  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.
