  It is shown by microscopic calculations for trapped imbalanced Fermi
superfluids that the gap function has always sign changes, i.e., the
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state like, up to a critical imbalance
$P_c$, beyond which normal state becomes stable, at temperature T=0. A phase
diagram is constructed in $T$ vs $P$, where the BCS state without sign change
is stable only at $T\neq 0$. We reproduce the observed bimodality in the
density profile to identify its origin and evaluate $P_c$ as functions of $T$
and the coupling strength. These dependencies match with the recent
experiments.
