Wilkinson Microwave Anisotropy Probe (WMAP) First Year Observations: TE Polarization

Transcription

1 Accepted by the Astrophysica Journa Wikinson Microwave Anisotropy Probe (WMAP) First Year Observations: TE Poarization A. Kogut 2, D. N. Sperge 3, C. Barnes 4, C. L. Bennett 2, M. Hapern 5, G. Hinshaw 2, N. Jarosik 4, M. Limon 2,4,6, S. S. Meyer 7, L.Page 3, G. S. Tucker 2,6,8, E. Woack 2, E. L. Wright 9 arxiv:astro-ph/ v3 22 Ju 2003 Aan.J.Kogut@nasa.gov ABSTRACT The Wikinson Microwave Anisotropy Probe (WMAP) has mapped the fu sky in Stokes I, Q, and U parameters at frequencies 23, 33, 41, 61, and 94 GHz. We detect correations between the temperature and poarization maps significant at more than 10 standard deviations. The correations are inconsistent with instrument noise and are significanty arger than the upper imits estabished for potentia systematic errors. The correations are present in a WMAP frequency bands with simiar ampitude from 23 to 94 GHz, and are consistent with a superposition of a CMB signa with a weak foreground. The fitted CMB component is robust against different data combinations and fitting techniques. On sma anguar scaes (θ < 5 ), the WMAP data show the temperature-poarization correation expected from adiabatic perturbations in the temperature power spectrum. The data for > 20 agree we with the signa predicted soey from the temperature power spectra, with no additiona free parameters. We detect excess power on arge anguar scaes (θ > 10 ) compared to predictions based on the temperature power spectra aone. The excess power is we described by reionization at redshift 11 < z r < 30 at 95% confidence, depending on the ionization history. A mode-independent fit to reionization optica depth yieds resuts consistent with the 1 WMAP is the resut of a partnership between Princeton University and NASA s Goddard Space Fight Center. Scientific guidance is provided by the WMAP Science Team. 2 Code 685, Goddard Space Fight Center, Greenbet, MD Dept of Astrophysica Sciences, Princeton University, Princeton, NJ Dept. of Physics, Jadwin Ha, Princeton, NJ Dept. of Physics and Astronomy, University of British Coumbia, Vancouver, BC Canada V6T 1Z1 6 Nationa Research Counci (NRC) Feow 7 Depts. of Astrophysics and Physics, EFI and CfCP, University of Chicago, Chicago, IL Dept. of Physics, Brown University, Providence, RI UCLA Astronomy, PO Box , Los Angees, CA

2 2 best-fit ΛCDM mode, with best fit vaue τ = 0.17±0.04 at 68% confidence, incuding systematic and foreground uncertainties. This vaue is arger than expected given the detection of a Gunn-Peterson trough in the absorption spectra of distant quasars, and impies that the universe has a compex ionization history: WMAP has detected the signa from an eary epoch of reionization. Subject headings: cosmic microwave background, cosmoogy: observations, instrumentation: poarimeters 1. INTRODUCTION Linear poarization of the cosmic microwave background(cmb) resuts from anisotropic Thomson scattering of CMB photons by free eectrons. By symmetry, an isotropic radiation fied can not generate a net poarization. Any net poarization resuts from the quadrupoe moment of the CMB temperature distribution seen by each scatterer. Mutipe scattering suppresses poarization by damping the temperature anisotropy; hence, CMB poarization originates primariy from epochs when the opacity was of order unity or ess. Standard cosmoogica modes predict two such epochs, corresponding to two characteristic anguar scaes. The first is the decouping surface at redshift z 1089, when the ionization fraction x e abrupty fas from near unity to near zero. The acoustic horizon at decouping subtends an ange θ 1 ; poarization on these scaes refects conditions in the photon-baryon fuid just prior to recombination. Poarization data from decouping compement measurements of the temperature anisotropy. Astrophysica sources generate additiona poarization as ionizing radiation from the first coapsed objects reionizes the intergaactic medium. For reionization at redshift z < 50 the horizon is on arge anguar scaes, θ > 5. Poarization on these scaes directy probes the poory-understood process of reionization. Since CMB poarization originates at modest opacity, the underying temperature anisotropy is not heaviy damped and remains observabe today. Precise predictions can be made of the average poarization pattern expected from a given power spectrum of temperature anisotropy (Rees (1968); Kaiser (1983); Bond & Efstathiou (1984); Couson et a. (1994); Kamionkowski et a. (1997); Zadarriaga & Sejak (1997); Hu & White (1997); for recent reviews, see Kosowsky (1996); Hu & Dodeson (2002)). The pattern of poarization on the sky is a vector fied with both an ampitude and direction at each point, and can be separated into two scaar fieds, one giving the cur and the other the gradient component (caed B and E modes in anaogy with eectromagnetic fieds). The DASI coaboration has detected CMB poarization on anguar scaes 0. 5 (Kovac et a. 2002). DASI reports an E mode signa significant at 4.9σ and a TE temperature-poarization correation significant at 2σ. Both signas are consistent with the concordance cosmoogica mode (spatiay fat mode dominated by a cosmoogica constant and cod dark matter; see, e.g., Hu & Dodeson (2002)) and support an adiabatic origin for the CMB temperature anisotropy. TheMicrowave Anisotropy ProbehasmappedthefuskyintheStokesI, Q, andu parameters

3 3 on anguar scaes θ > 0. 2 in 5 frequency bands centered at 23, 33, 41, 61, and 94 GHz (Bennett et a. 2003a). WMAP was not designed soey as a poarimeter, in the sense that none of its detectors are sensitive ony to poarization. Incident radiation in each differencing assemby (DA) is spit by an orthomode transducer (OMT) into two orthogona inear poarizations (Page et a. 2003b; Jarosik et a. 2003). Each OMT is oriented so that the eectric fied directions accepted in the output rectanguar waveguides ie at ±45 with respect to the yz symmetry pane of the sateite (see Bennett et a. (2003a) Fig. 2 for the definition of the sateite coordinate system). The two orthogona poarizations from the OMT are measured by two independent radiometers. Each radiometer differences the signa in the accepted poarization between two positions on the sky (the A and B beams), separated by 140. The signa from the sky in each direction ˆn can be decomposed into the Stokes parameters T(ˆn) = I(ˆn)+Q(ˆn)cos2γ +U(ˆn)sin2γ, (1) where we definethe ange γ from a meridian through the Gaactic poes to the projection on the sky of the E-pane of each output port of the OMT (Fig. 1). In principe, by tracking the orientation of the OMTs on the sky as the sateite scan pattern observes each sky pixe in different orientations, each radiometer coud independenty produce a map of the Stokes I, Q, and U parameters. In practice, the non-uniform coverage of γ at each pixe woud generate significant correations between the fitted Stokes parameters, aowing eakage of the dominant temperature anisotropy into the much fainter poarization maps. We avoid this probem by differencing the outputs of the two radiometers in each differencing assemby in the time-ordered data. Denoting the two radiometers by subscripts 1 and 2, the instantaneous outputs are and T 1 = I(ˆn A )+Q(ˆn A )cos2γ A +U(ˆn A )sin2γ A I(ˆn B ) Q(ˆn B )cos2γ B U(ˆn B )sin2γ B (2) T 2 = I(ˆn A ) Q(ˆn A )cos2γ A U(ˆn A )sin2γ A I(ˆn B )+Q(ˆn B )cos2γ B +U(ˆn B )sin2γ B. The sum T I 1 2 ( T 1 + T 2 ) = I(ˆn A ) I(ˆn B ) (3) is thus proportiona to the unpoarized intensity, whie the difference T P 1 2 ( T 1 T 2 ) = Q(ˆn A )cos2γ A +U(ˆn A )sin2γ A Q(ˆn B )cos2γ B U(ˆn B )sin2γ B. (4) is proportiona ony to the poarization. We produce fu-sky maps of the Stokes I, Q, and U parameters from the sum and difference time-ordered data using an iterative mapping agorithm. Since the poarization is faint, the Q and U maps are dominated by instrument noise and converge rapidy (Hinshaw et a. 2003a).

4 4 The Stokes Q and U components depend on a specific choice of coordinate system. For each pair of pixes, we define coordinate-independent quantities Q = Qcos(2φ)+U sin(2φ) U = U cos(2φ) Qsin(2φ), (5) where the ange φ rotates the coordinate system about the outward-directed norma vector to put the meridian aong the great circe connecting the two positions on the sky (Kamionkowski et a. 1997; Zadarriaga & Sejak 1997). A of our anayses use these coordinate-independent inear combinations of the Q and U sky maps. Simuations of the mapping agorithm demonstrate that WMAP can accuratey recover the poarization pattern on the sky, even after aowing for residua caibration uncertainty in the individua radiometer channes. However, non-idea instrumenta signas affect the Q and U sky maps to a greater extent than the unpoarized I maps. The spacecraft spin about its z axis sweeps the beams across the sky in a direction 45 from the OMT orientation, preferentiay couping signas not fixed on the sky into the U map. Residua striping exists to a esser extent in the I and Q maps. Systematic errors in the individua Q and U maps are not yet fuy assessed; consequenty, we defer detaied anaysis of the Q or U maps to a ater paper. Cross-correations between maps are argey unaffected by striping or any other channe-specific signa, aowing much simper anaysis of the faint poarization signa than woud be possibe for the individua Q or U maps. This paper discusses the temperature-poarization (TE) correation in the WMAP one-year sky maps. We compute the temperature-poarization cross-correation using three different techniques: the two-point correation function, a quadratic estimator for the power spectrum, and a tempate comparison in pixe space between the poarization maps and the predicted poarization given the observed pattern of temperature anisotropy. A three methods yied simiar resuts despite disparate treatments of the data. 2. CORRELATION FUNCTION The simpest measure of temperature-poarization cross-correation is the two-point anguar correation function C IQ (θ) = I iq j w iw j ij w, (6) iw j whereiandj are pixe indices and w are the weights. To avoid possibeeffects of 1/f noise, we force the temperature map to come from a different frequency band than the poarization maps, and thus use the temperature map at 61 GHz (V band) for a correations except the V-band poarization maps, which we correate against the 41 GHz (Q band) temperature map. Since WMAP has a high signa-to-noise ratio measurement of the CMB temperature anisotropy, we use unit weight (w i = 1) for the temperature maps and noise weight (w j = N j /σ 2 0 ) for the poarization maps, where N j is

5 5 the effective number of observations in each pixe j and σ 0 is the standard deviation of the white noise in the time-ordered data (Tabe 1 of Bennett et a. (2003b)). We compare the correation functions to Monte Caro simuations of a nu mode, which simuates the temperature anisotropy using the best-fit ΛCDM mode (Sperge et a. 2003) but forces the poarization signa to zero. Each reaization generates a CMB sky in Stokes I, Q, and U parameters, convoves this simuated sky with the beam pattern for each differencing assemby, then adds uncorreated instrument noise to each pixe in each map. We then co-add the simuated skies in each frequency band and compute C IQ (θ) using the same software for both the WMAP data and the simuations. A anaysis uses ony pixes outside the WMAP Kp0 foreground emission mask(bennett et a. 2003c), approximatey 76% of the fu sky. Figure 2 shows C IQ (θ) derived by co-adding the individua correation functions for the frequencies41, 61, and94ghz(q,v, andwbands)east ikey tobeaffected bygaactic foregrounds. The grey band shows the 68% confidence interva for the nu simuations. It is cear that WMAP detects a temperature-poarization signa at high statistica confidence, and that signas exist on both arge and sma anguar scaes. We define a goodness-of-fit statistic χ 2 = ab [C IQ MAP CIQ sim ] a M 1 ab [CIQ MAP CIQ sim ] b, (7) where C IQ MAP is the co-added correation function from WMAP data, CIQ sim is the mean from the Monte Caro simuations, and M is the covariance matrix between anguar bins a and b derived from the simuations. We find χ 2 = 207 for 78 degrees of freedom when comparing WMAP to the nu mode: WMAP detects temperature-poarization correations significant at more than 10 standard deviations Systematic Error Anaysis Having detected a significant signa in the data, we must determine whether this signa has a cosmoogica origin or resuts from systematic errors or foreground sources. We test the convergence of the mapping agorithm using end-to-end simuations, comparing maps derived from simuated time-ordered data to the input maps used to generate the simuated time series. The simuations incude a major instrumenta effects, incuding beam eipticity, radiometer performance, and instrument noise (incuding 1/f component), and are processed using the same map-making software as the WMAP data (Hinshaw et a. 2003a). The Q and U maps converge rapidy, within the 30 iterations required to derive the caibration soution. Correations in the time-ordered data introduce an anti-correation in the U map at anges corresponding to the beam separation, with ampitude 0.5% of the noise in the map. This effect is independent for each radiometer and does not affect temperature-poarization cross-correations. Simiary, residua 1/f noise in the time series can create faint striping in the maps, but does not affect cross-correations. The argest potentia systematic error in the temperature-poarization cross-correation resuts

6 6 from bandpass mismatches in the ampification/detection chains. We caibrate the WMAP data in thermodynamic temperature using the Dopper dipoe from the sateite s orbit about the Sun as a beam-fiing caibration source (Hinshaw et a. 2003a). Astrophysica sources with a spectrum other than a 2.7 K backbody are thus sighty mis-caibrated. The ampitude is dependent on the product of the source spectrum with the unique bandpass of each radiometer. If the bandpasses in each radiometer were identica, the effect woud cance for any frequency spectrum, but differences in the bandpasses between the two radiometers in each DA generate a non-zero residua in the difference signa used to generate poarization maps (Eq. 4). This signa is spatiay correated with the unpoarized foreground intensity but is independent of the orientation of the radiometers on the sky (poarization ange γ). In the imit of uniform samping of γ this term drops out of the sky map soution. However, the WMAP scan pattern does not view each pixe in a orientations; unpoarized emission with a non-cmb spectrum can thus be aiased into poarization if the bandpasses of the two radiometers in each DA are not identica. This is a significant probem ony at 23 GHz (K band), where the foregrounds are brightest and the bandpass mismatch is argest. We quantify the effect of bandpass mismatch using end-to-end simuations. For each timeordered sampe, we compute the signa in each radiometer using an unpoarized foreground mode and the measured pass bands in each output channe (Jarosik et a. 2003). We then generate maps from the simuated data using the WMAP one-year sky coverage and compute C IQ (θ) using the output I, Q, and U maps from the simuation. Figure 3 shows the predicted signa at K band. We treat this as an anguar tempate and compute the east-squares fit of the WMAP data to this bandpass tempate to determine the ampitude of the effect in the observed correation functions. We correct the WMAP correation functions C IQ (θ) and C IU (θ) at K and Ka bands by subtracting the best-fit tempate ampitudes. The fitted signa has peak ampitude of 8 µk 2 at 23 GHz and 5 µk 2 at 33 GHz. No other channe has a statisticay significant detection of this effect. Sideobe pickup of poarized emission from the Gaactic pane can aso produce spurious poarization at high atitudes in the Q and U maps. We estimate this effect using the measured farsideobe response for each beam in each poarization (Barnes et a. 2003). The simpest approach woud be to estimate the signa in each time-ordered sampe, convoving the fu sky sideobe response with the Stokes I, Q, and U maps given the instantaneous orientation of the beams for each sampe. Such an approach is computationay expensive. We instead approximate the signa in each pixe by convoving the fu sky sideobe response with the one-year Q and U maps. For each pixe, we fix one beam on that pixe whie sweeping the other beam through a orientations achieved in fight. The average from the convoution yieds the sideobe contribution for the pixe in question. Detais of the sideobe maps are presented in Barnes et a. (2003). We correate the sideobe maps with the temperature anisotropy maps in each channe to estimate the systematic error in the temperature-poarization correation. Sideobe pickup of poarized structure in the Gaactic pane is ess than 1 µk 2 in C IQ (θ) at 23 GHz and beow 0.1 µk 2 in a other bands. The effect of bandpass mismatch in the far sideobes (as opposed to the main beam) is simiary weak, with imits 1.3 µk 2 at 23 GHz and ess than 0.05 µk 2 in a other bands. We correct the poarization maps

7 7 for the estimated sideobe signa and propagate the associated systematic uncertainty throughout our anaysis. Note that a of these systematic errors depend on the Gaactic foregrounds, and have different frequency dependence than CMB poarization. Other instrumenta effects are negigibe. We measure poarization by differencing the outputs of the two radiometers in each differencing assemby (Eq. 4). Caibration errors (as opposed to the bandpass effect discussed above) can aias temperature anisotropy into a spurious poarization signa. We have simuated the uncertainty in the caibration soution using both reaistic gain drifts and drifts ten times arger than observed in fight (Hinshaw et a. 2003a). Gain drifts (either intrinsic or thermay-induced) contribute ess than 1 µk 2 to C IQ (θ) in the worst band. Nu tests provide an additiona check for systematic errors. Thomson scattering of scaar temperature anisotropy produces a cur-free poarization pattern. A non-zero cosmoogica signa is thus expected ony for the IQ (TE) correation, whereas systematic errors or foreground sources can affect both the IQ and IU (TB) correations. We aso test inear combinations of radiometer maps which cance the poarization signa but which test for systematic effects. We compute the IQ and IU correation functions by correating the Stokes I sum map from the Q- or V-band (as noted above) with the poarization difference maps (Q1 Q2)/2, (V1 V2)/2, (W1 W2)/2, and (W3 W4)/2. We then co-add the resuts with their noise weights, and compare the co-added resut for the poarization difference maps to a simiar computation for the poarization sum maps. The temperature (Stokes I) map in a cases is a sum map; the test is thus primariy sensitive to systematic errors in the poarization data. Tabe 1 shows resuts of the nu tests. We compare C IQ (θ) and C IU (θ) for the sum and difference maps to a nu hypothesis that the data consist of Stokes I and instrument noise, with no poarization in the Stokes Q or U maps. We break the data into 2 anguar regimes to differentiate between signas at decouping vs reionization. We find a cear signa detection for C IQ (θ) in the sum map for both anguar scaes. A other tests are consistent with instrument noise there is no evidence for additiona systematic errors in the temperature-poarization cross-correation Foregrounds Gaactic emission is not a strong contaminant for CMB temperature anisotropy, but coud be significant in poarization. WMAP measurements of unpoarized foreground emission show synchrotron, free-free, and therma dust emission a sharing significant spatia structure (Bennett et a. 2003c). Of these components, ony synchrotron emission is expected to generate significant poarization; other sources such as spinning dust are imited to ess than 5% of the tota intensity at 33 GHz. Synchrotron emission from eectrons acceerated in the Gaactic magnetic fied is the dominant unpoarized foreground at frequencies beow 50 GHz. Athough it is known to be ineary poarized, previous radio surveys provide itte guidance for the high-atitude poarization at mm

8 8 waveengths. Extrapoation of radio poarization maps (Brouw & Spoestra 1976) to miimeter waveengths indicate a poarization fraction between 10% and 50% depending on Gaactic atitude (Lubin & Smoot 1981). The unpoarized component has anguar power spectrum c 2, whie the CMB power spectrum rises to a set of peaks on anguar scaes θ 1 (cf Fig 10(b) of Bennett et a. (2003c)). The anguar dependence of the poarized foreground component is expected to be even steeper (Baccigaupi et a. 2001; Bruscoi et a. 2002; Tucci et a. 2002), suggesting that foreground poarization is most ikey to affect temperature-poarization correations on arge anguar scaes. Radio maps at ow Gaactic atitude, however, demonstrate that the poarization intensity is not necessariy we correated with the unpoarized intensity, compicating tempate anaysis for temperature-poarization cross-correations (Uyanıker et a. 1998, 1999). We thus use the frequency dependence of the measured temperature-poarization cross-correation to separate cosmic from foreground signas. Foreground poarization above 40 GHz is faint: fitting the correation functions at 41, 61, and 94 GHz (Q, V, and W bands) to a singe power-aw C IQ (θ,ν) = C IQ 0 (θ) (ν/ν 0) β yieds spectra index β = 0.4 ± 0.4, consistent with a CMB signa (β = 0) and inconsistent with the spectra indices expected for synchrotron (β 3), spinning dust (β 2), or therma dust (β 2). The measured signa can not be produced soey by a singe foreground emission component (uness the fractiona poarization of the foreground emission has a compensating frequency dependence, which seems unikey). A two-component fit ( ) C IQ (θ,ν) = C IQ ν β CMB (θ) + CIQ Ga (θ) (8) ν 0 tests for the superposition of a CMB component with a singe foreground component. Figure 4 shows the resuting decomposition into CMB and foreground components. We obtain a margina detection of foreground component with best-fit spectra index β = 3.7 ± 0.8 consistent with synchrotron emission. We test for consistency or possibe residua systematic errors by repeating the fit using different temperature maps and different combinations of WMAP poarization channes. The fitted CMB component (eft panes of Fig. 4) is robust against a combinations of frequency channes and fitting techniques. Note the agreement in Fig. 4 between neary independent data sets: the co-added QVW data (uncorrected for foreground emission) and the KKaQ data (corrected for foreground emission). We obtain additiona confirmation by repacing the V-band temperature map in the cross-correation (Eq. 6) with the interna inear combination temperature map designed to suppress foreground emission (Bennett et a. 2003c). The fitted CMB component does not change. We test for systematic errors by repacing the temperature map with the COBE-DMR map of the CMB temperature(bennett et a. 1996), excuding any instrumenta correation between the temperature and poarization data. Again, the resuts are unchanged. We further constrain foreground contributions by computing the cross-correation between the WMAP poarization data and temperature maps dominated by foregrounds. We repace the

9 9 temperature map in Eq. 6 with either the WMAP maximum-entropy foreground mode (Bennett et a. 2003c) or a residua foreground map created by subtracting the interna inear combination CMB map from the individua WMAP temperature maps. We then correate the foreground temperature map against the WMAP poarization data in each frequency band, and fit the resuting correation functions to CMB and foreground components (Eq. 8). The two foreground maps provide neary identica resuts. The fitted CMB component has neary zero ampitude, consistent with the instrument noise. The fitted foreground has ampitude 0.5 ± 0.1 µk 2 at ν 0 = 41 GHz, with best-fit index β = 3.4 consistent with synchrotron emission. 3. POLARIZATION CROSS-POWER SPECTRA In a second anaysis method, we compute the anguar power spectrum of the temperaturepoarization correations using a quadratic estimator (Appendix A). The power spectrum is the Legendre transform of the two-point correation function, and is more commony encountered for theoretica predictions. We compute c TE and c TB individuay for the each WMAP frequency band, using uniform weight for the temperature map and noise weight for the poarization maps. We then combine the anguar power spectra, using noise-weighted QVW data for > 21 where foregrounds are insignificant, and a fit to CMB pus foregrounds using a 5 frequency bands for 21. Since foreground contamination is weak, we gain additiona sensitivity in this anaysis by using the Kp2 sky cut retaining 85% of the sky. We estimate the uncertainty in each bin using the covariance matrix M for the poarization cross-power spectrum. Based on our anaysis of the c TT covariance matrix (Hinshaw et a. 2003b), the c TE covariance matrix has the form aong the diagona of M = < c TE c TE > < c TE (ctt +n TT /w )(c EE > 2 (9) +n EE /w )+(c TE ) 2 (2 +1)f sky f eff sky where n TT and n EE are the TT and EE noise bias terms, w is the effective window function for the combined maps (Page et a. 2003a), c TT and c EE are the temperature and poarization anguar power spectra, f sky = 0.85 is the fractiona sky coverage for the Kp2 mask, and fsky eff = f sky/1.14 for noise weighting. We take the c TT term from the measured temperature power spectra (Hinshaw et a. 2003b) andthec EE term predicted by the best-fit ΛCDM mode(sperge et a. 2003) (aowing c EE to vary as a function of optica depth in the ikeihood anaysis). Figure 5 compares the anaytic expression for the diagona eements of the covariance matrix to the mean derived from 7500 Monte Caro simuations. The anaytic form (Eq. 10) accuratey describes the simuations. We approximate the off-diagona terms using the geometric mean of the covariance matrix terms (10)

10 10 for uniform and noise weighting (Hinshaw et a. 2003b), 10 M ( M M )0.5 r. (11) Figure 6 shows the off-diagona terms r measured from Monte Caro simuations. The argest contribution, 2.8%, is at = 2 from the symmetry of our sky cut and noise coverage. The tota anticorreation is 0 r = Because of this anti-correation, the error bars for the binned c TE are sighty smaer than the naive estimate. A second method of estimating the errors reies on end-to-end simuations derived from simuated time-ordered data consisting soey of instrument noise (incuding the estimated contribution from 1/f fuctuations). We have generated 11 noise sky maps each in Stokes I, Q, and U and compute the variance in TE directy from the variance in the simuated signa. These two approaches yied errors that are consistent to better than 5%. Since there are 2 +1 mutipoes at each vaue, the fractiona uncertainty expected in the Monte Caro variance is [2/(11(2 +1)f sky )] 0.5, in agreement with this resut. Figure 7 shows the poarization cross-power spectra for the WMAP one-year data. The soid ine shows the predicted signa for adiabatic CMB perturbations, based ony on a fit to the measured temperature anguar power spectrum c TT (Sperge et a. 2003; Hinshaw et a. 2003b). Two features are apparent. The TE data on degree anguar scaes ( > 20) are in exceent agreement with a priori predictions of adiabatic modes (Couson et a. 1994). Other than the specification of adiabatic perturbations, there are no free parameters the soid ine is not a fit to c TE. The χ 2 of 24.2 for 23 degrees of freedom indicates that the CMB anisotropy is dominated by adiabatic perturbations. On arge anguar scaes ( < 20) the data show excess power compared to adiabatic modes, suggesting significant reionization. The WMAP detection of the acoustic structure in the TE spectrum confirms severa basic eements of the standard paradigm. The ampitudes of the peak and anti-peak are a measure of the thickness of the decouping surface, whie the shape confirms the assumption that the primordia fuctuations are adiabatic. Adiabatic fuctuations predict a temperature/poarization signa anticorreated on arge scaes, with TE peaks and anti-peaks ocated midway between the temperature peaks Hu & Sugiyama (1994). The existence of TE correations on degree anguar scaes aso provides evidence for super-horizon temperature fuctuations at decouping, as expected for infationary modes of cosmoogy (Peiris et a. 2003) 4. TEMPLATE POWER SPECTRA Figure 7 demonstrates that the power spectrum of temperature-poarization correations on degree anguar scaes can be predicted using the power spectrum of the temperature fuctuations 10 Note that Hinshaw et a. (2003b) define off-diagona eements in terms of the inverse covariance matrix, which differs from r by a sign.

11 11 aone. We use this for a third derivation of the TE cross-power spectrum, based on tempate matching in pixe space. For pixe sizes of a few degrees, the signa-to-noise ratio for the temperature maps is much arger than one per mutipoe, whie the S/N ratio in the poarization maps is much ess than one. The ikeihood function for the poarization measurement then has the simpe form ogl = (ˆP α ˆP pred ) T N 1 (ˆP α ˆP pred ), (12) where ˆP is the measured poarization signa (a 2 N pixe vector), α = c TE /c TT is the poarization fraction at each, N is the pixe noise correation matrix (a 2N pixe 2N pixe matrix) and Q pred (ˆn) = m U pred (ˆn) = i m a m ( 2 Y m (ˆn)+ 2 Y m (ˆn)) a m ( 2 Y m (ˆn) 2 Y m (ˆn)). (13) Here ±2 Y m (ˆn) are the spin harmonics, whie a m are the measured coefficients for an a-sky map of the CMB temperature. Imposing a cut to mask the Gaactic pane introduces additiona correations; we avoid this by using the interna inear combination temperature map (Bennett et a. 2003c) without imposing a sky cut. The maps Q pred and U pred represent the predicted poarization pattern based on the observed pattern of temperature anisotropy. We fit these tempate maps to the observed Q and U poarization maps to derive the poarization fraction α and thus the c TE poarization cross-power spectrum. Minimizing the ikeihood function yieds the norma equations where and K α = y, (14) y = ˆPN 1 ˆP pred (15) K = ˆP pred N 1 ˆP pred. (16) These equations show the advantages of this approach. We compare the data with a tempate in pixe space, making it straightforward to incude a spatiay varying noise signa. We directy compare the measured poarization maps to a prediction based on the measured temperature maps, yieding a measurement of the TE poarization cross-power spectrum in the observed sky unaffected by cosmic variance. We can thus more easiy compute the errors on the measured poarization fraction. The input temperature map (Stokes I) is aready corrected for foreground emission (much simper in pixe space where the unpoarized foregrounds are more easiy measured), greaty reducing the foreground contribution to the cross-power spectra. We thus compute the temperature-poarization cross-correation using three disparate techniques: the two-point anguar correation function, a quadratic estimator for the power spectrum in Fourier space, and a tempate fit in pixe space. A methods are in good agreement despite their very dissimiar treatment of the data. A methods show a significant excess of power for < 10.

12 12 5. REIONIZATION WMAP detects statisticay significant correations between the CMB temperature and poarization. The signa on degree anguar scaes ( > 20) agrees with the signa expected in adiabatic modes based soey on the temperature power spectrum, without any additiona free parameters. We aso detect power on arge anguar scaes ( < 10) we in excess of the signa predicted by the temperature power spectrum aone. This signa can not be expained by data processing, systematic errors, or foreground poarization, and has a frequency spectrum consistent with a cosmoogica origin. The signa on arge anguar scaes has a natura interpretation as the signature of eary reionization. 11 Both the temperature and temperature-poarization power spectra can be reated to the power spectrum of the radiation fied during scattering (Zadarriaga 1997). Thomson scattering damps the temperature anisotropy and regenerates a poarized signa on scaes comparabe to the horizon. The existence of poarization on scaes much arger than the acoustic horizon at decouping impies significant scattering at more recent epochs Reionization in a ΛCDM Universe If we assume that the ΛCDM mode is the best description of the physics of the eary universe, we can fit the observed temperature-poarization cross-power spectrum to derive the optica depth τ. We assume a step function for the ionization fraction x e and use the CMBFAST code (Sejak & Zadarriaga 1996) to predict the mutipoe moments as a function of optica depth. Whie this assumption is simpistic, our concusions on optica depth are not very sensitive to detais of the reionization history or the background cosmoogy. Figure 8 compares the poarization cross-power spectrum c TE derived from the quadratic estimator to ΛCDM modes with and without reionization. The rise in power for < 10 is ceary inconsistent with no reionization. We quantify this using a maximum-ikeihood anaysis L exp( 1 2 χ2 ) M 1/2. (17) Figure 9 shows the reative ikeihood L/Max(L) for the optica depth τ assuming a ΛCDM cosmoogy, with a other parameters fixed at the vaues derived from the temperature power spectrum aone (Sperge et a. 2003). The ikeihood for the 5-band data corrected for foreground emission peaks at τ = 0.17 ± 0.03 (statistica error ony): WMAP detects the signa from reionization at high statistica confidence. 11 Athough tensor modes can aso generate TE correations at arge anguar scaes, tensor-to-scaar ratios r arge enough to fit the WMAP TE data are rued out by the WMAP TT data (Sperge et a. 2003).

13 13 A fu error anaysis for τ must account for systematic errors and foreground uncertainties. We propagate these effects by repeating the maximum ikeihood anaysis using different combinations of WMAP frequency bands and different systematic error corrections. We correct C IQ (θ) in each frequency band not for the best estimate of the systematic error tempates, but rather the best estimate pus or minus one standard deviation. We then fit the mis-corrected C IQ (θ,ν) for a CMB piece pus a foreground piece (Eq. 8) and use the CMB piece in a maximum-ikeihood anaysis for τ. The change in the best-fit vaue for τ as we vary the systematic error corrections propagates the uncertainties in these corrections. Systematic errors have a negigibe effect on the fitted optica depth; atering the systematic error corrections changes the best-fit vaues of τ by ess than The argest non-random uncertainty is the foreground separation. We assess the uncertainty in the foreground separation by repeating the entire systematic error anaysis (using both standard and atered systematic error corrections) with the foreground spectra index β = 3.7 ± 0.8 shifted one standard deviation up or down from the best-fit vaue. Tabe 2 shows the fitted optica depth τ and goodness-of-fit statistic χ 2 for different data combinations and foreground spectra indices derived from the anaysis of the two-point correation function C IQ (θ). The first set of rows shows vaues derived by simpy co-adding the WMAP frequency channes, without any correction for foregrounds. Data at 41, 61, and 94 GHz (Q, V, and W bands) where foregrounds are negigibe show simiar vaues for τ; the χ 2 66 for 57 degrees of freedom indicates that the data are in agreement with reionized modes. Adding additiona ow-frequency channes reduces the forma statistica uncertainty, but introduces non-zero foreground contamination as shown by the marked increase in χ 2. The next three sets of rows show the resuts when the data are separated into CMB and foreground components (Eq. 8). A data combinations are now in agreement; we obtain neary identica vaues for τ when fitting either the highest-frequency data set QVW or the owest-frequency set KKaQ. The fitted optica depth is insensitive to the spectra index: varying the spectra index from -2.9 to -4.5 changes the fitted vaues by 0.02 or ess. We adopt τ = 0.17±0.04 as the best estimate for the optica depth to reionization, where the error bar refects a 68% confidence eve interva incuding statistica, systematic, and foreground uncertainties. Sperge et a. (2003) incude the TE data in a maximum-ikeihood anaysis combining WMAP data with other astronomica measurements. The resuting vaue, τ = 0.17±0.06, is consistent with the vaue derived from the TE data aone. The arger uncertainty refects the effect of simutaneousy fitting mutipe parameters. The TE anaysis propagates foreground uncertainties by re-evauating the ikeihood using different foreground spectra index. Since foreground affect ony the owest mutipoes, the combined anaysis propagates foreground uncertainty by doubing the statistica uncertainty in c TE for 2 4 to account for this effect Mode-Independent Estimate An aternative approach avoids assuming any cosmoogica mode and uses the measured temperature anguar correation function to determine the radiation power spectrum at recombination.

14 14 This approach assumes that the best estimate of the three dimensiona radiation power spectrum is the measured anguar power spectrum rather than a mode fit to the anguar power spectrum. Given the observed temperature power spectrum c TT, we derive the predicted poarization cross-power ( 4), which we then fit to the observed TE spectrum as a function of optica depth τ. We obtain τ = 0.16 ± 0.04, in exceent agreement with the vaue derived assuming a ΛCDM cosmoogy. We emphasize that the mode-independent technique makes no assumptions about the cosmoogy. The fact that it agrees we with the best-fit mode from the combined temperature and poarization data (Sperge et a. 2003) is an additiona indication that the observed temperaturepoarization correations on arge anguar scaes represent the imprint of physica conditions at reionization. The dependence on the underying cosmoogy is sma. spectrum c TE 5.3. Eary Star Formation Reionization can aso be expressed as a redshift z r assuming an ionization history. We consider two simpe cases. For instantaneous reionization with ionization fraction x e = 1 at z < z r, the measured optica depth corresponds to redshift z r = 17±3. This conficts with measurements of the Gunn-Peterson absorption trough in spectra of distant quasars, which show neutra hydrogen present at z 6 (Becker 2001; Djorgovski et a. 2001; Fan 2002). Reionization ceary did not occur through a singe rapid phase transition. However, since absorption spectra are sensitive to even sma amounts of neutra hydrogen, modes with partia ionization x e 1 can have enough neutra coumn density to produce the Gunn-Peterson trough whie sti providing free eectrons to scatter CMB photons and produce arge-scae poarization. Direct Gunn-Peterson observations ony impy a neutra hydrogen fraction 1% (Fan 2002). Accordingy, we modify the simpest mode to add a second transition: a jump from x e = 0 to x e = 0.5 at redshift z r, foowed by a second transition from x e = 0.5 to x e = 1 at redshift z = 7. Fitting this mode to the measured optica depth yieds z r 20. In reaity, reionization is more compicated than simpe step transitions. Aowing for mode uncertainty, the measured optica depth is consistent with reionization at redshift 11 < z r < 30, corresponding to times 100 < t r < 400 Myr after the Big Bang (95% confidence). Extrapoations of the observed ionizing fux to higher redshift ead to predicted CMB optica depth between (Mirada-Escude 2002), ower than our best fit vaues. The measured optica depth thus impies additiona sources of ionizing fux at high redshift. An eary generation of very massive (Pop III) stars coud provide the required additiona heating. Tegmark (1997) estimate that 10 3 of a baryons shoud be in coapsed objects by z = 30. If these baryons form massive stars, they woud reionize the universe. However, photons beow the hydrogen ionization threshod wi destroy moecuar hydrogen (the principa vehice for cooing in eary stars), driving the effective mass threshod for star formation to 10 8 soar masses and impeding subsequent star formation (Haiman et a. 1997; Gnedin & Ostriker 1997; Tegmark 1997). X-ray heating and ionization (Venkatesan et a. 2001; Oh 2001) may provide a oophoe to this argument by enhancing

15 15 the formation of H 2 moecues (Haiman et a. 2000). Cen (2003) provides a physicay-motivated mode of doube reionization that resembes the two-step mode above. A first generation of massive Pop III stars initiay ionizes the intergaactic medium. The increased metaicity of the intergaactic medium then produces a transition to smaer Pop II stars, after which the reduced ionizing fux aows regeneration of a neutra hydrogen fraction. The ionization fraction remains at x e 0.6 unti the goba star formation rate surpasses the recombination rate at z = 6, restoring x e = 1. The predicted vaue τ = 0.10 ±0.03 shoud be increased somewhat to refect the higher WMAP vaues for the baryon density Ω b and normaization σ 8 (Sperge et a. 2003). The contribution from ionized heium wi aso serve to increase τ (Venkatesan et a. 2003; Wyithe & Loeb 2003). The WMAP determination of the optica depth indicates that ionization history must be more compicated than a simpe instantaneous step function. Whie physicay pausibe modes can reproduce the observed optica depth, reionization remains a compex process and can not be fuy characterized by a singe number. A more compete determination of the ionization history requires evauation of the detaied T E and EE power spectra (Kapinghat et a. 2003; Hu & Hoder 2003). 6. CONCLUSIONS WMAP detects statisticay significant correations between the temperature and poarization maps. The correations are inconsistent with instrument noise and are significanty arger than the upper imits estabished for potentia systematic errors. The correations are present in a WMAP frequency bands with simiar ampitude from 23 to 94 GHz; fitting the data to a singe power-aw in frequency yieds a spectra index β = 0.4 ± 0.4, consistent with a CMB signa (β = 0) and inconsistent with the measured spectra indices for Gaactic foreground emission. A two-component fit to a superposition of CMB and Gaactic foregrounds yieds a positive foreground detection in both cur- and cur-free modes, with best-fit spectra index β = 3.7 ± 0.8 consistent with synchrotron emission of ampitude 0.5±0.1 µk 2 antenna temperature at 41 GHz. The fitted CMB component is robust against different data combinations and fitting techniques. On sma anguar scaes (θ < 5 ), the WMAP data show the temperature-poarization expected from adiabatic perturbations in the temperature power spectrum. The data for > 20 agree we with the signa predicted soey from the temperature power spectra, with no additiona free parameters. The data show excess power on arge anguar scaes (θ > 10 ) compared to the predictions based on the temperature power spectrum aone. The excess power is we described by eary reionization at redshift z r = , corresponding to times t r = Myr after the Big Bang (95% confidence). A mode-independent fit to reionization optica depth yieds resuts consistent with the ΛCDM mode. Our best estimate for the optica depth is τ = 0.17±0.04 (68% confidence) where the error terms incude statistica, systematic, and foreground uncertainties. This vaue is

16 16 arger than expected given the detection of a Gunn-Peterson trough in the absorption spectra of distant quasars, and impies that the universe has a compex ionization history. The WMAP detection of eary reionization opens a new frontier to expore the universe at redshift 6 < z < 30. WMAP s sensitivity to reionization is currenty imited by instrument noise, both as direct statistica uncertainty and in the abiity to better mode and remove faint poarized foregrounds. Instrumenta effects do not imit anaysis of temperature-poarization correations. The TE power spectrum and covariance matrix are avaiabe at We are currenty performing a more compete set of systematic error anayses in the individua Q and U maps. A future data reease wi incude fu-sky poarization maps and poarization power spectra. The WMAP mission is made possibe by the support of the Office of Space Sciences at NASA Headquarters and by the hard and capabe work of scores of scientists, engineers, technicians, machinists, data anaysts, budget anaysts, managers, administrative staff, and reviewers.

17 17 A. Quadratic Estimator for Temperature-Poarization Power Spectrum We estimate the temperature-poarization power spectrum from pixeized sky maps using the foowing formaism. We begin by expanding the temperature and poarization fuctuations in generaized spherica harmonics T(ˆn) = a m Y m (ˆn) (A1) m Q(ˆn)±iU(ˆn) = m a 2,m 2 Y m (ˆn) (A2) We then decompose the poarization fuctuations into E and B ike pieces a ±2,m = E m ±ib m. (A3) We can use the basic properties of the spherica harmonics NY m = ( 1) N NY m dˆn N Y m (ˆn) N Y m (ˆn) = δ δm m (A4) (A5) to derive E m = 1 2 dˆn [ Q(ˆn)( 2 Y m (ˆn)+ 2Y m (ˆn)) B m = 1 2 dˆn i U(ˆn)( 2 Y m (ˆn) 2Y m (ˆn)) ] [ U(ˆn)( 2 Y m (ˆn)+ 2Y m (ˆn)) +i Q(ˆn)( 2 Ym (ˆn) 2Ym (ˆn)) ]. (A6) We can now generaize the approach of Hivon et a. (2002) to estimate the couping terms. We mutipy the temperature and poarization maps by a weighting function T m = dˆn w T (ˆn) T(ˆn)Y m (ˆn) (A7) Ẽ m = 1 dˆn w P (ˆn) [Q(ˆn)( 2 Ym 2 (ˆn)+ 2Ym (ˆn)) iu(ˆn) ( 2 Ym (ˆn) 2Ym (ˆn))] (A8) B m = 1 dˆn w P (ˆn) [U(ˆn)( 2 Ym 2 (ˆn)+ 2Ym (ˆn)) +iq(ˆn)( 2 Ym (ˆn) 2Ym (ˆn))]. (A9) We expand the weighting function in spherica harmonics w(ˆn) = m w m Y m (ˆn), (A10)

18 18 and combine with equations A1 A3 to yied T m = dˆn Y m (ˆn)Y m (ˆn)Y m (ˆn) Ẽ m = 1 2 B m = 1 2 m m w T m T m m m w P m +ib m m m w P m ie m [E m dˆn Y m (ˆn)( 2Y m (ˆn) 2Y m (ˆn)+ 2Y m (ˆn) 2Y m (ˆn)) ] dˆn Y m (ˆn)( 2Y m (ˆn) 2Ym (ˆn) 2Y m (ˆn) 2Ym (ˆn)) [B m dˆn Y m (ˆn)( 2Y m (ˆn) 2Ym (ˆn)+ 2Y m (ˆn) 2Ym (ˆn)) dˆn Y m (ˆn)( 2Y m (ˆn) 2Y m (ˆn) 2Y m (ˆn) 2Y m (ˆn)) ]. (A11) We can then use [ (2 +1)(2 dˆn N Ym (ˆn) N Y m (ˆn) N Y m (ˆn) = +1)(2 +1) ( 1)N+m 4π ( )( N N N m m m ] 1/2 ) (A12) to compute c TT c TE c TB c EE c BB = M ab c TT c TE c TB c EE c BB. (A13) These expressions can be reduced using the symmetry and orthogonaity properties of 3-j symbos, as given in Eqs. 1.8 and 1.14 of Rotenberg et a. (1959). In particuar, imaginary terms drop out, and summations over products of 3-j symbos with m, m and m in the bottom row evauate to 1/(2 +1). After some agebra, the couping terms reduce to ( M TT,TT = (2 +1) W TT 4π M TE,TE M EE,EE = M TB,TB = (2 +1) 8π = M BB,BB = (2 +1) 16π ( W TP W PP [( ) )[( ) ( ) ( )] (A14) )] (A15)

19 19 [( ) + ( )] M EE,BB = M BB,EE [( = (2 +1) W PP 16π [( ) ( ) )] ( )] (A16) (A17) where W ab = m w a m wb m, (A18) with a and b referring to either T or P. A of the other couping terms are zero. Note that if we use different weighting functions for T, Q and U, we increase the couping between E and B modes. B. Uniform Temperature Weighting If we use the fu sky to compute the temperature spherica harmonic terms, then the crosscorreation term and its error matrix becomes particuary simpe. For this case, w00 T = 1/ 4π and a other couping terms are 0. In this imit, the measured c TE is just a constant times the true c TE where c TE f = = cte f The covariance matrix for these terms are diagona. w E (ˆn) dˆn 4π (B1) (B2) M = ctt c EE (2 +1)f 2 (B3)

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