Transcript The Accelerating Universe

Open page
L. Perivolaropoulos
http://leandros.physics.uoi.gr
Department of Physics
University of Ioannina
Talk Made in Corfu-Greece
Summer 2006
Dark Energy Probes include
-SnIa (Gold sample and SNLS),
-CMB shift parameter (WMAP 3-year),
-Baryon Acoustic Oscillation Peak in LSS surveys,
-Cluster gas mass fraction,
-Linear growth rate from 2dF (z=0.15)
Some of these probes mildly favor
an evolving w(z) crossing the phantom divide w=-1 over ΛCDM
Minimally Coupled Quintessence is inconsistent with such crossing
Scalar Tensor Quintessence is consistent with w=-1 crossing
Boisseau, Esposito-Farese, Polarski, Starobinsky 2000
LP 2005
Extended Gravity Theories (DGP, Scalar Tensor etc) predict unique
signatures in the perturbations growth rate
m 
Friedmann Equation
Flat
8aG
8 G
H
H aa   m  a   maa
3
3a 
2
22
Directly
Observable
No
Yes
m
3
~ a t 
V
Not
Consistent
Directly Dark Energy
Observable (Inferred)
 dL 

5 log empty
 dL 
 dL 

5 log empty
 dL 
z~0.5: Acceleration starts
from Spergel et. al. 2006
157 SnIa
a
1  d  d L ( z) 
 z   H  z    
 
c  dz  1  z   
Q: What causes this accelerating expansion? a
Flat
1
w
Friedman eqn I:
pX


a
4 G

m    1  3w 



a
3
w 
1
3
 Negative
a
d

3
a
   pX d

p X  w 
 ~ e
 a 

 
3

1
da '
(1 w ( a '))
a'
3




Pressure 
~ a
31 w 
Friedman eqn II:
2
H ( z)
 H
0 m 
2
0
2
a

a2

0m
3

8 G
 a0 



0m 
  

3 
 a 
1  z 
0 m
 0.2  0.3
crit
3
 X
 z 

 
 
 crit 



a


(from large scale structure observations)
0m    k  1

2
a
3
2
2
2
H ( z )  2  H 0 0 m 1  z     k 1  z 
a

 L 
d L ( z )obs
2.5 log10 
  m( z )  M  25  5 log10
Mpc
 l ( z) 
d L ( z )th 
c 1  z 
H0
m   
N
 2  m ,     
i 1

sin 
1


m   

H 0 dz 
 1
0 H  z ; 0 m , 
z

5 log10  d L ( zi )obs   5 log  d L ( zi ; m ,  )th  

 i2





2
 min
SNLS
Truncated
Gold
Gold
Sample
S. Nesseris, L.P.
Phys. Rev. D72:123519, 2005
astro-ph/0511040
z
Physical Model  H  z; a1 , a2 ,..., an 
 dz
0
ansatz 


z
 dz
Data: d Lobs  zi 
th
0

 d L  z; a1 , a2 ,..., an  

 2  2
min
 H  z; a1 , a2 ,..., an 

d L  z; a1 , a2 ,..., an 
Data: d Lobs  zi 

  a1 , a2 ,..., an   
2
 2   min
 w  z; a1 , a2 ,..., an 
2
  min
 a1 , a2 ,..., an 

  ( z )  a0 1  z 
31 w 
Constant w
w( z)  w0  w1 z
w( z )  w0  w1
w( z )  
i
Weller-Albrecht 2002
z
1 z
  ( z )  a0  a1 1  z   a2 1  z 
Chevalier-Polarski 2001, Linder 2003
2
wi
  z  zi     zi 1  z  
2 
 ( z)  a0  a1 cos  a2 z  a3 
w( z ) 
2
d ln H
1 z
1

pX ( z)
3
dz
w z 

2
 X ( z)
3
 H0 
1 
  0 m 1  z 
 H 
 z  zT 
w  w w  w

tanh 

2
2
 z 
Sahni et. al. 2003
Huterer-Cooray 2004
Nesseris-LP 2004
Pogosian et. al. 2005
• All best fit parameterizations cross the phantom divide at z~0.25
• The parametrization with the best χ2 is oscillating
m  0.3
CP  L
2
2
 CDM   177.1
min
OA  171.7 min
Lazkoz, Nesseris, LP 2005
Espana-Bonet, Ruiz-Lapuente
astro-ph/0503210
Wang, Lovelace 2001
Huterer, Starkman 2003
Saini 2003
Wang, Tegmark 2005
Espana-Bonet, Ruiz-Lapuente 2005
Q: Do other SnIa data confirm this trend?
w( z )  w0  w1
z
1 z
SNLS (115 points z<1)
m  0.24
Trunc. Gold (140 points, z<1)
Full Gold (157 points, z<1.7)
SNLS data show no trend for crossing the phantom divide w=-1!
S. Nesseris, L.P.
Phys. Rev. D72:123519, 2005
astro-ph/0511040
R
Definition:
l1TT ~
l1TT ~
1
1
l1


TT
l1

TT
TT
1
TT
1
d A  zrec 

d A  zrec 
rs  zrec 

rs  zrec 
rs  z rec  d A  z rec 
rs  zrec  d A  zrec 
rs arec 
1
: Peak Location of considered model or data
d A  arec 
1
: Peak Location of Corresponding SCDM model:
1
m  1, b h2  b h2 , m h2  m h2
rs 
arec

0
cs  a  da arec cs  a  da  r h2



a


a 2 H  a  0 H 0 1/m 2  m h 2

d A  zrec   arec
1

arec
c da
 c arec
2
a H a
zrec

0
1
2
rs 
arec

0
cs  a  da arec cs  a  da  r h2



a


a 2 H  a  0
H 0  m h 2

1
2
5000
dz
H0 E  z 
m  0.27, b  0.043, bh2  0.022, mh2  0.14
 m  1, b  0.157, bh2  0.022, mh2  0.14
R
1
1
2 c arec 
1  r  2   r  arec  2 


H 0 
rs  zrec  d A  zrec 

rs  zrec  d A  zrec 
2
m
zrec

0
l l 1 Cl TT 2
d A  zrec  
K ^2
3000
1    12     a  12   l1
r
r
rec
 l1TT
dz ' 
E  z '
2000
1500
TT
1000
l1TT  220
5
10
50
mult. number l
100
l '1TT  246
500
1000
5000
m  0.27, b  0.043, b h2  0.022, mh2  0.14
 m  1, b  0.157, b h2  0.022, mh2  0.14
l l 1 Cl TT 2
K ^2
3000
R
l '1TT 246

 1.123 
l1TT 220
2
m
zrec

0
1    12     a  12 
r
r
rec

dz ' 
E  z '
 0.965
2000
R '  m
zrec
dz '
 E  z '  1.7
0
1500
Q: Does R contain all the info
about H(z) in the CMB Spectrum?
1000
l1TT  220  0.8
5
10
50
mult. number l
100
l '1TT  246
500
1000
R  1.7, b h2  0.022, mh2  0.142
m  0.27, w0  0.8, w1  0.0
5000
m  0.27, w0  0.9, w1  0.3
l l 1 Cl TT 2
K ^2
m  0.27, w0  0.8, w1  0.0
m  0.15, w0  1.32, w1  0.0
3000
m  0.50, w0  0.3, w1  0.02
2000
w( z )  w0  w1
1500
z
1 z
 1


3 wunaffected
1 
1
CMB Spectrum
practically
3
31
w0  w1 
 1 z

H ( z )  H  0 m 1  z   1  0 m 1  z 
e





2
2
0
1000
All the useful H(z) related info coming from
the CMB spectrum is contained in R.
5
10
50
mult. number l
100
500
1000
2
d ln H
1 z 
1

pDE ( z )
3
dz
w z 

2
 DE ( z )
3
 H0 
1 
 0 m 1  z 
 H 
1.5
1.5
1.5
Gold dataset
Riess -et. al. (2004)
1
w(z )
SNLS dataset
Astier -et. al. (2005)
1
0.5
0.5
0.5
0
0
0
0.5
0.5
0.5
1
1
1.5
1.5
0m  0.2
1
w( z )  w0  w1
z
1 z
1.5
0
0.25
0.5
0.75
1
1.25
1.5
1.75
0
z
0.25
0.5
0.75
1
1.25
1.5
0
1.75
Gold dataset
Riess -et. al. (2004)
SNLS dataset
Astier -et. al. (2005)
0
 R 

0.5
0.5
m , w1 , w2   1.70 
0.032
1
Wang, Mukherjee 2006
0.5
0.75
0.5
2
2
2
 CMB
 m , w1 , w2   0BAO
 m , w1 , w2    cl2  w1 , w2    LSS
 w1 , w2  0
0m  0.3
Minimize:
0 m  0.2
0.25
1
Other data:
CMB, BAO, LSS, Clusters
1
0.5
0
S. Nesseris,
L.P.
in prep.
1.25
1.5
1.75
0.75
1.5
1
0.5
1.5
0.5
z
1.5
w(z )
0.25
z
1.5
1
Other data:
Other BAO,
data: LSS, Clusters
CMB,
CMB, BAO, LSS, Clusters
1
1
1.25
z
1.5
1.75
2
 A

1
m , w1 , w2   0.469 
2
0.017 2
26

i 1
1.5
f
 zi ; w1 , w2   f gas i 
SCDM0.5
gas
0.5
0.75
1
1.25
z
1.5
1.75
2
0.112
1

 aD' (a)
g  z   1  0.15 
 0.51 0.11
a

 D( a )
Eisenstein et. al. 2005
0.25
 g  z  0.15; w , w   0.51

1
2
 gas
i
1
1.5
0
2
0
0.25
0.5
Allen et. al. 2004
0.75
1
z
1.25
1.5
2dF:Verde
et. al.
1.75
MNRAS 2002
2
CMB  BAO  Clusters  LSS m  0.2
0m  0.2
CMB  BAO  Clusters  LSS m  0.3
0m  0.3
What theory produces crossing of the w=-1?
1 2
+: Quintessence
L     V  
2
-: Phantom
 Quint   1
1 2
   V  
p
 0
w  2

1
  1  2  V   Phant  < 1
 
2
To cross the w=-1 line the kinetic energy term
must change sign
(impossible for single phantom or quintessence field)
Generalization for k-essence:
Non-minimal Coupling
1
8 Geff
    1
F    , U  Φ
 F   1
 , p
1 
1 2

H 
  m    U  3HF 
3F 
2

1
2
H 



p


 F  HF 

m
m
2F
2
L.P. astro-ph/0504582, JCAP 0510:001,2005, S. Nesseris, L.P. astro-ph/0602053, Phys.Rev.D73:103511,2006
JCAP 0511:010,2005
F(Φ)
U(Φ)
Minimum: Generic feature
Φ
Φ
Growth Factor:
Growth Factor Evolution
(Linear-Fourier Space):
 a 
 a
m
a, D a 

  a0  1
 3 H ' a  
3 0 m
 D' k , a  
D' ' k , a    
f k , a Dk , a   0
2
5


a
H
a
2
a H a 


D(a)  a
General Relativity:
a0
f (k , a)  1  D(k , a)  D(a)
Koyama and Maartens (2006)
DGP:
Scalar Tensor:
Modified Poisson:
f (k , a)  1 
1
3  a 
  1
,
f (k, a)  G(a)  G0 1   1  a 
f (k , a )  1  
H (a)  H '(a)a 
1 

H 0rc 
3H (a) 
Boisseau, Esposito-Farese,
Polarski Staroninski (2000)
Uzan (2006)
1
kr
1   s 
 a
2
Sealfon et. al. (2004)
1
Flat Matter Only
0.9
ΛCDM (SnIa best fit, Ωm=0.26)
0.8
ga
g a 
aD '(a)
D(a )
Scalar Tensor (α=-0.5, Ωm=0.26)
0.7
DGP
SnIa best fit
+
Flat Constraint
0.6
1

 aD' (a)
g  z   1  0.15 
 0.51 0.11
a

 D( a )
Verde et. al.
MNRAS 2002
Hawkins et. al.
MNRAS 2003
0.5
0.4
0
0.2
0.6
0.4
a
0.8
1
• Interesting probes of the dark energy evolution include:
- SnIa (Gold sample, SNLS)
- CMB shift parameter
- Baryon Acoustic Oscillations (BAO) Peak of LSS correlation (z=0.35)
- Clusters X-ray gas mass fraction
- Growth rate of perturbations at z=0.15 (from 2dFGRS)
• All recent data indicate that w(z) is close to -1.
Thus w(z) may be crossing the w=-1 line.
• Minimally Coupled Scalar predicts no crossing of w=-1 line
• Scalar Tensor Theories are consistent with crossing of w=-1
• Extended Gravity Theories (DGP, Scalar Tensor etc) predict unique
signatures in the growth rate of cosmological perturbations
F
G r 
 0
G0
F r 
SnIa peak luminosity:
SnIa Absolute Magnitude Evolution:
SnIa Apparent Magnitude:
with:
Parametrizations:
1
m  0.27, w0  0.8, w1  0.0
m  0.27, w0  0.9, w1  0.3
0.8
Models degenerate in ISW are also
degenerate in linear growth factor.
m  0.27, w0  0.8, w1  0.0
m  0.15, w0  1.32, w1  0.0
0.6
Da
m  0.50, w0  0.3, w1  0.02
w( z )  w0  w1
0.4
z
1 z
 a
m

a

a
,
D
a

 
Growth Factor:     
  a0  1
 3 H 'a 
3 0 m
D ''  a    
D a  0
 D '  a  
2
5
a
H
a
2


a H a


0.2
D(a)  a
0
0.2
0.4
0.6
a
0.8
a0
1
Hubble free luminosity Distance
Apparent Magnitude:
χ2 depends on M:
Expandin M :
Minimize:
where
Gold Sample
SNLS
Uniform Analysis of Data
Uniform Analysis of Data
(light curves) by one Group (light curves) by one Group
Combination of Data from
Various Instruments
Use of a single ground based
instrument (megaprime of
CFH 3.6m telescope)
Redshift Range 0<z<1.7
Redshift Range 0<z<1
157 datapoints
73 new datapoints
 i 
Data  d L  zi  
 d Ls  z ,     d L  zi K  z , zi ,  
K z , z ,

1  d  d ( z , )  
 H  z ,     
 
c  dz  1  z   
d
dz
s
s
L
1
smoothing scale
d ln H  z ,  
2
1 z 
1
d

dz
dz

 ws  z ,    3
2
3
 H0 
1 
 0 m 1  z 
 H 
s
Wang, Lovelace 2001
Huterer, Starkman 2003
Saini 2003
Wang, Tegmark 2005
Espana-Bonet, Ruiz-Lapuente 2005
Fisher Matrix:
1  2  2 w1 , w2 
 Aij w1 , w2   Aij w1 , w2   Aij1 w1 , w2   Cij w1 , w1 
2 wi w j
Parameter Estimation:
wi  wi  Cii  w1 , w2 
w(z) plot with error regions:
z
1 z
w( z )  w0  w1
w  z 
w1 ( z )  w  z   
i , j 1 wi
w
2
i , j  wi , j
Covariance Matrix
w  z 
w j w
i , j  wi , j
Cij  w1 , w2 
from Max Tegmark's home page
Effective Scale:
1/ 3
2
 c z  z dz 


1/
3
z  z
2
DV  z    x  x  

  c 
  

H
z
H
z
    0  
 

Correlation function:
z 
  , z  H
  r   rpeak 
LCDM
DV z  0.35
rpeak  rsound
DVLCDM z  0.35
x  AB  c
z
H z 
 z dz  
x  CD  x    c 
 

H
z


 0

DV z  0.35  1370 64 Mpc
DV  z  0.35 m H 02
A
 0.469  0.017
0.35 c
w( z )  w0  w1
Assume:
z
1 z
2
2
Minimize: CMB
 m , w1, w2   BAO
 m , w1, w2  
1.5
 R  m , w1, w2  1.70
0.032

 A  m , w1, w2   4.69
0.172
1.5
m  0.25
1
m  0.3
1
0.5
0.5
wz 
0
0
0.5
0.5
1
1
1.5
1.5
wz 
2
0
0.25
0.5
0.75
1
z
1.25
1.5
1.75
0
0.25
0.5
0.75
1
z
1.25
1.5
1.75
2
Define Cluster Baryon Gas Mass fraction:
f gas 
M b  gas
M tot

b
m
Global Mass Fraction vs Baryon Gas Mass fraction:
M b gas
b
Mb
b

 1   
 1    f gas
m M tot
M tot
Isothermal Gas Model:
5
1
r
 ...  M b  gas   R   B  Te , c ,   rc3/ 2 LX   R  2  C Te ,c , , l X , z  d A  z  2
R 

 c d A z 
4 d L  z  l X   R 
2
Cluster
Hydrostatic Equilibrium:  ...  Mtot   R  D Te  d A  z 
O
c
Cluster Baryon Gas Mass fraction:
f gas 
M gas   R 
M tot   R 

3
3
C
d A  z  2  Q c , , Te ... d A  z  2
D
Observed
Connect to Global Mass fraction:
3

b b  1    f gas  1    Qi d A  zi  2
m
SCDM
f gas
 zi 
SCDM
SCDM
f
z

Q
d


 zi   Qi  SCDM
gas
i
i
A
Define:
dA
 zi 
f gas 
3
1 b
 Qi d A  zi  2
1   m
SCDM
b b  d ASCDM zi  
SCDM


f gas zi  
1    m  d A zi  
Data
LCDM
3
2
w( z )  w0  w1
Assume:
z
1 z
26
Minimize:
cl2  w1 , w2   
f
SCDM
gas
i 1
 zi ; w1 , w2   f gas i 
2
 gas
i
1
1
m  0.25
0
wz 
2
1
1
2
2
wz 
3
3
4
4
5
5
6
6
0
0.25 0.5 0.75
1
1.25 1.5 1.75
z
m  0.3
0
0
0.25 0.5 0.75
1
1.25 1.5 1.75
z
1.5
1.5
1
1
0.5
wz 
CMB  BAO  Clusters m  0.25
CMB  BAO m  0.25

2
CMB

0.5
wz 
2
BAO
0
0
0.5
0.5
1
1
1.5
1.5
0
0.25
0.5
0.75
1
z
1.25
1.5
1.75
2
2
CMB
 BAO
 cl2
0
0.25
0.5
0.75
1
z
1.25
1.5
1.75
Growth Factor:
Growth Factor Evolution
(Linear-Fourier Space):
 a 
 a
m
a, D a 

  a0  1
 3 H ' a  
3 0 m
 D' k , a  
D' ' k , a    
f k , a Dk , a   0
2
5


a
H
a
2
a H a 


D(a)  a
General Relativity:
f (k , a)  1  D(k , a)  D(a)
a0
2
w( z )  w0  w1
z
1 z
m  0.25, w1  0.8, w2  0.0
1.75
m  0.25, w1  0.9, w2  0.3
m  0.25, w1  1.0, w2  0.59
1.5
m  0.25, w1  3, w2  0.0
aD '(a)
1.25
ga
g a 
D(a )
m  0.25, w1  0.5, w2  0.0
1

 aD' (a)
g  z   1  0.15 
 0.51 0.11
a

 D( a )
Verde et. al.
MNRAS 2002
Hawkins et. al.
MNRAS 2003
1
0.75
0.5
0.25
0.2
0.4
0.6
a
0.8
1
w( z )  w0  w1
Assume:
2
 LSS
 w1, w2  
Minimize:
 g  z  0.15; w , w   0.51
1
0.112
1.5
1
1
CMB  BAO m  0.25

2
CMB

2
2
2
CMB
 BAO
 LSS
0
0.5
0.5
1
1
1.5
1.5
0.25
0.5
0.75
CMB  BAO  LSS m  0.25
0.5
2
BAO
0
0
2
2
1.5
0.5
wz 
z
1 z
1
z
1.25
1.5
1.75
0
0.25
0.5
0.75
1
1.25
1.5
1.75
'
d
dz
positive energy of gravitons
For U(z)=0 there is no acceptable F(z)>0 in 0<z<2 consistent with
the H(z) obtained even from a flat LCDM model.
1
0.75
0.5
F
0.25
0
0.25
0.5
0.75
0
0.2
0.4
0.6
z
0.8
1
SNLS
Truncated
Gold
Full
Gold
S. Nesseris, L.P.
Phys. Rev. D72:123519, 2005
astro-ph/0511040
Minimize:
N
 2  m ,    
i 1

5 log10  d L ( zi )obs   5 log  d L ( zi ; m ,  )th  

 i2
2
Fisher Matrix:
1  2  2 w1 , w2 
 Aij w1 , w2   Aij w1 , w2   Aij1 w1 , w2   Cij w1 , w1 
2 wi w j
Parameter Estimation:
wi  wi  Cii  w1 , w2 
w(z) plot with error regions:
z
1 z
w( z )  w0  w1
w  z 
w1 ( z )  w  z   
i , j 1 wi
w
2
i , j  wi , j
Covariance Matrix
w  z 
w j w
i , j  wi , j
Cij  w1 , w2 
0.078 0.189 0.011
0.088 0.184 0.011
0.143 0.167 0.019
0.188 0.169 0.011
0.206 0.180 0.015
0.2
0.18
0.16
0.14
0.12
0.1
0.08
0.2
0.4
0.6
0.8