Diffusing non-local inflation: Solving the field equations as an initial value problem

There has been considerable recent interest in solving non-local equations of motion which contain an infinite number of derivatives. Here, focusing on inflation, we review how the problem can be reformulated as the question of finding solutions to a diffusion-like partial differential equation with non-linear boundary conditions. Moreover, we show that this diffusion-like equation, and hence the non-local equations, can be solved as an initial value problem once non-trivial initial data consistent with the boundary conditions is found. This is done by considering linearised equations about any field value, for which we show that obtaining solutions using the diffusion-like equation is equivalent to solving a local but infinite field cosmology. These local fields are shown to consist of at most two canonically normalized or phantom fields together with an infinite number of quintoms. We then numerically solve the diffusion-like equation for the full non-linear case for two string field theory motivated models.