Black Hole Universe

Yoo, Chulmoon （ YITP)

Hiroyuki Abe (Osaka City Univ.)Ken-ichi Nakao (Osaka City Univ.) Yohsuke Takamori (Osaka City Univ.)

Chulmoon Yoo

2Cluster 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～～

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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

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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” ?)

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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?

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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

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Part 1A recipe for the BH

universe

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…

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

ExpandingBH

…

…

…

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PunctureBoundary

Infinity of the other world

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Constraint Eqs.

We assume

Setting trK by hand, we solve these eqs.How should we choose trK?

We construct the initial data.

where

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tr K must be a finite value around the boundary

Expand

finite Hubble parameter HH =-tr K / 3

→Swiss-cheese case

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CMC (constant mean curvature) Slicetr K = const. ⇔ ∇ana=const.

induced metric

isotropic coordinate

CMC slice

?

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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

Difficulty to use CMC slice

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trK

CMC sliceMaximal slice

trK

/Kc

R

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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 Xi

1

＊ f=0 at the boundary

r=∞

R=0

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Equations

xy

z

L

R:=(x2+y2+z2)1/2

Source terms must vanish by integrating in a box

Poisson equation with periodic boundary condition

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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

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18Effective Hubble Equation

Integrating in a box, we have

Hubble parameter H

effective mass density

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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

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Part 2Structure of the BH

universe

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Parameter Settings

L/M 2 3 4 5

σ/M 1.3 2 2.7 3.4

l/M=0.6 (horizon is at R～ 0.5)

trK

/Kc

R

0.6

hori

zon

σ

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Numerical Solutions(1)

xy

z

L

ψ(x,y,L) for L=2M

ψ(x,y,0) for L=2M

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Numerical Solutions(2)Z(x,y,L) for L=2M

Z(x,y,0) for L=2M

xy

z

L

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Numerical Solutions(3)

Xx(x,y,L) for L=2M

Xx(x,y,0) for L=2M

xy

z

L

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Convergence Test

◎2nd order convergence has been checked for some cases

◎We are now checking the other cases...

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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

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Expansion ◎parameter : L=2M

χ+

χ-

◎Horizons are almost spherically symmetric

◎BH and WH horizons are almost identical in our settings, i.e., bifurcation point

R

exp

an

sio

n

horizon

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Time slice◎BH horizon always exists outside WH horizon

BH horizon

“WH horizon”Bifurcation point

We would have this case changing the trK profile but it’s relatively numerically unstable and hasn’t passed the convergence test

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Inhomogeneity

(x,y,L) for L=2M

(x,y,0) for L=2M

xy

z

L

◎Square of the traceless part of 3-dim Ricci curvature

homogeneous ⇔

homogeneous and empty⇒Milne universe (ΩK=1)

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Inhomogeneity(x,y,L) for L=2M

(x,y,L) for L=4M(x,y,L) for L=5M0.6

0.6

0.7

Not homogeneous around the center of a boundary face

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Effect of Xi

(x,y,L) for L=2M

(x,y,L) for L=4M(x,y,L) for L=5M0.08

0.05

0.4

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An Averaging◎Effective density

xy

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”

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Effective Hubble◎Effective Hubble parameter

◎It asymptotically agrees with the expected value!

H2M

2

S/M2

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Conclusion◎We constructed initial data for the BH universe

◎What about the evolution...? future work...?

◎BH and WH horizon are are almost identical in our settings, i.e., bifurcation point

◎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!

◎Around vertices, it is Milne universe

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Thank you very much!