Hot Rocks Survey III: A deep eclipse for LHS 1140c and a new Gaussian process method to account for correlated noise in individual pixels

Time-series photometry at mid-infrared wavelengths is becoming a common technique to search for atmospheres around rocky exoplanets. This method constrains the brightness temperature of the planet to determine whether heat redistribution is taking place - indicative of an atmosphere - or whether the heat is reradiated from a low albedo bare rock. By observing at 15$\mu$m we are also highly sensitive to CO$_2$ absorption. We observed three eclipses of the rocky super-Earth LHS 1140c using MIRI/Imaging with the F1500W filter. We found significant variation in the initial settling ramp for these observations and identify a potential trend between detector settling and the previous filter used by MIRI. We analysed our data using aperture photometry but also developed a novel approach which joint-fits pixel light curves directly using a shared eclipse model and a flexible multi-dimensional Gaussian process which models changes in the PSF over time. We demonstrate using simulated data that our method has the ability to weight away from particular pixels which show increased systematics, allowing for the recovery of eclipse depths in a more robust and precise way. Both methods and an independent analysis detect the eclipse at $>5\sigma$ and are highly consistent with a low albedo bare rock. We recover a dayside brightness temperature of $T_\mathrm{day} = 561\pm44$ K, close to the theoretical maximum of $T_\text{day; max} = 537\pm9$ K. We rule out a wide range of atmospheric forward models to $>3\sigma$ including pure CO$_2$ atmospheres with surface pressure $\ge10$ mbar and pure H$_2$O atmospheres with surface pressure $\ge1$ bar. Our strict constraints on potential atmospheric composition, in combination with future observations of the exciting outer planet LHS 1140b, could provide a powerful benchmark to understand atmospheric escape around M dwarfs.