CORRELATIONS IN THE (SUB)MILLIMETER BACKGROUND FROM ACT × BLAST

The beam-corrected power spectrum of the sky is a superposition, depending on wavelength, of the following terms:

$\begin{eqnarray} C_{\ell }^{\rm sky} &=& C_{\ell }^{\rm cirrus} + C_{\ell }^{\rm CMB} + C_{\ell }^{\rm radio} \nonumber \\ &&+\, C_{\ell }^{\rm DSFG} + C_{\ell }^{\rm SZ} + C_{\ell }^{\rm ff} + N_{\ell }, \end{eqnarray} \tag{ 1 }$

where C_ℓ represents the angular power spectrum in multipole space, ℓ, and N_ℓ is the noise. Here "cirrus" refers to emission from Galactic dust (Section 2.6), "ff" refers to free–free emission, and "radio" refers to radio sources, GHz-peaked sources and similar objects, whose fluxes increase at longer wavelengths (Section 2.4). It is assumed that diffuse synchrotron emission is negligible. We assume that the various components in Equation (1) are uncorrelated when in reality, correlations among various components likely exist. Such correlations should typically be small and can therefore be reasonably neglected. Furthermore, we define the power from the extragalactic sky as $C_l^{\rm CSB} \equiv C_l^{\rm sky} - C_l^{\rm cirrus} -C_{\ell }^{\rm ff}$. The C^ff_ℓ component is assumed to be negligible for the area of the sky we are dealing with. In what follows we report power spectra as both C_ℓ and P(k_θ), with k_θ the angular wavenumber. To convert from multipole ℓ to k_θ, or from μK² to Jy² sr⁻¹, see Appendix A.

In order to isolate the spectra of one or more contributors to the background requires removal, or adequate modeling, of the unwanted power. Since the contributors have distinct spectral signatures (i.e., their flux densities vary from band to band differently), multi-frequency observations make decomposition of the signal possible. For discrete sources, the ratio of flux densities from band to band is

$\begin{equation} \frac{S_{\nu _1}}{S_{\nu _2}} = \left(\frac{\nu _1}{\nu _2} \right) ^{\alpha _{\nu _1 - \nu _2}}, \end{equation} \tag{ 2 }$

where $\alpha _{\nu _1 - \nu _2}$ is the "spectral index" and is a function of the rest-frame spectral energy distributions (SEDs) of the sources that make up the galaxy population, and their redshift distributions. Consequently, measurements of the spectral indices can place powerful constraints on source population models (e.g., Marsden et al. 2011; Béthermin et al. 2011).

At wavelengths longer than ∼1 mm (ν = 350 GHz) the CMB dominates the power spectrum on scales greater than ∼8'. Multiple peaks in the spectrum have been measured most recently by Brown et al. (2009), Friedman et al. (2009), Reichardt et al. (2009a, 2009b), Sayers et al. (2009), Lueker et al. (2010), Sharp et al. (2010), Fowler et al. (2010), Das et al. (2011), and Nolta et al. (2009). Secondary anisotropies include CMB lensing, which acts to smooth out the peaks and add excess power to the damping tail; and the SZ effect, which distorts the primordial CMB signal.

In the present analysis, the CMB power in ACT maps, which dominates on large scales, does not correlate with signal in the BLAST maps; however, it does act to increase the noise on those scales (see Appendix B).

DSFGs, as their name implies, are galaxies undergoing vigorous star formation, much of which is optically obscured by dust. They have average flux densities of 5 mJy (at 250 μm; Marsden et al. 2009), star formation rates (SFRs) of ∼100–200 M_☉ yr⁻¹ (Pascale et al. 2009; Moncelsi et al. 2011), number densities of ∼2 × 10⁻⁴ Mpc⁻³ (e.g., van Dokkum et al. 2009), and typically lie at redshifts 0–4, with the peak in the distribution at z ∼ 2 (Amblard et al. 2010). They are distinguished from "submillimeter galaxies" (SMGs) discovered by SCUBA (Smail et al. 1997; Hughes et al. 1998; Eales et al. 1999), which are 10 times less abundant (Coppin et al. 2006), lie at slightly higher redshifts (Chapman et al. 2005), have SFR  ∼  1000 M_☉ yr⁻¹, and are thought to be triggered largely by major mergers (Engel et al. 2010). They are of course related: SMGs comprise the extreme, high-redshift end of the DSFG population.

The dust in DSFGs absorbs starlight and re-emits it in the IR/submm, with an SED phenomenologically well approximated by a modified blackbody,

$\begin{equation} S_{\nu } \propto \nu ^{\beta } B(\nu), \end{equation} \tag{ 3 }$

where B(ν) is the Planck function and β is the emissivity index (whose value typically spans 1.5–2; e.g., Draine & Lee 1984). The SED of a typical DSFG with temperature ∼30 K and β = 2 (Chapin et al. 2011) peaks at rest-frame λ ≃ 100 μm (which redshifts into the submm at z ∼ 1–10). A property of this shape is that with increasing redshifts, observations in the (sub)mm bands continue to sample at a rest-frame wavelength close to the peak of the SED, so that even though sources become more distant, their apparent flux remains roughly constant. This so-called negative K-correction makes observations at longer wavelengths more sensitive to higher-redshift sources (Blain et al. 2003). As a result, sources at z ≳ 1 have a significant impact on the power spectrum at (sub)mm bands.

The cross-frequency power spectrum between bands 1 and 2 arising from unclustered sources is related to the number counts—i.e., the surface density N as a function of flux density S—as follows:

$\begin{equation} C_{\ell }^{\rm Poisson}=\int ^{S_{\rm cut_1}}_0 \int ^{S_{\rm cut_2}}_0S_1 S_2 \frac{d^2N}{dS_1 dS_2} dS_1 dS_2, \end{equation} \tag{ 4 }$

where dN/dSΔS is the number of sources per unit solid angle in a flux bin of width ΔS, at frequency bands 1 and 2, and S_cut is the flux density at which the counts are truncated. When the slope of the counts is steeper than −3, the power diverges at low flux densities, and the power spectrum is dominated by the contribution to the background from faint sources. In the case of DSFGs, the strong evolution of the source counts with redshift results in a steep slope at the faint end (Devlin et al. 2009), so that after masking local sources (with S_cut ≲ 500 mJy) the DSFG component of the CIB at λ > 250 μm is dominated by faint sources and remains finite.

Since DSFGs are spatially correlated (Viero et al. 2009, hereafter V09), their power spectrum has both Poisson and clustered components:

$\begin{equation} C_{\ell }^{\rm DSFG}= C_{\ell }^{\rm DSFG, Poisson} + C_{\ell }^{\rm DSFG, clustered}. \end{equation} \tag{ 5 }$

The measured strength of the clustered component is such that it dominates over the Poisson on scales ≳ 3' (V09; Marsden et al. 2009; Hall et al. 2010; Dunkley et al. 2011; Shirokoff et al. 2011), with the exact value depending on frequency and flux cut. How the strengths of the Poisson and clustering terms scale with wavelength, and if they evolve together or independently, remain open questions.

We compare to the models of Béthermin et al. (2011, hereafter B11) and Marsden et al. (2011, hereafter M11) for DSFGs at the BLAST and ACT wavebands. These are phenomenological models which are specifically tailored to constrain the evolution of the rest-frame far-infrared peak of galaxies at redshifts up to z ∼ 4.5. The models use similar Monte Carlo fitting methods but differ in a few ways, e.g., the B11 model is fit using data (predominantly number counts) from a very wide range of IR wavelengths (15 μm to 1.1 mm) while M11 uses only data that constrains the evolution of the far-IR (FIR) peak and was not fit to any observations with λ < 70 μm. B11 also divides galaxies into two distinct populations based on luminosity and attempts to account for the strong lensing of high-redshift galaxies; M11 does neither of the above.

Synchrotron, and to a lesser extent free–free emission, dominates the SEDs of radio galaxies at rest-frame λ ≳ 1 mm. Thus radio galaxies become an increasingly important contribution to the CSB at wavelengths greater than ∼1.5 mm (ν ≲ 200 GHz). Their number counts are relatively shallow (e.g., de Zotti et al. 2010), meaning that their contribution to the power spectrum is dominated by the brighter sources, resulting in primarily Poisson noise.

While radio sources are a significant source of power at 2030 and 1380 μm, they do not feature prominently in the cross-frequency correlation of ACT and BLAST maps, so we do not include them in our models for the cross-spectra. They do, however, contribute to the uncertainties in the cross-power spectrum, as is described in Appendix B.

The SZ effect (Sunyaev & Zeldovich 1972) is the distortion of the microwave background due to interaction of CMB photons with free electrons in clusters, and consists of two main physical mechanisms, referred to as "thermal " and "kinetic." The thermal term (tSZ) is the inverse Compton scattering of CMB photons as they pass through the intracluster medium in galaxy clusters. The result is that the intensity of the CMB spectrum longward of 1380 μm (218 GHz) decreases (i.e., a decrement), while that shortward of 1380 μm increases (i.e., an increment), and 1380 μm is the null for the non-relativistic case. The kinetic term (kSZ) is the Doppler shift of scattered CMB photons by the bulk motion of galaxy clusters. The strength of the signal is proportional to the product of the free electron density and line-of-sight velocity.

On large angular scales, a significant source of fluctuation power is emission from Galactic cirrus. Although the south ecliptic pole (SEP) field is among the least contaminated by cirrus in the sky (I_mean = 1.16 MJy sr⁻¹), contributions from Galactic cirrus must still be accounted for.

The power spectrum of Galactic cirrus has been shown in many studies to exhibit power-law behavior. Its amplitude varies over the sky, but its slope is always between −2.6 and −3 (e.g., Gautier et al. 1992; Boulanger et al. 1996; Miville-Deschênes et al. 2007; Bracco et al. 2011). In the FIR/submm bands, the SED of Galactic cirrus is well described by a modified blackbody (Equation (3)), with T = 17.5 K and β ∼ 1.9, peaking at λ ∼ 150 μm (V09; Bracco et al. 2011). As a result, bands closest to the SED peak (in our case 250 μm) are most susceptible to contamination.

As one moves far from the SED peak, Finkbeiner et al. (1999) show that the modified blackbody approximation breaks down, and a multi-component fit is a much better description of the data. Therefore, for this analysis we measure the power spectrum at 100 μm for the SEP region and adopt model 8 of Finkbeiner et al. (1999) to estimate the amplitude of the power in our bands.

ACT is a 6 m off-axis Gregorian telescope (Fowler et al. 2007) situated at an elevation of 5190 m on Cerro Toco in the Atacama desert in northern Chile. ACT has three frequency bands centered at 148 GHz (2.0 mm), 218 GHz (1.4 mm), and 277 GHz (1.1 mm) with angular resolutions of roughly 14, 10, and 09, respectively. The high altitude site in the arid desert is excellent for mm observations due to low precipitable water vapor and stability of the atmosphere. The tropical location of ACT permits observations of both the northern and southern celestial hemispheres. Further details on the instrument are presented in Swetz et al. (2010), Fowler et al. (2010), and references therein.²⁶ The ACT maps used in this paper are made from data taken during the 2008 observing season (at 148 GHz and 218 GHz, or 2030 and 1380 μm, respectively) and are identical to the maps used in Hajian et al. (2010) and Das et al. (2011). The beam full widths at half-maxima (FWHM) are 14 and 10 at 148 GHz and 218 GHz, respectively (Hincks et al. 2010). The maps have mean 1σ sensitivities which vary slightly across the maps, ranging from 2.4–3.5 mJy beam⁻¹ (median ≈2.7 mJy beam⁻¹) and 3.2–5.4 mJy beam⁻¹ (median ≈3.7 mJy beam⁻¹), at 148 GHz and 218 GHz, respectively (Das et al. 2011). The map projection used is cylindrical equal area (CEA) with square pixels, 05 on a side. The ACT data set is divided into four equal subsets in time, such that the four independent maps generated from these subsets cover the same area and have similar depths. We call these "sub-maps." As described in Hajian et al. (2010), the ACT maps are directly calibrated to the Wilkinson Microwave Anisotropy Probe (WMAP). This results in a 2% fractional temperature uncertainty for the 148 GHz maps. The calibration error for the 218 GHz maps is 7%.

Because the ACT maps have poorly measured modes on the largest angular scales, we filter them using a high-pass filter F_c(ℓ) in Fourier space. The high-pass filter is a smooth sine-squared function in Fourier space given by

$\begin{equation} F_{\rm c}(\ell) = \sin ^2{x(\ell)} \Theta (\ell -\ell _{\mathrm{min}}) \Theta (\ell _{\mathrm{max}}-\ell) + \Theta (\ell -\ell _{\mathrm{max}}), \end{equation} \tag{ 6 }$

where x(ℓ) = (π/2)(ℓ − ℓ_min)/(ℓ_max − ℓ_min) and Θ is the Heaviside function. We choose ℓ_min = 100 and ℓ_max = 500. Moreover, the large-scale CMB in the ACT maps acts as noise in cross-correlations with the BLAST maps, since the CMB is absent in the latter. If not filtered, the large angular scale and CMB noise terms contaminate the real-space cross-frequency correlations described in Section 3.4. Therefore we use a filter with ℓ_max = 2200 when dealing with real-space cross-frequency correlations. The analyzed power spectra are corrected for this filter as well as for the effects of the beam and pixel window functions.

BLAST flew for 11 days from Antarctica in 2006 December. As a pathfinder for the SPIRE instrument (Griffin et al. 2003), it made observations at 250, 350, and 500 μm, of a number of targets, both Galactic and extragalactic. Its 1.8 m underilluminated primary resulted in beams with FWHM of 36'', 45'', and 60''. For a detailed description of the instrument see Pascale et al. (2008) and Truch et al. (2009).²⁷

Among the fields BLAST observed is an 8.6 deg² rectangle near the SEP—chosen because it is a relatively low-cirrus window through the Galaxy (see Section 2.6)—whose corners lie at [(5^h02^m, −52°50'); (4^h57^m, −51°35'); (4^h25^m, −54°19'); (4^h30^m, −55°41')] (see Figure 1). Further studies of the SEP field, including BLAST catalogs and 24 μm maps, can be found in Valiante et al. (2010) and Scott et al. (2010). The maps have mean 1σ sensitivities of 36.7, 27.2, and 19.1 mJy beam⁻¹ at 250, 350, and 500 μm, respectively. Furthermore, confusion noise, due to multiple point sources occupying a single beam element, is estimated to be 7.6, 6.0, and 4.4 mJy beam⁻¹ at 250, 350, and 500 μm, respectively. The 1σ uncertainty on the absolute calibration is accurate to 9.5%, 8.7%, and 9.2% at 250, 350, and 500 μm, respectively.

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The BLAST time-ordered data are divided into four sets— covering the same region of the sky to the same depth—from which we make four unique sub-maps. The number of subsets is chosen to maximize the number of maps that can be made while maintaining uniformity in hits and providing as much cross-linking as possible. The maps are made with the iterative map maker SANEPIC (Patanchon et al. 2009) resulting in a transfer function of unity on the scales of interest. These sub-maps are unique to this study and are publicly available at http://blastexperiment.info/results.php.

Due to poor cross-linking, however, large-scale noise, resembling waves in the map, is present. This noise is easily dealt with by filtering in Fourier space, as described in Section 4. Maps are made in tangent-plane projection, with 10'' pixels. In order to cross-correlate with ACT, BLAST maps are re-binned to ACT resolution and reprojected to CEA projection using Montage.²⁸ We confirm the alignment by analyzing the real-space cross-frequency correlation, described in detail in Section 3.4.

To estimate the contribution from Galactic cirrus we use three IRIS (reprocessed IRAS; Miville-Deschênes & Lagache 2005) HCON²⁹ maps at 100 μm. These maps are consistent with the Finkbeiner et al. (1999, hereafter FDS) maps used for estimating the contribution from Galactic cirrus in Das et al. (2011), but they are at a higher resolution. Since we are most interested in large-scale modes, the power spectrum is measured for a 30 deg² field surrounding the SEP (see Figure 2).

We test that the maps are properly aligned by inspecting their real-space cross-frequency correlations. This is done by inverse Fourier transforming the two-dimensional cross-frequency correlation of the Fourier components of the maps:

$\begin{equation} M_{a\times b}({\bf x}) = \sum _{{\mathbf {\ell }}} a({\bf \ell })b^*({\bf \ell })F_{\rm c}^2({\bf \ell }) {\rm exp}(i{\bf \ell \,\mbox{}\,x}), \end{equation} \tag{ 7 }$

where ℓ is a vector in Fourier space, and a(ℓ) and b(ℓ) are the ACT and BLAST maps in Fourier space, respectively. F_c(ℓ) denotes the high-pass filtering in Fourier space described in Equation (6). The resulting measured real-space cross-frequency correlation function M encodes the celestial correlation function between the bands as

$\begin{equation} M_{a\times b}({\bf x}) = C({\bf x}) \otimes B({\bf x}) + n({\bf x}), \end{equation} \tag{ 8 }$

where ⊗ denotes a convolution in real space, B(x) is the effective beam between the two maps, and n(x) is the noise. Perfectly aligned maps would result in a two-dimensional cross-frequency correlation function whose peak lies at x = 0. The shape of M_{a × b}(x) depends on the correlation length of the field, the high-pass filtering, and the beams. (See Hajian et al. 2010 for a comparison between measurements of a real-space cross-frequency correlation function and simulations for a related example.) As illustrated in Figure 3, we measure a cross-correlation with a peak at x = 0 (zero lag), indicating that the maps are correlated and properly aligned.

We use a flat-sky approximation for computing the power spectra, which are measured from Fourier transforms of the maps. We use three distinct types of power spectra: the BLAST auto-band spectra, the BLAST cross-band spectra, and the ACT/BLAST cross-band spectra.

The maps can be represented as

$\begin{equation} \Delta T(\bf x) = \Delta T_{\rm sky}{(\bf x)}\otimes B{(\bf x)} + N{(\bf x)}, \end{equation} \tag{ 9 }$

where $\Delta T_{\rm sky}{(\bf x)}$ is the sky temperature signal, N is the noise, B is the instrument beam and we use ⊗ to represent a convolution in real space. Both the ACT and BLAST maps are made with unbiased iterative map makers, whose transfer functions are approximately unity on the angular scales of interest in this study, and can thus be safely neglected.

All power spectra, both auto- and cross-frequency spectra, are computed using cross-correlations of sub-maps (described in Sections 3.1 and 3.2), the advantage being that the noise between sub-maps is uncorrelated and thus averages to zero in the cross-spectra. A cross-power spectrum computed this way provides an unbiased estimator of the true underlying power spectrum. The power spectrum methods used in this paper closely follow those used for cross-correlating ACT and WMAP in Hajian et al. (2010).

BLAST × BLAST power spectra are computed using the average of the six cross-correlations between the four BLAST sub-maps (in each band), such that

$\begin{equation} C_{\ell } = \frac{1}{6} \sum ^{1 \le \beta \le 4}_{\alpha, \beta ; \alpha < \beta } C_{\ell }^{\alpha \beta }, \end{equation} \tag{ 10 }$

where α and β index the four sub-maps. The reason for six cross-correlations, rather than nine, is that cross-frequency cross-correlations of sub-maps made from the same scans are not used, in order to avoid introducing correlated noise or other systematic effects.

Since the noise in all ACT and BLAST sub-maps is uncorrelated, we co-add the sub-maps for each frequency before computing the ACT×BLAST power spectra. The ACT × BLAST power spectra are computed from these maps. This is identical to cross-correlating all ACT sub-maps with all BLAST sub-maps and averaging them.

Several components contribute to the cross-spectrum uncertainties. An analytic approach to computing the uncertainties is described in Appendix B.

We adopt techniques developed in Hajian et al. (2010) in order to isolate and remove or downweight instrumental and systematic noise. This is done in two stages, in real space and in Fourier space, before reducing the two-dimensional spectrum into the familiar one-dimensional angular power spectrum binned in radius (ℓ). The ACT × BLAST cross-spectra are limited by the area of the BLAST maps, which are 15 × 57 rectangles (∼8.6 deg²), rotated by approximately 30°, and with noisy edges (see Figure 1). To apodize the sharp edges in the maps, we use the first Slepian taper (properties of which are described in Das et al. 2009) defined on the BLAST region. The gradual fall-off of this taper at the edges reduces the mode–mode coupling in the measured power spectrum. Any residual mode coupling caused by this weighting is corrected in the end by deconvolving the window function (i.e., the power spectrum of the taper function) from the measured power spectrum using the algorithm described in Hivon et al. (2002) and Das et al. (2009). Large-scale noisy modes are best treated in Fourier space. The statistical isotropy of the universe leads to an isotropic two-dimensional power spectrum from extragalactic sources, on average. Anisotropic power in two-dimensional Fourier space is caused by noise and is optimally dealt with using inverse noise weighting, which involves dividing the two-dimensional spectra by our best estimate of the two-dimensional noise power spectrum for each map. At each frequency, the noise model is computed from the difference between the average two-dimensional auto- and cross-spectra for each pair of maps, as described in Hajian et al. (2010). For every cross-correlation, noise weights are computed from the inverse of the square root of the product of the two noise power spectra corresponding to the two frequency bands. The weights are whitened by dividing by their angular averaged value with a fine binning. Using simulations we confirm that this weighting does not bias the signal power spectrum. Anisotropic, large-scale noise is evident at the center of the two-dimensional power spectrum, as shown in Figure 4. Noise weighting downweights the noisy vertical stripe that passes through the origin, which is predominantly due to large-scale unconstrained modes in the map (see also Fowler et al. 2010). To further ensure that our results are not contaminated by this stripe, we mask a vertical band spanning |ℓ_x| < 500 before averaging the two-dimensional power spectra in annuli. Our results are invariant under further widening of this mask. In order to be consistent with the V09 analysis, we make a mask to remove all point sources that have a flux greater than 0.5 Jy in the BLAST 250 μm map (six sources). We use that mask for computing BLAST power spectra only. For cross-correlations of ACT and BLAST, we instead mask just the brightest source in BLAST maps, which happens to be a local spiral galaxy. We confirm that masking more point sources does not have an effect on the cross-spectra. We also mask the known radio galaxies (Marriage et al. 2011) in the ACT 2030 μm map to reduce the uncertainties in the cross-spectra (see Appendix B).

IRIS maps at 100 μm contain three potential sources of power—diffuse Galactic emission (or cirrus), as well as the Poisson and clustered terms of the DSFG power spectra. Cirrus dominates the power spectrum on angular scales ℓ ≳ 800, but varies depending on the observed patch of sky. The Poisson level is highly sensitive to the adopted flux cut. Thus, in order to detect the signal from clustering, both the cirrus and Poisson noise must be sufficiently low. This is achieved by observing in a clean patch of sky, and cutting bright point sources. We realize these criteria by considering only the SEP (I_mean = 1.16 MJy sr⁻¹) and by cutting sources with S_cut > 1 Jy.

Miville-Deschênes et al. (2007) show that the power spectrum of Galactic cirrus can be approximated by a power law,

$\begin{equation} P_{\mathrm{cirrus}}(k_{\theta }) = P_0\left(\frac{k_{\theta }}{k_0} \right) ^{\alpha }, \end{equation} \tag{ 11 }$

where k_θ is the angular wavenumber in inverse arcminutes and P₀ is the power spectrum value at k₀ ≡ 0.01 arcmin⁻¹.

We are only concerned with the modes that affect our measurement, and since the cirrus power spectrum falls steeply with increasing k_θ, we focus our attention on the larger-scale modes. In order to probe these modes with maximum resolution in Fourier space, we measure the cirrus component of a ∼30 deg² region of the IRIS data surrounding the SEP field as indicated in Figure 2. We filter the IRIS maps using the high-pass filter of Equation (6). The power spectrum is computed from the mean of the three cross-spectra from the three HCON maps (using one taper at resolution 1; see Das et al. 2009 for details).

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We attempt to fit the data in two ways, where in both cases we fix the slope of the cirrus power spectrum at α = −2.7 (adopting the properties of region 5 of Bracco et al. 2011, whose mean flux density most resembles the SEP). The first is a two-parameter fit, where the free variables are the Poisson level and the amplitude of the cirrus power. For this we find P₀ = (0.47 ± 0.06) × 10⁶ Jy² sr⁻¹ and χ² = 21.9 (dof = 29). The measured power spectrum is shown in Figure 5. The power spectrum uncertainties are calculated in a manner analogous to that described in Appendix B.

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The second fitting procedure uses a three-parameter fit in which the free variables are the Poisson level, cirrus amplitude, and clustering amplitude of the DSFGs. The shape of the clustering component is simply that of a linear dark matter spectrum. In this case we find cirrus values P₀ = (0.19 ± 0.15) × 10⁶ Jy² sr⁻¹ and a clustering amplitude of ∼720 Jy² sr⁻¹ at ℓ = 3000, with χ² = 18.1(dof = 28).

The two approaches estimate consistent Poisson levels, but the fit with a clustered component appears to describe the data better than without, with Δχ² = 3.8. While not yet statistically significant, future studies with Photodetector Array Camera and Spectrometer at 100 μm (Poglitsch et al. 2010) should be able to measure the clustered signal to high significance. The cirrus power spectrum is assumed to continue to smaller angular scales and is estimated at the ACT and BLAST bands using the average dust emission color $(I_{\mathrm{\rm (sub)mm}}/I_{100})^2$, which is estimated using model 8 of Finkbeiner et al. (1999).

In principle the bias, b, is a function of scale and redshift, as well as environmental factors such as the host halo mass of DSFGs. Here we adopt a simple redshift-dependent bias of the form (Bond et al. 1991; Hui & Parfrey 2008)

$\begin{equation} b(z) = 1+ (b_0 -1) \frac{D(z_0)}{D(z)}, \end{equation} \tag{ 14 }$

where D(z) is the linear growth function and b₀ is an initial bias at some formation redshift, z₀. This parameterization assumes that DSFGs are members of a single population, which formed at the same epoch (z₀) and under the same conditions.

Our parameter space consists of the 12 Poisson levels plus b₀ and z₀. However, just as the Poisson contribution is a sum over the galaxy distribution, weighted by the square of the fluxes (Equation (4)), the average bias is a sum weighted linearly by the flux (e.g., Bond 1993). Thus, in any physical model for the star-forming objects and their bias, the two terms would be correlated in a model-dependent way; we therefore choose to adopt an independent bias for simplicity. Decoupling the Poisson level and the clustered component means the interpretation of our derived b is not straightforward. We find that moderate changes in z₀ (in the range 6 < z₀ < 10) have virtually no effect on the quality of the fit or best-fit b(z), and also find that b₀ and z₀ are almost degenerate, and so we fix z₀ = 8. We explore the remaining 13-parameter space using a Markov Chain Monte Carlo (MCMC) method with uniform priors on each parameter. We fit the ACT×BLAST data in the range ℓ = 950 (k_θ = 0.04 arcmin⁻¹) and ℓ = 9000.

We find a best-fit b₀ = 18.2^+2.3_{− 1.7} with χ² = 107 for 101 degrees of freedom. This corresponds to b(z = 1) = 5.0^+0.6_{− 0.4} and b(z = 2) = 6.8^+0.8_{− 0.5}. This Poisson-plus-clustering model is preferred to the null case with no ACT×BLAST correlation at over 25σ. The best-fit clustering and Poisson levels are reported in Table 1 and are plotted as blue dotted lines in Figures 6 and 9.

We also try fitting just the ACT×BLAST data with no clustering power. In this case we obtain a best-fit χ² = 64.3 with 48 degrees of freedom. After adding linear clustering to the model, we find χ² = 43.6 with 47 degrees of freedom, corresponding to Δχ² of 20.7 (with one fewer degree of freedom), so that the model including clustering is preferred to one with only Poisson and cirrus at greater than 4σ. Additionally, the Poisson levels are lower, and more consistent with expectations from number counts, when clustering is included.

Lastly, we try fitting a single-value bias, independent of redshift, to the entire range of power spectra. We find a best-fit single-value bias of 5.0 ± 0.4 with χ² = 110.8 for 101 degrees of freedom; worse than a redshift-dependent bias by Δχ² = 3.8. The single-value bias thus provides a good fit to the ACT×BLAST data however, when we include the measured 2030 μm (AR1) and 1380 μm (AR2) clustered power from Dunkley et al. (2011) in the likelihood calculation, we find the redshift-dependent bias is preferred at ∼2σ. Furthermore, the ACT clustering measurements are reproduced well with b(z) (see Figure 10, right panel) but underpredicted with the single-value bias.

Figure 8 shows b(z) calculated using Equation (14) for three realizations of the B11 model which span the entire range of results. The bias appears high compared to the linear bias estimates in V09 (who adopted the Lagache et al. 2004 model), although given the spread in b(z) from different realizations of the B11 model, the measurements are not inconsistent.

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Our choice of source model and bias parameterization is likely affecting b(z). The B11 model contains two distinct classes of IR sources, "normal" and "starburst," with substantially different luminosities, which is not accounted for in our b(z) parameterization. Also, the B11 model does not match observational constraints equally well across the whole wavelength range probed by ACT and BLAST; for instance, it underpredicts Poisson power and number counts compared to BLAST and South Pole Telescope (SPT) measurements at 500 and 1360 μm (see Section 5.6 of Béthermin et al. 2011, as well as Figure 9 of this paper). An underprediction of dI/dz would result in a higher bias to compensate.

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