  Inflationary mechanisms for generating primordial fluctuations ultimately
compute them as the leading contributions in a derivative expansion, with
corrections controlled by powers of derivatives like the Hubble scale over
Planck mass: $H/M_p$. At face value this derivative expansion breaks down for
models with a small sound speed, $c_s$, to the extent that $c_s \ll 1$ is
obtained by having higher-derivative interactions like $\mathfrak{L}_{\rm eff}
\sim (\partial \Phi)^4$ compete with lower-derivative propagation. This concern
arises more generally for models whose lagrangian is given as a function $P(X)$
for $X = -\partial_\mu \Phi \partial^\mu \Phi$ --- including in particular DBI
models for which $P(X) \propto \sqrt{1-kX}$ --- since these keep all orders in
$\partial \Phi$ while dropping $\partial^n \Phi$ for $n > 1$. We here find a
sensible power-counting scheme for DBI models that gives a controlled expansion
in powers of three types of small parameters: $H/M_p$, slow-roll parameters
(possibly) and $c_s \ll 1$. We do not find a similar expansion framework for
generic small-$c_s$ or $P(X)$ models. Our power-counting result quantifies the
theoretical error for any prediction (such as for inflationary correlation
functions) by fixing the leading power of these small parameters that is
dropped when not computing all graphs (such as by restricting to the classical
approximation); a prerequisite for meaningful comparisons with observations.
The new power-counting regime arises because small $c_s$ alters the kinematics
of free fluctuations in a way that changes how interactions scale at low
energies, in particular allowing $1-c_s$ to be larger than derivative-measuring
quantities like $(H/M_p)^2$.
