FIVE-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE OBSERVATIONS: ANGULAR POWER SPECTRA

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1 The Astrophysica Journa Suppement Series, 180: , 2009 February c The American Astronomica Society. A rights reserved. Printed in the U.S.A. doi: / /180/2/296 FIVE-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE OBSERVATIONS: ANGULAR POWER SPECTRA M. R. Nota 1, J. Dunkey 2,3,4,R.S.Hi 5,G.Hinshaw 6, E. Komatsu 7, D. Larson 8, L. Page 3, D. N. Sperge 2,9, C. L. Bennett 8, B. God 8, N. Jarosik 3, N. Odegard 5, J. L. Weiand 5, E. Woack 6, M. Hapern 10, A. Kogut 6,M.Limon 11, S. S. Meyer 12, G. S. Tucker 13, and E. L. Wright 14 1 Canadian Institute for Theoretica Astrophysics, 60 St. George St., University of Toronto, Toronto, ON M5S 3H8, Canada; nota@cita.utoronto.ca 2 Department of Astrophysica Sciences, Peyton Ha, Princeton University, Princeton, NJ , USA 3 Department of Physics, Jadwin Ha, Princeton University, Princeton, NJ , USA 4 Astrophysics, University of Oxford, Kebe Road, Oxford, OX1 3RH, UK 5 Adnet Systems, Inc., 7515 Mission Dr., Suite A100, Lanham, MD 20706, USA 6 Code 665, NASA/Goddard Space Fight Center, Greenbet, MD 20771, USA 7 Department of Astronomy, University of Texas, Austin, 2511 Speedway, RLM , Austin, TX 78712, USA 8 Department of Physics & Astronomy, The Johns Hopkins University, 3400 N. Chares St., Batimore, MD , USA 9 Princeton Center for Theoretica Physics, Princeton University, Princeton, NJ 08544, USA 10 Department of Physics and Astronomy, University of British Coumbia, Vancouver, BC V6T 1Z1, Canada 11 Coumbia Astrophysics Laboratory, 550 W. 120th St., Mai Code 5247, New York, NY , USA 12 Departments of Astrophysics and Physics, KICP and EFI, University of Chicago, Chicago, IL 60637, USA 13 Department of Physics, Brown University, 182 Hope St., Providence, RI , USA 14 UCLA Physics & Astronomy, P.O. Box , Los Angees, CA , USA Received 2008 March 4; accepted 2008 June 20; pubished 2009 February 11 ABSTRACT We present the temperature and poarization anguar power spectra of the cosmic microwave background derived from the first five years of Wikinson Microwave Anisotropy Probe data. The five-year temperature spectrum is cosmic variance imited up to mutipoe = 530, and individua -modes have signa-to-noise ratio S/N > 1 for <920. The best-fitting six-parameter ΛCDM mode has a reduced χ 2 for = of χ 2 /ν = 1.06, with a probabiity to exceed of 9.3%. There is now significanty improved data near the third peak which eads to improved cosmoogica constraints. The temperature-poarization correation is seen with high significance. After accounting for foreground emission, the ow- reionization feature in the EE power spectrum is preferred by Δχ 2 = 19.6 for optica depth τ = by the EE data aone, and is now argey cosmic variance imited for = 2 6. There is no evidence for cosmic signa in the BB, TB, or EB spectra after accounting for foreground emission. We find that, when averaged over = 2 6, ( +1)C BB /(2π) < 0.15 μk 2 (95% CL). Key words: cosmic microwave background cosmoogica parameters cosmoogy: observations eary universe arge-scae structure of universe space vehices: instruments 1. INTRODUCTION The Wikinson Microwave Anisotropy Probe (WMAP) sateite (Bennett et a. 2003) has measured the temperature and poarization of the microwave sky at five frequencies from 23 to 94 GHz. Hinshaw et a. (2009) present our new, more sensitive temperature and poarization maps. After removing a mode of the foreground emission from these maps (God et a. 2009), we obtain our best estimates of the temperature and poarization anguar power spectra of the cosmic microwave background (CMB). This paper presents our statistica anaysis of these CMB temperature and poarization maps. Our basic anaysis approach is simiar to the approach described in our first-year WMAP temperature anaysis (Hinshaw et a. 2003) and poarization anaysis (Kogut et a. 2003) and in the three-year WMAP temperature (Hinshaw et a. 2007) and poarization anaysis (Page et a. 2007). Whie most of the WMAP anaysis pipeine has been unchanged from our three-year anaysis, there have been a number of improvements that have reduced the systematic errors and increased the precision of the derived power spectra. Hinshaw et a. (2009) describe the WMAP data processing with an emphasis on these changes. Hi et a. (2009) present our more 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. compete anaysis of the WMAP beams based on five years of Jupiter data and physica optics fits to both the A- and B-side mirror distortions. The increase in main beam soid ange eads to a revision in the beam function that impacts our computed power spectrum by raising the overa ampitude for >200 by roughy 2%. God et a. (2009) introduce a new set of masks that are designed to remove regions of free free emission that were a minor (but detectabe) contaminant in anayses using the previous Kp2 mask used in the one- and three-year anaysis. Wright et a. (2009) update the point source cataog presented by Hinshaw et a. (2007) finding 67 additiona sources. Section 2 describes these changes and their impications for the measured power spectra. Section 3 presents the temperature anguar power spectrum (TT). WMAP has made a cosmic variance imited measurement of the anguar power spectrum to = 530 and we now report resuts into the third peak region. The WMAP resuts, combined with recent ground-based measurements of the TT anguar power spectrum (Readhead et a. 2004; Jones et a. 2006; Reichardt et a. 2008), resut in accurate measurements we into the fifth peak region. For WMAP, point sources are the argest astrophysica contaminant to the temperature power spectrum. We present estimates for the point source contamination based on muti-frequency data, source counts, and estimates from the bispectrum. The poarization observations are decomposed into E and B mode components (Kamionkowski et a. 1997; Sejak & 296

2 No. 2, 2009 WMAP FIVE-YEAR OBSERVATIONS: ANGULAR POWER SPECTRA 297 Zadarriaga 1997). Primordia scaar fuctuations generate ony E modes, whie tensor fuctuations generate both E and B modes. With T, E, and B maps, we compute the anguar autopower spectra of the three fieds, TT, EE, and BB, and the anguar cross-power spectra of these three fieds, TE, TB, and EB. If the CMB fuctuations are Gaussian random fieds, then these six anguar power spectra encode a of the statistica information in the CMB. Uness there is a preferred sense of rotation in the universe, symmetry impies that the TB and EB power spectrum are zero. In Section 4 we present both the TE and TB temperature-poarizatio cross power spectra. The WMAP measurements of the TE spectrum now ceary see mutipe peaks. The arge ange TE anti-correation is a distinctive signature of superhorizon fuctuations (Sperge & Zadarriaga 1997a). Komatsu et a. (2009) discuss how the TB measurements constrain parity-vioating interactions. Section 5 presents both the EE and BB poarization power spectra. The EE power spectrum now shows a cear 5σ signature of cosmic reionization. Dunkey et a. (2009) show that the ampitude of the signa impies that the cosmic reionization was an extended process. Dunkey et a. (2009) and Komatsu et a. (2009) discuss the cosmoogica impications of the anguar power spectrum measurements. 2. CHANGES IN THE FIVE-YEAR ANALYSIS The methodoogy used for the five-year power spectra anaysis is simiar to as that used for the three-year anaysis. In this section we ist the significant changes and their impact on the resuts: 1. Hinshaw et a. (2009) describe the changes in the map processing and the resutant reduction in the absoute caibration uncertainty from 0.5% to 0.2%. 2. The temperature mask used to compute the power spectrum has been updated, removing sighty more sky near the gaactic pane, and more high-atitude point sources (God et a. 2009). The gaactic mask used in the three- and oneyear reeases (Kp2) was constructed by seecting a pixes whose K-band emission exceeded a certain threshod. This procedure worked we in identifying areas contaminated by synchrotron emission; however, it missed a few sma regions contaminated by free free, particuary around ρ Oph, the Gum nebua, and the Orion/Eridanus Bubbe. For the five-year anaysis we have constructed a new gaactic mask to remove these contaminated areas. Wright et a. (2009) updated the WMAP point source cataog, finding 390 sources in the five-year data, 67 more sources than in the three-year cataog (Hinshaw et a. 2007). Of these 67 new sources, 32 were previousy unmasked, and therefore added to the three-year source mask to create the five-year source mask. 15 A tod, the new five-year temperature power spectrum mask (KQ85) retains 81.7% of the sky, whie the three-year mask (Kp2) retained 84.6% of the sky. 3. The five-year poarization mask is the same as the threeyear P06 mask described by Page et a. (2007), except that an additiona 0.27% of the sky has been removed due to combining P06 with the new processing mask (Hinshaw et a. 2009). 15 Six of the sources in the five-year cataog were not added to the mask; they were found in a ate update to the cataog after the mask had been finaized. 4. In addition to masking, the maps are further ceaned of gaactic foreground emission using externa tempates. The ceaning procedure is very simiar to that of the threeyear anaysis; see God et a. (2009) for detais. For the temperature map, three tempates are used: a synchrotron tempate (the WMAP K Ka difference map), an Hα tempate as a proxy for free free (Finkbeiner 2003), and a therma dust tempate (Finkbeiner et a. 1999). For the poarization maps, two tempates are used (since free free is unpoarized): the poarized K-band map and a poarized dust tempate constructed from the unpoarized dust tempate, a simpe mode of the gaactic magnetic fied, and poarization directions deduced from staright. 5. A great dea of work has gone into improving the determination of the beam maps and window functions (Hi et a. 2009). The main beam soid anges are arger than the three-year estimates by 1% 2% in V and W band. Increased soid ange (i.e., greater map smoothing) reduces the vaue of the transfer function b, raising the deconvoved CMB power spectra. The ratio of the three-year to five-year transfer functions can be seen in Figure 13 of Hi et a. (2009); the net effect is to raise the TT power spectrum by 2% for >200, which is within the three-year beam 1σ confidence imits. The beam-transfer function uncertainty is smaer than the three-year uncertainty by a factor of 2. The window function uncertainty is now 0.6% in ΔC /C for 200 << TEMPERATURE SPECTRUM The five-year 32 spectrum is described by Dunkey et a. (2009). At ow- the ikeihood function is no onger we approximated by a Gaussian so that we expicity sampe the ikeihood function to evauate the statistica distribution of each mutipoe. We construct the five-year TT spectrum for >32 in the same fashion as the three-year spectrum; we refer the reader to Hinshaw et a. (2007) for detais, and ony briefy summarize the process as foows: 1. We start with the singe year V1, V2, W1 W4 resoution- 10 maps, 16 masked by the KQ85 mask, and further ceaned via foreground tempate subtraction. 2. The pseudo-c cross power spectra are computed for each pair of maps. Two weightings are used: fat weighting and inverse noise variance (N obs ) weighting. 3. The year/da cross power spectra are combined by band, forming the V V, V W, and W W spectra. The auto-power spectra are not incuded in the combination, eiminating the need to subtract a noise bias. 4. A mode of the unresoved point source contamination with ampitude A ps = ± μk 2 sr is subtracted from the band-combined spectra. See Section 3.1 for more detais. 5. The V V, V W, and W W spectra are optimay combined -by- to create the fina CMB spectrum. As in the three-year anaysis, the diagona eements of the Ĉ covariance matrix are cacuated as (ΔĈ ) 2 = 16 12,582,912 pixes (N side = 1024). 2 (2 +1)f 2 sky ()(C + N ) 2 (1)

3 298 NOLTA ET AL. Vo. 180 Figure 1. WMAP five-year temperature (TT) power spectrum. The red curve is the best-fit theory spectrum from the ΛCDM/WMAP chain (Dunkey et a. 2009, Tabe 2) based on WMAP aone, with parameters (Ω b h 2, Ω m h 2, Δ 2 R,n s,τ,h 0 ) = (0.0227, 0.131, 2.41, 0.961, 0.089, 72.4). The uncertainties incude both cosmic variance, which dominates beow = 540, and instrumenta noise which dominates at higher mutipoes. The uncertainties increase at arge due to WMAP s finite resoution. The improved resoution of the third peak near = 800 in combination with the simutaneous measurement of the rest of the spectrum eads to the improved resuts reported in this reease. where C is the cosmic variance term and N is the noise term. The vaue of f sky (), the effective sky fraction, is caibrated from simuations: 17 { (/500) 2, 500; f sky () = (2) (500/), > 500. The five-year TT spectrum is shown in Figure 1. With the greater signa-to-noise ratio (S/N) of the five-year data the third acoustic peak is beginning to appear in the spectrum. The spectrum is cosmic variance imited up to = 530, and individua -modes have S/N >1for<920. In a fit to the best cosmoogica ΛCDM mode, the reduced χ 2 for = is χ 2 /ν = 1.06, with a probabiity to exceed of 9.3%. Figure 12 compares the unbinned five-year TT spectrum with the three-year resut. Aside from the sma upward shift of the five-year spectrum reative to that of the three-year, due to the new beam-transfer function, they are identica at ow-. Figure 13 shows the unbinned TT spectrum broken down into its frequency components (V V, V W, W W), demonstrating that the signa is independent of frequency. How much has the determination of the third acoustic peak improved with the five-year data? Over the range = , which approximatey spans the rise and fa of the third peak (from the bottom of the second trough to the point on the opposite side of the peak), the fiducia spectrum is preferred over a fat mean spectrum by Δχ 2 = 7.6. For the three-year data it was Δχ 2 = 3.6. With a few more years of data, WMAP shoud detect the curvature of third peak to greater than 3σ. In Figure 2 we compare the WMAP five-year TT power spectrum aong with recent resuts from other experiments (Readhead et a. 2004; Jones et a. 2006; Reichardt et a. 2008), showing great consistency between the various measurements. Severa on-going and future ground-based CMB experiments 17 The Markov chains in Dunkey et a. (2009) and Komatsu et a. (2009)were run with a version of the WMAP ikeihood code with oder and sighty arger vaues for f sky. The change in f sky increased the TT errors by on average 2%. Rerunning the ΛCDM chain with the new f sky eads to parameter shifts of at most 0.1σ. Figure 2. WMAP five-year TT power spectrum aong with recent resuts from the ACBAR (Reichardt et a. 2008, purpe), Boomerang (Jones et a. 2006, green), and CBI (Readhead et a. 2004, red) experiments. The other experiments caibrate with WMAP or WMAP s measurement of Jupiter (CBI). The red curve is the best-fit ΛCDM mode to the WMAP data, which agrees we with a datasets when extrapoated to higher. pan on caibrating themseves off their overap with WMAP at the highest- s; improving WMAP s determination of the third peak wi have the added benefit of improving their caibrations Unresoved Point Source Correction A popuation of point sources, Poisson-distributed over the sky, contributes an additiona source of white noise to the measured TT power spectrum, C TT C TT + C ps.givena known source distribution N(> S), the number of sources per steradian with fux greater than S, the point-source-induced signa is Sc C ps = g(ν) 2 ds dn 0 ds S2 (μk 2 sr) (3) where S is the source fux, S c is the fux cutoff (above which sources are masked and removed from the map), and g(ν) = (c 2 /2kν 2 )r(ν) converts fux density to thermodynamic temperature, with r(ν) = (ex 1) 2, x hν/kt x 2 e x CMB (4) converting antenna to thermodynamic temperature. At the frequencies and fux densities reevant for WMAP, source counts are dominated by fat-spectrum radio sources, which have fux spectra that are neary constant with frequency (S ν α with α 0). Wright et a. (2009) find the average spectra index of sources bright enough to be detected in the WMAP five-year data to be α = 0.09, with an intrinsic dispersion of σ α = Since a source with fux S ν α has a thermodynamic temperature T ν α 2 r(ν), we mode the frequency dependence of C ps as ( ) α 2 C ps ν i ν j (ν i,ν j ) = A ps r(ν i )r(ν j ) νq 2, (5) where ν i,j are the frequencies of the two maps used to cacuate the TT spectrum, A ps is an unknown ampitude, and ν Q = 40.7 GHz is the Q-band centra frequency. In this section, we estimate the vaue of A ps needed to correct the TT power spectrum, finding A ps = ± μk 2 sr,

4 No. 2, 2009 WMAP FIVE-YEAR OBSERVATIONS: ANGULAR POWER SPECTRA 299 Tabe 1 Unresoved Point Source Contamination Bands Mask A ps (α = 0) (10 3 μk 2 sr) A ps (α = 0.09) (10 3 μk 2 sr) QVW KQ ± ± 0.9 KQ ± ± 0.9 KQ ± ± 1.0 VW KQ ± ± 3.5 KQ ± ± 3.8 KQ ± ± 4.1 Note. A resuts are for = and discuss incorporating its uncertainty into the ikeihood function Estimating the Correction For a fixed beam size, fat-spectrum radio sources are much fainter in the W-band temperature maps than in Q- or V-band, aowing us to use the frequency dependence of the TT spectrum at high- to constrain the vaue of A ps. As in previous reeases, the estimator we use is Figure 3. Unresoved point source contamination A ps, measured in bins of Δ = 100 evauated at 40.7 GHz (Q band). For a source popuation whose fuxes are independent of frequency A ps scaes roughy as ν 2 in the WMAP data. The red data points are from the anaysis of V and W bands aone and the bue points are from the anaysis of Q, V, and W bands. The horizonta dashed green ines, at and 0.012, show the 1σ bounds for our adopted vaue of A ps. Note that the QVW ampitude is independent of. Â ps = h γ = s γ αβ Cα (Σ 1 ) αβ h β αβ sα (Σ 1 ) αβ h β, (6) αβ sα (Σ 1 ) αβ αβ (Σ 1 ) αβ, (7) where Greek etters represent a pair of frequencies (e.g., VW), C α is the measured TT cross-power spectrum, Σ αβ is the C αcβ covariance matrix incuding cosmic variance and detector noise, and s α = ( +1)C ps (α)/2π. The inverse estimator variance ([δâ ps ] 2 ) is given by the denominator of (6). Whie Σ αβ does not incude the off-diagona couping due to the mask, the diagona eements are renormaized to account for the oss of sky coverage. Measured vaues for A ps are isted in Tabe 1 for various frequency combinations (QVW and VW) and gaactic masks (KQ85, KQ80, and KQ75). The QVW estimates are insensitive to the gaactic mask; the VW estimate increases somewhat as more of the sky is masked. Both the QVW and VW estimates prefer the same vaue ( μk 2 sr) of A ps when the KQ75 mask is used. Whie we restrict the data to = , the QVW estimate is ony a weak function of the chosen -range; Figure 3 shows A ps estimated in bins of width Δ = 100. We adopt A ps = ± μk 2 sr as our correction to the fina combined TT spectrum. The consistency between -bins and between QVW and VW seen in Figure 3 is an important nu test for the anguar power spectrum. A ps (VW) is proportiona to the power in the V W map in a given -range. Figure 4 shows no evidence for any detectabe residua signa in the VW maps after point source subtraction. Because radio sources can ony have positive fux they introduce a positive skewness to the maps, which can be detected in searches for non-gaussianity. Komatsu et a. (2009) estimated the bispectrum induced by sources, finding b ps = (4.3 ± 1.3) 10 5 μk 3 sr 2 at Q band. Is this consistent with the vaue of A ps measured from the power spectrum? Given a theoretica mode for the source number counts N(> S), one can predict the measured vaues of C ps and b ps. Severa modes exist in the iterature; we tested our resuts against two, Toffoatti Figure 4. TT V W nu spectrum. After correcting for unresoved point source emission, the individua power spectra are subtracted in power spectrum space. The resut is consistent with zero and thus there is no evidence of point source contamination. In these units, point source contamination woud be evident as a horizonta offset from zero. At = 500, the TT power spectrum is C TT 0.06; thus the contamination is imited to roughy 3% in power. et a. (1998, Tof98) 18 and de Zotti et a. (2005, dez05). C ps is cacuated via (3), and b ps from Sc b src = g 3 (ν) ds dn 0 ds S3, (8) where g(ν) and S c are defined in (3). The comparison is compicated by the fact that S c is unknown. We mask out not ony the sources detected in WMAP data, but aso undetected sources from externa cataogues that are ikey to contribute contaminating fux. However, a singe vaue of S c predicts both C ps and b ps, so we can in principe tune S c to match one, and see if it agrees with the other. In Tabe 2 we compare our measured vaues of C ps and b ps with the rescaed Tof98 and dez05 predictions for severa vaues of S c. There is some tension 18 In Bennett et a. (2003) we found that the Tof98 mode needed to be rescaed by a factor of 0.66 to match the WMAP one-year number counts; Wright et a. (2009) refined the rescaing factor to 0.64 to match the WMAP five-year source counts.

5 300 NOLTA ET AL. Vo. 180 Tabe 2 Unresoved Point Source Contamination S c (Jy) C ps (10 3 μk 2 sr) b ps (10 5 μk 3 sr 2 ) WMAP5 (KQ75) 11.7 ± 1.1 a 4.3 ± 1.3 b Toffoatti et a. (1998) de Zotti et a. (2005) Notes. A numbers are evauated at 40.7 GHz (Q band). a By Equation (5), C ps (Q) = A ps r(q) 2 = A ps,wherea ps is the QVW/ KQ75 resut from Tabe 1. b From Komatsu et a. (2009), using the Q-band map and KQ75 mask. between the measured vaues and the mode predictions. Given our measured vaue for b ps the modes woud prefer a smaer vaue for A ps, in the range μk 2 sr. For the Tof98 mode, the S c 0.52 predictions are within 1σ of both C ps and b ps. However, the dez05 mode appears to be discrepant, and a singe vaue for S c cannot match both C ps and b ps. Other groups have independenty estimated the unresoved source contamination, and their resuts are in genera agreement with ours. When the three-year data were initiay reeased the correction was A ps = ± μk 2 sr. Huffenberger et a. (2006) reanayzed the data and caimed A ps = ± 0.001, noticing that A ps was sensitive to the choice of gaaxy mask; using the Kp0 mask instead of Kp2 reduced the vaue of A ps. Revisiting our origina estimate for the three-year anaysis, we reduced the correction to 0.014±0.003 for the pubished papers. In a subsequent paper, Huffenberger et a. (2008), the same group corrected their origina estimate after finding a sma error, finding ± 0.001, consistent with our pubished resut. 4. TEMPERATURE-POLARIZATION SPECTRA The standard mode of adiabatic primordia density fuctuations predicts a correation between the temperature and poarization fuctuations. The temperature traces primariy the density, and E-mode poarization the veocity, of the photon baryon pasma at recombination. The correation was seen in earier WMAP data by Kogut et a. (2003) and Page et a. (2007). The anti-correation near = 30 provides evidence that fuctuations exist on superhorizon scaes, as is observed on an anguar scae arger than the acoustic horizon at decouping (Sperge & Zadarriaga 1997b). No significant changes have been made in the five-year TE anaysis. We continue to use the method described by Page et a. (2007) to compute the TE power spectrum. The inputs are the KaQV poarization maps (God et a. 2009) and the VW temperature maps. For high mutipoes >23, the ikeihood can be approximated as a Gaussian, and we continue to use the ansatz given in Appendix C of Page et a. (2007) to compute the covariance matrix. At ow mutipoes, 23, the ikeihood of the poarization data is evauated directy from the maps, foowing Appendix D in Page et a. (2007). Figure 5 shows the TE spectrum. At ow- the spectrum and error bars are approximated using the Gaussian form, athough these are not used for cosmoogica anaysis. With five years of data the anti-correation at = 140 is ceary seen in the data, and the correation at = 300 is measured with higher accuracy. The second anti-correation at 450 is now better characterized, and is consistent with predictions of the ΛCDM mode. The structure tests the consistency of the simpe mode, which fits both the TT and TE spectra with ony six Figure 5. WMAP five-year TE power spectrum. The green curve is the best-fit theory spectrum from the ΛCDM/WMAP Markov chain (Dunkey et a. 2009). For the TE component of the fit, χ 2 = 415, and there are 427 mutipoes and six parameters; thus the number of degrees of freedom is ν = 421, eading to χ 2 /ν = The partice horizon size at decouping corresponds to 100. The cear anticorreation between the primordia pasma density (corresponding approximatey to T) and veocity (corresponding approximatey to E) in causay disconnected regions of the sky indicates that the primordia perturbations must have been on a superhorizon scae. Note that the vertica axis is ( +1)C /(2π), and not ( +1)C /(2π). Figure 6. WMAP five-year TB power spectrum, showing no evidence of cosmoogica signa. The nu reduced χ 2 for = is Note that the vertica axis is ( +1)C /(2π), and not ( +1)C /(2π). parameters. The best-fit ΛCDM mode has χ 2 = 415 for the TE component, with 421 degrees of freedom, giving χ 2 /ν = The consistency confirms that the fuctuations are predominanty adiabatic, and constrains the ampitude of isocurvature modes. The signa at the owest mutipoes, evauated using the exact ikeihood, is used to provide additiona constraints on the reionization history. Athough sma, the measurement is consistent with the EE signa, and consistent with the three-year WMAP observations of Page et a. (2007). Dunkey et a. (2009) discuss constraints on reionization. No correation is expected between the temperature and the B-mode poarization. The TB spectrum is therefore primariy used as a nu test, and is shown in Figure 6. It is consistent with no signa, as expected; over = the reduced nu χ 2 is This measurement is used in Komatsu et a. (2009) to pace constraints on the presence of any parity vioating terms couped to photons, that coud produce a TB correation. We now incude the TB spectrum at high- as an optiona modue for the ikeihood code.

6 No. 2, 2009 WMAP FIVE-YEAR OBSERVATIONS: ANGULAR POWER SPECTRA 301 Figure 7. Conditiona ikeihoods for the = 2 7 EE mutipoe moments (back curves), computed using the WMAP ikeihood code by varying the mutipoe in question, with a other mutipoes fixed to their fiducia vaues. For exampe, in the = 4 pane, the back curve is f (x) L(d...,C3 EE,CEE 4 = x,c5 EE,...). For comparison, naïve pseudo-c estimates are aso shown with Gaussian errors (red curves). The pseudo-c errors are noise ony, whie the conditiona distributions incude cosmic variance. Tabe 3 Beam/Source Likeihood Treatment Effect on Parameters Likeihood Treatment n s σ 8 Standard ± ± SRCMARG ± ± SRCMARG ± ± BEAMMARG ± ± BEAMMARG ± ± Notes. One-dimensiona marginaized vaues for n s and σ 8 for various treatments of the unresoved point source and beam uncertainty in the WMAP ikeihood code. See Appendix A for descriptions of SRCMARG and BEAMMARG. Here, 5 and 20 indicate the error has been increased by a factor of 5 and 20, respectivey. 5. POLARIZATION SPECTRA Due to its therma stabiity (Jarosik et a. 2007) and wecharacterized gain, WMAP can measure poarization signas even though the scan pattern was not optimized for doing so. The poarization signa is manifested in the time-ordered data (TOD) differenty from the temperature signa. As a resut, some of the ow- poarization mutipoes are we samped and other mutipoes are poory samped and have arge statistica errors (Hinshaw et a. 2007; Page et a. 2007). This is a rather different situation than from that of the temperature spectrum, and the data must be anayzed with some care. When we anayze the = 2 temperature power spectrum, we use the ikeihood function rather than Gaussian errors, as the Gaussian approximation starts to break down with ony 4 effective modes measured in the map (the reduction is due to f sky 0.7). For poarization, this effect is even more dramatic, as our scan pattern significanty owers the effective number of mutipoes measured, particuary for EE = 2, 5, 7, and 9 and BB = 3 (the peaks seen in Figure 16 in Page et a. 2007). Figure 8 demonstrates the importance of using the fu ikeihood description. The figure shows both the pseudo-c estimates of the = 2 7 BB mutipoes and the conditiona ikeihoods computed using the WMAP ikeihood code by varying the mutipoe in question, keeping the rest of the spectrum fixed to the fiducia best-fit ΛCDM mode. From the pots it is cear

7 302 NOLTA ET AL. Vo. 180 Figure 8. Conditiona ikeihoods for the = 2 7 BB mutipoe moments (back curves), computed using the WMAP ikeihood code by varying the mutipoe in question, with a other mutipoes fixed to their fiducia vaues. For comparison, naïve pseudo-c estimates are aso shown with Gaussian errors (red curves). The pseudo-c errors are noise ony, whie the conditiona distributions incude cosmic variance. Note the arge difference between the ikeihood code and the pseduo-c vaue for = 3; this mode is sensitive to the time-ordered data baseine and is extremey poory measured by WMAP, iustrating the compicated noise structure of the poarization data on arge scaes. that the best estimates of the mean and the uncertainty are not attained with the pseudo-c estimates. We next consider the ow- EE and BB power spectra in more detai. The ow- EE power spectrum is shown in Figure 9.The uncertainties are obtained from the conditiona ikeihood and incude cosmic variance; thus one cannot doube the error fags to get the 95% confidence imits. If we zero out the <10 portion of the fiducia EE and TE spectra the χ 2 increases by 22.3, of which 2.7 is due to TE. Thus, the reionization feature in the EE power spectrum is preferred by Δχ 2 = The = 2, 3, 4, and 6 mutipoes are cosmic variance imited, and the S/N for the combined = 2 7 bandpower is 11. Considerabe effort has gone into understanding the W-band = 7 EE signa. Because of the apparent anomaousy high- = 7 EE vaue computed by the pseudo-c agorithm, we have avoided using the W-band maps in cosmoogica anaysis and use them ony as an additiona check on various modes. Figure 8 of Hinshaw et a. (2009) shows that the = 7 vaue, whie high, appears to be consistent with being in the tai of a propery computed ikeihood distribution. The W-band = 7 probem may be a signature of poor statistics rather than a systematic. However, more data are needed to understand this potentia anomay. The = 3 BB signa gives perhaps the cearest exampe of the importance of using the fu ikeihood code. Whie the pseudo-c estimate impies a significant detection of power, the fu ikeihood code shows this to not be the case. The physica cause of the arge uncertainty is that with our scan strategy an = 3 BB signa resembes an offset in the data and thus is not we separated from the baseine (Page et a. 2007; Hinshaw et a. 2009). We see no evidence for a B-mode signa at ow-, imiting the possibe eve to ( +1)C=2 6 BB /(2π) < 0.15 μk2 (95% CL), incuding cosmic variance. With τ = 0.1 and r = 0.2, a typica estimate for currenty favored modes of infation, ( +1)C=2 6 BB /(2π) μk2. Since a signa of 0.15 μk 2 corresponds roughy to r 20, one can see that WMAP s imit

8 No. 2, 2009 WMAP FIVE-YEAR OBSERVATIONS: ANGULAR POWER SPECTRA 303 Figure 9. WMAP five-year EE power spectrum at ow-. The error bars are the 68% CL of the conditiona ikeihood of each mutipoe, with the other mutipoes fixed at their fiducia theory vaues; the diamonds mark the peak of the conditiona ikeihood distribution. The error bars incude noise and cosmic variance; the point at = 7 is the 95% CL upper imit. The pink curve is the fiducia best-fit ΛCDM mode (Dunkey et a. 2009). is not based on the BB data, but on the tensor contribution to the TT and EE spectra as discussed by Komatsu et a. (2009). For EE at > 10, there are hints of signa in the data consistent with the standard ΛCDM mode. However, the significance is not great enough to contribute to knowedge of the cosmoogica parameters. The five-year high- EE spectrum is shown in Figure 10, aong with recent resuts from ground-based experiments (Leitch et a. 2005; Montroy et a. 2006; Sievers et a. 2007). For = , χ 2 = assuming C EE = 0, and drops by 8.4, or amost 3σ, assuming the standard ΛCDM mode. For the three-year data the equivaent change in χ 2 was 6.2. The high- BB spectrum is consistent with no signa, having a reduced χ 2 of 1.02 over = for the QV data. The ack of any signa in the ow- and high- BB data is a necessary check of the foreground subtraction. As seen in Page et a. (2007), foreground emission produces E-modes and B-modes at simiar eves; thus the absence of a B-mode signa suggests that the eve of contamination in the E-mode signa is ow. This is quantified by Dunkey et a. (2009). 6. SUMMARY AND CONCLUSIONS We have presented the temperature and poarization anguar power spectra of the CMB derived from the first five years of WMAP data. With greater integration time our determination of the third acoustic peak in the TT spectrum has improved. The ow- reionization feature in the EE spectrum is now detected at neary 5σ. The TB, EB, and BB spectra show no evidence for cosmoogica signa. The spectra are in exceent agreement with the best-fit ΛCDM mode. Our knowedge of the power spectrum is improving both due to more detaied anayses, better modeing and understanding of the foreground emission, and more integration time. A of the five-year WMAP data products are being made avaiabe through the Legacy Archive for Microwave Background Data Anaysis (LAMBDA 19 ), NASA s CMB Thematic Data Center. The temperature and poarization anguar power spectra presented here are avaiabe, as is the WMAP ikeihood 19 Figure 10. WMAP five-year EE power spectrum, compared with resuts from the Boomerang (Montroy et a. 2006, green), CAPMAP (Bischoff et a. 2008, orange), CBI (Sievers et a. 2007, red), DASI (Leitch et a. 2005, bue), and QUAD (Ade et a. 2008, purpe) experiments. The pink curve is the best-fit theory spectrum from the ΛCDM/WMAP Markov chain (Dunkey et a. 2009). Note that the y-axis is C EE, not ( +1)CEE /(2π). code which incorporates our estimates of the Fisher matrix, point sources, and beam uncertainties. The WMAP mission is made possibe by the support of the Science Mission Directorate Office at NASA Headquarters. This research was additionay supported by NASA grants NNG05GE76G, NNX07AL75G S01, LTSA , ATPNNG04GK55G, and ADP E.K. acknowedges support from an Afred P. Soan Research Feowship. This research has made use of NASA s Astrophysics Data System Bibiographic Services. We acknowedge the use of the CAMB, CMBFAST, CosmoMC, and HEALPix (Gorski et a. 2005) software packages. APPENDIX LIKELIHOOD TREATMENT OF SOURCE/BEAM UNCERTAINTIES In this appendix, we test the treatment of the unresoved source correction and beam uncertainties in the WMAP ikeihood code, and show that it produces the correct resuts for cosmoogica parameters. We adopt the same ikeihood treatment of the unresoved point source correction uncertainty for the five-year ikeihood code as used in the three-year code (Hinshaw et a. 2007, Appendix A), updated for the five-year vaue of A ps. Briefy, a correction to the ogarithmic ikeihood, L 2nL = L 0 +L 1, where L 0 is the standard ikeihood and L 1 is the combined source and beam correction, is cacuated assuming the C are normay distributed, a reasonabe assumption at high-. Huffenberger et a. (2007, Huf08) disagreed with the source and beam ikeihood modue used in the three-year anaysis, pointing out that the uncertainty in n s (the index of primordia scaar perturbations) was unchanged even if the uncertainty in A ps was increased by a factor of 100 (Figure 2 in their paper). They proposed an aternative approach, integrating the beam/point source covariance matrix into the cosmic variance/noise/mask covariance matrix and inverting the resut in order to compute L directy, instead of cacuating L 1 as a separate correction. Using this form of the ikeihood, as δa ps was increased, the uncertainty in n s increases (abeit modesty; δn s increased by 38% when δa ps 100 δa ps ).

9 304 NOLTA ET AL. Vo. 180 Figure 11. Left: one-dimensiona marginaized ikeihood distributions of σ 8 for various treatments of the source uncertainty in the ikeihood code: the standard ikeihood function (back), the aternative treatment of the source uncertainty described in Equation (A5; bue), the aternative treatment, but with the unresoved point source error increased by 5 (cyan). The agreement between back and bue curves shows that the standard treatment is producing the correct answer. Right: one-dimensiona marginaized ikeihood distributions of n s for various treatments of the beam uncertainties: the standard ikeihood function (back), the aternative treatment of the beam uncertainty described in Equation (A5; red), the aternative treatment, but with the beam error increased by a factor of 20 (orange). The agreement between the back and red curves shows that the standard treatment is producing the correct answer. Figure 12. Unbinned WMAP five-year temperature (TT) power spectrum (back), compared with the WMAP three-year resut (red). The sight upward shift of the five-year spectrum reative to the three-year spectrum is due to the change in the beam-transfer function. The pink curve is the best-fit ΛCDM mode to the WMAP5 data. However, whie we agree that it is striking that the error in n s is seemingy unaffected by the uncertainty in A ps,wehavesome concerns regarding the Huf08 approach. To quote Huf08, the errors on the source measurement do not make much difference, as ong as [δa ps ] < 0.003[μK 2 sr], and their Figure 2 impies the same hods true when δa ps = This vaue is significant because it is the uncertainty adopted for the three-year WMAP anaysis. When Huf08 adopted the same uncertainty, they found the same absoute uncertainty in n s as the WMAP team, but their centra vaue was shifted higher by This shift persisted as δa ps 0, and thus was seemingy not due to the point source uncertainty. The concusion we draw is that they found the same vaue of δn s as the WMAP three-year anaysis, but their vaue of n s was biased high because of the way they treat the beam uncertainties. We beieve the Huf08 vaue of n s woud be in agreement with that found in WMAP3, but that it is biased high due to their treatment of beam uncertainties. Huf08 quoted the vaue of L 1 computed with their aternative ikeihood modue for a particuar CMB spectrum distributed with the WMAP three-year ikeihood code test program, finding L 1 = 2.64, whereas the WMAP vaue is L 1 = As a check, we numericay marginaize the L 0 portion of the ikeihood over beam and point source errors, to see if we can reproduce their vaue. The desired integra is exp( L 1 /2) = where 1 L 0 (d C ) C (x, y) ( C TT + xσ ptsrc dxd ye (x2 + y T y)/2 L 0 (d C (x, y)), ) ( 1+ i y i σ beam (i) ) (A1) (A2) ( ) is the theoretica mode C TT perturbed by point source and beam errors. With 10 dimensions to integrate over (nine beam modes and one point source mode), norma grid-based quadrature is impractica, so we turn to Monte Caro integration Figure 13. Unbinned WMAP five-year temperature (TT) power spectrum as a function of frequency, divided by the best-fit ΛCDM mode to the WMAP data. instead: exp ( L MC / ) 1 N MC 2 e n L(d C (x (i), y (i) )) n L(d C ), (A3) N MC i=1 where x (i) and y (i) j are independent unit-variance norma deviates. With N MC = 10 4 points, we find L MC 1 = 1.29 ± 0.04, consistent with the WMAP resut of 1.22, but not the Huf08 resut of As a further test of whether our cosmoogica parameter estimates fuy capture the point source uncertainty, we have run a Markov chain with a modified form of the point source ikeihood modue, dubbed SRCMARG. The point source correction is cacuated via a simpe numerica integration, exp ( L ptsrc / ) 1 2 = dα 1 e α2 /2 ( L 0 d C + ασ ptsrc ) 2π (A4) Δ 2π N i= N w i e (iδ)2 /2 ( L 0 d C + iδσ ptsrc ) (A5) with N = 25, Δ = 0.2, and w i = 1 except at the endpoints where w N = 1/2 (the trapezoida rue). The resuting onedimensiona marginaized distribution for σ 8, shown in the eft pane of Figure 11, is indistinguishabe from our standard resut.

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