arXiv subjects

Enrico Pajer

Publications and source records attributed to Enrico Pajer.

At least 19 recordsLinked to original sources

arXiv2026

Lectures on Open Systems and Cosmology

Open systems are ubiquitous in physics. Many realistic systems interact, at least weakly, with environmental degrees of freedom that may be too numerous, too complicated, inaccessible, or unknown. When only a subset of degrees of freedom is observed, its reduced dynamics can differ qualitatively from that of a closed system, displaying dissipation, noise, decoherence, memory effects, or loss of information into unobserved sectors. When the microscopic description is also unknown, one is led to an open effective description, in which the relevant degrees of freedom are treated systematically while the environment and the microscopics are parametrized rather than solved for explicitly. This perspective is especially important in gravity and cosmology. The main open problems of cosmology, including inflation, dark matter, and dark energy, involve spacetime-filling sectors whose microscopic nature is unknown and whose observed effects are primarily gravitational. At the same time, gravitational systems often lack a preferred notion of conserved energy because they are time dependent, and naturally display out-of-equilibrium dynamics. These lecture notes introduce the operator formalism and the Schwinger-Keldysh path-integral as tools to study open systems, with emphasis on open effective field theories and inflation. They are aimed at master students, PhD students, and researchers approaching these topics for the first time.

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arXiv2026

Counting Degrees of Freedom in Open Effective Theories

Open effective field theories provide a systematic framework for describing physical systems interacting with an environment whose microscopic details are unknown, unobservable, or uncalculable. A basic step in constructing any effective field theory is the identification of the relevant degrees of freedom. For open effective theories, however, this step is subtle: their dynamics is generically dissipative and non-Hamiltonian, so the standard Hamiltonian and Lagrangian algorithms are not directly applicable. To overcome this limitation, we develop an algorithm to count degrees of freedom directly from the equations of motion. Restricting to classical, linearised dynamics on homogeneous and isotropic backgrounds, our method applies to non-Lagrangian systems with unequal numbers of fields and equations, including systems with constraints and gauge redundancies. As a by-product, our algorithm identifies constraints, gauge identities, gauge redundancies, and consistency conditions on stochastic sources. The central ingredient is the introduction of a dual set of ``advanced'' equations, or equivalently an auxiliary Martin-Siggia-Rose functional. We show that gauge redundancies of the original fields are associated with gauge identities of the dual advanced equations, and vice versa. We illustrate our procedure in examples ranging from coupled scalar systems to electromagnetism in a medium and gravitational effective theories relevant for cosmology. Our results will prove useful in the study of stochastic dynamics, non-equilibrium statistical systems and the semi-classical limit of open quantum systems on the Schwinger-Keldysh path integral.

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arXiv2026

Phenomenology of an Open Effective Field Theory of Dark Energy

All observational evidence for dark matter and dark energy is so far exclusively gravitational. Hence, the dark sector may be equivalently described by a theory of the spacetime metric whose dynamics is affected by interactions with an unknown environment. Adapting open-system techniques, we have recently constructed such a general theory of open gravitational dynamics. Here we study a minimal and concrete realization of this theory that describes the late-time acceleration of the Universe. Our model provides a good fit to recent baryon acoustic oscillation measurements by construction, while avoiding violations of the null energy condition. Moreover, it leads to a set of correlated and observationally testable predictions. Studying the modified cosmological perturbation theory and compared to the $\Lambda$CDM model we find: a dissipative suppression of the gravitational-wave luminosity distance relative to the electromagnetic one; a modification in the evolution of the Bardeen potentials with a clear signal in the gravitational slip; and an enhancement of structure formation at low redshift. We present semi-analytical estimates of the magnitude of these effects and show that they lie within the reach of current constraints while providing clear targets for upcoming cosmological surveys.

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arXiv2026

All-Loop Renormalization and the Phase of the de Sitter Wavefunction

Cosmological observables of the primordial universe are encoded in the late-time field-theoretic wavefunction. For shift-symmetric scalars in de Sitter, a good approximation for many inflationary models, the wavefunction must be purely real at tree-level. This property is violated by a quantum anomaly in the process of renormalization. As a result, we show that the imaginary part of the wavefunction is fixed by its dependence on the renormalization scale to all loop orders in perturbation theory. This follows from unitarity, locality, dilation isometry and a Bunch-Davies state. The compact relation we uncover for the wavefunction implies an infinite set of relations among correlators of massless fields and their conjugate momenta, which we exemplify at one-loop order.

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arXiv2025

Unitary and Analytic Renormalisation of Cosmological Correlators

Loop contributions to cosmological correlators and to the associated wavefunction are of key theoretical and phenomenological interest. Here, we investigate and compare different renormalisation schemes proposed in the literature to handle ultraviolet divergences and develop new schemes adapting $\eta$ regulators to de Sitter spacetime. We focus on one-loop contributions to the quadratic wavefunction coefficient of a shift-symmetric massless scalar in de Sitter spacetime, which is a good toy model of primordial curvature perturbations. We show that different implementations of dimensional regularisation agree with each other and with unitarity and scale invariance in the final renormalised result. Imposing unitarity in the form of the cosmological optical theorem, we define a class of unitary and analytic $\eta$ regulators that agree with dim reg but feature considerable technical and conceptual simplifications. We show that the imaginary part of all one-loop wavefunction coefficients is universally fixed in terms of the logarithmic running of the real part, under the assumptions of scale invariance, Bunch-Davies vacuum and light bulk fields. Our work resolves discrepancies in the literature, establishes regulator-independent predictions for the imaginary part at one loop, and provides a practical framework for computing quantum contributions to cosmological correlators.

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arXiv2025

An Open System Approach to Gravity

Several major open problems in cosmology, including the nature of inflation, dark matter, and dark energy, share a common structure: they involve spacetime-filling media with unknown microphysics, and can be probed so far only through their gravitational effects. This observation motivates a systematic open-system approach to cosmology, in which gravity evolves in the presence of a generic, unobservable environment. In this work, we develop a general framework for open gravitational dynamics based on general relativity and the Schwinger-Keldysh formalism, carefully addressing the nontrivial constraints imposed by diffeomorphism invariance. At the quantum level, our path integral formulation computes the gravitational density matrix in perturbation theory around a semi-classical spacetime. As illustrative applications, we study inflation and the propagation of gravitational waves in classical regimes where environmental interactions are non-negligible. In the inflationary context, our framework reproduces the known Open Effective Field Theory of Inflation in the decoupling limit and extends it to include gravitational interactions. For gravitational waves, we derive the most general conservative and dissipative corrections to propagation. Remarkably, we find that the leading-order gravitational birefringence is dissipative in nature, whereas conservative birefringence appears only at higher derivative order, opposite to the electromagnetic case. Our results pave the way to modeling dissipative effects in the late universe.

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arXiv2025

On the Asymptotic Nature of Cosmological Effective Theories

Much of our intuition about Effective Field Theories (EFTs) stems from their formulation in flat spacetime, yet EFTs have become indispensable tools in cosmology, where time-dependent backgrounds are the norm. In this work, we demonstrate that in spacetimes undergoing significant expansion-such as accelerated FLRW and de Sitter backgrounds-the contributions of operators with mass dimension $\Delta$ to physical observables grow factorially with $\Delta$ at fixed couplings. This behavior stands in stark contrast to static flat spacetime. As a result, the cosmological EFT expansion is generally asymptotic rather than convergent, even at tree level. To illustrate this phenomenon, we analyze simple toy models involving a massless or conformally coupled scalar field interacting with a heavy scalar with zero or infinite sound speed. We demonstrate that meaningful EFT predictions can still be extracted via appropriate resummation techniques, performed in both Fourier and Mellin-momentum space. In the infinite sound speed limit, where the heavy field is effectively non-dynamical, the resummed EFT reproduces the exact result of the full theory. In other cases, the EFT captures only the local part of the dynamics, omitting nonlocal terms, which are exponentially suppressed in the large-mass limit for the Bunch-Davies state. Our results provide detailed and quantitative expectations for more general cosmological EFTs.

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arXiv2024

An Open Effective Field Theory for light in a medium

In many scenarios of interest, a quantum system interacts with an unknown environment, necessitating the use of open quantum system methods to capture dissipative effects and environmental noise. With the long-term goal of developing a perturbative theory for open quantum gravity, we take an important step by studying Abelian gauge theories within the Schwinger-Keldysh formalism. We begin with a pedagogical review of general results for open free theories, setting the stage for our primary focus: constructing the most general open effective field theory for electromagnetism in a medium. We assume locality in time and space, but allow for an arbitrary finite number of derivatives. Crucially, we demonstrate that the two copies of the gauge group associated with the two branches of the Schwinger-Keldysh contour are not broken but are instead deformed by dissipative effects. We provide a thorough discussion of gauge fixing, define covariant gauges, and calculate the photon propagators, proving that they yield gauge-invariant results. A notable result is the discovery that gauge invariance is accompanied by non-trivial constraints on noise fluctuations. We derive these constraints through three independent methods, highlighting their fundamental significance for the consistent formulation of open quantum gauge theories.

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arXiv2024

The UV Sensitivity of Axion Monodromy Inflation

We revisit axion monodromy inflation in the context of UV-inspired models and point out that its cosmological observables are sensitive to heavy fields with masses far above the Hubble scale, such as the moduli of flux compactifications. By studying a string-inspired two-field extension of axion monodromy with a small turning rate, we reveal that the oscillatory modulation of the axion potential leads to continuous excitation of heavy fields during inflation when the modulation frequency exceeds the field masses. This finding challenges the conventional single-field description, heavy moduli cannot be simply integrated out. Using a full bootstrap analysis, we demonstrate that this mechanism produces cosmological collider signals that bypass the usual Boltzmann suppression for heavy masses. Specifically, we identify detectably large signatures of heavy moduli in the primordial bispectrum, offering a promising avenue for probing high-energy physics through cosmological observations.

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arXiv2024

The Open Effective Field Theory of Inflation

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.

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arXiv2024

Perturbative unitarity bounds from momentum-space entanglement

Physical theories have a limited regime of validity and hence must be accompanied by a breakdown diagnostic to establish when they cease to be valid as parameters are varied. For perturbative theories, estimates of the first neglected order offer valuable guidance, but one is often interested in sharp bounds beyond which perturbation theory necessarily fails. In particle physics, it is common to employ the bounds on partial waves imposed by unitarity as such a diagnostic. Unfortunately, these bounds don't extend to curved spacetime, where scattering experiments are challenging to define. Here, we propose to use the growth of entanglement in momentum space as a breakdown diagnostic for perturbation theory in general field theories. This diagnostic can be readily used in cosmological spacetimes and does not require any flat spacetime limit. More in detail, we consider the so-called purity of the reduced density operator constructed by tracing out all but one of the Fourier modes in an effective theory and we present a diagrammatic technique to compute it perturbatively. Constraints on the regime of validity of perturbation theory are then derived when the perturbative purity violates its unitarity bounds. In flat spacetime, we compare these purity bounds to those from partial waves. We find general qualitative agreement but with remarkable differences: purity bounds can be sometimes weaker, but other times they exist when no partial wave bounds exist. We then derive purity bounds for scalar field theories in de Sitter spacetime for a variety of interactions that appear in inflationary models. Importantly, our bounds make no reference to time evolution and in de Sitter they depend exclusively on scale-invariant ratios of the physical kinematics.

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arXiv2024

Records from the S-Matrix Marathon: A Timeless History of Time

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.

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arXiv2024

Cosmological cutting rules for Bogoliubov initial states

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.

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arXiv2024

A Cosmological Bootstrap for Resonant Non-Gaussianity

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.

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arXiv2024

The In-Out Formalism for In-In Correlators

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 Poincaré 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.

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arXiv2024

A Cosmological Tachyon Collider: Enhancing the Long-Short Scale Coupling

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 $μ$-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.

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arXiv2023

Leading Loops in Cosmological Correlators

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.

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arXiv2023

Parity violation in the scalar trispectrum: no-go theorems and yes-go examples

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.

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