Gravitational Lensing and the Dark...

Gravitational Lensing and the Dark Energy

Future ProspectsWayne Hu

SLAC, August 2002

1

0.5

0

-0.5

0.6 0.65 0.7-1

ΩDE

w'

-0.9

-1

-1.1

0.6 0.65 0.7

ΩDE

w

Outline• Standard model of cosmology• Gravitational lensing• Lensing probes of dark energy

Outline

Collaborators

• Standard model of cosmology• Gravitational lensing• Lensing probes of dark energy

• Charles Keeton• Takemi Okamoto• Max Tegmark• Martin White

Outline

Collaborators

• Standard model of cosmology• Gravitational lensing• Lensing probes of dark energy

• Charles Keeton• Takemi Okamoto• Max Tegmark• Martin White

Microsoft

http://background.uchicago.edu("Presentations" in PDF)

Dark Energyand the

Standard Model of Cosmology

If its not dark, it doesn't matter!• Cosmic matter-energy budget:

Dark EnergyDark MatterDark Baryons

Visible MatterDark Neutrinos

Making Light of the Dark Side• Visible structures and the processes that form them are our only cosmological probe of the dark components

• In the standard, well-verified, cosmological model, structures grow through gravitational instability from small-fluctuations (perhaps formed during inflation)

15 Gyr

COBE

FIRSTen

SP MAX

IAB

OVROATCAIAC

Pyth

BAM

Viper

WD BIMA

MSAM

ARGO SuZIE

RING

QMAP

TOCO

Sask

BOOM

CAT

BOOM

Maxima

BOOM

CBI

VSA

DASI

CMB: high redshift anchor

10 100 1000

20

40

60

80

100

l

∆T (µ

K)

W. Hu 5/02

Power Spectra of Maps•Original

•Band Filtered

64º

Photon-Baryon Plasma• Before z~1000 when the CMB was T>3000K, hydrogen ionized

• Free electrons act as "glue" between photons and baryons by Compton scattering and Coulomb interactions• Nearly perfect fluid

Peak Location• Fundmental physical scale, the distance sound travels, becomes an

angular scaleby simple projection according to the angulardiameter distanceDA

θA = λA/DA

`A = kADA



θA = λA/DA

`A = kADA

• In a flat universe, the distance is simply DA= D ≡ η 0− η∗ ≈ η0, thehorizon distance, andkA = π/s∗ =

√3π/η∗ so

θA ≈η∗η0



θA = λA/DA

`A = kADA

• In a flat universe, the distance is simply DA= D ≡ η 0− η∗ ≈ η0, thehorizon distance, andkA = π/s∗ =

√3π/η∗ so

θA ≈η∗η0

• In amatter-dominateduniverseη ∝ a1/2 soθA ≈ 1/30 ≈ 2 or

`A ≈ 200

Angular Diameter Distance Test

500 1000 1500

20

40

60

80

l (multipole)

DT

(mK

)

Boom98CBI

DASIMaxima-1

l1~200

Curvature• In acurved universe, the apparent orangular diameter distanceis

no longer the conformal distance DA= R sin(D/R) 6= D

DA

D=Rθ

λ

α

• Objects in aclosed universearefurtherthan they appear!gravitationallensingof the background...

Curvature in the Power Spectrum•Angular location of harmonic peaks

•Flat = critical density = missing dark energy

Accelerated Expansion from SNe• Missing energy must also accelerate the expansion at low redshift

compilation from High-z team

Supernova Cosmology Project

34

36

38

40

42

44

Ωm=0.3, ΩΛ=0.7

Ωm=0.3, ΩΛ=0.0

Ωm=1.0, ΩΛ=0.0

m-M

(mag

)

High-Z SN Search Team

0.01 0.10 1.00z

-1.0

-0.5

0.0

0.5

1.0

∆(m

-M) (

mag

)

Acceleration Implies Negative Pressure• Role ofpressurein the background cosmology

• HomogeneousEinstein equationsGµν = 8πGTµν imply the twoFriedman equations(flat universe, or associating curvatureρK = −3K/8πGa2)(

1

a

da

dt

)2

=8πG

3ρ

1

a

d2a

dt2= −4πG

3(ρ+ 3p)

so that the total equation of statew ≡ p/ρ < −1/3 for acceleration

Acceleration Implies Negative Pressure• Role ofpressurein the background cosmology

• HomogeneousEinstein equationsGµν = 8πGTµν imply the twoFriedman equations(flat universe, or associating curvatureρK = −3K/8πGa2)(

1

a

da

dt

)2

=8πG

3ρ

1

a

d2a

dt2= −4πG

3(ρ+ 3p)

so that the total equation of statew ≡ p/ρ < −1/3 for acceleration

• Conservation equation∇µTµν = 0 implies

ρ

ρ= −3(1 + w)

a

a

• so thatρ must scale more slowly thana−2

Dark Mystery?• Coincidence: given different scalings with a, why are dark matter and energy densities comparable now?

10-5 10-4 10-3 10-2 10-1

1

0.8

0.6

0.4

0.2

scale factor a

frac

tion

of c

ritic

al Ω

radiation

matter

dark energy

w=-1w=-1/3

Dark Mystery?• Coincidence: given different scalings with a, why are dark matter and energy densities comparable now?• Stability: why doesn't negative pressure imply accelerated collapse? or why doesn't the vacuum suck?

Gravity

Pressure

Dark Mystery?• Coincidence: given different scalings with a, why are dark matter and energy densities comparable now?• Stability: why doesn't negative pressure imply accelerated collapse? or why doesn't the vacuum suck? pressure gradients, not pressure, establish stability

Dark Mystery?• Coincidence: given different scalings with a, why are dark matter and energy densities comparable now?• Stability: why doesn't negative pressure imply accelerated collapse? or why doesn't the vacuum suck? pressure gradients, not pressure, establish stability • Candidates:

Cosmological constant w=-1, constant in space and time, but >60 orders of magnitude off vacuum energy prediction

Ultralight scalar field, slowly rolling in a potential, Klein-Gordon equation: sound speed cs2=δp/δρ=1

Tangled defects w=-1/3, -2/3 but relativistic sound speed ("solid" dark matter)

Dark Energy Probes• (Comoving)distance-redshift relation: a = (1 + z)−1

D =

∫ 1

a

da1

a2H(a)=

∫ z

0

dz1

H(a)

H2(a) =8πG

3(ρm + ρDE)

in a flat universe, e.g.angular diameter distance, luminositydistance, number counts (volume)...

Dark Energy Probes• (Comoving)distance-redshift relation: a = (1 + z)−1

D =

∫ 1

a

da1

a2H(a)=

∫ z

0

dz1

H(a)

H2(a) =8πG

3(ρm + ρDE)

in a flat universe, e.g.angular diameter distance, luminositydistance, number counts (volume)...

• Pressuregrowth suppression: δ ≡ δρm/ρm ∝ aφ

d2φ

d ln a2+

[5

2− 3

2w(z)ΩDE(z)

]dφ

d ln a+

3

2[1− w(z)]ΩDE(z)φ = 0 ,

wherew ≡ pDE/ρDE andΩDE ≡ ρDE/(ρm + ρDE)

e.g. galaxycluster abundance, gravitational lensing...

Large-Scale Structureand

Gravitational Lensing

Making Light of the Dark Side• Visible structures and the processes that form them are our only cosmological probe of the dark components

• In the standard, well-verified, cosmological model, structures grow through gravitational instability from small-fluctuations (perhaps formed during inflation)

15 Gyr

Structure Formation Simulation• Simulation (by A. Kravstov)

Galaxy Power Spectrum Data Galaxy clustering tracks the dark matter – but bias depends on type•

k (h Mpc-1)

P(k)

(h-

1 M

pc)3

0.1 10.01

103

104

105

2dFPSCz<1994

A Fundamental Problem• All cosmological observables relate to theluminous matter:

photon-baryon plasma, galaxies, clusters of galaxies, supernovae

• Implications for the dark energy or cosmology in general dependon modelling theformationandevolutionof luminous objects

• Success of CMB anisotropy is in large part based on thesolidtheoretical groundingof its formation and evolution – wellunderstood linear gravitational physics

A Fundamental Problem• All cosmological observables relate to theluminous matter:

photon-baryon plasma, galaxies, clusters of galaxies, supernovae

• Implications for the dark energy or cosmology in general dependon modelling theformationandevolutionof luminous objects

• Success of CMB anisotropy is in large part based on thesolidtheoretical groundingof its formation and evolution – wellunderstood linear gravitational physics

• Distortion of the images of luminous objects bygravitationallensing is equally well understood

• Problem: image distortion is typically at %-level – verydemanding for the control ofsystematic errors– but recall CMB is10-5 level! (Tyson, Wenk & Valdes 1990)

Example of Weak Lensing• Toy example of lensing of the CMB primary anisotropies

• Shearing of the image

Lensing Observables• Image distortiondescribed byJacobian matrixof the remapping

A =

1− κ− γ1 −γ2

−γ2 1− κ+ γ1

,

whereκ is theconvergence, γ1, γ2 are theshearcomponents

Lensing Observables• Image distortiondescribed byJacobian matrixof the remapping

A =

1− κ− γ1 −γ2

−γ2 1− κ+ γ1

,

whereκ is theconvergence, γ1, γ2 are theshearcomponents

• related to thegravitational potentialΦ by spatial derivatives

ψij(zs) = 2∫ zs

0dzdD

dz

D(Ds −D)

Ds

Φ,ij ,

ψij = δij − Aij, i.e. viaPoisson equation

κ(zs) =3

2H2

0Ωm

∫ zs

0dz

dD

dz

D(Ds −D)

Ds

δ/a ,

Gravitational Lensing by LSS

• Shearing of galaxy images reliably detected in clusters

• Main systematic effects are instrumental rather than astrophysical

Colley, Turner, & Tyson (1996)

Cluster (Strong) Lensing: 0024+1654

Instrumental Systematics• Raw data has instrumental systematics (PSF anisotropy) larger than signal, removed by demanding stars be round

Jarvis et al. (2002) 1% ellipticity

Cosmic Shear Data• Shear variance as a function of smoothing scale

compilation from Bacon et al (2002)

Shear Power Modes

• Alignment of shear and wavevector defines modes

ε

β

Shear Power Spectrum

• Lensing weighted Limber projection of density power spectrum

• ε−shear power = κ power

10 100 1000l

10–4

10–5

10–7

10–6

l(l+

1)C

l /2

πεε

k=0.02 h/Mpc

Limber approximation

Linear

Non-linear

Kaiser (1992)Jain & Seljak (1997)

Hu (2000)

zs=1

zs

PM Simulations

• Simulating mass distribution is a routine exercise

6° × 6° FOV; 2' Res.; 245–75 h–1Mpc box; 480–145 h–1kpc mesh; 2–70 109 M

Convergence Shear

White & Hu (1999)

Dark Energy and

Gravitational Lensing

Degeneracies• All parameters of initial condition, growth and distance redshift relation D(z) enter• Nearly featureless power spectrum results in degeneracies

10 100 1000l

10–4

10–5

10–7

10–6l(l+1

)Cl

/2

πεε

zs=1

Degeneracies• All parameters of initial condition, growth and distance redshift relation D(z) enter• Nearly featureless power spectrum results in degeneracies

• Combine with information from the CMB: complementarity (Hu & Tegmark 1999)

• Crude tomography with source divisions (Hu 1999; Hu 2001)

• Fine tomography with source redshifts (Hu & Keeton 2002; Hu 2002)

10 100

20

40

60

80

100

l (multipole)

∆T

CMB

MAPPlanck

Error Improvement: 1000deg2

9.6% 14% 0.60 1.94 0.18 0.075 19% 0.28

Ωmh2Ωbh2 w nS T/SδζΩΛ τ

Cosmological Parameters

MAP

MAP +Er

ror I

mpr

ovem

ent

1

3

10

30

100

zmed=1

Hu & Tegmark (1999) Hu (2001)

Crude Tomography

• Divide sample by photometric redshifts

Hu (1999)

1 2

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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

100 1000 104

10–5

10–4

l

Pow

er

25 deg2, zmed=1

Crude Tomography

• Divide sample by photometric redshifts

• Cross correlate samples

• Order of magnitude increase in precision even after CMB breaks degeneracies

Hu (1999)

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2

2

D0 0.5

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1 1.5 2.0

g i(D

) n

i(D

)

(a) Galaxy Distribution

(b) Lensing Efficiency

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11

12

100 1000 104

10–5

10–4

l

Pow

er

25 deg2, zmed=1


9.6% 14% 0.60 1.94 0.18 0.075 19% 0.28



MAP

MAP +E

rror

Im

prov

emen

t

1

3

10

30

100

3-zno-z

Hu (1999; 2001)

Dark Energy & Tomography

• Both CMB and tomography help lensing provide interesting constraints on dark energy

0 0.5 1.0

0

1

2

ΩDE

w

l<3000; 56 gal/deg2

MAPno–z

1000deg2

Hu (2001)



0 0.5 1.0

0

1

2

ΩDE

w

l<3000; 56 gal/deg2

MAPno–z3–z

1000deg2

Hu (2001)



0 0.5 1.0

0.55–1

–0.9

–0.8

–0.7

–0.6

0.6 0.65 0.7

0

1

2

ΩDE

w

l<3000; 56 gal/deg2

MAPno–z3–z

1000deg2

Hu (2001)

Hidden Dark Energy Information• Most of the information on thedark energyis hidden in the

temporalor radialdimension

• Evolution ofgrowth rate(dark energy pressure slows growth)

• Evolution ofdistance-redshiftrelation

Hidden Dark Energy Information• Most of the information on thedark energyis hidden in the

temporalor radialdimension

• Evolution ofgrowth rate(dark energy pressure slows growth)

• Evolution ofdistance-redshiftrelation

• Lensing is inherentlytwo dimensional: all mass along the line ofsight lenses

• Tomography implicitly or explicitlyreconstructs radial dimensionwith source redshifts

• Photometric redshift errors currently∆z < 0.1 out toz ~ 1 andallow for ”fine” tomography

Fine Tomography• Convergence– projection of∆ = δ/a for eachzs

κ(zs) =3

2H2

0Ωm

∫ zs

0

dzdD

dz

D(Ds −D)

Ds

∆ ,


κ(zs) =3

2H2

0Ωm

∫ zs

0

dzdD

dz

D(Ds −D)

Ds

∆ ,

• Data islinear combinationof signal + noise

dκ = Pκ∆s∆ + nκ ,

[Pκ∆]ij =

32H2

0ΩmδDj(Di+1−Dj)Dj

Di+1Di+1 > Dj ,

0 Di+1 ≤ Dj ,


κ(zs) =3

2H2

0Ωm

∫ zs

0

dzdD

dz

D(Ds −D)

Ds

∆ ,

• Data islinear combinationof signal + noise

dκ = Pκ∆s∆ + nκ ,

[Pκ∆]ij =

32H2

0ΩmδDj(Di+1−Dj)Dj

Di+1Di+1 > Dj ,

0 Di+1 ≤ Dj ,

• Well-posed (Taylor 2002) but noisy inversion (Hu & Keeton 2002)

• Noise properties differ from signal properties→ optimal filters

Hidden in Noise• Derivatives of noisy convergence isolate radial structures

Hu & Keeton (2001)

400

0 0.5 1 1.5

200

z

∆

0.6

0.4

0.2

κ

(a) Density

(b) Convergence

Fine Tomography• Tomography can produce direct 3D dark matter maps, but realistically only broad features (Hu & Keeton 2002)

Radial density field

1

0.5

0

–0.5

–1

0 0.5 1 1.5z

Fine Tomography• Tomography can produce direct 3D dark matter maps, but realistically only broad features (Taylor 2002; Hu & Keeton 2002)

Radial density fieldWiener reconstruction

1

0.5

0

–0.5

–1

0 0.5 1 1.5z

Fine Tomography• Tomography can produce direct 3D dark matter maps, but realistically only broad features (Hu & Keeton 2002)

Radial density fieldWiener reconstruction

1

0.5

0

–0.5

–1

0 0.5 1 1.5z

Growth Function• Localized constraints (fixed distance-redshift relation)

4000 sq. deg

z0.5 1 1.5 2

windows

1.4

1.2

1.0

0.8

0.6

0.60.40.2

mtot=0.2eV

Hu (2002)

z0.5 1 1.5 2

1.5

1.0

0.5

0.50

-0.5

ρ DE/

ρ cr0

windows

w=–0.8

Dark Energy Density• Localized constraints (with cold dark matter)

Hu (2002)

4000 sq. deg

Dark Energy Parameters• Three parameter dark energy model (ΩDE, w, dw/dz=w')

Hu (2002)

1

0.5-0.9

-1

-1.1

0

-0.5

0.6 0.65 0.7-1

ΩDE0.6 0.65 0.7

ΩDE

w'

w

σ(ΩDE)=0.01

σ(w')=0

0.03

nonenone

4000 sq. deg.

ISW Effect

• Gravitational blueshift on infall does not cancel redshift on climbing out

• Contraction of spatial metric doubles the effect: ∆T/T=2∆Φ

• Effect from potential hills and wells cancel on small scales

ISW Effect and Dark Energy

• Raising equation of state increases redshift of dark energydomination and raises the ISW effect

• Lowering the sound speed increases clustering and reducesISW effect at large angles

w=–1

w=–2/3

ceff =1

ceff=1/3

10–10

10–11

10–12

Pow

er

10 100l

ISW Effect

Total

Hu (1998); Hu (2001)Coble et al. (1997)

Caldwell et al. (1998)

Direct Detection of Dark Energy?

• In the presence of dark energy, shear is correlated with CMB

temperature via ISW effect

1000deg2

65% sky

10–9

10–10

10–9

10–10

10–9

10–10

l(l+

1)C

l /2

πΘ

ε

10 100 1000l

z<1

1<z<1.5

z>1.5

Hu (2001)

Lensing of a Gaussian Random Field• CMB temperature and polarization anisotropies are Gaussian

random fields – unlike galaxy weak lensing

• Average over many noisy images – like galaxy weak lensing

Lensing by a Gaussian Random Field• Mass distribution at large angles and high redshift in

in the linear regime

• Projected mass distribution (low pass filtered reflectingdeflection angles): 1000 sq. deg

rms deflection2.6'

deflection coherence10°

Lensing in the Power Spectrum• Lensing smooths the power spectrum with a width ∆l~60

• Convolution with specific kernel: higher order correlations between multipole moments – not apparent in power

Pow

er

10–9

10–10

10–11

10–12

10–13

lensedunlensed

∆power

l10 100 1000

Seljak (1996); Hu (2000)

Reconstruction from the CMB• Correlation betweenFourier momentsreflectlensing potentialκ = ∇2φ

〈x(l)x′(l′)〉CMB = fα(l, l′)φ(l + l′) ,

wherex ∈ temperature, polarization fieldsandfα is a fixed weightthat reflects geometry

• Each pair forms anoisy estimateof the potential or projected mass- just like a pair of galaxy shears

• Minimum variance weightall pairs to form an estimator of thelensing mass

Quadratic Reconstruction• Matched filter (minimum variance) averaging over pairs of

multipole moments

• Real space: divergence of a temperature-weighted gradient

Hu (2001)

originalpotential map (1000sq. deg)

reconstructed1.5' beam; 27µK-arcmin noise

Ultimate (Cosmic Variance) Limit• Cosmic variance of CMB fields sets ultimate limit

• Polarization allows mapping to finer scales (~10')

Hu & Okamoto (2001)

100 sq. deg; 4' beam; 1µK-arcmin

mass temp. reconstruction EB pol. reconstruction

Matter Power Spectrum• Measuring projected matter power spectrum to cosmic vari-

ance limit across whole linear regime 0.002< k < 0.2 h/Mpc

Hu & Okamoto (2001)

10 100 1000

10–7

10–8

defle

ctio

n po

wer

"Perfect"

Linear

Lσ(w)∼0.06

Matter Power Spectrum• Measuring projected matter power spectrum to cosmic vari-

ance limit across whole linear regime 0.002< k < 0.2 h/Mpc

Hu & Okamoto (2001)

10 100 1000

10–7

10–8

defl

ectio

n po

wer

"Perfect"Planck

Linear

Lσ(w)∼0.06; 0.14

Cross Correlation with Temperature

• Any correlation is a direct detection of a smooth energy density component through the ISW effect

• 5 nearly independent measures in temperature & polarization

Hu (2001); Hu & Okamoto (2001)

10 100 1000

10–9

10–10

10–11

"Perfect"Planck

cros

s po

wer

L

Cross Correlation with Temperature

• Any correlation is a direct detection of a smooth energy density component through the ISW effect

• Show dark energy smooth >5-6 Gpc scale, test quintesence

Hu (2001); Hu & Okamoto (2001)

10 100 1000

10–9

10–10

10–11

"Perfect"Planck

cros

s po

wer

L

Summary• Standard model of cosmology well-established• Dark energy indicated, a mystery


• Dark matter distribution and its dependence on dark energy well-understood• Luminous tracers (supernovae/galaxies/clusters) require modelling of formation/evolution• Gravitational lensing avoids ambiguity, utilizes luminous objects only as background image


• Dark matter distribution and its dependence on dark energy well-understood• Luminous tracers (supernovae/galaxies/clusters) require modelling of formation/evolution• Gravitational lensing avoids ambiguity, utilizes luminous objects only as background image

• Evolution of dark energy can be extracted tomographically• Clustering of dark energy (test of scalar field paradigm) extractable from wide-field CMB lensing

Gravitational Lensing and the Dark...

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