On the stochastic approach to inflation and the initial conditions in the universe

Abstract

By the application of stochastic methods to a theory in which a potential V( ø) causes a period of quasi-exponential expansion of the universe, an expression for the probability distribution P( V) appropriate for chaotic inflation has recently been derived. The method was developed by Starobinsky and by Linde. Beyond some critical point ø c, long-wavelength quantum fluctuations δø ≈ H/2 π cannot be ignored. The effect of these fluctuation in general relativity for values of ø such that V( ø)> V( ø) has been considered by Linde, who concluded that most of the present universe arises as a result of expansion of domains with a domains with a maximum possible value of ø, such that V( ø max ≈ m p 4. We obtain the corresponding expression for P in a broken-symmetry theory of gravity, in which the newtonian gravitational constant is replaced by G = (8 πϵø 2) −1, and also for a theory which includes higher-derivative terms R 2 = γR 2 + βR 2 1n(R/μ 2) , so that the trace anomaly is T anom ≈ βR 2 , in which an effective inflation field ø e can be defined as ø e 2 = 24 γR. Conclusions analogous to those of Linde can be drawn in both these theories.