arXiv subjects

Paolo Benincasa

Publications and source records attributed to Paolo Benincasa.

At least 19 recordsLinked to original sources

arXiv2025

The Asymptotic Structure of Cosmological Integrals

We provide a general analysis of the asymptotic behaviour of perturbative contributions to observables in arbitrary power-law FRW cosmologies, indistinctly the Bunch-Davies wavefunction and cosmological correlators. We consider a large class of scalar toy models, including conformally-coupled and massless scalars in arbitrary dimensions, that admits a first principle definition in terms of (generalised/weighted) cosmological polytopes. The perturbative contributions to an observable can be expressed as an integral of the canonical function associated to such polytopes and to weighted graphs. We show how the asymptotic behaviour of these integrals is governed by a special class of nestohedra living in the graph-weight space, both at tree and loop level. As the singularities of a cosmological process described by a graph can be associated to its subgraphs, we provide a realisation of the nestohedra as a sequential truncation of a top-dimensional simplex based on the underlying graph. This allows us to determine all the possible directions -- both in the infrared and in the ultraviolet --, where the integral can diverge as well as their divergence degree. Both of them are associated to the facets of the nestohedra, which are identified by overlapping tubings of the graph: the specific tubing determines the divergent directions while the number of overlapping tubings its degree of divergence. This combinatorial formulation makes straightforward the application of sector decomposition for extracting both leading and subleading divergences from the integral, as the sectors in which the integration domain can be tiled are identified by the collection of compatible facets of the nestohedra, with the latter that can be determined via the graph tubings. Finally, the leading divergence can be interpreted as a restriction of the canonical function of the relevant polytope onto a special hyperplane.

hep-th↗

arXiv2025

Cosmological Infrared Subtractions & Infrared-Safe Computables

Cosmological observables in perturbation theory turn out to be plagued with infrared divergences, which represents both a conceptual and computational challenge. In this paper we present a proof of concept for a systematic procedure to remove these divergences in a large class of scalar cosmological integrals and consistently define an infrared safe computable in perturbation theory. We provide diagrammatic rules which are based on the nestohedra underlying the asymptotic structure of such integrals.

hep-th↗

arXiv2024

On one-loop corrections to the Bunch-Davies wavefunction of the universe

Understanding the loop corrections to cosmological observables is of paramount importance for having control on the quantum consistency of a theory in an expanding universe as well as for phenomenological reasons. In the present work, we begin with a systematic study of such corrections in the context scalar toy models whose perturbative Bunch-Davies wavefunction enjoys an intrinsic definition in terms of cosmological polytopes, focusing on one-loop graphs. Owing to the underlying twisted period integral representation they admit, their combinatorial structure along with their vector space structure, emerging from polynomial ideals algebra and intersection theory, are exploited to set-up and analyse the differential equations that the two- and three-site one-loop corrections have to satisfy upon variation of the external kinematic variables. We find that, while the two-site contribution can be written in terms of multiple-polylogarithms, this is no longer true for the three-site case, for which elliptic structures appear. As a non-trivial check, we consider the scattering amplitude limit, recovering the known result in terms of polylogarithms only.

hep-th↗

arXiv2024

Records from the S-Matrix Marathon: Observables in Expanding Universes

Observables in expanding universes are crucial to understand the physics of the early universe. In these lectures, we review some recent progress in understanding their mathematical structure and extract the physics encoded in them. After discussing the most salient features of an expanding background and their consequences for defining an observable, we focus on the so-called Bunch--Davies wavefunctional. We analyze its analytic properties on general grounds and introduce an integral representation for it in perturbation theory for a special class of scalar toy models. We discuss both the diagrammatics associated to the usual Feynman rules and combinatorial rules on the graphs, which generate a representation free of spurious poles. Such combinatorial rules find their origin in the combinatorics of the cosmological polytopes of which we provide a gentle introduction to its definition and its main features. Finally, the combinatorics of the cosmological polytopes turns out to determine the combinatorics of a special class of nestohedra that encode the asymptotic behaviour of the cosmological integrals. We provide a general description of such structures and behaviour, which is of crucial importance to understand the infrared divergences which plague observables in an expanding background. These notes are based on a series of lectures held during the S-Matrix Marathon workshop at the Institute for Advanced Study on 11--22 March 2024.

hep-th↗

arXiv2024

Perturbative Unitarity and the Wavefunction of the Universe

Unitarity of time evolution is one of the basic principles constraining physical processes. Its consequences in the perturbative Bunch-Davies wavefunction in cosmology have been formulated in terms of the cosmological optical theorem. In this paper, we re-analyse perturbative unitarity for the Bunch-Davies wavefunction, focusing on: 1) the role of the $iε$-prescription and its compatibility with the requirement of unitarity; 2) the origin of the different "cutting rules"; 3) the emergence of the flat-space optical theorem from the cosmological one. We take the combinatorial point of view of the cosmological polytopes, which provide a first-principle description for a large class of scalar graphs contributing to the wavefunctional. The requirement of the positivity of the geometry together with the preservation of its orientation determine the $iε$-prescription. In kinematic space it translates into giving a small negative imaginary part to all the energies, making the wavefunction coefficients well-defined for any value of their real part along the real axis. Unitarity is instead encoded into a non-convex part of the cosmological polytope, which we name optical polytope. The cosmological optical theorem emerges as the equivalence between a specific polytope subdivision of the optical polytope and its triangulations, each of which provides different cutting rules. The flat-space optical theorem instead emerges from the non-convexity of the optical polytope. On the more mathematical side, we provide two definitions of this non-convex geometry, none of them based on the idea of the non-convex geometry as a union of convex ones.

hep-th↗

arXiv2024

The Geometry of Cosmological Correlators

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 {\it 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.

hep-th↗

arXiv2022

Wavefunctionals/S-matrix techniques in de Sitter

Flat-space physics is highly constrained by basic principles such as Lorentz invariance, locality, unitarity and causality. This is neatly seen in the structure of scattering amplitudes. For processes occurring in an expanding background we do not have the same level of understanding, not even in the case of de Sitter space. In this talk I provide a brief account of some of the recent efforts which aim to fill this gap. I will discuss some novel results in the understanding of the analytic structure of the Bunch-Davies wavefunction(al) of the universe in the perturbative regime, some fundamental constraints that it ought to satisfy as well as some general insights coming from an underlying combinatorial structure.

hep-th↗

arXiv2022

Amplitudes meet Cosmology: A (Scalar) Primer

We review the most recent progress in our understanding of quantum mechanical observables in cosmology in the perturbative regime. It relies on an approach that considers them directly as functions of the data at the space-like boundary at future infinity prescinding from the explicit time evolution. It takes inspiration from the on-shell formulation of perturbative scattering amplitudes developed in the past 20 years: starting with the requirement of consistency with some fundamental principles such as causality, unitarity and locality, it provides different ways of phrasing and extracting predictions. In this review, we aim to provide a pedagogical treatment of the most recent insights about the analytic structure of the perturbative quantum mechanical observables in cosmology, its relation to fundamental principles as well as physical processes, and how such observables and their features emerge from novel well-defined mathematical objects with their own first principle definition. The review is divided in three parts: Part 0 discusses the definition of quantum mechanical observables in cosmology and some general principles; Part I reviews the boundary approach to the analysis and computation of the perturbative wavefunction of the universe; Part II provides an introduction to the combinatorial-geometrical description of cosmological processes in terms of cosmological polytopes.

hep-th↗

arXiv2022

Physical Representations for Scattering Amplitudes and the Wavefunction of the Universe

The way we organise perturbation theory is of fundamental importance both for computing the observables of relevance and for extracting fundamental physics out of them. If on one hand the different ways in which the perturbative observables can be written make manifest different features (e.g. symmetries as well as principles such as unitarity, causality and locality), on the other hand precisely demanding that some concrete features are manifest lead to different ways of organising perturbation theory. In the context of flat-space scattering amplitudes, a number of them are already known and exploited, while much less is known for cosmological observables. In the present work, we show how to systematically write down both the wavefunction of the universe and the flat-space scattering amplitudes, in such a way that they manifestly show physical poles only. We make use of the invariant definition of such observables in terms of {\it cosmological polytopes} and their {\it scattering facet}. In particular, we show that such representations correspond to triangulations of such objects through hyperplanes identified by the intersection of their facets outside of them. All possible triangulations of this type generate the different representations. This allows us to provide a general proof for the conjectured all-loop causal representation of scattering amplitudes. Importantly, all such representations can be viewed as making explicit a subset of compatible singularities, and our construction provides a way to extend Steinmann relations to higher codimension singularities for both the flat-space scattering amplitudes and the cosmological wavefunction.

hep-th↗

arXiv2020

Steinmann Relations and the Wavefunction of the Universe

The physical principles of causality and unitarity put strong constraints on the analytic structure of the flat-space S-matrix. In particular, these principles give rise to the Steinmann relations, which require that the double discontinuities of scattering amplitudes in partially-overlapping momentum channels vanish. Conversely, at cosmological scales, the imprint of causality and unitarity is in general less well understood---the wavefunction of the universe lives on the future space-like boundary, and has all time evolution integrated out. In the present work, we show how the flat-space Steinmann relations emerge from the structure of the wavefunction of the universe, and derive similar relations that apply to the wavefunction itself. This is done within the context of scalar toy models whose perturbative wavefunction has a first-principles definition in terms of cosmological polytopes. In particular, we use the fact that the scattering amplitude is encoded in the scattering facet of cosmological polytopes, and that cuts of the amplitude are encoded in the codimension-one boundaries of this facet. As we show, the flat-space Steinmann relations are thus implied by the non-existence of codimension-two boundaries at the intersection of the boundaries associated with pairs of partially-overlapping channels. Applying the same argument to the full cosmological polytope, we also derive Steinmann-type constraints that apply to the full wavefunction of the universe. These arguments show how the combinatorial properties of cosmological polytopes lead to the emergence of flat-space causality in the S-matrix, and provide new insights into the analytic structure of the wavefunction of the universe.

hep-th↗

arXiv2020

Positive Geometries and Differential Forms with Non-Logarithmic Singularities I

Positive geometries encode the physics of scattering amplitudes in flat space-time and the wavefunction of the universe in cosmology for a large class of models. Their unique canonical forms, providing such quantum mechanical observables, are characterised by having only logarithmic singularities along all the boundaries of the positive geometry. However, physical observables have logarithmic singularities just for a subset of theories. Thus, it becomes crucial to understand whether a similar paradigm can underlie their structure in more general cases. In this paper we start a systematic investigation of a geometric-combinatorial characterisation of differential forms with non-logarithmic singularities, focusing on projective polytopes and related meromorphic forms with multiple poles. We introduce the notions of covariant forms and covariant pairings. Covariant forms have poles only along the boundaries of the given polytope; moreover, their leading Laurent coefficients along any of the boundaries are still covariant forms on the specific boundary. Whereas meromorphic forms in covariant pairing with a polytope are associated to a specific (signed) triangulation, in which poles on spurious boundaries do not cancel completely, but their order is lowered. These meromorphic forms can be fully characterised if the polytope they are associated to is viewed as the restriction of a higher dimensional one onto a hyperplane. The canonical form of the latter can be mapped into a covariant form or a form in covariant pairing via a covariant restriction. We show how the geometry of the higher dimensional polytope determines the structure of these differential forms. Finally, we discuss how these notions are related to Jeffrey-Kirwan residues and cosmological polytopes.

hep-th↗

arXiv2019

Cosmological Polytopes and the Wavefuncton of the Universe for Light States

We extend the investigation of the structure of the late-time wavefunction of the universe to a class of toy models of scalars with time-dependent masses and polynomial couplings, which contains general massive scalars in FRW cosmologies. We associate a universal integrand to each Feynman diagram contributing to the wavefunction of the universe. For certain (light) masses, such an integrand satisfies recursion relations involving differential operators, connecting states with different masses and having, as a seed, the massless scalar (which describes a conformally coupled scalar as a special case). We show that it is a degenerate limit of the canonical form of a generalisation of the cosmological polytopes describing the wavefunction for massless scalars. Intriguingly, the flat-space scattering amplitude appears as a higher codimension face: it is encoding the leading term in the Laurent expansion as the total energy is taken to zero, with the codimension of the face providing the order of the total energy pole. The same connection between the other faces and the Laurent expansion coefficients holds for the other singularities of the wavefunction of the universe, all of them connectable to flat-space processes. As the degenerate limit is taken, some of the singularities of the canonical form of the polytope collapse onto each other generating higher order poles. Finally, we consider the mass as a perturbative coupling, showing that the contribution to the wavefunction coming from graphs with mass two-point couplings can be identified with a degenerate limit of the canonical form of the cosmological polytope, if the perturbative expansion is done around the conformally coupled state; or as double degenerate limit of the canonical form of the extension of the cosmological polytopes introduced in the present paper, if the perturbative expansion is done around minimally coupled states.

hep-th↗

arXiv2018

From the flat-space S-matrix to the Wavefunction of the Universe

The physical information encoded in the cosmological late-time wavefunction of the universe is tied to its singularity structure and its behaviour as such singularities are approached. One important singularity is identified by the vanishing of the total energy, where the wavefunction reduces to the physics of scattering in flat space. In this paper, we discuss the behaviour of the perturbative wavefunction as its other singularities are approached and the role played by the flat-space scattering, in the simplified context of the class of toy models admitting a first principle definition in terms of cosmological polytopes. The problems then translates into the analysis of the structure of its facets, one of which -- the scattering facet -- beautifully encodes the flat-space S-matrix. We show that all the boundaries of the cosmological polytope encode information about the flat-space physics. In particular, a subset of its facets turns out to have a similar structure as the scattering facet, with the vertices which can be grouped together to form lower dimensional scattering facets. The other facets admit one (and only one) triangulation in terms of products of lower dimensional scattering facets. As a consequence, the whole perturbative wavefunction can be represented as a sum of product of flat-space scattering amplitudes. Finally, we turn the table around and ask whether the knowledge of the flat-space scattering amplitudes suffices to reconstruct the wavefunction of the universe. We show that, at least for our class of toy models, this is indeed the case at tree level if we are also provided with a subset of symmetries that the wavefunction ought to satisfy. Once the tree cosmological polytopes are reconstructed, the loop ones can be obtained as a particular projection of them.

hep-th↗

arXiv2018

On the Emergence of Lorentz Invariance and Unitarity from the Scattering Facet of Cosmological Polytopes

The concepts of Lorentz invariance of local (flat space) physics, and unitarity of time evolution and the S-matrix, are famously rigid and robust, admitting no obvious consistent theoretical deformations, and confirmed to incredible accuracy by experiments. But neither of these notions seem to appear directly in describing the spatial correlation functions at future infinity characterizing the "boundary" observables in cosmology. How then can we see them emerge as {\it exact} concepts from a possible ab-initio theory for the late-time wavefunction of the universe? In this letter we examine this question in a simple but concrete setting, for the perturbative wavefunction in a class of scalar field models where an ab-initio description of the wavefunction has been given by "cosmological polytopes". Singularities of the wavefunction are associated with facets of the polytope. One of the singularities -- corresponding to the "total energy pole" -- is well known to be associated with the flat-space scattering amplitude. We show how the combinatorics and geometry of this {\it scattering facet} of the cosmological polytope straightforwardly leads to the emergence of Lorentz invariance and unitarity for the S-matrix. Unitarity follows from the way boundaries of the scattering facet factorize into products of lower-dimensional polytopes, while Lorentz invariance follows from a contour integral representation of the canonical form, which exists for any polytope, specialized to cosmological polytopes.

hep-th↗

arXiv2017

Cosmological Polytopes and the Wavefunction of the Universe

We present a connection between the physics of cosmological time evolution and the mathematics of positive geometries, roughly analogous to similar connections seen in the context of scattering amplitudes. We consider the wavefunction of the universe in a class of toy models of conformally coupled scalars (with non-conformal interactions) in FRW cosmologies. The contribution of each Feynman diagram to the wavefunction of the universe is associated with a certain universal rational integrand, which we identify as the canonical form of a "cosmological polytope", which have an independent, intrinsic definition, making no reference to physics. The singularity structure of the wavefunction for this model of scalars is common to all theories, and is geometrized by the cosmological polytope. Natural triangulations of the polytope reproduce the path-integral and "old-fashioned perturbation theory" representations of the wavefunction, and we also find new representations of the wavefunction with no extant physical interpretation. We show in suitable examples how symmetries of the cosmological polytope descend to symmetries of the wavefunction, (such as conformal invariance). In cases such as $ϕ^3$ theory in $dS_4$, the final wavefunction obtained from integration of the rational functions gives rise to polylogarithms associated with every graph. We give an explicit expression for the symbol of these polylogs, which record the geometry of sequential projections of the cosmological polytope.

hep-th↗

arXiv2017

An SLE approach to four dimensional black hole microstate entropy

In this note, we model the Bekenstein-Hawking entropy of a four dimensional extremal black hole in terms of classifying particles moving in its near horizon AdS_2 geometry. We use the framework of SLE curves in AdS_2 to classify these particle trajectories in terms of their boundary conditions. These turn out to be related to singular vectors in two-dimensional conformal quantum gravity theory in AdS_2 and the dynamics of these particles are governed by the Hamiltonians of the integrable Calogero-like models, for these boundary conditions. We use this classification to count the leading order Bekenstein-Hawking entropy of the black hole and arrive at a first principle microscopic computation of black hole degeneracy.

hep-th↗

arXiv2016

On-shell diagrams and the geometry of planar N < 4 SYM theories

We continue the discussion of the decorated on-shell diagrammatics for planar N < 4 Supersymmetric Yang-Mills theories started in arXiv:1510.03642. In particular, we focus on its relation with the structure of varieties on the Grassmannian. The decoration of the on-shell diagrams, which physically keeps tracks of the helicity of the coherent states propagating along their edges, defines new on-shell functions on the Grassmannian and can introduce novel higher-order singularities, which graphically are reflected into the presence of helicity loops in the diagrams. These new structures turn out to have similar features as in the non-planar case: the related higher-codimension varieties are identified by either the vanishing of one (or more) Plucker coordinates involving at least two non-adjacent columns, or new relations among Plucker coordinates. A distinctive feature is that the functions living on these higher-codimenson varieties can be thought of distributionally as having support on derivative delta-functions. After a general discussion, we explore in some detail the structures of the on-shell functions on Gr(2,4) and Gr(3,6) on which the residue theorem allows to obtain a plethora of identities among them.

hep-th↗

arXiv2015

On-shell diagrammatics and the perturbative structure of planar gauge theories

We discuss the on-shell diagrammatic representation of theories less special than maximally supersymmetric Yang-Mills. In particular, we focus on planar $\mathcal{N}\,\le\,2$ gauge theories, including pure Yang-Mills. For such a class of theories, the on-shell diagrammatics is endowed with a decoration which carries the information on the helicity of the coherent states. In the first part of the paper we extensively discuss the properties of this decorated diagrammatics. Particular relevance have the helicity flows that the decoration induces on the diagrams, which allows to identify the different classes of singularities and, consequentely, the singularity structure of the on-shell processes. The second part of the paper establishes a link between the decorated on-shell diagrammatics and the scattering amplitudes for the theories under examination. We prove that an all-loop recursion relation at integrand level holds also for $\mathcal{N}\,=\,1,\,2$, while for $\mathcal{N}\,=\,0$ we are able to set up a preliminary analysis at one loop. In both supersymmetric and non-supersymmetric case, the treatment of the forward limit is subtle. We provide a fully on-shell analysis of it which is crucial for the proof of the all-loop recursion relation and for the analysis of pure Yang-Mills.

hep-th↗