(b) Inertia-dominated / resonant regime. If the vortex has appreciable mass and the susceptibility features a resonance at 0=K/mv\omega_0=\sqrt{K/m_v}, then ()2 |\chi(\omega)|^2 is sharply peaked near 0\omega_0. For a white-ish bath or one with weight near 0\omega_0,
Var()G2S(0)mv202eff,\mathrm{Var}(\delta\Phi) \;\propto\; |\mathbf{G}|^2\,\frac{S_\xi(\omega_0)}{m_v^2\omega_0^2\Gamma_{\rm eff}},
where eff/mv\Gamma_{\rm eff}\sim \eta/m_v is an effective linewidth. Rewriting gives the approximate scaling
Var()G2mv0S(0).(6)\mathrm{Var}(\delta\Phi) \;\propto\; \frac{|\mathbf{G}|^2}{m_v \, \omega_0}\;\frac{S_\xi(\omega_0)}{\eta}. \tag{6}
Qualitatively: increasing mvm_v shifts the resonance 0\omega_0 down and can either reduce variance (by decoupling from high-frequency bath noise) or increase it (if the bath has strong weight at the shifted 0\omega_0). Thus the dependence on mvm_v is nontrivial and regime-dependent.
3. Effect on interference visibility
Recall the standard relation linking phase noise to fringe visibility:
V=Videalexp(12Var()).(7)V \;=\; V_{\rm ideal}\,\exp\!\Big(-\tfrac12 \mathrm{Var}(\delta\Phi)\Big). \tag{7}
Combining with (4) we obtain a direct, quantitative chain:
mv,,K,S()Var()V.m_v,\eta,K,\;S_\xi(\omega) \quad\Rightarrow\quad \mathrm{Var}(\delta\Phi) \quad\Rightarrow\quad V.
From the scaling cases above:
