neoclassical probes in of the dark energy - university of...
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Wayne HuCOSMO04 Toronto, September 2004
of the Dark Energy
NeoClassical Probes
-
Structural Fidelity• Dark matter simulations approaching the accuracy of CMB calculations
WMAP Kravtsov et al (2003)
-
Equation of State Constraints• NeoClassical probes statistically competitive already CMB+SNe+Galaxies+Bias(Lensing)+Lyα
• Future hinges on controlling systematicsSeljak et al (2004)
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
0 0.2 0.4 0.6 0.8 1 1.2 1.4
w
z
smooth: w=w0+(1-a)wa+(1-a)2waa
68%
95%
-
Dark Energy Observables
-
Making Light of the Dark Side• Line element:
ds2 = a2[(1− 2Φ)dη2 − (1 + 2Φ)(dD2 + D2AdΩ)]
where η is the conformal time, D comoving distance, DA thecomoving angular diameter distance and Φ the gravitationalpotential
• Dark components visible in their influence on the metric elementsa and Φ – general relativity considers these the same: consistencyrelation for gravity
• Light propagates on null geodesics: in the background ∆D = ∆η;around structures according to lensing by Φ
• Matter dilutes with the expansion and (free) falls in thegravitational potential
-
Friedmann and Poisson Equations• Friedmann equation:(
dln a
dη
)2=
8πG
3a2∑
ρ ≡ [aH(a)]2
implying that the densities of dark components are visible in thedistance-redshift relation (a = (1 + z)−1)
D =
∫dη =
∫d ln a
aH(a)=
∫dz
H
• Poisson equation:
∇2Φ = −4πGa2δρ
implying that the (comoving gauge) density field of the darkcomponents is visible in the potential (lensing, motion of tracers).
-
Growth Rate• Relativistic stresses in the dark energy can prevent clustering. If
only dark matter perturbations δm ≡ δρm/ρm are responsible forpotential perturbations on small scales
d2δmdt2
+ 2H(a)dδmdt
= 4πGρm(a)δm
• In a flat universe this can be recast into the growth rateG(a) ∝ δm/a equation which depends only on the properties ofthe dark energy
d2G
d ln a2+
[5
2− 3
2w(a)ΩDE(a)
]dG
d ln a+
3
2[1− w(a)]ΩDE(a)G = 0
with initial conditions G =const.
• Comparison of distance and growth tests dark energy smoothnessand/or general relativity
-
Fixed Deceleration Epoch
-
High-z Energy Densities• WMAP + small scale temperature and polarization measures provides self-calibrating standards for the dark energy probes
WMAP: Bennett et al (2003)
Multipole moment (l)
l(l+
1)C
l/2π
(µK
2 )
Angular Scale
00
1000
2000
3000
4000
5000
6000
10
90 2 0.5 0.2
10040 400200 800 1400
� ModelWMAPCBIACBAR
TT Cross PowerSpectrum
-
High-z Energy Densities• Relative heights of the first 3 peaks calibrates sound horizon and matter radiation equality horizon
Multipole moment (l)
l(l+
1)C
l/2π
(µK
2 )
00
1000
2000
3000
4000
5000
6000
10 10040 400200 800 1400
baryons(sound speed)
-
High-z Energy Densities• Relative heights of the first 3 peaks calibrates sound horizon and matter radiation equality horizon
Multipole moment (l)
l(l+
1)C
l/2π
(µK
2 )
00
1000
2000
3000
4000
5000
6000
10 10040 400200 800 1400
dark matter(horizon, equality)
leading sourceof error!
-
High-z Energy Densities• Cross checked with damping scale (diffusion during horizon time) and polarization (rescattering after diffusion): self-calibrated and internally cross-checked
WMAP: Bennett et al (2003)
Multipole moment (l)
l(l+
1)C
l/2π
(µK
2 )
00
1000
2000
3000
4000
5000
6000
10 10040 400200 800 1400
cross checked
-
Standard Ruler• Standard ruler used to measure the angular diameter distance to recombination (z~1100; currently 2-4%) or any redshift for which acoustic phenomena observable
WMAP: Bennett et al (2003)
Multipole moment (l)
l(l+
1)C
l/2π
(µK
2 )
00
1000
2000
3000
4000
5000
6000
10 10040 400200 800 1400
angular scalev. physical scale
-
Standard Amplitude• Standard fluctuation: absolute power determines initial fluctuations in the regime 0.01-0.1 Mpc-1
WMAP: Bennett et al (2003)
Multipole moment (l)
l(l+
1)C
l/2π
(µK
2 )
00
1000
2000
3000
4000
5000
6000
10 10040 400200 800 1400
k~0.02Mpc-1
k~0.06Mpc-1
-
Standard Amplitude• Standard fluctuation: precision mainly limited by reionization which lowers the peaks as e-2τ; self-calibrated by polarization, cross checked by CMB lensing in future
WMAP: Bennett et al (2003)
Multipole moment (l)
l(l+
1)C
l/2π
(µK
2 )
00
1000
2000
3000
4000
5000
6000
10 10040 400200 800 1400
e-2τsee saturday/astrophCAPMAP, CBI, DASI
-
σ8(z)• Determination of the normalization during the acceleration epoch,
even σ8, measures the dark energy with negligible uncertaintyfrom other parameters
• Approximate scaling (flat, negligible neutrinos: Hu & Jain 2003)
σ8(z) ≈δζ
5.6× 10−5
(Ωbh
2
0.024
)−1/3 (Ωmh
2
0.14
)0.563(3.12h)(n−1)/2
×(
h
0.72
)0.693G(z)
0.76,
• δζ , Ωbh2, Ωmh2, n all well determined; eventually to ∼ 1%precision
• h =√
Ωmh2/Ωm∝ (1− ΩDE)−1/2 measures dark energy density
• G measures dark energy dependent growth rate
-
Reionization• Biggest surprise of WMAP: early reionization from temperature polarization cross correlation (central value τ=0.17)
Kogut et al (2003)
Multipole moment (l)0
(l+1)
Cl/2
π� (µ
K2 )
-1
0
1
2
3
10 10040 400200 800 1400
TE Cross PowerSpectrumReionization
-
Leveraging the CMB• Standard fluctuation: large scale - ISW effect; correlation with large-scale structure; clustering of dark energy; low multipole anomalies? polarization
WMAP: Bennett et al (2003)
Multipole moment (l)
l(l+
1)C
l/2π
(µK
2 )
00
1000
2000
3000
4000
5000
6000
10 10040 400200 800 1400
-
Standard Deviants[H0 ↔ w(z = 0.4)][σ8 ↔ dark energy]
-
Dark Energy Sensitivity• Fixed distance to recombination DA(z~1100) • Fixed initial fluctuation G(z~1100)• Constant w=wDE; (ΩDE adjusted - one parameter family of curves)
∆DA/DA
∆G/G
∆H-1/H-1
∆wDE=1/30.1
0
0.1
0.2
-0.2
-0.1
1z
-
Dark Energy Sensitivity• Other cosmological test, e.g. volume, SNIa distance constructed as linear combinations
decelerationepoch measuresHubble constant
∆DA/DA
∆(H0DA)/H0DA
∆H0/H0∆G/G
∆H-1/H-1
∆wDE=1/30.1
0
0.1
0.2
-0.2
-0.1
1z
-
Amplitude of Fluctuations• Recent determinations:
compilation: Refrigier (2003)
-
Dark Energy Sensitivity• Three parameter dark energy model: w(z=0)=w0; wa=-dw/da; ΩDE • wa sensitivity; (fixed w0 = -1; ΩDE adjusted)
decelerationepoch measuresHubble constant
∆DA/DA
∆(H0DA)/H0DA
∆H0/H0∆G/G
∆H-1/H-1
∆wa=1/2; ∆w0=00.1
0
0.1
0.2
-0.2
-0.1
1z
-
Pivot Redshift
• Any deviation from w=-1 rules out a cosmological constant
Seljak et al (2004)
-2
-1.8
-1.6
-1.4
-1.2
-1
-0.8
-0.6
0 0.2 0.4 0.6 0.8 1 1.2 1.4
w
z
smooth: w=w0+(1-a)wa+(1-a)2waa
68%
95%
• Equation of state best measured at z~0.2-0.4 = Hubble constant
-
Dark Energy Sensitivity• Three parameter dark energy model: w(z=0)=w0; wa=-dw/da; ΩDE • H0 fixed (or ΩDE); remaining w0-wa degeneracy• Note: degeneracy does not preclude ruling out Λ (w(z)≠-1 at some z)
decelerationepoch measuresHubble constant
∆DA/DA
∆(H0DA)/H0DA
∆G/G
∆H-1/H-1
∆wa=1/2; ∆w0=-0.150.1
0
0.01
0.02
-0.02
-0.01
1z
-
Rings of Power
-
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
Eisenstein, Hu & Tegmark (1998)k (Mpc–1)
0.05
2
4
6
8
0.1
Pow
er
h
Observedh Mpc-1
CMB provided
-
Cosmological Distances• Modes perpendicular to line of sight measure angular diameter distance
Cooray, Hu, Huterer, Joffre (2001)k (Mpc–1)
0.05
2
4
6
8
0.1
Pow
er
Observedl DA-1
CMB provided
DA
-
Cosmological Distances• Modes parallel to line of sight measure the Hubble parameter
Eisenstein (2003); Seo & Eisenstein (2003) [also Blake & Glazebrook 2003; Linder 2003; Matsubara & Szalay 2002]k (Mpc–1)
0.05
2
4
6
8
0.1
Pow
er
ObservedH/∆z
CMB provided
H
-
0.02
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.04 0.06 0.08 0.10 0.12 0.14k⊥ (Mpc-1)
k (
Mpc
-1)
0.02 0.04 0.06 0.08 0.10 0.12 0.14k⊥ (Mpc-1)
Hu & Haiman (2003)
• Information density in k-space sets requirements for the redshifts• Purely angular limit corresponds to a low-pass k|| redshift survey in the fundamental mode set by redshift resolution
Projected Power
(a) DA (b) H
angularLimber limit
-
Galaxies and Halos• Recent progress in the assignment of galaxies to halos and halo substructure reduces galaxy bias largely to a counting problem
Kravtsov et al (2003)SDSS: Zehavi et al (2004)
-
Gravitational Lensing
-
Lensing Observables• Correlation of shear distortion of background images: cosmic shear• Cross correlation between foreground lens tracers and shear: galaxy-galaxy lensing
cosmicshear
galaxy-galaxy lensing
-
Halos and Shear
-
Lensing Tomography
• Divide sample by photometric redshifts• Cross correlate samples
• Order of magnitude increase in precision even after CMB breaks degeneracies
Hu (1999)
1
1
2
2
D0 0.5
0.1
1
2
3
0.2
0.3
1 1.5 2.0
g i(D
) n
i(D
)
(a) Galaxy Distribution
(b) Lensing Efficiency
22
11
12
100 1000 104
10–5
10–4
l
Pow
er
25 deg2, zmed=1
-
Galaxy-Shear Power Spectra• Auto and cross power spectra of galaxy density and shear in multiple redshift bins
g1g1
ε1ε1
ε1ε2ε2ε2g 1ε 2
g 1ε 1
g 2ε 2g2ε 1
g2g2
100
10-4
0.001
0.01
0.1
1000l
l2C
l /2π
Hu & Jain (2003) [also geometric constraints: Jain & Taylor (2003); Bernstein & Jain (2003); Zhang, Hui, Stebbins (2003); Knox & Song (2003)]
shear-shearcross correlationand B modescheck systematics
Hoekstra et al (2002); Guzik & Seljak (2001)
-
Mass-Observable Relation
• With bias determined, galaxy spectrum measures mass spectrum• Use galaxy-galaxy lensing to determine relation with halos or bias
0.001
0.01
0.1
1
10
1011 1012 1013 1014 1015
Mh (h-1M )
M*
ρ0
dnhd ln Mh N(Mh)
msAs
Mth
σlnM
z=0
-
Cluster Abundance
-
Counting Halos• Massive halos largely avoid N(Mh) problem of multiple objects
0.001
0.01
0.1
1
10
1011 1012 1013 1014 1015
Mh (h-1M )
M*
ρ0
dnhd ln Mh N(Mh)
msAs
Mth
σlnM
z=0
-
Counting Halos for Dark Energy• Number density of massive halos extremely sensitive to the growth of structure and hence the dark energy
• Potentially %-level precision in dark energy equation of state
Haiman, Holder & Mohr (2001)
-
Mass-Observable Relation• Relationship between halos of given mass and observables sets mass threshold and scatter around threshold
• Leading uncertainty in interpreting abundance; self-calibration?
0.001
0.01
0.1
1
10
1011 1012 1013 1014 1015
Mh (h-1M )
M*
ρ0
dnhd ln Mh N(Mh)
msAs
Mth
σlnM
z=0
Pierpaoli, Scott & White (2001); Seljak (2002); Levine, Schultz & White (2002)
-
Dark Energy Clustering
-
ISW Effect• If dark energy is smooth, gravitational potential decays with expansion• Photon receives a blueshift falling in without compensating redshift• Doubled by metric effect of stretching the wavelength
Sachs & Wolfe (1967); Kofman & Starobinski (1985)
-
ISW Effect• ISW effect hidden in the temperature power spectrum by primary anisotropy and cosmic variance
[plot: Hu & Scranton (2004)]
10-12
10-11
10-10
10-9
l(l+
1)C
l /2
πΘ
Θ
10 100l
total
ISW
w=-0.8
-
ISW Galaxy Correlation• A 2-3σ detection of the ISW effect through galaxy correlations
Boughn & Crittenden (2003); Nolte et al (2003); Fosalba & Gaztanaga (2003); Fosalba et al (2003); Afshordi et al (2003)
0 10 20 30θ (degrees )
- 0.4
- 0.2
0
0.2
0.4
<X
T>
(T
OT
cn
ts s
-1µK
)
Boughn & Crittenden (2003)
-
Dark Energy Smoothness• Ultimately can test the dark energy smoothness to ~3% on 1 Gpc
Hu & Scranton (2004)
(Φ−Φ
s)/Φ
s
0.011 10
0.03
0.10
ceη0 (Gpc)
total
z10
fsky=1
-
Summary• NeoClassical probes based on the evolution of structure• CMB fixes deceleration epoch observables: expansion rate, energy
densities, growth rate, absolute amplitude, volume elements
• General relativity predicts a consistency relation betweendeviations in distances and growth due to dark energy
• Leading dark energy distance observable is H0 or equivalentlyw(z ∼ 0.4)
• Leading growth observable σ8 measures dark energy (andneutrinos)
• Many NeoClassical tests can measure the evolution of w but allwill require a control over systematic errors at the ∼ 1%
• Promising probes are beginning to bear fruit: cluster abundance,cosmic shear, galaxy clustering, galaxy galaxy lensing,
-
Dark Energy Sensitivity• H0 fixed (or ΩDE); remaining wDE-ΩT spatial curvature degeneracy• Growth rate breaks the degeneracy anywhere in the acceleration regime
decelerationepoch measuresHubble constant
∆DA/DA∆(H0DA)/H0DA
∆G/G
∆H-1/H-1
0.1
0
0.01
0.02
-0.02
-0.01
1z
∆wDE=0.05; ∆ΩT=-0.0033
-
Keeping the High-z Fixed• CMB fixes energy densities, expansion rate and distances to deceleration epoch fixed• Example: constant w=wDE models require compensating shifts in ΩDE and h
-1
0.7
0.6
0.5
-0.8 -0.6 -0.4wDE
ΩDEh
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