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Black Hole Universe

Black Hole Universe. Yoo, Chulmoon （ YITP). Hiroyuki Abe (Osaka City Univ.) Ken-ichi Nakao (Osaka City Univ.) Yohsuke Takamori (Osaka City Univ.). Cluster of Many BHs ～ Dust Fluids?. ～. dust fluid. ～. Naively thinking, we can treat the cluster of

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Black Hole Universe

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

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

  3. Lattice Universe “Dynamics of a Lattice Universe by the Schwarzschild-Cell Method” [Lindquist and Wheeler(1957)] 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 The maximum radius asymptotically agrees with the dust universe case Chulmoon Yoo

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

  5. Some Aspects of This Work 1. “Cosmological Numerical Relativity (CNR)” In which situation, CNR may be significant? 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. (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? 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 Chulmoon Yoo

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

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

  9. Puncture Boundary Infinity of the other world Chulmoon Yoo

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

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

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

  13. Difficulty to use CMC slice r=∞ r=∞ R=Rc 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 Chulmoon Yoo

  14. trK trK/Kc R Maximal slice CMC slice Chulmoon Yoo

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

  16. z y x Equations R:=(x2+y2+z2)1/2 L Poisson equation with periodic boundary condition Source terms must vanish by integrating in a box Chulmoon Yoo

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

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

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

  20. Part 2Structure of the BH universe Chulmoon Yoo

  21. Parameter Settings σ 0.6 trK/Kc horizon R l/M=0.6 (horizon is at R～0.5) Chulmoon Yoo

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

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

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

  25. Convergence Test ◎2nd order convergence has been checked for some cases ◎We are now checking the other cases... Chulmoon Yoo

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

  27. Expansion ◎parameter : L=2M χ+ expansion horizon χ- R ◎Horizons are almost spherically symmetric ◎BH and WH horizons are almost identical in our settings, i.e., bifurcation point Chulmoon Yoo

  28. Time slice ◎BH horizon always exists outside WH horizon Bifurcation point “WH horizon” BH horizon We would have this case changing the trK profile but it’s relatively numerically unstable and hasn’t passed the convergence test Chulmoon Yoo

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

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

  31. Effect of Xi (x,y,L) for L=2M 0.4 (x,y,L) for L=4M (x,y,L) for L=5M 0.05 0.08 Chulmoon Yoo

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

  33. Effective Hubble ◎Effective Hubble parameter H2M2 S/M2 ◎It asymptotically agrees with the expected value! Chulmoon Yoo

  34. Conclusion ◎We constructed initial data for the BH universe ◎BH and WH horizon are are almost identical in our settings, i.e., bifurcation point ◎Around vertices, it is Milne universe ◎If the box size is much 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! ◎What about the evolution...? future work...? Chulmoon Yoo

  35. Thank you very much! Chulmoon Yoo

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