Mean maps for cosmic web structures in cosmological initial conditions

Halos, filaments, sheets and voids in the cosmic web can be defined in terms of the eigenvalues of the smoothed shear tensor and a threshold $\lambda_{\rm th}$. Using analytic methods, we construct mean maps centered on these types of structures for Gaussian random fields corresponding to cosmological initial conditions. Each map also requires a choice of shear at the origin; we consider three possibilities. We find characteristic sizes, shapes and other properties of the central objects in these mean maps and explore how these properties change with varying the threshold and smoothing scale, i.e. varying the separation of the cosmic web into different kinds of components. The mean maps become increasingly complex as the threshold $\lambda_{\rm th}$ decreases to zero. We also describe scatter around these mean maps, subtleties which can arise in their construction, and some comparisons between halos in the maps and collapsed halos at final times.