We study the differences and equivalences between the non-perturbative description of the evolution of cosmic structure furnished by the Szekeres dust models (a non-spherical exact solution of Einsteins equations) and the dynamics of Cosmological Perturbation Theory (CPT) for dust sources in a $Lambda$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 linearised exact fluctuations are specific (deterministic) realisations 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 non-linear regime. We prove two important results: (i) the conservation of the curvature perturbation (at all scales) also holds for the appropriate approximation of the exact Szekeres fluctuations in a $Lambda$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 linearised 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.
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