Note for the time evolution sample of an adiabatic spherical case - cmyoo/cosmos GitHub Wiki

Initial condition and parameter settings

Here we consider the adiabatic perturbation generated by the initial curvature perturbation $\zeta$. As is described in Ref. [1], once the spatial profile of $\zeta$ is specified, the growing mode solution can be described up through the next-to-leading order of the long-wavelength approximation. We use the analytic expressions for the geometrical variables with the constant-mean-curvature and zero-shift gauge up through the next-to-leading order as the initial data for the sample code.

In this sample code, we consider the numerical domain given by $-L\leq X\leq L$, $0\leq Y\leq L$, and $0\leq Z\leq L$ with $X$, $Y$ and $Z$ being the reference Cartesian coordinates. The specific profile of the initial curvature perturbation is the same as in Ref. [2]:

$$ \zeta=-\mu\exp\left(-\frac{1}{6}k^2R^2\right)W(R), $$

where $R^2=X^2+Y^2+Z^2$ and the function $W(R)$ is given by [3]

$$ W(R;R_{\rm W},L)=\left\{ \begin{array}{ll} 1&{\rm for}~0\leq R \leq R_{\rm W} \ 1-\frac{\left(\left(R_{\rm W}-L\right)^6-\left(L-R\right)^6\right)^6}{\left(R_{\rm W}-L\right)^{36}}&{\rm for}~R_{\rm W} \leq R \leq L \ 0&{\rm for}~L \leq R \end{array}\right. $$

with $R_{\rm W}=0.8L$. The parameters $k$ and $\mu$ are set as $k=10/L$ and $\mu=0.83$. The amplitude $\mu$ is slightly larger than the threshold value $\mu_{\rm th}\simeq0.805$ for PBH formation reported in Ref. 2(#ref-2).

The background universe is assumed to be radiation-dominated, and the Hubble parameter at the initial time $H_{\rm i}$ is set to be $H_{\rm i}=50/L$. The coordinate length $L$ is covered by 36 grid points with grids on both ends. Since this sample code is just for demonstration, we do not seriously consider the constraint violations.

Non-Cartesian coordinate

We employ the non-Cartesian scale-up coordinates explained in Scale-up non-Cartesian coordinates with the value of $\eta$ being $10$.

Mesh refinement

On top of the non-Cartesian coordinate system, we implement fixed mesh refinement explained in Fixed mesh refinement. In the sample code, 2 additional layers for the mesh refinement are introduced when the value of the lapse function at the origin reduces to 0.3 and 0.15. The region covered by one higher layer is the region covered by 9 grids in the lower layer. These parameters for the mesh refinement can be controlled by the parameter file par_fmr.d.

Results of the time evolution

In the default setting, the time evolution starts from the data given by the data file ini_all.dat, which describes the system soon after the horizon formation. When the numerical code is executed, first the data file ini_all.dat, in which 2 higher layers are already introduced, is loaded. Then the apparent horizon finder starts, and a horizon will be found after some time:

Figure of Apparent horizon generated from out_AHfig.dat.

The time evolution starts after the apparent horizon finder. After 3 steps of the evolution, all the data will be restored into out_all.dat, and the calculation stops. The value of the lapse function on the $x$-$y$ plane is shown below:

Lapse function on the x-axis generated from 1st and 2nd columns in out_xkl.dat(left) and x-y plane generated from 1st, 2nd, and 3rd columns in out_xyl.dat(right).

To start the time evolution from the initial data described in the previous sections, One has to set the value of "maximum step of the main loop" in par_ini.d to a sufficiently large number and set the value of the line indicated by "0:no continue 1:continue" to 0. Then one finds PBH formation at around the coordinate time $t\simeq 70L$. It takes a very long time.

Excision

When an apparent horizon is found, the excision prescription is introduced (see Apparent horizon finder and excision). The region excised is a cubic region whose size can be controlled by the parameter "excision grid" in par_ini.d.

References

[1]: T. Harada, C.-M. Yoo, T. Nakama, and Y. Koga, "Cosmological long-wavelength solutions and primordial black hole formation," Phys. Rev. D 91, 084057 (2015), arXiv:1503.03934 [gr-qc].

[2]: C.-M. Yoo, T. Harada, and H. Okawa, "Threshold of Primordial Black Hole Formation in Nonspherical Collapse," Phys. Rev. D 102, 043526 (2020), arXiv:2004.01042 [gr-qc], [Erratum: Phys.Rev.D 107, 049901 (2023)].

[3]: C.-M. Yoo, T. Ikeda, and H. Okawa, "Gravitational Collapse of a Massless Scalar Field in a Periodic Box," Class. Quant. Grav. 36, 075004 (2019), arXiv:1811.00762 [gr-qc].

[4]: C.-M. Yoo, "Primordial black hole formation from a nonspherical density profile with a misaligned deformation tensor," Phys. Rev. D 110, 043526 (2024), arXiv:2403.11147 [gr-qc].

[5]: M. Alcubierre and B. Bruegmann, "Simple excision of a black hole in (3+1)-numerical relativity," Phys. Rev. D 63, 104006 (2001), arXiv:gr-qc/0008067.