HIGH-RESOLUTION CMB POWER SPECTRUM FROM THE COMPLETE ACBAR DATA SET

We performed a series of tests to search for and constrain potential systematic errors in the power spectrum results. As described in K04, the data can be divided into two halves based on whether the chopping mirror is moving to the left or right. The "left minus right" jackknife is a sensitive test for errors in the transfer function correction, microphonic vibrations excited by the chopper motion, or the effects of wind direction. Maps with bright sources such as RCW38 can provide particularly sensitive tests of the transfer function (see Runyan et al. 2003, for a description of ACBAR's transfer functions). Similarly, the data can be split based on the time that the observation occurred. A nonzero signal could be produced in the "first half minus second half" jackknife by variation in the calibration, pointing, beam and sidelobe, or any other time-dependent variations in the instrument. In addition, the band powers of each jackknife constrain the mis-estimation of noise during that period.

We applied the left–right jackknife to the 2005 CMB data and found the band powers were inconsistent with zero at 2.5σ at high ℓ (ℓ>2100). We reran a set of left–right jackknives, dropping individual channels, and found that two channels stood out. With both channels excluded, the discrepancy in the left–right jackknife band powers disappeared. We were unable to find evidence for unusual microphonic lines or transfer functions in the two problematic channels, but hypothesize that these two channels have subtle microphonic response in the signal bandwidth that are detectable only in a deep integration. Both channels are excluded from the 2005 data for all results reported in this paper.

We apply the first–second half jackknife test to the joint CMB power spectrum with the exclusion of the bad channels from the 2005 data. The power spectrum of the chronologically differenced maps is compared to the band powers of a set of Monte Carlo realizations of simulated difference maps in order to account for a number of effects, such as the small filtering differences due to different scan patterns and the temporal uncertainty in the beam sidelobes (see Section 5). We find that the jackknife band powers are consistent with the predictions of the Monte Carlo above ℓ = 400. There is a 4σ residual of ∼ 15 μK² in the first bin. Because the combined statistical and cosmic variance uncertainty in this bin is a factor of six larger, we assume that the band-power estimate will not be significantly biased.

We also perform the left–right jackknife on the joint CMB power spectrum found from the complete data set. The results are consistent with zero for ℓ>900. Statistically, the probability to exceed the measured χ² for ℓ>900 is 15%. The results are inconsistent with zero at a very low (∼ 4 μK²) level (Figure 3) on larger angular scales. This residual could be due to a small noise mis-estimate at low ℓ, possibly caused by neglected atmospheric correlations. The jackknife failure of ∼ 4 μK² is much smaller than the band-power uncertainties (90–300 μK²) in these ℓ bins, which are dominated by cosmic variance. The first–second half jackknife is insensitive to discrepancies of this magnitude due to the greater uncertainties introduced by small pointing and filtering differences. We conclude that the complete ACBAR data set shows no significant residuals in the jackknife tests, and we expect no significant systematic contamination of the resulting power spectrum.

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The potential contribution of foreground emission must be considered in the interpretation of CMB temperature anisotropies. There are three foregrounds with significant emission at 150 GHz on the relevant angular scales: galactic dust, extragalactic radio sources, and dusty protogalaxies. As an effectively single-frequency instrument, ACBAR depends on data from other experiments to construct and constrain foreground models. We use the methodology described in K04 to remove templates for radio sources and dust emission from the CMB maps without making assumptions about their flux. The contribution from dusty protogalaxies is less certain; however, we do not expect the combined residual foreground emission to significantly impact the power spectrum for ℓ < 2400.

We remove modes from the CMB maps corresponding to radio sources in the 4.85 GHz Parkes-MIT-NRAO (PMN) survey (Wright et al. 1994). Of the 1601 PMN sources with a flux greater than 40  mJy that lie in the ACBAR fields, we detect 37 sources including the guiding quasars at greater than 3σ with the application of an optimal-matched filter. There are less than 2.2 false detections expected with this detection threshold. The measurement errors are estimated through sampling the distribution of pixels in a set of 100 Monte Carlo realizations of the CMB+noise for each field. Table 2 lists the parameters of the detected PMN sources. Except for the few bright sources detected at 150 GHz, removing the PMN point sources does not significantly affect the band powers.

It is possible that faint radio sources, undetected at 150 GHz, could contribute to the observed band powers. We parameterize the contribution as

$$\begin{equation} {\cal D}^{\rm src}_\ell = q_{\rm src}\left(\frac{\ell }{2600}\right)^2 \, \mu {\rm K}^2,\, \end{equation} \tag{ 3 }$$

which is appropriate for unclustered point sources. We compare the 150 GHz ACBAR point source number counts to the model in White & Majumdar (2004), based on WMAP Q-band data,

$$$ \frac{dN}{dS_\nu } = \frac{80\, {\rm deg}^{-2}}{1\, {\rm mJy}} \left(\frac{S_\nu }{1\, {\rm mJy}}\right)^{-2.3}, $$$

to constrain the residual power contribution at 150 GHz. We can use this model to estimate the residual band-power contribution due to sources too faint to be included in the PMN catalog.

Following the convention in that work, the spectral dependence of the fluxes is parameterized as S_ν ∝ ν^β. The number of PMN sources detected at 150 GHz in a logarithmic flux bin, n^obs_B, are compared to the predicted number counts from the same population of sources with a given β, n^(β)_B + n^noise_B. Here, n^(β)_B is the modeled number counts and n^noise_B is the expected number of false detections due to ACBAR's measurement uncertainty. The number counts are assumed to follow a Poisson distribution. Sources with estimated measurement errors greater than 140 mJy in the ACBAR maps are cut to reduce the n^noise_B term. The modeled number counts n^(β)_B are scaled by (N_tot − N_cut)/N_tot to compensate. All other sources with measured amplitudes greater than 350 mJy at 150 GHz are included in the calculation without consideration of the signal to noise. If the sources found by ACBAR at 150 GHz are the same population found by WMAP, this implies a uniform spectral index of β = 0.14 ± 0.15. However, this small sample selected for high flux at 150 GHz is heavily biased toward sources with flat or rising spectra. We increase ACBAR's sensitivity to dimmer sources by binning all sources within a given PMN flux range, and look at the ratio of the average flux at 150 GHz to the average flux at 4.85 GHz within each bin. We find the ratio (S₁₅₀/S_4.85) increases with PMN flux from 0.07 for sources below 400 mJy to 0.41 for sources above 1600 mJy at 4.85 GHz. This implies that the sources in the PMN catalog have a flux-dependent spectral index where dimmer objects typically have a more steeply falling spectrum. The band-power contribution of the low-flux sources depends sensitively on the extrapolation of the 40 mJy flux cutoff in the 4.85 GHz PMN catalog to 150 GHz. Based on the observed flux ratios for PMN sources with S_4.85 < 400 mJy, we conservatively assume S₁₅₀/S_4.85 = 0.1 for a flux cutoff at 4 mJy at 150 GHz. This flux ratio corresponds to a spectral index of β = −0.67, well below β = 0.14 ± 0.15 estimated from the ACBAR detected sources and the WMAP Q-band source model. Estimating the residual radio source band-power contribution at 150 GHz with a flux cutoff of 4 mJy gives q^radio_src = 2.2. At this level, the residual contribution from radio sources will be negligible in the ACBAR data.

The ACBAR fields are positioned in the "Southern Hole," a region of exceptionally low Galactic dust emission (Figure 1). Finkbeiner et al. (1999, FDS99) constructed a multicomponent dust model that predicts thermal emission at CMB frequencies from the combined observations of IRAS, COBE/DIRBE, and COBE/FIRAS. Taking into account the ACBAR filtering, the FDS99 model¹³ predicts an RMS dust signal at the few μK level, primarily on large angular scales. The ACBAR maps can be decomposed as the sum of the CMB and dust signals T_CMB + ξT_FDS. The dust amplitude parameter ξ is predicted to equal unity by the FSD99 model. The ACBAR maps are cross-correlated with the dust templates T_FDS to calculate the amplitude in each field. The errors are estimated by applying the same procedure to 100 CMB+noise map realizations for each field. The uncertainty in ξ is dominated by CMB fluctuations. The best-fit amplitude from combining all the fields is ξ = 0.1 ± 0.5. The estimated amplitudes of the individual fields are shown in Figure 4. The reduced χ² of the measured amplitudes ξs of the eight fields analyzed is 0.75 for the no-dust assumption of 〈ξ〉 = 0 and increases to χ² = 1.12 for the FDS99 model amplitude of 〈ξ〉 = 1. Therefore, the ACBAR data slightly favor a lower amplitude than predicted by the FDS99 model. The dust signal is not detectable in any of the ACBAR fields, and removing the dust template has a negligible impact on the measured power spectrum.

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Dusty IR galaxies are the third and least-constrained foreground in the ACBAR fields. This population of high redshift, star-forming galaxies has been studied by several experiments at higher frequencies (Coppin et al. 2006; Laurent et al. 2005; Maloney et al. 2005; Greve et al. 2004). However, as discussed in K07, extrapolating the expected signal to 150 GHz remains highly uncertain, and there remain significant uncertainties in the number counts dN/dS and spatial clustering of the sources. The frequency dependence can be empirically determined by comparing the measured number counts in overlapping fields observed at different frequencies. This comparison has been done with MAMBO (1.2 mm) and SCUBA (850 μm), leading to a spectral dependence of S_ν ∝ ν^2.65 (Greve et al. 2004). A second method for estimating the index used nearby galaxy data to obtain S_ν ∼ ν^{2.6 ± 0.3} (Knox et al. 2004). The uncertainty in the spectral dependence significantly affects the extrapolation of the flux of dusty galaxies to 150 GHz. We use estimates of the source number counts from the SHADES survey (Coppin et al. 2006) and Bolocam Lockman Hole Survey (Maloney et al. 2005). We apply the formulas in Scott & White (1999) to estimate the expected power spectrum for the source number counts, ignoring the clustering terms. In this limit, ${\cal D}_{\ell }$ will have the form in Equation (3). Scaling the results to 150 GHz with the MAMBO/SCUBA prescription of S_ν ∝ ν^2.65 leads to an estimated contribution of q^dusty_src = 17–29. This range reflects the differences between the measured number counts, but does not include the uncertainty in the spectral dependence of the fluxes. Combining the median index with earlier SCUBA data fit by two power laws in S from (Borys et al. 2003), we find an excess of 22 μK² at ℓ = 2600, within that range. This level is only a factor of two smaller than the instrumental noise of ACBAR and might influence the interpretation of high ℓ band powers. We tentatively assume that contamination from dusty protogalaxies does not significantly affect the resulting power spectrum. The implications of relaxing this assumption for cosmological parameter estimation are explored in Section 8.3.

The power spectrum presented in Figure 5 is produced by the application of the analysis algorithm outlined in Section 3 to ACBAR data from the 2001, 2002, and 2005 austral winters. The resulting power spectrum is compared to the WMAP5 and B03 spectra in Figure 6. The zero curvature, ΛCDM "ACBAR+WMAP5" best fit model, is shown in each figure for reference. The decorrelated band powers are tabulated in Table 3. Our choice of the decorrelation transformations follows Tegmark (1997). The band powers can be compared to a theoretical model using the window functions (Knox 1999). As in K04, we sample the likelihood function ${\cal L}(\Delta) = \frac{1}{\sqrt{C}}e^{-(\Delta ^t C^{-1} \Delta)/2}$ near the maximum and fit the results with offset lognormal functions (Bond et al. 2000). The fit parameters σ, x are listed in Table 3 as well. The band powers, likelihood-fit parameters, and window functions are available for download from the ACBAR website.¹⁴

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The ACBAR data extend the measurement of the temperature anisotropies well into the damping tail with S/N > 5 for ℓ ≲ 2300. The fourth and fifth acoustic peaks are detected for the first time in the ACBAR band powers, providing additional support for the coherent origin of anisotropy (Albrecht et al. 1996). The position of the third acoustic peak is consistent with previous detections of the feature by CBI (Readhead et al. 2004a), B03 (Jones et al. 2006), ACBAR (K07), and QUaD (Ade et al. 2007). The ACBAR band powers are in excellent agreement with the cosmological models constrained by observations on larger angular scales. The probability to exceed the reduced χ² between the ACBAR band powers and the WMAP3-only best-fit ΛCDM model is 17%. This probability increases to 62% with the WMAP5-only best-fit model. This serves as both a powerful confirmation of our basic cosmological model and an indication of the quality of the ACBAR data set.

Several theoretical calculations (Cooray et al. 2000; Komatsu & Seljak 2002) and hydrodynamical simulations (Bond et al. 2005; White et al. 2002) suggest that the thermal Sunyaev–Zel'dovich effect power spectrum will exceed that of the primary CMB temperature anisotropies for ℓ ≳ 2500 at 150 GHz. The amplitude of the SZE power spectrum is closely related to the amplitude of matter perturbations, which is commonly parameterized as σ₈; the SZE power spectrum is expected to scale as σ⁷₈ (Zhang et al. 2002). To a lesser extent, the level of the SZE will also depend on details of cluster gas physics and thermal history. The nonrelativistic thermal SZE (ΔT_SZ) has a unique frequency dependence

$$\begin{equation} \frac{\Delta T_{\rm SZ}}{ T_{\rm CMB}} = y \left(x\frac{e^x + 1}{e^x - 1} - 4\right),\, \end{equation} \tag{ 4 }$$

where $x = \frac{h\nu }{ k T_{\rm CMB}}$ = ν/56.8 GHz. The variable y is the Compton parameter and is proportional to the integrated electron pressure along the line of sight. The CBI extended mosaic observations (Readhead et al. 2004a) detected more power above ℓ = 2000 than is expected from primary CMB anisotropies. This excess could be the first detection of the SZE power spectrum (Mason et al. 2003; Readhead et al. 2004a; Bond et al. 2005). However, there are alternative explanations for the observed power ranging from an unresolved population of low-flux radio sources to nonstandard inflationary models (Cooray & Melchiorri 2002; Griffiths et al. 2003; Subramanian et al. 2003) that produce higher than expected CMB anisotropy power at small angular scales. The frequency dependence of the excess power can be exploited to help discriminate between the SZE and other potential explanations.

The ACBAR band powers reported in this paper are slightly larger at ℓ>2000 than expected for the "ACBAR+WMAP5" best-fit model. We subtract the predicted band powers at ℓ>1950 from the measured band powers in Table 3, and find an excess of $22 \pm 20\ \mu \rm K^2$ in a flat band power from 1950 < ℓ < 3000. This estimate ignores the band-power contribution from dusty protogalaxies, which is expected to be comparable (see Section 6.2). The ACBAR band powers at 150 GHz can be used to place constraints on frequency spectrum of the larger CBI excess measured at 30 GHz. We parameterize the excess power for ℓ>1950 at the two frequencies as P₃₀ = αP₁₅₀, and sample the likelihood surface for α ∈ [0, 10] and P₁₅₀ ∈ [0, 300] $\mu \rm K^2$. The beam uncertainty and the calibration error for both experiments are taken into account by Monte Carlo techniques. The likelihood function is averaged over 1000 realizations under the assumption that each of the errors has a normal distribution. The resulting likelihood function for α (after P₁₅₀ is marginalized) is shown in Figure 7. From the ACBAR and CBI frequency bands, we expect α = 4.3 for power originating from the SZE. If the excess is due to primary CMB anisotropies, we expect α = 1. We conclude that it is more than five times as likely that the excess seen by CBI and ACBAR is caused by the thermal SZE than a primordial source. We expect the contribution of radio sources to CMB power to be at least a factor of ten higher at 30 GHz than at 150 GHz (α ⩾ 10). Because of the relatively weak detection of excess power by ACBAR, flat spectrum radio sources are determined to be ∼ 10% more likely than the SZE to be the source of the excess. The lower level of excess power seen by ACBAR argues against the CBI excess having a primordial origin, but is consistent with either the SZE or radio source foregrounds. When we include the expected contribution of dusty protogalaxies to the ACBAR excess band power, the likelihood of the CBI excess being due to radio sources increases with respect to thermal SZE.

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In this section, we estimate cosmological parameters for a minimal inflation-based, spatially flat, tilted, gravitationally lensed, ΛCDM model characterized by six parameters, and then investigate models with additional parameters to test extensions of the theory. For our base model, the six parameters are: the physical density of baryonic and dark matter, Ω_bh² and Ω_ch²; a uniform spectral index n_s and amplitude ln A_s of the primordial power spectrum, the optical depth to last scattering, τ; and θ, the ratio of the sound horizon at last scattering to the angular diameter distance. The primordial comoving scalar curvature power spectrum is expressed as ${\cal P}_s(k) = A_s (k/k_\star)^{(n_s-1)}$, where the normalization (pivot-point) wavenumber is chosen to be k_⋆ = 0.05 Mpc⁻¹. The parameter θ maps angles observed at our location to comoving spatial scales at recombination; changing θ shifts the entire acoustic peak/valley and damping pattern of the CMB power spectra. Additional parameters are derived from this basic set. These include: the energy density of a cosmological constant in units of the critical density, Ω_Λ; the age of the universe; the energy density of nonrelativistic matter, Ω_m; the rms (linear) matter fluctuation level in 8 h⁻¹Mpc spheres, σ₈; the redshift to reionization, z_re; and the value of the present day Hubble constant, H₀, in units of km s⁻¹Mpc⁻¹.

Single-field models of inflation predict the existence of a gravitational wave background characterized by a primordial power law ${\cal P}_t \sim k^{n_t}$. We characterize the strength by the tensor-to-scalar ratio $r={\cal P}_t/{\cal P}_s$ evaluated at a pivot point 0.002 Mpc⁻¹. We relate the tilt to r using the approximate consistency relation n_t ≈ −r/8/(1 − r/16). (We find little difference in the parameters if we just fix n_t to be zero, as has often been assumed when r is included, but using this relation is superior since it is motivated by inflation physics.)

A small running of the spectral index is also expected in slow-roll inflation and we test for this by extending the basic ΛCDM power law model to include a scale dependence of the scalar spectral tilt, dn_s/dln(k).

We have also added nonzero curvature Ω_k to our basic six parameters. The results are consistent with the flat case, but with the standard geometrical degeneracy relating Ω_k and Ω_Λ expressed through θ leading to a near-degenerate tail to Ω_k < 0.

The ACBAR spectrum includes band powers at ℓ>2000 where the signal due to secondary CMB anisotropies associated with postrecombination nonlinear effects should become significant. In particular, the thermal Sunyaev–Zel'dovich effect and the contribution of unresolved radio sources and dusty galaxies will ultimately dominate over the primary anisotropy damping tail; the only question is at what multipole crossover occurs. Without including such secondary effects, the parameters we derive from the primary anisotropy power spectrum could be biased. To account for this, we have added to the primary anisotropy power spectrum (1) the SZE template power spectrum ${\hat{\cal D}}_\ell ^{\rm SZ}$ used in K07 and (Bond et al. 2005; Goldstein et al. 2003), which was derived from cosmological hydrodynamics simulations, and (2) an unclustered point source template as in Equation (3). Each template is scaled by an overall amplitude parameter, q_SZ and q_src, which we assume have uniform prior measures with a range much larger than required by the ACBAR data. The white-noise form for ${\cal D}^{\rm src}_\ell$ given by Equation (3) is appropriate for the statistically averaged power of a distribution of unclustered sources. The clustering of radio sources is not a large effect, but we do expect submm sources associated with dusty galaxies at lower flux levels to be clustered. As mentioned in Section 6.2, in spite of great strides in submm observations in recent years, significant uncertainties remain in source fluxes and clustering at 150 GHz. Theoretical models suggest both will be important for a complete treatment, but the approximation adopted here should be sufficient for the ACBAR data set. In the parameter tables below, we show results including these secondary templates. We find the basic parameter central values and uncertainties change little whether we marginalize over either of the two template amplitudes or set them to zero.

The parameter constraints are obtained using a Monte Carlo Markov Chain (MCMC) sampling of the multidimensional likelihood as a function of model parameters. The pipeline is based on the publicly available CosmoMC¹⁵ package (Lewis & Bridle 2002). CMB angular power spectra and matter power spectra are computed using the CAMB code (Lewis et al. 2000). As described in Section 7, we approximate the full non-Gaussian band-power likelihoods with an offset lognormal distribution (Bond et al. 2000). Our standard CosmoMC results include the effects of weak gravitational lensing on the CMB (Seljak 1996; Lewis et al. 2000). Lensing effects in the temperature spectrum are expected to become significant at scales ℓ>1000, hence it is important to include this effect when interpreting the ACBAR results. The major effect of lensing is a scale-dependent smoothing of the angular power spectrum that diminishes the peaks and valleys of the spectrum. Inclusion of lensing in the model improves the fit to the data for all experiment combinations.

The typical computation consists of eight separate chains, each having different initial random parameter choices. The chains are run until the largest eigenvalue of the Gelman–Rubin test is smaller than 0.01 after accounting for burn-in. Uniform priors with very broad distributions are assumed for the basic parameters. The standard run also includes a weak prior on the Hubble constant (45 < H₀ < 90 km s⁻¹ Mpc⁻¹) and on the age of the universe (>10 Gyr), but these have negligible effects. We also investigate the influence of adding large-scale structure (LSS) data from the 2 degree Field Galaxy Redshift Survey (2dFGRS) (Cole et al. 2005) and the Sloan Digital Sky Survey (SDSS) (Tegmark et al. 2006). When including the LSS data, we use only the band powers for length scales larger than $k \sim 0.1\, \rm h$ Mpc⁻¹ to avoid nonlinear clustering and scale-dependent galaxy biasing effects. We marginalize over a parameter b²_g that describes the (linear) biasing of the galaxy–galaxy power spectrum for L_⋆ galaxies relative to the underlying mass density power spectrum. We adopt a Gaussian prior on b²_g centered around b_g = 1.0 with a very large width. We have also tried restricting the width to δb_g = 0.3, but the cosmic parameters are insensitive to this width.

The results for the basic spatially flat tilted ΛCDM parameters are presented in Table 4. The confidence limits are obtained by marginalizing the multidimensional likelihoods down to one dimension. The median value is obtained by finding the 50% integral of the resulting likelihood function while the lower and upper error limits are obtained by finding the 16% and 84% integrals, respectively. The CMBall data combination includes the ACBAR results presented here and other CMB data sets with published band powers and window functions: the WMAP 5-year angular power spectra (Nolta et al. 2008), and for comparison the WMAP 3-year spectra (Hinshaw et al. 2006); the CBI extended mosaic results (Readhead et al. 2004a) and polarization results (Readhead et al. 2004b; Sievers et al. 2005) combined in the manner described in Sievers et al. (2005);¹⁶ the DASI two-year results (Halverson et al. 2002); the DASI EE and TE band powers (Leitch et al. 2005); the VSA final results (Dickinson et al. 2004); the MAXIMA 1998 flight results (Hanany et al. 2000); and the TT, TE, and EE results from the BOOMERANG 2003 flight (Jones et al. 2006; Piacentini et al. 2006; Montroy et al. 2006). Only ℓ>350 band powers are included for BOOMERANG because of overlap with WMAP (although inclusion of the lower ℓ results leaves the parameter results essentially unchanged). While ACBAR and BOOMERANG are both calibrated through WMAP, this is a small contribution to the total uncertainty in the ACBAR calibration and we treat the calibration uncertainties as independent in our parameter analysis. Although the DASI, CBI, and BOOMERANG 2003 EE and TE results for high ℓ polarization are included, they have little impact on the values of the parameters we obtain.

Table 4. Basic Six Parameter Constraints

	WMAP5	WMAP5+ACBAR	CMBall	CMBall+LSS	CMBall+qSZ	CMBall+SZ+qsrc
Ωbh2	0.0226+0.0006−0.0006	0.0227+0.0006−0.0006	0.0227+0.0005−0.0005	0.0228+0.0005−0.0005	0.0227+0.0006−0.0005	0.0227+0.0005−0.0005
Ωch2	0.110+0.006−0.006	0.111+0.006−0.006	0.111+0.005−0.005	0.107+0.004−0.004	0.109+0.005−0.005	0.111+0.005−0.005
θ	1.041+0.003−0.003	1.042+0.003−0.003	1.042+0.002−0.002	1.042+0.002−0.002	1.042+0.002−0.002	1.042+0.002−0.002
τ	0.086+0.008−0.008	0.086+0.008−0.008	0.086+0.008−0.008	0.087+0.008−0.008	0.086+0.008−0.008	0.085+0.008−0.008
ns	0.967+0.015−0.015	0.967+0.014−0.014	0.964+0.013−0.013	0.968+0.012−0.012	0.962+0.013−0.013	0.962+0.013−0.013
${\cal A}_s$	3.07+0.04−0.04	3.07+0.04−0.04	3.07+0.04−0.04	3.05+0.04−0.04	3.05+0.04−0.04	3.06+0.04−0.03
ΩΛ	0.74+0.03−0.03	0.74+0.03−0.03	0.74+0.02−0.03	0.76+0.02−0.02	0.75+0.02−0.03	0.74+0.02−0.03
$\rm Age$	13.7+0.1−0.1	13.7+0.1−0.1	13.6+0.1−0.1	13.6+0.1−0.1	13.7+0.1−0.1	13.7+0.1−0.1
Ωm	0.26+0.03−0.03	0.26+0.03−0.03	0.26+0.03−0.02	0.24+0.02−0.02	0.25+0.03−0.02	0.26+0.03−0.02
σ8	0.80+0.04−0.04	0.80+0.03−0.03	0.80+0.03−0.03	0.78+0.03−0.02	0.79+0.03−0.03	0.80+0.03−0.03
zre	11.0+1.4−1.4	11.0+1.5−1.4	11.0+1.4−1.4	11.0+1.4−1.4	10.9+1.4−1.4	10.9+1.4−1.4
H0	71.8+2.7−2.7	72.0+2.6−2.5	72.1+2.4−2.4	73.6+1.9−1.9	72.5+2.4−2.4	72.0+2.4−2.3
qSZ	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	0.69+0.11−0.11	⋅⋅⋅
qsrc	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	29+12−28
σSZ8	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	0.93+0.04−0.05	⋅⋅⋅

Notes. Results for the basic parameter set. All runs assume flat cosmologies, uniform, and broad priors on each of the basic six parameters, and a weak prior on the Hubble constant (45 < H₀ < 90 km s⁻¹ Mpc⁻¹) and the age of the universe (>10 Gyr). Here ${\cal A}_s\equiv \log [10^{10}A_s]$. All runs include the effect of weak gravitational lensing on the CMB. Column 5 presents the results when an SZ template characterized by the overall SZ template-power q_SZ is included. The values of σ^SZ₈ = q^1/7_SZ(Ω_bh)^2/7 are higher than the σ₈ derived from the primary anisotropies. Column 6 shows the results when the point source contribution scaled by q_src is included in combination with the SZ template with q_SZ set to σ⁷₈(Ω_bh)². The marginalization over either extra high ℓ contribution does not significantly shift the results for the basic parameters.

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The latest WMAP likelihood code found at http://lambda.gsfc.nasa.gov/ has been used in our analyses. When this ACBAR paper was submitted, all parameter analyses were done using WMAP3 (Hinshaw et al. 2006); the new WMAP5 results came out a few months later. The WMAP5 data allowed for an improved cross-calibration of the ACBAR and WMAP band powers, and resolved the (<1 − σ) parameter tensions that existed between the WMAP3 and ACBAR results. We note that the WMAP5 team's parameter analysis (Dunkley et al. 2008) made use of the ACBAR band powers with the 1.4% higher temperature calibration from the WMAP3 and ACBAR cross-calibration. The marginalized one-dimensional likelihood distributions for the basic parameter set we obtain are shown in Figure 8. Note the contrast between the parameter determinations for WMAP3 and WMAP5. The primary improvement of WMAP5 over WMAP3 was a better understanding of the beam, which resulted in an improved measurement of the third acoustic peak. Other improvements in WMAP5 were updated point source correction, stimulated by (Huffenberger et al. 2006), and foreground marginalization on large angular scales.

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The addition of LSS data has little impact on the mean values and errors in the cosmic parameters. The largest shift is <1 − σ in Ω_ch², from 0.111^+0.005_−0.005 to 0.107^+0.004_−0.004. Our LSS results only include information on the shape of the density power spectrum, not its overall amplitude since we marginalize over the galaxy bias factor. If results from weak lensing are included in the LSS data, then there is a slight increase in σ₈, but it depends somewhat on which lensing results are included. The effect is to slightly increase the CMB+LSS result and improve the consistency with the CMBall results. The third peak is well determined by both ACBAR and WMAP5, and this defines the dark matter density; the inclusion of the LSS data does not significantly improve the constraints. We have also found that the results do not significantly change from CMBall+LSS when SN1a data are included (in this case, from the Riess et al. (2004) gold set), so we have not included a separate column. Including SN1a data would be crucial if we were attempting to constrain the equation of state of dark energy.

All parameter results listed in Tables 4 and 5 include the effects of weak lensing of the CMB on the resulting power spectrum. For every case including CMB lensing, we have performed an identical calculation, neglecting the effects of lensing. Including lensing improves the fit of the model to the observed band powers compared for all data combinations. This can be quantified by the log-ratio of the lensed to no-lensed Bayesian evidence, $\Delta \ln {\cal E}=\ln [P({\rm lens}|{\rm data,theory})$/P(no-lens|data, theory)]. The evidence P(lens|data, theory) is an integral of the product of the a priori probability (the parameters' measure) and the likelihood of data given those parameters; it appears in the denominator in the Bayesian chain to ensure the a posteriori probability has unity normalization. The resulting number is a conditional probability given the data and the assumptions about the parameters. The parameters and their measures are exactly the same, so the ratio is a robust indicator of preference. For WMAP5 alone it is $\Delta \ln {\cal E}=2.04$; it increases to 2.89 with ACBAR included; and is 2.63 for CMBall. Naively relating this to a Gaussian translates to a significance of ∼2.3σ for CMBall. From Figure 5, it is clear that the difference in the power spectra of lensing and no-lensing is small; for this plot, the best-fit parameters from the lensed analysis were fixed and used to compute a lensed and unlensed spectrum. The difference, ΔC^lens_ℓ ≡ C^lens_ℓ − C^no-lens_ℓ, is explicitly shown in the inset of Figure 10. We find only small shifts in the median value of the cosmic parameters when lensing is included; e.g., for the ACBAR+WMAP5 data combination, we find σ₈ = 0.79^+0.03_−0.03 → 0.80^+0.03_−0.03 and Ω_ch² = 0.109^+0.006_−0.006 → 0.111^+0.006_−0.006 when going from nonlensed to lensed models, respectively.

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Table 5. Running Spectral Index Parameter Constraints

	WMAP5	WMAP5+ACBAR	CMBall	CMBall+LSS	CMBall+qSZ	CMBall+SZ+qsrc
Ωbh2	0.0219+0.0009−0.0008	0.0219+0.0007−0.0007	0.0220+0.0006−0.0006	0.0225+0.0006−0.0006	0.0217+0.0006−0.0006	0.0217+0.0007−0.0006
Ωch2	0.117+0.009−0.009	0.119+0.008−0.008	0.120+0.008−0.008	0.109+0.005−0.005	0.122+0.008−0.008	0.124+0.008−0.008
θ	1.040+0.003−0.003	1.041+0.003−0.003	1.042+0.002−0.002	1.042+0.002−0.002	1.041+0.003−0.002	1.041+0.002−0.002
τ	0.090+0.008−0.008	0.091+0.009−0.008	0.092+0.009−0.008	0.091+0.009−0.008	0.092+0.009−0.009	0.094+0.009−0.009
ns	0.923+0.043−0.041	0.918+0.033−0.031	0.914+0.030−0.029	0.948+0.026−0.025	0.894+0.031−0.029	0.896+0.030−0.030
dns/dln(k)	−0.031+0.029−0.028	−0.037+0.023−0.022	−0.039+0.021−0.021	−0.016+0.019−0.018	−0.052+0.022−0.021	−0.052+0.022−0.022
${\cal A}_s$	3.09+0.05−0.05	3.10+0.05−0.04	3.10+0.05−0.04	3.07+0.04−0.04	3.10+0.05−0.04	3.11+0.05−0.04
ΩΛ	0.70+0.05−0.06	0.69+0.04−0.05	0.69+0.04−0.05	0.75+0.02−0.03	0.68+0.04−0.05	0.67+0.04−0.05
$\rm Age$	13.8+0.2−0.2	13.8+0.1−0.1	13.8+0.1−0.1	13.7+0.1−0.1	13.8+0.1−0.1	13.8+0.1−0.1
Ωm	0.30+0.06−0.05	0.31+0.05−0.04	0.31+0.05−0.04	0.25+0.03−0.02	0.32+0.05−0.04	0.33+0.05−0.04
σ8	0.81+0.04−0.04	0.83+0.03−0.03	0.83+0.03−0.03	0.79+0.03−0.03	0.83+0.03−0.03	0.84+0.03−0.03
zre	11.7+1.7−1.7	11.9+1.7−1.6	12.0+1.7−1.6	11.5+1.6−1.6	12.2+1.8−1.7	12.4+1.8−1.7
H0	68.4+4.2−3.8	67.9+3.5−3.3	67.7+3.3−3.2	72.4+2.4−2.4	66.7+3.3−3.1	66.2+3.2−3.2
qSZ	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	0.77+0.12−0.11	⋅⋅⋅
qsrc	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	30+11−29
$\sigma _8^{q_{\rm SZ}}$	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	⋅⋅⋅	0.93+0.04−0.05	⋅⋅⋅

Notes. Marginalized parameter constraints for models with a running of the spectral index. The tendency for the low ℓ data to prefer negative values continues with the higher ℓ data, but at less than two sigma for CMB only. The last two columns include marginalization over extra high ℓ contributions from SZ and point sources. The basic parameters are stable with respect to this marginalization, while the median value for dn_s/dln k(k_⋆) is increased slightly when the SZ or point source contribution is included.

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We now test whether the strength of the lensing modification is consistent with expectations for lensing, by multiplying the lensing template ΔC^lens_ℓ, which varies with cosmic parameters, by a variable strength q_lens

$$\begin{equation} C_\ell ^{\rm lens} = C_\ell ^{\rm no-lens} + q_{\rm lens} \Delta C_\ell ^{\rm lens} \,. \end{equation} \tag{ 5 }$$

The normalization is such that q_lens = 1 gives the normal lensed CMB spectrum, while q_lens = 0 gives the no-lensing case. An accurate determination of this subtle effect requires highly accurate window functions for the bands. We use a flat prior probability for q_lens, allowing it to vary from 0 to 10. With WMAP5 alone, we obtain q_lens = 1.34^+0.27(+1.51)_{−0.26(−0.85)}; WMAP5+ACBAR gives q_lens = 1.23^+0.21(+0.83)_{−0.23(−0.76)}; CMBall gives q_lens = 1.21^+0.24(+0.82)_{−0.24(−0.76)}. We have also listed the two-sigma errors (in parentheses), which are far from twice the one-sigma values, reflecting the highly non-Gaussian nature of the marginalized likelihoods evident in the figure. Although we emphasize that q_lens is not in any sense an independent parameter, it does illustrate that lensing of the expected strength is preferred.¹⁷ The significance of the detection is somewhat less than the WMAP3/NVSS/SDSS cross-correlation results of Smith et al. (2007) and Hirata et al. (2008). In Figure 10, we show the marginalized distribution for the q_lens parameter using various combinations of CMB data.

We have also run a limited set of nonflat model chains. The models in this case do not include the effect of weak lensing and we keep the same weak prior on H₀. When Ω_k = 0 is not enforced, the weak prior on H₀ has a significant effect on the result as it restricts the extent of the geometrical degeneracy, which is present in this case. For WMAP5-only, we obtain Ω_k = −0.018^+0.027_−0.026, which becomes Ω_k = −0.013^+0.019_−0.029 when ACBAR is included.

As discussed in Section 6.2, the ACBAR band powers at ℓ>1950 marginally exceed the predictions of the best-fit models for the primary CMB. We have repeated the basic parameter runs including an unclustered source contribution described by Equation (3) and marginalize over a wide uniform prior in q_src from 0 to $4600 \,\mu \rm K^2$, i.e., more than 100 times the power required to fit the high ℓ points. The results for runs including this marginalization are shown in Tables 4 and 5. The template amplitude q_src is not well constrained by the fit, and the effect of the marginalization on the basic parameters is negligible. The addition of a source template improves the fit to the high ℓ points, and therefore increases, albeit modestly, the best-fit likelihood: for the ACBAR+WMAP5 combination the change in likelihood is Δln L = 0.24. The value and uncertainty for q_src given in Table 4 spans the range of predictions for the contribution from dusty protogalaxies.

Contributions from point sources or the SZE are nearly degenerate in the high ℓ ACBAR band powers. However, the CMBall combination is potentially sensitive to an SZE contribution because of the SZE frequency dependence. The CBI data have a more significant excess at high ℓ; however, through substantial modeling and vetoing, the CBI band powers are expected to be free of radio source contamination. We assume that the residual contribution of the unresolved source background to the CBI band powers is sufficiently low that we do not require a source template for that data, only the SZE template.

The amplitude of the SZE signal depends strongly on the overall matter fluctuation amplitude, σ₈. We have modified our parameter fitting pipeline to allow for extra frequency dependent contributions to the CMB power spectrum and have implemented it in a simple analysis using a fixed template ${\hat{\cal D}}_\ell ^{\rm SZ}$ for the shape of the thermal SZE power spectrum. The template was obtained from two large, hydrodynamical simulations of a scale-invariant (n_s = 1) ΛCDM model with Ω_bh = 0.029 and with σ₈ = 0.9 and σ₈ = 1.0 (see Bond et al. 2005, for a detailed description of the simulations). Recently, the WMAP team have used a different SZE template based on analytic estimations of the power spectrum (Spergel et al. 2006). It is characterized by a slower rise in ℓ than the simulation-based template, which cut nearby clusters out of the power spectrum. There has been no fine-tuning of either spectra to agree with all of the X-ray and other cluster data. This may have an effect on shape, especially at high ℓ.

The SZE contribution, ${\cal D}_\ell ^{\rm SZ} = (q_{\rm SZ})f_\nu { \hat{\cal D}}_\ell ^{\rm SZ}$, added to the base six-parameter model spectrum has a frequency-dependent SZE prefactor f_ν. Including this SZE template with all model parameters free to vary is complementary to the analysis of Section 7.2, which directly compared the residual CBI and ACBAR band powers at ℓ>1950 for the best fit WMAP5+ACBAR model power spectrum. In that more restrictive analysis, the primary power spectrum is fixed and f_ν is allowed to vary as well as a broadband excess power. We found the ratio of excess power seen by CBI and ACBAR to be compatible with the ratio of the frequency prefactors f_ν at 30 GHz and 150 GHz for the Sunyaev–Zel'dovich effect, although foreground contamination of both experiments could also contribute to the observed excess. In this section, we assume that all the excess power seen by the CBI experiment is due to the SZE. In the hydrodynamical simulations used to derive the SZE template, the amplitude was shown to scale as q_SZ = (σ₈/0.9)⁷(Ω_bh/0.029)². We consider two cases: (1) the scaling parameter q_SZ is slaved to the results from the hydrodynamical simulations, which means that it is primarily determined by the primary CMB data; (2) q_SZ is allowed to float freely and an independent σ^SZ₈ is derived, to be compared with the σ₈ that is derived from the basic six parameters. We use a uniform prior in q_SZ in this case with limits 0 ⩽ q_SZ ⩽ 4.0.

Regardless of the data combination, we find that including an SZE component in the model has little effect on the values of most basic cosmological parameters (see Table 4), whether q_SZ is related to cosmic parameters through q_SZ = (σ₈/0.9)⁷(Ω_bh/0.029)² or is allowed to float freely. Note that the SZE results break the A_se^−2τ near-degeneracy (as does weak lensing, though not as strongly).

With the combination of the ACBAR and WMAP5 data, for which there is only a weak indication of excess power, we find a freely floating SZE amplitude results in q_SZ = 0.94^+0.35_−0.93, with no effective lower bound. We can use the above relation of q_SZ(σ₈, Ω_bh) to estimate a corresponding σ^(SZ)₈ = 0.97^+0.09(+0.13)_{−0.13(−0.31)}, where in parentheses we have indicated the one-sigma errors. This result is higher than, but compatible within the uncertainties, values obtained from the primary CMB fits alone: the ACBAR+WMAP5 fits in Table 4 give σ₈ = 0.80^+0.03_−0.03. The confidence limits of the derived σ^(SZ)₈ depend strongly on the choice of measure, which is here taken to be uniform in the amplitude q_SZ. (This is evident in the translation of the relative flatness of the likelihood at low q_SZ to a "one-sigma detection" in σ^(SZ)₈.) When the SZE contribution is slaved to the σ₈ and Ω_bh values from the primary spectrum, there is no effect on σ₈. The high ℓ excess power is not significant enough to change the well-constrained value of σ₈ to the weakly preferred higher value. The effect of slaved and unslaved fits can be seen in Figure 12.

When the high ℓ band powers of CBI and BIMA are included in the analysis, there is a significant detection of excess power. Both the CBI and BIMA band powers are from 30 GHz interferometric observations and have higher f_ν values than ACBAR. For the slaved case, the σ₈ value and errors are unchanged. For the floating case, we find q_SZ = 0.69^+0.11_−0.11, which maps to σ^(SZ)₈ = 0.93^+0.04_−0.05.

These central values and uncertainties are computed by transforming integrals of the likelihood L(q_SZ, Ω_bh, ...) over the prior measures to confidence limits for $\sigma _{8}^{(\rm SZ)}$. Exactly what measure to place on q_SZ and therefore on $\sigma _{8}^{(\rm SZ)}$ is debatable. A measure uniform in α_SZ = q^1/2_SZ, as we used in K07 and (Goldstein et al. 2003) translates into a measure ∝q^−1/2_SZdq_SZ that favors lower values of $\sigma _{8}^{(\rm SZ)}$ than the measure uniform in q_SZ that we have adopted.

As the cosmological parameters vary, the SZE template may depend on σ₈ and Ω_bh, and certainly depends on the spectral index n_s and astrophysical issues such as the history of energy injection into the cluster system. In a more complete treatment than that presented here, the shape should be modified along with the base cosmological parameters in the MCMC runs.

We caution that the derived σ^(SZ)₈ depends on the SZE template shape, its extension into the higher ℓ regime probed by BIMA, and the prior measure placed upon q_SZ.¹⁸ The modest decrease in log likelihood for the fit to the model when the SZE is taken into account is Δln L = 0.26 for the ACBAR+WMAP5 combination and increases to Δln L = 1.91 for CMBall+BIMA.

Regardless of the assumed prior, the interpretation of excess power as being due to the SZE results in a nonzero $\sigma _{8}^{(\rm SZ)}$ for the CMBall+BIMA combination. Uncertainties for σ^(SZ)₈ are about a factor of two larger than for the σ₈ determined from the primary CMB data, and there is a tension at about the one-sigma level between the two median values. A visual summary of the results is shown in Figure 13 where we plot both σ₈ and σ^(SZ)₈ against the spectral index for a number of data combinations. The addition of LSS data does not significantly change these results. When the ACBAR dusty point source or SZE template marginalization is included, σ₈ decreases slightly for the CMBall data set.

Recent weak lensing results are in basic agreement with the primary σ₈ values when Ω_m = 0.26 ± 0.03 from Table 4 is used. With 100 deg² of lensing data from the combination of the CFHT weak lensing legacy, RCS, Virmos-Descart, and GaBaDos surveys, Benjamin et al. (2007) get σ₈(Ω_m/0.26)^0.59 = 0.80 ± 0.05. From the CFHT weak lensing legacy survey alone, Fu et al. (2007) get σ₈(Ω_m/0.26)^0.64 = 0.753 ± 0.043, and get σ₈(Ω_m/0.26)^0.53 = 0.82 ± 0.084 if only large-scale linear-regime results are used. These weak lensing numbers are lower than past published results because of improved treatments of the redshift distribution of the lensed sources.

As shown in Figure 12, the marginal excess power in ACBAR is consistent with the combination of a point source contribution at the upper limit of the 150 GHz extrapolations and an SZE template at the level predicted from the primary anisotropy σ₈ value. In this scenario, the cosmological parameters are virtually unchanged, but no explanation is provided for the CBI excess. The SZE analyses with a free-floating amplitude have not included additional foreground sources for CBI, BIMA, or ACBAR. The effect of radio sources extrapolated to 30 GHz was included in the original CBI and BIMA results, and is unlikely to be an important contaminant for ACBAR. Dusty protogalaxies should not affect CBI and BIMA, but will contribute to the high ℓ ACBAR band powers. Given the uncertainty in the source contamination for ACBAR, the weak detection of excess power does not significantly support the SZE interpretation of the CBI+BIMA excess.

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There has been interest in the running of the spectral index dn_s/dln k since the first release of WMAP data, which showed evidence for a significant negative running when combined with LSS and Lyman alpha forest observations (Spergel et al. 2003). With the precision of the new ACBAR data, we might expect improved constraints on running of the spectral index. To the basic six parameters in the minimal model, we add running of the spectral index dn_s/dln k(k_⋆) around the pivot point k_⋆ = 0.05 Mpc⁻¹. We adopt the conventionally used uniform prior in dn_s/dln k(k_⋆), although in usual slow-roll-inflation models, the spectral index fluctuation δn_s ∝ ln(k/k_⋆)dn_s/dln k(k_⋆) is typically restricted by |1 − n_s|. Table 5 summarizes the parameter values when running is allowed and demonstrates that its inclusion has only a small effect on the other parameters. For WMAP5 only, we find dn_s/dln k(k_⋆) = −0.031^+0.029_−0.028. The main tendency for the negative value comes from the low ℓ WMAP5 data. When the ACBAR data is added, the median value is similar, dn_s/dln k(k_⋆) = −0.037^+0.023_−0.022, with a reduced error. The results are nearly identical for CMBall, but more compatible with no-running when LSS is added, −0.016^+0.019_−0.018. The precise measurement of the high ℓ CMB power spectrum provided by ACBAR is a potentially powerful constraint on running, but may also include significant contributions from secondary anisotropies and foregrounds. Therefore, in the last two columns, we have included the effect of marginalizing over the SZE or point source templates. In both cases, the mean value of the running becomes more negative and more significant at the one-sigma level. A visual representation of the impact of adding the ACBAR data is given in Figure 11, which shows the correlation between n_s and dn_s/dln k. The scalar spectral index, n_s(k_⋆) = 0.918 ± 0.032 for the WMAP5 + ACBAR combination, depends on the choice of pivot point k_⋆; a smaller value would yield a higher result, while a higher one would give an even lower result.

We have run a limited number of cases including tensor modes, characterizing their strength relative to scalar perturbations by r and fixing the tensor tilt by n_t ≈ −r/8(1 − r/16). The most stringent upper limit for r is given by the CMBall combination which yields r < 0.40 (95% confidence). This assumes a uniform prior measure for r, as is conventional in parameter estimation, although without much justification except that it is conservative. Adding LSS tightens this limit. Our result can be compared with those obtained by Dunkley et al. (2008), r < 0.43 (95% confidence) for WMAP5 alone.