Weak Lensing and Dark Energy Cosmology Tong-Jie Zhang[ 张同杰 ] Department of Astronomy, Beijing Normal University Cosmology Workshop Institute of High Energy Physics, Chinese Academy of Sciences 2008/12/08

3-D : Accelerating Universe WMAP 3-D Universe: 3 dark sides

(1). Our Universe — Dark energy

(2). Dark Matter [halo]( 暗物质 [ 晕 ])

(3). Dark ages( 黑暗时代 )

Outline 0. Basic of Gravitational lensing0. Basic of Gravitational lensing 1. Dark Energy and Neutrino Mass Constraints from WL, SN Ia and RGA1. Dark Energy and Neutrino Mass Constraints from WL, SN Ia and RGA 2. The signatures of BAOs on the convergence2. The signatures of BAOs on the convergence power spectrum of weak lensing power spectrum of weak lensing. 3. Application of wavelet on Weak lensing3. Application of wavelet on Weak lensing

0. Basic of Gravitational lensing SchematicDiagram of Gravitational Lensing ( 引力透镜示意图 ) Schematic Diagram of Gravitational Lensing ( 引力透镜示意图 )

Physics of Gravitational Lensing (GL) Bending of Light under Gravity Light will follow the straightest possible path through flat space time. If spacetime is curved near a massive object, so the trajectory of light is also curved.

Observational Event of of Gravitational Lensing Einstein’s Cross an Einstein ring galaxy directly behind a galaxy

HST Image of a gravitational lens in galaxy cluster

Category of GL Strong gravitational lensingStrong gravitational lensing Weak gravitational lensingWeak gravitational lensing

Gravitational lens Theory — Sketch of a typical gravitational lens system

Deflection angle General Relativity: for a point mass MGeneral Relativity: for a point mass M

Lensing equation or ray-trace equation Position of source Position of image

Lensing equation Tong-Jie Zhang ApJ 602, L5-8(2004) [ Multiple images can be produced if lens is strong Position of source Position of image

Convergence and shear K>=1  strong K =1  strong K<<1  weak Deflection potential

Distortion and Magnification Critical curves in lens plane; Caustics in source plane Magnification : Shear : Det

Strong lensing Sources are close to the caustic lines.Sources are close to the caustic lines. K >=1 and |r|>=1: The convergence and shear are strong enough to produce giant arcs and multiple images.K >=1 and |r|>=1: The convergence and shear are strong enough to produce giant arcs and multiple images.

The probability for strong lensing E(z) and f (M, z): dependent on cosmological model

CLASS observation The Cosmic Lens All-Sky Survey CLASS): An international (USA, UK and Netherlands) collaborative project to map more than 10,000 radio sources in order to create the largest and best studied statistical sample of gravitationally lensed systems. Sample: Well-defined statistical sample: 8958 Multiply imaged sourses: 13 P_ob=N(>\theta)/8958

Lensing models SISSIS GNFWGNFW

Image separation probability for GNFW model Tong-Jie Zhang ApJ 602, L5-8(2004)

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

Weak lensing (cosmic shear) Cosmic shear is the distortion of the shapes of background galaxies due to the bending of light by the potentials associated with large-scale structure in the universe. Cosmic shear is the distortion of the shapes of background galaxies due to the bending of light by the potentials associated with large-scale structure in the universe. Wek lensing regime: K <<1 and |r|<<1

Distortion of background images: shape and correlation Before lensed After lensed

Measurement The ellipticity of galaxy and the intrinsic ellipticity and shearThe ellipticity of galaxy and the intrinsic ellipticity and shear

The mean expectation of source ellipticities and alignment Weak lensing shear: spin-2 polarization field Φ x y b a

Shear component The tangential shear and the 45 degree rotated shear in the local frame defined by the line connecting the pair of galaxiesThe tangential shear and the 45 degree rotated shear in the local frame defined by the line connecting the pair of galaxies xixi xjxj θ a b a b a

Shear correlation function

Two-point cosmic shear statistics 2. the top-hat filtered variance of the shear 3. the variance of the aperture-mass 1. shear correlation

Power spectrum of convergence OCDM  CDM (linear)

Observational 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 CMB Contaldi et al, PRL, 90, 2003 Contaldi et al, PRL, 90, 2003

1. Dark Energy and Neutrino Mass Constraints from WL, SN Ia and RGA Yan, Gong, Tong-Jie Zhang, Tian Lan and Xue-Lei ChenYan, Gong, Tong-Jie Zhang, Tian Lan and Xue-Lei Chen Sumitted to ) Sumitted to ApJ

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 WMAP5:

WMAP5 Results on neutrino WMAP5:WMAP5:

Weak Leasing and Neutrino Mass 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: SN Ia:SN Ia: RGA:RGA: BAO:BAO:,

Data sets: Weak lensing dataWeak lensing data CFHLST-wide, 22 deg^2 (Fu et al. 2008); CFHLST-wide, 22 deg^2 (Fu et al. 2008); RCS, 53 deg^2 (Hoekstra et al. 2002) RCS, 53 deg^2 (Hoekstra et al. 2002) SN Ia dataSN Ia data SCP “ Union ” data, 307 samples (Kowalski et al. 2008) SCP “ Union ” data, 307 samples (Kowalski et al. 2008) RGA (relative galaxy ages)RGA (relative galaxy ages) H(z) from GDDS, 9 samples (Simon et al. 2005) H(z) from GDDS, 9 samples (Simon et al. 2005) BAO dataBAO data A at z=0.35 (Eisenstein et al. 2005) A at z=0.35 (Eisenstein et al. 2005)

[1]. Weak Lensing Constraints on w: Weak constraint on w for current WL data WL+SN+RGA+BAO: w = at 95.5% C.L. ( w = for WMAP5 ) The similar degeneracy direction and constraint ability for SN Ia and RGA wCDM Results:

[2]. Weak Lensing Constraints on Σ m v Σmv<=0.4eVΣmv<=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. The signatures of BAOs on the convergence power 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 after the epoch of recombination [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 the Early Universe and Today Christopher J. Miller,1 Robert C. Nichol,1 David J. Batuski2 22 JUNE 2001 VOL 292 SCIENCE

BAO 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). Tgravitaional lensing: strong and weak(c). The power spectrum gravitaional lensing: strong and weak

(a). BAO on Correlation function of galaxies(z=0): Sound Waves in Matter Each initial overdensity (in DM & gas) is an overpressure that launches a spherical sound wave. This wave travels outwards at 57% of the speed of light.Each initial overdensity (in DM & gas) is an overpressure that launches a spherical sound wave. This wave travels outwards at 57% of the speed of light. Pressure-providing photons decouple at recombination. CMB travels to us from these spheres.Pressure-providing photons decouple at recombination. CMB travels to us from these spheres. Sound speed plummets. Wave stalls at a radius of about 100 Mpc.Sound speed plummets. Wave stalls at a radius of about 100 Mpc. Overdensity in shell (gas) and in the original center (DM) both seed the formation of galaxies. Preferred separation of 100 Mpc.Overdensity in shell (gas) and in the original center (DM) both seed the formation of galaxies. Preferred separation of 100 Mpc. 100 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 tgravitational lensing: weak (c). BAO on the power spectrum gravitational lensing: weak The matter power spectrum Tong-Jie Zhang, Qiang Yuan, Tian Lan

The convergence power spectrum of weak lensing Tong-Jie Zhang, Qiang Yuan, Tian Lan arXiv:

The statistical errors in the measurements of weak lensing power spectrum Tong-Jie Zhang, Qiang Yuan, Tian Lan

Conlusions [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 Weak lensing Construction of ConvergenceConstruction of Convergence Theoretical expression

N-body simulation parameters Itanium Beowulf cluster at CITAItanium Beowulf cluster at CITA 1024^3 mesh resolution1024^3 mesh resolution 512^3 particles512^3 particles output periodic surface density maps at 2048^2 resolutionoutput periodic surface density maps at 2048^2 resolution an initial redshift z_i=50, 1000 stepsan initial redshift z_i=50, 1000 steps comoving box size L=200h^{-1} Mpccomoving box size L=200h^{-1} Mpc

Parameters a Hubble constant h=0.7a Hubble constant h=0.7 A scale invariant n=1 initial power spectrumA scale invariant n=1 initial power spectrum A flat cosmological model with \Omega_m + \Lambda = 1 A flat cosmological model with \Omega_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 example ) Produce

Wavelet Pls see papers written by Prof.Fang Li-ZhiPls see papers written by Prof.Fang Li-Zhi such as: such as: 1. Fang Li-zhi and W. Thews Wavelet in Physics. Would Scientific Singapore 1. Fang Li-zhi and W. Thews Wavelet in Physics. Would Scientific Singapore 2. Fang Li-zhi et al ’ s papers appeared in ApJ. 2. Fang Li-zhi et al ’ s papers appeared in ApJ.

Non-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 Non- Gaussianity 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- Gaussianity in the distribution of matter fluctuations.

Non-Gaussianity using wavelet Skewness and Kurtosis Non-Gaussianity using wavelet : Skewness and Kurtosis Jes ú s Pando, David Valls-Gabaud, and Li-Zhi Fang, PRL, Vol. 81, p ( 1998) No significant non-Gaussianity can be identified from the third and fourth order cumulants.

Weak lensing R=3*60/2^j [arcmins]; J_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 Lensing strong 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 (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 (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 (08/2003)Thanks!