Out of Equilibrium Fields in Self-consistent Inflationary Dynamics. Density Fluctuations

TL;DR

This work treats inflation as a non-equilibrium quantum-field problem and develops a self-consistent, non-perturbative framework based on the large-N limit of an O(N) inflaton coupled to gravity in a semiclassical setting. The authors derive renormalized dynamical equations for the inflaton mean field and its quantum fluctuations, showing that spinodal and parametric instabilities drive rapid growth of long-wavelength modes, assemble into an effective zero mode, and eventually terminate inflation through backreaction. The analysis yields a gauge-invariant calculation of scalar and tensor perturbations, predicting a red-tilted scalar spectrum with a small tensor amplitude, and provides a consistent criterion to separate large non-perturbative fluctuations from the small fluctuations that seed cosmological perturbations. The results offer a coherent mechanism for graceful exit from inflation and a tangible link between non-equilibrium quantum dynamics and observable primordial fluctuations.

Abstract

The physics during the inflationary stage of the universe is of quantum nature involving extremely high energy densities. Moreover, it is out of equilibrium on a fastly expanding dynamical geometry.We present in these lectures non-perturbative out of equilibrium field theoretical methods in cosmological universes. We then study the non-linear dynamics of quantum fields in matter and radiation dominated FRW and de Sitter universes. We investigate the explosive particle production due to spinodal instabilities and parametric amplification in FRW and deSitter universes with and without symmetry breaking. We show how the particle production is sensitive to the expansion of the universe.We present a complete renormalization scheme for the equation of motion and the energy momentum tensor in flat cosmologies. We then consider an O(N) inflaton model coupled self-consistently to gravity in the semiclassical approximation, with `new inflation' type initial conditions. We study the dynamics self-consistently and non-perturbatively with non-equilibrium field theory methods in the large N limit. We find that spinodal instabilities drive the growth of non-perturbatively large quantum fluctuations which shut off the inflationary growth of the scale factor. A very specific combination of these large quantum fluctuations plus the inflaton zero mode assemble into a new effective field. This new field behaves classically and it is the object which actually rolls down. The metric perturbations during inflation are computed using this effective field and the Bardeen variable for superhorizon modes during inflation. We compute the amplitude and index for the spectrum of scalar density and tensorperturbations and find for these models that the spinodal instabilities are responsible for a `red' primordial spectrum.

Figures (10)

• Figure 1: Symmetry broken, slow roll, large $N$, matter dominated evolution of (a) the zero mode $\eta(\tau)$ vs. $\tau$, (b) the quantum fluctuation operator $g\Sigma(\tau)$ vs. $\tau$, (c) the number of particles $gN(\tau)$ vs. $\tau$, (d) the particle distribution $gN_k(\tau)$ vs. $k$ at $\tau=149.1$ (dashed line) and $\tau=398.2$ (solid line), and (e) the ratio of the pressure and energy density $p(\tau)/\varepsilon(\tau)$ vs. $t$ for the values, $\eta(\tau_0) = 10^{-7}, \; \dot{\eta}(\tau_0)=0, \; g = 10^{-12},\; h(\tau_0) = 0.1$.
• Figure 2: Symmetry broken, $\eta(\tau) \equiv 0$, matter dominated evolution of (a) the quantum fluctuation operator $g\Sigma(\tau)$ vs. $\tau$, (b) the number of particles $gN(\tau)$ vs. $\tau$, (c) the particle distribution $gN_k(\tau)$ vs. $k$ at $\tau=150.1$ (dashed line) and $\tau =397.1$ (solid line), and (d) the ratio of the pressure and energy density $p(\tau)/\varepsilon(\tau)$ vs. $\tau$ for the values $\eta(\tau_0) = 0, \; \dot{\eta}(\tau_0)=0, \; g = 10^{-12}, \; h(\tau_0) = 0.1$.
• Figure 3: Symmetry broken, chaotic, large $N$, radiation dominated evolution of (a) the zero mode $\eta(\tau)$ vs. $\tau$, (b) the quantum fluctuation operator $g\Sigma(\tau)$ vs. $\tau$, (c) the number of particles $gN(\tau)$ vs. $\tau$, (d) the particle distribution $gN_k(\tau)$ vs. $k$ at $\tau=76.4$ (dashed line) and $\tau=392.8$ (solid line), and (e) the ratio of the pressure and energy density $p(\tau)/\varepsilon(\tau)$ vs. $\tau$ for the values $\eta(\tau_0) = 4, \; \dot{\eta}(\tau_0)=0, \; g = 10^{-12}, h(\tau_0) = 0.1$.
• Figure 4: Symmetry broken, chaotic, large $N$, matter dominated evolution of (a) the zero mode $\eta(\tau)$ vs. $\tau$, (b) the quantum fluctuation operator $g\Sigma(\tau)$ vs. $\tau$, (c) the number of particles $gN(\tau)$ vs. $\tau$, (d) the particle distribution $gN_k(\tau)$ vs. $k$ at $t=50.8$ (dashed line) and $\tau=399.4$ (solid line), and (e) the ratio of the pressure and energy density $p(\tau)/\varepsilon(\tau)$ vs. $\tau$ for the parameter values $m^2=-1$, $\eta(\tau_0) = 4$, $\dot{\eta}(\tau_0)=0$, $g = 10^{-12}$, $h(\tau_0) = 0.1$.
• Figure 5: Symmetry unbroken, chaotic, large $N$, matter dominated evolution of (a) the zero mode $\eta(\tau)$ vs. $\tau$, (b) the quantum fluctuation operator $g\Sigma(\tau)$ vs. $\tau$, (c) the number of particles $gN(\tau)$ vs. $\tau$, (d) the particle distribution $gN_k(\tau)$ vs. $k$ at $\tau=77.4$ (dashed line) and $\tau=399.7$ (solid line), and (e) the ratio of the pressure and energy density $p(\tau)/\varepsilon(\tau)$ vs. $\tau$ for the parameter values $m^2=+1$, $\eta(\tau_0) = 5$, $\dot{\eta}(\tau_0)=0$, $g = 10^{-12}$, $H(\tau_0) = 0.1$.