Download A New Method To Determine Large Scale Structure From

A New Method To Determine
Large Scale Structure From The
Luminosity Distance
Hsu-Wen Chiang
in collaboration with
Antonio Enea Romano and Pisin Chen
Leung Center for Cosmology and Particle Astrophysics (LeCosPA)
National Taiwan University
Class. Quan. Grav. Vol.31 115008, arXiv:1312.4458
Accelerating Expansion from Large
Scale Inhomogeneity
• A homogeneous
• An inhomogeneous
universe (FRW model) universe (Void model)
Expansion Contraction
Expansion Contraction
Accelerating Expansion from Large
Scale Inhomogeneity
• Early time:
• Now:
Denser
Expansion Contraction
Looser
As Time Goes By
Accelerating Expansion!
Accelerating Expansion from Large
Scale Inhomogeneity
• Early time:
• Now:
Denser
Expansion
Looser
Compatible
With
Experiments?
Contraction
As Time Goes By
Accelerating Expansion!
Outline
• Accelerating Expansion from Large Scale
Inhomogeneity: LTB Model
• Mimicking ΛCDM Model: Central Spatial
Curvature as “Free Parameter”
• Climbing over Apparent Horizon: An Unique
Solution
(R = Areal Radius)
Modeling Large Scale Inhomogeneity
• Assuming spherical symmetry (LTB metric):
2


2


r

a
t
,
r


dr
2
2
2
,
ds 2  dt 2  a  t , r  1  r

r
d


2
a  t , r   1  k  r  r




2
k  r  2M  r 
2 r M
 t a 
2
 2 2
, H 

 3 3

2
a r  r r a  a 
a
ar
 a 
1
3
M
r


r


• Fix gauge freedom of
r
by
setting
0
t
6
d


• Conformal time
0 a  , r   tb  r 
0 

a  , r  
1

cos
k
r





 From now on we set tb=0
6k  r  
1
0 
t  , r  

 k  r  sin  k  r    tb  r 

6k  r  
(R = Areal Radius)
Fixing Initial Condition
• Assuming a central observer, we have
r  z  0   0, a  z  0   a0 ,   z  0   0 , k  z  0   k0
 ln a  t , r 
LTB
LTB
and H  z  0  H0 ,where H 
.
t
LTB
• Since a   0 , r  0  a0 and H   0 , r  0  H0
are given,0 , k0 and 0 can be determined up
to k0 .
• Central spatial curvature as “free parameter”
• Luminosity distance on past null geodesic
2
DLobs  z   1  z  a  t  z  , r  z   r  DLFRW  z  is the input.
(R = Areal Radius)
Numerical Results
• Not so stable around apparent horizon at
• Overcome AH through extrapolation?!
r
redshift-blueshift
transition
best-fit
z 1.6
R  ra
AH
negative
density
k0  0.9376
k0  0.937613784686 1
k0  0.93762
z
z
(R = Areal Radius)
Numerical Results
• Not so stable around apparent horizon at
Only One Valid k0, Why?
• Overcome AH through extrapolation?!
r
redshift-blueshift
transition
best-fit
z 1.6
R  ra
AH
negative
density
k0  0.9376
k0  0.937613784686 1
k0  0.93762
z
z
Expansion Around Apparent Horizon
• Staring at the geodesic equations, we found a
common denominator r k 1  s 2   s 1  kr 2
1
, where s  H 0ka 1  k0  .
Unstable when R  1  k0  r 3 H0 or s  kr 2
• Expand the numerator around R  1  k0  r 3 H0
dR
z
dk Bk
dz


dz Ak A s  kr 2
k
Ck

• Needs s 
dR
 z
dr Br
dz


dz Ar A s  kr 2
r
Cr

kr 2
,

and
dR
 z  0
dz

dR
 z
d B
dz


dz A A s  kr 2

C
,
at same spot


Expansion Around Apparent Horizon
dR
 z  0
dz
• Needs s  kr and
• Also s 2  1   z  z AH 
2
at same spot
kr
dR
8.4
 z
dz
s
kr
2
1
d
s
dR
Cs
z

2
B
kr  s 
dz
dz
As A s  kr 2
s

z

Expansion Around Apparent Horizon
r
rk0  z   r*  z   hyperbolic correction
best-fit
k0  0.93761378
k0  0.937613784686 1
k0  0.9376138
Validation of Extrapolation
z
Uniqueness of k0
• Sudden jump happens only at RAH  1  k0  rz 3 H0
• The existence of solution extended beyond AH
is indicated by transit of cause of stop of
integrator between dr  0 and dz  0 .
dz
dr
• We scanned over parameter space k0   1,  
and found 1 solution.
• Not a rigorous proof yet
AH
Conclusion
• There exists 1 to N correspondence between
ΛCDM metric with certain parameter, and LTB
metrics with specific setups that mimic the
luminosity distance of that ΛCDM metric.
• But only 1 LTB metric can go beyond apparent
horizon without hazards like negative density.
ΛCDM Metric
m ,  
1 to N LTB Metrics kk  r  , ak t , r 
k0   1,  
DL
Hazard-free
k*  r  , a*  t , r 

1 to 1
Hazardous
0
0
Conclusion
• The error of extrapolation used to overcome
apparent horizon is marginal as long as k 0
used in simulation is close to the best fit k 0
value.
• Best extrapolation method is 1st order Taylor
expansion.
The End