Weak Lensing Weak Lensing and and

Dark Energy CosmologyDark Energy Cosmology

Tong-Jie Zhang[Tong-Jie Zhang[张同杰张同杰 ]]

Department of Astronomy, Beijing Normal UniversityDepartment of Astronomy, Beijing Normal University

Cosmology WorkshopCosmology Workshop

Institute of High Energy Physics, Chinese Academy of Institute of High Energy Physics, Chinese Academy of Sciences Sciences

2008/12/082008/12/08

3-D : Accelerating 3-D : Accelerating UniverseUniverse

WMAPWMAP

3-D Universe: 3 dark sides 3-D Universe: 3 dark sides

(1). Our Universe—Dark (1). Our Universe—Dark energyenergy

(2). Dark Matter [halo]((2). Dark Matter [halo]( 暗物暗物质质 [[ 晕晕 ])])

(3). Dark ages((3). Dark ages( 黑暗时代黑暗时代 ))

OutlineOutline

• 0. Basic of Gravitational lensing0. Basic of Gravitational lensing• 1. Dark Energy and Neutrino Mass 1. Dark Energy and Neutrino Mass

Constraints Constraints from WL, SN Ia and RGA from WL, SN Ia and RGA

• 2. The signatures of BAOs on the 2. The signatures of BAOs on the convergence convergence

power spectrum of weak lensingpower spectrum of weak lensing.• 3. Application of wavelet on Weak lensing3. Application of wavelet on Weak lensing

0. Basic of Gravitational 0. Basic of Gravitational lensinglensing

SchematicSchematic Diagram of Gravitational Lensing (Diagram of Gravitational Lensing ( 引引力透镜示意图力透镜示意图 ))

Physics of Gravitational Physics of Gravitational Lensing (GL)Lensing (GL)

Bending of Light under GravityBending of Light under Gravity

Light will follow the Light will follow the straightest possible path straightest possible path through flat space time.through flat space time.

If spacetime is curved near a If spacetime is curved near a massive object, so the massive object, so the trajectory of light is also trajectory of light is also curved.curved.

Observational Event of of Observational Event of of Gravitational LensingGravitational Lensing

Einstein’s Crossan Einstein ring

galaxy directly behind a galaxy

HST Image of a gravitational lens HST Image of a gravitational lens in galaxy clusterin galaxy cluster

Category of GLCategory of GL• Strong Strong

gravitational gravitational lensinglensing

• Weak gravitational Weak gravitational lensinglensing

Gravitational lens Theory—Sketch of a typical Gravitational lens Theory—Sketch of a typical gravitational lens systemgravitational lens system

Deflection angleDeflection angle

• General Relativity: for a point mass General Relativity: for a point mass MM

Lensing equation or ray-trace Lensing equation or ray-trace equationequation

Position of sourcePosition of source Position of imagePosition of image

Lensing equationLensing equation

Tong-Jie Zhang ApJ 602, L5-8(2004) [Tong-Jie Zhang ApJ 602, L5-8(2004) [astro-astro-ph@0401040]ph@0401040]

Multiple images can be Multiple images can be produced produced if lens is strongif lens is strong

Position of sourcePosition of source

Position of imagePosition of image

Convergence and shearConvergence and shear

K>=1 K>=1 strong strongK<<1 K<<1 weak weak

Deflection potentialDeflection potential

Distortion and Distortion and Magnification Magnification

Critical curves in lens plane; Critical curves in lens plane; Caustics in source planeCaustics in source plane

MagnificationMagnification::

ShearShear::

DetDet

Strong lensingStrong lensing

• Sources are close to the caustic Sources are close to the caustic lines.lines.

• K >=1 and |r|>=1: The K >=1 and |r|>=1: The convergence and shear are strong convergence and shear are strong enough to produce giant arcs and enough to produce giant arcs and multiple images.multiple images.

The probability for strong The probability for strong lensinglensing

E(z) and f (M, z): dependent on E(z) and f (M, z): dependent on cosmological modelcosmological model

CLASS observationCLASS observation

The Cosmic Lens All-Sky Survey CLASS): The Cosmic Lens All-Sky Survey CLASS): An international (USA, UK and Netherlands) An international (USA, UK and Netherlands)

collaborative collaborative project to map more than 10,000 radio sources in project to map more than 10,000 radio sources in

order to create order to create the largest and best studied statistical sample of the largest and best studied statistical sample of

gravitationally lensed systems. gravitationally lensed systems.

Sample: Sample: Well-defined statistical sample: 8958Well-defined statistical sample: 8958Multiply imaged sourses: 13Multiply imaged sourses: 13P_ob=N(>\theta)/8958P_ob=N(>\theta)/8958

Lensing modelsLensing models

• SISSIS

• GNFWGNFW

Image separation probability for Image separation probability for GNFW modelGNFW model

Tong-Jie Zhang ApJ 602, L5-8(2004) [Tong-Jie Zhang ApJ 602, L5-8(2004) [astro-ph@0401040]astro-ph@0401040]

Constraint on potentialConstraint on potential

Kyu-Hyun Chae et al ApJ 607, L71-Kyu-Hyun Chae et al ApJ 607, L71-74(2004)74(2004)

Weak lensing (cosmic Weak lensing (cosmic shear)shear)

Cosmic shearCosmic shear is the distortion of the shapes of is the distortion of the shapes of background galaxiesbackground galaxies due to the bending of light due to the bending of light by the potentials associated with by the potentials associated with large-scale large-scale structurestructure in the universe. in the universe.

Wek lensing regime: K <<1 and |r|<<1Wek lensing regime: K <<1 and |r|<<1

Distortion of background images: shape and correlationDistortion of background images: shape and correlation

Before lensed After lensedBefore lensed After lensed

MeasurementMeasurement

• The ellipticity of galaxy and the The ellipticity of galaxy and the intrinsic ellipticity and shear intrinsic ellipticity and shear

The mean expectation of source ellipticities and alignmentThe mean expectation of source ellipticities and alignment

Weak lensing shear: spin-2 polarization fieldWeak lensing shear: spin-2 polarization field

Φx

y

ba

Shear componentShear component

• The tangential shear and the 45 The tangential shear and the 45 degree rotated shear in the local degree rotated shear in the local frame defined by the line connecting frame defined by the line connecting the pair of galaxiesthe pair of galaxies

xi xj

θ

a

ba

ba

Shear correlation functionShear correlation function

Two-point cosmic shear Two-point cosmic shear statisticsstatistics

2. the top-hat filtered variance of the shear

3. the variance of the aperture-mass

1. shear correlation

Power spectrum of convergence

OCDMOCDM

CDMCDM

CDMCDM(linear)(linear)

Observational Constraint on cosmologyObservational Constraint on cosmology

•H. Hoekstra, Y. Mellier, L. van Waerbeke, E. Semboloni, L. Fu et al, The Astrophysical Journal, 647:116–127, 2006

Joint constraint using WL and CMBJoint constraint using WL and CMB

Contaldi et al, PRL, Contaldi et al, PRL, 90, 200390, 2003

1. 1. Dark Energy and Neutrino Mass Dark Energy and Neutrino Mass Constraints Constraints

from WL, SN Ia and RGA from WL, SN Ia and RGA • Yan, Gong, Tong-Jie Zhang, Tian Lan and Xue-Yan, Gong, Tong-Jie Zhang, Tian Lan and Xue-

Lei ChenLei Chen

• (astro-ph@arXiv: 0810.3572) Sumitted to Sumitted to ApJApJ

The existence of non-zero neutrino masses

• has been established firmly by the experiments detecting

• [1]. atmospheric neutrinos,• [2]. solar neutrinos • [3]. reactor neutrinos • [4]. accelerator beam neutrinos

• The neutrinos were still relativistic at the decoupling epoch.

• However, they are definitely non-relativistic at the present epoch, as the neutrino oscillation experiments have shown.

• Therefore, the matter density must contain the neutrino contribution when they are non-relativistic,

Current constraints on neutrino mass:

F.D.Bernardis et al. 2008WMAP5:

WMAP5 Results on WMAP5 Results on neutrinoneutrino

• WMAP5: WMAP5:

Weak Leasing and Neutrino Weak Leasing and Neutrino MassMass

W. Hu & D. J. Eisenstein, 1998, ApJ

Free streaming effect

,

• The massive neutrinos could suppress the matter power spectrum on small scales, due to their free streaming, thus reducing the convergence power spectrum of the weak lensing, which is sensitive to the small scale matter distribution.

• Weak lensing is therefore a powerful measurement for both the dark energy and the massive neutrinos.

The Likelyhood of WL:

Shear correlation function (Crittenden et al. 2002):

The likelyhood:

Other Likelyhood:Other Likelyhood:

• SN Ia:SN Ia:

• RGA:RGA:

• BAO:BAO:

,

Data sets:Data sets:

• Weak lensing dataWeak lensing data CFHLST-wideCFHLST-wide, 22 deg^2 (Fu et al. 2008);, 22 deg^2 (Fu et al. 2008); RCSRCS, 53 deg^2 (Hoekstra et al. 2002), 53 deg^2 (Hoekstra et al. 2002)

• SN Ia dataSN Ia data SCP SCP “Union” data, 307 samples (Kowalski et al. “Union” data, 307 samples (Kowalski et al.

2008)2008)

• RGA RGA (relative galaxy ages)(relative galaxy ages)

H(z) from H(z) from GDDSGDDS, 9 samples (Simon et al. 2005), 9 samples (Simon et al. 2005)

• BAO dataBAO data AA at z=0.35 (Eisenstein et al. 2005) at z=0.35 (Eisenstein et al. 2005)

[1]. Weak Lensing Constraints [1]. Weak Lensing Constraints on w:on w:

Weak constraint on

w for current WL data

WL+SN+RGA+BAO:

w = -1.0 +0.19 -0.21

at 95.5% C.L.

(w = -1.0 +0.14 -0.11 for WMAP5)

The similar degeneracy

direction and constraint

ability for SN Ia and RGA

wCDM

Results:

[2]. Weak Lensing Constraints [2]. Weak Lensing Constraints

on on ΣΣmmvv

Σmv<=0.4eV Σmv<=0.8eV

at 95.5% C.L.

Weak degeneracy

between w andΣmv

[3]. Constraints on w and Σmv:

Compatible and

comparable with the

results of WMAP5

2. 2. The signatures of BAOs on the convergence The signatures of BAOs on the convergence power spectrum of weak lensingpower spectrum of weak lensing

• In the early universe prior to recombination, the free electrons couple the baryons to the photons through Compton interactions, so these three species move together as a single fluid.

• The primordial cosmological perturbations on small scales excite sound waves in this relativistic plasma, which results in the pressure-induced oscillations and acoustic peak.

•The memory of these baryon acoustic oscillations (BAOs) still remain after the epoch of recombination.

Two Effect of BAO Two Effect of BAO after the epoch of after the epoch of recombinationrecombination

• [1]. The BAOs leave their imprints through the propagating of photons on the last scattering surface and produce a harmonic series of maxima and minima in the anisotropy power spectrum of the cosmic microwave background (CMB) at z=1000.

• [2]. Due to the significant fraction of baryons in the universe, BAOs can also be imprinted onto the latetime power spectrum of the non-relativistic matter.

Acoustic Oscillations in theAcoustic Oscillations in theEarly Universe and TodayEarly Universe and Today

Christopher J. Miller,1 Robert Christopher J. Miller,1 Robert C. Nichol,1 David J. Batuski2C. Nichol,1 David J. Batuski2

22 JUNE 2001 VOL 292 22 JUNE 2001 VOL 292 SCIENCESCIENCE

BAO on the latetime power spectrum of the non-relativistic matterBAO on the latetime power spectrum of the non-relativistic matter

BAOs can give rise to the wiggles in the matter power spectrum:

• (a). Correlation function of galaxies (z=0)(a). Correlation function of galaxies (z=0)

• (b). The power spectrum of 21 cm emission generated from the neutral hydrogen from the epoch of reionization through the underlying density perturbation

• (c). T(c). The power spectrum gravitaional lensing: gravitaional lensing: strong and weakstrong and weak

(a). BAO on Correlation function of galaxies(z=0): (a). BAO on Correlation function of galaxies(z=0): Sound Waves in MatterSound Waves in Matter

• Each initial overdensity (in DM & gas) is an overpressure Each initial overdensity (in DM & gas) is an overpressure that launches a spherical sound wave. This wave travels that launches a spherical sound wave. This wave travels outwards at 57% of the speed of light.outwards at 57% of the speed of light.

• Pressure-providing photons decouple at recombination. Pressure-providing photons decouple at recombination. CMB travels to us from these spheres.CMB travels to us from these spheres.

• Sound speed plummets. Wave stalls at a radius of about Sound speed plummets. Wave stalls at a radius of about 100 Mpc.100 Mpc.

• Overdensity in shell (gas) and in the original center (DM) Overdensity in shell (gas) and in the original center (DM) both seed the formation of galaxies. Preferred separation both seed the formation of galaxies. Preferred separation of 100 Mpc.of 100 Mpc.

100 Mpc100 Mpc

D. J. Eisenstein et al., Astrophys. J. 633, 560 (2005)

(b). BAO on the power spectrum of 21 cm emission

Xiao-Chun Mao and Xiang-Ping Wu, ApJ, 673: L107–L110, 2008

(c). BAO on t(c). BAO on the power spectrum gravitational lensing: weakgravitational lensing: weak

The matter power spectrum

Tong-Jie Zhang, Qiang Yuan, Tian Lan astro-ph@arXiv:0812.0521

The convergence power spectrum of weak lensing

Tong-Jie Zhang, Qiang Yuan, Tian Lan arXiv:0812.0521

The statistical errors in themeasurements of weak lensing power

spectrum

Tong-Jie Zhang, Qiang Yuan, Tian Lan astro-ph@arXiv:0812.0521

ConlusionsConlusions• [1]. The BAOs wiggles can be found in both of

the linear and nonlinear convergence power spectra of weak lensing at about 40 <= l<= 600, but they are weaker than that of matter power spectrum.

• [2]. Although the statistical error for LSST are greatly smaller than that of CFHT and SNAP survey especially at about 30 < l < 300, they are still larger than the their maximum variations of BAOs wiggles.

• [3]. Thus, the detection of BAOs with the ongoing and upcoming surveys such as LSST, CFHT and SNAP survey confront a technical challenge.

3. Application of wavelet on 3. Application of wavelet on Weak lensingWeak lensing

• Construction of ConvergenceConstruction of Convergence Theoretical expressionTheoretical expression

N-body simulation N-body simulation parametersparameters

• Itanium Beowulf cluster at CITAItanium Beowulf cluster at CITA• 1024^3 mesh resolution 1024^3 mesh resolution • 512^3 particles512^3 particles• output periodic surface density maps at output periodic surface density maps at

2048^2 resolution2048^2 resolution• an initial redshift z_i=50, 1000 steps an initial redshift z_i=50, 1000 steps • comoving box size L=200h^{-1} Mpccomoving box size L=200h^{-1} Mpc

ParametersParameters

• a Hubble constant h=0.7a Hubble constant h=0.7• A scale invariant n=1 initial power A scale invariant n=1 initial power

spectrumspectrum• A flat cosmological model with \A flat cosmological model with \

Omega_m + \Lambda = 1Omega_m + \Lambda = 1• \Omega_m=0.3\Omega_m=0.3• \sigma_8=0.82\sigma_8=0.82

Stacking of map (here just a Stacking of map (here just a exampleexample))

Produce

WaveletWavelet

• Pls see papers written by Prof.Fang Pls see papers written by Prof.Fang Li-ZhiLi-Zhi

such as: such as:

1. Fang Li-zhi and W. Thews 1. Fang Li-zhi and W. Thews Wavelet in Physics. Would Scientific Wavelet in Physics. Would Scientific SingaporeSingapore

2. Fang Li-zhi et al’s papers 2. Fang Li-zhi et al’s papers appeared in ApJ.appeared in ApJ.

Non-GaussianityNon-Gaussianity• In the standard model of cosmology,

fluctuations start off small, symmetric, and Gaussian. Even in

some non-Gaussian models such as topological defects, initial fluctuations are still symmetric: positive and negative fluctuations occur with equal probability.

• As fluctuations grow by gravitational instability, this

symmetry can no longer be maintained: overdensities can be arbitrarily large, while underdense regions can never have less than zero mass. This leads to Non-GaussianityNon-Gaussianity in the distribution of matter fluctuations.

Non-Gaussianity using waveletNon-Gaussianity using wavelet : Skewness and Kurtosis Skewness and Kurtosis

Jesús Pando, David Valls-Gabaud, and Li-Zhi Fang, PRL, Vol. 81, p. 4568-4571 ( 1998)

No significant non-Gaussianity can be identified from the third and fourth order cumulants.

Weak lensingWeak lensingR=3*60/2^j R=3*60/2^j [arcmins]; [arcmins]; J_max=11 for 2048J_max=11 for 2048

Tong-Jie Zhang, Ue-Li Pen, Li-Zhi Fang, in preparation for submitting to ApJ

The significant non-Gaussianity can be identified on small scale.

My appeared papers related to My appeared papers related to LensingLensing

strong or weakstrong or weak• (1). Reconstruction of the One-Point Distribution of Convergence from Weak Lensing by Large-Scale Structure

Zhang Tong-Jie; Pen Ue-Li The Astrophysical Journal [ApJ], Volume 635, Issue 2, pp. 821-826 (12/2005) • (2) Gravitational Lensing by Dark Matter Halos with Nonuniversal Density

Profiles Zhang, Tong-Jie The Astrophysical Journal [ApJ], Volume 602, Issue 1, pp. L5-L8.(02/2004)

• (3). Optimal Weak-Lensing Skewness Measurements Zhang, Tong-Jie; Pen, Ue-Li; Zhang, Pengjie; Dubinski, John The Astrophysical Journal [ApJ], Volume 598, Issue 2, pp. 818-826. (12/2003)

• (4). Detection of Dark Matter Skewness in the VIRMOS-DESCART Survey: Implications for Omega0

Pen, Ue-Li; Zhang, Tongjie; van Waerbeke, Ludovic; Mellier, Yannick; Zhang, Pengjie; Dubinski, John

The Astrophysical Journal [ApJ], Volume 592, Issue 2, pp. 664-673. (08/2003)

Thanks!Thanks!