Power Spectra Tests
8
Power
6
Growth
4
Distance
2
0.05
k (Mpc–1)
0.1
of the Dark Energy
Wayne Hu
NASA/Goddard Space Flight Center
Outline
Power spectrum tests of the dark energy in light of the CMB
From crude to precision measurements
• Variance of cluster counts in cells
• Angular power spectra
• 3D power spectra with δz < 0.01
Collaborators:
• Zoltan Haiman
• Bhuvnesh Jain
• Andrey Kravtsov
• Marcos Lima
Dark Energy Sensitivity
•
•
Growth: G=(growth rate)/a; normalized to high z
Distances: D=comoving distance; D*=to recombination
1
0.8
G
0.6
0.4
D/D*
0.2
0
1
2
z
3
4
5
Dark Energy Sensitivity
•
•
•
Growth: G=(growth rate)/a; normalized to high z
Distances: D=comoving distance; D*=to recombination
Sensitivity to wDE: comparable
0
∆G/G
-0.05
∆D/D
-0.1
-0.15
0
∆wDE=1/3
1
2
z
3
4
5
Initial Power Spectrum
•
•
Initial power spectrum on scales relevant for clusters determined
by CMB
Improvement goes with the reionization optical depth τ
1
10-1
10-2
∆m2(k,z=3) e-2τ
10-3
10-4
σ8
10-5
0.001
WMAPext; M. White
0.01
k (Mpc-1)
0.1
1
σ8
• Determination of the normalization during the acceleration epoch,
even σ8 , measures the dark energy with negligible uncertainty
from other parameters
• Approximate scaling (flat, negligible neutrinos:
σ8
2
−1/3 Ωb h
δζ
≈
5.6 × 10−5 0.024
0.693
G0
h
,
×
0.76
0.72
2
Ωm h
0.14
Hu & Jain 2003)
0.563
(3.12h)(n−1)/2
• δζ , Ωb h2 , Ωm h2 , n all well determined; eventually to ∼ 1%
precision
p
• h = Ωm h2 /Ωm ∝ (1 − ΩDE )−1/2 measures dark energy density
• G0 measures dark energy dependent growth rate
Cluster Abundance
•
•
Power spectrum controls halo abundance; exponential sensitivity
above M*; σ(M*)=1
Abundance evolution determines dark energy evolution
10
1
0.1
M * dnh
ρ 0 d ln Mh
0.01
M*
0.001
1011
1012
Mh
1013
1014
(h-1M )
1015
Idealized Constraints
•
•
Fixed high-z power inverts redshift trends and eliminates σ8 as
parameter
Sampling errors neither dominant nor negligible
-0.8
Evolution fixed
wDE
-0.85
-0.9
-0.95
4000 deg2
Mth=1014.2 h-1 M
1% CMB Priors
0.6
0.65
Hu & Kravtsov (2002) [Haiman et al 2001]
0.7
ΩDE
0.75
0.8
Sample Variance
•
•
Cluster mass halos are rare, strongly clustered
Sample variance increases noise on abundance but also provides
a handle on the mass threshold
10
b(Mh)
1
0.1
M * dnh
ρ 0 d ln Mh
0.01
M*
0.001
1011
1012
Mh
1013
1014
(h-1M )
1015
Mass-Observable Relation
•
•
Relationship between halos of given mass and observables sets
mass threshold and scatter around threshold
Clusters largely avoid N(Mh) problem with multiple objects in halo
10
1
0.1
σlnM
M * dnh
ρ 0 d ln Mh
ms
As
Mth
N(Mh)
0.01
z=0
0.001
1011
1012
Mh
1013
1014
(h-1M )
1015
Mass-Observable Evolution
Jointly fitting an unknown normalization and power law evolution
degrades constraints (Levine et al 2003; Hu 2003; Majumdar & Mohr 2003)
-0.8
Evolution fixed
-0.85
wDE
•
Evolution marginalized
-0.9
-0.95
4000 deg2
Mth=1014.2 h-1 M
1% CMB Priors
0.6
0.65
0.7
ΩDE
0.75
0.8
Mass-Observable Evolution
•
•
(Sampling) noise is signal!
Excess variance of counts in cells measures mass-dependent bias
in clustering (requires only photometric redshifts ∆z~0.1)
-0.8
Evolution fixed
wDE
-0.85
Evolution marginalized
Sample variance info
-0.9
-0.95
400 10 deg2 cells
Mth=1014.2 h-1 M
1% CMB Priors
0.6
Lima & Hu (2004)
0.65
0.7
ΩDE
0.75
0.8
Angular Power Spectrum
•
•
Match mass threshold to counts for given cosmology
Angular power spectra including cluster weak lensing (cluster mass
correlation) gives dark energy
-0.8
distance
(efficiency ratio)
wDE
-0.85
-0.9
joint angular
power spectra
-0.95
Hu & Jain (2003)
0.6
0.65
0.7
ΩDE
0.75
0.8
4000 deg2 l<3000
Mth=1014.2 h-1 M
1% CMB Priors
Standard Rulers
•
CMB peaks provide standard ruler for distance measures to
recombination and potentially lower redshifts: sound horizon
and particle horizon at equality imprinted in transfer function
Angular Scale
90
6000
2
0.5
0.2
TT Cross Power
Spectrum
l(l+1)C l /2π (µK2 )
5000
Model
WMAP
CBI
ACBAR
4000
3000
2000
1000
0
0
10
40
100
200
400
Multipole moment (l)
WMAP: Bennett et al (2003)
800
1400
Dark Energy Sensitivity
•
•
•
Angular diameter distance to recombination fixed to 4%
Follow degeneracy line in wDE, ΩDE for flat universe
Sensitivity to wDE changes:
0.1
∆D/D
0
∆G/G
-0.1
0
∆wDE=1/3
1
2
z
3
4
5
•
•
Local Test: H0
Locally DA = ∆z/H0, and the observed power spectrum
is isotropic in h Mpc-1 space
Template matching the features yields the Hubble constant
CMB provided
8
Observed
h Mpc-1
Power
6
4
h
2
0.05
Eisenstein, Hu & Tegmark (1998)
k (Mpc–1)
0.1
Cosmological Distances
•
Modes perpendicular to line of sight measure angular diameter
distance
CMB provided
8
Observed
l DA-1
Power
6
4
2
DA
0.05
Cooray, Hu, Huterer, Joffre (2001)
k (Mpc–1)
0.1
Acoustic Rings
•
Baryon oscillations appear as rings in a 2D power spectrum with
modes parallel and perpedicular to the line of sight
0.14
k
(Mpc-1)
0.12
0.10
0.08
0.06
0.04
~10% peak to trough
0.02
0.02 0.04 0.06 0.08 0.10 0.12 0.14
-1)
(Mpc
k
⊥
Seo &Eisenstein (2003); Blake & Glazenbrook (2003)
D(z) and H(z)
•
Galaxy survey similar to SDSS Main + LRG; cluster survey
similar to SPT but with redshift followup
1.2
(a) Angular DA-1
(b) Radial H
shift
1.1
1.0
0.9
0.8
w=-1
w=-1
w=-2/3
w=-2/3
clusters
galaxies
0.1
Hu & Haiman (2003)
clusters
galaxies
z
1
0.1
z
1
Dark Energy
•
Even marginalizing bias evolution b(z) and redshift space distortions
β, dark energy constraints possible with a reasonable extrapolation
from current CMB power spectrum priors
w
-0.9
-0.95
Galaxies
-1
Hu & Haiman (2003)
Clusters
0.7
0.72
ΩDE
0.74
Summary
• Amplitude and shape of the power spectrum known precisely from
the CMB at high z and will continue to improve
• Determination of amplitude (σ8 ) or distance (H0 ) at z = 0
measures well defined dark energy parameters
• Intermediate redshifts probe the dark energy evolution
• Cluster abundance for known mass-observable relation
• Variance of counts in cells to calibrate mass-observable relation
• Angular power spectra at fixed counts
• 3D power spectra to measure D(z) and H(z)