Weak Lensing Analysis of Galaxy Clusters

Transcription

1 Zao Cluster Workshop (4/Oct/007) Weak Lensing Analysis of Galaxy Clusters Umetsu,, Keiichi (Academia Sinica IAA) 梅津敬一 ( 中央研究院天文所 )

2 Contents 0. Examples of Gravitational Lensing 1. Gravitational Lensing Theory. Cluster Weak Lensing Theory 3. Weak Lensing Analysis: KSB Method 4. Practical Applications 5. Summary

3 (Fort & Mellier) Lens Effects by a Galaxy Cluster Images of background galaxies can be distorted coherently by the gravitational field of a foreground cluster: Unlensed Lensed (simulated) Weak Strong

4 Cluster Strong Lensing: Multiple Imaging Subaru/Suprime-Cam Lens: cluster at z=0.83 Source: galaxy at z=3.9 (t ~ 1.5Gyr) Image positions and parities constrain the cluster mass distribution. HST/ACS Umetsu, Tanaka, Kodama et al. 005

5 Special Example of Cluster Weak Lensing Weak Shearing = tangential shape distortion of background galaxy images

6 Weak Shear Field Observable shear: a b γ = a + b γ = γ cos(φ ) γ 1 = γ sin(φ ) Cosmic shear: a few % Cluster shear: 10-0% Cluster z = 0.77; Arc z = 4.89: Photo from H. Yee (HST/ACS) Simulated 3x3 degree field (Hamana 0)

7 1. Gravitational Lens Theory 1.1. Lens equation 1.. Image distortion 1.3 Convergence & Shear 1.4 Strong to Weak Lensing Regimes 1.5 Geometric Scaling of Lensing Signal References: Bartelmann & Schneider (001) Ph.D. thesis of T. Hamana Ph.D. thesis of M. Takada Hattori, Kneib, & Makino (1999) Umetsu, Tada, & Futamase (1999)

8 1.1. Gravitational Lensing Gravitational deflection angle in the weak-field limit ( Φ /c^<<1) Lens equation weak field ( Φ <<1), or small angle scattering (b>>m) limit Born approximation simplify the various calculation δ ˆ α β δp p = c θ ψ ( θ) Φ( x, x source D = θ ds dαˆ observer D s ) δx D: angular diameter distances - Ds: observer to source - Dd: observer to lens - Dds: lens to source Ψ: effective D lensing potential

9 Cosmological Lens Equation Solve the linearized geodesic eq. in the weak field limit (Futamase 95): θ( λ ) s θ( λ ) o = r( λ ) s λ dλ Φ( λ) c null r ( λ ) s Functions defined with the affine parameter: dλ = a dχ Cosmological Poisson eq.: ( Δ (3) + 3K) Φ( χ) Φ( χ = 4πGa 4πGa With Born approximation (Futamase 1995): χs r( χs χ // ) β = θ χ // (, χ // ) d Φ χ c 0 r ( χ ) = θ c χ 0 s dχ // D D ds s s, χ ) // [ ρ( χ) ρ ] ρ δ ( χ) formally reduced to the conventional lens eq. β:=θ(λs), θ:=θ(0) Functions defined with co-moving coordinates

10 r θ r β image source 1.. Image Distortion Differential deflection causes a distortion in the area and shape of an image. Image distortions by weak lensing deflector observer z s z d 0 Deformation of the shape/area of an image For an infinitesimal light source: d r r β = A d θ δβ i (1) 1 () = Dij δθ j + Dijk δθ j k with A D (1) 1 δθ +... A: Jacobian matrix of the lens equation

11 1.3 Lensing Convergence and Shear Symmetric x Distortion Matrix described by 1+ components A ij (θ) = δ ψ (1 κ) δ ij i j ij Γ ij (1) Convergence κ Isotropic area distortion, or flux magnification () Gravitational shear matrix κ Γ + γ 1 ψ = γ Anisotropic quadruple (elliptical) shape distortion: Complex Shear: γ=γ 1 +iγ ij 1 Δ ψ, i j 1 δijδ θ γ γ 1 D effective potential, as a L.o.S. projection of Newtonian potential ψ ( θ) = c χ s 0 dλ D D D d ds s Φ

12 Effect of Shear = Quadruple Distortion Spin- Complex Shear γ = γ 1 + iγ γ e i φ γ e e 1 Image quadrupoles: Q = ij x i Complex ellipticity, e=e 1 +ie : x j e = a a b + b e iφ Q + iu e 1 = Q Q Q + Q, e = Q1 Q + Q 11 In the weak limit (κ, γ <<1): e e ( s ) + γ e(s): source intrinsic ellipticity Assuming that background sources are statistically circular: e γ + O σ e N

13 κ = = Physical Meaning of κ Lensing Convergence: weighted-projection of 3-D density contrast 3H 0 Ω c dλ( ρ m m source observer c ρ ) 4πG dχ g( χ, χ ) a Ds D D Critical surface mass density of gravitational lensing d ds s 1 1 δρm ρ = dσ m Σ 1 crit Σ crit ( zd, zs; Ωm, ΩDE, H 0) = c 4πG Ds D D d ds Strong lensing Weak lensing Cosmic shear κ high density regions (e.g., cluster cores) r = 100kpc a few Mpc large-scale structure (~10Mpc) probability

14 1.4 Strong to Weak Lensing Regimes Tangential critical curve: Einstein rings, Tangential arcs, Giant Luminous Arcs θ E 4GM ( θe ) = c Dd D D ds M ( θe ) 0" h M s sun 1/ 1/ dds ddd s 1/ Radial critical curve: Radial arcs SL: r <~ 0.5 arcmin WL: r =~ 1 1 to

15 1.5 Geometric Scaling of Lensing Signal g( a, a r( χ, χ s ) r( χ) r( χ ) s ) = = s DdsD D a s d g: : geometric scaling function In optical observations, the median redshift of background galaxies is: r( χ, χ s ) r( χ) g( a, a ) = r( χ ) s = s DdsD D a s d 0 texp.9 30 min z s with the mean galaxy number density of For a background source at zs=1 n g 40arcmin (Hamana) t exp 30min 0.3 FWHM 0.8" 1

16 Systematic Uncertainties in Mass Inverse critical density as a func of zd Deviation from the true (zs=1) Signal Σ 1 crit In flat 3-space, 3 ( z, z ) D D / D 10% deviation Signal χd (1 χd / χ s ) d s d ds s Cluster redshift Conversion from shear & convergence to physical masses requires geometrical information of lense/sources If χ d <<χ s (z d <~ 0.), the strength of gravitational lensing is insensitive to the source distance, i.e., no accurate information of redshift distribution is required. Figure from Okabe & Umetsu (007)

17 . Cluster Weak Lensing.1 Weak Lensing E/B-mode Patterns. Tangential Shear Profile.3 Signal to Noise.4 Shear to Mass.5 Mass Reconstruction

18 .1 E and B Mode Patterns Formal similarities btw. shear and Stokes Q,U of linearly polarized light Observable x shear tensor: Γ ij = + γ 1 γ γ γ 1 + Q U U Q Shear matrix in terms of potential: Δκ Δκ E B = i = ε ij j i Γ k Γ ij ij Γ = jk E i B j 1 δ ij Δ ψ E,B-fields (rotation invariant) by projection of shear tensor B-mode = 0 in Weak Lensing B-mode signal can be used to monitor systematics in shape measurements E + 1 ( ε ) kj i k + ε ki j k ψ B

19 . Tangential Shear Profile For a given point on the sky selected as a center, one can form a tangential (E-mode) projection of the spin- shear field: Then, measure the azimuthally averaged tangential shear profile as a function of angular radius This observable is related with differential information of mass properties: mean κ within θ M ( < θ ) θ At large radii, κ(r)~0, so that γ + ( θ ) ~ κ ( < θ )

20 Example: A1689 (Subaru/S-Cam) Tangential (E-mode) shear measurements (Top) from a red background sample, showing a significant coherent signal B-mode shear measurements (Bottom), showing no significant systematics S/N = sqrt[s^/n^] = 14 S/N = sqrt[s^/n^] = Measurements beyond the virial radius (~15 ) The errorbars are independent (no correlation) V-i selected red-galaxies (i <5.5), n g ~10 arcmin -

21 Practical shear estimate: γ obs.3 S/N in Shear Estimates = Cγ + ( γ + Δγ ) noise int γ Cγ = obs C: bias factor, need to be calibrated out γ int : intrinsic ellipticity Δγ noise : shape measurement error Δγ noise Now assume C=1 (no systematic bias, or perfect calibration): σ g int = σ + σ noise from Subaru/S-Cam observations S/N of amplitude of averaged shear in a pixel from a local sample of N-galaxies S/N γ γ σ Δθ ( ) g g.4 / arcmin σ N 1' = g γ 1 n 1/ S/N of azimuthally averaged tangential shear for an SIS halo S/N γ + ~ γ + σ g / N σ v km/s σ g ng 30arcmin 1/ ln( θmax / θ ln10 min ) 1/ D ds / D 0.7 s

22 S/N in Mass Maps Sensitivity in the reconstructed mass map is: σ σ 0.4 n 30arcmin 1 g g FWHM g σ κ = = N 1' 1/ To have an S/N of, say, ν:=κ/σ κ =5, for an unresolved mass halo with κ=0., FWHM 1 1 ν κ σ = 0.85" g n g FWHM: Gaussian FWHM in the mapmaking arcmin In weak lensing mapmaking from ground-based observations, the effective angular resolution is of the order of 1 arcmin To resolve substructures, (1) Study nearby clusters (Okabe & Umetsu 007, to appear in PASJ) () Go to space (HST) (3) Use weak lensing Flexion (Okura,Umetsu,Futamase( 007ab) 1/

23 Observable shear (image ellipticity) ) field to a mass map κ i κ = Δ = j Γ ij.4 Shear to Mass r ( u γ ) ( ) ( r ) i j Γ = Δ u 1 1 ij i r γ Solve the integral eq from measured shear field for a given survey geometry (boundary condition + constant mass sheet) Kaiser & Squires 1993 ^ r ^ r ^ r r κ l ) = cos(φ ) γ ( l ) + sin(φ ) γ ( l ), l ( l 1 l 0 On large scales (>10 Mpc), <δ matter > = 0. Uses the Green function for the D Poisson equation to solve the inversion equation; usually, l=0 mode is taken to be zero Justification of using Kaiser & Squires method with <κ> = 0

24 Weak Shear Map E-mode Pattern Observable shear: a b γ = a + b γ = γ cos(φ ) γ 1 = γ sin(φ ) Cosmic shear: a few % Cluster shear: 10-0% Cluster z = 0.77; Arc z = 4.89: Photo from H. Yee (HST/ACS) WL Cluster Survey (Hamana-san s talk): Schneider 96, Hamana et al. 004; Umetsu & Futamase 000, Miyazaki et al. 00 Simulated 3x3 degree field (Hamana 0)

25 3. Weak Lensing Analysis 3.1 Practical Approaches: KSB, Shapelets 3. Observable Image Ellipticity 3.3 Star/Galaxy Separation 3.4 PSF Corrections 3.5 Weak Lensing Dilution Effect References: Kaiser, Squires, Broadhurst (KSB 1995) Erben et al. (001) Hoekstra et al. (1998) Bartelmann & Schneider (001) Refregier (003) Shear Testing Programme (STEP)

26 3.1 Practical Approaches Practical observations suffer from: Noise in shape measurements of faint, small distant galaxies PSF anisotropy (5-10%) varying from position to position Smearing of the lensing signal due to atmospheric smearing How to improve the shape measurement? How to correct for the PSF effects? How to obtain the unbiased shear estimate? KSB method based on the moment approach (KSB95) Quantify the shapes of images by Gaussian-weighted quadrupole shape moments IMCAT :: KSB-implementation for object detection, shape measurements KSB (1995) made Weak Lensing a reliable observational tool!!! Shapelets based on a complete, orthonormal set of D basis functions constructed from Hermite polynomials weighted by a Gaussian (Refregier 003)

27 3. Observable Image Ellipticity Observable ellipticity is expressed as a linear sum of (1) intrinsic, () gravitational shear, (3) PSF contributions in the limit of weak shearing both in lensing and PSF (KSB 1995, Bartelmann & Schneider 001): Ellipticity measured for a background galaxy e^(s): intrinsic ellipticity of the background q: quadrupole PSF anisotropy (CCD, stacking error etc.) P^γ: : response for gravitational shearing P^sm: : response for PSF anisotropy Ellipticity for a foreground star since e^(s)=γ=0

28 3.3 Star/Galaxy Separation Sample of foreground + cluster + background galaxies 3σ limiting mag Sample of bright, unsaturated stars (a few stars per arcmin^) for measuring PSF size/shape properties cf. number of usable field galaxies in S-Cam data (Hamana) n g 40arcmin t exp 30min 0.3 FWHM 0.8" 1 A1689 field (Subaru/S-Cam)

29 3.4 PSF Anisotropy Correction raw residual Subaru/S-Cam data of A1689 (i -band) Seeing FWHM=0.8

30 Isotropic Shear Calibration Mean shear calibration factor, as a function of object size Anisotropy correction for individual galaxy ellipticities: Large correction Isotropic correction: Small object Small objects are more affected by atmospheric smearing 1pixel = 0.0 arcsec (S-Cam)

31 3.5 Weak Lensing Dilution Effect Averaged shear over a local sample of Background + Cluster galaxies: γ diluted = γ BG N BG N + BG N Cluster Dilution factor: 1+ f dilution 1+ N N cluster BG In the central region of a rich cluster, the degree of dilution is as large as f~4!! Broadhurst, Takada, Umetsu et al. (005) A secure background selection is crucial!!

32 Background Color Selection Color-Magnitude Diagram for A1689 (z=0.183) Red background Red sample <zs>=0.87, /D s >=0.69 <D ds /D Cluster galaxies Blue background Medezinski, Broadhurst, Umetsu et al. (007) Faint blue sample <zs>=.0 <D ds /D s >=0.83 After color-selection, n g (red) ~ 5-10 galaxies / arcmin^ in z= clusters

33 4. Weak Lensing Observations 4.1 1D Mass Profile Reconstruction of A D Mass Reconstruction of Clusters

34 4.1 1D Mass Reconstruction (A1689) Wide-field imaging of Subaru/Suprime-Cam Strong+Weak lensing analysis 34 (4.4Mpc/h) Broadhurst, Takada, Umetsu et al. 005 Revealing ~100 lensed multiple images of ~30 background galaxies Superb angular resolution of HST/ACS 7 (3.5Mpc/h) ~3 (400kpc/h)

35 Weak Shearing + Magbias Measurements combined with ACS Strong Lensing Combines 1D Weak (1) shear and () magnification bias measurements to derive a model-independent, mass density profile, κ(θ) Distortion alone mass-sheet degeneracy (no constraint on l=0) Magnification bias breaking the degeneracy (noisier in general) Then, combines Strong and Weak Lensing mass profiles κ(θ) to test the CDM / NFW model over 10<r<000 kpc/h (!!) Magnification Bias δn / n0 ( α 1) κ, α < 1 lensed Ω survey unlensed # counts n > F) F 0 ( α ~5-6 unlensed

36 Mass 0.005r vir to r vir Projected mass density profile CDM acceptable χ min dof. = 13./ M vir = vir =.0 ± 0.1Mpc h 1. 4 cvir = r / data points used 1 M sun Cored Pow-Law preferred χ min dof. = 4.5/ n = r c = kpc / h SIS/Cored-IS Strongly rejected (>10σ!!)

37 4. D Mass Reconstructions Gravitational shear field Magnification bias measurement Subaru/S-Cam i -data of A1689 (z=0.183) Broadhurst, Takada, Umetsu et al. (005), Umetsu, Broadhurst, Takada (007), Umetsu (007 in prep)

38 Mass and Light in A1689 Subaru i -image i image + Mass (yellow)) + Light (white) contours in the central 15 x15 x15 region of A1689 (Umetsu et al. 007) Flexion reconstruction in a 4 x44 x4 region (Okura, Umetsu,, & Futamase 007ab)

39 Full D-based Mass Profile in A1689 D-based mass profile (blue)) obtained by a 1D projection of MEM reconstructed mass map of A1689 Error bars are correlated 1D and D reconstructions are consistent with each other, supporting the assumed quasi circular symmetry in projected mass distribution UTB007

40 Distributions of Mass and Baryons Subaru/S-Cam R-band image Okabe & Umetsu (007, PASJ accepted) Systematic study of 7 nearby merging clusters based on Subaru/S-Cam observations Mass contours R-band luminosity Chandra contours Chandra image AMiBA T-SZE Mass contours Mass contours AMiBA team

41 Summary Weak lensing has become a reliable observational tool (not just a theoretical tool) in the past decade thanks to successful developments in practical observational techniques (KSB/IMCAT, Shapelets etc.) as well as in instrumental technology (Subaru, HST, CFHT etc.). Cluster weak lensing is now a very hot research topic: - Mass distributions in merging clusters:: cluster physics, dark matter nature - Dark matter distributions out to cluster virial radii (or to filaments) - Strong + Weak lensing to probe the entire mass profile, constraining the nature of dark matter The dilution effect by cluster contamination is significant in the cluster central region, leading to systematic underestimations of NFW concentration and virial mass parameters. A secure background selection with color/photo-z information is crucial for cluster mass measurements. Thanks to high image quality, Subaru observations allow masssubstructure mapping in z= clusters (see also Okura, Umetsu, Futamase 007ab for the use of the higher-order WL effect, Flexion)

42 FIN

43 .1. Cosmological weak lensing past z=z s Arises from total matter clustering Not affected by galaxy bias uncertainty Well modeled based on simulation Tiny 1-% level effect Intrinsic ellipticity per galaxy, ~30% Needs numerous number (10^8) of galaxies for the precise measurement >1000deg^ Large-scale structure z=z l present z=0

44 Subaru/Suprime Suprime-Cam and Weak Lensing Subaru telescope / Suprime-Cam Large photon collecting area, D=8.3m Wide ~30 x 30 FoV From NAOJ Small, stable PSF anisotropy; subarcsec seeing Hu & Cowie 006 Nature Ideal instrument for weak lensing (large AΩ and high image quality)

45 HST and Subaru Telescope HST/ACS Subaru/Suprime Suprime-Cam From STSCI From NAOJ D=.4m Superb angular resolution ~ 3 x 3 Field-of-View (FoV) Ideal instrument for strong lensing in the innermost region Large photon collecting area, D=8.3m Wide ~30 x 30 FoV Small, stable PSF anisotropy; subarcsec seeing Ideal instrument for weak lensing, detectable out to virial radii

46 Validity of weak Validity of weak-field approximation field approximation Validity of the weak-lensing approximation in a cosmological situation? 1 / Φ c << a H Ga m δ ρδ π Ω = ΔΦ = Ω Ω = Ω Φ Mpc / ~ δ λ δ λ λ δ λ m m H m h a a c H c The potential field can be smooth and small even if the underlying density perturbations are peaky and non-linear Poisson Eq.

47 Shape Deformation δβ i (1) 1 () = Dij δθ j + Dijk δθ j k δθ +... D D (1) ij () ijk = = A ij A = δ ij, k ij = i i j ψ j k ψ Jacobian matrix (1 st order) nd order size-dependent deformation [angle^-1], called Flexion (e.g., Okura, Umetsu, Futamase 007ab) Deformation matrix modified including higher-order terms, corresponding to highly stretched arcs etc. δβi δθ j (1) () = Dij + Dijk k δθ +...

48 (Appendix) Multiple (Appendix) Multiple Lensing Lensing p n p n p n p n p n p n p n r χ r c r χ r c ψ χ χ χ χ χ = = Φ Δ 1 1 (1) 1 1 (1) ) ( ) ( ) ( ) ( ) ( ) ( θ χ θ θ Lens Lens-equation with ( equation with (discretized discretized) multiple ) multiple-lens approximation lens approximation +Δ Φ χ χ χ χ χ ψ p p d p ), ( // // χ ) ( 1 1 (1) ) ( ) ( ), ( n n p p p n p ij n n g = = Ψ I A H I θ θ A χ χ j i p ij H p χ χ ψ =, Jacobian Jacobian matrix of matrix of lensing lensing: + + = + Ψ Ψ Ψ Ψ Ψ + Ψ = 1 1 '' ' ', ' ) ( γ κ ω γ ω γ γ κ p p p p p p p p p n I A

49 (Appendix) Cumulative Mass Estimator Aperture densitometry, or so-called z-statistic mass estimator (Fahlmann et al. 1994; Clowe et al. 000), in terms of observable galaxy shear estimates: θ max be taken as large as possible (θ( max <~ for Subaru/Scam) ζ(r) ) is measured from outside of R, allowing to avoid the inner strong lensing regime!!! The following gives a lower bound to the enclosed mass: > mean background, similar to the concept of background subtraction in photometry

50 Example: A1689 (Subaru/S-Cam) ζ-statistic cumulative mass measurements assuming <zs>=1 for the red-background sample; θ max =19 arcmin, while the projected virial radius is θ vir ~ 15 arcmin Note the errorbars are correlated!

51 Measurement (I): tangential shear Significant S/N of 1σ Signal being diluted by a factor of -5 without a secure background selection (x) cf., Clowe & Schneider 001; Bardeau et al. 004 Good agreement with r < 3 Signal r > 3 is weaker than expected from ACS Null-detection of B-mode signal, g X No significant systematics ( )

52 Measurement (II): magnification bias Number counts of the Background No gravitational lensing n0 = (1.6 ± 0.3) arcmin Significant detection of a depletion of red galaxy counts (9.3 σ) r>3 is weaker than expected from ACS Masking effect by member galaxies corrected ( )

53 Dilution vs. Lensing Magbias Effect δn n g g 0 δ magbias ( α 1) κ < = 1+ δ + δ 0 dilution magbias Depression of background # counts due to gravitational magnification bias κ δ magbias ( α 1) κ ~ α = α<1 found at fainter magnitudes of redder optical bands cf. α~0. at i =5.5 (BTU05) Contamination by cluster galaxies (dilution of lensing signal) δ dilution = n cluster n g 0 GL δ dilution ~ several in the central region of rich clusters

54 Weak Lensing Halo Surveys Schneider 1996, Erben et al. 000, Umetsu & Futamase 000, Miyazaki et al. 003 etc.. On-going/Future surveys: CFHTLS, GaBoDS, / Pan-STARRS, Subaru HSC etc.. Mass concentrations Foreground galaxy concentrations MS (z=0.83) observed with the Subaru telescope