Time evolution (geometry and scalar field) - cmyoo/cosmos GitHub Wiki

We employ the standard scheme called the BSSN formalism [1,2] for solving the Einstein equations with scalar field equations. Some details are shown below.

Time evolution scheme

4th-order Runge-Kutta method

Spatial derivatives

4th order central difference and an asymmetric finite difference for advection terms

Geometry

We basically follow the evolution equations shown in Ref. [3]. We do not explain all the details and the definitions of variables, but the basic equations are briefly summarized as follows:

• Spatial metric

  $$\left(\partial_t-\beta^i\partial_i\right)\psi=\frac{1}{6}\psi\left(\partial_i\beta^i-\alpha K\right)$$

  $$\left(\partial_t-\beta^k\partial_k\right)\tilde \gamma_{ij}=-2\alpha\tilde A_{ij}+\tilde \gamma_{ik}\partial_j\beta^k+\tilde\gamma_{jk}\partial_i\beta^k-\frac{2}{3}\partial_k\beta^k\tilde \gamma_{ij}$$

• Extrinsic curvature

  $$\left(\partial_t-\beta^k\partial_k\right){\rm tr}K= \alpha\left(\tilde A_{ij}\tilde A^{ij}+\frac{2}{3}{\rm tr}K^2\right)-\triangle\alpha$$

  $\left(\partial_t-\beta^k\partial_k\right)\tilde A_{ij}=$ functions of $[\psi,\tilde \gamma, {\rm tr}K, \tilde A, \tilde \Gamma,\alpha,\beta,\partial\psi,\triangle\psi,\cdots]$

• Auxiliary variables for numerical stability $\tilde \Gamma:=- \mathcal D_j \tilde \gamma^{ij}$

  $\left(\partial_t-\beta^k\partial_k\right)\tilde \Gamma^i =$ functions of $[\psi,\tilde \gamma, {\rm tr}K, \tilde A, \tilde \Gamma,\alpha,\beta,\partial\psi,\triangle\psi,\cdots]$

• Equations for gauge fixing, e.g.,

  • Modified version of the ``1+log slice" specialized for cosmological settings with the background Hubble $H_b$

  $$\left(\partial_t-\beta^i\partial_i\right)\alpha=-2\alpha({\rm tr}K+3H_b)$$

  • Shift gauge conditions

  $$\left(\partial_t-\beta^k\partial_k\right)\beta^i=\frac{3}{4}B^i$$

  $$\left(\partial_t-\beta^k\partial_k\right)B^i= \partial_t\tilde \Gamma^i-3H_bB^i$$

• Totally 24 variables for geometry

Scalar field [4]

$(\partial_t-\beta^i\partial_i)\phi=-\alpha \Pi$

$(\partial_t-\beta^i\partial_i)\Pi=-\alpha \triangle \phi -\gamma^{\mu\nu}\partial_\mu\alpha \partial_\nu\phi+\alpha K \Pi+\alpha V'(\phi)$

References

[1]: M. Shibata and T. Nakamura, Phys. Rev. D 52, 5428 (1995), Evolution of three-dimensional gravitational waves: Harmonic slicing case.

[2]: T. W. Baumgarte and S. L. Shapiro, Phys. Rev. D 59, 024007 (1998), arXiv:gr-qc/9810065, On the numerical integration of Einstein’s field equations.

[3]: E. Gourgoulhon, (Springer, Berlin New York, 2012).

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