Prospects of detecting rotational flatness of exoplanets from space-based photometry

In the era of photometry with space-based telescopes, such as CHEOPS (CHaracterizing ExOPlanets Satellite), JWST (James Webb Space Telescope), PLATO (PLAnetary Transits and Oscillations of stars), and ARIEL (Atmospheric Remote-sensing Infrared Exoplanet Large-survey), the road has opened for detecting subtle distortions in exoplanet transit light curves -- resulting from their non-spherical shape. We investigate the prospects of retrieval of rotational flatness (oblateness) of exoplanets at various noise levels. We present a novel method for calculating the transit light curves based on the Gauss-Legendre quadrature. We compare it in the non-rotating limit to the available analytical models. We conduct injection-and-retrieval tests to assess the precision and accuracy of the retrievable oblateness values. We find that the light curve calculation technique is about $25$\% faster than a well-known analytical counterpart, while still being precise enough. We show that a $3 \sigma$ oblateness detection is possible for a planet orbiting bright enough stars, by exploiting a precise estimate on the stellar density obtained e.g. from asteroseismology. We also show that for noise levels $\geq 256$ ppm (expressed as point-to-point scatter with a $60$~s exposure time) detection of planetary oblateness is not reliable.