  We describe the effects of disorder on the critical temperature of $s$-wave
superfluids from the BCS to the BEC regime, with direct application to
ultracold fermions. We use the functional integral method and the replica
technique to study Gaussian correlated disorder due to impurities, and we
discuss how this system can be generated experimentally. In the absence of
disorder, the BCS regime is characterized by pair breaking and phase coherence
temperature scales which are essentially the same allowing strong correlations
between the amplitude and phase of the order parameter for superfluidity. As
non-pair breaking disorder is introduced the largely overlapping Cooper pairs
conspire to maintain phase coherence such that the critical temperature remains
essentially unchanged, and Anderson's theorem is satisfied. However in the BEC
regime the pair breaking and phase coherence temperature scales are very
different such that non-pair breaking disorder can affect dramatically phase
coherence, and thus the critical temperature, without the requirement of
breaking tightly-bound fermion pairs simultaneously. In this case, Anderson's
theorem does not apply, and the critical temperature can be more easily reduced
in comparison to the BCS limit. Lastly, we find that the superfluid is more
robust against disorder in the intermediate region near unitarity between the
two regimes.
