# Hyperbolic field space and swampland conjecture for DBI scalar

## TL;DR

This paper explores a hyperbolic two-scalar field model reducing to a single-field DBI model, applying the swampland conjecture, and proposing a new conjecture condition that remains consistent in the limit, with implications for multi-field models.

## Contribution

It introduces a new swampland conjecture condition for DBI scalar fields derived from a hyperbolic two-field model and extends the conjecture to general scalar fields with non-standard kinetic terms.

## Key findings

- The swampland condition remains well-defined in the single-field limit.
- The proposed condition differs from existing literature and passes a key consistency check.
- An extension of the conjecture to $P(X,\varphi)$ scalar fields is proposed.

## Abstract

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 $X=-(\partial\varphi)^2/2$.