The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities (1811.00024v3)

Abstract: Scattering amplitudes at weak coupling are highly constrained by Lorentz invariance, locality and unitarity, and depend on model details only through coupling constants and particle content. In this paper, we develop an understanding of inflationary correlators which parallels that of flat-space scattering amplitudes. Specifically, we study slow-roll inflation with weak couplings to extra massive particles, for which all correlators are controlled by an approximate conformal symmetry on the boundary of the spacetime. After classifying all possible contact terms in de Sitter space, we derive an analytic expression for the four-point function of conformally coupled scalars mediated by the tree-level exchange of massive scalars. Conformal symmetry implies that the correlator satisfies a pair of differential equations with respect to spatial momenta, encoding bulk time evolution in purely boundary terms. The absence of unphysical singularities completely fixes this correlator. A spin-raising operator relates it to the correlators associated with the exchange of particles with spin, while weight-shifting operators map it to the four-point function of massless scalars. We explain how these de Sitter four-point functions can be perturbed to obtain inflationary three-point functions. We reproduce many classic results in the literature and provide a complete classification of all inflationary three- and four-point functions arising from weakly broken conformal symmetry. The inflationary bispectrum associated with the exchange of particles with arbitrary spin is completely characterized by the soft limit of the simplest scalar-exchange four-point function of conformally coupled scalars and a series of contact terms. Finally, we demonstrate that the inflationary correlators contain flat-space scattering amplitudes via a suitable analytic continuation of the external momenta.

• The paper presents a novel bootstrap method that derives inflationary correlators purely from symmetry principles and singularity constraints, avoiding conventional Lagrangian dynamics.
• It utilizes conformal symmetry in de Sitter space to construct four-point functions and uncover non-Gaussian features that signal massive particle exchanges.
• Results include both perturbative and non-perturbative solutions, revealing analytic continuations to flat-space limits that guide templates for observational cosmology.

Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities

In the paper of cosmological correlation functions, the paper "The Cosmological Bootstrap: Inflationary Correlators from Symmetries and Singularities" introduces a method akin to the bootstrapping techniques used in the analysis of scattering amplitudes in flat space-time. The authors present a comprehensive framework for determining cosmological correlators purely from considerations of symmetries and the absence of unphysical singularities, without a reliance on Lagrangian dynamics or explicit time evolution.

Key Concepts and Methodology

1. Conformal Symmetries in de Sitter Space: The main focus is on the interplay between the conformal symmetries of de Sitter (dS) space and the analytic structure of inflationary correlators. By considering the boundary at future infinity, where these correlate are conformally invariant, the authors parallel the well-trodden route laid out by the flat-space S-matrix approach.
2. Four-Point Functions and Analytic Structure: The paper systematically builds four-point correlators of scalar fields in dS space, leveraging symmetries to reduce the problem to simpler components. Contact interactions in dS space manifest as poles in the total energy, analogous to the analytic terms in scattering amplitudes. Meanwhile, tree-level exchanges are described by differential equations connecting the internal dynamics of the theory to the behavior on the boundary.
3. Treatment of Spinning Particles: The paper extends beyond scalar exchanges to include massive particles with spin, using spin-raising operators to generate these more complex correlation functions from scalar ones. This elegantly simplifies the inclusion of particles with more intricate angular momentum properties within the inflationary universe.
4. Non-Gaussian Features and Particle Signatures: By examining the properties of correlators in specific momentum limits, the authors elucidate how observables may encode information about massive particle spectra during inflation. Oscillatory features in these correlators hint at particle masses and spins, akin to resonance peaks in scattering experiments.
5. Inflationary Perturbations and Real Observables: The paper computes inflationary bispectra, paying particular attention to slow-roll parameters and their interplay with the previously derived dS correlators. These insights bridge the theoretical constructs with potential observational signatures in CMB and large-scale structure surveys.

Results and Implications

The paper presents several significant results:

• Formal Solutions: The authors derive both perturbative series and non-perturbative corrections for four-point functions, showing how singularities inform the physical spectrum of produced particles in the early universe.
• Flat-Space Limit Connections: A notable achievement of this work is demonstrating that the correlators possess a particular singularity when analytically continued to the flat-space limit, closely resembling known scattering amplitudes.
• Inflationary Templates: The research culminates in a proposed template of inflationary non-Gaussianity that could guide observational efforts: identifying particle signatures through characteristic oscillatory behavior provides a unique window into high-energy physics scales otherwise inaccessible.

Future Directions

This cosmological correlator bootstrap opens various avenues for advancing both theoretical understanding and practical applications:

• Higher-Order Corrections: Extending the framework to encompass one-loop or higher-order corrections could yield richer structures and refine the understanding of inflationary particle processes.
• Generalizing Beyond Weak Conformal Breaking: Exploring models with stronger deviations from conformal symmetry could provide insights relevant to scenarios beyond the minimal inflationary paradigm, possibly intersecting with effective field theories predicting large non-Gaussianities.
• Geometric Interpretations and Dualities: Inspired by developments in scattering amplitude research, further work may uncover geometric structures or dual descriptions that encapsulate these correlators' behavior, enhancing the analytical power available.

In summary, this paper presents a novel approach to inflationary perturbation theory, channeling the robust principles of the amplitude bootstrap to unravel the complex but structured nature of cosmological correlators, potentially ushering in a new era of connecting high-energy physics with cosmological observations.