Predicting the bubble cutoff and microphase threshold in active phase separation

Active model $B+$ predicts reverse Ostwald ripening, which can arrest coarsening and produce a steady population of gas bubbles inside the dense phase; large $2\mathrm{D}$ simulations of active Brownian particles (ABPs) show bubbles with a power-law size distribution up to a cutoff and, at large system size $L$ or density, microphase separation. Derive the active model $B+$ coefficients $(\lambda,\zeta)$ from ABP parameters and predict the bubble cutoff scale and the size or density threshold for microphase separation in $d=2\ \text{and}\ 3$.

A derivation of the active model $B+$ coefficients from particle parameters whose predicted bubble cutoff and microphase threshold match particle simulations over a wide range of $L$ in $d = 2\ \text{and}\ 3$.