Search

By directly probing the initial conditions of our universe, cosmological surveys offer us a unique observational handle on quantum field theory in curved spacetime with dynamical gravity and might even allow us to glean information about a full theory of quantum gravity. Here we report on recent progress to study the natural observables in the problem, namely cosmological correlators. After setting the stage, we review results from three different approaches. First, we present the in-out formalism as an interesting alternative to the well-known in-in formalism and stress some of its advantages, such as the derivation of recursion relations, correlators cutting rules and a proposal for a de Sitter scattering matrix. Second, we tackle the important open problem of constructing effective theories in curved spacetime, which generally requires an open quantum system approach. Third, we provide an executive summary of general properties of the field-theoretic wavefunction that follow from symmetries, unitarity, causality and locality. We describe how these properties can be leveraged to bootstrap all tree-level results and we discuss loop contributions. 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., Comment: A chapter to be published by Springer Lecture Notes in Physics

The field theoretic wavefunction in cosmological spacetimes has received much attention as a fundamental object underlying the generation of primordial perturbations in our universe. Assuming an initial Bunch-Davies state, unitary time evolution implies an infinite set of cutting rules for the wavefunction to all orders in perturbation theory, collectively known as the cosmological optical theorem. In this work, we generalise these results to the case of Bogoliubov initial states, accounting for both parity-even and parity-odd interactions. We confirm our findings in a few explicit examples, assuming IR-finite interactions. In these examples, we preserve scale invariance by adiabatically turning on interactions in the infinite past rather than imposing a Bogoliubov state at some finite initial time. Finally, we give a prescription for computing Bogoliubov wavefunction coefficients from the corresponding Bunch-Davies coefficients for both n-point contact and four-point exchange diagrams.

In our quest to understand the generation of cosmological perturbations, we face two serious obstacles: we do not have direct information about the environment experienced by primordial perturbations during inflation, and our observables are practically limited to correlators of massless fields, heavier fields and derivatives decaying exponentially in the number of e-foldings. The flexible and general framework of open systems has been developed precisely to face similar challenges. Building on previous work, we develop a Schwinger-Keldysh path integral description for an open effective field theory of inflation, describing the possibly dissipative and non-unitary evolution of the Goldstone boson of time translations interacting with an unspecified environment, under the key assumption of locality in space and time. Working in the decoupling limit, we study the linear and interacting theory in de Sitter and derive predictions for the power spectrum and bispectrum that depend on a finite number of effective couplings organised in a derivative expansion. The smoking gun of interactions with the environment is an enhanced but finite bispectrum close to the folded kinematical limit. We demonstrate the generality of our approach by matching our open effective theory to an explicit model. Our construction provides a standard model to simultaneously study phenomenological predictions as well as quantum information aspects of the inflationary dynamics., Comment: 56 pages without appendices (81 pages in total), 12 figures, 2 tables

Cosmological correlators, the natural observables of the primordial universe, have been extensively studied in the past two decades using the in-in formalism pioneered by Schwinger and Keldysh for the study of dissipative open systems. Ironically, most applications in cosmology have focused on non-dissipative closed systems. We show that, for non-dissipative systems, correlators can be equivalently computed using the in-out formalism with the familiar Feynman rules. In particular, the myriad of in-in propagators is reduced to a single (Feynman) time-ordered propagator and no sum over the labelling of vertices is required. In de Sitter spacetime, this requires extending the expanding Poincar\'e patch with a contracting patch, which prepares the bra from the future. Our results are valid for fields of any mass and spin but assuming the absence of infrared divergences. We present three applications of the in-out formalism: a representation of correlators in terms of a sum over residues of Feynman propagators in the energy-momentum domain; an algebraic recursion relation that computes Minkowski correlators in terms of lower order ones; and the derivation of cutting rules from Veltman's largest time equation, which we explicitly develop and exemplify for two-vertex diagrams to all loop orders. The in-out formalism leads to a natural definition of a de Sitter scattering matrix, which we discuss in simple examples. Remarkably, we show that our scattering matrix satisfies the standard optical theorem and the positivity that follows from it in the forward limit., Comment: 38+10 pages, 4 figures. v2: added clarifying comments and references

The squeezed limit of the primordial curvature bispectrum is an extremely sensitive probe of new physics and encodes information about additional fields active during inflation such as their masses and spins. In the conventional setup, additional fields are stable with a positive mass squared, and hence induce a decreasing signal in the squeezed limit, making a detection challenging. Here we consider a scalar field that is temporarily unstable by virtue of a transient tachyonic mass, and we construct models in which it is embedded consistently within inflation. Assuming IR-finite couplings between the tachyon and the inflaton, we find an exchange bispectrum with an enhanced long-short scale coupling that grows in the squeezed limit parametrically faster than local non-Gaussianity. Our approximately scale-invariant signal can be thought of as a cosmological tachyon collider. In a sizeable region of parameter space, the leading constraint on our signal comes from the cross correlation of $\mu$-type spectral distortions and temperature anisotropies of the microwave background, whereas temperature and polarization bispectra are less sensitive probes. By including anisotropic spectral distortions in the analysis, future experiments such as CMB-S4 will further reduce the allowed parameter space., Comment: 31 pages, 5 figures

Recent progress has revealed a number of constraints that cosmological correlators and the closely related field-theoretic wavefunction must obey as a consequence of unitarity, locality, causality and the choice of initial state. When combined with symmetries, namely homogeneity, isotropy and scale invariance, these constraints enable one to compute large classes of simple observables, an approach known as (boostless) cosmological bootstrap. Here we show that it is possible to relax the restriction of scale invariance, if one retains a discrete scaling subgroup. We find an infinite class of solutions to the weaker bootstrap constraints and show that they reproduce and extend resonant non-Gaussianity, which arises in well-motivated models such as axion monodromy inflation. We find no evidence of the new non-Gaussian shapes in the Planck data. Intriguingly, our results can be re-interpreted as a deformation of the scale-invariant case to include a complex order of the total energy pole, or more evocatively interactions with a complex number of derivatives. We also discuss for the first time IR-divergent resonant contributions and highlight an inconsequential inconsistency in the previous literature., Comment: 33 pages, 4 figures; v2: published version, minor corrections and references added

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., Comment: 34 pages, 4 figures; added appendix C, typos corrected

The wavefunction in quantum field theory is an invaluable tool for tackling a variety of problems, including probing the interior of Minkowski spacetime and modelling boundary observables in de Sitter spacetime. Here we study the analytic structure of wavefunction coefficients in Minkowski as a function of their kinematics. We introduce an off-shell wavefunction in terms of amputated time-ordered correlation functions and show that it is analytic in the complex energy plane except for possible singularities on the negative real axis. These singularities are determined to all loop orders by a simple energy-conservation condition. We confirm this picture by developing a Landau analysis of wavefunction loop integrals and corroborate our findings with several explicit calculations in scalar field theories. This analytic structure allows us to derive new UV/IR sum rules for the wavefunction that fix the coefficients in its low-energy expansion in terms of integrals of discontinuities in the corresponding UV-completion. In contrast to the analogous sum rules for scattering amplitudes, the wavefunction sum rules can also constrain total-derivative interactions. We explicitly verify these new relations at one-loop order in simple UV models of a light and a heavy scalar. Our results, which apply to both Lorentz invariant and boost-breaking theories, pave the way towards deriving wavefunction positivity bounds in flat and cosmological spacetimes., Comment: 47+31 pages, 9 figures, 3 tables

We compute the tree-level late-time graviton four-point correlation function, and the related quartic wavefunction coefficient, for Einstein gravity in de Sitter spacetime. We derive this result in several ways: by direct calculation, using the in-in formalism and the wavefunction of the universe; by a heuristic derivation leveraging the flat space wavefunction coefficient; and by using the boostless cosmological bootstrap, in particular the combination of the cosmological optical theorem, the amplitude limit, and the manifestly local test. We find agreement among the different methods., Comment: 46 pages

We derive a set of no-go theorems and yes-go examples for the parity-odd primordial trispectrum of curvature perturbations. We work at tree-level in the decoupling limit of the Effective Field Theory of Inflation and assume scale invariance and a Bunch-Davies vacuum. We show that the parity-odd scalar trispectrum vanishes in the presence of any number of scalar fields with arbitrary mass and any parity-odd scalar correlator vanishes in the presence of any number of spinning fields with massless de Sitter mode functions, in agreement with the findings of Liu, Tong, Wang and Xianyu [1]. The same is true for correlators with an odd number of conformally-coupled external fields. We derive these results using both the (boostless) cosmological bootstrap, in particular the Cosmological Optical Theorem, and explicit perturbative calculations. We then discuss a series of yes-go examples by relaxing the above assumptions one at the time. In particular, we provide explicit results for the parity-odd trispectrum for (i) violations of scale invariance in single-clock inflation, (ii) the modified dispersion relation of the ghost condensate (non-Bunch-Davies vacuum), and (iii) interactions with massive spinning fields. Our results establish the parity-odd trispectrum as an exceptionally sensitive probe of new physics beyond vanilla inflation., Comment: 38 pages, 5 figures. v2 fixes typo in Eq. (2.8)

We summarize progress made in theoretical astrophysics and cosmology over the past decade and areas of interest for the coming decade. This Report is prepared as the TF09 "Astrophysics and Cosmology" topical group summary for the Theory Frontier as part of the Snowmass 2021 process., Comment: 57 pages

This white paper summarizes recent progress in the cosmological bootstrap, an approach to the study of the statistics of primordial fluctuations from consistency with unitarity, locality and symmetry assumptions. We review the key ideas of the bootstrap method, with an eye towards future directions and ambitions of the program. Focusing on recent progress involving de Sitter and quasi-de Sitter backgrounds, we highlight the role of singularities and unitarity in constraining the form of the correlators. We also discuss nonperturbative formulations of the bootstrap, connections to anti-de Sitter space, and potential implications for holography., Comment: 52 pages, 7 figures; Contribution to Snowmass 2021

Our understanding of quantum field theory rests largely on explicit and controlled calculations in perturbation theory. Because of this, much recent effort has been devoted to improve our grasp of perturbative techniques on cosmological spacetimes. While scattering amplitudes in flat space at tree level are obtained from simple algebraic operations, things are harder for cosmological observables. Indeed, computing cosmological correlation functions or the associated wavefunction coefficients requires evaluating a growing number of nested time integrals already at tree level, which is computationally challenging. Here, we present a new "differential" representation of the cosmological wavefunction in de Sitter spacetime that obviates this problem for a large class of phenomenologically relevant theories. Given any tree-level Feynman-Witten diagram, we give simple algebraic rules to write down a seed function and a differential operator that transforms it into the desired wavefunction coefficient for any scale-invariant, parity-invariant theory of massless scalars and gravitons with general boost-breaking interactions. In particular, this applies to large classes of phenomenologically relevant theories such as those described by the effective field theory of inflation or solid inflation. Trading nested bulk time integrals for derivatives on boundary kinematical data provides a great computational advantage, especially for processes involving many vertices., Comment: 41 pages

Gravitational interferometers and cosmological observations of the cosmic microwave background offer us the prospect to probe the laws of gravity in the primordial universe. To study and interpret these datasets we need to know the possible graviton non-Gaussianities. To this end, we derive the most general tree-level three-point functions (bispectra) for a massless graviton to all orders in derivatives, assuming scale invariance. Instead of working with explicit Lagrangians, we take a bootstrap approach and obtain our results using the recently derived constraints from unitarity, locality and the choice of vacuum. Since we make no assumptions about de Sitter boosts, our results capture the phenomenology of large classes of models such as the effective field theory of inflation and solid inflation. We present formulae for the infinite number of parity-even bispectra. Remarkably, for parity-odd bispectra, we show that unitarity allows for only a handful of possible shapes: three for graviton-graviton-graviton, three for scalar-graviton-graviton and one for scalar-scalar-graviton, which we bootstrap explicitly. These parity-odd non-Gaussianities can be large, for example in solid inflation, and therefore constitute a concrete and well-motivated target for future observations., Comment: 58 pages, 2 tables, 5 figures, added section on "Perturbativity, naturalness and strong coupling", matches JHEP version

Our understanding of quantum correlators in cosmological spacetimes, including those that we can observe in cosmological surveys, has improved qualitatively in the past few years. Now we know many constraints that these objects must satisfy as consequences of general physical principles, such as symmetries, unitarity and locality. Using this new understanding, we derive the most general scalar four-point correlator, i.e., the trispectrum, to all orders in derivatives for manifestly local contact interactions. To obtain this result we use techniques from commutative algebra to write down all possible scalar four-particle amplitudes without assuming invariance under Lorentz boosts. We then input these amplitudes into a contact reconstruction formula that generates a contact cosmological correlator in de Sitter spacetime from a contact scalar or graviton amplitude. We also show how the same procedure can be used to derive higher-point contact cosmological correlators. Our results further extend the reach of the boostless cosmological bootstrap and build a new connection between flat and curved spacetime physics., Comment: 31 pages

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., Comment: 36 + 18 pages, 4 figures

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., Comment: 48+11 pages, 5 figures

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 bootstrap tools to efficiently compute cosmological correlators/wavefunctions. 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., Comment: Published version

The observation of primordial correlators by cosmological surveys is a very promising avenue to probe high energies and the perturbative regime of quantum gravity. Hence, it is imperative that we understand how these observables are shaped by the pillars of fundamental physics, namely unitarity, locality and symmetries. To this end, we study the three-point correlators of gravitons and scalar curvature perturbations around a quasi de Sitter spacetime. We identify a set of Bootstrap Rules that fully fix the form of these correlators in the asymptotic future, i.e. at the boundary, and make no reference to bulk time evolution. Importantly, our Boostless Bootstrap accounts for the ubiquitous (spontaneous) breaking of de Sitter boosts caused by any inflationary background. We show how all bispectra involving gravitons in single-clock, canonical inflation can be easily derived in this approach. We also derive for the first time the scalar bispectrum in the Effective Field Theory of inflation to any order in derivatives. In many cases, our derivation is computationally simpler than the corresponding explicit calculation, and makes particularly transparent the implications of locality, the choice of vacuum, and the underlying symmetries., Comment: 34+10 pages, 1 figure

The unitarity of time evolution, or colloquially the conservation of probability, sits at the heart of our descriptions of fundamental interactions via quantum field theory. The implications of unitarity for scattering amplitudes are well understood, for example through the optical theorem and cutting rules. In contrast, the implications for in-in correlators in curved spacetime and the associated wavefunction of the universe, which are measured by cosmological surveys, are much less transparent. For fields of any mass in de Sitter spacetime with general local interactions, which need not be invariant under de Sitter isometries, we show that unitarity implies an infinite set of relations among the coefficients $ \psi_{n} $ of the wavefunction of the universe with $ n $ fields, which we name Cosmological Optical Theorem. For contact diagrams, our result dictates the analytic structure of $ \psi_{n} $ and strongly constrains its form. For example, any correlator with an odd number of conformally-coupled scalar fields and any number of massless scalar fields must vanish. For four-point exchange diagrams, the Cosmological Optical Theorem yields a simple and powerful relation between $ \psi_{3} $ and $ \psi_{4} $, or equivalently between the bispectrum and trispectrum. As explicit checks of this relation, we discuss the trispectrum in single-field inflation from graviton exchange and self-interactions. Moreover, we provide a detailed derivation of the relation between the total-energy pole of cosmological correlators and flat-space amplitudes. We provide analogous formulae for sub-diagram singularities. Our results constitute a new, powerful tool to bootstrap cosmological correlators., Comment: 37+19 pages, 4 figures

Poincar\'e invariance is a well-tested symmetry of nature and sits at the core of our description of relativistic particles and gravity. At the same time, in most systems Poincar\'e invariance is not a symmetry of the ground state and is hence broken spontaneously. This phenomenon is ubiquitous in cosmology where Lorentz boosts are spontaneously broken by the existence of a preferred reference frame in which the universe is homogeneous and isotropic. This motivates us to study scattering amplitudes without requiring invariance of the interactions under Lorentz boosts. In particular, using on-shell methods and assuming massless, relativistic and luminal particles of any spin, we show that the allowed interactions around Minkowski spacetime are severely constrained by unitarity and locality in the form of consistent factorization. The existence of an interacting massless spin-2 particle enforces (analytically continued) three-particle amplitudes to be Lorentz invariant, even those that do not involve a graviton, such as cubic scalar couplings. We conjecture this to be true for all n-particle amplitudes. Also, particles of spin S > 2 cannot self-interact nor can be minimally coupled to gravity, while particles of spin S > 1 cannot have electric charge. Given the growing evidence that free gravitons are well described by massless, luminal relativistic particles, our results imply that cubic graviton interactions in Minkowski must be those of general relativity up to a unique Lorentz-invariant higher-derivative correction of mass dimension 9. Finally, we point out that consistent factorization for massless particles is highly IR sensitive and therefore our powerful at-space results do not straightforwardly apply to curved spacetime., Comment: 73 pages. An extra assumption added and discussed

The space of inflationary models is vast, containing wide varieties of mechanisms, symmetries, and spectra of particles. Consequently, the space of observational signatures is similarly complex. Hence, it is natural to look for boundaries of the space of models and their signatures. In this paper, we explore the possible symmetries associated with the primordial cosmological perturbations and their correlators in the asymptotic future. Assuming the observed homogeneity, isotropy and (approximate) scale invariance, we prove three main results. First, correlation functions of scalar metric fluctuations are uniquely characterized by soft theorems and are free from ambiguity under field redefinitions. Second, whatever the particle content and interactions, when the standard soft theorems apply, invariance under de Sitter boosts (linearly realized conformal invariance) is only possible if all connected correlators vanish identically, i.e. if the theory is free. Third, conformal invariance is the largest set of linearly realized (bosonic) symmetries of the correlators of any single scalar, irrespectively of any soft theorems or particle content., Comment: 33 pages; v2: minor corrections

The effect of spatial curvature on primordial perturbations is controlled by $ \Omega_{K,0}/c_{s}^{2} $, where $ \Omega_{K,0} $ is today's fractional density of spatial curvature and $ c_{s} $ is the speed of sound during inflation. Here we study these effects in the limit $ c_{s}\ll 1 $. First, we show that the standard cosmological soft theorems in flat universes are violated in curved universes and the soft limits of correlators can have non-universal contributions even in single-clock inflation. This is a consequence of the fact that, in the presence of spatial curvature, there is a gap between the spectrum of residual diffeomorphisms and that of physical modes. Second, there are curvature corrections to primordial correlators, which are not scale invariant. We provide explicit formulae for these corrections to the power spectrum and the bispectrum to linear order in curvature in single-clock inflation. We show that the large-scale CMB anisotropies could provide interesting new constraints on these curvature effects, and therefore on $ \Omega_{K,0}/c_{s}^{2} $, but it is necessary to go beyond our linear-order treatment., Comment: 42 pages, 8 figures

We provide a complete classification of Poincar\'e-invariant scalar field theories with an enlarged set of classical symmetries to leading order in derivatives, namely for the so-called $P(X,\phi)$ theories, in two or more spacetime dimensions. We find only three possibilities: Dirac-Born-Infeld, Cuscuton and Scaling theories. The latter two classes of actions involve an arbitrary function of the scalar field. As an application, we use the scaling symmetry to derive an infinite set of constraints on the Wilsonian coefficients of the low-energy Effective Field Theory. Furthermore, we study the extension of these results to cosmological (FLRW) and (Anti-)de Sitter spacetimes. We find in particular that the Cuscuton action has a generic set of symmetries around any background spacetime that possesses Killing vector fields, while the DBI actions have well-known analogues that we summarize explicitly., Comment: 26 pages

Our current understanding of the Universe is established through the pristine measurements of structure in the cosmic microwave background (CMB) and the distribution and shapes of galaxies tracing the large scale structure (LSS) of the Universe. One key ingredient that underlies cosmological observables is that the field that sources the observed structure is assumed to be initially Gaussian with high precision. Nevertheless, a minimal deviation from Gaussianityis perhaps the most robust theoretical prediction of models that explain the observed Universe; itis necessarily present even in the simplest scenarios. In addition, most inflationary models produce far higher levels of non-Gaussianity. Since non-Gaussianity directly probes the dynamics in the early Universe, a detection would present a monumental discovery in cosmology, providing clues about physics at energy scales as high as the GUT scale., Comment: 5 pages + references; Submitted to the Astro2020 call for science white papers. This version: fixed author list

We derive cosmological soft theorems for solids coupled to gravity. To this end, we first derive all cosmological adiabatic modes for solids, which display the interesting novelty of non-vanishing anisotropic stresses on large scales. Then, from the corresponding symmetries of the action of perturbations we compute the leading order related soft theorems using the operator product expansion. For the scalar bispectrum, we re-derive the result that Maldacena's consistency relation is recovered only upon angular averaging over the long mode direction. In addition, we find theorems for soft tensor and vector perturbations. In passing, we also clarify the derivation of these soft theorems in gauges where no residual diffeomorphisms exist.

We present a complete classification of symmetric superfluids, namely shift-symmetric and Poincar\'{e} invariant scalar field theories that have an enlarged set of classically conserved currents at leading order in derivatives. These theories arise in the decoupling limit of the effective field theory of shift-symmetric, single-clock cosmologies and our results pick out all models with couplings fixed by additional symmetry. Remarkably, in $ D\geq 2 $ spacetime dimensions there are only two possibilities: the Dirac-Born-Infeld theory and Scaling Superfluids with Lagrangian $ (-\partial_{\mu}\phi\partial^{\mu}\phi)^{\alpha}$, for some real $ \alpha$. The scaling symmetry present for any $ \alpha $ is further enhanced to the full conformal group only for $ \alpha=D/2 $, and to infinitely many additional generators for $ \alpha=1/2 $. We discuss the stability of Scaling Superfluids and point out that all coupling constants are determined by the speed of sound., Comment: Replaced to match published version

Distortions of the Cosmic Microwave Background energy spectrum of the $\mu$ type are sensitive to the primordial power spectrum through the dissipation of curvature perturbations on scales $k\simeq 50$ - $10^4\,\mathrm{Mpc}^{-1}$. Their angular correlation with large-scale temperature anisotropies is then sensitive to the squeezed limit of the primordial bispectrum. For inflationary models obeying the single-field consistency relation, we show that the observed $\mu T$ angular correlation that would correspond to the local shape vanishes exactly. All leading non-primordial contributions, including all non-linear production and projection effects, are of the "equilateral shape", namely suppressed by $k^2/{\cal H}_f^2$, where ${\cal H}_f\simeq 10^{-1}\,{\rm Mpc}^{-1}$ is the Hubble radius at the end of the $\mu$-era. Therefore, these non-primordial contributions are orthogonal to a potential local primordial signal (e.g. from multi-field inflation). Moreover, they are very small in amplitude. Our results strengthen the position of $\mu$ distortions as the ultimate probe of local primordial non-Gaussianity., Comment: 34 pages (21+13), 2 figures, matches JCAP version

Long-short mode coupling during inflation, encoded in the squeezed bispectrum of curvature perturbations, induces a dependence of the local, small-scale power spectrum on long-wavelength perturbations, leading to a scale-dependent halo bias. While this scale dependence is absent in the large-scale limit for single-field inflation models that satisfy the consistency relation, certain models such as resonant non-Gaussianity show a peculiar behavior on intermediate scales. We reconsider the predictions for the halo bias in this model by working in Conformal Fermi Coordinates, which isolate the physical effects of long-wavelength perturbations on short-scale physics. We find that the bias oscillates with scale with an envelope similar to that of equilateral non-Gaussianity. Moreover, the bias shows a peculiar modulation with the halo mass. Unfortunately, we find that upcoming surveys will be unable to detect the signal because of its very small amplitude. We also discuss non-Gaussianity due to interactions between the inflaton and massive fields: our results for the bias agree with those in the literature., Comment: 27+15 pages, 6 figures, matches JCAP version

A shift symmetry is a ubiquitous ingredient in inflationary models, both in effective constructions and in UV-finite embeddings such as string theory. It has also been proposed to play a key role in certain Dark Energy and Dark Matter models. Despite the crucial role it plays in cosmology, the observable, model independent consequences of a shift symmetry are yet unknown. Here, assuming an exact shift symmetry, we derive these consequences for single-clock cosmologies within the framework of the Effective Field Theory of Inflation. We find an infinite set of relations among the otherwise arbitrary effective coefficients, which relate non-Gaussianity to their time dependence. For example, to leading order in derivatives, these relations reduce the infinitely many free functions in the theory to just a single one. Our Effective Theory of shift-symmetric cosmologies describes, among other systems, perfect and imperfect superfluids coupled to gravity and driven superfluids in the decoupling limit. Our results are the first step to determine observationally whether a shift symmetry is at play in the laws of nature and whether it is broken by quantum gravity effects., Comment: 38 pages

We derive soft theorems for single-clock cosmologies that enjoy a shift symmetry. These so-called consistency conditions arise from a combination of a large diffeomorphism and the internal shift-symmetry and fix the squeezed limit of all correlators with a soft scalar mode. As an application, we show that our results reproduce the squeezed bispectrum for Ultra-slow-roll inflation, a particular shift-symmetric, non-attractor model which is known to violate Maldacena's consistency relation. Similar results have been previously obtained by Mooij and Palma using background-wave methods. Our results shed new light on the infrared structure of single-clock cosmological spacetimes., Comment: 4 pages, v2: citation added, v3: citations added and edited in accordance with published version

Adiabatic modes are cosmological perturbations that are locally indistinguishable from a (large) change of coordinates. At the classical level, they provide model independent solutions. At the quantum level, they lead to soft theorems for cosmological correlators. We present a systematic derivation of adiabatic modes in spatially-flat cosmological backgrounds with asymptotically-perfect fluids. We find several new adiabatic modes including vector, time-dependent tensor and time-dependent scalar modes. The new vector and tensor modes decay with time in standard cosmologies but are the leading modes in contracting universes. We present a preliminary derivation of the related soft theorems. In passing, we discuss a distinction between classical and quantum adiabatic modes, we clarify the subtle nature of Weinberg second adiabatic mode and point out that the adiabatic nature of a perturbation is a gauge dependent statement.

We make progress towards an analytical understanding of the regime of validity of perturbation theory for large scale structures and the nature of some non-perturbative corrections. We restrict ourselves to 1D gravitational collapse, for which exact solutions before shell crossing are known. We review the convergence of perturbation theory for the power spectrum, recently proven by McQuinn and White, and extend it to non-Gaussian initial conditions and the bispectrum. In contrast, we prove that perturbation theory diverges for the real space two-point correlation function and for the probability density function (PDF) of the density averaged in cells and all the cumulants derived from it. We attribute these divergences to the statistical averaging intrinsic to cosmological observables, which, even on very large and "perturbative" scales, gives non-vanishing weight to all extreme fluctuations. Finally, we discuss some general properties of non-perturbative effects in real space and Fourier space., Comment: 34 pages + appendices, 11 figures. Version 2: updated to match the published version. Restructured the paper to make it more readable. Added new Figure 9 to support Figure 8. Corrected typo in equation (5.4). Added some minor clarifications of notation

Non-Gaussianities of dynamical origin are disentangled from primordial ones using the formalism of large deviation statistics with spherical collapse dynamics. This is achieved by relying on accurate analytical predictions for the one-point probability distribution function (PDF) and the two-point clustering of spherically-averaged cosmic densities (sphere bias). Sphere bias extends the idea of halo bias to intermediate density environments and voids as underdense regions. In the presence of primordial non-Gaussianity, sphere bias displays a strong scale dependence relevant for both high and low density regions, which is predicted analytically. The statistics of densities in spheres are built to model primordial non-Gaussianity via an initial skewness with a scale-dependence that depends on the bispectrum of the underlying model. The analytical formulas with the measured nonlinear dark matter variance as input are successfully tested against numerical simulations. For local non-Gaussianity with a range from $f_{\rm NL}=-100$ to $+100$ they are found to agree within 2\% or better for densities $\rho\in[0.5,3]$ in spheres of radius 15 Mpc$/h$ down to $z=0.35$. The validity of the large deviation statistics formalism is thereby established for all observationally relevant local-type departures from perfectly Gaussian initial conditions. The corresponding estimators for the amplitude of the nonlinear variance $\sigma_8$ and primordial skewness $f_{\rm NL}$ are validated using a fiducial joint maximum likelihood experiment. The influence of observational effects and the prospects for a future detection of primordial non-Gaussianity from joint one- and two-point densities-in-spheres statistics are discussed., Comment: 18 pages, 10 figures

We forecast the scientific capabilities to improve our understanding of cosmic inflation of CORE, a proposed CMB space satellite submitted in response to the ESA fifth call for a medium-size mission opportunity. The CORE satellite will map the CMB anisotropies in temperature and polarization in 19 frequency channels spanning the range 60-600 GHz. CORE will have an aggregate noise sensitivity of $1.7 \mu$K$\cdot \,$arcmin and an angular resolution of 5' at 200 GHz. We explore the impact of telescope size and noise sensitivity on the inflation science return by making forecasts for several instrumental configurations. This study assumes that the lower and higher frequency channels suffice to remove foreground contaminations and complements other related studies of component separation and systematic effects, which will be reported in other papers of the series "Exploring Cosmic Origins with CORE." We forecast the capability to determine key inflationary parameters, to lower the detection limit for the tensor-to-scalar ratio down to the $10^{-3}$ level, to chart the landscape of single field slow-roll inflationary models, to constrain the epoch of reheating, thus connecting inflation to the standard radiation-matter dominated Big Bang era, to reconstruct the primordial power spectrum, to constrain the contribution from isocurvature perturbations to the $10^{-3}$ level, to improve constraints on the cosmic string tension to a level below the presumptive GUT scale, and to improve the current measurements of primordial non-Gaussianities down to the $f_{NL}^{\rm local} < 1$ level. For all the models explored, CORE alone will improve significantly on the present constraints on the physics of inflation. Its capabilities will be further enhanced by combining with complementary future cosmological observations., Comment: Latex 107 pages, revised with updated author list and minor modifications

Gravity is a non-linear theory, and hence, barring cancellations, the initial super-horizon perturbations produced by inflation must contain some minimum amount of mode coupling, or primordial non-Gaussianity. In single-field slow-roll models, where this lower bound is saturated, non-Gaussianity is controlled by two observables: the tensor-to-scalar ratio, which is uncertain by more than fifty orders of magnitude; and the scalar spectral index, or tilt, which is relatively well measured. It is well known that to leading and next-to-leading order in derivatives, the contributions proportional to the tilt disappear from any local observable, and suspicion has been raised that this might happen to all orders, allowing for an arbitrarily low amount of primordial non-Gaussianity. Employing Conformal Fermi Coordinates, we show explicitly that this is not the case. Instead, a contribution of order the tilt appears in local observables. In summary, the floor of physical primordial non-Gaussianity in our Universe has a squeezed-limit scaling of $k_\ell^2/k_s^2$, similar to equilateral and orthogonal shapes, and a dimensionless amplitude of order $0.1\times(n_\mathrm{s}-1)$., Comment: 26 + 18 pages, 2 figures. References added and minor typos corrected. Matches published version

We argue that the non-detection of primordial tensor modes has taught us a great deal about the primordial universe. In single-field slow-roll inflation, the current upper bound on the tensor-to-scalar ratio, $r < 0.07$ $(95 \% ~CL)$, implies that the Hubble slow-roll parameters obey $\varepsilon \ll \eta$, and therefore establishes the existence of a new hierarchy. We dub this regime the conformal limit of (slow-roll) inflation, and show that it includes Starobinsky-like inflation as well as all viable single-field models with a sub-Planckian field excursion. In this limit, all primordial correlators are constrained by the full conformal group to leading non-trivial order in slow-roll. This fixes the power spectrum and the full bispectrum, and leads to the "conformal" shape of non-Gaussianity. The size of non-Gaussianity is related to the running of the spectral index by a consistency condition, and therefore it is expected to be small. In passing, we clarify the role of boundary terms in the $\zeta$ action, the order to which constraint equations need to be solved, and re-derive our results using the Wheeler-deWitt formalism., Comment: 29 pages (45 with appendix and references), 2 figures

We study inflation in a random multifield potential, using techniques developed by Marsh et al. The potential is a function of a large number of fields, and we choose parameters so that inflation only occurs in regions where the potential is accidentally flat. Using an improved estimate for the dynamics of eigenvalue repulsion, we are able to describe the steepening of the potential as inflation progresses. We provide suggestive arguments, but not a proof, that the resulting scalar power spectrum generically disagrees with observations. We also point out two problematic aspects of the model: there is no well-defined probability distribution for the gradient of the potential, and the evolution of the potential over small distances in field space is unphysical., Comment: 24 pages, 3 figures

Primordial non-Gaussianity (PNG) in Large Scale Structures is obfuscated by the many additional sources of non-linearity. Within the Effective Field Theory approach to Standard Perturbation Theory, we show that matter non-linearities in the bispectrum can be modeled sufficiently well to strengthen current bounds with near future surveys, such as Euclid. We find that the EFT corrections are crucial to this improvement in sensitivity. Yet, our understanding of non-linearities is still insufficient to reach important theoretical benchmarks for equilateral PNG, while, for local PNG, our forecast is more optimistic. We consistently account for the theoretical error intrinsic to the perturbative approach and discuss the details of its implementation in Fisher forecasts., Comment: 21 pages + appendices, 10 figures, 3 tables

We use the recent observations of Cosmic Microwave Background temperature and polarization anisotropies provided by the Planck satellite experiment to place constraints on the running $\alpha_\mathrm{s} = \mathrm{d}n_{\mathrm{s}} / \mathrm{d}\log k$ and the running of the running $\beta_{\mathrm{s}} = \mathrm{d}\alpha_{\mathrm{s}} / \mathrm{d}\log k$ of the spectral index $n_{\mathrm{s}}$ of primordial scalar fluctuations. We find $\alpha_\mathrm{s}=0.011\pm0.010$ and $\beta_\mathrm{s}=0.027\pm0.013$ at $68\%\,\mathrm{CL}$, suggesting the presence of a running of the running at the level of two standard deviations. We find no significant correlation between $\beta_{\mathrm{s}}$ and foregrounds parameters, with the exception of the point sources amplitude at $143\,\mathrm{GHz}$, $A^{PS}_{143}$, which shifts by half sigma when the running of the running is considered. We further study the cosmological implications of such preference for $\alpha_\mathrm{s},\beta_\mathrm{s}\sim0.01$ by including in the analysis the lensing amplitude $A_L$, the curvature parameter $\Omega_k$, and the sum of neutrino masses $\sum m_{\nu}$. We find that when the running of the running is considered, Planck data are more compatible with the standard expectations of $A_L = 1$ and $\Omega_k = 0$ but still hint at possible deviations. The indication for $\beta_\mathrm{s} > 0$ survives at two standard deviations when external datasets such as BAO and CFHTLenS are included in the analysis, and persists at $\sim 1.7$ standard deviations when CMB lensing is considered. We discuss the possibility of constraining $\beta_\mathrm{s}$ with current and future measurements of CMB spectral distortions, showing that an experiment like PIXIE could provide strong constraints on $\alpha_\mathrm{s}$ and $\beta_\mathrm{s}$., Comment: 10+1 pages, 9 figures, 10 tables. Matches published version

We discuss the implications of a PIXIE-like experiment, which would measure $\mu$-type spectral distortions of the CMB at a level of $\sigma_{\mu}=(1/n)\times 10^{-8}$, with $n\geq1$ representing an improved sensitivity (e.g. $n=10$ corresponds to PRISM). Using Planck data and considering the six-parameter $\Lambda$CDM model, we compute the posterior for $\mu_8\equiv\mu\times 10^{8}$ and find $\mu_8=1.57^{+0.11}_{-0.13}$ ($68\%\,\mathrm{CL}$). This becomes $\mu_{8} = 1.28^{+0.30}_{-0.52}$ ($68\%\,\mathrm{CL}$) when the running $\alpha_\mathrm{s}$ of the spectral index is included. We point out that a sensitivity of about $3\times$ PIXIE implies a guaranteed discovery: $\mu$-distortion is detected or $\alpha_\mathrm{s}\geq 0$ is excluded (both at $95\%\,\mathrm{CL}$ or higher). This threshold sensitivity sets a clear benchmark for CMB spectrometry. For a combined analysis of PIXIE and current Planck data, we discuss the improvement on measurements of the tilt $n_\mathrm{s}$ and the running $\alpha_\mathrm{s}$ and the dependence on the choice of the pivot. A fiducial running of $\alpha_\mathrm{s}=-0.01$ (close to the Planck best-fit) leads to a detection of negative running at $2\sigma$ for $5\times$ PIXIE. A fiducial running of $\alpha_\mathrm{s}=-0.02$, still compatible with Planck, requires $3\times$ PIXIE to rule out $\alpha_\mathrm{s} = 0$ (at $95\%\,\mathrm{CL}$). We propose a convenient and compact visualization of the improving constraints on the tilt, running and tensor-to-scalar ratio., Comment: 10+2 pages, 8 figures, 2 tables. Matches published version

A perturbative description of Large Scale Structure is a cornerstone of our understanding of the observed distribution of matter in the universe. Renormalization is an essential and defining step to make this description physical and predictive. Here we introduce a systematic renormalization procedure, which neatly associates counterterms to the UV-sensitive diagrams order by order, as it is commonly done in quantum field theory. As a concrete example, we renormalize the one-loop power spectrum and bispectrum of both density and velocity. In addition, we present a series of results that are valid to all orders in perturbation theory. First, we show that while systematic renormalization requires temporally non-local counterterms, in practice one can use an equivalent basis made of local operators. We give an explicit prescription to generate all counterterms allowed by the symmetries. Second, we present a formal proof of the well-known general argument that the contribution of short distance perturbations to large scale density contrast $\delta$ and momentum density $\mathbf\pi(\mathbf k)$ scale as $k^2$ and $k$, respectively. Third, we demonstrate that the common practice of introducing counterterms only in the Euler equation when one is interested in correlators of $ \delta$ is indeed valid to all orders., Comment: 28+14

We develop the effective theory of large-scale structure for non-Gaussian initial conditions. The effective stress tensor in the dark matter equations of motion contains new operators, which originate from the squeezed limit of the primordial bispectrum. Parameterizing the squeezed limit by a scaling and an angular dependence, captures large classes of primordial non-Gaussianity. Within this parameterization, we classify the possible contributions to the effective theory. We show explicitly how all terms consistent with the symmetries arise from coarse graining the dark matter equations of motion and its initial conditions. We also demonstrate that the system is closed under renormalization and that the basis of correction terms is therefore complete. The relevant corrections to the matter power spectrum and bispectrum are computed numerically and their relative importance is discussed., Comment: 59 pages, 33 figures

Cosmological observations have provided us with the measurement of just three numbers that characterize the very early universe: $ 1-n_{s} $, $ N $ and $\ln\Delta_R^2$. Although each of the three numbers individually carries limited information about the physics of inflation, one may hope to extract non-trivial information from relations among them. Invoking minimality, namely the absence of ad hoc large numbers, we find two viable and mutually exclusive inflationary scenarios. The first is the well-known inverse relation between $1- n_{s} $ and $ N $. The second implies a new relation between $ 1-n_{s} $ and $\ln\Delta_R^2$, which might provide us with a handle on the beginning of inflation and predicts the intriguing $\textit{lower}$ bound on the tensor-to-scalar ratio $ r> 0.006 $ ($ 95\% $ CL)., Comment: 5 pages, 3 figures