Maps of the Southern Millimeter-wave Sky from Combined 2500 deg² SPT-SZ and Planck Temperature Data

The SPT was used to observe each field in a series of observations composed of successive scans across the sky along azimuth. Adjacent scans are separated by a small step in elevation. As the telescope is located at the South Pole, the scan direction is along R.A., with steps in declination. Time-varying emission from the atmosphere leads to increased noise on large angular scales. The raw time stream data in each scan are filtered to reduce this noise. For more details on SPT-SZ filtering, see Schaffer et al. (2011).

The time stream data from each scan are fit with a seventh-order polynomial and set of low-order sines and cosines, which are subtracted from these data. This results in an effective scan-direction high-pass filter with a cutoff of ℓ ≃ 270. Sources measured to be brighter than 50 mJy at 150 GHz are excluded from the fitting (with a 5 arcmin masking radius). Fainter sources are not masked before filtering, which gives them wing-like features (aligned with the scan direction) in the resulting maps.

The six modules of the SPT-SZ camera each contain 160 detectors. Each module is equipped with filters determining their observing frequency (95 GHz, 150 GHz, or 220 GHz). At every time sample and separately for each module, the mean and two spatial gradients of the data from all detectors in a module are subtracted from their data. This reduces atmospheric noise at large angular scales. A low-pass filter is applied in Fourier space to the data from each detector to avoid aliasing when the data are sampled into map pixels.

Note that the transfer function t_ℓm as defined in Equation (4) is an approximation when applied to SPT-SZ data, in that the actual filtering is performed on individual scans in the time domain and in Fourier space, which does not transform perfectly into a simple convolution. However, as we will show in Section 6.3.1, this is a very good approximation.

We use the "Hierarchical Equal Area isoLatitude Pixelation of a sphere" (HEALPix; Górski et al. 2005)³⁸ scheme to pixelize our maps. The filtered SPT maps of each field are initially in the Lambert azimuthal equal-area projection. We match the response of each field from the year-varying beams into a common beam in two-dimensional Fourier space. The common beam is chosen to be a ≃2 arcmin full width at half maximum (FWHM) Gaussian; however, this is not the final resolution of the combined maps. This common beam is replaced at the end of the combining pipeline with a 1.85 arcmin FWHM Gaussian.

The fields are beam-matched by multiplying the two-dimensional FFT of the coadded temperature map of each field with the ratio of the common beam to the beam for that year and frequency. Then, in position space we perform bilinear interpolation of these beam-matched fields onto the nearest HEALPix pixel locations (with resolution ${N}_{\mathrm{side}}=8192$). We interpolate the weights for each field onto the same HEALPix grid, and compute the weighted sum of the temperature values using these weights. The final combined maps are in HEALPix format with resolution ${N}_{\mathrm{side}}=8192$.

We mask the bright sources (flux densities greater than 50.0 mJy at 150 GHz) that are masked during time stream filtering, and regions close to the SPT-SZ boundary. This mask is constructed by cutting holes with a radius of 5 arcmin in an SPT-SZ boundary mask. We apodize the mask outside the holes and at the boundary with a Gaussian with σ = 5 arcmin. This mask is applied to SPT and Planck data, noise, and simulations.

The filter transfer functions for each of the 2500 deg² SPT-SZ maps are computed as follows. We create 100 full-sky simulated input maps per band consisting of the lensed CMB, Gaussian foregrounds (which are correlated between bands), and the Poisson-distributed population of point sources with 150 GHz flux densities in the range 6.4 mJy ≤ F₁₅₀ ≤ 50.0 mJy (W. B. Everett et al. 2018, in preparation). Simulated lensed CMB maps are generated by running LensPix (Lewis 2005) on temperature power spectra derived from the Planck TT + lowP + lensing cosmology³⁹ (Planck Collaboration et al. 2016f). We simulate SZ-detected galaxy clusters with statistical significance ξ ≥ 5.0 in the 150 GHz band from the Bleem et al. (2015) catalog. The clusters are simulated in their observed positions in the 95 GHz and 150 GHz bands (there is negligible SZ signal at 220 GHz). Using the integrated Comptonization ${Y}_{\mathrm{SZ}}$ (over an 0.75 arcmin radius disk) and core radius θ_c taken from the Bleem et al. (2015) catalog, the radial temperature profile of each cluster ΔT(θ) is computed assuming a projected isothermal β-model (Cavaliere & Fusco-Femiano 1976) with β = 1:

$$\begin{eqnarray}&&{\rm{\Delta }}T(\theta )={y}_{0}{T}_{\mathrm{CMB}}{g}_{\nu }{(1+{\theta }^{2}/{\theta }_{c}^{2})}^{\tfrac{1-3\beta }{2}},\end{eqnarray} \tag{ 3 }$$

where y₀ is the peak Compton y-parameter, the CMB temperature ${T}_{\mathrm{CMB}}=2.7255$ K (Fixsen 2009), g_ν is a function of observing frequency ν, θ is the angular distance from the cluster center, and θ_c is the cluster core radius (Bleem et al. 2015).

We create 100 simulated maps (one per input sky) for every individual observation of each field at each band. We create mock timestreams from each input map, and these mock timestreams are made into single-observation simulated maps using the same filtering and processing as used on the real data. The individual observation maps of each field at each band are then coadded, and the single-field maps are combined into full 2500 deg² maps in the same way as the real data. This results in 100 simulated 2500 deg² maps in each band; these maps have the same statistical properties as our best estimate of the observed sky.

The input maps and observed maps (in HEALPix format using the method described in Section 4.1.2) are masked, and their spherical harmonic coefficients are computed (${\tilde{a}}_{{\ell }m}^{\mathrm{in}}$ and ${\tilde{a}}_{{\ell }m}^{\mathrm{out}}$, respectively). The filter transfer function is then computed by

$$\begin{eqnarray}&&{t}_{{\ell }m}\equiv \displaystyle \frac{1}{{B}_{{\ell }}}\displaystyle \frac{\langle {\tilde{a}}_{{\ell }m}^{\mathrm{out}}{({\tilde{a}}_{{\ell }m}^{\mathrm{in}})}^{* }\rangle }{\langle {\tilde{a}}_{{\ell }m}^{\mathrm{in}}{({\tilde{a}}_{{\ell }m}^{\mathrm{in}})}^{* }\rangle },\end{eqnarray} \tag{ 4 }$$

where $\langle \ldots \rangle$ denotes the ensemble average over all simulations, and B_ℓ is the common beam introduced in Section 4.1.2. Averaging over 100 simulations per band, we compute the transfer function using this equation. The filter transfer functions for each SPT-SZ band are plotted in Figure 3. Applying the property of spherical harmonic coefficients ${a}_{{\ell }(-m)}={a}_{{\ell }(+m)}^{* }$ to t_ℓm, whose imaginary part is negligible, implies ${t}_{{\ell }(-m)}={t}_{{\ell }(+m)}$. We therefore only need to plot half of the total number of modes (e.g., m ≥ 0 or m ≤ 0). One can interpret t_ℓm with ℓ on the x-axis and m on the y-axis as follows:

• 1.  
  Larger values of m correspond to smaller angular scales along the SPT scan direction in equatorial coordinates.
• 2.  
  Filtering in the time domain leads to the localized strip of suppressed modes in t_ℓm at m ≲ 300.
• 3.  
  Applying a mask with a restricted range of declination and positioned away from the equator (such as the SPT-SZ survey mask) to a full-sky map leads to a triangle-shaped region of suppressed modes at high m. This feature is not due to filtering; it is purely from the geometry of the survey mask when viewed in spherical harmonic space. High-m spherical harmonics have most of their power at the equator, so the survey mask suppresses these modes.
• 4.  
  The ℓ-dependent trend in t_ℓm is the smoothing due to the 1 arcmin pixelization, and is the same for all three frequencies. Note that we are showing t_ℓm exactly as in Equation (4), so the ≃2 arcmin beam from beam-matching B_ℓ is not included.

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The units of CMB maps are differential temperatures relative to the CMB temperature,

$$\begin{eqnarray}&&{\rm{\Delta }}T\equiv T-{T}_{\mathrm{CMB}},\end{eqnarray} \tag{ 5 }$$

denoted K_CMB. The calibration accuracies of the Planck maps in the 100, 143, and 217 GHz bands are ≃0.1% (Table 6 of Planck Collaboration et al. 2016c). We use this highly precise Planck calibration to calibrate the SPT-SZ maps. We use the 150 GHz calibration factor derived from comparison to Planck 143 GHz data in the SPT-SZ footprint from Hou et al. (2018). We then inter-compare the SPT-SZ 95, 150, and 220 GHz maps to obtain calibration factors at 95 and 220 GHz. The uncertainties of these relative calibrations are 0.22, 0.15, and 0.43% (in map units) at 95, 150, and 220 GHz respectively.

Upon calculating the spherical harmonic coefficients of the SPT data ${\tilde{a}}_{{\ell }m}^{\mathrm{SPT}}$ we divide out the filter transfer function t_ℓm and the common beam B_ℓ. For the small subset of modes at low m that are heavily suppressed due to filtering in the time domain, the transfer function is close to zero and there is negligible signal. We do not deconvolve t_ℓm for modes where m < 300. These low-m modes get filled in with Planck data later on. The transfer function also becomes exponentially small at high m due to a low-pass filter. This filter is not the reason for the triangle-shaped wedge of modes at high m—that is due to the application of the 2500 deg² survey mask—but it causes the transfer function to fall off at ℓ ≃ 8000, m ≃ 5000. Modes with m >5000 remain filtered in the final maps.

A half-difference map is calculated for each observation of each field, such that the left-going scans (along increasing R.A.) and right-going scans (along decreasing R.A.) are differenced and divided by 2. This nulls the signal in each scan while preserving the statistics of the noise. The mean of all of the half-difference maps for a field, with a randomly selected half of them multiplied by −1, gives us an estimate of the noise in that field. Using random ±1's allows us to produce more realizations of the noise. We make 100 noise realizations of each field and each frequency, and construct a HEALPix map out of each of them.

To estimate the two-dimensional noise power N_ℓm in each band, we apply the boundary and point-source mask to each noise realization, compute their harmonic transforms ${\tilde{n}}_{{\ell }m}$, deconvolve the beam and filter response functions, and then compute their variance:

$$\begin{eqnarray}&&{N}_{{\ell }m}={({F}_{{\ell }m})}^{-2}\langle {| {\tilde{n}}_{{\ell }m}| }^{2}\rangle .\end{eqnarray} \tag{ 6 }$$

The auto-spectrum of the SPT noise realizations for each frequency, with their beam and filter transfer functions deconvolved, are shown by the dotted lines in Figure 4.

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As a test of our algorithm, we have compared the angular power spectrum of simulated combined maps against the input power spectrum. We mentioned in Section 4.1.1 that t_ℓm as defined in Equation (4) is an approximation to the true filtering. Now we test this approximation by measuring how accurately we can recover the input power spectra from mock-observed maps after deconvolving t_ℓm. We also test that the combining algorithm does not bias the power spectrum.

The first part of this test consists of processing simulated noise-free maps in the same way as the real data. Then, using the "Spatially Inhomogeneous Correlation Estimator for Temperature and Polarization" (PolSpice; Chon et al. 2004) code⁴¹ we compute the angular power spectrum of each simulated map C_ℓ, which is usually plotted as ${D}_{{\ell }}\equiv {\ell }({\ell }+1){C}_{{\ell }}/2\pi$.

We use the following procedure to correct for the effect of M_ℓm on measured power spectra. Using a separate set of simulated maps where the true power spectrum is known (${C}_{{\ell }}^{\mathrm{true}}$), we compute the estimated power spectrum of the simulated maps (${C}_{{\ell }}^{\mathrm{est}}$) after applying the mask to the maps, computing the spherical harmonic transform, applying the filter M_ℓm, and then inverting the harmonic transform. We compute the ratio of the average biased spectrum to the true spectrum

$$\begin{eqnarray}&&{m}_{{\ell }}\equiv \displaystyle \frac{{C}_{{\ell }}^{\mathrm{true}}}{\langle {C}_{{\ell }}^{\mathrm{est}}\rangle },\end{eqnarray} \tag{ 11 }$$

and multiply the estimated power spectrum of the combined maps (simulated and real data) by m_ℓ. We have checked using the separate set of simulated maps that the uncertainty contributed by m_ℓ is negligible compared to other sources of uncertainty.

Using this procedure we compute auto- and cross-power spectra averaged over a set of 100 simulations. The average power spectra are binned and divided by their corresponding input spectra. These ratios are plotted in Figure 11. A ratio of 1.0 indicates perfect agreement between the spectra of simulated output and input. For the majority of the ℓ range, especially where CMB is the dominant source of fluctuations, the agreement is better than 1 percent in power. This test shows excess power in the simulations at high values of ℓ (most notably ℓ ≳ 6000). The excess power has similar ℓ dependence in auto-spectra, cross-spectra, and across bands. If our calculated transfer functions were noisy we would see excess power in the output simulations. However, we found that the excess power is unaffected by increasing the number of simulations that go into the transfer functions. We also ran this test on each field separately and found that the result does not change significantly from field to field. Together, these findings have led us to believe that the departure from 1.0 at high ℓ is likely due to spatial variation in the data weights. We recommend caution when using these maps above ℓ = 6000.

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We compare auto- and cross-spectra of our SPT-Planck data maps with theoretical power spectra and previously published SPT-SZ power spectra. Specifically, we compare to the power spectra published in S13, which presented the 150 GHz auto-spectrum in the multipole range 650 < ℓ < 3000, and in George et al. (2015, hereafter G15), which presented auto and cross-spectra of SPT-SZ data in all three bands in the multipole range 2000 < ℓ < 11000. The theory spectra we compare to are the Planck 2015 best-fit CMB plus best-fit model foreground spectra from G15. These power spectrum comparisons are intended to validate the maps in terms of the transfer functions and noise, not to estimate cosmological parameters.

We note that in this work, the target multipole range, and hence the filtering of SPT-SZ data and masking of point sources, are matched to those in S13 and not G15, but we expect to be able to make meaningful comparisons between the two analyses regardless. Sources with 150 GHz flux densities greater than 6.4 mJy were masked in the G15 mapmaking and power spectrum calculations. The maps presented here were made with sources above 50 mJy at 150 GHz masked. For the comparison to G15, we compute auto and cross power spectra using the same 6.4 mJy mask as in that work.

The auto- and cross-spectra are shown in Figures 12 and 13, respectively. The auto-spectra have not had noise bias-subtracted; to compare to G15 and S13 spectra (which were computed using individual observation cross-spectra and hence do not suffer noise bias), we have added the SPT-Planck noise power spectra to the published G15 and S13 auto-spectra and to the theoretical spectra. We have also applied a small correction to the S13 auto-spectrum to account for the different point-source mask used in S13. The error bars are the standard deviation of the binned D_ℓ of noisy simulated maps.

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We find that our cross-spectra are in good agreement with the theoretical and G15 cross-spectra. The difference between our observed auto-spectra (signal plus noise) and the expected spectra (theory plus noise spectra, or G15 auto-spectra plus noise spectra) is found to be 3% of our noise power spectra.