JWST/NIRSpec Observations of Ly$\alpha$ Emission in Star Forming Galaxies at $6.5\lesssim z\lesssim13$

We present an analysis of JWST Ly$\alpha$ spectroscopy of $z\gtrsim6.5$ galaxies, using observations in the public archive covering galaxies in four independent fields (GOODS-N, GOODS-S, Abell 2744, EGS). We measure Ly$\alpha$ emission line properties for a sample of $210$ $z\simeq6.5-13$ galaxies, with redshifts confirmed independently of Ly$\alpha$ in all cases. We present $3$ new detections of Ly$\alpha$ emission in JWST spectra, including a large equivalent width (EW $=143\ \r{A}$) Ly$\alpha$ emitter with strong CIV emission (EW $=21\ \r{A}$) at $z=7.1$ in GOODS-N. We measure the redshift-dependent Ly$\alpha$ EW distribution across our sample. We find that strong Ly$\alpha$ emission (EW $>25\ \r{A}$) becomes increasingly rare at earlier epochs, suggesting that the transmission of Ly$\alpha$ photons decreases by $4\times$ between $z\simeq5$ and $z\simeq9$. We describe potential implications for the IGM neutral fraction. There is significant field to field variance in the Ly$\alpha$ emitter fraction. In contrast to the three other fields, the EGS shows no evidence for reduced transmission of Ly$\alpha$ photons at $z\simeq7-8$, suggesting a significantly ionized sightline may be present in the field. We use available NIRCam grism observations from the FRESCO survey to characterize overdensities on large scales around known Ly$\alpha$ emitters in the GOODS fields. The strongest overdensities appear linked with extremely strong Ly$\alpha$ detections (EW $>50\ \r{A}$) in most cases. Future Ly$\alpha$ spectroscopy with JWST has the potential to constrain the size of ionized regions around early galaxy overdensities, providing a new probe of the reionization process.