Fitting pseudo-S${\rm \acute{e}}$rsic(Spergel) light profiles to galaxies in interferometric data: the excellence of the $uv$-plane

Modern (sub)millimeter interferometers, such as ALMA and NOEMA, offer high angular resolution and unprecedented sensitivity. This provides the possibility to characterize the morphology of the gas and dust in distant galaxies. To assess the capabilities of current softwares in recovering morphologies and surface brightness profiles in interferometric observations, we test the performance of the Spergel model for fitting in the $uv$-plane, which has been recently implemented in the IRAM software GILDAS (uv$\_$fit). Spergel profiles provide an alternative to the Sersic profile, with the advantage of having an analytical Fourier transform, making them ideal to model visibilities in the $uv$-plane. We provide an approximate conversion between Spergel index and Sersic index, which depends on the ratio of the galaxy size to the angular resolution of the data. We show through extensive simulations that Spergel modeling in the $uv$-plane is a more reliable method for parameter estimation than modeling in the image-plane, as it returns parameters that are less affected by systematic biases and results in a higher effective signal-to-noise ratio (S/N). The better performance in the $uv$-plane is likely driven by the difficulty of accounting for correlated signal in interferometric images. Even in the $uv$-plane, the integrated source flux needs to be at least 50 times larger than the noise per beam to enable a reasonably good measurement of a Spergel index. We characterise the performance of Spergel model fitting in detail by showing that parameters biases are generally low (<10%) and that uncertainties returned by uv$\_$fit are reliable within a factor of two. Finally, we showcase the power of Spergel fitting by re-examining two claims of extended halos around galaxies from the literature, showing that galaxies and halos can be successfully fitted simultaneously with a single Spergel model.