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Observational constraints and cosmological parameters Antony Lewis Institute of Astronomy, Cambridge

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

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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

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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 (Lewis, Bridle: astro-ph/ ) also CMBEASY AnalyzeThis

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

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Local parameters When is now (Age or T CMB, H 0, Ω m etc. ) Background parameters and geometry Energy densities/expansion rate: Ω m h 2, Ω b h 2,a(t), w.. Spatial curvature (Ω K ) Element abundances, etc. (BBN theory -> ρ b /ρ γ ) Neutrino, WDM mass, etc… Astrophysical parameters Optical depth tau Cluster number counts, etc..

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General regular perturbation Scalar Adiabatic (observed) Matter density Cancelling matter density (unobservable in CMB) Neutrino density (contrived) Neutrino velocity (very contrived) Vector Neutrino vorticity (very contrived) Tensor Gravitational waves General perturbation parameters -isocurvature- Amplitudes, spectral indices, correlations…

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

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Main WMAP3 parameter results rely on polarization

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

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

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

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Black: SZ marge; Red: no SZSlightly LOWERS n s SZ Marginazliation Spergel et al. Secondaries that effect adiabatic spectrum n s constraint

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

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

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

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Current 95% indirect limits for LCDM given WMAP+2dF+HST+z re >6 CMB Polarization Lewis, Challinor : astro-ph/ WMAP1extWMAP3ext

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

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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 Gordon, Lewis: astro-ph/ WMAP3+2df+CMB -0.53**
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Assume Flat, w=-1 WMAP3 only Degenerate CMB parameters Use other data to break remaining degeneracies

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Galaxy lensing Assume galaxy shapes random before lensing 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

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

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SDSS Lyman-alpha white: LUQAS (Viel et al) SDSS (McDonald et al) SDSS, LCDM no tensors: n s = ± s 8 = 0.86 ± 0.03 n s < 1 at 2 sigma LUQAS The Lyman-alpa plots I showed were wrong

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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 n s <1, or something interesting No evidence for running, esp. using small scale probes

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