Arthur Hebecker, Heidelberg University

Inflationary Correlation Functions without Infrared Divergences

The definition of correlation functions relies on measuring distances on some late surface of equal energy density. If invariant distances are used, the curvature correlation functions of single-field inflation are free of any IR sensitivity. By contrast, conventional correlation functions, defined using the coordinate distance between pairs of points, receive large IR corrections if measured in a "large box" and if inflation lastet for a sufficiently long period. The underlying large logarithms are associated with long-wavelength fluctuations of both the scalar and the graviton background. This effect is partially captured by the familiar delta-N-formalism. Conventional, IR-sensitive correlation functions are related to their IR-safe counterparts by simple and very general formulae. In particular, the coefficient of the leading logarithmic correction to any n-point function is controlled by the first and second logarithmic derivatives of this function with respect to the overall momentum scale. This allows for a simple evaluation of corrections to leading and higher-order non-Gaussianity parameters.

Emil Akhmedov, Moscow ITEP 

On radiation and IR divergences in dS space

It is well known that there should be a total cancellation of the IR divergences in closed systems described by interacting quantum field theories, such as QED and gravity. I am going to show that such a cancellation does not happen in de Sitter space.

Atsushi Higuchi, University of York

On the Equivalence between Euclidean and In-In Formalisms in de Sitter QFT

We study the relation between two sets of correlators in interacting quantum field theory on de Sitter space.  The first are correlators computed using in-in perturbation theory in the region of de Sitter space to the future of a cosmological horizon (also known as the expanding cosmological patch, the conformal patch, or the Poincare patch), and for which the free propagators are taken to be those of the free Euclidean vacuum.  The second are correlators obtained by analytic continuation from interacting QFT on Euclidean de Sitter; i.e., they are correlators in the Hartle-Hawking vacuum.  We give an analytic argument that these correlators coincide for interacting massive scalar fields with any positive mass.  We also verify this result via direct analytical and numerical calculation in two simple examples.  The correspondence holds diagram by diagram, and at any finite value of a Pauli-Villars regulator mass M.  Along the way, we note interesting connections between various prescriptions for perturbation theory in general static spacetimes with bifurcate Killing horizons.

Donald Marolf, University of California Santa Barbara

Singular gauges and dS invariance of the graviton vacuum

There has been a long-running discussion as to whether free gravitons on dS have a dS-invariant state. On the one hand, de Sitter invariant states are clearly singular in gauges favored by cosmologists; e.g. transverse traceless synchronous gauge associated with the k=0 slicing of dS. However, Higuchi has constructed a dS-invariant state using a different gauge. We resolve this tension by showing that the above “cosmologists gauge” is in fact singular on global de Sitter space. This observation may prove useful in understanding the physics of calculations indicating large IR effects involving gravity in dS.

Alexander Dolgov, University of Ferrara

Infrared instability of massless scalars in De Sitter space-time.

Vacuum expectation value of the square of the quantum field operator of massless or light scalar field is calculated in the De Sitter space-time. The suggested method of calculation is different from the standard one used in the 80th. The calculations are heavily based on the De Sitter covarinace of the relevant quantities. The found result is significantly different for the old one for the massless field but coincides with the classical result for light massive field. Possible explanation of the discrepancy by a spontaneous breaking of De Sitter invariance or by finite duration of the (quasi) De Sitter stage is discussed.

Ugo Moschella, University of Insubria

Particle decay in the de Sitter universe

We study particle decay in the de Sitter spacetime as given by first order perturbation theory in an interacting quantum field theory. We discuss first a general construction of bosonic two-point functions, including a recently discovered class of tachyonic theories that do exist in the de Sitter spacetime at discrete negative values of the squared mass parameter and have no Minkowskian counterpart. We show then that for fields with masses above a critical mass $m_c$ there is no such thing as particle stability, so that decays forbidden in flat space-time do occur there. The lifetime of such a particle also turns out to be independent of its velocity when that lifetime is comparable with de Sitter radius. 

For particles with lower mass is yet not completely solved. We show however that the masses of their decay products should obey quantification rules.

Dan Boyanovsky, University of Pittsburgh

Anomalous scaling dimensions and particle decay during inflation

I will discuss the emergence of anomalous scaling dimensions for superhorizon fluctuations and the main ideas and concepts of particle decay during inflation, via the resummation of secular terms with the dynamical renormalization group. There are loops, IR effects and (lots of...) issues.

Yuko Urakawa, Waseda University

IR divergence problem in single-field models of inflation

We clarify the origin of IR divergence in single-field models of inflation and provide the correct way to calculate the observable fluctuations.  First, we show the presence of gauge degrees of freedom in the frequently used gauges such as the comoving gauge and the flat gauge. These gauge degrees of freedom are responsible for the IR divergences that appear in loop corrections of primordial perturbations. We propose, in this talk, one simple but explicit example of gauge-invariant quantities. Then, we explicitly calculate such a quantity to find that the IR divergence is absent in the slow-roll approximation. In this formalism, we revisit the consistency relation that connects the three-point function in the squeezed limit with the spectral index.

Tomislav Prokopec, Utrech University

Going beyond de Sitter

De Sitter space is a maximally symmetric space, and any (time dependent) backreaction necessarily breaks the symmetry. On the other hand, the backreaction in FLRW spaces (which, from the cosmological perspective, are more realistic spaces) respects the symmetries of the background space. One could therefore argue that it is more natural to study the backreaction on quasi-de Sitter spaces, and more generally on FLRW spaces. As examples, I will present our results on the one loop (Hubble) effective potential calculation, and the one loop stress energy calculation. We find that the backreaction from infrared modes can be important if a massless scalar couples nonminimally to the Ricci with a negative coupling, and if the Universe expands faster than exponentially.  

Richard Woodard, University of Florida

Interacting Quantum Fields in de Sitter Space

Infrared logarithms are factors of the logarithm of the inflationary scale factor which arise in quantum field theoretic loop corrections that involve either massless, minimally coupled scalars or gravitons. They have been found by myself and collaborators in 1PI functions and by Steven Weinberg in the power spectrum of primordial perturbations. Because the inflationary scale factor grows so rapidly, infrared logarithms enhance loop corrections far beyond expectations based upon the coupling constant. They also inject time dependence into what are usually static results. For a very long period of inflation this enhancement can grow so large that weak field perturbation theory breaks down. In this talk I explain the physical reasons why infrared logarithms occur, I review the computations in which they have been seen and I describe Starobinskii's technique for evolving past the breakdown of perturbation theory. I also comment on the potential of infrared logarithms to change our understanding of inflationary cosmology.

Emanuela Dimastrogiovanni, Istituto Nazionale di Fisica Nucleare

One-loop corrections in slow-roll and in more general theories of inflation

I will present our work on loop corrections to the power spectrum of curvature fluctuations in single-field inflationary models. We consider both standard slow-roll (where the interactions between gravitons and the scalar are included for the first time) and non-canonical Lagrangians. We show that the tensor modes cannot be neglected since, in some models, they produce one loop contributions with an amplitude that is comparable to the one coming from the scalar sector. Our study of loop corrections in non-canonical theories characterized by a small speed of sound (c_s) provides quantitative bounds on c_s, to be compared with similar constraints derived from CMB observations.

Takahiro Tanaka, Yukawa Institute at Kyoto University

Effects of decoherence on IR divergence

We propose one way to regularize the fluctuations generated during inflation. We show that, as long as we consider the case that the non-linear interaction acts for a finite duration, observable fluctuations are free from IR divergences not only in the single field models but also in the multi field model. In contrast to the single field model, to discuss observables, we need to take into account the effects of quantum decoherence which pick up a unique history of the universe from various possibilities contained in initial quantum state set naturally in the early stage of the universe.

Albert Roura, Max Planck Institute for Gravitational Physics (Albert Einstein Institute)

One-loop Riemann correlators and dS invariance

I will start with a brief qualitative discussion of the construction of a dS-invariant state for interacting theories using Euclidean methods and its real-time evolution within the closed-time-path formalism, as well as of the closely related in-in formalism. Next, I will focus on the two-point quantum correlation function for the Riemann tensor of the metric perturbations around dS including the one-loop correction from matter fields. A key object is the stress tensor two-point function, from which the one-loop Ricci correlator follows straightforwardly. We have obtained the exact result for minimally coupled fields with arbitrary mass in terms of maximally symmetric bitensors, which makes dS invariance manifest. Long range correlations are present for sufficiently small (but nonvanishing) masses, and the discontinuity of the massless limit can be understood in a simple way. Finally, I will comment on the implications for the tensorial power spectrum and on the calculation of the Weyl correlations.

Steven Weinberg, University of Texas at Austin

Ultraviolet Divergences in Cosmological Correlations

A method is developed for dealing with ultraviolet divergences in calculations of cosmological correlations, which does not depend on dimensional regularization.  An extended version of  the WKB approximation is used to analyze the divergences in these calculations, and these divergences are controlled by the introduction of  Pauli--Villars regulator fields.  This approach is illustrated in the theory of a scalar field with arbitrary self-interactions in a fixed flat-space Robertson--Walker metric with arbitrary scale factor $a(t)$.  Explicit formulas are given for the counterterms needed to cancel all dependence on the regulator properties, and an explicit prescription is given for calculating finite correlation functions.

Emil Mottola, LANL

New Horizons in Cosmology: The Trace Anomaly, Cosmological Horizon Modes and Dynamical Dark Energy

General Relativity receives quantum corrections relevant at macroscopic distance scales and near event horizons. These arise from the conformal scalar degrees of freedom in the extended effective field theory of gravity generated by the trace anomaly of massless quantum fields in curved space. Linearized perturbations of the Bunch-Davies state in de Sitter space show that these new scalar degrees of freedom are associated with macroscopic changes of state on the cosmological horizon scale, with potentially large stress tensors that can lead to substantial backreaction effects in cosmology. In the extended effective theory the cosmological ``constant"  is a state dependent condensate whose value is scale dependent and which possesses an infrared stable conformal fixed point at zero. These considerations suggest that the observed dark energy of our universe may be a macroscopic finite size effect whose value depends not upon Planck scale physics but upon extreme infrared physics on the cosmological horizon scale.

Nikolaos Tsamis, University of Crete

Gravity-Driven Cosmology

A simple phenomenological model for early cosmological evolution is constructed. Its motivation is the physics of quantum infrared effects in a de Sitter geometry.

Steve Giddings, University of California Santa Barbara

The problems of quantum gravity: from high-energy scattering to black holes and cosmology

Much work on quantum gravity has focussed on short-distance problems such as non-renormalizability and singularities.  However, quantization of gravity raises important long-distance issues, which may be more important guides to the conceptual advances required.  These include the problems of black hole information and gauge invariant observables, and those of inflationary cosmology.  An overview of aspects of these problems, and apparent connections, will be given.

Arvind Rajaraman, UC Irvine

Massless fields in Euclidean de Sitter space

In theories with light or massless fields, loop diagrams can develop infrared divergences. We demonstrate how these can be resolved for a theory of massless interacting scalar fields in Euclidean de Sitter space. We also comment on applications to the in-in formalism in Lorentzian de Sitter space.

Louis Leblond, Perimeter Institute 

Resumming late time divergences and comparing thermal vs dS

I will argue that the dynamical renormalization group can be used to resum late time divergences appearing in loop computations in de Sitter. In the case of a scalar field with quartic interactions, the resummed  propagator is the massive one. Standard mean field theory techniques can then be used to estimate the mass. This is analogous to the thermal field theory story but with some notable differences. We discuss whether a critical point can exist in dS where mean field methods fail.

Jose Roberto Vidal, Universidad Autonoma de Madrid

Unitarity and vacuum decay

The stability of spacetime has been related to the production of particles and also to the imaginary parts of the perturbative series. The unitarity relations of the quantum theory impose relations between these two phenomena.

Martin Sloth, CERN

Semiclassical approaches to IR issues in quasi de Sitter universes

Using simple semiclassical relations it is possible to show that the conventional cosmological correlation functions  are affected by significant IR corrections in quasi de Sitter space-times when averaged over very large volumes (in the "large box"). The IR effects apparently imply a breakdown of perturbation theory in the large box on sufficiently long time scales, for example the time between self-reproduction and reheating in chaotic inflation. An interpretation of the apparent breakdown of the perturbative expansion of gravity will also be briefly discussed.

Jason Kumar, University of Hawaii at Manoa

The Feynman Propagator and Correlation Functions in an Inflating Spacetime

We discuss the definition of the Feynman propagator in de Sitter space.  We show that the ambiguities in the propagator zero-mode can be used to make sense of the behavior of low-momentum modes in an inflating space-time.  We use this tool to calculate loop corrections to non-Gaussian correlation functions, and show that there are limits where the loop terms dominate.  These models can be probed with the Planck satellite.

Alexander Polyakov, Princeton University

Infrared sensitivity of  unstable vacua

Leonardo Senatore, IAS/Stanford

On IR effects in single field inflation