Quadratic density-phase action ($S_2=\int [g(\delta \rho {)}^2/2+i\delta \rho {\partial }_{\tau }\varphi +|\nabla \delta \rho {|}^2/(8m{\rho }_0)+{\rho }_0|\nabla \varphi {|}^2/(2m)]$)

Expand the density-phase action of a Bose gas around ${\rho }_0=\mu /g>0$ and a constant phase. Density and phase are coupled by $i\delta \rho {\partial }_{\tau }\varphi$, and retaining density gradients gives the full quadratic Bogoliubov spectrum. Dropping those gradients is a long-wavelength approximation. A small-fluctuation Gaussian extension around ${\rho }_0$ does not exactly remove the original nonnegative-density constraint.

In the smooth zero-winding long-wave sector, integrate the density Gaussian integral by completing $g(\delta \rho +i{\partial }_{\tau }\varphi /g{)}^2/2$. Its field-independent normalization leaves the displayed action. The real-frequency pole is ${\omega }^2=g{\rho }_0{k}^2/m$, so the phonon speed is $\sqrt{g{\rho }_0/m}$. Keeping density gradients yields ${\omega }^2=g{\rho }_0{k}^2/m+k^4/(4m^2)$, matching the Bogoliubov spectrum.