Cosmological Parameters from the QUaD CMB polarization experiment
QUaD collaboration, P. G. Castro, P. Ade, J. Bock, M. Bowden, M. L. Brown, G. Cahill, S. Church, T. Culverhouse, R. B. Friedman, K. Ganga, W. K. Gear, S. Gupta, J. Hinderks, J. Kovac, A. E. Lange, E. Leitch, S. J. Melhuish, Y. Memari, J. A. Murphy, A. Orlando, C. Pryke, R. Schwarz, C. O'Sullivan, L. Piccirillo, N. Rajguru, B. Rusholme, A. N. Taylor, K. L. Thompson, A. H. Turner, E. Y. S. Wu, M. Zemcov
aa r X i v : . [ a s t r o - ph . C O ] J a n S UBMITTED TO A P J Preprint typeset using L A TEX style emulateapj v. 02/07/07
COSMOLOGICAL PARAMETERS FROM THE QUAD CMB POLARIZATION EXPERIMENT QU A D COLLABORATION – P. G. C
ASTRO , P. A DE , J. B OCK , M. B
OWDEN , M. L. B
ROWN , G. C
AHILL , S. C HURCH ,T. C ULVERHOUSE , R. B. F RIEDMAN , K. G ANGA , W. K. G EAR , S. G UPTA , J. H INDERKS , J. K
OVAC , A. E. L ANGE ,E. L EITCH , S. J. M
ELHUISH , Y. M EMARI , J. A. M URPHY , A. O RLANDO
C. P
RYKE , R. S CHWARZ , C. O’ S ULLIVAN ,L. P ICCIRILLO , N. R AJGURU , B. R
USHOLME , A. N. T AYLOR , K. L. T HOMPSON , A. H. T URNER , E. Y. S. W U AND
M. Z
EMCOV
Submitted to ApJ
ABSTRACTIn this paper we present a parameter estimation analysis of the polarization and temperature power spectrafrom the second and third season of observations with the QUaD experiment. QUaD has for the first timedetected multiple acoustic peaks in the E -mode polarization spectrum with high significance. Although QUaD-only parameter constraints are not competitive with previous results for the standard 6-parameter Λ CDM cos-mology, they do allow meaningful polarization-only parameter analyses for the first time.In a standard 6-parameter Λ CDM analysis we find the QUaD TT power spectrum to be in good agreementwith previous results. However, the QUaD polarization data shows some tension with Λ CDM. The origin of this1 - σ tension remains unclear, and may point to new physics, residual systematics or simple random chance.We also combine QUaD with the five-year WMAP data set and the SDSS Luminous Red Galaxies 4 th datarelease power spectrum, and extend our analysis to constrain individual isocurvature mode fractions, constrain-ing cold dark matter density, α cdmi < .
11 (95% CL), neutrino density, α ndi < .
26 (95% CL), and neutrinovelocity, α nvi < .
23 (95% CL), modes. Our analysis sets a benchmark for future polarization experiments.
Subject headings:
CMB, anisotropy, polarization, cosmology INTRODUCTION
The anisotropies of the Cosmic Microwave Background(CMB) radiation are among the most important tests of cos-mology. The large-angle Sachs-Wolfe effect, multiple acous-tic oscillations and the Silk damping tail in the tempera-ture power spectrum have now been confirmed by a rangeof experiments from the largest angular scales down to an-gular scales of a few arcminutes (Dunkley et al. (2008);Reichardt et al. (2008)). The full repository of CMB dataavailable, in conjunction with other cosmological observ-ables, such as data coming from the large-scale distribu- Scottish Universities Physics Alliance (SUPA), Institute for Astronomy,University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh EH93HJ, UK. Current address : CENTRA, Departamento de Física, Edifício Ciência,Instituto Superior Técnico, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugal. School of Physics and Astronomy, Cardiff University, Queen’s Build-ings, The Parade, Cardiff CF24 3AA, UK. Jet Propulsion Laboratory, 4800 Oak Grove Dr., Pasadena, CA 91109,USA. California Institute of Technology, Pasadena, CA 91125, USA. Kavli Institute for Particle Astrophysics and Cosmology and Departmentof Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305,USA. Kavli Institute for Cosmological Physics, Department of Astronomy &Astrophysics, Enrico Fermi Institute, University of Chicago, 5640 South El-lis Avenue, Chicago, IL 60637, USA. Current address : Cavendish Laboratory, University of Cambridge, J.J.Thomson Avenue, Cambridge CB3 OHE, UK. Department of Experimental Physics, National University of IrelandMaynooth, Maynooth, Co. Kildare, Ireland. APC UMR 7164 (Univ. Paris Diderot-Paris 7 - CNRS - CEA - Obs. deParis), 10, rue Alice Domon et Léonie Duquet, 75205 Paris Cedex 13, France. Current address : NASA Goddard Space Flight Center, 8800 GreenbeltRoad, Greenbelt, Maryland 20771, USA. School of Physics and Astronomy, University of Manchester, Manch-ester M13 9PL, UK. Current address : Department of Physics and Astronomy, UniversityCollege London, Gower Street, London WC1E 6BT, UK. tion of galaxies or Supernova type Ia observations, are ex-tremely well described by the spatially flat Λ CDM cosmolog-ical model.A generic prediction of cosmology is that the CMB pho-tons should be polarized at the 10% level. The polarizationfield can be decomposed into two components: primary even-parity, curl-free E -modes are generated at the last scatteringsurface by both scalar and tensor metric perturbations (grav-itational waves); primary odd-parity B -modes are generatedonly by tensor perturbations due to gravitational waves pass-ing through the primordial plasma. Secondary anisotropiesin both the E and B -mode polarization arise at the epoch ofreionization, while E and B -modes are mixed by gravitationallensing by intervening large-scale structure along the line ofsight (see eg Hu & White (1997)). Observations of this lin-early polarized component provide an important consistencycheck of the standard model and a detection of primordialgravitational waves in the odd-parity B -mode on large angularscales would be strong evidence for inflation.After the DASI experiment (Kovac et al. (2002)) madethe first measurement of E -mode power, other experi-ments have provided us with further measurements ata wide range of angular scales (Barkats et al. (2005);Readhead et al. (2004); Montroy et al. (2006); Sievers et al.(2007); Page et al. (2003); Bischoff et al. (2008); Nolta et al.(2008)). Despite this we were still lacking precision measure-ments of its power spectra down to arcminute scales, as wehave for the temperature. The B -mode polarization has not yetbeen detected and only upper limits have been determined.The QUaD experiment is at the forefront of this small-scale polarization quest, and after three years of observationshas delivered the highest resolution E -mode spectrum and the DASI stands for “Degree Angular Scale Interferometer”. QUaD stands for “QUEST and DASI”. In turn, QUEST is “Q & UExtragalactic Survey Telescope”. The two experiments merged to becomeQUaD in 2003.
QUaD collaborationtightest upper limits on the B -modes yet measured. This is asignificant improvement over the first season of data results,previously reported by Ade et al. (2008). In particular, thesensitivity of QUaD has allowed us to see, for the first time,four acoustic oscillations in the E -mode spectrum and all sig-nificant oscillations in the TE spectrum to ℓ = 2000. The over-all consistency of peak phases and spacings between the tem-perature and QUaD EE data was shown in Pryke et al. (2008)(hereafter referred to as the "Power Spectra Paper").In this paper we concentrate on using the QUaD tempera-ture and polarization power spectra to constrain the standardcosmological model. Using this baseline model, we analyzethe different contributions coming from each of the QUaDspectra. We also go beyond the standard LCDM model us-ing the QUaD data in combination with WMAP and SDSS toconstrain an isocurvature contribution. COSMOLOGICAL PARAMETER ESTIMATION: METHODOLOGY
Monte Carlo Markov Chain
The Monte Carlo Markov Chain (MCMC) method is amethod designed to efficiently explore an unknown Probabil-ity Distribution Function (PDF) by sequentially drawing sam-ples from it according to a proposal probability function inour case the Metropolis algorithm (Metropolis et al. (1953)among others). The ensemble of these samples constitute aMarkov Chain whose distribution corresponds to that of theunknown PDF. We adopted the Gelman and Rubin R -statisticto verify that our chains are properly mixed and converged(Gelman & Rubin (1992); Verde et al. (2003)), and comparesets of 4 chains of around 100 000 steps from which we re-move a burn-in period during which our criterium is not met.We use at least 80 000 steps after burn-in.The rate of convergence of the Markov chain is sloweddown by degeneracies between parameters, and the choice ofthe step size in the Metropolis algorithm. Therefore we applya standard partial re-parametrization of the parameter space assuggested in Kosowsky et al. (2002). To further reduce the re-maining degeneracies between parameters, we apply a changeof basis in parameter space described in Tegmark et al. (2004)which uses a covariance matrix to take account of all the cor-relations between parameters. In the new basis, the new pa-rameters have zero average and unit variance.When we have a fair sample of the underlying distribution,the MCMC method trivializes marginalization to a simpleprojection of the points of the chain. The mean marginalizedvalue of each parameter will hereafter be called "the mean re-covered model". Note that some authors refer to the meanrecovered model as the “best-fit model”.To obtain constraints on the mean parameters values onesimply produces the 1D histograms of the chain values foreach parameter, and calculate confidence intervals using the p th and (1 - p th ) quantiles of the histograms as in Verde et al.(2003). Normally we use 68% equivalent to a nominal 1- σ constraint. However if the constraint as defined above hits theprior boundary on one end we instead choose the level whichcontains 95% of the total probability and quote an upper (orlower) limit.We shall also plot 2-D marginalized parameter distributionswith 68% and 95% contours estimated at ∆ ln L = - . - .
17 from the peak values. Assuming a Gaussian distribu-tion we quote χ values corresponding to our mean recov-ered model, and the Probability To Exceed (PTE), P ( > χ | ν ),which gives the random probability to have found the mea-sured value of χ or greater by chance, for ν degrees of free- dom. The Likelihood and Nuisance Marginalization
The likelihood for the measured C ℓ bandpowers is well ap-proximated by a Gaussian distribution, given by P ( ˆ C b | C th b ) ∝ exp (cid:20) - ∆ C b ( p ) M - bb ′ ∆ C † b ′ ( p ) (cid:21) , (1)where ∆ C b ( p ) = ˆ C b - C th b ( p ), ˆ C b are the measured QUaDbinned C ℓ bandpowers, C th b ( p ) are the theoretical power spec-tra which depend on the cosmological parameters, p , andwhich have been transformed to predictions of the binnedspectra by means of the experimental bandpower windowfunction (BPWF), and M bb ′ = h ∆ C b ∆ C † b ′ i is the measured ˆ C b bandpower covariance matrix (BPCM).The BPCM is estimated from an ensemble of simulations ofthe CMB sky, assuming a fixed fiducial Λ CDM, run throughthe QUaD analysis pipeline (see e.g. Brown et al. (2005)). Inour analysis, this BPCM remains independent of the cosmo-logical parameters, but in principle we should vary it as wemove around parameter space. This re-scaling of the samplevariance with C th is equivalent to saying that P ( C th b ( p ) | ˆ C b )is log-normally distributed, and in the case where noise ispresent, offset log-normally distributed. Bond et al. (1998)propose a transformation of the bandpowers and BPCMwhich accounts for this effect and allows use of a fixedBPCM. We have tested the impact of this transformation onour full data set and find that the difference in parameter esti-mates and uncertainties is insignificant.The Gaussian likelihood has the added benefit that we cansimplify the marginalization over beam and calibration nui-sance parameters. Indeed, in this case, one can directly applyan analytic marginalization scheme as in Bridle et al. (2002).The resulting marginalization results in extra terms added tothe BPCM of our Gaussian likelihood that acts as a source ofextra noise. The likelihood function is then given byln L = - ∆ C b M ′ - bb ′ ∆ C † b ′ -
12 Tr ln M ′ , (2)where M ′ bb ′ = M bb ′ + σ C b C † b ′ + σ b δℓ b C b ℓ b ′ C † b ′ , (3)is the marginalized bandpower covariance matrix, where σ is the variance on the calibration, σ b = θ FWHM / √ θ FWHM is the effective beam size, δ is the fractional beam error and ℓ b is the average multipole in a bin.In assessing the goodness-of-fit of our mean recoveredmodels, and when comparing the measured ˆ C b bandpowerswith the WMAP5 Λ CDM model, we shall use the χ -statistic,introduced previously in Section 2.1, and which we define by χ = ∆ C b M ′ - bb ′ ∆ C † b ′ , (4)where we use the nuisance marginalized BPCM, M ′ bb ′ , as de-fined in the previous equation. The Standard Cosmological Model
We parameterize our flat Λ CDM cosmological model withthe following standard set of 6 cosmological parameters: theHubble constant, H = 100 h kms - Mpc - ; the physical matterdensity, Ω m h ; the physical baryon density, Ω b h ; the ampli-tude of scalar fluctuations, A s ; the scalar spectral index, n s ;osmological Parameters from QUaD 3 F IG . 1.— 2-D projected basic parameter likelihood surfaces with two-parameter 1- and 2-sigma contours for QUaD only constraints using the TT / TE / EE / BB data set (TP: the blue contours), using the TE / EE / BB data set (P: the red & magenta contours) and using the TT spectrum (T: the yellow & orange contours)versus the WMAP5 constraints (black/empty contours). Pivot scale used is k p = 0 .
05 Mpc - . and the optical depth, τ . When using QUaD data by itself, wepresent the combination A s e - τ as our individual constraintson the degenerate parameters A s and τ are prior driven andthus biased, as explained in Appendix A. Initial conditions aretaken to be purely adiabatic with an initial power-law mass-density perturbation spectrum. Due to the range of angularscales probed by QUaD, the pivot point we use when ana-lyzing QUaD data by itself is k p = 0 .
05 Mpc - (note this isindependent of h ). When comparing our QUaD results withWMAP we regenerate WMAP best-fit values based on thispivot value using our own pipeline, however when addingQUaD data to other data sets for a combined analysis we re-vert to the WMAP preferred pivot scale of k p = 0 .
002 Mpc - .To generate our theoretical spectra we use the publiclyavailable CAMB code (Lewis et al. (2000)), including the ef-fects of reionization, and gravitational lensing by foregroundstructure. We impose the following flat priors in the likeli-hood analysis: 0 ≤ Ω c h ≤
1, 0 ≤ Ω b h ≤
1, 0 . ≤ θ ≤ . ≤ τ ≤ .
8, 0 ≤ A s ≤ . ≤ n s ≤
2. The parameter θ isthe angular sound horizon, and Ω c h is the physical cold darkmatter density. Note that the partial re-parametrization of theparameter space as suggested in Kosowsky et al. (2002) intro-duces an implicit prior on the h parameter. RESULTS: BASIC 5-PARAMETER CONSTRAINTS
QUaD Only Constraints
The QUaD data set we use to constrain cosmological pa-rameters are the optimally combined spectra obtained fromthe 100 GHz, 150 GHz and frequency-cross temperature andpolarization E- and B- power spectra, measured in 23 band-powers over angular multipoles from 200 < ℓ < . The data set is publicly available online at http://quad.uchicago.edu/quad
In parameter estimation we use the diagonal and the firsttwo off-diagonal terms of the BPCM for TT - TT , TE - TE , EE - EE , and BB - BB covariances, but only the diagonal and firstoff-diagonal terms in the TT - TE and TE - EE covariances. Thisis motivated by the need to avoid excessive noise in the off-diagonal terms of the BPCM, due to its estimation from nu-merical simulations. We also ignore the covariance between TT and EE , which is much smaller than the other terms.We explore various sets of combinations of the QUaD tem-perature and polarization data in order to understand the newinformation each spectrum brings to parameter estimation. InTable 1 we present the mean recovered models. Figure 1shows the corresponding 2-D marginalized contour projec-tions of the likelihood in the 5-parameter space.All of the statistics shown verify that the QUaD TT tem-perature power spectrum is compatible with the results fromWMAP5. This is a non-trivial test, since the overlap of scalesmeasured by QUaD and WMAP5 is only in the range ℓ ≈ ℓ ≈ ℓ ≈ TE / EE / BB combination wehave Ω b h = 0 . ± . Ω b h = 0 . ± . χ of the WMAP5 best-fit modelfor this data set has a PTE of 7% indicating a modest degreeof tension. The spectrum responsible for this behaviour seemsto be the TE spectrum.Clearly the TE only constraints are weak and most param-eters are prior driven, but surprisingly we obtain constraintson Ω b h and Ω m h that are not influenced by their choice ofpriors. Figure 2 shows the 2-D projected likelihood surfacefor the ( Ω b h , Ω m h ) parameter space. To illustrate the dif-ference with the TT only contours we overplot them. In ad-dition we show the results from WMAP5 and the Big BangNucleosynthesis (BBN) constraint of Ω b h = 0 . ± . TE QUaD collaboration
TABLE 1B
ASIC COSMOLOGICAL MEAN PARAMETER CONSTRAINTS USING QU A D BANDPOWER SPECTRA FOR VARIOUS DATACOMBINATIONS .Symbol Q08 TT / TE / EE / BB Q08 TE / EE / BB Q08 TT Q08 TE Q08 EE / BB WMAP5 Ω b h . + . - . . ± . . + . - . . ± . . + . - . . + . - . Ω m h . + . - . . ± .
017 0 . + . - . . ± .
025 0 . + . - . . + . - . h . ± .
09 0 . ± .
10 0 . + . - . . ± .
11 0 . ± .
18 0 . + . - . A s e - τ a . ± .
08 0 . ± .
09 0 . + . - . . ± .
13 0 . ± .
21 0 . + . - . n s a . ± .
078 0 . ± .
152 0 . + . - . . + . - . . + . - . . ± . χ ( ν ) b . . . . . PTE : P ( ≥ χ | ν ) 40 .
26% 14 .
72% 75 .
38% 29 .
14% 76 . χ ( WMAP | Q c . . . . . PTE ( WMAP | Q
08) 11 .
36% 7 .
07% 91.24% 11 .
24% 65 . a The pivot point for A s and n s is k p = 0 .
05 Mpc - for both the QUaD data and WMAP5 data. b χ for the 6-parameter mean recovered model against QUaD data, with the number of degrees of freedom in brackets. c χ for WMAP5 mean recovered model given the QUaD data set, with the number of degrees of freedom in brackets.F IG . 2.— 2-D marginalized contours of the parameters Ω b h versus Ω m h obtained from QUaD TE data only. Also plotted are the contoursfrom QUaD TT data only, the results from WMAP5, and the BBN constraint(Kirkman et al. (2003)). QUaD bandpower spectrum versus its mean recovered modeland the WMAP5 full data set best-fit model. This figure vi-sually illustrates the differences of the mean recovered mod-els between the two data sets in terms of height and locationof the peaks. The main reason for the higher baryon densityparameter seems to be larger acoustic oscillations at highermultipole in TE , as well as a shift to higher multipoles of thepeaks, resulting in a slight degeneracy with h , which may ex-plain its high value. The origin of this source of tension isunclear, but could be due to a new physical mechanism, resid-ual systematics or random chance.Another interesting result comes from the EE and BB spec-tra. As expected they provide very little information on pa-rameters, and the χ of the WMAP5 best-fit model for thisdata set has an acceptable PTE of 65 . n s = 0 . + . - . ). So although TT and TE share the majority of the constraining power, the EE and BB spectra exert an influence in combination with the remainingspectra by restricting the n s -range to low values.If we combine all the spectra together then the polarizationdata dominate the constraints. The majority of parameters areconsistent with the WMAP5 results, but the spectral index n s is lower, influenced by the EE / BB contribution, and the Hub- th]F IG . 3.— Plot of the QUaD TE data bandpower spectrum (in red) versusthe QUaD TE mean recovered model (in solid black) together with WMAP5best-fit model (blue). For the TE mean recovered model, we assumed theWMAP5 best-fit value for the optical depth ( τ = 0 . A s = 0 .
75, given our A s e - τ constraint. ble parameter h and Ω b h are higher, driven by the polariza-tion data in particular the TE spectrum. Compared to the BBNvalue Ω b h is almost 3- σ away. Figure 4 shows a comparisonbetween our TT / TE / EE / BB data and models and the WMAP5data and best-fit model. The χ of the WMAP5 best-fit modelindicates that there is an 11 .
36% chance that the combinedQUaD spectra are a realization of this model.
Combining QUaD with Other Data Sets
In this section we will add to the QUaD spectra the WMAP5-year TT , TE , EE and BB data set. We use the WMAP5 like-lihood code, publicly available on the LAMBDA website ,and their methodology (Dunkley et al. (2008)), but do not in-clude the Sunyaev Zel’dovich (SZ) marginalization. We shallassume the two data sets are independent, as the QUaD dataonly covers a small fraction of the WMAP5 sky, and the over-lap in multipole range is only partial.We will further add large-scale structure data from theSDSS Luminous Red Galaxies (LRG) fourth data releaseusing publicly available likelihood code, measurements andwindow functions (Tegmark et al. (2006)). Results from theSDSS LRG and the main SDSS galaxy samples are consis-tent, but the former provides higher signal-to-noise ratio. We LAMBDA website: http://lambda.gsfc.nasa.gov/ osmological Parameters from QUaD 5
TABLE 2B
ASIC MEAN PARAMETERS FOR QU A D TT / TE / EE / BB , SDSS LRG AND
WMAP5
DATA .Symbol Q08 TT / TE / EE / BB +WMAP5 Q08 TT / TE / EE / BB +WMAP5+SDSS WMAP5 Ω b h . + . - . . + . - . . + . - . Ω m h . ± . . + . - . . ± . h . ± .
027 0 . ± .
019 0 . + . - . τ . ± .
017 0 . ± .
016 0 . ± . A s a . ± .
038 0 . + . - . . ± . n s a . + . - . . + . - . . + . - . The pivot point for A s and n s is k p = 0 .
002 Mpc - for QUaD, WMAP5 and SDSS LRG data.F IG . 4.— The QUaD power spectra used in this analysis (red points) for TT , TE , EE and BB (top to bottom). The blue data points are WMAP 5-year power spectra data. The blue line shows the basic WMAP5 best-fitmodel as defined in Dunkley et al. (2008), while the black solid line showsour TT / TE / EE / BB mean recovered model with values given in Table 1. Weassumed the WMAP5 best-fit value for the optical depth ( τ = 0 . A s = 0 .
79, given our A s e - τ constraint. use the SDSS LRG matter power spectrum over wavenum-bers 0 . h Mpc - < k SDSS < . h Mpc - so that we do not haveto consider any nonlinear correction. We marginalize overthe amplitude of the galaxy power spectrum which removesany dependence on the galaxy bias parameter, b g , and linearredshift-space distortion.As can be seen in Table 2, the QUaD TT / TE / EE / BB powerspectra have little impact on the baseline 6-parameter meanparameter fit when combined with WMAP5. This is per-haps unsurprising, given the accuracy of the WMAP5 mea- surement of the first acoustic peak in the TT spectrum, andits low- ℓ power in TT and TE . The impact it does have is totighten the error bars on parameters determined from the rela-tive heights of acoustic peaks, i.e. on the baryon density, Ω b h ,and the matter density, Ω m h , as QUaD data adds a substantialamount of well-defined peak information at high- ℓ .When we combine the SDSS LRG and WMAP5 data withthe QUaD data we see an improvement compared to theQUaD and WMAP5 combination, as expected. This improve-ment is mostly due to the extra constraining power on Ω m h and Ω b h coming from the break-scale in the SDSS galaxypower spectrum, and the baryon acoustic oscillations. How-ever the QUaD data still reduces the uncertainty on Ω b h . BEYOND THE STANDARD 6-PARAMETER MODEL:ISOCURVATURE MODES
Theoretical predictions of isocurvature modes and theirevolution, and the role of CMB polarization observa-tions in constraining them, has been an active fieldover the past few years (Kawasaki & Sekiguchi (2007);Keskitalo et al. (2007); Bean et al. (2006); Beltran et al.(2004); Moodley et al. (2004) among many others). Pureisocurvature perturbations have been ruled out (Stompor et al.(1996); Langlois & Riazuelo (2000); Enqvist et al. (2000);Amendola et al. (2002)) although the presence of a subdom-inant isocurvature fraction has been claimed (Keskitalo et al.(2007)). Observationally, isocurvature modes have a phasedifference from adiabatic modes, which provides a distinctsignature.We can completely characterize the primordial perturba-tions by one adiabatic and several isocurvature components.The adiabatic component is given by the associated curva-ture perturbation R corresponding to an initial overdensity δ = δρ/ρ . The non-adiabatic components are given by en-tropy perturbations S x = δ x - (3 / δ γ between photons and adifferent species, x . These correspond to four possible nonde-caying isocurvature modes: baryon density, cold dark matterdensity (cdmi), neutrino density (ndi) and neutrino velocity(nvi). Bucher et al. (2000) have presented a thorough analysisof these components.We parameterize the contribution of adiabatic and isocur-vature modes to the total temperature and polarization powerspectra by C X ℓ = A s (cid:2) (1 - α ) ˆ C X , Ad ℓ + α ˆ C X , Iso ℓ (cid:3) , (5)where α is the isocurvature fraction. The adiabatic spectra, C X , Ad ℓ , and the isocurvature spectra, C X , Iso ℓ , are defined withunit amplitude and the same spectral index. In this analysiswe shall assume there is no correlation between adiabatic andisocurvature modes, and will constrain one isocurvature modeat a time. Also we do not present results for the baryon density QUaD collaboration TABLE 3CDM I
SOCURVATURE MEAN PARAMETER CONSTRAINTS FOR QU A D TT / TE / EE / BB , WMAP5 AND
SDSS LRG
DATA .Symbol Q08 TT / TE / EE / BB +WMAP5 Q08 TT / TE / EE / BB +WMAP5+SDSS WMAP5 Ω b h . + . - . . ± . . + . - . Ω m h . + . - . . + . - . . + . - . h . ± .
039 0 . ± .
022 0 . ± . τ . ± .
017 0 . + . - . . ± . A s a . ± .
037 0 . + . - . . ± . n s a . ± .
023 0 . + . - . . + . - . α cdmi (95 % cl) < . < . < . a The pivot point for A s and n s for all isocurvature constraints is k p = 0 .
002 Mpc - for QUaD, WMAP and SDSS data.TABLE 4NDI I SOCURVATURE MEAN PARAMETER CONSTRAINTS FOR QU A D TT / TE / EE / BB , WMAP5 AND
SDSS LRG
DATA .Symbol Q08 TT / TE / EE / BB +WMAP5 Q08 TT / TE / EE / BB +WMAP5+SDSS WMAP5 Ω b h . ± . . ± . . + . - . Ω m h . + . - . . ± . . + . - . h . ± .
030 0 . + . - . . + . - . τ . ± .
017 0 . ± .
016 0 . ± . A s . ± .
058 0 . + . - . . ± . n s . + . - . . + . - . . + . - . α ndi (95 % cl) < . < . < . SOCURVATURE MEAN PARAMETER CONSTRAINTS FOR QU A D TT / TE / EE / BB , WMAP5 AND
SDSS LRG
DATA .Symbol Q08 TT / TE / EE / BB +WMAP5 Q08 TT / TE / EE / BB +WMAP5+SDSS WMAP5 Ω b h . ± . . + . - . . ± . Ω m h . ± . . ± . . ± . h . ± .
029 0 . ± .
018 0 . + . - . τ . ± .
017 0 . + . - . . ± . A s . + . - . . + . - . . ± . n s . + . - . . + . - . . + . - . α nvi (95% cl ) < . < . < . isocurvature mode as these have a very similar signature to thecold dark matter mode.We analyze the QUaD TT / TE / EE / BB power spectra com-bined with WMAP5, and combined with the WMAP5 plusthe SDSS LRG data. The shape of the galaxy power spec-trum is sensitive to an isocurvature contribution, and hasbeen used in the past to improve on isocurvature constraints(e.g. Beltran et al. (2004, 2005); Crotty et al. (2003)). The re-sults we obtain are given in Table 3 for the cdmi mode, inTable 4 for the ndi mode and in Table 5 for the nvi mode.Our analysis shows a small improvement in the isocurva-ture cold dark matter constraint when we add the QUaD tothe WMAP5 data, from α cdmi < .
21 to α cdmi < .
19 (95%confidence limits). In addition we find an improvement in the Ω b h and Ω m h constraints. There is a similar improvementfor the neutrino density isocurvature constraints: we go from α ndi < .
38 to α ndi < .
37. For the neutrino velocity isocur-vature modes there is no improvement, the constraint stayingat α nvi < . α cdmi < .
11 (95 % CL). The largestimprovement is for the neutrino density isocurvature mode, α ndi < .
26, while the smallest improvement is for the neu-trino velocity isocurvature mode, α nvi < . SUMMARY AND CONCLUSIONS
We have presented a standard cosmological parameter con-straint analysis, and its extension to include isocurvaturemodes, using QUaD TT , TE , EE and BB bandpower spectra.This is the first CMB experiment to detect with confidence theacoustic oscillations in the EE spectrum and so the first to beable to provide significant constraints on cosmological param-eters from individual CMB polarization spectra — in fact ourpolarization-only constraints are superior to our temperature-only constraints. In combination with the WMAP5 data setQUaD offers a small improvement in the constraints on thebaryon and matter densities.We find our QUaD temperature data is in good agreementwith the results from WMAP5, which is a non trivial testof LCDM as the QUaD data extends to ℓ ≈ TE , EE and BB )data is in less good agreement yielding a higher baryon den-sity value of Ω b h = 0 . ± . Ω b h =0 . + . - . from our TT data and 0 . + . - . from ourre-analysis of WMAP5. A χ test shows there is a 7% proba-osmological Parameters from QUaD 7 F IG . 5.— 2D marginalized plot, showing the scattered values (green) of the 5 parameters mean recovered basic cosmological model obtained from 50simulations of QUaD TT , TE , EE and BB data generated from the WMAP3 best-fit model (in red) from table 2 of Spergel et al. (2007). The mean over the 50simulations is shown as a blue point. bility of the QUaD polarization results arising by chance, as-suming the WMAP5 Λ CDM model is correct. Although notof high significance, this modest level of tension, that seems tooriginate from the TE spectrum, could be due to new physicsin polarization, residual systematics in the data, or randomchance. It will be interesting to see if this trend continues infuture polarization experiments.We also investigate isocurvature cold dark matter den-sity, neutrino density and neutrino velocity modes. We findQUaD provides a marginal improvement on the fractionalcold dark matter density mode parameter, α cdmi , from < . < . APPENDIXSIMULATING PARAMETER ESTIMATION
We test our MCMC pipeline by running it on a set of 50 simulations of the QUaD TT , TE , EE and BB bandpower spectra.These were generated by simulating the signal and noise properties of the time-ordered data and passing these through the QUaDpipeline in the same way as the data (see the Power Spectra Paper for details). The input cosmological model for these simulationswas the WMAP3 mean recovered model (see Table 2 of Spergel et al. (2007)). The scatter in the values of the mean recoveredmodel obtained from each one of the 50 simulations can be seen in Figure 5. We also overplot the average values for eachparameter calculated from the 50 simulations (see blue points). We have verified that the scatter in the simulated mean parameterresults is close to the size of the contours produced by our MCMC code when using real data, indicating that our code accuratelyestimates the parameter uncertainties.We can also compare the mean parameter values and the scatter about them with the input WMAP3 best-fit model (red crosses).The average over the simulations closely matches the input model indicating that our parameters are not biased. If constrainedindependently, the scalar amplitude, A s , and the optical depth, τ , parameters are biased, their values being systematically higherthan the input values. This is due to the combination of the large degeneracy between the amplitude, A s , and optical depth, τ , andthe parameter priors. To break this degeneracy requires large-scale polarization measurements probing the re-ionization bumpsat lower ℓ -modes. As can be seen in the Figure, this problem can be avoided if we combine A s and τ into the parameter A s e - τ along the line of degeneracy, which is the approach followed in Section 3.1. REFERENCESAde, P. et al. 2008, ApJ, 674, 22, arXiv:0705.2359 Amendola, L., Gordon, C., Wands, D., & Sasaki, M. 2002, Phys. Rev. Lett.,88, 211302, astro-ph/0107089