the contact and exchange diagrams are given by (generalised) Hypergeometric functions, which reduce to existing expressions available in the literature for d = 3 and external scalars which are either simultaneously conformally coupled or massless. This approach can also be used for the perturbative bulk evaluation of momentum space boundary correlators in (d + 1)-dimensional anti-de Sitter space (Witten diagrams).

We provide a first principle definition of cosmological correlation functions for a large class of scalar toy models in arbitrary FRW cosmologies, in terms of novel geometries we name weighted cosmological polytopes. Each of these geometries encodes a universal rational integrand associated to a given Feynman graph. In this picture, all the possible ways of organising, and computing, cosmological correlators correspond to triangulations and subdivisions of the geometry, containing the in-in representation, the one in terms of wavefunction coefficients and many others. We also provide two novel contour integral representations, one connecting higher and lower loop correlators and the other one expressing any of them in terms of a building block. We study the boundary structure of these geometries allowing us to prove factorisation properties and Steinmann-like relations when single and sequential discontinuities are approached. We also show that correlators must satisfy novel vanishing conditions. As the weighted cosmological polytopes can be obtained as an orientation-changing operation onto a certain subdivision of the cosmological polytopes encoding the wavefunction of the Universe, this picture allows us to sharpen how the properties of cosmological correlators are inherited from the ones of the wavefunction. From a mathematical perspective, we also provide an in-depth characterisation of their adjoint surface.

A bstract Cosmological correlators encode rich information about physics at the Hubble scale and may exhibit characteristic oscillatory signals due to the exchange of massive particles. Although many 1-loop processes, especially those that break de Sitter (dS) boosts, can generate significant leading signals for various particle models in cosmological collider physics, the precise results for these correlators or their full signals remain unknown due to the lack of symmetry. In this work, we apply the method of partial Mellin-Barnes (PMB) representation to the calculation of cosmological correlators at the loop level. As a first step, we use the PMB representation to calculate four-point cosmological correlators with bubble topology. We find that both the nonlocal and local signals arise from the factorized part, validating the cutting rules proposed in previous work, and are free from ultraviolet (UV) divergence. Furthermore, the UV divergence originates solely from the background piece and can be manifestly canceled by introducing the appropriate counterterm, similar to the procedure in flat spacetime. We also demonstrate how to renormalize the 1-loop correlators in Mellin space. After a consistency check with known results for the covariant case, we provide new analytical results for the signals generated from a nontrivial dS-boost-breaking bubble.

Abstract Cosmological correlators from inflation are often generated at tree level and hence loop contributions are bounded to be small corrections by perturbativity. Here we discuss a scenario where this is not the case. Recently, it has been shown that for any number of scalar fields of any mass, the parity-odd trispectrum of a massless scalar must vanish in the limit of exact scale invariance due to unitarity and the choice of initial state. By carefully handling UV-divergences, we show that the one-loop contribution is non-vanishing and hence leading. Surprisingly, the one-loop parity-odd trispectrum is simply a rational function of kinematics, which we compute explicitly in a series of models, including single-clock inflation. Although the loop contribution is the leading term in the parity-odd sector, its signal-to-noise ratio is typically bounded from above by that of a corresponding tree-level parity-even trispectrum, unless instrumental noise and systematics for the two observables differ. Furthermore, we identify a series of loop contributions to the wavefunction that cancel exactly when computing correlators, suggesting a more general phenomenon.

Abstract We study the fundamentals of quantum field theory on a rigid de Sitter space. We show that the perturbative expansion of late-time correlation functions to all orders can be equivalently generated by a non-unitary Lagrangian on a Euclidean AdS geometry. This finding simplifies dramatically perturbative computations, as well as allows us to establish basic properties of these correlators, which comprise a Euclidean CFT. We use this to infer the analytic structure of the spectral density that captures the conformal partial wave expansion of a late-time four-point function, to derive an OPE expansion, and to constrain the operator spectrum. Generically, dimensions and OPE coefficients do not obey the usual CFT notion of unitarity. Instead, unitarity of the de Sitter theory manifests itself as the positivity of the spectral density. This statement does not rely on the use of Euclidean AdS Lagrangians and holds non-perturbatively. We illustrate and check these properties by explicit calculations in a scalar theory by computing first tree-level, and then full one- loop-resummed exchange diagrams. An exchanged particle appears as a resonant feature in the spectral density which can be potentially useful in experimental searches.