Cosmological nonlinear structure formation in full general relativity

We perform numerical evolutions of cosmological scenarios using a standard general relativistic code in spherical symmetry. We concentrate on two different situations: initial matter distributions that are homogeneous and isotropic, and perturbations to those that respect the spherical symmetry. As matter models we consider the case of a pressureless perfect fluid, i.e. dust, and the case of a real massive scalar field oscillating around the minimum of the potential. Both types of matter have been considered as possible dark matter candidates in the cosmology literature, dust being closely related to the standard cold dark matter paradigm. We confirm that in the linear regime the perturbations associated with these types of matter grow in essentially the same way, the main difference being that in the case of a scalar field the dynamics introduce a cutoff in the power spectrum of the density perturbations at scales comparable with the Compton wavelength of the field. We also follow the evolutions well beyond the linear regime showing that both models are able to form structure. In particular we find that, once in the nonlinear regime, perturbations collapse faster in a universe dominated by dust. This is expected to delay the formation of the first structures in the scalar field dark matter scenario with respect to the standard cold dark matter one.

Figure 1

Evolution of the scale factor $a$ (top panel) and Hubble parameter $H$ (bottom panel), as functions of coordinate time $t$, for an expanding homogeneous and isotropic universe dominated by a perfect fluid with no pressure (dust). Here we have chosen ${\rho }_0=2\times 10^{-6}$ and $\delta {\rho }_{*}=0$. We show both the numerical evolution (dotted-line), and the analytic solution of Eq. (30) (solid-line). The scale is zoomed in near the origin for the bottom panel in order to appreciate the detail and for comparison below.

Figure 2

Snapshots of the evolution of the gauge-invariant perturbations in the metric tensor, ${\mathrm{\Phi }}_{\mathrm{g}.\mathrm{i}}$. and ${\mathrm{\Psi }}_{\mathrm{g}.\mathrm{i}}$., as functions of the comoving radial coordinate, $r$. Here we have chosen initial data of the form in Eq. (38) with ${\overline{\rho }}_0=2.0\times 10^{-6}$, $\delta {\rho }_{*}=1.2\times 10^{-8}$ and $L=50$. We show the results of the numerical evolution, ${\mathrm{\Psi }}_{\mathrm{g}.\mathrm{i}}$. (dotted-line), ${\mathrm{\Phi }}_{\mathrm{g}.\mathrm{i}}$. (dashed-line), together with the analytic solution obtained in the linear regime, Eq. (31a), (solid-line).

Figure 3

Snapshots of the contrast in the energy density, $\delta {\rho }_{\mathrm{g}.\mathrm{i}.}/\overline{\rho }$, for the same initial data as in Fig. 2. We show the results of the numerical evolution (dashed-line), together with the analytic solution obtained in the linear regime, Eq. (31b), (solid-line).

Figure 4

Similar to Fig. 2 but now for initial data with ${\overline{\rho }}_0=2.0\times 10^{-6}$, $\delta {\rho }_{*}=1.2\times 10^{-7}$ and $L=50$ in Eq. (38). Note that, after some transient time $t\sim 200$, deviations from the linearized predictions become apparent.

Figure 5

Similar to Fig. 3 but now for the same initial data of Fig. 4. Note again that, after some transient time $t\sim 200$, deviations from the linearized predictions become apparent.

Figure 6

Central value of the potentials ${\mathrm{\Psi }}_{\mathrm{g}.\mathrm{i}}$. and ${\mathrm{\Phi }}_{\mathrm{g}.\mathrm{i}}$. as a function of the central value of the contrast in the energy density for the same evolution as in Figs. 4 and 5. The dashed line corresponds to the results of the numerical evolution, while the solid line shows the analytic solution obtained in the linear regime. The dotted line represents the asymptotic value for the linearized approximation at the origin, $C_1(r=0)$.

Figure 7

Evolution of the scale factor $a$ (top panel) and Hubble parameter $H$ (bottom panel), as functions of coordinate time $t$, for an expanding homogeneous and isotropic universe dominated by the coherent oscillations of a massive scalar field (dotted line). Here we have chosen ${\phi }_0=2\times 10^{-3}$ and $\delta {\phi }_{*}=0$ in order to have the same initial configuration for the scale factor as in Fig. 1. We also show the analytic expression for the case of dust, Eq. (30) (solid line). The scale is zoomed in near the origin for the bottom panel in order to appreciate the oscillations of the Universe containing the scalar field.

Figure 8

Snapshots of the gauge-invariant perturbations in the metric tensor, ${\mathrm{\Phi }}_{\mathrm{g}.\mathrm{i}}$. and ${\mathrm{\Psi }}_{\mathrm{g}.\mathrm{i}}$., as a function of the comoving radial coordinate, $r$, at different instants of time in the cosmological evolution. Here we have chosen an initial data of the form in Eq. (44) with ${\overline{\phi }}_0=2.0\times 10^{-3}$, $\delta {\phi }_{*}=10^{-5}$, and $L=2$. Note that the inhomogeneity is composed mostly of short wavelength modes that disperse away.

Figure 9

Snapshots of the energy density contrast, $\delta {\rho }_{\mathrm{g}.\mathrm{i}.}/\overline{\rho }$, for the same initial data as in Fig. 8.

Figure 10

Similar to Fig. 8 but now for an initial data with $L=150$. We show the results of the numerical evolution of the gauge-invariant quantities ${\mathrm{\Psi }}_{\mathrm{g}.\mathrm{i}}$. (dotted-line) and ${\mathrm{\Phi }}_{\mathrm{g}.\mathrm{i}}$. (dashed-line), together with the analytic solution obtained in the linear regime for a universe dominated by dust, Eq. (31a) (solid-line).

Figure 11

Similar to Fig. 9 but now for an initial data with $L=150$. We show the results of the numerical evolution of the density contrast (dashed-line), together with the analytic solution obtained in the linear regime for a universe dominated by dust, Eq. (31b) (solid-line).

Figure 12

Initial density profiles used for the evolutions starting in the full nonlinear regime, Figs. 13 and 14 below. The parameters were adjusted in order to have initially the same central value of the energy density in dust (solid-line) and in the scalar field (dashed-line).

Figure 13

Evolution of the central energy density measured by the Eulerian observers as a function of proper time $\tau$, for both a universe dominated by dust (solid-line) and one dominated by a scalar field (dashed-line). For comparison we also show the value of the energy density on the boundary of the computational domain (dotted-line). In both cases the central energy density deviates from the asymptotic background evolution, reaching a turnaround point and finally undergoing full gravitational collapse. Meanwhile, the asymptotic region evolves as a true background through the whole simulation.

Figure 14

Central value of the trace of the extrinsic curvature $K$ (essentially the negative of the expansion) as a function of proper time for both dust (solid-line) and scalar field (dashed-line) evolutions. For comparison, the value on the background asymptotic region is also plotted (dotted-line). After a transient when local effects dominate, the value of the expansion catches on with that of the background and deviates again dramatically when reaching a state of full gravitational collapse.