Cosmic shear and cosmology

Transcription

1 Cosmic shear and cosmology Overview Second-order cosmic shear statistics Shear tomography (2 1/2 D lensing) Third-order cosmic shear statistics 3D lensing Peak statistics Shear-ratio geometry test (Flexion) Weak Lensing and Cosmology 25 / 126

2 Second-order cosmic shear statistics Shear components Recall: complex shear γ = γ 1 + iγ 2 = γ exp(2iφ) is measure of an object s ellipticity Tangential and cross-component γ t = R ( γe 2iϕ) and γ = I ( γe 2iϕ) ϑ 2 ϑ 2 θ ϑ π/4 π/2 γ t γ ϑ ϑ ϕ ϑ ϑ 1 ϑ 1 Shear is polar/spin-2 quantity! Weak Lensing and Cosmology 26 / 126

3 Second-order cosmic shear statistics Shear in apertures Aperture mass: weighted convergence/shear in a circle M ap (θ) = d 2 ϑ U θ (ϑ )κ(ϑ ) = d 2 ϑ Q θ (ϑ )γ t (ϑ ), U θ is a compensated filter dϑ ϑ U θ (ϑ) = 0 Filter functions are related Q θ (ϑ) = 2 ϑ ϑ 2 dϑ ϑ U θ (ϑ ) U θ (ϑ). 0 Weak Lensing and Cosmology 27 / 126

4 Convergence and shear field κ γ N-body simulation and ray-tracing from T.Hamana

5 Second-order cosmic shear statistics U θ (ϑ) Q θ (ϑ) Û(η) { Aperture filter functions polynomial ( ) ( ) 9 πθ 1 ϑ2 1 2 θ 2 3 ϑ2 θ 2 ϑ < θ 0 { else 1 ϑ2 ϑ < θ ( 6 ϑ 2 πθ 2 θ 2 θ 2 ) 0 else Gaussian ( ) 1 2πθ 1 ϑ2 2 2θ exp 2 ( ϑ 2 4πθ exp 4 ) ϑ2 2θ 2 ( 24J 4(η) η 2 η 2 2 exp η 2 2 ( ) ϑ2 2θ 2 ) arbitrary units ϑ/θ U θ (ϑ) poly Q θ (ϑ) poly U θ (ϑ) Gauss Q θ (ϑ) Gauss Weak Lensing and Cosmology 29 / 126

6 Second-order cosmic shear statistics Second-order statistics Correlation of the shear at two points yields four quantities > 0 < 0 γ t γ t γ γ γ t γ t < 0 γt γ, γ γ t Parity conservation γ t γ = γ γ t = 0 Shear two-point correlation function (2PCF) ξ + (ϑ) = γ t γ t (ϑ) + γ γ (ϑ) ξ (ϑ) = γ t γ t (ϑ) γ γ (ϑ) Weak Lensing and Cosmology 30 / 126

7 Second-order cosmic shear statistics Relation to the power spectrum Two-point correlation function ξ + (θ) = 1 2π ξ (θ) = 1 2π dl lj 0 (lθ)p κ (l) dl lj 4 (lθ)p κ (l), Aperture-mass variance/dispersion Map (θ) 2 = 1 dl l P κ (l)û 2 (θl) 2π Top-hat-variance γ 2 (θ) = 1 πθ 2 d 2 ϑ γ(ϑ)γ (ϑ) = 1 [ 2J1 (lθ) dl l P κ (l) 2π lθ ] 2 Weak Lensing and Cosmology 31 / 126

8 Second-order cosmic shear statistics Filter functions 10 1 ξ + ξ + 2 <M ap> (polynomial) 2 <M ap> (Gaussian) < γ 2 > 0.1 filter(η) e-04 1e-05 1e η Weak Lensing and Cosmology 32 / 126

9 Second-order cosmic shear statistics Second-order shear statistics second-order shear statistics 1e-04 1e-05 1e-06 ξ + ξ - 2 <M ap> (polynomial) 2 <M ap> (Gaussian) < γ 2 > 1e θ [arcmin] M 2 ap is narrow band-pass filter of P κ localized probe ξ +, γ 2 are low-pass filter of P κ high S/N, sensitive to large scales Weak Lensing and Cosmology 33 / 126

10 Second-order cosmic shear statistics growth of structures, initial conditions P κ (l) = G(w) = 3 2 Dependence on cosmology ( ) l dw G 2 (w)p δ f K (w) ) 2 Ω wlim m dw p(w ) f K(w w) c a(w) w f K (w ) ( H0 cosmological Parameters redshift distribution of source galaxies geometrical factors Weak Lensing and Cosmology 34 / 126

11 Second-order cosmic shear statistics Parameter degeneracies 2π/l [arcmin] l 2 Pκ(l) amplitude tilt default Ω m σ8 z 0β default Ω mh = Γ n sλ l Cosmological parameters from weak lensing show high level of near-degeneracies. P κ relatively featureless because of projection and non-linear growth. Weak Lensing and Cosmology 35 / 126

12 Second-order cosmic shear statistics Cosmology from cosmic shear Probes Universe at low medium redshifts (z ). That s where dark energy is important! Probes LSS at small scales (R 0.3h 1 (θ/1 ) Mpc): non-linear & non-gaussian structure formation Independent of relation between dark & luminous matter (e.g. galaxy bias) Most sensitive to Ω m and power spectrum normalization σ 8 Complementary & independent method Weak Lensing and Cosmology 36 / 126

13 Weak lensing and cosmology Weak lensing and cosmology CFHTLS Wide σ8 Ω0.6 m const Ωm = 0.3 fixed, flat Universe: σ8 = 0.85 ± 0.06 [Hoekstra et al. 2006] Weak Lensing and Cosmology Second-order cosmic shear statistics Ωm σ8 CTIO lensing survey flat Universe [Jarvis, Jain & Bernstein 2006] 37 / 126

14 Weak lensing and cosmology Weak lensing and cosmology Second-order cosmic shear statistics Ωm w CFHTLS Wide [Hoekstra et al. 2006] Weak Lensing and Cosmology CTIO lensing survey [Jarvis, Jain & Bernstein 2006] 38 / 126

15 Second-order cosmic shear statistics Lift degeneracies Lifting near-degeneracies by combining weak lensing with other experiments (CMB, SNIa,...) shear tomography combining second- and third-order statistics Weak Lensing and Cosmology 39 / 126

16 Second-order cosmic shear statistics Scatter in σ 8 galaxy clusters cosmic shear GaBoDS cosmic shear WMAP3 [Hetterscheidt et al. 2006] Scatter in σ 8 from WL larger than error bars? Problem with systematics, e.g. calibration of shear amplitude? STEP project Weak Lensing and Cosmology 40 / 126

17 Second-order cosmic shear statistics Redshift distribution p(z)!"# Even with 4 deg 2 (CFHTLS Deep, calibrated with VVSD): cosmic variance 2 8 larger than statistical (Poisson) error [van Waerbeke et al. 2007]... until recently from HDF. Cosmic variance: wrong p(z) biases measured σ 8. Weak Lensing and Cosmology 41 / 126

18 Second-order cosmic shear statistics Determination of parameters Likelihood function (posterior) Gaussian likelihood L(d; p) = 1 (2π) n det C exp[ χ2 (d; p)/2] Log-likelihood χ 2 (d; p) = (d(p) d obs) t ( C 1 d(p) d obs) d : data vector, e.g. d i = ξ(ϑ i ), M 2 ap (θ i ) C : covariance matrix, C = dd t d d t p : vector of cosmological parameters, e.g. Ω m, σ 8, h, w... Weak Lensing and Cosmology 42 / 126

19 Second-order cosmic shear statistics The E- and the B-mode Convergence κ and shear γ are both second derivatives of the lensing potential ψ. Relation exists ( ) 1 γ κ = γ 2 = u 2 γ 1 1 γ 2 The vector u is the gradient of potential κ, therefore u = 0 Gravitational lensing produces only gradient component (E-mode). But: Measured u from data will not be curl-free due to measurement errors, systematics, noise, second-order effects, intrinsic shape correlations. Use this curl-component (B-mode) to assess data quality! Weak Lensing and Cosmology 43 / 126

20 Second-order cosmic shear statistics Separating the E- and B-mode E mode B mode mass peak mass trough Local measure for E- and B-mode: M 2 ap Remember: M ap (θ) = d 2 ϑ Q θ (ϑ)γ t (ϑ). Define: M (θ) = d 2 ϑ Q θ (ϑ)γ (ϑ). Dispersion M 2 is only sensitive to B-mode, i.e., vanishes if there is no B-mode. Weak Lensing and Cosmology 44 / 126

21 Second-order cosmic shear statistics VIRMOS survey, CFHT, 6.5 deg 2, I AB = 24.5 [van Waerbeke et al. 2001] Weak Lensing and Cosmology 45 / 126

22 Second-order cosmic shear statistics RCS survey, CFHT+CTIO, 53 deg 2, R C = 24 [Hoekstra et al. 2002] Weak Lensing and Cosmology 46 / 126

23 Second-order cosmic shear statistics VIRMOS survey, CFHT, 8.5 deg 2, I AB = 24.5 [van Waerbeke, Mellier & Hoekstra 2005] Weak Lensing and Cosmology 47 / 126

24 Second-order cosmic shear statistics 2.5e-05 2e e-05 E-mode B-mode WMAP + σ 8 =0.75 1e-05 2 <M ap >(θ) 5e-06 0 < γ 2 >(θ) -5e-06-1e e e-05 6e-05 4e-05 2e e θ [arcmin] θ [arcmin] E-mode B-mode WMAP + σ 8 =0.75 CFHTLS W3, 18 deg 2, I AB = 24.5 [Fu et al (in prep.)] Weak Lensing and Cosmology 48 / 126

25 Shear tomography (2 1/2 D lensing) Shear tomography (2 1/2 D lensing) If redshifts of source galaxies are known... Divide galaxies into i = 1... n redshift bins Measure power spectrum (shear statistics) from different bins Pκ ii and cross-spectra Pκ ij [Jain, Connnolly & Takada 2007] Different projections of LSS, different redshift ranges evolution of structure growth, dark energy evolution, lift parameter degeneracies Weak Lensing and Cosmology 49 / 126

26 Shear tomography (2 1/2 D lensing) Redshift binning Requirements Redshifts do not have to be very accurate for individual galaxy but: systematics have to be well controlled! photometric redshifts using a few (3-10) broad-band filters are sufficient (more later) Redshift bins can be broad and overlap, but distribution has to be known fairly accurately! (E.g. bias of mean z bias and dispersion σ z. Higher moments?) Small number of redshift bins sufficient, n = 2 already huge improvement Weak Lensing and Cosmology 50 / 126

27 Shear tomography (2 1/2 D lensing) zupper improvement two Improvement on parameter constraints three upper fraction Improvement from shear tomography on error of Ω Λ p α σ α f 1/2 sky Error Improvement 1 2( 1 2 ) 2( 1 4 ) 2( 1 8 ) 3( 1 3 ) 3( 1 4 ) 3( 1 8 ) Ω Λ Ω K m ν ln A [Hu 1999] Weak Lensing and Cosmology 51 / 126

28 Results on shear tomography so far... not many CFHTLS Wide Shear tomography using one band (magnitude binning) [Semboloni et al. 2006] COSMOS: 1.6, observed by HST, Spitzer, GALEX, XMM, Chandra, Subaru, VLA, VLT, UKIRT, NOAO, CTHT,... from radio to X-ray Many bands from UV to IR [Massey et al. 2007]

29 Lensing tomography with clusters CFHTLS Deep Weak lensing signal from a cluster using ugriz [Gavazzi & Soucail 2007] COSMO17 17 bands (5 broad, 12 medium) [Taylor et al. 2004]

30 Shear tomography (2 1/2 D lensing) Growth of structure COSMOS, [Massey et al. 2007] Weak Lensing and Cosmology 54 / 126

31 Third-order cosmic shear statistics Second-order shear statistics probes power spectrum P κ (l) Third-order statistics probes bispectrum B κ (l 1, l 2, l 3 ) = B κ (l 1, l 2, cos β) l 3 l 2 l 1 β Reduced bispectrum, depends on triangle configuration

32 Bispectrum: Probing the cosmic web θ k 1 2 k 1 1

33 Bispectrum: Probing the cosmic web θ k 1 1 k 1 2

34 Bispectrum: Probing the cosmic web θ k 1 2 k 1 1

35 Bispectrum: Probing the cosmic web k 1 2 θ k 1 1

36 Bispectrum: Probing the cosmic web k 1 2 θ k 1 1

37 Bispectrum: Probing the cosmic web k 1 2 θ k 1 1

38 Bispectrum: Probing the cosmic web k 1 2 θ k 1 1

39 Bispectrum: Probing the cosmic web

40 Third-order cosmic shear statistics Three-point correlation function (3PCF) γ γ t 8 components: γ t γ o γ γ t γ t γ t γ t γ t γ t γ γ t γ γ γ t γ γ t γ γ t γ γ γ t γ t γ γ γ t γ γ γ t and with respect to (some) center of triangle Natural components Γ (0), Γ (1), Γ (2), Γ (3) C = linear combinations of the γ µ γ ν γ λ [Schneider & Lombardi 2003] 3PCF has 8 (non-vanishing) components, depends on 3 quantities and is not a scalar [SL03, Takada & Jain 2003, Zaldarriaga & Scoccimarro 2003] Weak Lensing and Cosmology 57 / 126

41 Γ (0) Γ (1) y2 [arcmin] y 1 [arcmin] Γ (2) Γ (3)

42 Third-order cosmic shear statistics Flavors of 3 rd -order statistics Projected 3PCF, integrated over elliptical region [Bernardeau, van Waerbeke & Mellier 2002, 2003] [VIRMOS-DESCART] Measurement consistent with ΛCDM Weak Lensing and Cosmology 59 / 126

43 Third-order cosmic shear statistics Aperture-mass skewness M 3 ap (θ) probes convergence bispectrum B κ (l 1 1/θ, l 2 1/θ, l 3 1/θ) Generalized skewness M 3 ap (θ 1, θ 2, θ 3 ) = M ap (θ 1 )M ap (θ 2 )M ap (θ 3 ) probes bispectrum B κ (l 1 1/θ 1, l 2 1/θ 2, l 3 1/θ 3 ), cross-correlation or mode coupling of the large-scale structure on different scales [Schneider, MK & Lombardi 2005, MK & Schneider 2005] θ 1 θ 2 θ 3 Weak Lensing and Cosmology 60 / 126

44 Third-order cosmic shear statistics E- and B-mode components: M 3 ap, M ap M 2, M 2 apm, M 3 Quantities with odd power in M should vanish if shear field is parity-invariant [CTIO, Jarvis et al. 2004] [VIRMOS, Pen et al. 2003] Weak Lensing and Cosmology 61 / 126

45 Third-order cosmic shear statistics Properties of M 3 ap M 3 ap is scalar (3PCF: spin-2 and spin-6) separates E- & B-mode one can obtain M 3 ap from 3PCF M 3 ap contains same amount of information than 3PCF: 3PCF not sensitive to power on large scales Skewness of LSS (asymmetry between peaks and troughs) can be probed with aperture-mass skewness E mode mass peak mass trough Weak Lensing and Cosmology 62 / 126

46 Third-order cosmic shear statistics Third-order statistics and cosmology On small scales: Need non-linear model. E.g.: HEPT (Hyper-Extended Perturbation Theory) [Scoccimarro & Couchman 2001], halomodel Non-linear models not (yet) good enough for %-precision cosmology On large scales: Signal too small to measure? Source-lens clustering worrying (if not fatal) contamination to lensing skewness <M ap 3 (θ, θ, θ)> θ [arcmin] Ray tracings HEPT PT Weak Lensing and Cosmology 63 / 126

47 Predictions for CFHTLS Wide (very optimistic...) Γ 0.15 σ ns Ω m Ω m Ω m σ ns 1.00 ns M 2 ap M 3 ap combin Γ σ 8 Γ Fisher matrix with 5 parameters assuming flat CDM cosmology 1σ-ellipses σ(z 0 ) = 0.01 Γ = Ω m h

48 More predictions (even more optimistic...) [Takada & Jain 2004]

49 Primordial Non-Gaussianity from lensing? [Takada & Jain 2004]

50 3D lensing [Heavens 2003, Heavens et al. 2006] Principle of 3D lensing Spherical transformation of the 3D shear field, sampled at galaxy positions (ϑ i, w i ) (flat Universe) 2 ˆγ(l, k) = γ(ϑ i, w i ) j l (kw i ) exp( iϑl) π i Comoving distance w i from (photometric) redshift z ph and fiducial cosmological model Log-Likelihood χ 2 = l,k,k [ ln det Cl (k, k ) + ˆγ t (l, k) C 1 l (k, k ) ˆγ(l, k) ] assuming different l-modes are uncorrelated. Weak Lensing and Cosmology 67 / 126

51 3D lensing Covariance matrix is sum of signal and noise term, C = S + N Note: The data vector has zero expectation, ˆγ = 0! All information is contained in the (signal) covariance matrix C l which depends on the 3D power spectrum P δ. [C.f. CMB anisotropies] Applied to COMBO-17 survey (proof of concept) COMBO-17 5 broad-band filters (UBVRI) + 17 medium-band filters for excellent photo-zs 4 selected fields each using MPG/ESO 2.2m, R = 24 (for lensing) Weak Lensing and Cosmology 68 / 126

52 3D lensing 3D lensing: first results Solid: Dashed: 3D lensing (2 fields) 2D lensing (3 fields) COMBO-17 [Kitching et al. 2007] Weak Lensing and Cosmology 69 / 126

53 Peak statistics Peak statistics A shear-selected sample of halos (M > M ) can be used to constrain cosmological parameters by comparing to theoretical mass function n(m, z). Galaxy clusters: matter density, normalization σ 8, dark energy evolution and BAO can be measured Shear might be better proxy for mass than richness, σ v, L X, T X, SZ signal,.... Independent of morphology, dynamical state, galaxy formation. CDM N-body simulations for calibration [Hennawi & Spergel 2005] Weak Lensing and Cosmology 70 / 126

54 Peak statistics Detecting peaks Measure filtered γ t in annuli M(ζ, θ) = d 2 ϑ Q θ (ϑ)γ t (ϑ ϑ), Look for peaks in this M -map higher than some S/N-threshold ν. Choices for Q: compensated filter (M ap ), lower limit on mass matched filter (Q γ t (NF W )), high efficiency Weak Lensing and Cosmology 71 / 126

55 5 P(zd) zd 2.0 P(zd) zd 2.0 P(zd) zd 4 P(zd) zd

56 Peak statistics Main difficulty: Noise (intrinsic ellipticty and LSS/chance projections) increases n peak (ν)! Efficiency ε = n halos /n peaks 1 (from simulations) because of many false positives The higher ν, the higher ε, but the lower the completeness. Weak Lensing and Cosmology 73 / 126

57 n(e) [deg -2 ] No Tomography Noisy M ap Gaussian NFW Trunc NFW Noiseless 3 Redshift Bins e [efficiency] e [efficiency] 1.0 M h 1 M < z < completeness(z) z-bins no tomography ε 60%(S/N 3.5) completeness(m) ε 75%(S/N 4.5) z M [h 1 M sun ]

58 Peak statistics Cosmology with peak statistics Problem: Cannot just compare n peak with theoretical mass function n(m, z) because of false positives. Optical/X-ray follow-up to confirm galaxy cluster: introduces bias again, back to square one! Compare with n peak from simulations. To fit cosmological parameters, need a grid of N-body simulations, expensive! But: Correlations between peaks not needed, simple and fast simulations maybe sufficient Observations: Shear-selected samples from DLS [Wittman et al. 2006], GaBoDS [Schirmer et al. 2007, Maturi et al. 2007], BLOX [Dietrich et al. 2007] Weak Lensing and Cosmology 75 / 126

59 Peak statistics Cosmic shear & peak statistics Question: Can combining cosmic shear with peak statistics improve parameters constraints? Isn t it not just sampling of the high-end part of the power spectrum? Answer: No! [Takada & Bridle 2007] FIG. 12: Similar to the previous plot, but for the lensing-based cluster co clusters having the lensing signal greater than the threshold are included Weak Lensing and Cosmology in the upper panel of each plot is same as that in Fig. 11. A76 similar / 126 impro

60 Shear-ratio geometry test Shear-ratio geometry test [Jain & Taylor 2003, Taylor et al. 2007] The principle: The variation of the weak lensing signal with redshift around massive foreground objects depends solely on the angular diameter distances. Cross-correlation between tangential shear and halo (galaxy cluster) w t,h (θ) = 1 wlim dw ( ) l 2π 0 f K (w) n f(w)g(w) dl l P δh 0 f K (w), w J 2 (θl) [ c.f. ξ ± (θ) = 1 2π dw G 2 (w) ( ) l dl l P δ f K (w), w ] J 0,4 (θl) Weak Lensing and Cosmology 77 / 126

61 Shear-ratio geometry test Shear-ratio geometry test Lens efficiency G(w) = 3 2 ( H0 for a single source redshift z: w w(z) c ) 2 Ω wlim m dw p(w ) f K(w w) a(w) w f K (w ) G(w(z l )) f K[w(z) w(z l )] a[w(z l )]f K [w(z)] Plus single lens redshift z l : f K [w(z) w(z l )] w t,h (θ, z) f K [w(z)]a[w(z l )]f K [w(z)] dl l P δ [l, w(z l )]J 0,4 (θl) Weak Lensing and Cosmology 78 / 126

62 Shear-ratio geometry test Shear-ratio geometry test Ratio of shear at two source redshifts w t,h (z 1 ) w t,h (z 2 ) = f K[w(z 1 ) w(z l )]/f K [w(z 1 )] f K [w(z 2 ) w(z l )]/f K [w(z 2 )] is independent of halo details (mass, profile,...) and angular distance θ. Clean measure of angular diameter distance as functions of redshift geometry of the Universe. Simple signal-to-noise estimate: Assume only shot noise from intrinsic ellipticities: S N = γ ( rms Ng 6 σ ɛ n g arcmin 2 ) A 1/2 deg 2 Weak Lensing and Cosmology 79 / 126

63 Shear-ratio geometry test Shear-ratio geometry test Advantages of this method High shear values (1% 10%) around clusters First-order in γ, less sensitive to PSF effects, less stringent imaging requirements Detailed error analysis must include shot-noise photo-z errors contribution from large-scale structure (cosmic shear): First detection using three clusters (A901a, A901b, A902) in COMBO-17, γ t (θ, z) fitted to SIS profile [Kitching et al. 2007]. Weak Lensing and Cosmology 80 / 126