Large scale magnetogenesis from a non-equilibrium phase transition in the radiation dominated era

We study the generation of large scale primordial magnetic fields by a cosmological phase transition during the radiation dominated era. The setting is a theory of N charged scalar fields coupled to an abelian gauge field, that undergoes a phase transition at a critical temperature much larger than the electroweak scale. The dynamics after the transition features two distinct stages: a spinodal regime dominated by linear long-wavelength instabilities, and a scaling stage in which the non-linearities and backreaction of the scalar fields are dominant. This second stage describes the growth of horizon sized domains. We implement a recently introduced formulation to obtain the spectrum of magnetic fields that includes the dissipative effects of the plasma. We find that large scale magnetogenesis is very efficient during the scaling regime. The ratio between the energy density on scales larger than L and that in the background radiation r(L,T) = rho_B(L,T)/rho_{cmb}(T) is r(L,T) \sim 10^{-34} at the Electroweak scale and r(L,T) \sim 10^{-14} at the QCD scale for L \sim 1 Mpc. The resulting spectrum is insensitive to the magnetic diffusion length. We conjecture that a similar mechanism could be operative after the QCD chiral phase transition.Comment: LaTex, 25 pages, no figures, to appear in Phys. Rev.

Non-equilibrium dynamics in quantum field theory at high density: the tsunami

The dynamics of a dense relativistic quantum fluid out of thermodynamic equilibrium is studied in the framework of the Phi^4 scalar field theory in the large N limit. The time evolution of a particle distribution in momentum space (the tsunami) is computed. The effective mass felt by the particles in such a high density medium equals the tree level mass plus the expectation value of the squared field. The case of negative tree level squared mass is particularly interesting. In such case dynamical symmetry restoration as well as dynamical symmetry breaking can happen. Furthermore, the symmetry may stay broken with vanishing asymptotic squared mass showing the presence of out of equilibrium Goldstone bosons. We study these phenomena and identify the set of initial conditions that lead to each case. We compute the equation of state which turns to depend on the initial state. Although the system does not thermalize, the equation of state for asymptotically broken symmetry is of radiation type. We compute the correlation functions at equal times. The two point correlator for late times is the sum of different terms. One stems from the initial particle distribution. Another term accounts for the out of equilibrium Goldstone bosons created by spinodal unstabilities when the symmetry is asymptotically broken.Both terms are of the order of the inverse of the coupling for distances where causal signals can connect the two points. The contribution of the out of equilibrium Goldstones exhibits scaling behaviour in a generalized sense.Comment: LaTex, 49 pages, 15 .ps figure

Magnetic field generation from non-equilibrium phase transitions

We study the generation of magnetic fields during the stage of particle production resulting from spinodal instabilities during phase transitions out of equilibrium. The main premise is that long-wavelength instabilities that drive the phase transition lead to strong non-equilibrium charge and current fluctuations which generate electromagnetic fields. We present a formulation based on the non-equilibrium Schwinger-Dyson equations that leads to an exact expression for the spectrum of electromagnetic fields valid for general theories and cosmological backgrounds and whose main ingredient is the transverse photon polarization out of equilibrium. This formulation includes the dissipative effects of the conductivity in the medium. As a prelude to cosmology we study magnetogenesis in Minkowski space-time in a theory of N charged scalar fields to lowest order in the gauge coupling and to leading order in the large N within two scenarios of cosmological relevance. The long-wavelength power spectrum for electric and magnetic fields at the end of the phase transition is obtained explicitly. It follows that equipartition between electric and magnetic fields does not hold out of equilibrium. In the case of a transition from a high temperature phase, the conductivity of the medium severely hinders the generation of magnetic fields, however the magnetic fields generated are correlated on scales of the order of the domain size, which is much larger than the magnetic diffusion length. Implications of the results to cosmological phase transitions driven by spinodal unstabilities are discussed.Comment: Preprint no. LPTHE 02-55, 30 pages, latex, 2 eps figures. Added one reference. To appear in Phys. Rev.

Dynamical renormalization group approach to relaxation in quantum field theory

The real time evolution and relaxation of expectation values of quantum fields and of quantum states are computed as initial value problems by implementing the dynamical renormalization group (DRG).Linear response is invoked to set up the renormalized initial value problem to study the dynamics of the expectation value of quantum fields. The perturbative solution of the equations of motion for the field expectation values of quantum fields as well as the evolution of quantum states features secular terms, namely terms that grow in time and invalidate the perturbative expansion for late times. The DRG provides a consistent framework to resum these secular terms and yields a uniform asymptotic expansion at long times. Several relevant cases are studied in detail, including those of threshold infrared divergences which appear in gauge theories at finite temperature and lead to anomalous relaxation. In these cases the DRG is shown to provide a resummation akin to Bloch-Nordsieck but directly in real time and that goes beyond the scope of Bloch-Nordsieck and Dyson resummations. The nature of the resummation program is discussed in several examples. The DRG provides a framework that is consistent, systematic and easy to implement to study the non-equilibrium relaxational dynamics directly in real time that does not rely on the concept of quasiparticle widths.Comment: LaTex, 27 pages, 2 .ps figure

Effective theory for the soft fluctuation modes in the spontaneously broken phase of the N-component scalar field theory

The effective dynamics of the low-frequency modes is derived for the O(N) symmetric scalar field theory in the broken symmetry phase. The effect of the high-frequency fluctuations is taken into account at one-loop level exactly. A new length scale is shown to govern the long-time asymptotics of the linear response function of the Goldstone modes. The large time asymptotic decay of an arbitrary fluctuation is determined in the linear regime. We propose a set of local equations for the numerical solution of the effective non-linear dynamics. The applicability of the usual gradient expansion is carefully assessed.Comment: 21 pages, LaTeX; final version to appear in Phys. Rev.

An improved time-dependent Hartree-Fock approach for scalar \phi^4 QFT

The $\lambda \phi^4$ model in a finite volume is studied within a non-gaussian Hartree-Fock approximation (tdHF) both at equilibrium and out of equilibrium, with particular attention to the structure of the ground state and of certain dynamical features in the broken symmetry phase. The mean-field coupled time-dependent Schroedinger equations for the modes of the scalar field are derived and the suitable procedure to renormalize them is outlined. A further controlled gaussian approximation of our tdHF approach is used in order to study the dynamical evolution of the system from non-equilibrium initial conditions characterized by a uniform condensate. We find that, during the slow rolling down, the long-wavelength quantum fluctuations do not grow to a macroscopic size but do scale with the linear size of the system, in accordance with similar results valid for the large $N$ approximation of the O(N) model. This behavior undermines in a precise way the gaussian approximation within our tdHF approach, which therefore appears as a viable mean to correct an unlikely feature of the standard HF factorization scheme, such as the so-called ``stopping at the spinodal line'' of the quantum fluctuations. We also study the dynamics of the system in infinite volume with particular attention to the asymptotic evolution in the broken symmetry phase. We are able to show that the fixed points of the evolution cover at most the classically metastable part of the static effective potential.Comment: Accepted for publication on Phys. Rev.