Simple model for the inflationary expansion of the early Universe

The evolution of the early Universe is studied under the following assumptions. (1) The large-scale structure of the Universe is homogeneous and isotropic and governed by the Robertson-Walker metric in flat space with a spatial scale factor a(t). (ii) Matter can be described by a quantum-mechanical N-vector model with a single- or double-well potential. (iii) The system is initially in thermal equilibrium at some high temperature T relative to the Planck temperature. Quantitative progress is possible in the large-N limit where the phase structure of the N-vector model is known exactly. The evolution of the system is characterized by three time regimes. For the earliest times the main effect of gravity is to rapidly decrease the kinetic energy of the system, while the potential energy and the local ‘‘order’’-parameter fluctuation S(t) change very little. Thus the system is driven far from equilibrium. There is rapid growth of a(t) during this period. One enters the intermediate time period after a(t) is sufficiently large that the spectrum of the order-parameter correlation function ‘‘freezes’’ due to red-shifting. By this time the energy density ε is dominated by the potential energy and one enters a de Sitter phase where, approximately, the pressure P=-ε and a(t) is growing exponentially. In this regime the Hubble constant and S(t) are decaying exponentially with precisely the same slow rate. The amount of inflation depends most strongly on the initial temperature [lna≊(T/$T_P$ ${)}^2$] where $T_P$ is the Planck temperature. In the final stage the system crosses over to behavior in agreement with the standard model. Except for the ‘‘critical’’ case where the quadratic part of the potential vanishes, the system appears to be ‘‘matter’’ dominated: a(t)≊$t^{2/3}$ and the system grows the ordered state associated with a spontaneously broken symmetry if the quadratic part of the potential is negative. In the critical case the system appears to be radiation dominated and a(t)≊$t^{1/2}$.