FAUST: XXVI. The dust opacity spectral indices of protostellar envelopes bridge the gap between interstellar medium and disks

Context. The (sub-)millimetre dust opacity spectral index (β) is a critical observable for constraining dust properties, such as the maximum grain size of an observed dust population. It has been widely measured at Galactic scales and down to protoplanetary disks. Because of observational and analytical challenges, however, quite a gap exists in following the evolution of dust in the interstellar medium (ISM): we lack measures of the dust properties in the envelopes that feed newborn protostars and their disks. Aims. To fill this gap, we used sensitive dust continuum emission data at 1.2 and 3.1 mm from the ALMA FAUST Large Program and constrained the spectral index of the submillimetre dust opacity for a sample of protostars. Methods. Our high-resolution data, along with a method that was more refined than the methods in previous efforts, allowed us to distinguish the contributions from the disk and envelope in the uv-plane, and thus, to measure spectral indices for the envelopes that are not contaminated by the optically thick emission of the inner disk regions. Results. The FAUST sources (n = 13) include a variety of morphologies in continuum emission: compact young disks, extended collapsing envelopes, and dusty outflow cavity walls. Firstly, we found that the young disks in our sample are small (down to < 9 au) and optically thick. Secondly, we measured the dust opacity spectral index β at envelope scales for n = 11 sources: The β of n = 9 of these sources were not constrained before. We effectively doubled the number of sources for which the dust opacity spectral index β has been measured at these scales. Thirdly, by combining the available literature measurements with our own (a total n = 18), we showed the distribution of the envelope spectral indices between ISM-like and disk-like values. This bridges the gap in the inferred dust evolution. Finally, we statistically confirmed a significant correlation between β and the mass of protostellar envelopes, as previously suggested in the literature. Conclusions. Our findings indicate that the optical dust properties smoothly vary from the ISM (« 0.1 parsec) through envelopes (∼500- 2000 au) to protoplanetary disks (< 200 au). Multi-wavelength surveys including longer wavelengths and in controlled starforming regions are needed to further this study and make more general claims about the dust evolution in its pathway from the cloud to disks.