A waveform model for the missing quadrupole mode from black hole coalescence: memory effect and ringdown of the $(\ell=2,m=0)$ spherical harmonic

In this paper we describe a model for the $(\ell=2, m=0)$ spherical harmonic mode of the gravitational wave signal emitted by the coalescence of binary black holes, in particular, spin-aligned systems. This mode can be viewed as consisting of two components, gravitational wave memory and quasi-normal ringdown, which are both included in our model. Depending on the parameters of the binary and the sensitivity curve of the detector, but in particular for high masses, the ringdown part can contribute significantly to the signal-to-noise ratio. The model is constructed using the methods of the phenomenological waveforms program, and is calibrated to public numerical relativity data from the Simulating eXtreme Spacetimes (SXS) waveforms catalog, with the analytical results derived from the Bondi-Metzner-Sachs (BMS) balance laws. The code has been implemented as an extension to the computationally efficient IMRPhenomTHM model, it can therefore be used for computationally expensive applications such as Bayesian parameter estimation. The region of validity of our model in the parameter space is given by: $q\leq10$ and $\chi_{1},\chi_{2}\in[-1,1]$, and no restrictions apply in terms of the length of the waveforms.