Worlds Next Door. III. Indirect Evidence for Enhanced Atmospheric Metallicity and/or the Presence of Water Clouds in the Nearest Jupiter-analog $\epsilon$ Eri b
We present the most sensitive direct imaging search for the nearest ($d = 3.2$ pc) Jupiter-analog exoplanet, $\epsilon$ Eri b, with JWST/NIRCam coronagraphy between 4-5 $\mu$m (F444W). We achieve a 5$\sigma$ contrast sensitivity $\approx3.0\times10^{-7}$ ($\Delta \approx 16.3$ mag) in the F444W filter at the expected planet separation of $\approx$1". This is the deepest 4-5 $\mu$m contrast performance achieved for any JWST/NIRCam observation to date at these separations (and $>10\times$ better than ground-based limits). Yet, the planet remains elusive to imaging. We update the star's age to $1.1\pm0.1$ Gyr, older than previous age estimates, using the latest gyrochronology relations. This significantly impacts $\epsilon$ Eri b's inferred effective temperature ($T_{\rm eff}$), which is now expected to lie between 150-200 K based on evolutionary models for a 1 $M_{\rm Jup}$ planet. Using cloud-free Sonora Flame Skimmer models and custom PICASO patchy cloud models in the above $T_{\rm eff}$ range, we find that the F444W non-detection of $\epsilon$ Eri b can be explained by a metal-enriched atmosphere and/or an atmosphere containing water ice clouds. Both possibilities suggest that $\epsilon$ Eri b's atmosphere is strikingly similar to that of Jupiter in our Solar System. Alternatively, if we do not enforce the dynamical mass ($0.98 \pm 0.09\;M_{\rm Jup}$), a solar metallicity, cloud-free, $\lesssim0.81\;M_{\rm Jup}$ planet would be consistent with the NIRCam upper limit based on the Sonora Flame Skimmer evolutionary models. Finally, we place limits on the size of a potential ring system using the NIRCam/F210M data and discuss the opportunity to directly image $\epsilon$ Eri b with additional JWST observations, the Roman Coronagraph Instrument, the ExtraSolar Coronagraph on the Lazuli Observatory, and EELT/METIS.