Massless spinning correlators in cosmology are extremely complicated. In contrast, the scattering amplitudes of massless particles with spin are very simple. We propose that the reason for the unreasonable complexity of these correlators lies in the use of inconvenient kinematic variables. For example, in de Sitter space, consistency with unitarity and the background isometries imply that the correlators must be conformally covariant and also conserved. However, the commonly used kinematic variables for correlators do not make all of these properties manifest. In this paper, we introduce twistor space as a powerful way to satisfy all kinematic constraints. We show that conformal correlators of conserved currents can be written as twistor integrals, where the conservation condition translates into holomorphicity of the integrand. The functional form of the twistor-space correlators is very simple and easily bootstrapped. For the case of two- and three-point functions, we verify explicitly that this reproduces known results in embedding space. We also perform a half-Fourier transform of the twistor-space correlators to obtain their counterparts in momentum space. We conclude that twistors provide a promising new avenue to study conformal correlation functions that exposes their hidden simplicity.

Netherlands

Taiwan

Germany

Daniel Baumann

Switzerland

Grégoire Mathys

Italy

Guilherme Pimentel

Facundo Rost

A new twist on spinning (A)dS correlators

Effective field theories in flat space and in anti-de Sitter space are constrained by causality and unitarity, often in the form of positivity bounds. Similar bounds have been harder to demonstrate in cosmological backgrounds, where the roles of unitarity and causality are more obscure. Fortunately, the expansion of the universe ensures that late-time cosmological correlators are effectively classical and the role of unitarity is played by classical statistical inequalities. For multi-field inflation, the resulting positivity constraints have long been known in terms of the Suyama-Yamaguchi inequality. In this paper, we demonstrate that similar statistical bounds imply nontrivial constraints for massive fields in the early universe. We show that any real anomalous dimensions for principal series fields in de Sitter space must be positive. We also derive a limit on the amplitude of particular oscillatory signals from inflation, including those arising in cosmological collider physics. Finally, we demonstrate that these constraints manifest themselves directly in the two-point statistics of matter and galaxies that will be measured in upcoming surveys.

United States

Daniel Green

Yiwen Huang

Chia-Hsien Shen

Positivity from Cosmological Correlators

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.

Wei-Ming Chen

Carlos Duaso Pueyo

Austin Joyce

Hayden Lee

Linking the singularities of cosmological correlators

A key insight of the bootstrap approach to cosmological correlations is the fact that all correlators of slow-roll inflation can be reduced to a unique building block — the four-point function of conformally coupled scalars, arising from the exchange of a massive scalar. Correlators corresponding to the exchange of particles with spin are then obtained by applying a spin-raising operator to the scalar-exchange solution. Similarly, the correlators of massless external fields can be derived by acting with a suitable weight-raising operator. In this paper, we present a systematic and highly streamlined derivation of these operators (and their generalizations) using tools of conformal field theory. Our results greatly simplify the theoretical foundations of the cosmological bootstrap program.

Carlos Pueyo

The cosmological bootstrap: weight-shifting operators and scalar seeds

Scattering amplitudes at weak coupling are highly constrained by Lorentz invariance, locality and unitarity, and depend on model details only through coupling constants and the particle content of the theory. For example, four-particle amplitudes are analytic for contact interactions and have simple poles with appropriately positive residues for tree-level exchange. 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 correlation functions are controlled by an approximate conformal symmetry on the boundary of the spacetime. After systematically 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 (and the correct normalization of physical ones) completely fixes this correlator. Moreover, 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. Using our formalism, we reproduce many classic results in the literature, such as the three-point function of slow-roll inflation, and provide a complete classification of all inflationary three- and four-point functions arising from weakly broken conformal symmetry. Remarkably, 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, which can also be directly connected with the signals for particle production seen in the squeezed limit.

Nima Arkani-Hamed

The cosmological bootstrap: inflationary correlators from symmetries and singularities

Sum rules connecting low-energy observables to high-energy physics are an interesting way to probe the mechanism of inflation and its ultraviolet origin. Unfortunately, such sum rules have proven difficult to study in a cosmological setting. Motivated by this problem, we investigate a precise analogue of inflation in anti-de Sitter spacetime, where it becomes dual to a slow renormalization group flow in the boundary quantum field theory. This dual description provides a firm footing for exploring the constraints of unitarity, analyticity, and causality on the bulk effective field theory. We derive a sum rule that constrains the bulk coupling constants in this theory. In the bulk, the sum rule is related to the speed of radial propagation, while on the boundary, it governs the spreading of nonlocal operators. When the spreading speed approaches the speed of light, the sum rule is saturated, suggesting that the theory becomes free in this limit. We also discuss whether similar results apply to inflation, where an analogous sum rule exists for the propagation speed of inflationary fluctuations.

Thomas Hartman

Dynamical constraints on RG flows and cosmology

The representation theory of de Sitter space allows for a category of partially massless particles which have no flat space analog, but could have existed during inflation. We study the couplings of these exotic particles to inflationary perturbations and determine the resulting signatures in cosmological correlators. When inflationary perturbations interact through the exchange of these fields, their correlation functions inherit scalings that cannot be mimicked by extra massive fields. We discuss in detail the squeezed limit of the tensor-scalar-scalar bispectrum, and show that certain partially massless fields can violate the tensor consistency relation of single-field inflation. We also consider the collapsed limit of the scalar trispectrum, and find that the exchange of partially massless fields enhances its magnitude, while giving no contribution to the scalar bispectrum. These characteristic signatures provide clean detection channels for partially massless fields during inflation.

Garrett Goon

Hong Kong

Partially massless fields during inflation

We study the imprints of massive particles with spin on cosmological correlators. Using the framework of the effective field theory of inflation, we classify the couplings of these particles to the Goldstone boson of broken time translations and the graviton. We show that it is possible to generate observable non-Gaussianity within the regime of validity of the effective theory, as long as the masses of the particles are close to the Hubble scale and their interactions break the approximate conformal symmetry of the inflationary background. We derive explicit shape functions for the scalar and tensor bispectra that can serve as templates for future observational searches.

United Kingdom

Non-Gaussianity as a particle detector

In this paper, we explore a novel observational signature of gravitational corrections during slow-roll inflation. We study the coupling of the inflaton field to higher-curvature tensors in models with a minimal breaking of conformal symmetry. In that case, the most general correction to the tensor two-point function is captured by a coupling to the square of the Weyl tensor. We show that these scenarios lead to a correction to the tilt of the tensor power spectrum and hence a violation of the tensor consistency condition. We arrive at the same conclusion through an analysis in conformal perturbation theory.

High-scale inflation and the tensor tilt

Observations of the cosmic microwave background do not yet determine whether inflation was driven by a slowly-rolling scalar field or involved another physical mechanism. In this paper we discuss the prospects of using the power spectra of scalar and tensor modes to probe the nature of inflation. We focus on the leading modification to the slow-roll dynamics, which entails a sound speed cs for the scalar fluctuations. We derive analytically a lower bound on cs in terms of a given tensor-to-scalar ratio r, taking into account the difference in the freeze-out times between the scalar and tensor modes. We find that any detection of primordial B-modes with rÂ &gt;Â 0.01 implies a lower bound on cs that is stronger than the bound derived from the absence of non-Gaussianity in the Planck data. For rÂ â³Â 0.1, the bound would be tantalizingly close to a critical value for the sound speed, (cs)âÂ =Â 0.47 (corresponding to (fNLequil)âÂ =Â -0.93), which we show serves as a threshold for non-trivial dynamics beyond slow-roll. We also discuss how an order-one level of equilateral non-Gaussianity is a natural observational target for other extensions of the canonical paradigm.