1. Numerical Relativity in Cosmology - my personal perspective - Yoo, Chulmoon （ Nagoya U. ） with Hirotada Okawa （ Lisbon, IST ） New Perspectives on Cosmology @ APCTP

2. Aim of This Talk New Perspectives on Cosmology@APCTPChulmoon Yoo 2 ◎ Aim of this workshop ◎ Aim of this talk To get hints for my future work with help of your ideas ◎ Contents - What we did and what we can do(Black hole universe) - What we are trying to do (GW universe, Dust collapse) - Possible future works (PBH, inhomogeneous inflation)

3. New Perspectives on Cosmology@APCTPChulmoon Yoo 3 BH universe ―What is the BH universe?―

4. New Perspectives on Cosmology@APCTPChulmoon Yoo 4 … What is BH Universe? ◎ Vacuum solution for the Einstein eqs. ◎ Spatially flat universe on average Periodic boundary Expanding BH … … … ◎ How is the expansion law? Same as dust universe? ◎ No symmetry, nonlinear → numerical relativity

5. New Perspectives on Cosmology@APCTPChulmoon Yoo 5 Spatial Structure Boundary Infinity of the other world ◎ We want to obtain “puncture structure”

6. New Perspectives on Cosmology@APCTPChulmoon Yoo 6 Puncture with Sch. BH ◎ Line element ◎ Isotropic coordinate flat(Euclid) metric

7. New Perspectives on Cosmology@APCTPChulmoon Yoo 7 Spatial Structure Boundary Infinity of the other world ◎ We want to obtain “puncture structure”

8. New Perspectives on Cosmology@APCTPChulmoon Yoo 8 BH universe ―Initial data construction―

9. New Perspectives on Cosmology@APCTPChulmoon Yoo 9 Constraint Eqs. ◎ Setting a functional form of trK, we solve these eqs. ◎ Hamiltonian constraint ◎ Momentum constraint ◎ Decomposition ◎ Assumptions conformally flat, ◎ 4 equations

10. New Perspectives on Cosmology@APCTPChulmoon Yoo 10 Static case we can immediately find a solution time symmetric slice of Sch. BH ◎ It does not satisfy the periodic boundary condition ◎ We adopt trK=0 and these form of Ψ and X i near the center of the box ◎ If trK=0, ◎ R=0 is the “puncture”

11. New Perspectives on Cosmology@APCTPChulmoon Yoo 11 Integrability Condition ◎ Since l.h.s. is positive, trK cannot be zero everywhere ◎ The volume expansion is necessary for the existence of a solution ◎ trK gives volume expansion rate i n the case of a homogeneous and isotropic universe, ◎ Integration of Hamiltonian constraint integration in a box except at the center

12. New Perspectives on Cosmology@APCTPChulmoon Yoo 12 Functional Form of trK W(R)W(R) R ◎ We need to solve X i because ∂ i K is not zero

13. New Perspectives on Cosmology@APCTPChulmoon Yoo 13 Typical scales ◎ We set H eff so that the following equation is satisfied ◎ This is just the integration of the constraint equation. We update the value of H eff at each step of the numerical iteration. H eff cannot be freely chosen. ◎ Non-dimensional free parameter is only L/M ・ BH Horizon radius ・ Box size ・ Hubble radius ◎ Typical lengths

14. New Perspectives on Cosmology@APCTPChulmoon Yoo 14 BH universe ―Time evolution scheme―

15. New Perspectives on Cosmology@APCTPChulmoon Yoo 15 Time Evolution ◎ Spatial metric ◎ Extrinsic curvature Einstein equations do not fix them Gauge fixing conditions are needed

16. Chulmoon Yoo 16 Evolution equations ◎ Spatial metric New Perspectives on Cosmology@APCTP ◎ Extrinsic curvature ◎ auxiliary variable for numerical stability

17. Chulmoon Yoo 17 Gauge Conditions ◎ Time slicing condition ◎ Spatial coordinates New Perspectives on Cosmology@APCTP total 24 variables should be evolved

18. Chulmoon Yoo 18 COSMOS code New Perspectives on Cosmology@APCTP ◎ Originally coded by Hirotada Okawa ◎ BSSN Einstein eq. solver(C++) based on SACRA code [Yamamoto-Shibata-Taniguchi(2008)] ◎ Specification that I used - Cartesian uni-grid - 4-th order finite differencing in space - 4-th order time integration with Runge-Kutta - Apparent horizon finder - OpenMP parallelization - Geodesic solver on the boundary ◎ Options

19. Chulmoon Yoo 19 Parameters ◎ Simulation region: 1/8 box x y z 80 New Perspectives on Cosmology@APCTP

20. Chulmoon Yoo 20 BH universe ―Results―

21. Chulmoon Yoo 21 Apparent Horizon New Perspectives on Cosmology@APCTP ◎ AHF fails to find the horizon because of low resolution ◎ Computation does not crash and we proceed ◎ AH shrinks with time

22. Chulmoon Yoo 22 Constraint Violation New Perspectives on Cosmology@APCTP ◎ Failure in AH search ⇒ error inside starts to propagate outward But, vicinity the boundary is reliable On x-axis

23. Chulmoon Yoo 23 Effective scale factor ◎ Which time slice? -Time slices should be well-defined by only boundary variables. (We don’t want to use information inside the cube) -The effective scale factors should be functions of a “time”, and the time should have natural relation with a time in EdS universe ◎ Area of faces→effective scale factor New Perspectives on Cosmology@APCTP ◎ Geodesic slice -Constant proper time slice (geodesic equation need to be solved) ◎ Fiducial(EdS) scale factor

24. Chulmoon Yoo 24 Results New Perspectives on Cosmology@APCTP ◎ Well fitted by EdS case! higher resolution -Two constants are fixed by fitting

25. Chulmoon Yoo 25 Deviation New Perspectives on Cosmology@APCTP higher resolution

26. Chulmoon Yoo 26 Conclusion ＆ Discussion ◎ The behaviour of the effective scale factor is well approximated by that of EdS universe New Perspectives on Cosmology@APCTP ◎ The number of black holes inside Hubble radius expect that the deviation becomes smaller as the universe expands →effect of the local inhomogeneity becomes negligible ◎ We find the same results with Λ (several technical things...) Preliminary

27. New Perspectives on Cosmology@APCTPChulmoon Yoo 27 What we are trying to do - GW universe - with Taishi Ikeda(student in Nagoya U.) Yasusada Nambu(Nagoya U.)

28. Chulmoon Yoo 28 Isaacson’s formula ◎ Effective stress-energy tensor for GW is radiation fluid New Perspectives on Cosmology@APCTP ◎ How accurate beyond the above approximations? - Nonlinear GW → we need numerical relativity ◎ Simplest situation → GW universe - Short-wavelength approximation - Small amplitude(perturbation) - Periodic Standing wave solution - How is the expansion law? Radiation dominated? oscillation

29. New Perspectives on Cosmology@APCTPChulmoon Yoo 29 Settings ◎ Assumptions for initial data ◎ Hamiltonian constraint - Conformal metric Ikeda-kun is trying to solve this equation... to be reported somewhere

30. New Perspectives on Cosmology@APCTPChulmoon Yoo 30 What we are trying to do - Dust collapse - with Tomohiro Harada(Rikkyo U.)

31. Chulmoon Yoo 31 PBH in matter dominated ◎ Tachyonic preheating model New Perspectives on Cosmology@APCTP ◎ Dust collapse in cosmological background - Enhances the power spectrum on small scales - PBH formation in matter dominated phase immediately after inflation may be important - Non-spherical effects are crucial in dust collapse → 3D numerical relativity ◎ Collisionless particle system

32. New Perspectives on Cosmology@APCTPChulmoon Yoo 32 Settings ◎ Assumptions for initial data ◎ Hamiltonian constraint - Conformally flat [NOTE] Decaying modes are included in this setting!! not appropriate for PBH formation - I succeeded to obtain initial data for given distributions of particles - I’m trying to implement collisionless particle system in COSMOS

33. Chulmoon Yoo 33 Possible future works New Perspectives on Cosmology@APCTP

34. Chulmoon Yoo 34 Non-spherical PBH ◎ Non-spherical effects on PBH formation New Perspectives on Cosmology@APCTP ◎ Are they rotating? - Suppress PBH formation rate??? - Existence of rotation may change the evaporation process... ◎ And... any idea...? - What about matter dominated phase???

35. Chulmoon Yoo 35 Inhomogeneous Inflation ◎ Does inhomogeneity prevent inflation? New Perspectives on Cosmology@APCTP - How does it have to be homogeneous at the beginning? - What kind of large scale inhomogeneity can remain? Observable? ◎ An old work(3-dim GR simulation before BSSN!)

36. Chulmoon Yoo 36 Kurki-Suonio, Laguna, Matzner ◎ Their conclusion New Perspectives on Cosmology@APCTP ◎ Are more systematic investigations possible? needed?

37. Chulmoon Yoo 37 These are my personal perspectives. Any suggestion? New Perspectives on Cosmology@APCTP Thank you for your attention

38. New Perspectives on Cosmology@APCTPChulmoon Yoo 38 Extraction of 1/R ◎ Extraction of 1/R divergence 1 ＊ f =0 at the boundary ◎ Near the center R=0 (trK=0) f R ◎ Equations