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Register a new userFirst Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Parameter Estimation Methodology
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L. Verde
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We describe our methodology for comparing the WMAP measurements of the cosmic microwave background (CMB) and other complementary data sets to theoretical models. The unprecedented quality of the WMAP data, and the tight constraints on cosmological parameters that are derived, require a rigorous analysis so that the approximations made in the modeling do not lead to significant biases. We describe our use of the likelihood function to characterize the statistical properties of the microwave background sky. We outline the use of the Monte Carlo Markov Chains to explore the likelihood of the data given a model to determine the best fit cosmological parameters and their uncertainties. We add to the WMAP data the l>~700 CBI and ACBAR measurements of the CMB, the galaxy power spectrum at z~0 obtained from the 2dF galaxy redshift survey (2dFGRS), and the matter power spectrum at z~3 as measured with the Ly alpha forest. These last two data sets complement the CMB measurements by probing the matter power spectrum of the nearby universe. Combining CMB and 2dFGRS requires that we include in our analysis a model for galaxy bias, redshift distortions, and the non-linear growth of structure. We show how the statistical and systematic uncertainties in the model and the data are propagated through the full analysis.
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Full sky maps are made in five microwave frequency bands to separate the temperature anisotropy of the CMB from foreground emission. We define masks that excise regions of high foreground emission. The effectiveness of template fits to remove foreground emission from the WMAP data is examined. These efforts result in a CMB map with minimal contamination and a demonstration that the WMAP CMB power spectrum is insensitive to residual foreground emission. We construct a model of the Galactic emission components. We find that the Milky Way resembles other normal spiral galaxies between 408 MHz and 23 GHz, with a synchrotron spectral index that is flattest (beta ~ -2.5) near star-forming regions, especially in the plane, and steepest (beta ~ -3) in the halo. The significant synchrotron index steepening out of the plane suggests a diffusion process in which the halo electrons are trapped in the Galactic potential long enough to suffer synchrotron and inverse Compton energy losses and hence a spectral steepening. The synchrotron index is steeper in the WMAP bands than in lower frequency radio surveys, with a spectral break near 20 GHz to beta < -3. The modeled thermal dust spectral index is also steep in the WMAP bands, with beta ~ 2.2. Microwave and H alpha measurements of the ionized gas agree. Spinning dust emission is limited to < ~5% of the Ka-band foreground emission. A catalog of 208 point sources is presented. Derived source counts suggest a contribution to the anisotropy power from unresolved sources of (15.0 +- 1.4) 10^{-3} microK^2 sr at Q-band and negligible levels at V-band and W-band.
The Wilkinson Microwave Anisotropy Probe (WMAP) has mapped the full sky in Stokes I, Q, and U parameters at frequencies 23, 33, 41, 61, and 94 GHz. We detect correlations between the temperature and polarization maps significant at more than 10 standard deviations. The correlations are present in all WMAP frequency bands with similar amplitude from 23 to 94 GHz, and are consistent with a superposition of a CMB signal with a weak foreground. The fitted CMB component is robust against different data combinations and fitting techniques. On small angular scales theta < 5 deg, the WMAP data show the temperature-polarization correlation expected from adiabatic perturbations in the temperature power spectrum. The data for l > 20 agree well with the signal predicted solely from the temperature power spectra, with no additional free parameters. We detect excess power on large angular scales (theta > 10 deg) compared to predictions based on the temperature power spectra alone. The excess power is well described by reionization at redshift 11 < z_r < 30 at 95% confidence, depending on the ionization history. A model-independent fit to reionization optical depth yields results consistent with the best-fit LambdaCDM model, with best fit value tau = 0.17 +- 0.04 at 68% confidence, including systematic and foreground uncertainties. This value is larger than expected given the detection of a Gunn-Peterson trough in the absorption spectra of distant quasars, and implies that the universe has a complex ionization history: WMAP has detected the signal from an early epoch of reionization.
We present cosmological parameter constraints based on the final nine-year WMAP data, in conjunction with additional cosmological data sets. The WMAP data alone, and in combination, continue to be remarkably well fit by a six-parameter LCDM model. When WMAP data are combined with measurements of the high-l CMB anisotropy, the BAO scale, and the Hubble constant, the densities, Omegabh2, Omegach2, and Omega_L, are each determined to a precision of ~1.5%. The amplitude of the primordial spectrum is measured to within 3%, and there is now evidence for a tilt in the primordial spectrum at the 5sigma level, confirming the first detection of tilt based on the five-year WMAP data. At the end of the WMAP mission, the nine-year data decrease the allowable volume of the six-dimensional LCDM parameter space by a factor of 68,000 relative to pre-WMAP measurements. We investigate a number of data combinations and show that their LCDM parameter fits are consistent. New limits on deviations from the six-parameter model are presented, for example: the fractional contribution of tensor modes is limited to r<0.13 (95% CL); the spatial curvature parameter is limited to -0.0027 (+0.0039/-0.0038); the summed mass of neutrinos is <0.44 eV (95% CL); and the number of relativistic species is found to be 3.84+/-0.40 when the full data are analyzed. The joint constraint on Neff and the primordial helium abundance agrees with the prediction of standard Big Bang nucleosynthesis. We compare recent PLANCK measurements of the Sunyaev-Zeldovich effect with our seven-year measurements, and show their mutual agreement. Our analysis of the polarization pattern around temperature extrema is updated. This confirms a fundamental prediction of the standard cosmological model and provides a striking illustration of acoustic oscillations and adiabatic initial conditions in the early universe.
We present the angular power spectrum derived from the first-year Wilkinson Microwave Anisotropy Probe (WMAP) sky maps. We study a variety of power spectrum estimation methods and data combinations and demonstrate that the results are robust. The data are modestly contaminated by diffuse Galactic foreground emission, but we show that a simple Galactic template model is sufficient to remove the signal. Point sources produce a modest contamination in the low frequency data. After masking ~700 known bright sources from the maps, we estimate residual sources contribute ~3500 uK^2 at 41 GHz, and ~130 uK^2 at 94 GHz, to the power spectrum l*(l+1)*C_l/(2*pi) at l=1000. Systematic errors are negligible compared to the (modest) level of foreground emission. Our best estimate of the power spectrum is derived from 28 cross-power spectra of statistically independent channels. The final spectrum is essentially independent of the noise properties of an individual radiometer. The resulting spectrum provides a definitive measurement of the CMB power spectrum, with uncertainties limited by cosmic variance, up to l~350. The spectrum clearly exhibits a first acoustic peak at l=220 and a second acoustic peak at l~540 and it provides strong support for adiabatic initial conditions. Kogut et al. (2003) analyze the C_l^TE power spectrum, and present evidence for a relatively high optical depth, and an early period of cosmic reionization. Among other things, this implies that the temperature power spectrum has been suppressed by ~30% on degree angular scales, due to secondary scattering.
We present limits to the amplitude of non-Gaussian primordial fluctuations in the WMAP 1-year cosmic microwave background sky maps. A non-linear coupling parameter, f_NL, characterizes the amplitude of a quadratic term in the primordial potential. We use two statistics: one is a cubic statistic which measures phase correlations of temperature fluctuations after combining all configurations of the angular bispectrum. The other uses the Minkowski functionals to measure the morphology of the sky maps. Both methods find the WMAP data consistent with Gaussian primordial fluctuations and establish limits, -58<f_NL<134, at 95% confidence. There is no significant frequency or scale dependence of f_NL. The WMAP limit is 30 times better than COBE, and validates that the power spectrum can fully characterize statistical properties of CMB anisotropy in the WMAP data to high degree of accuracy. Our results also validate the use of a Gaussian theory for predicting the abundance of clusters in the local universe. We detect a point-source contribution to the bispectrum at 41 GHz, b_src = (9.5+-4.4) X 1e-5 uK^3 sr^2, which gives a power spectrum from point sources of c_src = (15+-6) X 1e-3 uK^2 sr in thermodynamic temperature units. This value agrees well with independent estimates of source number counts and the power spectrum at 41 GHz, indicating that b_src directly measures residual source contributions.
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