The detection of small planets orbiting nearby stars is an important step towards the identification of Earth twins. In previous work using the Spitzer Space Telescope, we found evidence to support at least one sub-Earth-sized exoplanet orbiting the nearby mid-M dwarf star GJ 436. As a follow up, here we used the Hubble Space Telescope to investigate the existence of one of these candidate planets, UCF-1.01, by searching for two transit signals as it passed in front of its host star. Interpretation of the data hinges critically on correctly modeling and removing the WFC3 instrument systematics from the light curves. Building on previous HST work, we demonstrate that WFC3 analyses need to explore the use of a quadratic function to fit a visit-long time-dependent systematic. This is important for establishing absolute transit and eclipse depths in the white light curves of all transiting systems. The work presented here exemplifies this point by putatively detecting the primary transit of UCF-1.01 with the use of a linear trend. However, using a quadratic trend, we achieve a better fit to the white light curves and a reduced transit depth that is inconsistent with previous Spitzer measurements. Furthermore, quadratic trends with or without a transit model component produce comparable fits to the available data. Using extant WFC3 transit light curves for GJ436b, we further validate the quadratic model component by achieving photon-limited model fit residuals and consistent transit depths over multiple epochs. We conclude that, when we fit for a quadratic trend, our new data contradict the prediction of a sub-Earth-sized planet orbiting GJ 436 with the size, period, and ephemeris posited from the Spitzer data by a margin of 3.1{sigma}.
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