Fully relativistic non-linear cosmological evolution in spherical symmetry using the BSSN formalism

We present a fully relativistic numerical method for the study of cosmological problems using the Baumgarte-Shapiro-Shibata-Nakamura formalism on a dynamical Friedmann-Lemaître-Robertson-Walker background. This has many potential applications, including the study of the growth of structures beyond the linear regime. We present one such application by reproducing the Lemaître-Tolman-Bondi solution for the collapse of pressureless matter with arbitrary lapse function. The regular and smooth numerical solution at the center of coordinates proceeds in a natural way by relying on the partially implicit Runge-Kutta algorithm described in Montero and Cordero-Carrión [arXiv:1211.5930]. We generalize the usual radiative outer boundary condition to the case of a dynamical background and show the stability and convergence properties of the method in the study of pure gauge dynamics on a de Sitter background.

Figure 1

Evolution of a pure gauge pulse in time on a de Sitter background with $H_0=0.01$. The asymptotic value of $\alpha$ gets rescaled during the evolution.

Figure 2

Evolution of the scale factor $a(t)$ (upper panel) and background lapse function ${\alpha }_{\text{bkg}}$ (lower panel).

Figure 3

$L^2$ norm of the Hamiltonian (upper panel) and the momentum constraints (lower panel) in pure gauge dynamics ($\mathrm{\Delta }r=0.05$).

Figure 4

Value of the Hamiltonian constraint in pure gauge dynamics for three different resolutions at $t=10$. The rescaling of the curves shows good agreement with the expected second-order convergence of the numerical method.

Figure 5

Initial conformal factor in the case of a dust matter overdensity of central value ${\delta }_m^0=0.1$ (plain line). The solution agrees well with the asymptotic value imposed as a boundary condition (dotted line).

Figure 6

Metric components ${\gamma }_{rr}$ (top curves) and ${\gamma }_{\theta \theta }/r^2$ (bottom curves). The plain lines show the result of the evolution of the LTB variables, while crosses and circles are the evolution of the BSSN equations. The curves coincide at initial time (upper-left panel). The maximum of the relative difference between the curves is of the order $\sim 10^{-5}$.

Figure 7

$L^2$-norm of the Hamiltonian constraint for long-time evolution of the collapse of dust for different resolutions.

Figure 8

Evolution of the dust density contrast profile.

Figure 9

Long-term evolution of the central overdensity of dust matter [${\delta }_c≔\delta (t,r=0)$].

Figure 10

Evolution of the background scale factor in time (plain line) compared with the central conformal expansion factor (dotted line) defined as $a(t){\psi }^2(t,r=0)$.

Figure 11

$L^2$-norm of the Hamiltonian constraint as a function of the cosmic time for the evolution of dust using the Bona-Masso slicing with $f=0.333$ for different resolutions.

Figure 12

$L^2$-norm of the momentum constraint as a function of the cosmic time for the evolution of dust using the Bona-Masso slicing with $f=0.333$ for different resolutions.