Transcript Slide 1
Cosmological Parameters and the WMAP data Antony Lewis CfA, 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 Cosmological Parameters: combining CMB+Weak Lensing WMAP+ACBAR+CBI+VSA with RCS + weak BBN prior Contaldi, Hoekstra, Lewis: astro-ph/0302435 Vanilla Universe marginalized parameter constraints flat, massless neutrinos, cosmological constant, power law power spectrum, … mh2 ~ 0.144 0.006 large compared to WMAPext+2dF (0.134±0.006) Good agreement with more conservative independent CMB+2dF analysis WMAP TT power spectrum at low l compared to theoretical power law model (mean over realizations) Pseudo-Cl data points from http://lambda.gsfc.nasa.gov/ Low quadrupole? Foreground uncertainty: WMAP Pseudo-Cl: C2 = 123 Tegmark cleaned map: C2 = 184 (kp2 cut, Pseudo-Cl estimator on map from astro-ph/0302496) Likelihood modelling: Likelihood of theoretical value given observed value Observed Standard models 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? Bridle, Lewis, Weller, Efstathiou: astro-ph/0302306 P(k)=0 for k<kc ~ 3 x 10-4 Mpc-1 Slightly favoured by the data Does not give very low quadrupole because of ISW contribution from larger k>kc Contributions to the quadrupole C 2 ~ P ( k ) | k |2 Total ISW Δk Last scattering k MPc 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 model gives a very low quadrupole The low value is not that unlikely in 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 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? Measured and marginalized errors from simulations: Eigenmodes: