black hole universe yoo, chulmoon ( yitp) hiroyuki abe (osaka city univ.) ken-ichi nakao (osaka...
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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)
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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
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!