  In weakly coupled BCS superconductors, only electrons within a tiny energy
window around the Fermi energy, $E_F$, form Cooper pairs. This may not be the
case in strong coupling superconductors such as cuprates, FeSe, SrTiO$_3$ or
cold atom condensates where the pairing scale, $E_B$, becomes comparable or
even larger than $E_F$. In cuprates, for example, a plausible candidate for the
pseudogap state at low doping is a fluctuating pair density wave, but no
microscopic model has yet been found which supports such a state. In this work,
we write an analytically solvable model to examine pairing phases in the
strongly coupled regime and in the presence of anisotropic interactions.
Already for moderate coupling we find an unusual finite temperature phase,
below an instability temperature $T_i$, where local pair correlations have
non-zero center-of-mass momentum but lack long-range order. At low temperature,
this fluctuating pair density wave can condense either to a uniform $d$-wave
superconductor or the widely postulated pair-density wave phase depending on
the interaction strength. Our minimal model offers a unified microscopic
framework to understand the emergence of both fluctuating and long range pair
density waves in realistic systems.
