Abstract

We study how both the swampland distance conjecture and the Lyth bound affect the parameter space of multi-field models of inflation. A generic feature of multi-field inflation is that the geodesic distance [Delta phi](G) separating any two points laying along the inflationary trajectory differs from the non-geodesic distance [Delta phi](NG) traversed by the inflaton between those points. These distances must respect a relation of the form [Delta phi](G) = f ([Delta phi](NG)) <= [Delta phi](NG), where f is a function determined by the specific multi-field model under scrutiny. We show that this relation leads to important constraints on the parameter space characterizing the multi-field dynamics. Indeed, the swampland distance conjecture implies an upper bound on [Delta phi](G) set by the details of the ultraviolet completion of inflation, whereas the Lyth bound implies a lower bound on [Delta phi](NG) determined by the value of the tensor-to-scalar ratio. If future observations confirm the existence of primordial tensor perturbations, these two bounds combined lead to tight constraints on the possible values of the entropy mass of the isocurvature fields orthogonal to the inflationary trajectory and the rate of turn of the inflationary trajectory in multi-field space. We analyze the emerging constraints in detail for the particular case of two-field inflation in hyperbolic field spaces.