A note on the equivalence of a barotropic perfect fluid with a K-essence scalar field
Frederico Arroja, Misao Sasaki
TL;DR
The paper addresses when a noncanonical single-field model with Lagrangian $P(X,\phi)$ is exactly dual to a barotropic perfect fluid under irrotational flow. By equating the scalar perturbation speed $c_{ph}^2=\frac{P_{,X}}{\rho_{,X}}$ with the adiabatic speed $c_s^2=\frac{\dot P_0}{\dot \rho_0}$, the authors derive a second-order PDE for $P(X,\phi)$ and show its general solution is $P(X,\phi)=f(Xg(\phi))$, with $Y=Xg(\phi)$. After a field redefinition, the Lagrangian becomes a purely kinetic k-essence model $P(Y)$, and for this class the nonadiabatic pressure perturbation vanishes at all orders and scales, matching the barotropic-fluid behavior. This duality provides a robust bridge between fluid and scalar-field descriptions, enabling cross-checks of perturbative results and clarifying when canonical scalar fields fall outside this equivalence. Note that the canonical case $P(X,\phi)=X-V(\phi)$ is not included in the dual class.
Abstract
In this short note, we obtain the necessary and sufficient condition for a class of non-canonical single scalar field models to be exactly equivalent to barotropic perfect fluids, under the assumption of an irrotational fluid flow. An immediate consequence of this result is that the non-adiabatic pressure perturbation in this class of scalar field systems vanishes exactly at all orders in perturbation theory and on all scales. The Lagrangian for this general class of scalar field models depends on both the kinetic term and the value of the field. However, after a field redefinition, it can be effectively cast in the form of a purely kinetic K-essence model.