Weak Lensing Tomography

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1 Weak Lensing Tomography m tot =0.2eV windows z Wayne Hu Fermilab, October 2002

2 Collaborators Chuck Keeton Takemi Okamoto Max Tegmark Martin White

3 Collaborators Chuck Keeton Takemi Okamoto Max Tegmark Martin White Microsoft /~whu/presentations/ferminu.pdf

4 Massive Neutrinos Relativistic stresses of a light neutrino slow the growth of structure Neutrino species with cosmological abundance contribute to matter as Ω ν h 2 = m ν /94eV, suppressing power as P/P 8Ω ν /Ω m P P 0.6 ( mtot ev ) min. ~3% (atmospheric)

5 Absolute effect is large! Galaxy Surveys SDSS BRG P(k) (arbitrary norm.) m v = 0 ev m v = 1 ev nonlinear scale W. Hu Feb k (h Mpc -1 )

6 Galaxy Surveys But because of bias we can only use the shape information not the much larger overall suppression Current constraints from 2dF suggest m ν < 1.8eV SDSS BRG P(k) (arbitrary norm.) m v = 0 ev m v = 1 ev nonlinear scale W. Hu Feb k (h Mpc -1 )

7 Massive Neutrinos Current data from 2dF galaxy survey indicates m ν <1.8eV assuming a ΛCDM model with parameters constrained by the CMB. Elgaroy et al (2002) dF P(k) (h-1 Mpc) decorrelation by Tegmark, Hamilton, Xu (2002) k (h Mpc-1)

8 Example of Weak Lensing Toy example of lensing of the CMB primary anisotropies Shearing of the image

9 Lensing Observables Image distortion described by Jacobian matrix of the remapping A = 1 κ γ 1 γ 2 γ 2 1 κ + γ 1, where κ is the convergence, γ 1, γ 2 are the shear components

10 Lensing Observables Image distortion described by Jacobian matrix of the remapping A = 1 κ γ 1 γ 2 γ 2 1 κ + γ 1, where κ is the convergence, γ 1, γ 2 are the shear components related to the gravitational potential Φ by spatial derivatives ψ ij (z s ) = 2 zs 0 dz dd dz D(D s D) D s Φ,ij, ψ ij = δ ij A ij, i.e. via Poisson equation κ(z s ) = 3 2 H2 0Ω m zs 0 dz dd dz D(D s D) D s δ/a,

11 Gravitational Lensing by LSS Shearing of galaxy images reliably detected in clusters Main systematic effects are instrumental rather than astrophysical Colley, Turner, & Tyson (1996) Cluster (Strong) Lensing:

12 Cosmic Shear Data Shear variance as a function of smoothing scale compilation from Bacon et al (2002)

13 Shear Power Modes Alignment of shear and wavevector defines modes ε β

14 Shear Power Spectrum Lensing weighted Limber projection of density power spectrum ε shear power = κ power εε l(l+1)cl /2π k=0.02 h/mpc zs Linear Non-linear 10 7 Limber approximation l zs=1 Kaiser (1992) Jain & Seljak (1997) Hu (2000)

15 COMBO-17 Brown et al (2002) Recent Measurement h Mpc 5 h Mpc h Mpc σ 8 = (0.74 ± 0.09) A = 1.25deg 2 z m = 0.85 ± 0.05 ( Ωm 0.3 ) 0.5 iterated quadratic estimator a la Bond, Jaffe, Knox (1998); Hu & White (2001)

16 PM Simulations Simulating mass distribution is a routine exercise Convergence Shear 6 6 FOV; 2' Res.; h 1 Mpc box; h 1 kpc mesh; M White & Hu (1999)

17 Degeneracies All parameters of initial condition, growth and distance redshift relation D(z) enter Nearly featureless power spectrum results in degeneracies 10 4 εε l(l+1)cl /2π l zs=1

18 Degeneracies All parameters of initial condition, growth and distance redshift relation D(z) enter Nearly featureless power spectrum results in degeneracies T CMB MAP Planck l (multipole) Combine with information from the CMB: complementarity (Hu & Tegmark 1999) Crude tomography with source divisions (Hu 1999; Hu 2001) Fine tomography with source redshifts (Hu & Keeton 2002; Hu 2002)

19 Error Improvement Error improvements over MAP/CMB with a 1000 deg 2 survey 100 Error Improvement MAP Ω m h 2 Ω b h 2 m ν Ω Λ Ω K τ n S T/S A Hu & Tegmark (1999)

20 Crude Tomography Divide sample by photometric redshifts g i (D) ni(d) (a) Galaxy Distribution 1 2 (b) Lensing Efficiency 2 Power deg 2, z med = D l Hu (1999)

21 Crude Tomography Divide sample by photometric redshifts Cross correlate samples g i (D) ni(d) Order of magnitude increase in precision even after CMB breaks degeneracies Hu (1999) (a) Galaxy Distribution (b) Lensing Efficiency D Power deg 2, z med = l

22 Efficacy of Crude Tomography Error improvements over MAP/CMB with a 1000 deg 2 survey Error Improvement MAP + photo-z no-z 1 MAP Ω m h 2 Ω b h 2 m ν Ω Λ Ω K τ n S T/S A Hu & Tegmark (1999); Hu (2000)

23 Dark Energy & Tomography Both CMB and tomography help lensing provide interesting constraints on dark energy 2 MAP no z 1 w deg Ω DE l<3000; 56 gal/deg2 Hu (2001)

24 Dark Energy & Tomography Both CMB and tomography help lensing provide interesting constraints on dark energy 2 1 MAP no z 3 z w deg Ω DE l<3000; 56 gal/deg2 Hu (2001)

25 Dark Energy & Tomography Both CMB and tomography help lensing provide interesting constraints on dark energy 2 1 MAP no z 3 z w deg Ω DE l<3000; 56 gal/deg2 Hu (2001)

26 DarkSector and Radial Information Much oftheinformationonthe darksector ishiddeninthe temporal or radial dimension Evolution of growth rate (dark energy pressure slows growth) Evolution of distance-redshift relation Lensing is inherently two dimensional: all mass along the line of sight lenses Tomography implicitly or explicitly reconstructs radial dimension with source redshifts Photometricredshift errors currently z <0.1out toz~1and allow for fine tomography

27 Tomography in Practice Localization and selection of clusters γ t z phot z phot Wittman et al. (2001; 2002)

28 Hidden in Noise Derivatives of noisy convergence isolate radial structures (a) Density 0.6 κ (b) Convergence z Hu & Keeton (2001)

29 Fine Tomography Convergence projection of = δ/a for each z s κ(z s ) = 3 2 H2 0Ω m zs 0 dz dd dz D(D s D) D s,

30 Fine Tomography Convergence projection of = δ/a for each z s κ(z s ) = 3 2 H2 0Ω m zs 0 dz dd dz Data is linear combination of signal + noise d κ = P κ s + n κ, D(D s D) D s, [P κ ] ij = { 3 2 H2 0Ω m δd j (D i+1 D j )D j D i+1 D i+1 > D j, 0 D i+1 D j,

31 Fine Tomography Convergence projection of = δ/a for each z s κ(z s ) = 3 2 H2 0Ω m zs 0 dz dd dz Data is linear combination of signal + noise d κ = P κ s + n κ, D(D s D) D s, [P κ ] ij = { 3 2 H2 0Ω m δd j (D i+1 D j )D j D i+1 D i+1 > D j, 0 D i+1 D j, Well-posed (Taylor2002) but noisy inversion (Hu &Keeton2002) Noise properties differ from signal properties optimal filters

32 Fine Tomography Tomography can produce direct 3D dark matter maps, but realistically only broad features (Hu & Keeton 2002) Radial density field z

33 Fine Tomography Tomography can produce direct 3D dark matter maps, but realistically only broad features (Taylor 2002; Hu & Keeton 2002) Radial density field Wiener reconstruction z

34 Fine Tomography Tomography can produce direct 3D dark matter maps, but realistically only broad features (Taylor 2002; Hu & Keeton 2002) Radial density field Wiener reconstruction z

35 Fine Tomography Tomography can produce direct 3D dark matter maps, but realistically only broad features (Hu & Keeton 2002) Radial density field Wiener reconstruction z

36 Fine Tomography Tomography can produce direct 3D dark matter maps, but realistically only broad features (Hu & Keeton 2002) Radial density field Wiener reconstruction z

37 Growth Function Localized constraints (fixed distance-redshift relation) sq. deg m tot =0.2eV windows z Hu (2002) 3 degenerate mass ν's assumed e.g. Beacom & Bell (2002)

38 Dark Energy Density Localized constraints (with cold dark matter) sq. deg ρ DE /ρ cr w= windows z Hu (2002)

39 Dark Energy Parameters Three parameter dark energy model (Ω DE, w, dw/dz=w') none -0.9 σ(w')=0 none w' σ(ω DE )=0.01 w sq. deg Ω DE Ω DE Hu (2002)

40 Lensing of a Gaussian Random Field CMB temperature and polarization anisotropies are Gaussian random fields unlike galaxy weak lensing Average over many noisy images like galaxy weak lensing

41 Lensing by a Gaussian Random Field Mass distribution at large angles and high redshift in in the linear regime Projected mass distribution (low pass filtered reflecting deflection angles): 1000 sq. deg rms deflection 2.6' deflection coherence 10

42 Lensing in the Power Spectrum Lensing smooths the power spectrum with a width l~60 Convolution with specific kernel: higher order correlations between multipole moments not apparent in power Power lensed unlensed power rms deflection 2.6' coherence 10 Seljak (1996); Hu (2000) l

43 Reconstruction from the CMB Correlation between Fourier moments reflect lensing potential κ = 2 φ x(l)x (l ) CMB = f α (l, l )φ(l + l ), where x temperature, polarization fields and f α is a fixed weight that reflects geometry Each pair forms a noisy estimate of the potential or projected mass - just like a pair of galaxy shears Minimum variance weight all pairs to form an estimator of the lensing mass

44 Ultimate (Cosmic Variance) Limit Cosmic variance of CMB fields sets ultimate limit Polarization allows mapping to finer scales (~10') mass temp. reconstruction EB pol. reconstruction 100 sq. deg; 4' beam; 1µK-arcmin Hu & Okamoto (2001)

45 Matter Power Spectrum Measuring projected matter power spectrum to cosmic variance limit across whole linear regime 0.002< k < 0.2 h/mpc Linear P P deflection power 0.6 ( mtot ev ) Hu & Okamoto (2001) "Perfect" L σ(w) 0.06

46 Matter Power Spectrum Measuring projected matter power spectrum to cosmic variance limit across whole linear regime 0.002< k < 0.2 h/mpc Linear deflection power "Perfect" Planck Hu & Okamoto (2001) L σ(w) 0.06; 0.14

47 Summary Gravitational lensing is the only direct probe of the dark sector: composition of dark matter: massive neutrinos nature of the dark energy: scalar field? Λ?

48 Summary Gravitational lensing is the only direct probe of the dark sector: composition of dark matter: massive neutrinos nature of the dark energy: scalar field? Λ? With sources distributed in redshift, tomography possible Coarse radial resolution sufficient for recoving linear growth rate dark energy density evolution Requires good photometric redshifts, elimination of systematics, avoidance of intrinsic alignment contamination

49 Summary Gravitational lensing is the only direct probe of the dark sector: composition of dark matter: massive neutrinos nature of the dark energy: scalar field? Λ? With sources distributed in redshift, tomography possible Coarse radial resolution sufficient for recoving linear growth rate dark energy density evolution Requires good photometric redshifts, elimination of systematics, avoidance of intrinsic alignment contamination CMB provides ultimate high-z source for tomography; precision neutrino constraints in principle possible

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