Testing Tidal-Torque Theory: II. Alignment of Inertia and Shear and the Characteristics of Proto-haloes

We investigate the cross-talk between the two key components of tidal-torque theory, the inertia (I) and shear (T ) tensors, using a cosmological N -body simulation with thousands of well-resolved haloes. We find that the principal axes of I and T are strongly aligned, even though I characterizes the proto-halo locally while T is determined by the large-scale structure. Thus, the resultant galactic spin, which plays a key role in galaxy formation, is only a residual due to ∼10 per cent deviations from perfect alignment of T and I. The T -I correlation induces a weak tendency for the proto-halo spin to be perpendicular to the major axes of T and I, but this correlation is erased by non-linear effects at late times, making the observed spins poor indicators of the initial shear field. However, the T -I correlation implies that the shear tensor can be used for identifying the positions and boundaries of proto-haloes in cosmological initial conditions — a missing piece in galaxy formation theory. The typical configuration is of a prolate proto-halo lying perpendicular to a large-scale high-density ridge, with the surrounding voids inducing compression along the major and intermediate inertia axes of the proto-halo. This leads to a transient sub-halo filament along the large-scale ridge, whose sub-clumps then flow along the filament and merge into the final halo. The centres of proto-haloes tend to lie in ∼ 1σ over-density regions, but their association with linear density maxima smoothed on galactic scales is vague: only ∼ 40 per cent of the proto-haloes contain peaks within them. Several other characteristics distinguish proto-haloes from density peaks, e.g., they tend to compress along two principal axes while many peaks compress along three axes.