Shi Pi, Anxianyi Xiong

We revisit cosmological perturbations in multi-field inflation using the $δN$ formalism. By extending the analysis to directions transverse to the inflationary trajectory, we explicitly account for the geometry of the final hypersurface. Applying this framework to hybrid inflation, we identify an enhancement mechanism of the curvature perturbation driven by the growing isocurvature perturbation due to the tachyonic instability of the waterfall field. This amplification occurs during the trajectory's turn in field space, a process qualitatively distinct from the non-attractor solution in single-field inflation models such as ultra-slow-roll. The resulting power spectrum features a broad peak with a characteristic $k^3$ infrared growth and a ultraviolet spectral tilt that uniquely determines the nonlinear parameter $f_\mathrm{NL}$ of a logarithmic non-Gaussianity, all of which are primarily governed by the waterfall dynamics. We found that in hybrid inflation, the sign of $f_{\rm NL}$ is fixed by tachyonic waterfall geometry and is always positive, leading to a generic enhancement of primordial black hole formation. The enhanced curvature perturbation can simultaneously account for primordial black hole dark matter and a stochastic gravitational wave background detectable by LISA, Taiji, and TianQin.