  Nearly all template-based gravitational wave (GW) searches only include the
quasi-circular quadrupolar modes of the signals in their templates. Including
additional degrees of freedom in the GW templates corresponding to higher-order
harmonics, orbital precession, or eccentricity is challenging because: ($i$)
the size of template banks and the matched-filtering cost increases
significantly with the number of degrees of freedom, $(ii)$ if these additional
degrees are not included properly, the search can lose sensitivity overall (due
to an increase in the rate of background triggers). Here, we focus on including
aligned-spin higher harmonics in GW search templates. We use a new mode-by-mode
filtering approach, where we separately filter GW strain data with three
harmonics [namely $(\ell, |m|)=(2,2)$, $(3,3)$ and $(4,4)$]. This results in an
increase in the matched-filtering cost by only a factor of $3$ compared to that
of a $(2,2)$-only search. We develop computationally cheap trigger-ranking
statistics to optimally combine the different signal-to-noise ratios (SNR)
timeseries from different harmonics, which ensure only physically-allowed
combinations of the different harmonics are triggered on. We use an empirical
template-dependent background model in our ranking statistic to account for
non-Gaussian transients. In addition, we develop a tool called band eraser
which specifically excises narrow time-varying noisy bands in time-frequency
space (without having to excise entire time chunks in the data). New GW
candidate events that we detect using our search pipeline and the details of
our template banks are discussed in accompanying papers: Wadekar et al. [1] and
[2] respectively. Apart from higher harmonics, we expect our methodology to
also be useful for cheap and optimal searches including orbital precession and
eccentricity in GW waveforms.
