  We present highly accurate, dimensionally-reduced gravitational waveforms for
binary inspirals whose components have large spin-induced quadrupole moments.
The spin-induced quadrupole of a body first appears in the phase of a waveform
at the early inspiral stage of the binary coalescence, making it a relatively
clean probe of the internal structure of the body. However, for objects with
large quadrupolar deviations from Kerr, searches using binary black hole (BBH)
models would be ineffective. In order to perform a computationally-feasible
search, we present two dimensionally-reduced models which are derived from the
original six-dimensional post-Newtonian waveform for such systems. Our
dimensional reduction method is guided by power counting in the post-Newtonian
expansion, suitable reparameterizations of the source physics, and truncating
terms in the phase that are small in most physically well-motivated regions of
parameter space. In addition, we note that large quadrupolar deviations cause
the frequency at which a binary system reaches its minimum binding energy to be
reduced substantially. This minimum signals the end of the inspiral regime and
provides a natural cutoff for the PN waveform. We provide accurate analytic
estimates for these frequency cutoffs. Finally, we perform injection studies to
test the effectualness of the dimensionally reduced waveforms. We find that
over $80\%$ of the injections have an effectualness of $\varepsilon > 0.999$,
significantly higher than is typically required for standard BBH banks, for
systems with component spins of $|\chi_i| \lesssim 0.6$ and dimensionless
quadrupole of $\kappa_i \lesssim 10^3$. Importantly, these waveforms represent
an essential first step towards enabling an effective search for astrophysical
objects with large quadrupoles.
