Black Hole Universe -BH in an expanding box-

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Black Hole Universe-BH in an expanding box-
Yoo, Chulmoon(YITP)
Hiroyuki Abe (Osaka City Univ.) Ken-ichi Nakao
(Osaka City Univ.) Yohsuke Takamori (Osaka City
Univ.)
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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
maximum radius of lattice universe
number of BHs
The maximum radius asymptotically agrees with
the dust universe case
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Swiss-cheese Universe
Homogeneous dust universe
Expand
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
1. Cosmological Numerical Relativity (CNR)
In which situation, CNR may be significant?
If perturbations of metric components are small
enough, we dont 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.
(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
Periodic boundary

Expanding


BH

?Vacuum solution for the Einstein eqs.
?Expansion of the universe is crucial to avoid
the potential divergence
First, we construct the puncture initial data
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Puncture
Boundary
Infinity of the other world
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Constraint Eqs.
We construct the initial data.
We assume
where
Setting trK by hand, we solve these eqs. How
should we choose trK?
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Expansion of the universe
?Swiss-cheese case
Expand
finite Hubble parameter H
H -tr K / 3
tr K must be a finite value around the boundary
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CMC (constant mean curvature) Slice
tr K const. ? ?anaconst.
induced metric
isotropic coordinate
CMC slice
?
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Difficulty to use CMC slice
r8
r8
RRc
For K?0, we have a finite R at r8
We need to take care of the inner boundary
To avoid this, we choose K0 near the
infinity (maximal slice)
r8
R0
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trK
trK/Kc
R
Maximal slice
CMC slice
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Constraint Eqs.
r8
Near the center R0 (trK0)
R0
Extraction of 1/R divergence
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f0 at the boundary
? is regular at R0 Periodic boundary condition
for ? and Xi
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Equations
R(x2y2z2)1/2
L
Poisson equation with periodic boundary condition
Source terms must vanish by integrating in a box
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Integration of source terms
Vanishes by integrating in the box because
Kconst. at the boundary
vanishes by integrating in the box because ?x Z
and ?x K are odd function of x
Integration of this part also must vanish
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Effective 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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trK
0.1
trK/Kc
L-0.1
R
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Numerical Solutions(1)
?(x,y,L) for L2M
L
?(x,y,0) for L2M
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Numerical Solutions(2)
Z(x,y,L) for L2M
L
Z(x,y,0) for L2M
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Numerical Solutions(3)
Xx(x,y,L) for L2M
L
Xx(x,y,0) for L2M
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Convergence Test
?Beautiful quadratic convergence!
?We cannot find the solution for Llt1.4M
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Horizons
?To see Horizons, we calculate outgoing() and
ingoing(-) null expansions of spheres
unit normal vector to sphere
?Horizons (approximate position)
Black hole horizon
White hole horizon
?We plot the value of ? for three independent
directions (? is not spherically symmetric in
general)
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Expansion
?parameter L1.4M
?
expansion
?-
WH
BH
R
?Horizons are almost spherically symmetric
?BH horizon exists outside WH horizon or they
are almost identical
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Time slice
WH horizon
BH horizon
Bifurcation point
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Inhomogeneity
?Square of the traceless part of 3-dim Ricci
curvature
(x,y,L) for L2M
homogeneous ?
(x,y,0) for L2M
L
homogeneous and empty ?Milne universe (OK1)
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Inhomogeneity
(x,y,L) for L2M
0.7
(x,y,L) for L4M
(x,y,L) for L5M
0.6
0.6
Not homogeneous around the center of a boundary
face
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An Averaging
?Effective density
Area
Effective volume of a box ( )
Effective density
L
?Hubble parameter (defined by the boundary value
of trK)
This relation is nontrivial!
?We may expect (?)
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!
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Conclusion
?We constructed initial data for the BH universe
?BH horizon exists outside WH horizon or
they are almost identical
?Around vertices, it is well described by the
Milne 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, that is, the BH universe is the EdS
universe on Average!
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Including ?lt0
?We may have momentarily static initial data
integrate in a box
?Negative ? can compensate the mass term
?Probably, It will collapse when we consider
the time evolution because essentially it is
dust negative ? universe
?It seems very difficult to get stable solution
without exotic matter other than negative ?
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Thank you very much!