Nonspherical Szekeres models in the language of cosmological perturbations

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.

Figure 1

The linear suppression factor. Logarithmic graph of the Szekeres linear growth factor $f_1$ obtained in Eq. (87) vs the $\mathrm{\Lambda }\mathrm{CDM}$ background parameter ${\overline{\mathrm{\Omega }}}^m$, both expressed as functions of $(\overline{a},{\overline{\mathrm{\Omega }}}_0^m)$ for the values ${\mathrm{\Omega }}_0^m=0.2$, 0.35. The line with slope $6/11$ is represented by the circles. The figure shows how $f_1\sim [{\overline{\mathrm{\Omega }}}^m{]}^{6/11}$ is a good approximation for $f_1$ in the ranges ${\overline{\mathrm{\Omega }}}^m\approx 1$.