The refined de Sitter conjecture

Let $\Phi=(\phi^1,\dots,\phi^N)$ be scalar fields with potential $V(\Phi)$ in a low-energy effective field theory coupled to dynamical $D$-dimensional spacetime. Let $\nabla_i$ denote the covariant derivative for the field-space metric, and let $\nabla_i\nabla_jV$ be evaluated in an orthonormal frame. Refined de Sitter conjecture. At least one of the inequalities

$$|\nabla_iV|\geq\frac{c_1}{M_{\mathrm{P}}}V$$

and

$$\min(\nabla_i\nabla_jV)\leq-\frac{c_2}{M_{\mathrm{P}}^2}V$$

must hold, where $c_1$ and $c_2$ are positive constants of order one.

This conjecture is a proposed obstruction to stable de Sitter vacua in quantum gravity and refines the original de Sitter conjecture by allowing a sufficiently tachyonic direction. Its status is not resolved here.