Prospects for Characterizing the Haziest Sub-Neptune Exoplanets with High Resolution Spectroscopy

Observations to characterize planets larger than Earth but smaller than Neptune have led to largely inconclusive interpretations at low spectral resolution due to hazes or clouds that obscure molecular features in their spectra. However, here we show that high-resolution spectroscopy (R $sim$ 25,000 to 100,000) enables one to probe the regions in these atmospheres above the clouds where the cores of the strongest spectral lines are formed. We present models of transmission spectra for a suite of GJ1214b-like planets with thick photochemical hazes covering 1 - 5 $mu$m at a range of resolutions relevant to current and future ground-based spectrographs. Furthermore, we compare the utility of the cross-correlation function that is typically used with a more formal likelihood-based approach, finding that only the likelihood based method is sensitive to the presence of haze opacity. We calculate the signal-to-noise of these spectra, including telluric contamination, required to robustly detect a host of molecules such as CO, CO$_{2}$, H$_{2}$O, and CH$_{4}$, and photochemical products like HCN, as a function of wavelength range and spectral resolution. Spectra in M band require the lowest S/N$_{res}$ to detect multiple molecules simultaneously. CH$_{4}$ is only observable for the coolest models ($T_{rm{eff}} =$ 412 K) and only in the L band. We quantitatively assess how these requirements compare to what is achievable with current and future instruments, demonstrating that characterization of small cool worlds with ground-based high resolution spectroscopy is well within reach.