Strong rapid turn inflation and contact Hamilton-Jacobi equations

Abstract: We consider the consistency condition for strong'' sustained rapid turn inflation with third order slow roll (SRRT) in two-field cosmological models with oriented scalar manifold as a geometric PDE which constrains the metric and potential of such models. When supplemented by appropriate boundary conditions, the equation determines one of these objects in terms of the other and hence selectsfiducial'' models for strong SRRT inflation. When the scalar potential is given, the equation can be simplified by fixing the conformal class of the scalar field metric (equivalently, fixing a complex structure which makes the scalar manifold into a complex Riemann surface). Then the consistency equation becomes a contact Hamilton-Jacobi PDE which determines the scalar field metric within the given conformal class. We analyze this equation with standard methods of PDE theory, discuss its approximation near a nondegenerate critical point of the scalar potential and extract natural asymptotic conditions for its solutions at such points. We also give numerical examples of solutions to a simple Dirichlet problem. For the case of elliptic curves relevant to two-field axion cosmology, we determine the general symmetry-adapted solution of the equation for potentials with a single charge vector.