1. Constraining cluster abundances using weak lensing Håkon Dahle Institute of Theoretical Astrophysics, University of Oslo

2. Cluster abundances Cluster mass function n(>M, z) The number density of clusters above a certain mass threshold is sensitive to:   m - The average mass density of the universe  8 - The rms of density fluctuations (using a spherical tophat filter of radius 8h -1 Mpc) m  - neutrino mass w - equation of state parameter of dark energy (e.g. cosmological constant: w = -1) Complementary to CMB, SN Ia

3. Dark energy with w=-1 No dark energy Evolution of the cluster population (from simulations) (Borgani et al. 2003) Time

4. Requirements for “precision cosmology” 1.Precise mass measurements/ calibration of mass-observable relation 2.Well-understood selection criteria (survey volume, mass limit) 

5. A new approach to an old problem… Most previous studies use the X-ray temperature function (XTF) or the X-ray Luminosity function (XLF) to constrain n(>M) This requires good knowledge of the M-T x or M-L x relation (and the scatter around these!!) Calibrate M-T x using lensing masses ? Here (for the first time!) we derive n(>M) from lensing directly (i.e., not via XLF or XTF) X-ray data only enter through the cluster selection

6. Weak lensing survey of X-ray luminous clusters Initial data set: 38 clusters Dahle et al. 2002 Current data set: 53 clusters From RASS-based samples of X-ray luminous clusters Ebeling et al. 1996,1998,2000; Boehringer et al. 2000; Briel & Henry 1993 L X > 1.2x10 45 h 50 -2 erg/s, corresponds to M 180 > 7.5x10 14 h -1 M sun Complete, volume-limited sub-sample of 36 (e)BCS clusters (0.l5 0 o, |b| > 20 o )

7. Weak lensing cluster survey 2.56m NOT + UH2.2m

8. M 180 estimate from lensing Observable: reduced shear g T =  T  averaged in radial bins Fit to NFW profile with concentration parameter predicted by Bullock et al. (2001)  M 180c Significant extrapolation is required for ~2/3 of the clusters (those not observed with UH8k camera)

9. Projection effects Effect of correlated structures Metzler, White & Loken (2001) estimate M obs /M true dispersion of 0.26 about the mean, tail towards high M obs /M true Clowe, De Lucia & King (2004) find no net bias: = 1, when fitting the radial shear profile out to the virial radius Effect of uncorrelated structures Foreground and background structures do not produce a net bias, but add ~1.0x10 14 h -1 M sun to the mass uncertainty

10. L x cutoff + scatter  soft mass cutoff. Probablility of including cluster of mass M 180c ? Observed cumulative mass function Strong Incompleteness

11. M-L x normalization and scatter Weak lensing masses for 50 clusters; L x from RASS (Solid: fixed slope; dashed: arbitrary slope)

12. The mass-luminosity relationship evolution parameter Best fit slope and normalization from 50 clusters with weak lensing masses: Best fit normalization when fixing slope to theoretical value (  =0.75): Luminosity cutoff limit L X > 1.2x10 45 h 50 -2 erg/s  corresponds to mass cutoff

13. Procedure: Account for selection effects: - BCS/eBCS completeness estimate - Probability of including a cluster of intrinsic mass M 180c Only used clusters well above mass cutoff ( M 180c > 10 15 h -1 M sun ) Account for observed uncertainties: - Convolve theoretical mass function with set of observed uncertainties in M 180c Contribution to these from 2D projection effects should be better understood w.r.t. bias and scatter Include errors from cosmic variance Fit to theoretical/simulated mass function (Sheth & Tormen 1999; Warren et al. 2005)

14.  m -  8 from lensing-based cluster mass function  WMAP + SDSS WMAP (solid  dash  From fit to mass function predicted from simulations by Warren et al. (2005)    m 

15. Summary - First cluster mass function directly from weak lensing - Avoids the problem with M-T x calibration - We find  8 (  m =0.3) = (0.78+/-0.05) Future: 1. Combination with similar cluster sample at higher z (e.g. MACS)  evolution of mass fn. constrains w 2. More clusters with wide-field data (reduces M180 uncertainty by ~2) 3. Lensing measurements of SZ clusters (from Planck etc.)