  The most conventional mechanism for gravitational waves (gw) production
during inflation is the amplification of vacuum metric fluctuations. In this
case the gw production can be uniquely related to the inflationary expansion
rate $H$. For example, a gw detection close to the present experimental limit
(tensor-to-scalar ratio $r \sim 0.1$) would indicate an inflationary expansion
rate close to $10^{14} \, {\rm GeV}$. This conclusion, however, would be
invalid if the observed gw originated from a different source. We construct and
study one of the possible covariant formulations of the mechanism suggested in
[43], where a spectator field $\sigma$ with a sound speed $c_{s} \ll 1$ acts as
a source for gw during inflation. In our formulation $\sigma$ is described by a
so-called $P(X)$ Lagrangian and a non-minimal coupling to gravity. This field
interacts only gravitationally with the inflaton, which has a standard action.
We compute the amount of scalar and tensor density fluctuations produced by
$\sigma$ and find that, in our realization, $r$ is not enhanced with respect to
the standard result but it is strongly sensitive to $c_s$, thus breaking the
direct $r \leftrightarrow H$ connection.
