Cosmological Parameters and the WMAP data

Antony LewisCfA, Harvard / CITA, Toronto

http://cosmologist.info

• Standard assumptions – what are the parameters?

• Unexpected features, validity of assumptions?

• Low quadrupole, cut-off/running/dark energy• Asymmetries• ‘features’ of WMAP analysis

MCMC sampling for parameter estimation

• MCMC sample points in cosmological parameter space drawn from the posterior distribution given the data P(parameters|data)

• Each sample gives an equally likely set of parameters given the data. “possible universes”

• Number density of samples proportional to probability density

• Just requires a function to compute likelihood for each set of parameters

• CosmoMC code at http://cosmologist.info/cosmomc uses CAMB (http://camb.info) to generate Cl

• Lewis, Bridle: astro-ph/0205436

Samples.py

Cosmological Parameters:combining CMB+Weak Lensing

Contaldi, Hoekstra, Lewis: astro-ph/0302435

WMAP+ACBAR+CBI+VSA with RCS + weak BBN prior

Vanilla Universe marginalized parameter constraints

Good agreement with more conservative independent CMB+2dF analysis

006.0144.0~2 hm

large compared to WMAPext+2dF (0.134±0.006)

flat, massless neutrinos, cosmological constant, power law power spectrum, …

WMAP TT power spectrum at low l

Pseudo-Cl data points from http://lambda.gsfc.nasa.gov/

compared to theoretical power law model (mean over realizations)

Low quadrupole?

Standard models

Observed

Likelihood of theoretical value given observed value

WMAP Pseudo-Cl: C2 = 123Tegmark cleaned map: C2 = 184(kp2 cut, Pseudo-Cl estimator on map from astro-ph/0302496)

Foreground uncertainty:

Likelihood modelling:

Running ns?WMAP+CMB+2dF, with and without l =2,3,4 multipoles

Low quadrupole and octopole drive ~1 sigma evidence for running

Need small scale data more reliable than Lyman-α

Cut-off in initial power spectrum?

P(k)=0 for k<kc ~ 3 x 10-4 Mpc-1

Slightly favoured by the data

Does not give very low quadrupolebecause of ISW contribution from larger k>kc

Bridle, Lewis, Weller, Efstathiou: astro-ph/0302306

Contributions to the quadrupole

ISW

Last scattering

k MPc

Total

Δk

22 ||)(~ kkPC

Changing ISW is tricky…

E.g. Dark energy with w > -1,cs2 <1 or

w<-1, cs2≥1 give less ISW than

cosmological constant

Weller, Lewis: astro-ph/0307104 Bean, Doré: astro-ph/0307100

No simple theoretical modelgives a very low quadrupole

The low value is not that unlikelyin a realisation of a standard model

P(k) on smaller scales

Bridle, Lewis, Weller, Efstathiou: astro-ph/0302306

Asymmetry of low multipoles?• after Eriksen et al

astro-ph/0307507:

l <~31 shows unlikely asymmetry:evaluate binned Cl on half sky as a function of axis: the lowest ratio of power on opposite two halves is small compared to simulations. Low power in N ecliptic hemisphere.

• Also astro-ph/0307282 find quadrupole and octopole alignment is unlikely at 1/60 level

Samples.py

WMAP is great, but…• Foreground uncertainties significant at low l

– e.g. different analyses of TE power spectrum. Foreground uncertainties not included in likelihoods

• Pseudo-Cl estimators combined with maximum likelihood error bars not strictly correct

• Noise not included in TT likelihood at l<100, even though larger at l~<100 than l>~100

• Significant correlation between TT and TE power spectra neglected – bias on e.g.

• Likelihood approximation not valid for outlier points

• Is it valid to do parameter estimation with usual assumptions when Cl not consistent with Gaussian expectations? Do outliers bias results? ...

Versions of TE power spectrum

Conclusions

• Standard ΛCDM cosmology fits the overall shape of the WMAP power spectrum and is consistent with other data

• Low quadrupole is not that unlikely in standard models, but favours models predicting low values by factor <~ 10

• Outlier points/asymmetries – quite strong evidence for analysis problems, foregrounds, or new physics

• Parameter constraints from naïve analysis may be misleading – should really understand unexpected features first.

In two bins…

1<l<18

17<l<31

No power in northern hemisphere 3-point function?

Eigenmodes:

Measured andmarginalized errorsfrom simulations: