Hyperbolic field space and swampland conjecture for DBI scalar
  We study a model of two scalar fields with a hyperbolic field space and show
that it reduces to a single-field Dirac-Born-Infeld (DBI) model in the limit
where the field space becomes infinitely curved. We apply the de Sitter
swampland conjecture to the two-field model and take the same limit. It is
shown that in the limit, all quantities appearing in the swampland conjecture
remain well-defined within the single-field DBI model. Based on a consistency
argument, we then speculate that the condition derived in this way can be
considered as the de Sitter swampland conjecture for a DBI scalar field by its
own. The condition differs from those proposed in the literature and only the
one in the present paper passes the consistency argument. As a byproduct, we
also point out that one of the inequalities in the swampland conjecture for a
multi-field model with linear kinetic terms should involve the lowest mass
squared for scalar perturbations and that this quantity can be significantly
different from the lowest eigenvalue of the Hessian of the potential in the
local orthonormal frame if the field space is highly curved. Finally, we
propose an extension of the de Sitter swampland conjecture to a more general
scalar field with the Lagrangian of the form $P(X,\varphi)$, where
$X=-(\partial\varphi)^2/2$.
