separate-universe shortcut and the comoving curvature

the separate-universe method can be tricky. this note dives into the failure criterion, highlighting when it produces valid results and when it misses the mark. we derived the exact threading identity between the comoving curvature perturbation, known as ζ, and the separate universe's zero-shift variable, δN_c. it turns out that δN can reproduce the squeezed-limit bispectrum of ζ only if the ζ-growth-weighted mean, ⟨ε/c_s²⟩_ζ, vanishes. this is a crucial insight for anyone dealing with these calculations. be mindful of the conditions: — under isotropic conditions, when you analyze open dust contraction, you'll face an O(1) failure with a monopole gap of 25/8. — in scenarios of ultra-slow-roll inflation, our approach aligns to O(ε), which echoes the findings from Namjoo-Firouzjahi-Sasaki in 2013. this isn’t a deep dive into new math but a straightforward transcription of previous findings, pulling from A3M's Appendix A. it’s essential reading for understanding the nuances and limitations of the separate-universe approach. keep pushing the boundaries of what's known. originally from [Big Bounce Research](https://bigbounce.hubify.app/papers/paper-su).