Observational constraints and cosmological parameters

Antony LewisInstitute of Astronomy, Cambridge

http://cosmologist.info/

CMB PolarizationBaryon oscillationsWeak lensingGalaxy power spectrumCluster gas fractionLyman alphaetc…

+

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.

nn ddddataPdata ..)|...(| 2132111

• BUT: Large n-integrals very hard to compute!

• If we instead sample from P( {ө} | data) then it is easy:

)(11

1| i

iNdata

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 in6D parameterspace

Local parameters• When is now (Age or TCMB, H0, Ωm etc. )

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…

Astrophysical parameters

• Optical depth tau• Cluster number counts, etc..

General regular perturbation

Scalar

Vector

Tensor

Adiabatic(observed)

Matter density

Cancelling matter density(unobservable in CMB)

Neutrino vorticity(very contrived)

Gravitational waves

Neutrino density(contrived)

Neutrino velocity(very contrived)

General perturbation parameters

-iso

curv

atu

re-

Amplitudes, spectral indices, correlations…

WMAP 1 WMAP 3

ns < 1 (2 sigma)

CMB Degeneracies

TTAll

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+priorRed: full WMAP

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

Rule out naïve HZ model

Black: SZ marge; Red: no SZ Slightly LOWERS ns

SZ Marginazliation

Spergel et al.

Secondaries that effect adiabatic spectrum ns constraint

CMB lensing

For Phys. Repts. review see

Lewis, Challinor : astro-ph/0601594

Theory is robust: can be modelled very accurately

CMB lensing and WMAP3Black: withred: without

- increases ns

not included in Spergel et al analysisopposite effect to SZ marginalization

LCDM+Tensors

ns < 1or tau is highor there are tensorsor the model is wrongor we are quite unlucky

ns =1 So:

No evidence from tensor modes-is not going to get much betterfrom TT!

Current 95% indirect limits for LCDM given WMAP+2dF+HST+zre>6

CMB Polarization

Lewis, Challinor : astro-ph/0601594

WMAP1ext WMAP3ext

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

Gordon, Lewis: astro-ph/0212248

WMAP3+2df+CMB

-0.53<B<0.43

WMAP1+2df+CMB+BBN+HST

-0.42<B<0.25

Assume Flat, w=-1WMAP3 only

Degenerate CMB parameters

Use other data to breakremaining degeneracies

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

Contaldi, Hoekstra, Lewis: astro-ph/0302435

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

Spergel et al

CFTHLS

SDSS Lyman-alpha

white: LUQAS (Viel et al)SDSS (McDonald et al)

SDSS, LCDM no tensors:ns = 0.965 ± 0.015s8 = 0.86 ± 0.03

ns < 1 at 2 sigma

LUQAS

The Lyman-alpa plots I showed were wrong

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