How to convert from plus and cross polarization modes ($h_{+}$, $h_{×}$) to spin-weighted spherical harmonic $h_{lm}$? I was wondering if there is a method to express the $h_{+}$ & $h_{×}$ polarization modes to spin-weighted spherical harmonic $h_{lm}$. I ask this in the context of gravitational waves. We see that LIGO/VIRGO records the waveform in terms of $h_{+}$ & $h_{×ll}$ polarization modes but, numerical relativity binary merger simulations provide $h_{lm}$ data for the waveform.
 A: Yes. (This answer assumes you want to go from $h_{\ell m}$ to $h_+$ and $h_\times$; in other words, to go from NR data to a predicted waveform that a detector like LIGO might see. If you want to go the other way, @mmeent in the comments points out you would need to know $h_+$ and $h_\times$ everywhere on the celestial sphere).
For non-precessing binaries, there is a relatively simple relationship (see, for example Eq. 1-2 of [1] or Eq 2.14 of [2], these aren't the original references that derived this formula but they describe waveform models used in LIGO data analysis and explain how the waveform models were calibrated to and compared with numerical relativity simulations)
\begin{equation}
h_+ - i h_\times = \sum_{\ell \geq 2} \sum_{\ell=-m}^m {}_{-2}Y_{\ell m} h_{\ell m}
\end{equation}
where ${}_{-2}Y_{\ell m}$ are the spin-weighted spherical harmonics and $h_{\ell m}$ are the modes of the gravitational waveform. The references explain some of the conventions in more detail, which are important to understand before using this formula, and you if you dig into the papers they cite or turn to a book on gravitational waves, such as Maggiore's textbook, you can find more explanation as well.
For precessing systems, the above formula is still valid (again, see @mmeent's comment below). However, in phenomenological waveform modeling used in data analysis, things are more complicated. To relate $h_+$ and $h_\times$ to the gravitational waveform, it is conventional to modify the above formula by performing a time-dependent rotation so the $z$ axis is aligned with the instantaneous orbital angular momentum of the binary. I will leave the details of this to the references as well.
Incidentally, what a single LIGO interferometer actually measures is not $h_+$ or $h_\times$ directly, but the linear combination $h=F_+ h_+ + F_\times h_\times$, where $F_+$ and $F_\times$ are (time and sky-position dependent) antenna patterns, describing the response of the detector to gravitational waves coming from different sky locations.
[1] https://arxiv.org/abs/1911.06050
[2] https://arxiv.org/abs/2004.09442
