1. Black Hole Universe -BH in an expanding box- Yoo, Chulmoon （ YITP) Hiroyuki Abe (Osaka City Univ.) Ken-ichi Nakao (Osaka City Univ.) Yohsuke Takamori (Osaka City Univ.)

2. Chulmoon Yoo 2 Cluster of Many BHs ～ Dust Fluids? Naively thinking, we can treat the cluster of a number of BHs as a dust fluid on average In this work, as a simplest case, we try to construct “the BH universe” which would be approximated by the EdS universe on average But, it is very difficult to show it from the first principle. Because we need to solve the N-body dynamics with the Einstein equations. dust fluid ～ ～

3. Chulmoon Yoo 3 Lattice Universe “Dynamics of a Lattice Universe by the Schwarzschild-Cell Method” [Lindquist and Wheeler(1957)] The maximum radius asymptotically agrees with the dust universe case Putting N equal mass Sch. BHs on a 3-sphere, requiring a matching condition, we get a dynamics of the lattice universe maximum radius of lattice universe number of BHs

4. Chulmoon Yoo 4 Swiss-cheese Universe Expand Homogeneous dust universe Cutting spherical regions, put Schwarzschild BHs with the same mass Swiss-cheese universe We want to make it without cheese (“Swiss universe” ?)

5. Chulmoon Yoo 5 Some Aspects of This Work If perturbations of metric components are small enough, we don’t need to treat full GR but perturbation theory is applicable. Perhaps, even if the density perturbation is nonlinear in small scales, we could handle the inhomogeneities without full numerical relativity. 1. “Cosmological Numerical Relativity (CNR)” In which situation, CNR may be significant? (In this sense, for late time cosmology, CNR might not be significant.) CNR may play a role in an extreme situation where the metric perturbation is full nonlinear on cosmological scales (e.g. primordial BH formation) 2. BH simulation without asymptotic flatness -In higher-dimensional theory, compactified directions often exist, and they are not asymptotically flat. -BH physics might be applied to other fields (e.g. AdS/CFT,QCD,CMP) without asymptotic flatness Their dynamical simulations might have common feature?

6. Chulmoon Yoo 6 Contents ◎ Part 1 : “A recipe for the BH universe” How to construct the initial data for the BH universe ◎ Part 2 “Structure of the BH universe” - Horizons - Effective Hubble equation with an averaging

7. Chulmoon Yoo 7 Part 1 A recipe for the BH universe

8. Chulmoon Yoo 8 … What We Want to Do ◎ Vacuum solution for the Einstein eqs. First, we construct the puncture initial data ◎ Expansion of the universe is crucial to avoid the potential divergence Periodic boundary Expanding BH … … …

9. Chulmoon Yoo 9 Puncture Boundary Infinity of the other world

10. Chulmoon Yoo 10 Constraint Eqs. We assume Setting trK by hand, we solve these eqs. How should we choose trK? We construct the initial data. where

11. Chulmoon Yoo 11 Expansion of the universe tr K must be a finite value around the boundary Expand finite Hubble parameter H H =-tr K / 3 →Swiss-cheese case

12. Chulmoon Yoo 12 CMC (constant mean curvature) Slice tr K = const. ⇔ ∇ a n a =const. induced metric isotropic coordinate CMC slice ?

13. Chulmoon Yoo 13 r=∞ R=R c For K≠0, we have a finite R at r=∞ We need to take care of the inner boundary To avoid this, we choose K=0 near the infinity (maximal slice) r=∞ R=0 Difficulty to use CMC slice

14. Chulmoon Yoo 14 trK CMC slice Maximal slice trK/K c R

15. Chulmoon Yoo 15 Constraint Eqs. Extraction of 1/R divergence Near the center R=0 (trK=0) ψ is regular at R=0 Periodic boundary condition for ψ and X i 1 ＊ f=0 at the boundary r=∞ R=0

16. Chulmoon Yoo 16 Equations x y z L R :=( x 2 + y 2 + z 2 ) 1/2 Source terms must vanish by integrating in a box Poisson equation with periodic boundary condition

17. Chulmoon Yoo 17 Integration of source terms vanishes by integrating in the box because ∂ x Z and ∂ x K are odd function of x Vanishes by integrating in the box because K=const. at the boundary Integration of this part also must vanish

18. Chulmoon Yoo 18 Effective Hubble Equation Integrating in a box, we have Hubble parameter H effective mass density

19. Chulmoon Yoo 19 Parameters BH mass Box size (isotropic coord.) Hubble radius We set K c so that the following equation is satisfied This is just the integration of the constraint equation. We update the value of K c at each step of the numerical iteration. Free parameter is only other than and

20. Chulmoon Yoo 20 Part 2 Structure of the BH universe

21. Chulmoon Yoo 21 trK trK/K c R 0.1 L-0.1

22. Chulmoon Yoo 22 Numerical Solutions(1) x y z L ψ(x,y,L) for L=2M ψ(x,y,0) for L=2M

23. Chulmoon Yoo 23 Numerical Solutions(2) Z(x,y,L) for L=2M Z(x,y,0) for L=2M x y z L

24. Chulmoon Yoo 24 Numerical Solutions(3) X x (x,y,L) for L=2M X x (x,y,0) for L=2M x y z L

25. Chulmoon Yoo 25 Convergence Test ◎ Beautiful quadratic convergence! ◎ We cannot find the solution for L<1.4M

26. Chulmoon Yoo 26 Horizons ◎ To see Horizons, we calculate outgoing(+) and ingoing(-) null expansions of spheres ◎ We plot the value of χ for three independent directions (χ is not spherically symmetric in general) : unit normal vector to sphere ◎ Horizons (approximate position) : Black hole horizon : White hole horizon

27. Chulmoon Yoo 27 Expansion ◎ parameter : L = 1.4M χ+χ+ χ-χ- ◎ Horizons are almost spherically symmetric ◎ BH horizon exists outside WH horizon or they are almost identical R expansion WHBH

28. Chulmoon Yoo 28 Time slice BH horizon “WH horizon” Bifurcation point

29. Chulmoon Yoo 29 Inhomogeneity (x,y,L) for L=2M (x,y,0) for L=2M x y z L ◎ Square of the traceless part of 3-dim Ricci curvature homogeneous ⇒ homogeneous and empty ⇒ Milne universe (Ω K =1)

30. Chulmoon Yoo 30 Inhomogeneity (x,y,L) for L=2M (x,y,L) for L=4M (x,y,L) for L=5M 0.6 0.7 Not homogeneous around the center of a boundary face

31. Chulmoon Yoo 31 An Averaging ◎ Effective density x y z L Area: Effective volume of a box ( ) Effective density ◎ Hubble parameter (defined by the boundary value of trK) ◎ We may expect (?) This relation is nontrivial! No dust, No matter, No symmetry, but additional gravitational energy other than “the point mass”

32. Chulmoon Yoo 32 Effective Hubble ◎ Effective Hubble parameter ◎ It asymptotically agrees with the expected value!

33. Chulmoon Yoo 33 Conclusion ◎ We constructed initial data for the BH universe ◎ BH horizon exists outside WH horizon or they are almost identical ◎ When the box size is sufficiently larger than the Schwarzschild radius of the mass M, an effective density and an effective Hubble parameter satisfy Hubble equation of the EdS universe, that is, the BH universe is the EdS universe on Average! ◎ Around vertices, it is well described by the Milne universe

34. Chulmoon Yoo 34 Including Λ<0 ◎ We may have momentarily static initial data integrate in a box ◎ Probably, It will collapse when we consider the time evolution because essentially it is dust + negative Λ universe ◎ Negative Λ can compensate the mass term ◎ It seems very difficult to get stable solution without exotic matter other than negative Λ

35. Chulmoon Yoo 35 Thank you very much!