Scalar Field Dynamics in Friedman Robertson Walker Spacetimes

TL;DR

This paper analyzes non-equilibrium quantum field dynamics of scalar fields in expanding FRW spacetimes to understand preheating after inflation. It employs non-perturbative Hartree and leading-order large $N$ approximations within the closed time path formalism to capture backreaction and renormalization in a time-dependent background. Key results show explosive growth of quantum fluctuations via spinodal instabilities and parametric amplification, a renormalization scheme with time-independent counterterms, and late-time behavior that satisfies a Goldstone-consistent sum rule with $\langle\psi^2\rangle_r \sim 1/a^2(t)$ and a transition of the equation of state toward matter domination. The findings illuminate how the expansion rate influences particle production and the approach to a matter-dominated epoch, with implications for reheating and early-universe dynamics.

Abstract

We study the non-linear dynamics of quantum fields in matter and radiation dominated universes, using the non-equilibrium field theory approach combined with the non-perturbative Hartree and the large N approximations. We examine the phenomenon of explosive particle production due to spinodal instabilities and parametric amplification in expanding universes with and without symmetry breaking. For a variety of initial conditions, we compute the evolution of the inflaton, its quantum fluctuations, and the equation of state. We find explosive growth of quantum fluctuations, although particle production is somewhat sensitive to the expansion of the universe. In the large N limit for symmetry breaking scenarios, we determine generic late time solutions for any flat Friedman-Robertson-Walker cosmology. We also present a complete and numerically implementable renormalization scheme for the equation of motion and the energy momentum tensor in flat FRW cosmologies. In this scheme the renormalization constants are independent of time and of the initial conditions.

Figures (12)

• Figure 1: Figure 1: Symmetry broken, slow roll, large $N$, matter dominated evolution of (a) the zero mode $\eta(t)$ vs. $t$, (b) the quantum fluctuation operator $g\Sigma(t)$ vs. $t$, (c) the number of particles $gN(t)$ vs. $t$, (d) the particle distribution $gN_k(t)$ vs. $k$ at $t=149.1$ (dashed line) and $t=398.2$ (solid line), and (e) the ratio of the pressure and energy density $p(t)/\varepsilon(t)$ vs. $t$ for the parameter values $m^2=-1$, $\eta(t_0) = 10^{-7}$, $\dot{\eta}(t_0)=0$, $g = 10^{-12}$, $H(t_0) = 0.1$.
• Figure 2: Figure 2: Symmetry broken, slow roll, Hartree, matter dominated evolution of (a) the zero mode $\eta(t)$ vs. $t$, (b) the quantum fluctuation operator $g\Sigma(t)$ vs. $t$, (c) the number of particles $gN(t)$ vs. $t$, (d) the particle distribution $gN_k(t)$ vs. $k$ at $t=150.7$ (dashed line) and $t=396.1$ (solid line), and (e) the ratio of the pressure and energy density $p(t)/\varepsilon(t)$ vs. $t$ for the parameter values $m^2=-1$, $\eta(t_0) = 3^{1/2}\cdot 10^{-7}$, $\dot{\eta}(t_0)=0$, $g = 10^{-12}$, $H(t_0) = 0.1$.
• Figure 3: Figure 3: Symmetry broken, no roll, matter dominated evolution of (a) the quantum fluctuation operator $g\Sigma(t)$ vs. $t$, (b) the number of particles $gN(t)$ vs. $t$, (c) the particle distribution $gN_k(t)$ vs. $k$ at $t=150.1$ (dashed line) and $t=397.1$ (solid line), and (d) the ratio of the pressure and energy density $p(t)/\varepsilon(t)$ vs. $t$ for the parameter values $m^2=-1$, $\eta(t_0) = 0$, $\dot{\eta}(t_0)=0$, $g = 10^{-12}$, $H(t_0) = 0.1$.
• Figure 4: Figure 4: Symmetry broken, chaotic, large $N$, radiation dominated evolution of (a) the zero mode $\eta(t)$ vs. $t$, (b) the quantum fluctuation operator $g\Sigma(t)$ vs. $t$, (c) the number of particles $gN(t)$ vs. $t$, (d) the particle distribution $gN_k(t)$ vs. $k$ at $t=76.4$ (dashed line) and $t=392.8$ (solid line), and (e) the ratio of the pressure and energy density $p(t)/\varepsilon(t)$ vs. $t$ for the parameter values $m^2=-1$, $\eta(t_0) = 4$, $\dot{\eta}(t_0)=0$, $g = 10^{-12}$, $H(t_0) = 0.1$.
• Figure 5: Figure 5: Symmetry broken, chaotic, large $N$, matter dominated evolution of (a) the zero mode $\eta(t)$ vs. $t$, (b) the quantum fluctuation operator $g\Sigma(t)$ vs. $t$, (c) the number of particles $gN(t)$ vs. $t$, (d) the particle distribution $gN_k(t)$ vs. $k$ at $t=50.8$ (dashed line) and $t=399.4$ (solid line), and (e) the ratio of the pressure and energy density $p(t)/\varepsilon(t)$ vs. $t$ for the parameter values $m^2=-1$, $\eta(t_0) = 4$, $\dot{\eta}(t_0)=0$, $g = 10^{-12}$, $H(t_0) = 0.1$.