p-adic Inflation

We construct approximate inflationary solutions rolling away from the unstable maximum of p-adic string theory, a nonlocal theory with derivatives of all orders. Novel features include the existence of slow-roll solutions even when the slow-roll parameters, as usually defined, are much greater than unity, as well as the need for the Hubble parameter to exceed the string mass scale m_s. We show that the theory can be compatible with CMB observations if g_s^2 / (\ln p) \sim 10^{-15}, where g_s is the string coupling, and if m_s < 10^{-6} M_p. A red-tilted spectrum is predicted, and an observably large tensor component is possible. The p-adic theory is shown to have identical inflationary predictions to a local theory with superPlanckian parameter values, but with the advantage that the p-adic theory is ultraviolet complete