IR issues and loops in de Sitter space - 2010

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.