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**Observational constraints and cosmological parameters**

Antony Lewis Institute of Astronomy, Cambridge

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**Cosmological parameters**

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 WMAP1 CMB data alone color = optical depth

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**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..

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**General perturbation parameters**

-isocurvature- Amplitudes, spectral indices, correlations…

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

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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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**ns < 1 at ~3 sigma (no tensors)?**

Rule out naïve HZ model

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**Secondaries that effect adiabatic spectrum ns constraint**

SZ Marginazliation Spergel et al. Black: SZ marge; Red: no SZ Slightly LOWERS ns

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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 ns**

not included in Spergel et al analysis opposite effect to SZ marginalization

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**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

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**CMB Polarization WMAP1ext WMAP3ext**

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

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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 -0.53<B<0.43 -0.42<B<0.25 WMAP1+2df+CMB+BBN+HST WMAP3+2df+CMB Gordon, Lewis: astro-ph/

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**Degenerate CMB parameters**

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

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**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

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**CMB (WMAP1ext) with galaxy lensing (+BBN prior)**

CFTHLS Contaldi, Hoekstra, Lewis: astro-ph/ Spergel et al

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**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 = ± 0.015 s8 = 0.86 ± 0.03 ns < 1 at 2 sigma LUQAS

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

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