Observational constraints and cosmological parameters Antony Lewis Institute of Astronomy, Cambridge http://cosmologist.info/

Cosmological parameters CMB Polarization Baryon oscillations Weak lensing Galaxy power spectrum Cluster gas fraction Lyman alpha etc… + Cosmological parameters

Bayesian parameter estimation Can compute P( {ө} | data) using e.g. assumption of Gaussianity of CMB field and priors on parameters Often want marginalized constraints. e.g. BUT: Large n-integrals very hard to compute! If we instead sample from P( {ө} | data) then it is easy: Use Markov Chain Monte Carlo to sample

Markov Chain Monte Carlo sampling Metropolis-Hastings algorithm Number density of samples proportional to probability density At its best scales linearly with number of parameters (as opposed to exponentially for brute integration) Public WMAP3-enabled CosmoMC code available at http://cosmologist.info/cosmomc (Lewis, Bridle: astro-ph/0205436) also CMBEASY AnalyzeThis

WMAP1 CMB data alone color = optical depth Samples in 6D parameter space WMAP1 CMB data alone color = optical depth

Background parameters and geometry Energy densities/expansion rate: Ωm h2, Ωb h2,a(t), w.. Spatial curvature (ΩK) Element abundances, etc. (BBN theory -> ρb/ργ) Neutrino, WDM mass, etc… Local parameters When is now (Age or TCMB, H0, Ωm etc. ) Astrophysical parameters Optical depth tau Cluster number counts, etc..

General perturbation parameters -isocurvature- Amplitudes, spectral indices, correlations…

CMB Degeneracies WMAP 1 WMAP 3 All TT ns < 1 (2 sigma)

Main WMAP3 parameter results rely on polarization

CMB polarization Page et al. No propagation of subtraction errors to cosmological parameters?

WMAP3 TT with tau = 0.10 ± 0.03 prior (equiv to WMAP EE) Black: TT+prior Red: full WMAP

ns < 1 at ~3 sigma (no tensors)? Rule out naïve HZ model

Secondaries that effect adiabatic spectrum ns constraint SZ Marginazliation Spergel et al. Black: SZ marge; Red: no SZ Slightly LOWERS ns

CMB lensing For Phys. Repts. review see Lewis, Challinor : astro-ph/0601594 Theory is robust: can be modelled very accurately

CMB lensing and WMAP3 Black: with red: without - increases ns not included in Spergel et al analysis opposite effect to SZ marginalization

LCDM+ Tensors No evidence from tensor modes is not going to get much better from TT! ns < 1 or tau is high or there are tensors or the model is wrong or we are quite unlucky So: ns =1

CMB Polarization WMAP1ext WMAP3ext Current 95% indirect limits for LCDM given WMAP+2dF+HST+zre>6 WMAP1ext WMAP3ext Lewis, Challinor : astro-ph/0601594

Polarization only useful for measuring tau for near future Polarization probably best way to detect tensors, vector modes Good consistency check

Matter isocurvature modes Possible in two-field inflation models, e.g. ‘curvaton’ scenario Curvaton model gives adiabatic + correlated CDM or baryon isocurvature, no tensors CDM, baryon isocurvature indistinguishable – differ only by cancelling matter mode Constrain B = ratio of matter isocurvature to adiabatic -0.53<B<0.43 -0.42<B<0.25 WMAP1+2df+CMB+BBN+HST WMAP3+2df+CMB Gordon, Lewis: astro-ph/0212248

Degenerate CMB parameters Assume Flat, w=-1 WMAP3 only Use other data to break remaining degeneracies

Galaxy lensing Assume galaxy shapes random before lensing In the absence of PSF any galaxy shape estimator transforming as an ellipticity under shear is an unbiased estimator of lensing reduced shear Calculate e.g. shear power spectrum; constrain parameters (perturbations+angular at late times relative to CMB) BUT - with PSF much more complicated - have to reliably identify galaxies, know redshift distribution - observations messy (CCD chips, cosmic rays, etc…) - May be some intrinsic alignments - not all systematics can be identified from non-zero B-mode shear - finite number of observable galaxies

CMB (WMAP1ext) with galaxy lensing (+BBN prior) CFTHLS Contaldi, Hoekstra, Lewis: astro-ph/0302435 Spergel et al

SDSS Lyman-alpha white: LUQAS (Viel et al) SDSS (McDonald et al) The Lyman-alpa plots I showed were wrong SDSS, LCDM no tensors: ns = 0.965 ± 0.015 s8 = 0.86 ± 0.03 ns < 1 at 2 sigma LUQAS

Conclusions MCMC can be used to extract constraints quickly from a likelihood function CMB can be used to constrain many parameters Some degeneracies remain: combine with other data WMAP3 consistent with many other probes, but favours lower fluctuation power than lensing, ly-alpha ns <1, or something interesting No evidence for running, esp. using small scale probes