We study, both numerically and analytically, the development of equilibrium after preheating. We show that the process is characterized by the appearance of Kolmogorov spectra and the evolution towards thermal equilibrium follows self-similar dynamics. Simplified kinetic theory gives values for all characteristic exponents which are close to what is observed in lattice simulations. The resulting time for thermalization is long, and temperature at thermalization is low, $T\ensuremath{\sim}100\text{ }\text{ }\mathrm{e}\mathrm{V}$ in the simple $\ensuremath{\lambda}{\ensuremath{\Phi}}^{4}$ inflationary model. Our results allow a straightforward generalization to realistic models.

Relativistic Turbulence: A Long Way from Preheating to Equilibrium

We discuss the onset of symmetry breaking from the false vacuum in generic scenarios in which the mass squared of the symmetry breaking (Higgs) field depends linearly with time, as it occurs, via the evolution of the inflaton, in models of hybrid inflation. We show that the Higgs fluctuations evolve from quantum to classical during the initial stages. This justifies the subsequent use of real-time lattice simulations to describe the fully nonperturbative and nonlinear process of symmetry breaking. The early distribution of the Higgs field is that of a smooth classical Gaussian random field, and consists of lumps whose shape and distribution is well understood analytically. The lumps grow with time and develop into ``bubbles'' which eventually collide among themselves, thus populating the high momentum modes, in their way towards thermalization at the true vacuum. With the help of some approximations we are able to provide a quasianalytic understanding of this process.

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Symmetry breaking and false vacuum decay after hybrid inflation

The quantum time evolution of ${\ensuremath{\varphi}}^{4}$-field theory for a spatially homogeneous system in 2+1 space-time dimensions is investigated numerically for out-of-equilibrium initial conditions on the basis of the Kadanoff-Baym equations including the tadpole and sunset self-energies. Whereas the tadpole self-energy yields a dynamical mass, the sunset self-energy is responsible for dissipation and an equilibration of the system. In particular we address the dynamics of the spectral (``off-shell'') distributions of the excited quantum modes and the different phases in the approach to equilibrium described by Kubo-Martin-Schwinger relations for thermal equilibrium states. The investigation explicitly demonstrates that the only translation invariant solutions representing the stationary fixed points of the coupled equation of motions are those of full thermal equilibrium. They agree with those extracted from the time integration of the Kadanoff-Baym equations for $\stackrel{\ensuremath{\rightarrow}}{t}\ensuremath{\infty}.$ Furthermore, a detailed comparison of the full quantum dynamics to more approximate and simple schemes such as that of a standard kinetic (on-shell) Boltzmann equation is performed. Our analysis shows that the consistent inclusion of the dynamical spectral function has a significant impact on relaxation phenomena. The different time scales that are involved in the dynamical quantum evolution towards a complete thermalized state are discussed in detail. We find that far off-shell $1\ensuremath{\leftrightarrow}3$ processes are responsible for chemical equilibration, which is missed in the Boltzmann limit. Finally, we briefly address the case of (bare) massless fields. For sufficiently large couplings $\ensuremath{\lambda}$ we observe the onset of Bose condensation, where our scheme within symmetric ${\ensuremath{\varphi}}^{4}$ theory breaks down.

Quantum dynamics and thermalization for out-of-equilibrium phi^4-theory

PACS numbers: 98.80.Cq.-- 28 pages, 15 figures, ReVTeX.-- Submitted on 30 Apr 2003 (v1), last revised 16 Sep 2003 (this version, v2).-- With minor corrections, version to appear in Phys. Rev. D.

Chern-Simons production during preheating in hybrid inflation models

We consider a scenario in which the baryon asymmetry was created in the early universe during a cold electroweak transition. The spinodal instability of the Higgs field caused by a rapid change of sign of its effective mass-squared parameter induces tachyonic preheating. We study the development of Chern-Simons number in this transition by numerical lattice simulations of the SU(2)-Higgs model with an added effective CP-violating term. A net asymmetry is produced, and we study its dependence on the size of CP violation and the ratio of Higgs to W mass.

Baryon asymmetry from electroweak tachyonic preheating

For homogeneous initial conditions, Hartree (gaussian) dynamical approximations are known to have problems with thermalization, because of insufficient scattering. We attempt to improve on this by writing an arbitrary density matrix as a superposition of gaussian pure states and applying the Hartree approximation to each member of such an ensemble. Particles can then scatter via their back-reaction on the typically inhomogeneous mean fields. Starting from initial states which are far from equilibrium we numerically compute the time evolution of particle distribution functions and observe that they indeed display approximate thermalization on intermediate time scales by approaching a Bose-Einstein form. However, for very large times the distributions drift towards classical-like equipartition.

Thermalization in a Hartree ensemble approximation to quantum field dynamics

Staying thermal with Hartree ensemble approximations

The Hartree ensemble approximation is studied in the ``symmetric phase'' of $1+1$ dimensional $\ensuremath{\lambda}{\ensuremath{\varphi}}^{4}$ theory. In comparison with the ``broken phase'' studied previously, it is shown that the dynamical evolution of observables such as the particle distribution, energy exchange and auto-correlation functions is substantially slower. Approximate thermalization is found only for relatively large energy densities and couplings.

The Hartree ensemble approximation revisited: The "symmetric phase"

In numerical simulations studying preheating in the classical approximation there is the problem how to derive the classical initial conditions from the quantum vacuum fluctuations. In past treatments, the initial conditions often put an energy density into the classical field of order of the cutoff, leading to a divergent temperature after thermalization. We suggest a solution to the problem which follows naturally from a Hartree ensemble approximation, introduced recently as an improvement over the standard Hartree approximation. We study the effects on particle numbers of the various treatments, within the context of `tachyonic preheating' in 1+1 dimensional $\phi^4$ theory.

Initial conditions for simulated `tachyonic preheating' and the Hartree ensemble approximation

The topological defects of the lambda phi^4 theory, kink and antikink, are studied in the Hartree approximation. This allows us to discuss quantum effects on the defects in both stationary and dynamical systems. The kink mass is calculated for a number of parameters, and compared to classical, one loop and Monte Carlo results known from the literature. We discuss the thermalization of the system after a kink antikink collision. A classical result, the existence of a critical speed, is rederived and shown for the first time in the quantum theory. We also use kink antikink collisions as a very simple toy model for heavy ion collisions and discuss the differences and similarities, for example in the pressure. Finally, using the Hartree Ensemble Approximation allows us to study kink antikink nucleation starting from a thermal (Bose Einstein) distribution. In general our results indicate that on a qualitative level there are few differences with the classical results, but on a quantitative level there are some import ones.

M. Salle

Papers

Thermalization in a Hartree ensemble approximation to quantum field dynamics

Staying thermal with Hartree ensemble approximations

The Hartree ensemble approximation revisited: The "symmetric phase"

Initial conditions for simulated `tachyonic preheating' and the Hartree ensemble approximation

Kinks in the Hartree approximation