The role of energy shear in the collapse of protohaloes

Dark matter haloes form from the collapse of matter around special positions in the initial field, those where the local matter flows converge to a point. For such a triaxial collapse to take place, the energy shear tensor (the source of the evolution of the inertia tensor) must be positive definite. It has been shown that this is indeed the case for the energy shear tensor of the vast majority of protohaloes. At generic positions in a Gaussian random field, the trace and traceless parts of the tensor are independent of one another. Here we show that, on the contrary, in positive definite matrices they correlate strongly, and these correlations are very similar to those exhibited by protohaloes. Moreover, while positive-definiteness ensures that an object will collapse, it does not specify when. Previous work has shown that the trace of the energy tensor (the energy overdensity) exhibits significant scatter in its values, but must lie above a critical ‘threshold’ value for the halo to collapse by today. We show that suitable combinations of the eigenvalues of the traceless part are able to explain a substantial part of the scatter of the trace. These variables provide an efficient way to parametrise the initial value of the energy overdensity, allowing us to formulate an educated guess for the threshold of collapse. We validate our ansatz by measuring the distribution of several secondary properties of protohaloes, finding good agreement with our analytical predictions.