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

Aim of This Talk New Perspectives on 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)

New Perspectives on Yoo 3 BH universe ―What is the BH universe?―

New Perspectives on 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

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

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

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

New Perspectives on Yoo 8 BH universe ―Initial data construction―

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

New Perspectives on 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”

New Perspectives on 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

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

New Perspectives on 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

New Perspectives on Yoo 14 BH universe ―Time evolution scheme―

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

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

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

Chulmoon Yoo 18 COSMOS code New Perspectives on ◎ 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

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

Chulmoon Yoo 20 BH universe ―Results―

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

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

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 ◎ Geodesic slice -Constant proper time slice (geodesic equation need to be solved) ◎ Fiducial(EdS) scale factor

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

Chulmoon Yoo 25 Deviation New Perspectives on higher resolution

Chulmoon Yoo 26 Conclusion ＆ Discussion ◎ The behaviour of the effective scale factor is well approximated by that of EdS universe New Perspectives on ◎ 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

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

Chulmoon Yoo 28 Isaacson’s formula ◎ Effective stress-energy tensor for GW is radiation fluid New Perspectives on ◎ 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

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

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

Chulmoon Yoo 31 PBH in matter dominated ◎ Tachyonic preheating model New Perspectives on ◎ 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

New Perspectives on 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

Chulmoon Yoo 33 Possible future works New Perspectives on

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

Chulmoon Yoo 35 Inhomogeneous Inflation ◎ Does inhomogeneity prevent inflation? New Perspectives on - 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!)

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

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

New Perspectives on 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