= Solution
Put $\rho=\rho_0+\delta\rho$ and retain quadratic fluctuations in the smooth zero-winding sector. The <quadratic density-phase action> is
$$
S_2=\int\left[\frac g2(\delta\rho)^2+i\delta\rho\,\partial_\tau\phi+\frac{|\nabla\delta\rho|^2}{8m\rho_0}+\frac{\rho_0}{2m}|\nabla\phi|^2\right].
$$
At wavelengths long compared with the <healing length>, drop the density-gradient term. Completing the square gives
$$
\frac g2(\delta\rho)^2+i\delta\rho\,\partial_\tau\phi
=\frac g2\left(\delta\rho+\frac{i\partial_\tau\phi}g\right)^2+\frac{(\partial_\tau\phi)^2}{2g}.
$$
The real Gaussian density integral, or its equivalent contour shift, contributes only a field-independent <determinant>. Absorb that normalization and the uniform saddle action into $S_0$. To this quadratic long-wave accuracy,
$$
\boxed{Z\simeq e^{-S_0}\int\mathcal D\phi\,e^{-S_{\rm eff}},\qquad
S_{\rm eff}=\frac12\int_0^\beta d\tau\int d^dr\left[\frac1g(\partial_\tau\phi)^2+\frac{\rho_0}m|\nabla\phi|^2\right].}
$$
The Gaussian extension of $\delta\rho$ to the full real line is a fluctuation approximation around positive $\rho_0$; it is not an exact replacement of the global density constraint. The compact field's vortex/winding sectors also lie beyond this smooth-phonon integral.
The <Bose-gas phase-only action> is a continuum harmonic chain with Euclidean inverse propagator $\omega_n^2/g+\rho_0k^2/m$. Continuing to real frequency gives
$$
\boxed{E_k\sim c_s|k|,\qquad c_s=\sqrt{g\rho_0/m}=\sqrt{\mu/m}.}
$$
Restoring $\hbar$ gives $E_k\sim\hbar c_s|k|$ for a <wavenumber> $k$. This is the <phonon> branch of the <Bogoliubov spectrum>. Keeping the omitted density-gradient term replaces $g$ by $g+k^2/(4m\rho_0)$ and yields $E_k^2=c_s^2k^2+k^4/(4m^2)$ in the adopted units, so the full quadratic spectrum is $\sqrt{\epsilon_k(\epsilon_k+2g\rho_0)}$ with $\epsilon_k=k^2/(2m)$.
The phase-only action also tests the condensate assumption. At zero temperature its equal-time phase variance has an infrared contribution $\int d^dk/|k|$, logarithmically divergent in one dimension; at positive temperature the zero Matsubara mode gives $\int d^dk/k^2$, divergent in dimensions at or below two. Thus this same low-energy theory exposes the regimes where the mean-field condensate cannot describe true thermodynamic long-range order. It permits a finite infrared fluctuation at zero temperature for $d\ge2$ and at positive temperature for $d>2$, within the remaining weak-coupling assumptions.
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