Testing protostellar disk formation models with ALMA observations

Abridged: Recent simulations have explored different ways to form accretion disks around low-mass stars. We aim to present observables to differentiate a rotationally supported disk from an infalling rotating envelope toward deeply embedded young stellar objects and infer their masses and sizes. Two 3D magnetohydrodynamics (MHD) formation simulations and 2D semi-analytical model are studied. The dust temperature structure is determined through continuum radiative transfer RADMC3D modelling. A simple temperature dependent CO abundance structure is adopted and synthetic spectrally resolved submm rotational molecular lines up to $J_{\rm u} = 10$ are simulated. All models predict similar compact components in continuum if observed at the spatial resolutions of 0.5-1$"$ (70-140 AU) typical of the observations to date. A spatial resolution of $\sim$14 AU and high dynamic range ($> 1000$) are required to differentiate between RSD and pseudo-disk in the continuum. The peak-position velocity diagrams indicate that the pseudo-disk shows a flatter velocity profile with radius than an RSD. On larger-scales, the CO isotopolog single-dish line profiles are similar and are narrower than the observed line widths of low-$J$ lines, indicating significant turbulence in the large-scale envelopes. However a forming RSD can provide the observed line widths of high-$J$ lines. Thus, either RSDs are common or a higher level of turbulence ($b \sim 0.8 \ {\rm km \ s^{-1}}$ ) is required in the inner envelope compared with the outer part. Multiple spatially and spectrally resolved molecular line observations are needed. The continuum data give a better estimate on disk masses whereas the disk sizes can be estimated from the spatially resolved molecular lines observations. The general observable trends are similar between the 2D semi-analytical models and 3D MHD RSD simulations.