Self Similar Solutions of the Evolution Equation of a Scalar Field in an Expanding Geometry

We consider the functional Schrödinger equation for a self interacting scalar field in an expanding geometry. By performing a time dependent scale transformation on the argument of the field we derive a functional Schrödinger equation whose hamiltonian is time independent but involves a time-odd term associated to a constraint on the expansion current. We study the mean field approximation to this equation and generalize in this case, for interacting fields, the solutions worked out by Bunch and Davies for free fields. IPNO/TH 94-39 May 1994 Doctoral fellow of Coordenação de Aperfeiçoamento de Pessoal de Nı́vel Superior,Brasil Unité de Recherche des Universités Paris XI et Paris VI associée au CNRS 1 Several authors have pointed out that inflationary models based on effective potentials ignore the fact that the very concept of effective potential implies a static approximation which may be invalidated by dynamical effects [1]. The purpose of this present letter is to investigate this question using the functional Schrödinger equation for a scalar field in an expanding geometry: ih̄∂tΦ[φ] = 1 2 ∫ dx{− h̄ 2 a3(t) δΦ δφ(x)δφ(x) + [a(t)(∇φ(x)) + a(t)(m0φ(x) + λ 24 φ(x))]Φ}. (1) In this equation a(t) = exp(χt) specifies the metric ds = dt2−a2(t)dx2 and χ is the expansion parameter (Hubble’s constant). The constants m0 and λ are respectively the bare mass of the field and the bare coupling constant. The evolution equation (1) was considered by Guth and Pi [2] and by Eboli, Jackiw and SoYoung Pi [3]. In particular in reference [3] a numerical solution of this equation was attempted in the mean field approximation. However instabilities were found in the implementation of the method. Our aim in the present study is to perform a particular scale transformation on the wave functional which brings equation (1) into the form of a static functional Schrödinger equation. The method, although more elaborate, is reminiscent of the transformation used in nuclear physics to describe rotating nuclei [4]. Let us look for solutions of equation (1) in the form Φ[φ(x), t] = Ψ[ξα(x), t] 1