CONSTRAINTS ON THE HIGH-ℓ POWER SPECTRUM OF MILLIMETER-WAVE ANISOTROPIES FROM APEX-SZ

Primary anisotropy in the cosmic microwave background (CMB) radiation is produced by inhomogeneities in the hot baryon–photon plasma at the epoch of recombination. The amplitude of these anisotropies as a function of angular scale has been used to infer precise constraints on the parameters of the standard ΛCDM model (Dunkley et al. 2009; Komatsu et al. 2009). On angular scales below several arcminutes, the primary anisotropy is damped by photon diffusion and the observed power is expected to be dominated by sources of foreground emission and the interaction of the CMB with intervening structure. In particular, the inverse Compton scattering of CMB photons off hot plasma bound to clusters of galaxies (Sunyaev & Zel'dovich 1972) gives rise to a spectral distortion of the CMB known as the Sunyaev–Zel'dovich effect (SZE). At frequencies less than ∼220 GHz, the SZE produces a decrement in the CMB intensity in the direction of a galaxy cluster. Galaxy clusters produce (secondary) anisotropy power on arcminute scales corresponding to their angular size. The observed SZE power depends sensitively on the abundance of clusters and the history of structure formation. In particular, the amplitude of the SZE power scales as σ⁷₈, where σ₈ is the rms fluctuation of matter on scales of 8 h⁻¹ Mpc, and serves as an independent probe of the amplitude of density perturbations (Komatsu & Seljak 2002).

Evidence for small-scale power beyond that expected from the primary CMB has been reported by the Berkeley Illinois Maryland Association (BIMA) and Cosmic Background Imager (CBI) interferometers operating at 30 GHz. These measurements find a level of SZE anisotropy power consistent with a value of σ₈ somewhat greater than those preferred by other contemporary measurements, which favor σ₈ ∼ 0.8 (Vikhlinin et al. 2009; Komatsu et al. 2009). BIMA observations at 30 GHz covering a total of 0.1 deg² of sky produced a nearly 2σ detection of excess power in a flat band centered at a multipole ℓ = 5237 (Dawson et al. 2006). Due to the non-Gaussian distribution of the SZE on the sky and the low significance of the detection, this resulted in only weak constraints on the matter power spectrum normalization, σ₈ = 1.03^+0.20_−0.29 at 68% confidence. Observations with the CBI experiment at 30 GHz over a larger field were used to produce a >3σ detection of excess power on angular scales ℓ ∈ [1800, 4000] (Sievers et al. 2009). Interpreting this power as being due to the SZE, they find σ₈ = 1.015 ± 0.06 at 68% confidence. However, recent observations with the SZA experiment operating at 30 GHz and covering a total of 2 deg² of sky have determined an upper limit on the excess power in broad band centered at multipole ℓ = 4066 which is significantly lower than the band powers reported by CBI and appears to be consistent with more conventional values of σ₈ ∼ 0.8 (Sharp et al. 2009). The SZA team interprets the higher power measured by CBI as potentially being due to an unsubtracted population of radio sources.

At 150 GHz, we expect the SZE and emission from distant dusty galaxies to contribute significantly to the power on angular scales corresponding to ℓ>2500. The ACBAR experiment reported a ∼1σ excess power at ℓ>2000 that if interpreted as the SZE would be consistent with the higher value for σ₈ preferred by CBI (Reichardt et al. 2009). However, the ACBAR results are in excellent agreement with a more standard value σ₈ ∼ 0.8 when one considers the expected foreground emission from IR point sources. The QUaD Collaboration has recently released a 150 GHz power spectrum for ℓ 2000 (Friedman et al. (QUaD Collaboration) 2009). The QUaD band powers appear (numerical values have not yet been released) to be systematically lower than those produced by ACBAR with no evidence for contributions from secondary anisotropy or foreground emission. Bolocam has recently published new results for anisotropy power at 150 GHz on angular scales above ℓ = 3000 (Sayers et al. 2009). They report upper limits on the power of 1080 μK² at 95% confidence in a wide bin centered at ℓ = 5700 and determine that σ₈ < 1.57 at 90% confidence. New high resolution and sensitivity bolometer arrays operating at millimeter wavelengths such as those currently deployed on the APEX, SPT, and ACT telescopes have the capacity to drastically improve constraints on the SZE and point source contributions to the high-ℓ power spectrum.

This paper presents new measurements of small-scale anisotropy power made with the APEX-SZ bolometer array on the Atacama Pathfinder Experiment (APEX) telescope from its high elevation site in the Atacama Desert. This work significantly improves the constraints on excess power above that expected from primary CMB anisotropy at ℓ>3000 at a frequency of 150 GHz. In Section 2, we describe the APEX-SZ instrument and the observations used to produce the results presented in this paper. The beam and calibration of the instrument are described in Section 3. The algorithms used in the production of the temperature maps and the power spectrum are described in Section 4. In Section 5, we present the power spectrum results and address sources of foreground emission. Our conclusions are summarized in Section 6.

APEX-SZ is an array of 330 transition-edge superconducting (TES) bolometers operating at 150 GHz (Schwan et al. 2003; Dobbs et al. 2006; D. Schwan et al. 2009, in preparation). The bolometers are cooled to 280 mK via a three stage He sorption fridge and mechanical pulse tube cooler and instrumented with a frequency-domain multiplexed readout system. The array observes from the 12 m APEX telescope on the Atacama plateau in Chile (Güsten et al. 2006) and has approximately 1' FWHM beams with a 22' field of view (FOV). The extremely dry and stable atmospheric conditions make the Atacama one of the best sites for millimeter-wave astronomy.

The band powers reported in this work are derived from a single, 0.8 deg² field observed by APEX-SZ for 10 nights in August and September of 2007. This field is a subset of the XMM-LSS field (Pierre et al. 2004) and is centered on a moderately massive, X-ray detected cluster, XLSSU J022145.2−034614, with an X-ray temperature of 5 keV (Willis et al. 2005; Pacaud et al. 2007). A joint analysis of the X-ray and SZ data will be undertaken in a separate paper. A circular scan strategy was used instead of a raster scan to improve the observing efficiency. The scan strategy concentrates the integration time at the center of the map causing the time per pixel to increase steadily from the edges of the map to the center. The total integration time is 2.9k detector hours. The map center reached a depth of 12 μK per 1' pixel. More details on the instrument and scan strategy can be found in Halverson et al. (2009, hereafter H09) and D. Schwan et al. (2009, in preparation).

The average beam of the APEX-SZ bolometers is measured with daily observations of Mars. At 8'' diameter, Mars is nearly a point source for the 1' APEX-SZ beam, and it is sufficiently bright to map the beam near sidelobes to below −25 dB. The measured beam agrees well with the ZEMAX¹⁵ simulated beam profiles when optical cross talk is taken into account. The near sidelobes increase the real beam solid angle by 32% compared to the best-fit Gaussian beam. We divide the measurement uncertainty on the beam into two parts. The main lobe is well-fitted by a Gaussian, and we estimate the measurement uncertainty on the FWHM to be 2.5%. Due to the large angular scale of the sidelobe structure, a misestimation of beam sidelobe will effectively cause a miscalibration of the band powers. We include the uncertainty in the total beam area in the calibration error.

The observations of Mars are used to establish the absolute calibration of the APEX-SZ instrument. The temperature of Mars for our observation frequency and dates is taken from the Rudy model (Rudy et al. 1987; Muhleman & Berge 1991) that has been updated and maintained by Bryan Butler.¹⁶ The Rudy model is compared with measurements of the brightness temperature of Mars made with the WMAP satellite at 93 GHz during five periods across several years (Hill et al. 2009). The WMAP measurements of Mars have uncertainty of <1%, and we adjust the normalization of the Rudy Model down by 5.2% to bring it into agreement with the WMAP measurements. Combining the ∼1% uncertainty in the WMAP Mars measurements, the 1.0% scatter in the 93 GHz WMAP to Rudy Model comparison, and 0.9% for the uncertainty in the extrapolation from 93 GHz, where the Rudy Model is calibrated, to our observing frequency, we find the total uncertainty in the Mars brightness temperature to be 1.7%.

The calibration of each detector is set by comparing the peak amplitude in a map of Mars to the expected amplitude given the temperature and size of Mars and the size of the detector's beam. A correction factor for the atmospheric opacity is applied which is always less than 3%. The overall calibration uncertainty is estimated to be 5.5% in temperature, with the dominant errors due to a 4% uncertainty in the beam area, a 3% uncertainty to account for temporal variations between the model prediction and the temperature measured by APEX-SZ, a 1.7% uncertainty in the temperature of Mars, and a 1.4% uncertainty in the conversion of brightness temperature to CMB temperature for the measured APEX-SZ band. More details on the beam estimation and calibration can be found in H09.

4.1. Map Making

4.2. Power Spectrum Estimation

The band powers, q_B, are reported in CMB temperature units of μK² and parameterize the power spectrum according to

$$\begin{equation} \frac{\ell (\ell + 1)}{ 2 \pi } C_\ell \equiv {\it D}_\ell = \sum _B q_B \chi _{B\ell }, \end{equation} \tag{ 1 }$$

where χ_Bℓ are top hat functions, χ_Bℓ = 1 for ℓ ∈ B and 0 for $\ell \not\in B$. We use a pseudo-$\it {C}_\ell$ power spectrum estimator (Hivon et al. 2002). In this formalism, the map spectrum (also called the pseudo-C_ℓ or $\tilde{C}_\ell$) depends on the true spectrum (C_ℓ) as

$$\begin{equation} \tilde{C}_\ell = M_{\ell \ell ^\prime } T_{\ell ^\prime } B^2_{\ell ^\prime } C_{\ell ^\prime }. \end{equation} \tag{ 2 }$$

$\tilde{C}_\ell$ is calculated using the flat-sky approximation, in which the spherical harmonic transform of the sky reduces to a Fourier transform of the map. This is an excellent approximation for a subdegree-sized map. The experimental beam function is described by B_ℓ, and the mode coupling due to finite sky coverage is denoted by $M_{\ell \ell ^\prime }$. T_ℓ represents the transfer function of the map-making process which would ideally be equal to one. In practice, the HPF applied to the APEX-SZ timestreams eliminates power on scales ℓ < 1400, so T_ℓ = 0 on these scales and remains below one at all ℓ. Following the MASTER algorithm (Hivon et al. 2002), we measure the transfer function using a set of Monte Carlo sky realizations that have been passed through the full pipeline, from the timestreams to the maps. These simulated sky maps include a lensed WMAP5+ACBAR best-fit CMB model (Hill et al. 2009; Reichardt et al. 2009), realizations of the SZE signal (Shaw et al. 2009), and realizations of the point source populations (Negrello et al. 2007; Granato et al. 2004; de Zotti et al. 2005). This set of signal-only simulations is also used to estimate the cosmic variance contribution to the band power uncertainties.

The mode-coupling matrix $M_{\ell \ell ^\prime }$ is calculated analytically for the two sky masks used to estimate the APEX-SZ band powers. These masks describe the weighting applied to each pixel in the map before calculating the Fourier transform and are analogous to windowing data in a one-dimensional Fourier transform. We begin by applying an inverse-noise weighting to each pixel, based on the diagonal elements of the pixel–pixel noise covariance matrix. This effectively de-weights the noisy edges of the map, and is near-optimal in the low signal-to-noise per pixel regime. We modify this simple mask to exclude pixels near detected clusters or point sources. There are two X-ray detected clusters in the field: XLSSU J022145.2−034614 and XLSSU J022157.4−034001. The field was centered on the first and more massive of these clusters, which would introduce a bias into the determination of the SZE amplitude. The central cluster is masked to a diameter of 6' in the first mask, hereafter the Cluster mask. The second cluster is fainter, and was detected only in a joint analysis of X-ray and optical data (Andreon et al. 2005). We tested the effects of masking the second cluster as well and did not see a significant change in the band powers. The second mask removes the 27 point sources detected in the APEX-SZ map in addition to the central cluster. These sources are found by using SExtractor (Bertin & Arnouts 1996) to select sets of neighboring pixels above 3 σ in an optimally filtered map. This corresponds approximately to a detection threshold of 2 mJy. We discuss these sources further in Section 5.1. This mask will be referred to as the Cluster+Sources mask. We tested the effect of masking all NRAO VLA Sky Survey (NVSS) or bright Spitzer sources within the field, and observed no significant change in power.

The measured band powers will be the sum of the signal and noise band powers,

$$\begin{equation} {\it D}_\ell = {\it D}_\ell ^S + {\it D}_\ell ^N, \end{equation} \tag{ 3 }$$

and we must subtract the expected noise contribution to recover the underlying signal spectrum. The noise contribution to the APEX-SZ band powers is estimated from the average power in a set of 2200 jack-knife maps between two randomly selected half sets of the ∼1100 complete observations of the field. These difference maps effectively remove sky signal, while preserving correlated noise in the timestreams on timescales shorter than the few minute length of a scan with randomized phase. Noise on longer timescales has been removed by the 0.3 Hz HPF. The expectation value of the noise band powers, 〈D^N_ℓ〉, is taken to be the mean band powers measured across the set of 2200 jack-knives. The approach is similar to that used in the analysis of the Bolocam power spectrum (Sayers et al. 2009). We apply the same procedure to signal-only simulated maps and confirm that any residual signal power due to the small pointing, filtering, and weighting differences between observations is negligible.

The power spectrum presented in Figure 1 is the product of applying the analysis in Section 4 to APEX-SZ observations of the XMM-LSS field. The band powers for angular multipoles from 3000 to 10,000 are tabulated in Table 1. The band powers can be compared to a theoretical model using the window functions. The numerical values for the band powers and window functions can be downloaded from the APEX-SZ website.¹⁷

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The APEX-SZ band powers show a tendency to increase on smaller angular scales, suggestive of the ℓ² shape that a Poisson distribution of point sources will have in a plot of ℓ(ℓ + 1)C_ℓ/2π. The contribution of the primary CMB will be small at these small angular scales, and we subtract the estimated contribution for the best-fit WMAP5+ACBAR lensed ΛCDM power spectrum before fitting for the amplitude of a constant C_ℓ. The beam and calibration uncertainty is incorporated by averaging the likelihood function over a set of 200 Monte Carlo realizations. The best-fit power is C_ℓ = 1.0^+0.9_−0.6 × 10⁻⁵ μK², which includes both the point source and SZE contributions. Results for both masks are tabulated in the first row of Table 2. If the expected SZE power spectrum for σ₈ = 0.8 is subtracted in addition to the primary anisotropies, the average C_ℓ drops slightly to 0.9^+0.9_−0.6 × 10⁻⁵ μK². At the best-fit point source amplitude for σ₈ = 0.8, 92% of the astronomical power in the map is produced by point sources rather than the SZE. These results are obtained after masking the bright, central cluster. Leaving the central cluster unmasked increases the fit power by ∼1.0 ×10⁻⁵ μK². We also report the results after masking the sources internally detected in the map at >3σ in the second row of Table 2. These results are consistent with and improved over the previous ACBAR constraints from ℓ < 3000 of C_ℓ = 2.7^+1.1_−2.6 × 10⁻⁵ μK². However, these numbers should be compared with caution as the two experiments have different flux cuts for source masking, and the excess power will depend on the flux to which sources have been masked.

Table 2. Point Source Power and σ₈ Constraints

	Cluster Masked	Cluster+Sources Masked
Zero SZE power:
$\it {C}_\ell ^{PS}$ (10−5μK2)	1.0+0.9−0.6	1.2+1.0−0.8
Fixed σ8 = 0.8:
$\it {C}_\ell ^{PS}$ (10−5μK2)	0.9+0.9−0.6	1.1+0.9−0.8
Unconstrained σ8:
$\it {C}_\ell ^{PS}$ (10−5μK2)	0.9+0.9−0.6	1.1+0.9−0.8
σ8 (G) (95% CL)	0.94	0.94
σ8 (NG) (95% CL)	1.18	1.18
Flat excess:
(with ℓcenter = 4966)
$\it {D}_\ell$ (μK2)	33+37−24	36+39−26
95% CL	97	105

Notes. The constraint on point source power $\it {C}_\ell ^{PS}$ and the 95% CL upper limit on σ₈ derived from the APEX-SZ data set are tabulated for different assumptions about the SZE power and masks. The expected primary CMB anisotropy power has been subtracted from the measured band powers. We show the upper limit on σ₈ with (NG) and without (G) accounting for the non-Gaussian distribution of the SZE. Accounting for the non-Gaussianity in the expected SZE sky weakens the upper limit considerably. The first column (Cluster masked) shows the results when the massive, X-ray detected cluster in the field is masked to 6' diameter. The second column (Cluster+Sources masked) excludes the >3σ sources detected in the map as well as the cluster. More details on the masks can be found in Section 4.2. The measured point source power is in excellent agreement with the predicted amplitude of 1.1 × 10⁻⁵ μK² (1.7 Jy² sr⁻¹) for the sum of the radio and dusty galaxy models in all cases. We also show the results under the assumption that the power at high-ℓ above the primary CMB anisotropies can be modeled as a flat band power. The 95% CL upper limit on a flat excess of <105 μK² includes the beam and calibration uncertainties but does not include non-Gaussian contributions to cosmic variance.

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We also investigate the effects of allowing the amplitude of the SZE power spectrum to float freely. The SZE power spectrum template is based on the simulations in Shaw et al. (2009). The simulations are for a WMAP5 cosmology with σ₈ = 0.77. The amplitude of the SZE power spectrum is expected to scale approximately as σ⁷₈, so the derived SZE amplitude can be related to σ₈. In practice, the APEX-SZ data set lacks the sensitivity to make a detection of SZE power; however, the results can be used to place an upper limit on σ₈. The upper limits on σ₈ and the point source amplitudes for the joint fit are reported in Table 2. We assume a flat prior on σ₈. The exact amplitude of the SZE spectrum is only poorly understood, leading to a 10% systematic uncertainty on σ₈ (Komatsu & Seljak 2002). This systematic uncertainty is not included in the reported upper limits.

The non-Gaussian distribution of the SZE is very important on small patches of sky (Cooray 2001; Zhang et al. 2006). We incorporate the non-Gaussian statistics into our analysis by returning to the set of simulated SZ skys (Shaw et al. 2009). We extract 7500 independent realizations of the APEX-SZ map and calculate the power in each realization under the two masks. The maps have been convolved by the experimental beam and do not include noise. This process maps out the full, non-Gaussian cosmic variance of the expected SZE power for σ₈ = 0.77. We scale this to other cosmologies by assuming that the probability of measuring a power X will scale with σ₈ as

$$\begin{equation} P(X | \sigma _8) = \left(\frac{\sigma _8}{0.77} \right)^7 P\left(\left(\frac{0.77}{\sigma _8} \right)^7 X\,| \sigma _8=0.77\right). \end{equation} \tag{ 4 }$$

Bayes' theorem with a flat prior in σ₈ is applied to find a posterior probability density, P(σ₈|X). We determine the probability of a given SZE power from the data by marginalizing over a point source component as described in the last paragraph. Finally, the likelihood function of σ₈ given the APEX-SZ data d is calculated by

$$\begin{equation} P(\sigma _8 | {\bf d}) = \int {dX P(\sigma _8 | X) P(X | {\bf d}) } \end{equation} \tag{ 5 }$$

and integrated to find the 95% CL upper limit on σ₈. The limit rises to σ₈ < 1.18, substantially weaker than the limit of σ₈ < 0.94 derived under Gaussian assumptions (see the third row of Table 2, labeled "Unconstrained σ₈"). The marginalized likelihood function for both σ₈ and C^PS_ℓ are plotted in Figure 2, while the 2d likelihood surface for both parameters is shown in Figure 3. The upper limit is sensitive to the prior chosen since APEX-SZ does not make a detection of SZE power. A flat prior on power, σ⁷₈, strongly prefers higher values of σ₈ than the flat prior on σ₈, and raises the upper limit from 1.18 to σ₈ < 1.50 at 95% CL. This is entirely due to the weighting by the prior as the prior probability for σ₈ = 2 is 240 times the probability of σ₈ = 0.8.

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Finally, we combine the four APEX-SZ band powers into a single band to facilitate the comparison to other data sets. The resulting upper limit is ∼100 μK² after including the APEX-SZ calibration and beam uncertainty as shown in the last row of Table 2, "Flat Excess." We have assumed that D_ℓ is constant across the four bands and subtracted the contribution due to the primary CMB anisotropies. However, this upper limit does not include a non-Gaussian contribution to cosmic variance.

5.1. Radio and IR Source Contributions

Observations with the APEX-SZ instrument have been used to constrain the power in excess of the primary CMB temperature anisotropies at 150 GHz. This power is expected to be dominated by emission from submillimeter bright, dusty galaxies and the APEX-SZ band powers are consistent with this hypothesis. We find excellent agreement between the point source power in the APEX-SZ maps and model predictions based on observations at other frequencies. We estimate that the flux of these submillimeter bright galaxies scales with frequency as S_ν ∼ ν^2.64 by comparing the power measured by BLAST at 600 GHz to the best-fit point source power in the APEX-SZ maps at 150 GHz. Determining the contribution of these foreground sources not only constrains models for the population of dusty galaxies, but is important for planning current and future observations of secondary CMB anisotropies at these wavelengths.

We also place upper limits on σ₈ from fits to the amplitude of the SZE power spectrum while marginalizing over a Poisson point source contribution. We assume a template for the SZE power spectrum derived from simulations by Shaw et al. (2009) with the amplitude of the SZE power spectrum scaling as σ⁷₈. We find an upper limit of σ₈ < 1.18 at 95% confidence. This result is similar to the constraints from the CBI and BIMA interferometers operating at 30 GHz. The limits from SZA, QUaD, or ACBAR would likely be slightly lower, but they did not express their results in terms of upper limits on σ₈. At these frequencies and angular scales, the previous best limit comes from Sayers et al. (2009), who used observations with the Bolocam instrument to constrain σ₈ < 1.57 at 90% confidence.

A third of the APEX-SZ instrument was recently upgraded to more sensitive detectors with improved optical efficiencies. In the next year, the remainder of the focal plane will be upgraded resulting in significant improvements to the instrument's mapping speed. The instrument will continue observing in the next several years with a focus on developing a catalog of clusters with SZE, X-ray, and optical observations across the Southern Hemisphere.

We thank the staff at the APEX telescope site, led by David Rabanus and previously by Lars-Åke Nyman, for their dedicated and exceptional support. We also thank LBNL engineers John Joseph and Chinh Vu for their work on the readout electronics. APEX-SZ is funded by the National Science Foundation under Grant No. AST-0138348. Work at LBNL is supported by the Director, Office of Science, Office of High Energy and Nuclear Physics, of the US Department of Energy under Contract No. DE-AC02-05CH11231. Work at McGill is supported by the Natural Sciences and Engineering Research Council of Canada and the Canadian Institute for Advanced Research. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the US Department of Energy under Contract No. DE-AC02-05CH11231. N.H. acknowledges support from an Alfred P. Sloan Research Fellowship.