Measurements of the Cross-spectra of the Cosmic Infrared and Microwave Backgrounds from 95 to 1200 GHz

Between 2008 and 2011 the SPT-SZ camera on the SPT was used primarily to survey 2540 deg² of the sky at 95, 150, and 220 GHz. The survey region covers decl. between −40° and −65° and R.A. from 20^h to 7^h. This work is based on one of two deep fields that have $\sqrt{2}$ lower noise at 150 GHz than the full survey; deep field noise levels are ≲13 μK arcmin at 150 GHz. The field used in this work is approximately 10° × 10° centered at 23^h30^m, −55^d00^m, and was observed in 2008 and 2010. A full list of the SPT-SZ fields can be found in Story et al. (2013).

The SPT-SZ data, maps, and power spectrum analysis used in this work are described by G15. Briefly, the data are bandpass filtered to remove atmospheric emission at low frequencies and to avoid aliasing high-frequency noise into the signal band. The data are then binned into 05 pixels, and the power spectrum is estimated using a pseudo-C_ℓ method (Hivon et al. 2002; Polenta et al. 2005; Tristram et al. 2005). We emphasize that except for the calibration factor the CMB-only bandpowers in this work are identical to the bandpowers for this field in G15, and should therefore only be combined with the publicly available G15 bandpowers that do not include the ra23h30dec-55 field.

The SPT beams are determined as in G15. Briefly, the beam is measured using observations of Venus and Jupiter (see, e.g., Story et al. 2013). The approximate FWHMs of the beam mainlobes are $1\buildrel{\,\prime}\over{.} 7$, $1\buildrel{\,\prime}\over{.} 2$, and $1\buildrel{\,\prime}\over{.} 0$ at 95, 150, and $220\,\mathrm{GHz}$, respectively. The absolute calibration of the maps has been updated since G15. As described by Hou et al. (2018), the new calibration is determined by comparing the SPT maps to Planck maps over the same footprint. The uncertainty in the SPT power calibration at [95, 150, 220] GHz is [0.43%, 0.34%, 0.84%]. The correlated part of these uncertainties is 0.34%. The beam and calibration uncertainties are included as a correlation matrix in the data release.

For estimating the SPT × SPIRE bandpowers, we also coadd the O(100) complete, independent SPT maps used by G15 into a single map at each SPT frequency. Each individual map is assigned equal weight when coadding to match the weighting used in the SPT × SPT bandpowers. By coadding, we reduce the number of FFTs required in the bandpower estimation. Using a single coadded SPT map does not incur a noise bias as the SPT and SPIRE noise is uncorrelated. We will discuss the SPIRE × SPT bandpower estimation more in the next section.

The SPIRE maps at 600, 857, and 1200 GHz (500, 350, and 250 μm) used in this work are the same as those used in Holder et al. (2013) with minor modifications, described below. The maps were constructed from observations designed to target large scales by observing in fast-scan mode (60'' s⁻¹) to minimize the impact of 1/f noise. The full observation consists of twelve 10 × 3.3 degree scans, resulting in four complete passes of the entire region. The scan direction is alternated such that half of the scans are perpendicular to the other half. The 100 deg² map is constructed from a mosaic of these 12 scans.

Data are reduced with standard ESA software and the custom software package, SMAP (Levenson et al. 2010; Viero et al. 2013a). Maps are made using SMAP/SHIM (Levenson et al. 2010), an iterative map-maker designed to separate large-scale noise from signal. Initial processing uses the Herschel Interactive Processing Environment (HIPE³⁹ ) release version 13.0.4887, which includes that calibration tree version spire_cal_12_3. Practically, this represents a change in the calibration from maps used in previous SPT-SPIRE studies (e.g., Hanson et al. 2013; Holder et al. 2013) of 1.1%, 1.3%, and 1.3% at 600, 857, and 1200 GHz, respectively.

Two sets of maps with different pixel sizes are made: one with pixels 15'' on a side, and the other with pixels 30'' on a side. The smaller pixels are used for the SPIRE × SPIRE analysis, in which the target signal is detected strongly at scales smaller than 30''; this is not the case for SPIRE × SPT (because of the larger SPT beams), so we can save computing time and storage by using coarser maps for SPIRE × SPT.

The time-ordered data are split equally into two halves and made into two separate maps, each of which covers the full area. Hereafter, these half-data maps are referred to as jackknife maps or map pairs, and full-data maps as coadded. Note, 1 of the 12 scans is discarded because of a voltage bias swell (or "cooler-burp" Sauvage et al. 2014) that occurred during a cooler cycle, which if included would lead to large-scale striping. The result is that one of the jackknife maps is not fully sampled; this is taken into account in all stages of bandpower and uncertainty estimation. All sets of maps are made in Lambert azimuthal equal-area projection (also known as zenithal equal area, ZEA), with astrometry identical to that of the SPT map.

The SPIRE map noise levels are 5.6, 4.0, and 2.8 mJy arcmin, and the instrument effective beams are 366, 252, and 181 FWHM at 600, 857, and 1200 GHz, respectively. The maps are converted from their native units of Jy beam⁻¹ to Jy sr⁻¹ by dividing them by the effective beam areas (see below). The absolute calibration uncertainty is 5%, and is accounted for in the reported beam and calibration correlation matrix.

SPIRE beams are measured with Neptune; with an angular diameter of ≲25, Neptune is effectively a point source to SPIRE. Because the effective beam is mildly sensitive to the source spectrum through the passband, the effective solid angle of the beam must be color-corrected (Bendo et al. 2013; Griffin et al. 2013). Following the SPIRE Observers Manual⁴⁰ for a graybody spectrum representative of the CIB (T = 13.6 K and β = 1.4; e.g., Gispert et al. 2000; Planck Collaboration et al. 2014), and fully illuminated beam (passbands scaled by ν^1.7), the effective solid angles are 3.878, 1.873, and 1.080 × 10⁸ sr, at 600, 857, and 1200 GHz, respectively. Note that these differ slightly from the solid angles in Viero et al. (2013a), which were 3.688, 1.730, 1.053 × 10⁻⁸ sr.⁴¹ As described in detail in Viero et al. (2013a), the beam power spectra are estimated from the Neptune maps after masking background sources with flux densities greater than 30 mJy at 1200 GHz and all pixels beyond 10 × FWHM radius. We check that the exact flux density of masked sources and choice of masking radius do not significantly bias the measurement, and include the differences as part of the Monte Carlo procedure described in Section 2.3.3.

Lastly, the effective band centers for the SPIRE bands with fully illuminated beams and a CIB spectral energy distribution (SED) are 546.0, 796.0, and 1092.4 GHz (549.5, 376.9, and 274.6 μm). The SPT bands have effective CIB band centers of 99.0, 154.2, and 219.6 GHz (3.030, 1.946, and 1.367 mm), respectively.

Bandpowers are estimated using cross-spectrum-based pseudo-C_ℓ methods (Hivon et al. 2002; Tristram et al. 2005). There are minor implementation differences between the analysis of each of the data sets: SPT-only, SPIRE-only, and $\mathrm{SPT}\times \mathrm{SPIRE}$.

The SPT-only and SPIRE-only bandpowers (six frequency combinations each) are calculated as in previous SPT and SPIRE power spectrum papers (see George et al. 2015 for the former, and Viero et al. 2013a for the latter). To match George et al. (2015), the SPT-only bandpowers start at ℓ > 2000. At larger angular scales, the SPT data are dominated by the CMB. The SPT × SPIRE bandpowers are calculated similarly to the SPIRE only case; however, full coadded maps from each experiment are used in this case because the noise in SPT and SPIRE maps are uncorrelated. We outline these methods below.

2.3.1. Apodization Mask and Mode-coupling Matrix

2.3.2. Transfer Function and Beams

2.3.3. Covariance Matrix

The covariance matrix is estimated in one of two ways depending on the frequency combination. In all cases, we assume that the instrumental noise in each frequency band is independent. For the SPT × SPT bandpowers, the covariances are calculated exactly as in G15; sample variance is estimated from simulations, and the sum of noise and signal-cross-noise variance is estimated from the scatter in cross-spectra. The covariance estimator changes when SPIRE data appears in the covariance element. Covariance elements that include SPIRE data are calculated through a Monte Carlo simulation that creates many realizations of the sky signal and noise terms. These noise realizations are drawn from the null-map PSD for each frequency. The sky realization is divided into terms simulated as Gaussian fields (CMB, tSZ, kSZ, radio galaxies) and non-Gaussian fields (CIB). The Gaussian terms are fully described by G15. The CIB mock maps are made with correlated sources such that their clustering signal resembles that of the CIB, with fluxes drawn from the Béthermin et al. (2012) model. Note that the Gaussian and non-Gaussian terms are assumed to be uncorrelated. Each sky realization is filtered identically to the real data and added to a noise realization to create mock maps. The power spectrum of the mock maps is calculated with the same pipeline as that used for the real data.

The ensemble of estimated binned power spectra, P_b, are used to measure, V, the covariance matrix

$$\begin{eqnarray}&&{{\boldsymbol{V}}}_{{bb}^{\prime} }={\left\langle \left({P}_{b}-{\tilde{P}}_{b}\right)\left({P}_{b^{\prime} }-{\tilde{P}}_{b^{\prime} }\right)\right\rangle }_{\mathrm{MC}},\end{eqnarray} \tag{ 1 }$$

where the tilde denotes the mean over all realizations in bin b. The reported errors in the tables and plots are based on the diagonal entries of V:

$$\begin{eqnarray}&&{\sigma }_{{P}_{b}}=\sqrt{{{\boldsymbol{V}}}_{{bb}}}.\end{eqnarray} \tag{ 2 }$$

The non-Gaussianity of the CIB realizations leads to substantial off-diagonal terms in V. We follow Fowler et al. (2010) and check that the simulations are realistic by testing that the four-point function measured directly from data is consistent with that of the simulations. We find that the non-Gaussian contribution ranges are between 5% and 10% of the total error in the regime where the Poisson term dominates.

We also test the robustness of the covariance estimate by splitting the field into four quarters. One quarter is not used due to the voltage bias swell mentioned in Section 2.2. We estimate bandpowers from null maps created by differencing combinations of the other three quarters, and we calculate a χ² using the covariance matrix. We find a reasonable χ² for the number of degrees of freedom (303, 244, and 303, respectively, for 297 d.o.f), lending confidence to the covariance estimate.

The reported bandpower error bars and covariance matrix do not include the SPT or SPIRE absolute calibration uncertainties nor the beam uncertainties. The released data files do include a correlation matrix encapsulating these uncertainties that should be combined with the model bandpowers and added to the covariance matrix when evaluating the quality of fit to a model.

2.3.4. Handling Galactic Cirrus

The survey region was carefully chosen to minimize Galactic cirrus; nonetheless, it is important to quantify the small cirrus contribution. We adopt an approach similar to that taken by Viero et al. (2013a). As described in the A.1, this approach assumes that the cirrus can be described by a single map (i.e., 100% correlation between observing bands) with an amplitude that scales with frequency as a modified blackbody. While a modified blackbody spectrum has two free parameters, T_eff,c and β_c, the two are highly degenerate for the current set of observing frequencies. Therefore, as we are mostly interested in obtaining a good fit for the purposes of removal (as opposed to the exact qualities of the cirrus) we fix β_c = 1.8 (e.g., Martin et al. 2010). Furthermore, we assume that the cirrus spatial power spectrum follows a power law with index α_c. Under these assumptions, the cirrus power in all bands can be fit with two variables, α_c and T_eff,c.

To determine the power-law index, we take advantage of the fact that Galactic cirrus is well-traced by nearby H i in the Galactic disk (e.g., Boulanger et al. 1996), with velocities $| {v}_{\mathrm{LSR}}| \lt 30\,\mathrm{km}\,{{\rm{s}}}^{-1}$ with respect to the local standard of reference, and column densities N_H ≤ 5 × 10²⁰ cm⁻² (e.g., Viero et al. 2014). H₂ begins to form at higher column densities N_H ≤ 5 × 10²⁰ cm⁻² (Planck Collaboration et al. 2014), causing the correlation to break down. This field, however, is of low enough Galactic latitude and column density⁴² that this is not an issue. We take H i maps from Parkes Galactic All-Sky Survey survey (GASS; McClure-Griffiths et al. 2009; Kalberla et al. 2010), and measure the power spectrum of the local velocity component of H i across the 100 deg² field. We find that the H i exhibits a power-law behavior at ℓ < 600 (after which the beam attenuates the signal) with slope α_c = −3.1 ± 1.4. We assume that that behavior can be extrapolated to larger ℓ, noting that given the steep fall off there is very little cirrus power in practically all of the reported bandpowers. We also check that contributions from the higher velocity components of the velocity cube are negligible.

We then perform a Monte Carlo simulation to estimate the cirrus contribution to the SPIRE/SPT bands, and in effect the cirrus correction and its associated uncertainties. For each iteration, the cirrus slope and temperatures are drawn from normal distributions of α_c = −3.1 ± 1.4 and T_eff,c = 19.0 ± 1.5 K, respectively, and then the spectra are fit with a cirrus power law, templates for the clustered CIB power,⁴³ and a Poisson level. In addition to SPIRE and SPT we include spectra at 100 μm—where cirrus has a much higher contribution relative to CIB (Pénin et al. 2012)—from reprocessed IRAS (Neugebauer et al. 1984) observations⁴⁴ (IRIS; Miville-Deschênes & Lagache 2005). Note that we assume the IRIS transfer function on these scales is unity (Hajian et al. 2012; Pénin et al. 2012). The variation in the residual cirrus power (after subtracting the mean) is added to the bandpower covariance matrix.

The cirrus levels are shown as dashed gray lines in Figure 1. We find the corrections overall to be very small, with the only significant corrections in SPIRE × SPIRE spectra including 250 μm. Even in these spectra, the correction is only significant in the lowest-ℓ bin and peaks at 24% in the first bin of the 250 μm auto-spectrum.

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One question that may be addressed with these data is the degree to which the CIB is correlated across frequencies. We define a correlation coefficient between maps at different frequencies to be

$$\begin{eqnarray}&&{\xi }_{{\ell }}=\displaystyle \frac{{C}_{{\ell }}^{{\nu }_{1}\times {\nu }_{2},^{\prime} }}{\sqrt{{C}_{{\ell }}^{{\nu }_{1},^{\prime} }\cdot {C}_{{\ell }}^{{\nu }_{2},^{\prime} }}},\end{eqnarray} \tag{ 3 }$$

which are presented as red circles with black error bars in Figure 2. Note that C_ℓ' here represents "CIB-only" bandpowers—including only the CIB and any tSZ-CIB correlation. Thus the reported correlation coefficient will be sensitive to the magnitude of the tSZ-CIB correlation as well as the actual CIB correlations. Suppressing the other components is necessary because each component will have its own correlation structure. In particular, the CMB is extremely bright on large angular scales in the SPT bands and perfectly correlated across frequencies. Thus, including CMB power would tend to bring the correlation between the SPT bands to unity at low-ℓ. To create these CIB-only bandpowers, we subtract the best-fit values (including calibration factors from that point in the chain) from G15 for the power from the CMB, tSZ, kSZ, radio galaxies, and galactic cirrus from the measured bandpowers.

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Figure 2 shows the resulting correlation coefficients as a function of multipole. The observed correlations are largely independent of angular scale, although there is a suggestion of a slope in the terms involving 220 GHz and the SPIRE bands. There is a visual hint of a slope in the 95 GHz correlation coefficients as well, which is intriguing as the slope matches the expected direction for an anticorrelation between the tSZ and CIB power (i.e., the tSZ-CIB anticorrelation should reduce the cross-correlation coefficient as defined here around ℓ ∼ 3000). However, we stress that the uncertainties are large and the predicted variations in the G15 model on these scales is small compared to the current error bars. In the SPIRE-only maps, correlation coefficients are effectively independent of angular scale. The measured correlation coefficients, averaged across ℓ ∈ [2000, 7000], are reported in Table 2. For the SPIRE bands, we find correlation values of $\langle \xi \rangle =0.970\pm 0.010$, 0.861 ± 0.012, 0.9551 ± 0.0099 for 600 × 857, 600 × 1200, and 857 × 1200 GHz, respectively. These values agree with those found by Viero et al. (2013a). We report the average correlation between all frequency pairs in Table 4.

Our expectation is that the CIB should be highly correlated in neighboring frequency bands, with the correlation dropping as the frequency separation increases. Even if all DSFGs had identical SEDs in their local restframe, their observed SEDs would peak at different frequencies based on their individual redshifts. Effectively different frequency bands are weighted toward DSFGs at different redshifts. As the frequency separation increases, there is less overlap in which DSFGs are contributing to each band's power spectrum and more decorrelation. Variations in the restframe SED will also add to the decorrelation. We can measure this decoherence in the data, as shown in Figure 3. In this plot, the average (from ℓ = 2000 to 7000) cross-correlation coefficients are shown for each of the 15 unique frequency pairs. The colored symbol encodes the lower frequency in the pair, e.g., open purple circles for the five pairs involving the 95 GHz band and an open red triangle for the last pair between 857 and 1200 GHz. The x-axis encodes the logarithmic frequency difference between the two bands in the pair. An overly simplistic model for the decoherence is simply assuming each galaxy has a different β for its modified blackbody, drawn from a Normal distribution with some variance σ². The correlation factors are insensitive to T and β, so we fix these parameters to T = 18 K and β = 2.0. We conduct a fit to this toy model, and find σ² = 0.277 ± 0.009 (or σ = 0.53). The quoted error on σ² is based on the change that increases the χ² by one; this is likely to be optimistic given the poor overall fit (the best-fit χ² = 109). The dotted line in Figure 3 shows the predictions for the best-fit to this toy model. The data are qualitatively consistent with the degree of correlation depending only on the ratio of the frequencies, as would be expected for variations in the power-law spectral index between DSFGs. We stress that one should be very cautious in interpreting the toy model physically—the toy model does not include a DSFG redshift distribution and vacuums up the effects of different redshifts into variations in β.

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We can compare the measured cross-correlation coefficients to the predictions of the CIB modeling work in Béthermin et al. (2013). Table 1 in that work presents predicted correlation coefficients for the clustering component at ℓ = 1000 and the Poisson component at higher multipoles; the measurement in this work across ℓ = 2000 to 7000 should fall in between the two. There is also the caveat that the predictions are for frequencies close (but not identical) to the actual band centers. The agreement is reasonable at a qualitative level (a careful statistical comparison is difficult given the caveats). We measure a correlation from 600 to 857 GHz (recall the effective CIB band centers are 546 and 796 GHz) of 0.970 ± 0.010. Béthermin et al. (2013) predict the correlation between 545 and 857 GHz to be 0.916 (Poisson) to 0.971 (ℓ = 1000). The prediction for 220–857 GHz is 0.785 (Poisson) to 0.902 (ℓ = 1000); the actual observed correlation is 0.786 ± 0.016. The prediction for 100–857 GHz is 0.743 (Poisson) to 0.919 (ℓ = 1000); the actual observed correlation is 0.40 ± 0.32. There is a suggestion, especially in the 95 GHz band, that the observed correlations are lower than predicted; however, the significance is low at the current level of uncertainty. A low value would not necessarily point to a flaw in the CIB modeling either, because the model did not include the expected anticorrelation of the tSZ and CIB. The anticorrelation of the tSZ and CIB is most significant in the cross-frequency spectra including 95 GHz. We conclude that the observed correlations are in rough agreement with the predicted correlations of the Béthermin et al. (2013) CIB model.

It is interesting to compare the spectra from this work with other CIB studies. Figure 1 shows other recent measurements along with the SPIRE × SPT bandpowers. CIB measurements from SPIRE (Viero et al. 2013a, blue diamonds from 600 to 1200 GHz) and Planck (green squares from 600 to 857 GHz Planck Collaboration et al. 2014) are in excellent agreement. Note that we have scaled the Planck bandpowers at 660 and 857 GHz to account for differences in the passbands of SPIRE and Planck.

For the CIB × CMB measurements, we show measurements from BLAST × ACT (orange crosses over 150–220 × 600–1200 GHz; Hajian et al. 2012), and again from Planck (green squares over 150–220 × 600–857 GHz). The different data sets are in excellent agreement. As before, Planck values are adjusted to account for passband differences. It is notable how the SPIRE × SPT and Planck spectra complement each other, with the two data sets overlapping at ℓ ∼ 2000. This is because the Planck points are limited on smaller angular scales by the ∼5' beam, while the SPIRE × SPT points are limited on larger scales by the presence of CMB in the SPT maps—which acts as a noise term and, unlike in the Planck measurements, is not subtracted. The extra variance due to the CMB can be substantially larger than the expected CMB power in the SPIRE × SPT cross-spectrum because the variance includes terms proportional to the SPT × SPT auto-spectrum (which is dominated by the CMB). For instance, the variance of the 1200 × 150 GHz cross-spectrum has terms proportional to, C^150×150_ℓC^1200×1200_ℓ + (C^150×1200_ℓ)², and the CMB is extremely bright in C^150×150_ℓ despite being negligible in the other terms. The magnitude of this extra variance due to the CMB is shown by a red dashed line.

Diffuse Galactic cirrus introduces a foreground signal on large scales to the extragalactic submillimeter/millimeter-wave power spectrum. Other Galactic foregrounds such as synchrotron and free–free emission are negligible in the area of the sky and frequencies considered here and thus are ignored.

Galactic cirrus can be approximated as a power law

$$\begin{eqnarray}&&{C}_{{\ell }}^{\mathrm{cirrus}}={P}_{0,\lambda }{\left(\displaystyle \frac{{\ell }}{{{\ell }}_{0}}\right)}^{{\alpha }_{{\rm{c}}}},\end{eqnarray} \tag{ 5 }$$

with the amplitude P_0,λ dependent on the observing frequency λ, and the index α_c independent of observing frequency (e.g., Gautier et al. 1992; Miville-Deschênes et al. 2007). In the first studies of high Galactic latitude cirrus, the index had been found to have a value of around −3.0 (e.g., Gautier et al. 1992); however, more recent measurements have found that both the amplitude and index are quite direction-dependent, with values for α_c ranging from −1 to −4 (e.g., Roy et al. 2010; Boothroyd et al. 2011; Bracco et al. 2011).

We also need to consider the spectral dependence of the Galactic cirrus. At submillimeter/millimeter wavelengths, cirrus emission follows a modified blackbody, ${\nu }^{{\beta }_{c}}{B}_{{T}_{\mathrm{eff},{\rm{c}}}}(\nu )$. Here ${B}_{{T}_{\mathrm{eff},{\rm{c}}}}(\nu )$ is the Planck function for temperature T_eff,c and β_c is the emissivity index (Draine & Lee 1984). Past studies have found T_eff,c = 16–20 K and β_c ≈ 1.8 (e.g., Bracco et al. 2011).

The CIB power term in Equation (4) is sourced by DSFGs and can be further split into Poisson and clustering components. The Poisson (or shot noise) term results from the discrete sampling of dusty galaxies. The power spectrum of the Poisson term is independent of angular scale, with amplitude

$$\begin{eqnarray}&&{C}_{{\ell }}^{\mathrm{Poisson}}={\int }_{0}^{{S}_{\mathrm{cut}}}{S}_{\nu }^{2}\displaystyle \frac{{dN}}{{{dS}}_{\nu }}\left({S}_{\nu }\right){{dS}}_{\nu },\end{eqnarray} \tag{ 6 }$$

where dN/dS_ν is the distribution of flux densities of sources, and S_cut is the level at which sources with greater flux densities are masked.

The clustering term arises from overdensities in the background that trace the dark matter distribution. It can be described with a "halo model" approximation (e.g., Seljak et al. 2001), which consists of a linear (or two-halo) term on large scales (ℓ ≲ 3500 for the frequency bands in this work) and nonlinear (or one-halo) term on small scales (ℓ ≳ 3500). The linear term is simply the dark matter power spectrum multiplied by an effective bias (e.g., Kaiser 1984). The nonlinear term is more subtle, as the bias can be both scale- and mass-dependent largely because galaxy properties (stellar masses, star formation rates, bolometric luminosities, etc.) are tied to those of their host dark matter halos (halo mass, concentration, etc.; e.g., Shang et al. 2012; Viero et al. 2013a).

Radio galaxies also contribute to the Poisson term detailed in Equation (6). After masking sources above 6.4 mJy at 150 GHz (see Section 2.3.1), dusty galaxies are the dominant contributors to the Poisson term in all but the 95 GHz band. At 95 GHz, 80%–90% of the power is expected to be due to radio galaxies. As argued by G15 and others, we do not expect a detectable clustering term for the radio galaxies.

The galaxy clusters that give rise to the tSZ power spectrum similarly trace the dark matter distribution. Accurately modeling the tSZ power spectrum is challenging because of the complicated astrophysics affecting the intracluster medium in galaxy clusters; for instance, nonthermal pressure support due to merger shocks. Different prescriptions for the astrophysics can change the predicted power for a given cosmology by up to 50% (e.g., Battaglia et al. 2010, 2012; Shaw et al. 2010; Trac et al. 2011; Efstathiou & Migliaccio 2012).

Additionally, since both galaxies and clusters trace the same dark matter, correlations between the two populations reveal insights into their relationship. The first to demonstrate the power of this relationship was Addison et al. (2012), who showed that the strength and shape of the cross-correlation spectrum—which, again, appears as a negative signal—is largely influenced by the HOD of star-forming galaxies in the most massive dark matter halos. Furthermore, they show that, since shorter wavelengths favor lower redshifts, the strength of the effective anticorrelation is highest between 250 μm and CMB channels. The bandpowers reported in this work can be used to improve measurements of the tSZ-CIB correlation.

The bulk motion of electrons can impart a Doppler shift upon scattered photons, an effect known as the kSZ effect (Sunyaev & Zel'dovich 1970, 1980). The amplitude of the kSZ signal from a given volume scales as (v/c) δ_e, where c is the speed of light and δ_e is the overdensity of free electrons. As a result, significant kSZ power is sourced during the epoch of reionization (when velocities and mass overdensities are small, but there is a high contrast in the free electron density between ionized and neutral regions) and at late times (z < 2, when velocities and mass overdensities have grown larger). We expect current models for the post-reionization kSZ power spectrum to be more accurate than tSZ models because, unlike the tSZ, the kSZ signal is not weighted by gas temperature. Thus, the kSZ spectrum is less dependent on the nonlinear physics in dense halos. However, the kSZ power from reionization is highly uncertain, and offers a unique window into how reionization occurred (Mortonson & Hu 2010; Mesinger et al. 2012; Zahn et al. 2012; George et al. 2015).

Finally, the CMB is present in all of these maps. At SPIRE frequencies, the CMB is much fainter than the CIB and largely neglectable, but below 280 GHz the CMB quickly overwhelms the CIB at large angular scales. Although the two backgrounds are uncorrelated (lensing notwithstanding, e.g., Holder et al. 2013), CMB sample variance introduces a large noise signal to the cross-correlations between SPT and SPIRE frequency bands, and washes out the CIB cross-correlation signal (e.g., Hajian et al. 2012, Appendix B). The Planck Collaboration et al. (2014) overcame this limitation by subtracting the CMB directly from their maps, which they did by using their 100 GHz map as a template and scaling the signal as a perfect 2.73 K blackbody. The drawback of this technique is that cross-frequency correlation measurements with 100 GHz cannot be made—a measurement that arguably contains the most novel information for the tSZ-CIB correlation.