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Cosmic Acceleration
from the basics to the frontiers
Je-An Gu ( 顧哲安 )
National Center for Theoretical Sciences (NCTS)
2007/04/27 @ Academia Sinica
Accelerating Expansion
(homogeneous & isotropic)
Based on FRW Cosmology
Concordance: = 0.73 , M = 0.27
Supernova Observations
Supernova (SN) : mapping out the evolution herstory
Type Ia Supernova (SN Ia) : (standard candle)
– thermonulear explosion of carbon-oxide white dwarfs –
Correlation between the peak luminosity and the decline rate
absolute magnitude M
luminosity distance dL
(distance precision: mag = 0.15 mag dL/dL ~ 7%)
Spectral information redshift zSN Ia Data: dL(z) [ i.e, dL,i(zi) ] [ ~ x(t) ~ position (time) ]
2 4 LdLF F: flux (energy/areatime)
L: luminosity (energy/time)
Distance Modulus Mm 25)pc/( 5 10 MdlogMm L
(z)
history
3.0 ,7.0
model fiducial
)()()(
m
zmzmzm
(can hardly distinguish different models)
SCP(Perlmutter et. al.)
Distance Modulus Mm 25)pc/( 5 10 MdlogMm L
1998
Fig.4 in astro-ph/0402512 [Riess et al., ApJ 607 (2004) 665]Gold Sample (data set) [MLCS2k2 SN Ia Hubble diagram]
- Diamonds: ground based discoveries - Filled symbols: HST-discovered SNe Ia
- Dashed line: best fit for a flat cosmology: M=0.29 =0.71
2004
Riess et al. astro-ph/0611572200625)pc/( 5 10 Mdlog L
2006 Riess et al. astro-ph/0611572
Supernova / Acceleration Probe (SNAP)
z 0~0.2 0.2~1.2 1.2~1.4 1.4~1.7
# of SN 50 1800 50 15
observe ~2000 SNe in 2 years
statistical uncertainty mag = 0.15 mag 7% uncertainty in dL
sys = 0.02 mag at z =1.5
z = 0.002 mag (negligible)
Definition of Acceleration
Accelerating Expansion: Definition
00
00
LL
LL
:ration Accele; :onDecelerati
:Collapse ; :Expansion
22 L
LL
H
LLq
LLH
:parameter onDecelerati
length) pervelocity (~ :rate Expansion
Distance L
E.g. 1. Proper distance (Line Acceleration)
E.g. 2. L = VD1/3 (Domain Acceleration)
, at time t
0 :ration Accele; 0 :onDecelerati
0 :Collapse ; 0 :Expansion
HH
Accelerating Expansion : H > 0 , q < 0
VD
(Volume)
a large domain D (e.g. size ~ H01)
Friedmann-Lemaitre-Robertson-Walker Cosmology
Homogeneous & Isotropic Universe :
22
2
222
2222
;
1)( : distance proper
; 1
)( :metric Walker-Robertson
2
1
aaa
L
LLq
aa
LL
H
rk
drtaL
drrk
drtadtds
LL
r
r
0 , 0 :ration Accele; 0 , 0 :onDecelerati
0, :Collapse ; 0, :Expansion
qaqa
aHaH
Accelerating Expansion : H > 0 , q < 0 )0 , 0( aa
Friedmann-Lemaitre-Robertson-Walker (FLRW) Cosmology
22
2
2222
1 )( :metric (RW) Walker-Robertson dr
rk
drtadtds
Homogeneous & Isotropic Universe :
)( pG
a
a
G
a
k
a
a
33
4
3
82
2
)(
, costant with for
:onconservati momentum-energy
iwi
iiii
a
wwp
apdad
13
33
03
10 ap :onaccelerati ,
(Dark Energy)
Accelerating Expansion: Definition
Gauge Dependence of Acceleration ??
independent of gauge choice (coordinate choice) (frame choice)
Issues
Space Expansion or Particle Motion ??
Gauge-Independent Definition ?
Models
(from vacuum energy)
• Quintessence
Candidates: Dark Geometry vs. Dark Energy
Einstein Equations
Geometry Matter/Energy
Dark Geometry
↑Dark Matter / Energy
↑
Gμν = 8πGNTμν
• Modification of Gravity
• Averaging Einstein Equations
• Extra Dimensions
(Non-FLRW)
for an inhomogeneous universe(based on FLRW)
FLRW + CDM
22
2
2222
1dr
rk
drtadtds
)( :metric RW
)( mm
m
pG
a
a
G
a
k
a
a
33
4
3
3
8
32
2
0 a :onaccelerati enough large
1
88
00
pw
Gp
G
pwpm
mmm
,
, i.e. ,
Fine-tuning problems: cosmological constant () problem, coincidence problem
FLRW + CDM : fine-tuning problems
Coincidence problem
Cosmological constant problem
How to make vanish ?Pre-Dark-Energy
very huge if EWSUSYplcutoff MMMM ,,~4~ cutoffvac M
SSB Phase Transition:
Latent heat = vacuum energy (change) ~ TPT (eg. MEW)
How to make slightly deviate from 0 ?Post-Dark-Energy
411energydark eV103
+
Why ~ m NOW ?
Figure 1.1
3aMM
• Ratio changes rapidly with scale factor
• But at present time, M ~
• Why???
Why ΩΛ ~ ΩM now ?
Why accelerating now ?
Ωi ρi / ρc
Problem & Coincidence Problem
Why so small initially ?
FLRW + Quintessence
Quintessence: dynamical scalar field
VggdxS2
14Action :
01
3 222
2
Vat
Ht
Field equation:
Vat
p
Vat
2
2
2
2
2
2
6
1
2
1
2
1
2
1
energy densityand pressure :
Slow evolution and weak spatial dependence V() dominates w ~ 1 Acceleration
How to achieve it (naturally) ??
Non-Quinte: rapidly oscillating mode
: For 22
2
1 mV
22
22
0 ma
ktkx kk , :mode goscillatin an cos~
06
1
2
1 202
220
22
2
a
kpm
a
ktt
,
mode) -(small for ,
mode) -(large for ,
kma
k
kma
kp
wt
t
22
2
22
2
0
3
1
~ radiation
~ NR matter
time-averaged energy density and pressure :
Non-Quinte: ensemble of incoherent oscillators
: For 22
2
1 mV
)( , :soscillator of ensemble an 1210 Niiktkx ,,,cos~
06
1
2
1 202
220
22
2
a
kpm
a
kenen
,
mode) -(small for , 0
mode) -(large for , 3
1
22
2
22
2
kma
k
kma
kp
wen
en
~ radiation
~ NR matter
ensemble-averaged energy density and pressure :
(i : the phase of i-th oscillator)
Non-Quinte: oscillators
Thus, Oscillators
22
2
1 mV for least at
How about other potentials ??
?,,,, 643 V
Quinte: a slowly evolving mode or coherent state
: For 22
2
1 mV
22
22
0 ma
ktkx kk , :solution cos~
00
0
H
k
k ~
~
:evolutionslow
:dependence spatial weak
GeV480 10 ~Hm (unnaturally small !!)
G
HmV c
8
3
2
1 2022 ~~ [V() dominates.]
GeVenergy Planck 1921 10~~/G (unnaturally large !!)
Tracker Quintessence
V
n
nMV
4
MV exp
Inhomogeneous Cosmo. Model
(motivation & final goal: come to the reality)
-- Violating cosmological principle --
Is FLRW Cosmology a good approximation ??
Fundamental Question
If yes, then, WITHOUT DARK ENERGY,
there is NO WAY to generate Cosmic Acceleration.
Is FLRW Cosmology a good approximation ??
Fundamental Question
Acceleration from Inhomogeneity ??
Is FLRW Cosmology
a good approximation ??
…………
YES !YES !YES !
FLRW Cosmology
homogeneous & isotropic
Robertson-Walker (RW) metric
Friedmann-Lemaitre-Robertson-Walker (FLRW) Cosmology
jiij dxdxtadtds 222
Einstein equations: G = 8 G T
Representing the “real” situation of the energy contents of our universe
Is FLRW Cosmology a good approximation ??
(homogeneous & isotropic)
Apparently, our universe isNOT
homogeneous & isotropic.
NO
At large scales, after averaging,
the universe IS homogeneous & isotropic.)( , )(
spacespacetgt
In general, averaging/coarse graining is
NOT VALIDfor Einstein equations.
(due to the non-linearity)
YES
Einstein equations
abmnab GTgG 8
abmnabmnab TGgGgG 8
abmn Tg ,
abmn Tg ,
For which satisfy Einstein equations,
in general
DO NOT.
Is FLRW Cosmology a good approximation ??
NO YES
Contributions from metric perturbations are negligible.
Ishibashi & Wald [gr-qc/0509108]
jiij dxdxtadtds 2121 222
ji
jii
ii
i DDDDDDDDat
2
2
2
, 1
, 1
perturbed metric, non-perturb T
eff
peff
Averaged Einstein equations:
Gphhak
aa
aa
Ghhak
aa
881
81
2
883
815
3
2222
2222
kzkykxth coscoscos
Toy Model[ h(t) << 1 ]
Issues :
(1) Do these requirements fit the real situation of our universe ?
(2) (How much) Can we trust the perturbative analysis ?
Is FLRW Cosmology a good approximation ??
NO YES
jiij dxdxtadtds 2121 222
Newtonianly perturbed metric
terms keeping 22 xt ,
8G peff8G eff
23 t 2t
weff peff / eff
31 /
21
15
xa
21
xa
151 /
cannot generate acceleration
might be significant
Acceleration
from
Inhomogeneity ??
Acceleration from reality ?? -- Don’t know.(i.e. from the inhomogeneities of our universe)
General possibility ?? -- To be discussed
Do we really need
Dark Energy ??
NO YES
FLRW Cosmology:
Acceleration Dark Energy
homogeneous & isotropic
RW metric : jiij dxdxtadtds 222
Einstein equations: (G = 8 G T )
pG
a
a
G
a
k
a
a
33
43
82
2
3
103 pp i.e., , whenonaccelerati
Cosmic acceleration requires negative pressure (repulsive/anti gravity).
based on FLRW cosmology
could be model-dependent
Need Dark Energy ?? YES
Intuitively,
Normal matter
attractive gravity
slow down the expansion
CommonIntuition /
Consensus
We found
line/domain accel. Examples(generated by inhomog)(not by DE)
based on the LTB solution. (Lemaitre-Tolman-Bondi)
(exact solution)
(dust fluid) (spherical symmetry)
Chuang, Gu & Hwang [astro-ph/0512651]
( need dark energy )
Join the dark.
concentrate, balance.…
Do we really need Dark Energy ??
NO YES
Examples of Acceleration : q < 0
)( 0660.
Over-density
Under-density
Acceleration
Deceleration
Deceleration
Acceleration
Inhomogeneity
Examples of Line (Radial) Acceleration : qL < 0
)( 0660.
Acceleration from Inhomogeneity ??
Warning!! Be careful (!!)when connecting two regions.
E.g. FLRWdecel.
FLRWdecel.
Domain Acceleration !!
No physically observable effects of acceleration [regarding,e.g., dL(z)]
There could exist singularity which leads to strange pheno.
E.g. a lesson from Nambu & Tanimoto(incorrect accel. example) [gr-qc/0507057] (perhaps NOT exist at all !!)
Mr. Anderson, …
NOYES
Fake!? Illusion!?
3/1 , 0 DDD VLL
You are illusion !!
Acceleration from Inhomogeneity ??
Issues gauge-dep of acceleration
frame acceleration !?
NOYES
Fake!? Illusion!?
definition of acceleration
A system consisting of freely moving particles
(interacting only through gravity)
Frame Acceleration
Distance L
E.g. 1. Proper distance (Line Acceleration)
E.g. 2. L = VD1/3 (Domain Acceleration)
, at time t
VD
(Volume)
a large domain D
00
00
LL
LL
:ration Accele; :onDecelerati
:Collapse ; :Expansion
(e.g. size ~ H01)
A system consisting of freely moving particles
(interacting only through gravity)
Frame Acceleration
Distance L
E.g. 1. Proper distance (Line Acceleration)
E.g. 2. L = VD1/3 (Domain Acceleration)
, at time t
VD
(Volume)
a large domain D
00
00
LL
LL
:ration Accele; :onDecelerati
:Collapse ; :Expansion
(e.g. size ~ H01)
Frame Acceleration ??
independent of gauge choice (coordinate/frame choice)
Issues
Space Expansion or Particle Motion ??
Definition of Acceleration (revisit)
Gauge-Independent Definition ??
Definition of Acceleration (revisit)
Gauge-independent definition of accelerating expansion ?(maybe no)
Distance L
0 :onDecelerati
0 :onAccelerati
0 :Collapse
0 :Expansion
L
L
L
L
E.g. 1. Proper distance (Line Acceleration)
E.g. 2. L = VD1/3 (Domain Acceleration)
, at time t
VD
(Volume)
a large domain D (e.g. size ~ H01)
L
proper distance between two freely moving particles
constant particle number inside
Consider a system consisting of freely moving particlesInteracting with each other only through gravity
Avoid confusion from particle motion & frame acceleration ?
Benefits of Comoving/Synchronous Gauge
Universal time (?)
Avoiding frame acceleration.
Avoiding confusion about particle motion and space expansion.
) ( jiij dxdxgdtds 22
Definition of Acceleration (revisit)
proper distance between two freely moving particles(line)proper distance between two points fixed in space
constant particle number inside(domain)size of a domain with its boundary fixed in space
Summary and Perspectives
Model: FLRW + CDM
Fine-tuning problems:
cosmological constant problem
coincidence problem?
?
Model: FLRW + Quintessence
Oscillators
22
2
1 mV for least at
Other potentials ?? ? ? gV ,,,, 643
? Other fields ?
Slow evolution and weak spatial dependence V() dominates w ~ 1 Acceleration
GeV480 10 ~Hm (unnaturally small !!)
GeVenergy Planck 1921 10~~/G (unnaturally large !!)
Observations(SN Ia & others) 0RWa
ion AcceleratFLRW
Inhomogeneity Cosmic Acceleration
?
?
?
Acceleration? from DE? from Inhomogeneity?
Do we really need Dark Energy (DE) ??
(definition ?)
(Chuang, Gu & Hwang: mathematical examples)
(data fitting in LTB models)
Reality ? -- Don’t know.
General possibility ?? -- Yes.
Acceleration? from DE? from Inhomogeneity?
Do we really need Dark Energy (DE) ??
Difficulties & limitation stemming from the complexities of :
the complicated energy distribution of our universe the non-linear Einstein equations
Current approaches
Perturbative analysis approach (not convincing)
Utilizing exact solutions of the Einstein equations (toy model, maybe far away from the real situation)
Cannot deal with the full Einstein equations describing our universe with complicated energy distribution
Is FLRW Cosmology a good approximation ??
NO YES
Acceleration from Inhomogeneity ??
whowillwin???
NO
DON’T KNOW
YES
Is FLRW Cosmology a good approximation ??
NO YES
Acceleration from Inhomogeneity ??
whowillwin???
NO
April 30 (Mon)(early morning)
YES
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