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

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

3. 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

4. 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

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

6. 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..

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

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

9. Main WMAP3 parameter results rely on polarization

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

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

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

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

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

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

16. 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

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

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

19. 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/

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

21. 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

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

23. 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

24. 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