Keeping an Eye on DBI: Power-counting for small-$c_s$ Cosmology

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 $(H/M_p)^2$.