Download 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)
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
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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
What We Want to Do
…
Periodic boundary
…
BH
…
…
Expanding
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
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5
Einstein Eqs.
Einstein equations
10 equations
Some of these can be regarded as
wave equations for spatial metric
6 components
10-6=4
Initial data consist of
4 constraint
equations
and
~ time derivative of γij
6
+
6
= 12 components
We need to fix extra d.o.f giving appropriate assumptions
12
-5
(γis conformaly flat)
-2
=5
(TT parts of Kij=0)
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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
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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 adopt K=0 and these form of Ψ and Xi
only near the center of the box
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Extraction of 1/R
Near the center R=0 (trK=0)
Extraction of 1/R divergence
f
ψ is regular at R=0
Periodic boundary condition for ψ and Xi
1
R
＊f=0 at the boundary
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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
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K/Kc
Functional Form of K
10
R
We need to solve Xi because ∂iK is not zero
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Equations
R:=(x2+y2+z2)1/2
z
y
x
L
One component is enough
3 Poisson equations with periodic boundary condition
Source terms must vanish by integrating in the box
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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
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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
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Convergence Test
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◎Quadratic convergence!
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Numerical Solutions(1)
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ψ(x,y,L) for L=2M
z
x
y
L
ψ(x,y,0) for L=2M
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Numerical Solutions(2)
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Z(x,y,L) for L=2M
z
x
y
L
Z(x,y,0) for L=2M
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Numerical Solutions(3)
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Xx(x,y,L) for L=2M
z
x
y
L
Xx(x,y,0) for L=2M
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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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Effective Hubble Equation
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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
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Effective Hubble Eq.
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κ asymptotically vanishes
The Hubble Eq. of EdS is realized for L/M→∞
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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!
Chulmoon Yoo