Black Hole Universe

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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)

description

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). Cluster of Many BHs ~ Dust Fluids?. ~. dust fluid. ~. - PowerPoint PPT Presentation

Transcript of Black Hole Universe

Page 1: 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.)

Note that the geometrized units are used here (G=c=1)

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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

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

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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

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Einstein Eqs.

Some of these can be regarded aswave equations for spatial metric 6 components

10 equations

10-6=4

and

~ time derivative of γij

6 6+ = 12 components

12 - 5 - 2 = 5(γis conformaly flat) (TT parts of Kij=0)

We need to fix extra d.o.f giving appropriate assumptions

Einstein equations

4 constraint equations

Initial data consist of

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Constraint Eqs.

Ψ is the conformal factor K=γijKij and Xi gives remaining part of Kij

Setting the functional form of K, we solve these equations

4 equations

We still have 5 components to be fixed

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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 Xi only near the center of the box

If K=0,

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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

RPeriodic boundary condition for ψ and Xi

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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 ( )

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Functional Form of K

K/K

c

R

We need to solve Xi because ∂iK is not zero

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Equations

xy

z

L

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

Source terms must vanish by integrating in the box

3 Poisson equations with periodic boundary condition

One component is enough

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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

effective volume

integrating in the box

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Typical Lengths

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

・ Sch. radius

・ Box size

・Hubble radius

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Convergence Test

◎Quadratic convergence!

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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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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,

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19Effective 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

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Effective Hubble Eq.

The Hubble Eq. of EdS is realized for L/M→∞

κ asymptotically vanishes

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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)

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Thank you very much!