The Astrophysical Journal Supplement Series, 148:195–211, 2003 September # 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A. FIRST-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE (WMAP)1 OBSERVATIONS: PARAMETER ESTIMATION METHODOLOGY L. Verde,2,3 H. V. Peiris,2 D. N. Spergel,2 M. R. Nolta,4 C. L. Bennett,5 M. Halpern,6 G. Hinshaw,5 N. Jarosik,4 A. Kogut,5 M. Limon,5,7 S. S. Meyer,8 L. Page,4 G. S. Tucker,5,7,9 E. Wollack,5 and E. L. Wright10 Received 2003 February 11; accepted 2003 May 30 ABSTRACT We describe our methodology for comparing the Wilkinson Microwave Anisotropy Probe (WMAP)measure- ments 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 ‘e700 Cosmic Background Imager (CBI) and Arcminute Cosmology Bolometer Array Receiver (ACBAR) measurements of the CMB, the galaxy power spectrum at z 0obtainedfromtheTwo-Degree Field Galaxy Redshift Survey (2dFGRS), and the matter power spectrum at z 3asmeasuredwiththeLy 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 nonlinear growth of structure. We show how the statistical and systematic uncertainties in the model and the data are propagated through the full analysis. Subject headings: cosmic microwave background — cosmological parameters — cosmology: observations — methods: data analysis — methods: statistical 1. INTRODUCTION temperature-polarization angular cross-power spectrum (TE). In companion papers, we present the TT (Hinshaw Cosmic microwave background (CMB) experiments are et al. 2003a) and TE (Kogut et al. 2003) angular power spec- powerful cosmological probes because the early universe is tra and show that the CMB fluctuations may be treated as particularly simple and because the fluctuations over Gaussian (Komatsu et al. 2003). angular scales >0=2 are described by linear theory Our basic approach is to constrain cosmological (Peebles & Yu 1970; Bond & Efstathiou 1984; Zaldarriaga Wilkinson & Seljak 2000). Exploiting this simplicity to obtain precise parameters with a likelihood analysis first of the Microwave Anisotropy Probe (WMAP) TT and TE spectra constraints on cosmological parameters requires that we alone, then jointly with other CMB angular power spectrum accurately characterize the performance of the instrument (Jarosik et al. 2003b; Page et al. 2003b; Barnes et al. 2003; determinations at higher angular resolution, and finally of all CMB power spectra data jointly with the power spec- Hinshaw et al. 2003a), the properties of the foregrounds trum of the large-scale structure (LSS). In 2 we describe (Bennett et al. 2003a), and the statistical properties of the x the use of the likelihood function for the analysis of micro- microwave sky. wave background data. This builds on the Hinshaw et al. The primary goal of this paper is to present our approach (2003b) methodology for determining the TT spectrum and to extracting the cosmological parameters from the temper- its curvature matrix and Kogut et al. (2003), who describe ature-temperature angular power spectrum (TT) and the our methodology for determining the TE spectrum. In x 3 we describe our use of Markov Chain Monte Carlo (MCMC) techniques to evaluate the likelihood function of 1 WMAP is the result of a partnership between Princeton University and model parameters. While WMAP’s measurements are a the NASA Goddard Space Flight Center. Scientific guidance is provided by powerful probe of cosmology, we can significantly enhance the WMAP Science Team. their scientific value by combining the WMAP data with 2 Department of Astrophysical Sciences, Princeton University, Peyton Hall, Princeton, NJ 08544; [email protected]. other astronomical data sets. This paper also presents our 3 Chandra Fellow. approach for including external CMB data sets (x 4), LSS 4 Department of Physics, Princeton University, Jadwin Hall, P.O. Box data (x 5), and Ly forest data (x 6). When including exter- 708, Princeton, NJ 08544. nal data sets, the reader should keep in mind that the physics 5 NASA Goddard Space Flight Center, Code 685, Greenbelt, and the instrumental effects involved in the interpretation of MD 20771. 6 Department of Physics and Astronomy, University of British these external data sets (especially 2dFGRS and Ly ) are Columbia, 6224 Agricultural Road, Vancouver, BC V6T 1Z1, Canada. much more complicated and less well understood than for 7 National Research Council (NRC) Fellow. WMAP data. Nevertheless, we aim to match the rigorous 8 Departments of Astrophysics and Physics, EFI, and CfCP, University treatment of uncertainties in the WMAP angular power of Chicago, 5640 South Ellis Avenue, Chicago, IL 60637. 9 Department of Physics, Brown University, Providence, RI 02912. spectrum with the inclusion of known statistical and 10 Department of Astronomy, UCLA, P.O. Box 951562, Los Angeles, systematic effects (of the data and of the theory), in the CA 90095-1562. complementary data sets. 195 196 VERDE ET AL. Vol. 148 2. LIKELIHOOD ANALYSIS OF WMAP ANGULAR The likelihood function for the temperature fluctuations POWER SPECTRA observed by a noiseless experiment with full-sky coverage The first goal of our analysis program is to determine the has the form values and confidence levels of the cosmological parameters ÀÁ À1 th exp½ðTSffiffiffiffiffiffiffiffiffiffiffiTÞ=2 that best describe the WMAP data for a given cosmological L TjCl / p ; ð2Þ model. We also wish to discriminate between different det S classes of cosmological models, in other words, to assess whereP T denotes our temperature map and whether a cosmological model is an acceptable fit to WMAP th Sij ¼ ð2‘ þ 1ÞC P‘ðn^in^jÞ=ð4Þ, where the P‘ are the data. ‘ ‘ Legendre polynomials and n^i is the pixel position on the The ultimate goal of the likelihood analysis is to find a set map. If we expandP the temperature map in spherical har- of parameters that give an estimate of hC‘i, the ensemble monics, T n^ a Y , then the likelihood function 11 sky ð Þ¼ ‘m ‘m ‘m average of which the realization on our sky is C‘ . The for each a has a simple form: ^ th ‘m likelihood function, L½C‘jC‘ ðaÞ, yields the probability of ÀÁY 2 th the data given a model and its parameters (a). In our nota- exp½jja‘m =ð2C Þ sky th qffiffiffiffiffiffiffi ‘ ^ L TjC‘ / : ð3Þ tion C‘ denotes our best estimator of C‘ (Hinshaw et al. th ‘m Cth 2003a) and C‘ is the theoretical prediction for angular ‘ power spectrum. From Bayes’s theorem, we can split the expression for the probability of a model given the data as Since we assume that the universe is isotropic, the likelihood À Á  à function is independent of m. Thus, we can sum over m and P jC^ ¼ L C^jCthð Þ Pð Þ ; ð1Þ rewrite the likelihood function as ‘ ‘ ‘ "# ! X where PðaÞ describes our priors on cosmological parameters Cth C^ 2ln 2‘ 1 ln ‘ ‘ 1 4 and we have neglected a normalization factor that does not À L ¼ ð þ Þ þ th À ð Þ C^ C depend on the parameters . Once the choice of the priors is ‘ ‘ ‘ th specified, our estimator of hC‘i is given by C‘ evaluated at up to an irrelevant additive constant. Here,P for a full-sky, ^ 2 the maximum of PðajC‘Þ. noiseless experiment, we have identified jja‘m =ð2‘ þ 1Þ ^ m with C‘. Note that the likelihood function depends only on the angular power spectrum. In this limit, the angular power 2.1. Likelihood Function spectrum encodes all of the cosmological information in the One of the generic predictions of inflationary models is CMB. that fluctuations in the gravitational potential have Characteristics of the instrument are also included in the Gaussian random phases. Since the physics that governs the likelihood analysis. Jarosik et al. (2003a) show that the evolution of the temperature and metric fluctuations is detector noise is Gaussian (see their Fig. 6 and x 3.4); conse- linear, the temperature fluctuations are also Gaussian. If we quently, the pixel noise in the sky map is also Gaussian ignore the effects of nonlinear physics at z < 10 and the (Hinshaw et al. 2003b). The resolution of WMAP is quanti- effect of foregrounds, then all of the cosmological informa- fied with a window function, w‘ (Page et al. 2003a). Thus, tion in the microwave sky is encoded in the temperature and the likelihood function for our CMB map has the same form polarization power spectra. The leading-order low-redshift as equation (2), but with S replaced by C ¼ S^ þ N, where astrophysical effect is expected to be gravitational lensing of N isP the nearly diagonal noise correlation matrix12 and ^ th x the CMB by foreground structures. We ignore this effect Sij ¼ ‘ð2‘ þ 1ÞC‘ w‘P‘ðn^i n^jÞ=ð4Þ. here as it generates a less than 1% covariance in the TT If foreground removal did not require a sky cut and if the angular power spectrum on WMAP angular scales (Hu noise were uniform and purely diagonal, then the likelihood 2001; see also Spergel et al.