Positivity bounds from multiple vacua and their cosmological consequences

Positivity bounds - constraints on any low-energy effective field theory imposed by the fundamental axioms of unitarity, causality and locality in the UV - have recently been used to constrain various effective field theories relevant for cosmology. However, to date most of these bounds have assumed that there is a single Lorentz-invariant vacuum in which all fields have zero expectation value and in many cosmologically relevant models this is not the case. We explore ways to overcome this limitation by investigating a simple example model, the covariant Galileon, which possesses a one-parameter family of Lorentz-invariant vacua as well as multiple boost-breaking vacua. Each of these vacua has a corresponding set of positivity bounds, and we show how a particular (beyond-the-forward-limit) bound can be used to map out the parameter space according to which vacua may persist in the UV theory, finding that in general there are regions in which none, one or many of the effective field theory vacua can be consistent with unitarity, causality and locality in the UV. Finally, we discuss the interplay between this map and cosmological observations. We find that the observationally favoured region of parameter space is incompatible with a large class of vacua, and conversely that particular boost-breaking vacua would imply positivity bounds that rule out otherwise observationally favoured cosmologies. We also identify a specific boost-breaking vacuum which is `closest' to the cosmological background, and show that the particular positivity bound we consider reduces the otherwise cosmologically favoured region of Galileon parameter space by up to $70 \%$, ruling out the vast majority of cosmologies with a positive coefficient for the cubic Galileon in the process.