Black Hole Universe

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. 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 to show 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 The maximum radius asymptotically agrees with the dust universe case But this is based on an intuitive discussion and does not an exact solution for Einstein equations Chulmoon Yoo

4. 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 We need to solve Einstein equations as nonlinear wave equations We solve only constraint equations in this work Chulmoon Yoo

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

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

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

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

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

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

11. z y x Equations R:=(x2+y2+z2)1/2 L One component is enough 3 Poisson equations with periodic boundary condition Source terms must vanish by integrating in the box Chulmoon Yoo

12. Integration of source terms integrating in the box effective volume 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 Chulmoon Yoo

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

14. Convergence Test ◎Quadratic convergence! Chulmoon Yoo

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

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

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

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

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

22. Thank you very much! Chulmoon Yoo