Defect formation and critical dynamics in the early Universe

We study the nonequilibrium dynamics leading to the formation of topological defects in a symmetry-breaking phase transition of a quantum scalar field with $\lambda {\Phi }^4$ self-interaction in a spatially flat, radiation-dominated Friedmann-Robertson-Walker universe. The quantum field is initially in a finite-temperature symmetry-restored state and the phase transition develops as the universe expands and cools. We present a first-principles, microscopic approach in which the nonperturbative, nonequilibrium dynamics of the quantum field is derived from the two-loop, two-particle-irreducible closed-time-path effective action. We numerically solve the dynamical equations for the two-point function and we identify signatures of correlated domains in the infrared portion of the momentum-space power spectrum. We find that correlated domains formed during the phase transition scale in size as a power law with the expansion rate of the universe. We calculate the equilibrium critical exponents of the correlation length and relaxation time for this model and show that the power law exponent of the domain size, for both overdamped and underdamped evolution, is in good agreement with the “freeze-out” scenario proposed by Zurek. We introduce an analytic dynamical model, valid near the critical point, that exhibits the same power-law scaling of the size of correlated domains with the quench rate. The size of correlated domains provides an approximate measure of the initial scale of the topological defect density. By incorporating the realistic quench of the expanding universe our approach illuminates the dynamical mechanisms important for topological defect formation, and provides a preliminary step towards a complete and rigorous picture of defect formation in a second-order phase transition of a quantum field. The observed power law scaling of the size of correlated domains with the quench rate, calculated here in a quantum field theory context, provides evidence for the “freeze-out” scenario in three spatial dimensions.