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A strong low-temperature enhancement of the tunneling conductance betweengraphene bilayers has been reported recently, and interpreted as a signature ofequilibrium electron-hole pairing, first predicted in bilayers more than fortyyears ago but previously unobserved. Here we provide a detailed theory ofconductance enhanced by fluctuating electron-hole Cooper pairs, which are aprecursor to equilibrium pairing, that accounts for specific details of themulti-band double graphene bilayer system which supports several differentpairing channels. Above the equilibrium condensation temperature, pairs havefinite temporal coherence and do not support dissipationless tunneling.Instead, they strongly boost the tunneling conductivity via a fluctuationalinternal Josephson effect. Our theory makes predictions for the dependence ofthe zero bias peak in the differential tunneling conductance on temperature andelectron-hole density imbalance that capture important aspects of theexperimental observations. In our interpretation of the observations, cleanersamples with longer disorder scattering times would condense at temperatures$T_c$ up to $\sim 50 {\rm K}$, compared to the record $T_c \sim 1.5 $K achievedto date in the experiment.

Tunneling and Fluctuating Electron-Hole Cooper Pairs in Double Bilayer Graphene