Non-parametric Reconstruction of Dark Energy Equation of State

Non-parametric Reconstruction of Dark Energy Equation of State

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Energy Content of the Universe
Ujjaini Alam (COSMO11, Porto, Aug 25, 2011) – p. 1
SNe Ia & Dark Energy
High z SNe dimmer than expected ( 1997-98)
⇒ Expansion of Universe accelerating
⇒ Dominant energy component of Universe has negative pressure
= Dark Energy !!
Ujjaini Alam (COSMO11, Porto, Aug 25, 2011) – p. 2
SNe Ia & Dark Energy
High z SNe dimmer than expected ( 1997-98)
⇒ Expansion of Universe accelerating
⇒ Dominant energy component of Universe has negative pressure
= Dark Energy !!
Cosmological Constant : w = −1
Quiessence : −1 < w =constant < −1/3
Quintessence : L =
1
∂ φ
2 a
∂ a φ − V (φ)
V = V0 /φα
V = V0 exp(λφ2 )/φα
V = V0 (coshλφ − 1)p
Phantom fields with w < −1, Early Dark Energy Models
√
k-essence : L = −V (φ) 1 − ∂a φ ∂ a φ
(Chaplygin gas : P = −A/ρα )
Modified gravity models : f (r) theories, braneworld models....
Ujjaini Alam (COSMO11, Porto, Aug 25, 2011) – p. 2
Distance Measures for Dark Energy
r(z)
=
Z
0
µB (z)
=
dA (z)
rs (z⋆ )
=
R(zCMB )
=
z
dz
=
h(z)
Z
0
z
dz
r
Ω̃r (1 + z)4 + Ω0m (1 + z)3 + ΩΛ exp
M + 5log10 [(1 + z)r(z)]
r(z)
c
; H(z)rs (z⋆ )
H0 (1 + z)rs (z⋆ )
p
Ω0m r(zCMB )
← SNe
hR
z
3(1+w(u))du
0
1+u
i
← BAO
← CMB
Ujjaini Alam (COSMO11, Porto, Aug 25, 2011) – p. 3
Distance Measures for Dark Energy
r(z)
=
z
Z
dz
=
h(z)
0
dA (z)
rs (z⋆ )
=
R(zCMB )
=
dz
r
0
Ω̃r (1 + z)4 + Ω0m (1 + z)3 + ΩΛ exp
M + 5log10 [(1 + z)r(z)]
← SNe
r(z)
c
; H(z)rs (z⋆ )
H0 (1 + z)rs (z⋆ )
p
Ω0m r(zCMB )
0
hR
z
3(1+w(u))du
0
1+u
i
← BAO
← CMB
46
45
-0.2
44
-0.4
43
µ
=
z
w(z)
µB (z)
Z
42
-0.6
41
-0.8
40
39
-1
0.2
0.4
0.6
0.8
1
z
1.2
1.4
38
1.6
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
z
Ujjaini Alam (COSMO11, Porto, Aug 25, 2011) – p. 3
Distance Measures for Dark Energy
r(z)
=
Z
z
0
µB (z)
=
dA (z)
rs (z⋆ )
=
R(zCMB )
=
dz
=
h(z)
Z
z
dz
r
0
Ω̃r (1 + z)4 + Ω0m (1 + z)3 + ΩΛ exp
M + 5log10 [(1 + z)r(z)]
← SNe
r(z)
c
; H(z)rs (z⋆ )
H0 (1 + z)rs (z⋆ )
p
Ω0m r(zCMB )
0
hR
z
3(1+w(u))du
0
1+u
i
← BAO
← CMB
46
35000
12
45
-0.2
30000
44
10
-0.4
42
-0.6
Hrs
25000
DA/rs
µ
w(z)
43
8
20000
41
6
-0.8
40
15000
39
4
-1
38
0.2
0.4
0.6
0.8
1
z
1.2
1.4
1.6
0.2
0.4
0.6
0.8
1
z
1.2
1.4
1.6
0.2
0.4
0.6
0.8
1
z
1.2
1.4
10000
1.6
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
z
Ujjaini Alam (COSMO11, Porto, Aug 25, 2011) – p. 3
Gaussian Process Modeling
p(y|X)
=
1
1
2
(y
−
f
(x))
exp
−
2
2 )n/2
2σn
(2πσn
f (x)
=
GP(m(x), K(x, x′ ))
Ujjaini Alam (COSMO11, Porto, Aug 25, 2011) – p. 4
Gaussian Process Modeling
p(y|X)
=
1
1
2
(y
−
f
(x))
exp
−
2
2 )n/2
2σn
(2πσn
f (x)
=
GP(m(x), K(x, x′ ))
Ujjaini Alam (COSMO11, Porto, Aug 25, 2011) – p. 4
Application to distance measures
′ α
w(u) ∼ GP(−1, κ2 ρ|u−u | )
R s R s′
y(s) ∼ GP − ln(1 + s), κ2 0 0
′
ρu−u dudu′
(1+u)(1+u′ )
Joint GP for y(s) and w(u):


y(s)
w(u)


 ∼ MVN 
− ln(1 + s)
−1


Σ11
Σ12
Σ21
Σ22

 ,
Mean for y(s) given w(u) :
y(s)|w(u) = − ln(1 + s) + Σ12 Σ−1
22 (w(u) − (−1))
Obtain 2nd integral numerically, compute likelihood
Ujjaini Alam (COSMO11, Porto, Aug 25, 2011) – p. 5
Current Observations
SNe Union2 compilation → 557 SNe, σmB ∼ 0.15
BAO SDSS ⇒ rs (z⋆ )(H(z)/(1 + z)2 d2A cz)1/3 = 0.19 ± 0.0061(z = 0.2)
= 0.11 ± 0.0036(z = 0.35)
CMB WMAP7 ⇒ R = 1.719 ± 0.019
Ujjaini Alam (COSMO11, Porto, Aug 25, 2011) – p. 6
0.0
Current results
SNe+CMB
SNe+BAO
SNe+BAO+CMB
−1.0
−2.0
−1.5
w(z)
−0.5
SNe
0.2
0.4
0.6
0.8
1.0
1.2
0.2
0.4
0.6
0.8
1.0
1.2
0.2
0.4
0.6
z
0.8
1.0
1.2
0.2
0.4
0.6
z
0.8
1.0
1.2
z
10
5
0
Density
15
20
z
0.0
0.2
0.4
0.6
Ωm
0.8
0.0
0.2
0.4
0.6
Ωm
0.8
0.0
0.2
0.4
0.6
Ωm
0.8
0.0
0.2
0.4
0.6
0.8
1.0
Ωm
Ujjaini Alam (COSMO11, Porto, Aug 25, 2011) – p. 7
Future Predictions
0
-0.2
w(z)
-0.4
-0.6
-0.8
-1
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
z
Ujjaini Alam (COSMO11, Porto, Aug 25, 2011) – p. 8
Future Predictions
0
-0.2
w(z)
-0.4
-0.6
-0.8
-1
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
z
0.0
−0.5
−1.0
−2.0
−1.5
w(z)
−1.5
−2.0
w(z)
−1.0
−0.5
0.0
BIGBOSS (20 BAO)
0.5
1.0
z
1.5
0.5
1.0
1.5
z
Ujjaini Alam (COSMO11, Porto, Aug 25, 2011) – p. 8
Future Predictions
0.0
Union2 (557 SNe), BIGBOSS (20 BAO), WMAP7
SNe+CMB
SNe+BAO
SNe+BAO+CMB
−1.0
−2.0
−1.5
w(z)
−0.5
SNe
0.2
0.4
0.6
0.8
1.0
1.2
0.2
0.4
z
0.6
0.8
1.0
1.2
0.5
z
1.0
1.5
0.5
z
1.0
1.5
z
0.0
WFIRST (2298 SNe), BIGBOSS (20 BAO), WMAP7
SNe+CMB
SNe+BAO
SNe+BAO+CMB
−1.0
−1.5
−2.0
w(z)
−0.5
SNe
0.5
1.0
z
1.5
0.5
1.0
z
1.5
0.5
1.0
z
1.5
0.5
1.0
1.5
z
Ujjaini Alam (COSMO11, Porto, Aug 25, 2011) – p. 9
Future Predictions
0.0
Union2 (557 SNe), BIGBOSS (20 BAO), WMAP7
SNe+CMB
SNe+BAO
SNe+BAO+CMB
−1.0
−2.0
−1.5
w(z)
−0.5
SNe
0.2
0.4
0.6
0.8
1.0
1.2
0.2
0.4
z
0.6
0.8
1.0
1.2
0.5
z
1.0
1.5
0.5
z
1.0
1.5
z
0.0
WFIRST (2298 SNe), BIGBOSS (20 BAO), WMAP7
SNe+CMB
SNe+BAO
SNe+BAO+CMB
−1.0
−1.5
−2.0
w(z)
−0.5
SNe
0.5
1.0
z
1.5
0.5
1.0
z
1.5
0.5
1.0
z
1.5
0.5
1.0
1.5
z
Ujjaini Alam (COSMO11, Porto, Aug 25, 2011) – p. 10
Conclusions
SNe data alone– degeneracy between Ω0m and wDE
Combination of SNe, BAO, CMB consistent with ΛCDM
As data quality improves, parametric methods inadequate to find
subtle differences in wDE
Gaussian process modeling provides non-parametric, unbiased
estimation of wDE
GP may provide effective importance of different datasets
Phys.Rev.D82:103502,2010; Phys.Rev.Lett.105:241302,2010;
arXiv:1104.2041
Ujjaini Alam (COSMO11, Porto, Aug 25, 2011) – p. 11