Combining Lensing with SNIa and CMB Sampling the Posterior Martin Kilbinger IAP Paris Martin Kilbinger IAP Paris Toronto, 13 June 2008 Upcoming lensing surveys: Beyond the obvious Toronto, 13 June 2008 Upcoming lensing surveys: Beyond the obvious

Overview CFHTLS Weak lensing data [w/Fu, Semboloni, Hoekstra, van Waerbeke, Heymans, Mellier et al.] SNLS SNIa data [w/Guy, Astier, Benabed, Mellier et al.] CMB data [w/Benabed, Prunet] A new method to sample the posterior: Adaptive importance sampling [w/Wraith, Benabed, Cappé, Robert et al.]

Lensing: CFHTLS Wide T deg 2 (effective area 35 deg 2 ) 21.5 ≤ i AB ≤ ⋅ 10 6 galaxies for lensing, n eff =13.3 arcmin -2 Largest separation 470’ 8 degrees CFHTLS press release 21/02/2008 [Fu, Semboloni, Hoekstra, MK et al. 2008]

Shear correlations Shear correlation measured between 1 and 230 arc minutes B-modes consistent with zero on most scales (but: feature at 100’) Aperture-massCorrelation functionTop-hat variance

Cross-checks Shape measurements: KSB [Kaiser, Squires & Broadhurst 1995, Hoekstra et al. 1998] Calibrated with STEP simulations, ≤ 3% bias [Heymans et al. 2006, Massey et al. 2007] Systematics average out globally Problems for individual fields: PSF residuals, B- modes [van Waerbeke et al. in prep.] Two pipelinesW1, W2, W3PSF residuals

Current problems W1+1+1 L W1+1+3 K W1+2+2 J W1+3+2 K Variations between CFHTLS-Wide pointings Systematics in individual images average out over whole survey Shear does not increase with redshift as expected for z>1 [Curesy: L. Fu]

Source redshifts CFHTLS Deep: 318,776 galaxies with photo-zs from u*g’r’i’z’, calibrated with 3,241 VVDS spectra [Ilbert et al. 2006] Soon: photo-zs for T0004, 35 deg2 in 5 bands [Coupon, Ilbert, Arnouts, MK et al. in prep.] z-distribution for lensing Error bars: photo-z errors, Poisson, cosmic variance [Benjamin et al. 2007, Fu et al. 2008]

Constraints on Ω m and σ 8 Large linear scales not sensitive to non-linear and baryonic physics [Fu et al. 2008]

SNIa: SNLS sample 1st year data: 71 distant + 44 nearby SNIa Imaging: CFHTLS (4 deg 2 Deep fields) Spectroscopy: VLT, Gemini, Keck 3rd year data in prep. z=0.358 [Astier et al. 2006] Fits to two or more bands simultaneously using SALT [Guy et al. 2005] Returns rest-frame magnitude, stretch s and color c

SNIa: Cosmology and Systematics Distance estimator for i th SN: Take into account change in as function of photometric calibration errors: Calculate for zero-point shifts, add zero-points to Fit luminosity distance together with to data

Influence of zero-points flat Solid: Δzp=0. Dashed: Δzp≠0 ΛCDM (w=-1)

Bias from zero-points ΛCDM (w=-1) flat with sys no sys Bias 20% of error bars with sys no sys

Correlation between Systematic and Cosmological Parameters Correlation between M and ΔB, ΔV: = ΔU = ΔB = ΔV [MK et al. in prep.]

CMB: WMAP5 C l ’s: CAMB WMAP5 likelihood [Dunkley et al. 2008], no SZ (yet), no lensing

Lensing + N(z) + SNIa Model: non-flat wCDM

Sampling the posterior We need to calculate posterior = likelihood × prior some function e.g. parameter mean, marginals, evidence Problem: High-dimensional integral MCMC: Points of a Markov Chain, converges to the posterior. Asymtotically

Importance sampling (IS) importance weights Proposal distribution, easy to sample from e.g. mixture of Gaussians/student-t/... Proposal should cover posterior

Importance sampling: works poorly if proposal too far from posterior Solution: Adaptive IS = Population MonteCarlo (PMC) [Cappé et al. 2004] Iteratively improve proposal, update G i to G i+1 Updating mean, co- variance, component weights Population MonteCarlo (PMC)

Stop iterations when proposal ‘close enough’ to posterior Kullback-Leibler divergence distance between distributions and Calculate K using importance sampling Perplexity Maximize E[p] to get updated proposal

[Wraith, MK et al. in prep]

10dim posterior PMC MCMC

PMCMCMC Solid: PMC Dashed: MCMC Lensing + n(z) + SNIa

Components of the mixture proposal Component means

Summary: PMC vs. MCMC Very high dimensions challenging for all methods But: works for low-l CMB likelihood (n=40) [Benabed et al. in prep.] MCMCPMC Points correlated → thin-out, loose lots of points All points uncorrelated Acceptance rate ≈ 20%All points accepted (but: weights!) Serial (Markov chain) Parallelizable (independent samples), (N cpu ) speed-up Optimal: replace posterior by proposal Random starting pointNeed decent starting guess