Beyond relativistic Lagrangian perturbation theory: An exact-solution controlled model for structure formation

We develop a new nonlinear method to model structure formation in general relativity from a generalization of the relativistic Lagrangian perturbation schemes, controlled by Szekeres (and LTB) exact solutions. The overall approach can be interpreted as the evolution of a deformation field on an inhomogeneous reference model, obeying locally Friedmann-like equations. In the special case of locally one-dimensional deformations, the new model contains the entire Szekeres family of exact solutions. As thus formulated, this approach paraphrases the Newtonian and relativistic Zel’dovich approximations, having a large potential for applications in contexts where relativistic degrees of freedom are relevant. Numerical simulations are implemented to illustrate the capabilities and accuracy of the model.

Figure 1

Special subcases of GRZA. Other exact limits of Szekeres class I (Sze-I) and Szekeres class II (Sze-II) are not represented in the figure; see [27] for their classification.

Figure 2

Equatorial projection of the density contrast $\delta =\varrho /{\varrho }_b-1$ at the present cosmic time, $t_0$. Coordinates are defined as $X=A(t_0,z)z\cos \varphi$ and $Y=A(t_0,z)z\sin \varphi$, where $A(t_0,z)z$ is the areal radius. The plots correspond to different values of $\alpha$. For $\alpha =0$, the solution is exact, exhibiting the distinctive dipolar matter distribution of Szekeres models.

Figure 3

Estimation of the error from the violation of the Hamiltonian constraint at the present cosmic time. Equation (68) is evaluated along the radial rays defined by $\theta =\pi /2$ and $\varphi =[{\varphi }_{*}^{(1)},{\varphi }_{*}^{(2)},0,\pi /4,\pi /2,\pi ,3\pi /2]$ for the solution with $\alpha =4.8\times {10}^{-3}$. ${\varphi }_{*}^{(1)}$ and ${\varphi }_{*}^{(2)}$ denote the azimuthal angles of the two largest overdensities in the order seen by he reader in Fig. 2.

Figure 4

Foliation by 2-spheres plotted in terms of the radial proper length. Time slices of LTB and Szekeres solutions and of our numerical example model are leaves of a foliation by topological 2-spheres. Only in the LTB case, the 2-spheres are geometric and concentric. Overdense structures (or density maxima) arise at the loci of points where the proper distance between shells is minimal and becomes zero at shell-crossing. In the Szekeres and more general GRZA cases, these degeneracies also arise along the angular variables but have well-localized spatial positions at points of degeneracy of the deformation field, $\det ({{\eta }^a}_i)=0$. The regions marked in the figure are prone to forming shell-crossings.