  In a bilayer system consisting of a composite-fermion Fermi sea in each
layer, the tunnel current is exponentially suppressed at zero bias, followed by
a strong peak at a finite bias voltage $V_{\rm max}$. This behavior, which is
qualitatively different from that observed for the electron Fermi sea, provides
fundamental insight into the strongly correlated non-Fermi liquid nature of the
CF Fermi sea and, in particular, offers a window into the short-distance
high-energy physics of this state. We identify the exciton responsible for the
peak current and provide a quantitative account of the value of $V_{\rm max}$.
The excitonic attraction is shown to be quantitatively significant, and its
variation accounts for the increase of $V_{\rm max}$ with the application of an
in-plane magnetic field. We also estimate the critical Zeeman energy where
transition occurs from a fully spin polarized composite fermion Fermi sea to a
partially spin polarized one, carefully incorporating corrections due to finite
width and Landau level mixing, and find it to be in satisfactory agreement with
the Zeeman energy where a qualitative change has been observed for the onset
bias voltage [Eisenstein et al., Phys. Rev. B 94, 125409 (2016)]. For
fractional quantum Hall states, we predict a substantial discontinuous jump in
$V_{\rm max}$ when the system undergoes a transition from a fully spin
polarized state to a spin singlet or a partially spin polarized state.
