We examine the spatial extrema (local maxima, minima and saddle points) of the covariant scalars (density, Hubble expansion, spatial curvature and eigenvalues of the shear and electric Weyl tensors) of the quasispherical Szekeres dust models. Sufficient conditions are obtained for the existence of distributions of multiple extrema in spatial comoving locations that can be prescribed through initial conditions. These distributions evolve without shell crossing singularities at least for ever expanding models (with or without cosmological constant) in the full evolution range where the models are valid. By considering the local maxima and minima of the density, our results allow for setting up elaborated networks of “pancake” shaped evolving cold dark matter overdensities and density voids whose spatial distribution and amplitudes can be controlled from initial data compatible with standard early Universe initial conditions. We believe that these results have an enormous range of potential application by providing a fully relativistic nonperturbative coarse grained modeling of cosmic structure at all scales.

We study the differences and equivalences between the nonperturbative description of the evolution of cosmic structure furnished by the Szekeres dust models (a nonspherical exact solution of Einstein’s equations) and the dynamics of cosmological perturbation theory ($CPT$) for dust sources in a $\mathrm{\Lambda }\mathrm{CDM}$ background. We show how the dynamics of Szekeres models can be described by evolution equations given in terms of “exact fluctuations” that identically reduce (at all orders) to evolution equations of $CPT$ in the comoving isochronous gauge. We explicitly show how Szekeres linearized exact fluctuations are specific (deterministic) realizations of standard linear perturbations of $CPT$ given as random fields, but, as opposed to the latter perturbations, they can be evolved exactly into the full nonlinear regime. We prove two important results: (i) the conservation of the curvature perturbation (at all scales) also holds for the appropriate linear approximation of the exact Szekeres fluctuations in a $\mathrm{\Lambda }\mathrm{CDM}$ background, and (ii) the different collapse morphologies of Szekeres models yields, at nonlinear order, different functional forms for the growth factor that follows from the study of redshift space distortions. The metric-based potentials used in linear $CPT$ are computed in terms of the parameters of the linearized Szekeres models, thus allowing us to relate our results to linear $CPT$ results in other gauges. We believe that these results provide a solid starting stage to examine the role of non-perturbative general relativity in current cosmological research.

We examine the relation between the Szekeres models and relativistic Lagrangian perturbation schemes, in particular the relativistic Zel’dovich approximation (RZA). We show that the second class of the Szekeres solutions is exactly contained within the RZA when the latter is restricted to an irrotational dust source with a flow-orthogonal foliation of spacetime. In such a case, the solution is governed by the first principal scalar invariant of the deformation field, proving a direct connection with a class of Newtonian three-dimensional solutions without symmetry. For the second class, a necessary and sufficient condition for the vanishing of cosmological backreaction on a scale of homogeneity is expressed through integral constraints. Domains with no backreaction can be smoothly matched, forming a lattice model, where exact deviations average out at a given scale of homogeneity, and the homogeneous and isotropic background is recovered as an average property of the model. Although the connection with the first class of Szekeres solutions is not straightforward, this class allows for the interpretation in terms of a spatial superposition of nonintersecting fluid lines, where each world line evolves independently and under the RZA model equations, but with different associated “local backgrounds”. This points to the possibility of generalizing the Lagrangian perturbation schemes to structure formation models on evolving backgrounds, including global cosmological backreaction.

We examine the fully relativistic evolution of cosmic voids constituted by baryons and cold dark matter (CDM), represented by two non-comoving dust sources in a [equation]CDM background. For this...

We examine the gravitational collapse and black hole formation of multiple nonspherical configurations constructed from Szekeres dust models with positive spatial curvature that smoothly match to a Schwarzschild exterior. These configurations are made of an almost spherical central core region surrounded by a network of “pancake-like” overdensities and voids with spatial positions prescribed through standard initial conditions. We show that a full collapse into a focusing singularity, without shell crossings appearing before the formation of an apparent horizon, is not possible unless the full configuration becomes exactly or almost spherical. Seeking for black hole formation, we demand that shell crossings are covered by the apparent horizon. This requires very special fine-tuned initial conditions that impose very strong and unrealistic constraints on the total black hole mass and full collapse time. As a consequence, nonspherical nonrotating dust sources cannot furnish even minimally realistic toy models of black hole formation at astrophysical scales: demanding realistic collapse time scales yields huge unrealistic black hole masses, while simulations of typical astrophysical black hole masses collapse in unrealistically small times. We note, however, that the resulting time-mass constraint is compatible with early Universe models of primordial black hole formation, suitable in early dust-like environments. Finally, we argue that the shell crossings appearing when nonspherical dust structures collapse are an indicator that such structures do not form galactic mass black holes but virialize into stable stationary objects.