Transcript ai2004 4538
Weak Gravitational Lensing and Shapelets Alexandre Refregier (CEA Saclay) Collaborators: Richard Massey (Cambridge) David Bacon (Edinburgh) Tzu-Ching Chang (Columbia) Jason Rhodes (Caltech) Richard Ellis (Caltech) Jean-Luc Starck (CEA Saclay) Sandrine Pires (CEA Saclay) IPAM/UCLA – January 2004 Weak Gravitational Lensing Distortion Matrix: i 2 ij dz g ( z ) j i j Theory Direct measure of the distribution of mass in the universe, as opposed to the distribution of light, as in other methods (eg. Galaxy surveys) Weak Lensing Shear Measurement unlensed lensed background background galaxies galaxies mass and shear distribution Scientific Promise of Weak Lensing From the statistics of the shear field, weak lensing provides: • Mapping of the distribution of Dark Matter on various scales • Measurement of the evolution of structures • Measurement of cosmological parameters, breaking degeneracies present in other methods (SNe, CMB) • Explore models beyond the standard osmological model (CDM) Jain, Seljak & White 1997, 25’x25’, SCDM Cosmic Shear Surveys PSF anisotropy William Herschel Telescope La Palma, Canaries Deep Optical Images Correct for Bacon, Refregier & Ellis (2000) systematic effects: Bacon, Refregier, Clowe & Ellis (2001) Shear Measurement Method KSB Method: (Kaiser, Squires & Broadhurst 1995) Quadrupole moments: Qij d x xi x j w( x ) I ( x ) Ellipticity: Q11 Q22 2Q12 1 , 2 Q11 Q22 Q11 Q22 2 PSF Anisotropy correction: g 'g 2 1 Pgsm sm * P * PSF Smear & Shear Calibration: (P ) g 1 Cosmic Shear Measurements Shear variance in circular cells: 2 2 + Rhodes et al. ()=< > Bacon, Refregier & Ellis 2000* Bacon, Massey, Refregier, Ellis 2001 Kaiser et al. 2000* Maoli et al. 2000* Rhodes, Refregier & Groth 2001* Refregier, Rhodes & Groth 2002 van Waerbeke et al. 2000* van Waerbeke et al. 2001 Wittman et al. 2000* Hammerle et al. 2001* Hoekstra et al. 2002 * Brown et al. 2003 Hamana et al. 2003 * * not shown Jarvis et al. 2003 Casertano et al 2003* Rhodes et al 2004 Cosmological Constraints Hoekstra et al. 2002 E <Map2> B <Map2> (arcmin) E/B decomposition m 8 0.3 0.52 0.81 0.08 Normalisation of the Power Spectrum Rhodes et al. 2003 Moderate disagreement among cosmic shear measurements (careful with marginalisation) Non-linear clustering corrections (Cf. Smith et al.) This could be due to residual systematics (shear normalisation? Cluster physics?) Agreement on average with cluster and CMB+LSS constraints Future Surveys Survey start DLS Diameter FOV Area (m) (deg2) (deg2) 24 20.3 28 CFHTLS 3.6 1 172 2003 VST 2.6 1 x100 2004 VISTA 4 2 10000 2007 Pan-STARRS 41.8 44 31000 2008 LSST 8.4 7 30000 2012 SNAP/JDEM 2 (space) 0.7 300 2014 1999 The Shapelet Method | fnm = > < = f00 > > | | + f01 | > +… Refregier (2001) Refregier & Bacon (2001) also: Bernstein & Jarvis (2001) Decomposition of a galaxy image into shape components: f ( x ) f n1n2 Bn1n2 ( x ; ) n1n2 f n1n2 d x f ( x ) Bn1n2 ( x ; ) 2 Orthogonal Basis functions Gauss-Hermite Basis Functions Bn ( x; ) H n ( x)e 1 x2 / 2 2 Bn1n2 ( x, y; ) Bn1 ( x; ) Bn2 ( y; ) • Perturbations around a gaussian • Eigenfunctions of the Quantum Harmonic Oscillator • Coefficients are gaussianweighted multipole moments • Capture a range of scales: min / n 1, max n 1 m =rotational oscillations (c.f. QM Lr momn) Polar Shapelets n=radial oscillations (c.f. QM energy) n =radial oscillations (c.f. QM energy) m=rotational oscillations (c.f. QM Lr momn) HST galaxy Image Faithful description with a few shapelet coefficients Image Compression Keep the top largest coefficients Achieve compression factors of 40-90 (for well resolved HST galaxies) Fourier Transform and Convolution Convolution with a gaussian: Basis functions are invariant under Fourier transform (up to rescaling): ~ n 1 Bn (k , ) i Bn (k , ) Convolution: h( x ) ( f g )( x ) hn Cnml f m gl ml convolution tensor (analytic) Coordinate Transformations Transformations: • translations • rotations • shears • dilatations f f ' (1 i Sˆi ) f 1 2 2 2 2 ˆ ˆ ˆ ˆ ˆ S1 2 (a1 a2 a1 a2 ) Eg: effect of shear on a galaxy image: simple operations in shapelet space Difference Shear Measurement 1 = 0.1 2 = 0.1 Shear Estimators: ˆin fn fn S inm f m Combine estimators for minimum variance ***To be replaced Shear Measurement Shear recovery with ground based simulations: (Refregier & Bacon 2000) Advantages: • All shape information used • Deconvolution recovers all available coefficients • Linear estimator noise biases are minimised • Minimum variance estimator Lensing signal is maximised • Analytic and mathematically welldefined • Stable and accurate Simulating Space-Based Images • Decompose HDF galaxies into shape components (“shapelets”) • Simulated galaxies are drawn from same parameter space • Add noise, background, PSF, shear etc as required a12 a11 • Ensures simulated images have same statistical properties as true HDF • Realistic illustration of SNAP science Massey, Refregier, Conselice & Bacon 2002 Shapelet Parameter Space -functions representing every HDF galaxy are placed into an n-dimensional parameter space, with each axis corresponding to a (polar) shapelet coefficient or size/magnitude. The PDF is: • kernel-smoothed (assume a smooth underlying PDF exists) • Monte-Carlo sampled, to synthesise new ‘fake’ galaxies. Used in sims Used in sims param space smoothing param space smoothing Real HDF Real HDF Smoothing in Shapelet Space Massey et al. (2002) Simulated SNAP images Test of the Simulations Asymmetry Blind test: run Sextractor and morphology software on HDFs and simulated images. Concentration Weak Lensing Sensitivity Galaxies per arcmin2 RMS noise for the shear per galaxy RMS noise for the shear in 1 arcmin2 cell Prospects for SNAP zS > 1.0 zS < 1.0 SNAP wide survey Rhodes et al. 2003, Massey et al. 2003, Refregier et al. 2003 SNAP will measure the evolution of the lensing power spectrum and set tight constraints on dark energy input Wiener filter Dark Matter Mapping: Space observed Wavelets Starck, Refregier & Pires 2004 E: Lensing E/B Decomposition B: systematics FIRST Radio Survey Faint Images of the Radio Sky at Twenty-cm •VLA B-array at 1.4 GHz •10,000 deg2 area (~SDSS) •Resolution of 5”.4 • 90 Sources / deg at 1 mJy •<z>~1 Becker,White Helfand (1995) White,Becker,Helfand,Gregg (1997) Snapshot survey Sparse UV sampling Simulations Same observing condition as FIRST survey FWHM Beam Input Recovered Parallel code on COSMOS Origin2000: 23 Sources, ~ 200 parameters, ~18,000 visibilities ~1.5 GB memory, ~30 sec with 10 processors Cosmological Constraints Chang, Refregier & Helfand 2004 Constraints consistent with current measurements of 8 and current knowledge of the redshifts of radio sources Cosmic Shear with SKA Square Kilometer Array: • an international project planned to be constructed in 2010 • FOV ~ 1 deg2 at 1.4 GHz, PSF ~ 0”.1 Assuming 6 month’s observation: • 540 deg2 , beam FWH ~ 0”.1 • source number density: ~100 sources arcmin-2 (~ HDF) • < z > =1 • ~ 0.5 Conclusion • Weak Lensing provides a powerful measure of largescale structure and cosmological parameters • Shapelets provides a high-precision shape measurement method required for future surveys • Other applications of shapelets: astrometry & photometry, study of galaxy morphology, de-projection, multi-color morphology shapelet web page: http://www.ast.cam.ac.uk/~rjm/shapelets.html