Statistics of the Weak-lensing Convergence Field

Sheng Wang

Brookhaven National Laboratory

Columbia University

Collaborators:

Zoltán Haiman, Morgan May, and John Kehayias

Outline

• Shear-Selected Galaxy Clusters

• Projection Effects

• Alternative Method

• Systematic Errors

• Results and Conclusions

Cluster of Galaxies

• Abundance of galaxy clusters (redshift evolution) is

exponentially sensitive to matter density fluctuations thus

the growth of structure.

• X-ray or Sunyaev-Zel’dovich effect (SZE)

• Weak gravitational lensing (WL)

• Mass-observable relation is one potential problem.

• Self-calibration Majumdar & Mohr (2003)

• Scatter and bias Lima & Hu (2005)

Shear-Selected Galaxy Clusters

• WL — coherent distortion of background galaxy images

• Depends on gravity only (“cleanest” technique)

• Automatic mass estimates

• Projection Effects

• Efficiency — false detections

• Completeness — missing clusters

White, van Waerbeke, & Mackey (2002)Hennawi & Spergel (2004)

Hamana, Takada, & Yoshida (2004)

Projection Effects

WL is sensitive to all masses along the line of sight.

Hennawi & Spergel (2005)

Projection!

Press-Schecter Formalism

• Original P-S formalism

• Primordial matter density (δ) field — Gaussian p.d.f.

• Spherical collapse — δc ~ 1.69

• One-parameter family — σM

• Universal mass function (N-body simulation)

• ΛCDM, τCDM models Jenkins et al. (2001)

• w-dependence Linder & Jenkins

(2003)

• accuracy: 10% level

• Analogies

• 3D matter density field — 2D convergence (κ) field

• δc — S/N threshold

• Jenkins et al. mass function — universal p.d.f. for κ(κmin and σκ)

Valageas (2000)Munshi & Jain (2000)

Wang, Holz, & Munshi (2002)

Alternative Approach

Flat ΛCDMΩm=0.3zs = 1.0

One-point p.d.f. tail

• Fractional area of high S/N points

• Projection effects incorporated

Shear → Convergence

• Reconstruction of “mass map” (WL regime)

• Tangential shear (linear) Kaiser & Squires (1993)

• Maximum likelihood Bartelmann et al.

(1996)

• Intrinsic ellipticity noise

Gaussian random field (KS/maximum likelihood)

van Waerbeke (2000)

Systematic Errors

• Reduced shear (direct observable)

high κ — non-linear inversion Seitz & Schneider

(1995)

• Universality — Stable-clustering ansatz

valid for tail? (work in progress looking at simulations)

• Baryon effects

cooling — different density distribution

• Intrinsic ellipticity noise

Intrinsic ellipticity alignment / shear-ellipticity alignment?

Comparison of Technique

• Vs. shear-shear correlation (tomography)

• Simple one-point statistics yet extra information

• Different systematic errors

• Vs. number counts of galaxy clusters

• Closely related (galaxy clusters “mean” high S/N)

• Projection effects included as signals

Fisher Matrix

Covariance matrix estimated using log-normal approximation

(work in progress to estimate it from simulations)

ji

jji

i

p

zA

p

zAF

, ,),)(,(

1- ),()(

),(

Cov

• Formalism

• Background galaxy redshift bins (i, j)

• Signal-to-noise thresholds (μ,ν)

Fractional Area

ResultsLSST-like WL survey performed by a ground based telescope.

Sky coverage: 18000 deg2 ;

Background galaxies: ~ 50 /arcmin2.

Three background galaxy redshift bins (zs ~ 0.6, 1.1, 1.9)

with S/N thresholds = 2.0, 2.5, 3.0

+ future CMB anisotropy measurements (Planck):

Δ(w0) ~ 0.03, Δ(wa) ~ 0.1;

Constraints from CMB alone:

Δ(w0) ~ 0.3, Δ(wa) ~ 1.

Constraints from clusters:

Δ(w0) ~ 0.03, Δ(wa) ~ 0.09.

Conclusion

• Future galaxy cluster surveys using WL, such as LSST, will suffer from the projection effects when searching for clusters. To determine the selection function to ~ N -1/2 will be challenging.

• We propose an alternative, more robust one-point statistic using points in mass maps with high signal-to-noise. Compared with conventional statistics, they contain extra information and suffer from different systematics.

• This statistic, combined with future CMB anisotropy measurements, such as Planck, can place constraints on cosmological parameters, such as the evolution of dark energy, that are comparable to those from clusters.