1. Black Hole Universe Yoo, Chulmoon （ YITP) Hiroyuki Abe (Osaka City Univ.) Ken-ichi Nakao (Osaka City Univ.) Yohsuke Takamori (Osaka City Univ.) Note that the geometrized units are used here (G=c=1)

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 But this is based on an intuitive discussion and does not an exact solution for Einstein equations

4. Chulmoon Yoo 4 … What We Want to Do Vacuum solution for the Einstein eqs. Expansion of the universe is crucial to avoid the potential divergence Periodic boundary Expanding BH … … … We need to solve Einstein equations as nonlinear wave equations We solve only constraint equations in this work

5. Chulmoon Yoo 5 Einstein Eqs. Some of these can be regarded as wave equations for spatial metric 6 components 10 equations 10-6=4 and ~ time derivative of γ ij 66 + = 12 components 12 - 5 - 2 = 5 (γis conformaly flat) (TT parts of K ij =0) We need to fix extra d.o.f giving appropriate assumptions Einstein equations 4 constraint equations Initial data consist of

6. Chulmoon Yoo 6 Constraint Eqs. Ψ is the conformal factor K=γ ij K ij and X i gives remaining part of K ij Setting the functional form of K, we solve these equations 4 equations We still have 5 components to be fixed

7. Chulmoon Yoo 7 Constraint Eqs. we can immediately find a solution time symmetric slice of Schwarzschild BH It does not satisfy the periodic boundary condition We adopt K=0 and these form of Ψ and X i only near the center of the box If K=0,

8. Chulmoon Yoo 8 Extraction of 1/R Extraction of 1/R divergence ψ is regular at R=0 1 ＊ f =0 at the boundary Near the center R=0 (trK=0) f R Periodic boundary condition for ψ and X i

9. Chulmoon Yoo 9 Integrability Condition Since l.h.s. is positive, K cannot be zero everywhere Integrating in the box and using Gauss law in the Laplacian In the case of a homogeneous and isotropic universe, The volume expansion is necessary for the existence of a solution K gives volume expansion rate ( )

10. Chulmoon Yoo 10 Functional Form of K K/KcK/Kc R We need to solve X i because ∂ i K is not zero

11. Chulmoon Yoo 11 Equations x y z L R :=( x 2 + y 2 + z 2 ) 1/2 Source terms must vanish by integrating in the box 3 Poisson equations with periodic boundary condition One component is enough

12. Chulmoon Yoo 12 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 effective volume integrating in the box

13. Chulmoon Yoo 13 Typical Lengths 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. K c cannot be chosen freely. Non-dimensional free parameter is only L/M ・ Sch. radius ・ Box size ・ Hubble radius

14. Chulmoon Yoo 14 Convergence Test ◎ Quadratic convergence!

15. Chulmoon Yoo 15 Numerical Solutions(1) x y z L ψ(x,y,L) for L=2M ψ(x,y,0) for L=2M

16. Chulmoon Yoo 16 Numerical Solutions(2) Z(x,y,L) for L=2M Z(x,y,0) for L=2M x y z L

17. Chulmoon Yoo 17 Numerical Solutions(3) X x (x,y,L) for L=2M X x (x,y,0) for L=2M x y z L

18. Chulmoon Yoo 18 Rough Estimate Density Hubble parameter Number of BHs within a sphere of horizon radius We expect that the effective Hubble parameter and the effective mass density satisfy the Hubble equation of the EdS universe for L/M→∞ From integration of the Hamiltonian constraint,

19. Chulmoon Yoo 19 Effective Hubble Equation From integration of the Hamiltonian constraint, Does it vanish for L/M→∞?Hubble Eq. for EdS We plot κ as a function of L/M

20. Chulmoon Yoo 20 Effective Hubble Eq. The Hubble Eq. of EdS is realized for L/M→∞ κ asymptotically vanishes

21. Chulmoon Yoo 21 Conclusion ◎ We constructed initial data for the BH universe ◎ 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 ◎ We are interested in the effect of inhomogeneity on the global dyamics. We need to evolve it for our final purpose (future work)

22. Chulmoon Yoo 22 Thank you very much!