accelerating expansion from inhomogeneities ?
DESCRIPTION
Accelerating Expansion from Inhomogeneities ?. Je-An Gu (National Taiwan University). Collaborators: Chia-Hsun Chuang. (astro-ph/0512651). IRGAC2006, 2006/07/14. Accelerating Expansion. Based on FRW Cosmology. (homogeneous & isotropic). Supernova data ? Cosmic Acceleration. - PowerPoint PPT PresentationTRANSCRIPT
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Accelerating Expansion
from
Inhomogeneities ?
Je-An Gu(National Taiwan University)
Collaborators: Chia-Hsun Chuang
(astro-ph/0512651) IRGAC2006, 2006/07/14
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Accelerating Expansion
Based on FRW Cosmology
(homogeneous & isotropic)
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Based on FRW Cosmology
(homogeneous & isotropic)
Supernova data ? Cosmic Acceleration
However, apparently,
our universe is NOT homogeneous & isotropic.
At large scales, after averaging,the universe IS homogeneous & isotropic.
But, averaging !?
Is it legal ? Does it make sense ?
)(
)(
space
space
tg
t
Existence of cosmic acceleration Dark energy as a necessity of understanding acceleration
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Einstein equations
abmnab GTgG 8
abmnabmnab TGgGgG 8
abmn Tg ,
abmn Tg ,
For which satisfy Einstein equations,
in general
DO NOT.
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Effects of Inhomogeneities
through averaging Einstein equations
Toy Model:
ds2 = dt2 a2 (1 + h coskx cosky coskz) (dx2 + dy2 + dz2)
Einstein equations after averaging in space: (perturb: h << 1)
peff = eff / 3
Gpa
hk
a
a
a
a
Ga
hk
a
a
832
52
832
153
2
222
2
222
eff
peff
Dark Geometry
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Supernova data ? Cosmic Acceleration
Cosmic Acceleration requires Dark Energy ?
Questions
(or Inhomogeneity-induced Acceleration ?)
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Cosmic Acceleration requires Dark Energy ?
Normal matter attractive gravity
slow down the expansion
Need something abnormal :
e.g. cosmological constant, dark energy
-- providing anti-gravity (repulsive gravity)
Is This True ?
CommonIntuition /
Consensus
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Is This True ? Intuitively, YES ! (of course !!)
Normal matter attractive gravity slow down the expansion
Common Intuition / Consensus
** Kolb, Matarrese, and Riotto (astro-ph/0506534) :
Inhomogeneities of the universe might induce acceleration.
Mission Impossible ? or Mission Difficult ?
Two directions: 1. Prove NO-GO theorem.2. Find counter-examples. This is what we did.
We found counter-examples for a dust universe of spherical symmetry,described by the Lemaitre-Tolman-Bondi (LTB) solution.
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Lemaitre-Tolman-Bondi (LTB) Solution(exact solution in GR)
(unit: c = 8G = 1) Dust Fluid + Spherical Symmetry
k(r) = const., 0(r) = const., a(t,r) = a(t) FRW cosmology
arra
rrt
a
r
a
rk
a
a
drrrk
dr
a
artadtds
r
r
22
30
30
2
2
222
22222
6, :density energy
3 : equation Einstein
11 ,
/
Solution (parametric form with the help of ) adt
rtrkrkrk
rrt
rkrk
rra
b
sin
cos
1
6
16
0
0
,
,arbitrary functions of r :
k(r) , 0(r) , tb(r)
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Line (Radial) Acceleration
( qL < 0 )
Radial : Inhomogeneity AccelerationAngular : No Inhomogeneity No Acceleration
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What is Accelerating Expansion ? (I)
Line Acceleration
L
00
00
LL
LL
:ration Accele; :onDecelerati
:Collapse ; :Expansion
22
:parameter onDecelerati
:rate Expansion
L
LL
H
LLq
LLH
LL
L
homogeneous & isotropic universe: RW metric:
22
2
222
2222 2
1 11
aaaLLLqaaLLH
rk
drtaLdr
rk
drtadtds
LL
r
r
;
: distance proper ;
)(
)(
We found examples of qL < 0 (acceleration)in a dust universe described by the LTB solution.
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Line (Radial) Acceleration : qL < 0
Inhomogeneity the less smoother, the better
arbitrary functions of r : k(r) , 0(r) , tb(r)
0
1
11
00
rt
r
rr
rrkrk
b
nk
nkh
k
k
; const.
;
dr
rrk
rta
a
adrgL
LL r
r
rr
r rr
0 2
22
0 11
:distance Radial
,
parameters : (nk , kh , rk) , 0 , rL , t
1
kh
rk
k(r)
r0
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Examples of Line (Radial) Acceleration : qL < 0
arbitrary functions of r : k(r) , 0(r) , tb(r)
0
1
11
00
rt
r
rr
rrkrk
b
nk
nkh
k
k
const.
parameters : (nk , kh , rk) , 0 , rL , t
1
kh
rk
k(r)
r0
nk kh rk 0 rL t qL qD
20 1 0.7 1 1 1 0.8
Observations q ~ 1 (based on FRW cosmology)
Acceleration
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0.2 0.4 0.6 0.8 1
50
100
150
200
250
300
350
Examples of Line (Radial) Acceleration : qL < 0
nk kh rk 0 rL t qL qD
20 1 0.7 1 1 1 0.8
1, tr
r
k(r) = 0 at rk = 0.7
Over-density Under-density
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Examples of Line (Radial) Acceleration : qL < 0
rrt g2
r
nk kh rk 0 rL t qL qD
20 1 0.7 1 1 1 0.8 k(r) = 0 at rk = 0.7
)( 0660.
characterizing the accel/deceleration status of the radial line elements
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Examples of Line (Radial) Acceleration : qL < 0
rrt g2
r
Deceleration Deceleration
Acceleration
nk kh rk 0 rL t qL qD
20 1 0.7 1 1 1 0.8 k(r) = 0 at rk = 0.7
)( 0660.
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Examples of Line (Radial) Acceleration : qL < 0
)( 0660.
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Acceleration
Inhomogeneity
Examples of Line (Radial) Acceleration : qL < 0
)( 0660.
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Examples of Line (Radial) Acceleration : qL < 0
nk kh rk 0 rL t
(20) 1 0.7 1 1 1
Deceleration
AccelerationEasy to generate
nk=5larger nk
larger inhomogeneity
k
k
nk
nkh
rr
rrkrk
1
11
1
kh
rK
k(r)
r0
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Examples of Line (Radial) Acceleration : qL < 0
nk kh rk 0 rL t
20 1 0.7 1 1 (1)
Deceleration
Acceleration
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Domain Acceleration
( qD < 0 )
sphericaldomain r = 0
r = rD
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What is Accelerating Expansion ? (II)
Volume VD
a large domain D (e.g. size ~ H01)
2
31
DDDD
DDD
DD
aaaq
aaH
Va
NO-GO qD 0 > 0 (deceleration) in a dust universe(see, e.g., Giovannini, hep-th/0505222)
We found examples of qD < 0 (acceleration)in a dust universe described by the LTB solution.
[Nambu and Tanimoto (gr-qc/0507057) : incorrect example.]
Domain Acceleration
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Examples of Domain Acceleration : qD < 0
) ; ; ( 231DDDDDDDDD aaaqaaHVa
t
t
k
k
nt
ntbh
b
nk
nkh
rr
rrtrt
r
rr
rrkrk
1
1
11
00
const.
parameters : (nk , kh , rk), (nt , tbh , rt), 0 , rD , t
arbitrary functions of r : k(r) , 0(r) , tb(r)
nk kh rk nt tbh rt 0 rD t qD
40 40 0.9 40 10 0.9 105 1.1 0.1 1Acceleration
tb(r)
k(r)
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0.2 0.4 0.6 0.8 1
10
20
30
40
50
60
Examples of Domain Acceleration : qD < 0
nk kh rk nt tbh rt 0 rD t qD
40 40 0.9 40 10 0.9 105 1.1 0.1 1
1.0, tr
r
k(r) = 0 at r = 0.82
Over-density Under-density
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Examples of Domain Acceleration : qD < 0
0.2 0.4 0.6 0.8 1
500
1000
1500
2000
2500
rrt g2
r
Deceleration Deceleration
Acceleration
nk kh rk nt tbh rt 0 rD t qD
40 40 0.9 40 10 0.9 105 1.1 0.1 1
)( 51094 .
characterizing the accel/deceleration status of the radial line elements
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Examples of Domain Acceleration : qD < 0
nk kh rk nt tbh rt 0 rD t
(40) 40 0.9 40 10 0.9 105 1.1 0.1
Acceleration
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Examples of Domain Acceleration : qD < 0
nk kh rk nt tbh rt 0 rD t
40 (40) 0.9 40 10 0.9 105 1.1 0.1
Deceleration
Acceleration
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Examples of Domain Acceleration : qD < 0
nk kh rk nt tbh rt 0 rD t
40 40 (0.9) 40 10 (0.9) 105 1.1 0.1
Deceleration
Acceleration
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Examples of Domain Acceleration : qD < 0
Deceleration
Accelerationlarger nt
larger inhomogeneity
tb(r)
t
t
nt
ntbh
brr
rrtrt
1
nk kh rk nt tbh rt 0 rD t
40 40 0.9 (40) 10 0.9 105 1.1 0.1
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Examples of Domain Acceleration : qD < 0
Acceleration
nk kh rk nt tbh rt 0 rD t
40 40 0.9 40 (10) 0.9 105 1.1 0.1
Deceleration
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Examples of Domain Acceleration : qD < 0
Acceleration
nk kh rk nt tbh rt 0 rD t
40 40 0.9 40 10 0.9 (105) 1.1 0.1
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Examples of Domain Acceleration : qD < 0
Deceleration
Acceleration
nk kh rk nt tbh rt 0 rD t
40 40 0.9 40 10 0.9 105(1.1) 0.1
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Examples of Domain Acceleration : qD < 0
Deceleration
Acceleration
nk kh rk nt tbh rt 0 rD t
40 40 0.9 40 10 0.9 105 1.1 (0.1)
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Summary and Discussions
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Model: Inhomogeneous Universe (Reality?)
Against the common intuition and consensus : normal matter attractive gravity deceleration,
Counter-examples (acceleration) are found.
These examples support :
Inhomogeneity Acceleration
Toy Model: ds2 = dt2 a2 (1 + h coskx cosky coskz) (dx2 + dy2 + dz2)
peff = eff / 3
These examples raise two issues :
? Can inhomogeneities explain cosmic acceleration ?(cosmology issue)
? How to understand these counter-intuitive examples ?(GR issue)
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Can Inhomog. explain “Cosmic Acceleration”?
SN Ia Data Cosmic Acceleration
Inhomogeneities
?
?Mathematically, possible. In Reality ???
Can Inhomogeneities explain SN Ia Data?
IF YES
Does Cosmic Acceleration exist?
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How to understand the examples ?
Normal matter attractive gravity
slow down the expansion
CommonIntuition /
Consensus
Intuition for GR ? NO !?
(x)
(valid only for … ?)
)( )(
xgx 00 Newton? NO. GR? YES.
Intuition from Newtonian gravity, not from GR.
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Summary and Discussions
GR is still not fully understood
after 90 years !!
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Line (Radial) Acceleration : qL < 0
arra
rrt
a
r
a
rk
a
a
drrrk
dr
a
artadtds
r
r
22
30
30
2
2
222
22222
63
11
/
,
,
;
Sufficient and Necessary Condition: 02
r
ar
1
kh
rK
k(r)
r0
Sharp enough change in kh(r)
abmnab GTgG 8
Tuning/choosing the boundary condition
( For constant 0 )