http://arxiv.org/abs/1508.02670
In this paper, we analyze a Bianchi Type I model with a scalar field in a chaotic inflation potential, $V(\phi) = \frac{1}{2}\phi^2$ in the context of stochastic eternal inflation. We use the typical slow-roll approximation in combination with expansion-normalized variables in an orthonormal frame approach to obtain a dynamical system which describes the dynamics of the shear anisotropy and the inflaton field. We first show that the dynamics of the inflaton field can be decoupled from the dynamics of the shear anisotropy. We then use a fixed-points analysis in combination with global techniques from topological dynamical systems theory to prove that the cosmological model under consideration isotropizes irrespective of an inflationary epoch, which has also described by other authors who have investigated a Bianchi Type I model under similar configurations. We then show that for inflation to occur, the amount of anisotropy must be very small.
We also give a description of the stochastic dynamics of the inflaton field by using techniques from stochastic calculus. We show that the Klein-Gordon equation becomes a stochastic differential equation with a highly nonlinear drift term. In this case, the deceleration parameter itself becomes a random variable, and we give details regarding when such a model can undergo inflation. We finally derive the form of the long-term, stationary probability distribution of the inflaton field, and show that it has the form of a double-well potential. We then calculate the probability of inflation occurring based on this approach. We conclude the paper by performing some numerical simulations of the stochastic differential equation describing the dynamics of the inflaton field. We conjecture that even in the case of stochastic eternal inflation, one requires precise initial conditions for inflation to occur.
I. Kohli and M. Haslam
Wed, 12 Aug 15
50/50
Comments: N/A