Stochastic semiclassical cosmological models

We consider the classical stochastic fluctuations of spacetime geometry induced by quantum fluctuations of massless nonconformal matter fields in the early Universe. To this end, we supplement the stress-energy tensor of these fields with a stochastic part, which is computed along the lines of the Feynman-Vernon and Schwinger-Keldysh techniques; the Einstein equation is therefore upgraded to a so-called Einstein-Langevin equation. We consider in some detail the conformal fluctuations of flat spacetime and the fluctuations of the scale factor in a simple cosmological model introduced by Hartle, which consists of a spatially flat isotropic cosmology driven by radiation and dust.

Noise induced transitions in semiclassical cosmology

A semiclassical cosmological model is considered which consists of a closed Friedmann-Robertson-Walker spacetime in the presence of a cosmological constant, which mimics the effect of an inflaton field, and a massless, non-conformally coupled quantum scalar field. We show that the back-reaction of the quantum field, which consists basically of a nonlocal term due to gravitational particle creation and a noise term induced by the quantum fluctuations of the field, are able to drive the cosmological scale factor over the barrier of the classical potential so that if the universe starts near a zero scale factor (initial singularity), it can make the transition to an exponentially expanding de Sitter phase. We compute the probability of this transition and it turns out to be comparable with the probability that the universe tunnels from ``nothing'' into an inflationary stage in quantum cosmology. This suggests that in the presence of matter fields the back-reaction on the spacetime should not be neglected in quantum cosmology.

Stochastic semiclassical gravity

In the first part of this paper, we show that the semiclassical Einstein-Langevin equation, introduced in the framework of a stochastic generalization of semiclassical gravity to describe the back reaction of matter stress-energy fluctuations, can be formally derived from a functional method based on the influence functional of Feynman and Vernon. In the second part, we derive a number of results for background solutions of semiclassical gravity consisting of stationary and conformally stationary spacetimes and scalar fields in thermal equilibrium states. For these cases, fluctuation-dissipation relations are derived. We also show that particle creation is related to the vacuum stress-energy fluctuations and that it is enhanced by the presence of stochastic metric fluctuations.

Mode decomposition and renormalization in semiclassical gravity

We compute the influence action for a system perturbatively coupled to a linear scalar field acting as the environment. Subtleties related to divergences that appear when summing over all the modes are made explicit and clarified. Being closely connected with models used in the literature, we show how to completely reconcile the results obtained in the context of stochastic semiclassical gravity when using mode decomposition with those obtained by other standard functional techniques.

Stochastic semiclassical fluctuations in Minkowski spacetime

The semiclassical Einstein-Langevin equations which describe the dynamics of stochastic perturbations of the metric induced by quantum stress-energy fluctuations of matter fields in a given state are considered on the background of the ground state of semiclassical gravity, namely, Minkowski spacetime and a scalar field in its vacuum state. The relevant equations are explicitly derived for massless and massive fields arbitrarily coupled to the curvature. In doing so, some semiclassical results, such as the expectation value of the stress-energy tensor to linear order in the metric perturbations and particle creation effects, are obtained. We then solve the equations and compute the two-point correlation functions for the linearized Einstein tensor and for the metric perturbations. In the conformal field case, explicit results are obtained. These results hint that gravitational fluctuations in stochastic semiclassical gravity have a non-perturbative behavior in some characteristic correlation lengths.

Vacuum decay in quantum field theory

We study the contribution to vacuum decay in field theory due to the interaction between the long- and short-wavelength modes of the field. The field model considered consists of a scalar field of mass M with a cubic term in the potential. The dynamics of the long-wavelength modes becomes diffusive in this interaction. The diffusive behavior is described by the reduced Wigner function that characterizes the state of the long-wavelength modes. This function is obtained from the whole Wigner function by integration of the degrees of freedom of the short-wavelength modes. The dynamical equation for the reduced Wigner function becomes a kind of Fokker-Planck equation which is solved with suitable boundary conditions enforcing an initial metastable vacuum state trapped in the potential well. As a result a finite activation rate is found, even at zero temperature, for the formation of true vacuum bubbles of size M-1. This effect makes a substantial contribution to the total decay rate.

Induced quantum metric fluctuations and the validity of semiclassical gravity

We propose a criterion for the validity of semiclassical gravity (SCG) which is based on the stability of the solutions of SCG with respect to quantum metric fluctuations. We pay special attention to the two-point quantum correlation functions for the metric perturbations, which contain both intrinsic and induced fluctuations. These fluctuations can be described by the Einstein-Langevin equation obtained in the framework of stochastic gravity. Specifically, the Einstein-Langevin equation yields stochastic correlation functions for the metric perturbations which agree, to leading order in the large N limit, with the quantum correlation functions of the theory of gravity interacting with N matter fields. The homogeneous solutions of the Einstein-Langevin equation are equivalent to the solutions of the perturbed semiclassical equation, which describe the evolution of the expectation value of the quantum metric perturbations. The information on the intrinsic fluctuations, which are connected to the initial fluctuations of the metric perturbations, can also be retrieved entirely from the homogeneous solutions. However, the induced metric fluctuations proportional to the noise kernel can only be obtained from the Einstein-Langevin equation (the inhomogeneous term). These equations exhibit runaway solutions with exponential instabilities. A detailed discussion about different methods to deal with these instabilities is given. We illustrate our criterion by showing explicitly that flat space is stable and a description based on SCG is a valid approximation in that case.

Propagation in a thermal graviton background

It is well known that radiative corrections evaluated in nontrivial backgrounds lead to effective dispersion relations which are not Lorentz invariant. Since gravitational interactions increase with energy, gravity-induced radiative corrections could be relevant for the trans-Planckian problem. As a first step to explore this possibility, we compute the one-loop radiative corrections to the self-energy of a scalar particle propagating in a thermal bath of gravitons in Minkowski spacetime. We obtain terms which originate from the thermal bath and which indeed break the Lorentz invariance that possessed the propagator in the vacuum. Rather unexpectedly, however, the terms which break Lorentz invariance vanish in the high three-momentum limit. We also found that the imaginary part, which gives the rate of approach to thermal equilibrium, vanishes at one loop.

Cosmological perturbations from stochastic gravity

In inflationary cosmological models driven by an inflaton field the origin of the primordial inhomogeneities which are responsible for large-scale structure formation are the quantum fluctuations of the inflaton field. These are usually calculated using the standard theory of cosmological perturbations, where both the gravitational and the inflaton fields are linearly perturbed and quantized. The correlation functions for the primordial metric fluctuations and their power spectrum are then computed. Here we introduce an alternative procedure for calculating the metric correlations based on the Einstein-Langevin equation which emerges in the framework of stochastic semiclassical gravity. We show that the correlation functions for the metric perturbations that follow from the Einstein-Langevin formalism coincide with those obtained with the usual quantization procedures when the scalar field perturbations are linearized. This method is explicitly applied to a simple model of chaotic inflation consisting of a Robertson-Walker background, which undergoes a quasi-de Sitter expansion, minimally coupled to a free massive quantum scalar field. The technique based on the Einstein-Langevin equation can, however, deal naturally with the perturbations of the scalar field even beyond the linear approximation, as is actually required in inflationary models which are not driven by an inflaton field, such as Starobinsky¿s trace-anomaly driven inflation or when calculating corrections due to nonlinear quantum effects in the usual inflaton driven models.

Stability of de Sitter spacetime under isotropic perturbations in semiclassical gravity

A spatially flat Robertson-Walker spacetime driven by a cosmological constant is nonconformally coupled to a massless scalar field. The equations of semiclassical gravity are explicitly solved for this case, and a self-consistent de Sitter solution associated with the Bunch-Davies vacuum state is found (the effect of the quantum field is to shift slightly the effective cosmological constant). Furthermore, it is shown that the corrected de Sitter spacetime is stable under spatially isotropic perturbations of the metric and the quantum state. These results are independent of the free renormalization parameters.

Quantum metric spaces as a model for pregeometry

A new arena for the dynamics of spacetime is proposed, in which the basic quantum variable is the two-point distance on a metric space. The scaling dimension (that is, the Kolmogorov capacity) in the neighborhood of each point then defines in a natural way a local concept of dimension. We study our model in the region of parameter space in which the resulting spacetime is not too different from a smooth manifold.

Interaccion among systems of finite size in predictive relativistic mechanics I. General framework

We derive a Hamiltonian formulation for the three-dimensional formalism of predictive relativistic mechanics. This Hamiltonian structure is used to derive a set of dynamical equations describing the interaction among systems in perturbation theory.

Interaccion among systems of finite size in predictive relativistic mechanics -II. Electromagnetic and gravitational interactions

We explicitly construct a closed system of differential equations describing the electromagnetic and gravitational interactions among bodies to first order in the coupling constants, retaining terms up to order c-2. The Breit and Barker and O'Connell Hamiltonians are recovered by means of a coordinate transformation. The method used throws light on the meaning of these coordinates.

Interactions among systems of finite size en predictive relativistic mechanics III. Short-range interaction and second-order terms

We compute up to and including all the c-2 terms in the dynamical equations for extended bodies interacting through electromagnetic, gravitational, or short-range fields. We show that these equations can be reduced to those of point particles with intrinsic angular momentum assuming spherical symmetry.