PARAMETER ESTIMATION FROM IMPROVED MEASUREMENTS OF THE COSMIC MICROWAVE BACKGROUND FROM QUaD

The cosmic microwave background (CMB) radiation contains fluctuations in temperature and polarization which have specific spectral features that record the evolution and constituent properties of the universe. The radiation is predicted to be polarized at the 10% level due to Thomson scattering in the presence of velocity inhomogeneities in the photon–baryon fluid at last scattering.

The standard cosmological model predicts acoustic peaks in CMB intensity and polarization spectra. Polarized CMB radiation can be decomposed into even-parity E-modes which are generated by scalar and tensor perturbations and odd-parity B-modes which are generated by gravitational waves and gravitational lensing effects. In this paper, we examine the cosmological implication polarization spectra including the spectra produced by the 2009 improved analysis of QUaD second and third season observations presented in Brown et al. (2009). QUaD improves on the detections of E-modes made by the DASI (Kovac et al. 2002), CAPMAP (Hedman et al. 2002), Boomerang (Jones et al. 2006), Wilkinson Microwave Anisotropy Probe (WMAP; Dunkley et al. 2009), and CBI (Readhead et al. 2004) experiments, adding accurate polarization data at small angular scales.

We presented design and optics reports describing the QUaD¹⁵ experiment in Hinderks et al. (2009) and O'Sullivan et al. (2008). The first season of QUaD results appeared in Ade et al. (2008). We first reported measurements of temperature and polarization from the second and third season QUaD data in Pryke et al. (2009) and cosmological parameter analysis of the data was carried out in Castro et al. (2009). The second and third season data sets from the QUaD experiment have been re-analyzed using a ground template removal technique rather than field differencing to remove ground contamination. A description of the method and maps and spectra from two parallel and independent pipelines can be found in Brown et al. (2009). The re-analysis has effectively doubled the size of the QUaD field, increasing the precision of the CMB spectra measurements by ∼30% and constraining the amplitude of the lensing B-modes to <0.57 μK² at 95% confidence. In Brown et al. (2009), we also presented a cosmological parameter analysis which gauged the effect of using the improved spectra from QUaD in combination with WMAP and ACBAR data, demonstrating that QUaD data add significant power for constraining cosmological models which include tensors and running of the spectral index.

In this paper, we carry out a cosmological parameter estimation in which we explore separately the strengths of CMB temperature data and CMB polarization data including the two year data release by BICEP (Chiang et al. 2010), highlighting the contributions of each to a multi-experiment analysis.

The QUaD data set consists of 143 selected days of data measured during the Austral winters of 2006 and 2007, at 100 GHz and 150 GHz, in a region of approximately 60 deg², and has produced polarization spectra of unprecedented quality (Pryke et al. 2009; Brown et al. 2009). The QUaD experiment is the first experiment with the sensitivity to detect multiple acoustic oscillations in the E-mode spectrum and TE spectrum up to ℓ = 2000 in addition to providing the lowest upper limits on B-mode detection (Brown et al. 2009). The spectra we used for cosmological data analysis, Temperature–Temperature (TT), Temperature–E-mode (TE), E-mode–E-mode (EE), and B-mode–B-mode (BB), are an optimally weighted combination of three sets of spectra: 100 GHz, 150 GHz, and 100 GHz–150 GHz cross, each with 23 band power values in the range 200 < ℓ < 2000.

Our CosmoMC-based parameter analysis is broadly similar to the Monte Carlo Markov chain analysis of Castro et al. (2009), hereafter Parameter Paper 1. A single, constant covariance matrix was estimated from simulations within the QUaD pipeline. The covariance matrix is populated only in the diagonal, 1st off-diagonal, and 2nd off-diagonal terms of each of the sub-blocks of the spectra. TT–TE and TE–EE have no 2nd off-diagonal terms, and TT–EE and EE–BB have covariance terms up to the 12th and 4th band powers, respectively. Beyond these regions the covariance matrix would be dominated by noise from the numerical simulations. We also generated offset-lognormal x-factors from noise-only simulations. This enables us to model the offset-lognormal likelihood for our parameter estimation as suggested by Bond et al. (2000).

We make use of templates of the Sunyaev–Zel'dovich (SZ) amplitude derived from Komatsu & Seljak (2002), which model a frequency-dependent contribution to the temperature power spectrum from the thermal SZ effect.¹⁶ We apply a cut of spectral power above ℓ = 2000 to avoid the effects of residual point-source contamination and analytically marginalize over the SZ amplitude.¹⁷ The justification of a fit to SZ parameters on scales below ℓ = 2000 has not been confirmed by the high multipole temperature data obtained by QUaD (Friedman et al. 2009).

We estimate likelihoods from our CMB distribution using the publicly available CosmoMC (Lewis & Bridle 2002) Monte Carlo Markov chain algorithm. The theoretical CMB model at every stage is obtained from the publicly available CAMB Boltzmann code (Lewis et al. 2000). The choice of baseline parameter set and the shape of the priors on parameters fitted concurrently will impact on the one-dimensional (1D) marginalized parameters calculated using the Monte Carlo Markov chain. For the standard ΛCDM six-parameter cosmological model, we limit this effect by using the baseline parameter combination comprising the baryon density, cold dark matter density, acoustic peak scale which records the expansion history of the universe, scalar spectral amplitude, scalar spectral index, and optical depth: {h²Ω_b, h²Ω_c, θ, ln(10¹⁰A_s), n_s, τ}; the use of ln(10¹⁰A_s) also considerably increases the rate of convergence; H₀ = 100 h km s⁻¹ Mpc⁻¹ is the Hubble constant. This is the default parameter set of CosmoMC (Lewis & Bridle 2002). When presenting two-dimensional (2D) plots of likelihood space evaluated from a CMB spectrum that does not have information in the angular range below the first acoustic peak, we use the combined A_sexp(−2τ) parameter, a measure of the overall amplitude of the fluctuations in the spectra optimally sampled in the analysis using the baseline parameter set. The pivot scale used to evaluate the amplitude and spectral index is k_p = 0.05 Mpc⁻¹ in the case where we use QUaD data exclusively. The pivot scale used for multi-experiment CMB spectra analysis is k_p = 0.013 Mpc⁻¹ in common with our cosmological parameter results of Brown et al. (2009). We impose flat priors on the baseline parameter set (Table 1), make the assumption that the universe is flat, and include in all analyses the effects of weak gravitational lensing.

In addition, we make use of the CosmoMC utility to analytically marginalize over "nuisance" parameters, which is implemented for beam and calibration uncertainties (Bridle et al. 2002). The effective beam size for the combined QUaD spectra is 4.1 arcmin. The beam error is a function dominated by the sidelobes and varies with scale. The calibration uncertainty is 6.8% in power.

We provide parameter constraints on these baseline parameters and additionally include plots and constraints in the more traditional format of {h²Ω_b, h²Ω_c, H₀, A_s, n_s, τ}.

We present 68% and 95% confidence regions for cosmological parameters and 2D and 1D marginalized plots using our independent statistics code on 10 or more Monte Carlo Markov chains with at least 100,000 converged steps. Tests of convergence were carried out using the Gelman–Rubin statistic (Gelman & Rubin 1992). We also made use of the Getdist statistics package, which is bundled with CosmoMC, as a consistency check of our results.

We carried out a parameter fit from the products of both QUaD data analysis pipelines to confirm consistency.

2.1. Other Data

We combine QUaD with the following CMB data sets: ACBAR (Reichardt et al. 2009) including the 2% beam error on the 5 arcmin beam and calibration error 4.6% in power; CBI (Sievers et al. 2009) with a stated calibration error of 2.6% in power; BICEP (Chiang et al. 2010), 5.6% calibration error and beam error varying with scale; the WMAP 5 year release (Nolta et al. 2009) using the publicly available WMAP 5 year likelihood software.¹⁸ We describe the set of CMB data above, excluding QUaD, as CMBmany. In Figure 1, QUaD polarization spectra are plotted together with WMAP, CBI, Boomerang¹⁹, and BICEP polarization data.

Zoom In Zoom Out Reset image size

3.1. Parameter Constraints from QUaD Data

We present parameter constraints obtained from QUaD data in Table 2. The table presents cosmological parameters calculated from the QUaD TE/EE/BB spectra subset, QUaD EE/BB spectra, QUaD TT spectra, and the full QUaD data set. The parameters obtained from the subsets are consistent. Cosmological parameter limits from the CMBmany data set are also presented in Table 2.

Table 2. Cosmological Parameter Constraints Using QUaD Data

Parameter	QUaD TE/EE/BB	QUaD EE/BB	QUaD TT	QUaD	CMBmanya
Ωbh2	0.0233+0.0030−0.0030	0.0327+0.0097−0.0098	0.0218+0.0040−0.0040	0.0243+0.0025−0.0025	0.0225+0.0006−0.0006
Ωch2	0.124+0.030−0.030	0.162+0.032−0.031	0.117+0.036−0.034	0.119+0.0253−0.0250	0.114+0.006−0.006
θ	1.040 ± 0.006	1.032+0.011−0.010	1.045+0.011−0.011	1.041+0.005−0.005	1.041+0.003−0.003
τ b	<0.53 (95 % c.l.)	<0.46 (95 % c.l.)	<0.54 (95 % c.l.)	<0.54 (95 % c.l.)	0.087+0.017−0.017
ln(1010As)c	3.53+0.31−0.31	3.74+0.19−0.21	3.48+0.34−0.34	3.58+0.28−0.30	3.09+0.04−0.04
ns c	0.768+0.150−0.151	0.530+0.166−0.173	0.920+0.117−0.118	0.804+0.098−0.098	0.964+0.013−0.013
Age (Gyr)	13.7+0.4−0.4	13.1+0.93−0.93	13.6+0.6−0.6	13.5+0.4−0.4	13.7+0.1−0.1
ΩΛ	0.64 ± 0.18	0.45+0.26−0.27	0.68+0.18−0.20	0.68+0.14−0.14	0.72+0.03−0.03
Ωm	0.36+0.19−0.18	0.55+0.27−0.26	0.32+0.21−0.18	0.32+0.14−0.14	0.28+0.03−0.03
zre	23.1+9.0−10.0	18.4+9.3−9.8	24.3+10.3−11.2	23.9+8.1−9.0	10.5+1.4−1.4
H0	68.6+12.5−12.1	63.2+13.1−12.1	72.5+15.8−15.6	71.1+10.9−10.9	70.4+2.4−2.4
σ8 d	0.98 ± 0.17	1.07+0.14−0.15	0.98+0.21−0.21	0.98+0.15−0.15	0.82+0.03−0.03

Notes. ^aCMBmany describes the CMB data set including ACBAR (up to l= 2000), CBIPol, BICEP, and WMAP5 (see Section 2.1). ^bThe parameterization of reionization by CAMB has recently been changed. First ionization of helium and hydrogen are now modeled as simultaneous, affecting the returned values of τ and the time at which z_re is defined has changed at the 6% level. ^cThe pivot point for A_s and n_s used with all the data sets presented in this table is k_p = 0.05 Mpc⁻¹. ^dThe root mean squared linear mass perturbation is defined on a scale of 8 h^−l Mpc, where H₀ = 100 h km s⁻¹ Mpc⁻¹.

Download table as:  ASCIITypeset image

In Figure 2, we display the parameter constraints and contours obtained from QUaD polarization data (TE/EE/BB) alone. The resultant confidence regions are considerably smaller than the corresponding confidence regions from WMAP 5 year TE data, overplotted in Figure 2, for all parameters except A_s and τ. The parameter constraints arrived at using QUaD TE/EE/BB are within 95% agreement with the parameter constraints from the full WMAP data set and those from WMAP TE spectrum. We have carried out, in addition, a combined analysis using QUaD and BICEP polarization data; these latest generation ground-based polarization experiments detect polarization at both low- and high-ℓ. The preferred parameter regions for QUaD and BICEP TE/EE/BB data are also presented in Figure 2. We note that the center of the preferred range for the scalar spectral index from QUaD TE/EE/BB data is ∼0.77. We also note that this preferred range in n_s concurs with the QUaD polarization analysis of Parameter Paper 1, which found n_s = 0.766 ±  0.152.

Zoom In Zoom Out Reset image size

The best-fit spectral index value obtained from the QUaD EE/BB spectra subset is 0.53. We show in Figure 3 that the likelihood surface is skewed over the range of scalar spectral index values. We overplot the likelihood for WMAP TE to demonstrate the best-fit regions for the spectral index overlap. To supplement our QUaD EE/BB only parameter fits, we also carried out an analysis where we implement a Gaussian prior on the Hubble constant with the latest independent limits obtained by the Hubble Space Telescope (HST; Riess et al. 2009) 74.2 ± 3.6 km s⁻¹ Mpc⁻¹. The Hubble constant primarily impacts the width and separation of the spectral peaks, while the spectral index is evaluated from the tilt of the data. Applying the information we have on the Hubble constant improves our parameter confidence ranges, while not preferentially impacting the spectral index signal in the data. These contours are also plotted in Figure 3, confirming the robustness of the EE/BB analysis. The parameter constraints from QUaD EE/BB spectra improve on the corresponding constraints we presented in Parameter Paper 1.

Zoom In Zoom Out Reset image size

The constraint obtained on the acoustic peak scale, θ, is as tight from QUaD EE/BB only as it is from the QUaD temperature spectrum and the constraint on h²Ω_c is tighter.

The parameter ranges arrived at by using the complete QUaD data set agree with the values from the CMBmany data set within the 95% contour limits, as demonstrated in Figure 4. The preferred parameter range for the baryon density we presented in Parameter Paper 1 from the full Pryke et al. (2009) QUaD data set was ∼50% higher than that we now provide. The Hubble constant was also ∼30% higher than the value we return from the analysis of the new non-field-differenced data. The data analysis pipeline which produced the Brown et al. (2009) spectra differs from the Pryke et al. (2009) analysis in the implementation of a new noise-removal strategy and in modeling a new beam shape. This results in a material difference in the spectra in addition to a reduction of cosmic variance. The scalar spectral amplitude of Parameter Paper 1 (which we quoted as part of a combined A_sexp(−2τ) parameter) was significantly shifted relative to our current preferred range, the corollary being in extended parameter analysis isocurvature ratios were more constrained. However, as our parameter analysis has evolved, the confidence limits presented in Parameter Paper 1 cannot be directly compared to the limits in this paper. Repeating the parameter analysis of the second and third season QUaD data of Pryke et al. (2009), using our current CosmoMC-based pipeline, we find 68% confidence regions for A_sexp(−2τ) and the Hubble constant that are 30% greater than the corresponding values obtained using the improved data set of Brown et al. (2009).

Zoom In Zoom Out Reset image size

3.2. Parameter Constraints Combining Data Sets

We continue our parameter analysis by investigating the effect of adding QUaD data to the CMBmany temperature and polarization ensemble data set.

In order to investigate the influence of polarization in our cosmological analysis, we carried out cosmological parameter fits using the TT and TE/EE/BB subsets of the QUaD+CMBmany data set. The resulting best-fit regions, presented in Table 3 and Figure 5, are consistent, although the polarization spectra prefer lower values for the Hubble constant and acoustic peak scale as well as a lower spectral index. The multi-experiment TE/EE/BB data set contributes information on all six cosmological parameters.

Zoom In Zoom Out Reset image size

Table 3. Cosmological Parameter Constraints Using Combined CMB Temperature and Polarization

Parameter	QUaD+CMBmany TE/EE/BBa	QUaD+CMBmany TTb	QUaD + CMBmany
Ωbh2	0.0204 ± 0.0020	0.0231 ± 0.0008	0.0224 ± 0.0005
Ωch2	0.124 ± 0.014	0.109+0.007−0.007	0.113+0.005−0.005
θ	1.036 ± 0.005	1.042+0.003−0.002	1.041+0.003−0.002
τ c	0.073+0.017−0.017	0.127+0.079−0.079	0.086+0.017−0.017
ln(1010As) d	3.22+0.08−0.08	3.17+0.15−0.15	3.13+0.04−0.03
ns d	0.842 ± 0.055	0.983+0.026−0.025	0.960+0.013−0.013
Age (Gyr)	14.1+0.3−0.3	13.6+0.2−0.2	13.6+0.2−0.2
ΩΛ	0.63 ± 0.10	0.75+0.04−0.04	0.73+0.03−0.03
Ωm	0.37 ± 0.10	0.25+0.04−0.04	0.27+0.03−0.03
zre	10.1+1.5−1.5	12.6+5.5−5.7	10.4+1.4−1.4
H0	63.8+6.1−6.1	73.7+3.8−3.8	70.6+2.3−2.3
σ8	0.79+0.06−0.07	0.83+0.05−0.05	0.82+0.03−0.03

Notes. ^aCMBmany TE/EE/BB is the combination of WMAP, CBI, and BICEP polarization spectra. ^bCMBmany TT is the combination of WMAP, CBI, ACBAR, and BICEP temperature spectra. ^cThe parameterization of reionization by CAMB has recently been changed. First ionization of helium and hydrogen are now modeled as simultaneous, affecting the returned values of τ and the time at which z_re is defined has changed at the 6% level. ^dThe pivot point for A_s and n_s is k_p = 0.013 Mpc⁻¹.

Download table as:  ASCIITypeset image

We present best-fit regions for the standard ΛCDM six-parameter model using QUaD CMB spectra alone, and QUaD data in combination with CMBmany (BICEP, ACBAR, CBI, and WMAP).

QUaD data provide an unprecedented amount of independent information over four spectra in a multipole range of 200 < ℓ < 2000. This improves the ensemble of CMB data sets specifically providing independent cohesive information over an angular range broad enough to span the sparsely populated "hinge" multipole regions of the CMBmany data set.

Data from the QUaD experiment, as a single, discerning data set can provide excellent independent measures of the baryon density in the universe, the cold dark matter density, and the acoustic scale, which measures the universe's evolution history.

QUaD TE/EE/BB spectra provide considerably tighter constraints on four of the six standard ΛCDM parameters than can be obtained from 5 year WMAP polarization data. We present the best confidence ranges obtained from QUaD EE/BB spectra alone. The QUaD temperature spectrum alone also provides good parameter constraints in excellent agreement with the WMAP 5 year best-fit parameters.

QUaD is funded by the National Science Foundation in the USA, through grants ANT-0338138, ANT-0338335, & ANT-0338238, by the Science and Technology Facilities Council (STFC) in the UK, and by the Science Foundation Ireland. P.G.C. acknowledges funding from the Portuguese FCT. S.E.C. acknowledges support from a Stanford Terman Fellowship. J.R.H. acknowledges the support of an NSF Graduate Research Fellowship, a Stanford Graduate Fellowship, and a NASA Postdoctoral Fellowship. Y.M. acknowledges support from a SUPA Prize studentship. C.P. acknowledges partial support from the Kavli Institute for Cosmological Physics through the grant NSF PHY-0114422. E.Y.W. acknowledges receipt of an NDSEG fellowship. M.Z. acknowledges support from a NASA Postdoctoral Fellowship. We acknowledge the use of the CAMB (Lewis et al. 2000), CosmoMC (Lewis & Bridle 2002), and HEALPix (Górski et al. 2005) packages. We acknowledge the use of the Legacy Archive for Microwave Background Data Analysis (LAMBDA). Support for LAMBDA is provided by the NASA Office of Space Science. This work was performed using the computational facilities of the Advanced Research Computing @ Cardiff (ARCCA) Division, Cardiff University.