A measurement of the cosmic microwave background lensing potential and power spectrum from 500 deg of SPTpol temperature and polarization data
We present a measurement of the cosmic microwave background (CMB) lensing potential using 500 deg of 150 GHz data from the SPTpol receiver on the South Pole Telescope. The lensing potential is reconstructed with signal-to-noise per mode greater than unity at lensing multipoles , using a quadratic estimator on a combination of CMB temperature and polarization maps. We report measurements of the lensing potential power spectrum in the multipole range of from sets of temperature-only (T), polarization-only (POL), and minimum-variance (MV) estimators. We measure the lensing amplitude by taking the ratio of the measured spectrum to the expected spectrum from the best-fit CDM model to the Planck 2015 TT+lowP+lensing dataset. For the minimum-variance estimator, we find ; restricting to only polarization data, we find . Considering statistical uncertainties alone, this is the most precise polarization-only lensing amplitude constraint to date (), and is more precise than our temperature-only constraint. We perform null tests and consistency checks and find no evidence for significant contamination.
Gravitational potentials associated with large-scale structure deflect the paths of cosmic microwave background (CMB) photons as they propagate from the surface of last scattering – a process called gravitational lensing. Gravitational lensing breaks the statistical isotropy of the CMB and introduces correlations across CMB temperature and polarization fluctuations on different angular scales. These correlations are proportional to the projected gravitational potentials integrated along the line of sight and therefore can be used to reconstruct the lensing potential (Lewis & Challinor, 2006). The lensing potential is a probe of the growth of large-scale structure and the geometry of the universe between the epoch of recombination and today. Thus, from CMB observations alone, we can extract information about the universe at both the redshift of last scattering () and the redshifts of structure formation () and dark energy domination. This makes CMB lensing a powerful tool for pursuing some of the most ambitious goals in cosmology and particle physics today (e.g. CMB-S4 Collaboration et al., 2016), including constraining the sum of neutrino masses and the amplitude of primordial gravitational waves (Lesgourgues & Pastor, 2006; Kamionkowski & Kovetz, 2016).
While with the CMB lensing measurements from this work, we will not detect the sum of neutrino masses or significantly constrain primordial gravitational waves through delensing (Manzotti et al., 2017), these measurements will nevertheless provide relevant constraints to parameters of the standard cosmological model CDM. This is of particular interest currently because some optical probes of gravitational lensing are in mild tension with Planck’s CMB+lensing constraints on matter density and fluctuations (Abbott et al., 2018a; Hildebrandt et al., 2018; Hikage et al., 2019). The optical lensing measurements use the effect of gravitational lensing on the apparent shapes of background galaxies to measure the intervening gravitational potentials. Compared to using galaxies, an attractive characteristic of CMB lensing is that the source plane has nearly Gaussian statistics with a well-characterized angular power spectrum and is at a high and well-known redshift (Planck Collaboration et al., 2016a, 2018a). Owing to the high-redshift source plane, CMB lensing probes the integrated matter fluctuations to redshifts beyond optical surveys. Furthermore, CMB lensing measurements have different instrumental and astrophysical systematics compared to those from optical surveys. The use of independent probes will therefore help us investigate the source of the tension. It is thus an important goal for us to identify potential sources of instrumental and/or astrophysical systematics as the field advances the precision of CMB lensing measurements.
CMB lensing has entered the era of precision measurements in the last few years. This lensing effect was first detected through cross-correlations with radio sources (Smith et al., 2007). Since then, the CMB lensing potential power spectrum has been measured by multiple groups using temperature data only (T, Das et al. 2011; van Engelen et al. 2012; Planck Collaboration XVII 2013; Omori et al. 2017), polarization data only (POL, POLARBEAR Collaboration 2014; Keck Array et al. 2016), and combinations of temperature and polarization data (Story et al., 2015; Sherwin et al., 2017; Planck Collaboration et al., 2018b). The most precise lensing amplitude measurement, at 40 , comes from Planck’s minimum-variance (MV) estimator that combines both temperature and polarization estimators; in that measurement, the temperature reconstruction contributes most of the signal-to-noise ratio (S/N). More generally, in prior lensing measurements that used both temperature and polarization maps, the T estimator has always dominated the overall measurement precision. In this work, for the first time, the POL measurement is more constraining than the T measurement. Furthermore, if we consider only the statistical uncertainty, we have a constraint () on the lensing amplitude using polarization data alone – the tightest constraint of its kind to date.
In this work, we extend the lensing measurement to a 500 deg field from the 100 deg field of Story et al. (2015, hereafter S15). We build on the lensing pipeline presented in S15 with two main modifications. First, instead of including the Monte-Carlo (MC) bias – the difference between the recovered lensing spectrum from simulation and the input spectrum – as a systematic uncertainty, we identify its main contributor and correct for the bias using a multiplicative correction factor. Second, instead of treating extragalactic foreground biases as negligible, we subtract an expected foreground bias from the T and MV lensing spectra.
Because the input CMB maps are of similar depths as those used in S15, we also measure lensing modes with S/N better than unity for . However, in this analysis, we have 5 times more sky area and therefore are able to make a more precise measurement of the lensing spectrum. With both the T and POL lensing amplitudes well constrained, we improve the precision of the MV lensing amplitude measurement from S15’s 14% to . This is approaching the 3% precision of the lensing amplitude measurement from Planck (Planck Collaboration et al., 2018b). These two measurements arrive at their respective precisions from very different regimes: while the Planck measurement covers 67% of the sky, each lensing mode is measured with low S/N; our measurement covers only 1% of the sky, but many lensing modes are measured with high S/N. When our measurements are projected to cosmological parameter space, the constraint in the plane is only slightly weaker than those from Planck (Planck Collaboration et al., 2018b). This will be useful in illuminating the aforementioned mild tension between optical probes and Planck. In a future paper, we will present cosmological parameter constraints and comparisons with other lensing probes.
This paper is organized as follows: in Section 2, we describe the dataset used and the simulated skies generated for this analysis. In Section 3, we summarize the lensing analysis pipeline and describe new aspects. We present the lensing measurements in Section 4 and show that our measurements are robust against systematics in Section 5.1. In Section 5.2, we account for systematic uncertainties from sources that can bias the lensing measurement. We conclude in Section 6.
2. Data and Simulations
In this section, we describe the SPTpol survey, the data processing steps taken to generate the data maps, and the simulated skies generated for this analysis.
2.1. SPTpol 500 deg survey
The South Pole Telescope (SPT, Padin et al., 2008; Carlstrom et al., 2011) is a 10-meter diameter off-axis Gregorian telescope located at the Amundsen-Scott South Pole Station in Antarctica. In this work, we use data from the 150 GHz detectors from the first polarization-sensitive receiver on SPT, SPTpol. The SPTpol 500 deg survey spans 4 hours in right ascension (R.A.), from to , and 15 degrees in declination (dec.), from to . We use observations conducted between April 30, 2013 and October 27, 2015 (after the 100 deg survey for S15 was finished), which include 3491 independent maps of the 500 deg survey footprint. The field was observed using two strategies. Initially, we used a “lead-trail” scanning strategy, where the field was divided into two halves in R.A.. The telescope first scanned the lead half, and then switched to the trail half such that they were both observed over the same range in azimuth, and therefore the same patch of ground. In May 2014, the scanning strategy was switched to a “full-field” strategy, in which the full R.A. range of the field was covered in a single observation.
2.2. Data processing
The data reduction for this set of maps follows that applied to the TE/EE power spectrum analysis of the same field (Henning et al., 2018, hereafter H18). We therefore highlight here only aspects that are different or are particularly relevant for this analysis.
An observation is built from a collection of constant-elevation scans, where the telescope moves from one end of the R.A. range to the other. For every scan, we filter the timestreams, which corresponds in map space to mode removal along the scan direction. We subtract from each detector’s timestream a Legendre polynomial as an effective high-pass filter. We choose a 3rd order polynomial for the lead-trail scans and a 5th order polynomial for the full-field scans. We combine this with a high-pass filter with a cutoff frequency corresponding to angular multipole in the scan direction. During the polynomial and high-pass filtering step, regions within of point sources brighter than 50 mJy at 150 GHz are masked in the timestreams to avoid ringing artifacts. Finally, we low-pass the timstreams at an effective multipole of in the scan direction to avoid high frequency noise aliasing into the signal due to the adopted pixel resolution of in the maps.
Electrical cross-talk between detectors can bias our measurement. In S15, we accounted for this bias as a systematic uncertainty of 5% on the MV lensing amplitude. In this analysis, we correct the cross-talk between detectors at the timestream level as described in H18. With this correction, cross-talk is suppressed by more than an order of magnitude. With the 5% uncertainty from S15 being an upper limit before this suppression, we conclude that the residual cross-talk introduces negligible bias to our lensing amplitude measurement.
Before binning into maps, we calibrate the timestreams relative to each other using observations of the HII region RCW38 and an internal chopped thermal source. The per-detector polarization angles are calibrated based on measurements from a polarized thermal source (Crites et al., 2015). After that, the timestream data are binned into maps with pixels using the oblique Lambert azimuthal equal-area projection.
We apply the following corrections to the coadded map of the individual observations to obtain the final map: monopole subtraction, global polarization rotation, absolute calibration, and source and boundary masking. We measure leakage by taking a weighted average over multipole space of the cross-correlation of the temperature map with either the or the map. We find leakage factors of and . We obtain their uncertainties from the spread of leakage factors derived from 100 different half-splits of the data and find the fractional uncertainties to be 0.1% and 0.3% respectively. We deproject a monopole leakage term from the maps by subtracting a copy of the temperature map scaled by these factors from the and maps.
Assuming CDM cosmology, we expect the cross-spectra and to be consistent with zero. Therefore, we can estimate and apply the global polarization angle rotation needed to minimize the and correlations (Keating et al., 2013). We find the rotation angle to be and rotate the and maps accordingly.
The final absolute calibration is obtained by comparing the final coadded SPTpol map with Planck over the angular multipole range . Specifically, we take the ratio of the cross-spectrum of two half-depth SPTpol maps to the cross-spectrum of full-depth SPTpol and Planck maps and require the ratio to be consistent with 1 to determine the calibration factor. We estimate the polarization efficiency (or polarization calibration factor) similarly, by comparing a full-depth SPTpol -mode map with the Planck -mode map. The temperature and polarization calibration factors ( and ) as derived are and respectively (see H18 for details). We apply to the temperature map and to the polarization maps to obtain calibrated temperature and polarization maps. In the legacy Planck release, their polarization efficiency estimates are found to be potentially biased at the level (see Table 9 of Planck Collaboration et al., 2018c). To circumvent potential biases when we tie our polarization map calibration to Planck’s -mode map, we adjust by a multiplicative factor obtained from H18 without using Planck’s polarization maps. In H18, and were free parameters in the fits of the PlanckTT+SPTpol EETE dataset to the CDM + foregrounds + nuisance parameters model, with priors set by the and values and uncertainties derived from the comparison against Planck described above. In this work, we multiply by the best-fit parameter (1.01) from H18 (see Table 5 of H18). In Section 5.2, we use the posterior uncertainties of the and parameters to quantify their contributions to the systematic uncertainties of the lensing amplitude measurements.
We apply a mask that defines the boundary of the maps to downweight the noisy field edges. In addition, we mask bright point sources in the map. We use a radius to mask point sources from the SPT-SZ catalog by Everett et al. in prep. with flux density above 6 mJy at either 95 GHz or 150 GHz that are in the 500 deg footprint. We use a radius for point sources with flux density greater than 90 mJy at either 95 GHz or 150 GHz. Clusters detected in Bleem et al. (2015) within the 500 deg survey footprint are masked using a radius. The mask has a tophat profile for both the boundary and the sources, and mode-mixing due to the mask is suppressed by the inverse-variance map filtering step (Section 3.2). We compare this lensing measurement with one using a cosine-tapered apodization on the mask edges in Section 5.2 and find them to be consistent.
The noise levels in the coadded maps are 11.8 K–arcmin in temperature and 8.3 K–arcmin in polarization over the multipole range ***The temperature map noise is higher than the polarization map noise in this multipole range because of atmospheric noise contributions.. The map depth is similar to that of the 100 deg field used in S15†††The noise levels provided in S15 were estimated without the and corrections and are thus higher.. Since this map covers five times the sky area, the sample variance of the lensing power spectrum is reduced and hence the lensing spectrum measurement precision is improved.
We use simulated skies to estimate the mean-field bias to the lensing potential map (Section 3.1), to correct the analytical response of the lensing estimator (Section 3.1), to correct the expected biases (and , see Section 3.3) in the raw lensing power spectrum, and to estimate the uncertainties of the lensing measurements (Section 4).
The simulated skies contain the CMB, foregrounds, and instrumental noise. The input cosmology for generating the CMB is the best-fit CDM model to the Planck plikHM_TT_lowTEB_lensing dataset (second column in Table 4 of Planck Collaboration et al., 2016b). Using the best-fit cosmology, we use CAMB‡‡‡http://camb.info to generate theory spectra, which are then fed into Healpix (Górski et al., 2005) to generate the spherical harmonic coefficients for , and the lensing potential . The CMB are projected to maps and lensed by using LensPix (Lewis, 2005). The lensed maps are then converted back to , at which point foreground fluctuations are added, and the are multiplied by the -space instrument beam window function. The are subsequently projected on an equidistant cylindrical projection (ECP) grid for mock observation, which produces mock skies that are processed identically as the real sky.
The foreground components are generated as Gaussian realizations of model angular power spectra. We include power from thermal and kinematic Sunyaev-Zel’dovich effects (tSZ and kSZ), the cosmic infrared background (CIB), radio sources, and galactic dust. The amplitudes for tSZ, kSZ, and CIB components are taken from George et al. (2015), which has the same source masking threshold as this work. We use an amplitude = 5.66 K with a model shape from Shaw et al. (2010) for the sum of tSZ and kSZ components§§§Given that we mask all clusters in the Bleem et al. (2015) catalog, this tSZ level is high by 2 K. As a result, we non-optimally downweight high- modes in the temperature data map and slightly over-estimate the noise (Section 3.3). We compare the analytic between K vs K (keeping all other inputs equal) and find that the latter is 2% lower. Since there is both signal and noise variance in the uncertainty of the lensing amplitude measurement, and the T lensing amplitude uncertainty is (see Section 4), the T lensing amplitude would at most be reduced by 2% of , which is negligible.. We use K with for the radio source component. We use K with for the unclustered CIB component and K with for the clustered CIB. By modeling these terms as Gaussian, we have neglected the potential bias their non-Gaussianities can introduce to the lensing spectrum. We treat this potential bias explicitly later in the analysis (see Section 3.3). We assume a 2 polarization fraction for all the Poisson-distributed (unclustered) components to model extragalactic polarized emission (Seiffert et al., 2007). We model galactic dust power in temperature and polarization as a power law with , with K, K, and K (Keisler et al., 2015).
The instrumental noise realizations are generated by subtracting half of all the observations from the other half. By randomly grouping the observations into one of the halves, we create 500 different realizations of noise from the data themselves. The noise realizations are added to the simulated skies after mock observation.
We generate 500 lensed skies including CMB, foregrounds, and noise. All 500 skies are used for estimating the lensing spectrum bias . We use 100 skies to estimate the mean-field (, see Section 3.1): 50 for each of the two lensing potential estimates that enter the lensing spectrum calculation ( and in Eq. (6)). We use the other 400 to correct the analytical response of the lensing estimator, and to obtain the statistical uncertainty of the lensing spectrum. In addition, we generate 500 unlensed skies with all other inputs being identical. We use this set of simulations for testing the lensing pipeline and assessing the probability of detecting lensing from unlensed skies.
The bias is estimated using a different set of 200 lensed skies. We start with 200 realizations of unlensed skies and divide them into two groups. We then lens one sky from each group by the same (Section 3.3). We do not add instrumental noise nor foregrounds to this set of simulations, because comes from correlations between the CMB and the lensing modes. At the map-filtering step, we filter this set of simulations assuming the same level of foregrounds and noise as the other set of lensed skies.
3. Lensing Analysis
In this section, we describe the lensing pipeline that produces an unbiased estimate of the lensing potential and of the lensing power spectrum. This analysis follows that of S15 except for the treatment of the MC bias and foreground subtraction in the lensing spectrum. We summarize the steps here and refer the reader to S15 for details.
3.1. Estimating the lensing potential
The unlensed CMB sky is well-approximated by a statistically isotropic, Gaussian random field. Gravitational lensing breaks the statistical isotropy of the fluctuations, and introduces correlations between the otherwise uncorrelated Fourier modes of the CMB temperature and polarization maps. We use these correlations to estimate the underlying lensing potential with pairs of filtered maps using the quadratic estimator of Hu & Okamoto (2002):
where and are filtered , or fields as outlined in Section 3.2, is a weight function unique to each input pair of maps, are modes of the CMB and are modes of the lensing potential. We form lensing potentials from and modify from Hu & Okamoto (2002) by replacing the unlensed CMB spectra with lensed spectra to reduce a higher-order bias (, Hanson et al., 2011). As written, is a biased estimate of . To arrive at an unbiased estimate of , we remove an additive bias (the mean-field, MF) and normalize the estimator by the response (defined below).
The mean-field can arise from masking and inhomogeneous noise – sources that introduce mode-coupling across angular scales. These mode-couplings persist even when the CMB and realizations are different. We can therefore estimate the mean-field by averaging from many realizations of input lensed CMB maps.
We construct the response using an analytical calculation corrected by simulations. The analytic response of the estimator is:
where describes the diagonal approximation to the filter applied to the input CMB maps (Section 3.2). Because this filter assumes spatial stationarity of the statistics of the signal and the noise, it does not account for non-stationary effects from e.g. the boundary and source mask and causes the response to be slightly misestimated. To account for this, we use simulations to estimate the correction to the analytic response. The total response is thus , with denoting the correction estimated from simulations. We extract by first taking the cross-spectrum between the input and the intermediate estimate , which has been mean-field subtracted and normalized by the analytic response: . We then take the ratio of the average of the cross-spectra with the input spectrum over 400 simulation realizations. Similar to S15, we find the to be a correction.
The normalized and mean-field-corrected is,
for the individual estimators where . In this work, we use T to denote the lensing potential and spectrum constructed using only temperature data, i.e. and its spectrum. We use POL to denote the potential and spectrum constructed using only polarization data, and MV to denote the potential and spectrum constructed using both temperature and polarization data.
To construct the combined minimum-variance (MV) and polarization-only (POL) estimators, we first form a weighted average of the input estimators to form the intermediate . We use the inverse-noise variance of the input estimators as weights, which are approximated by their analytical responses. We then extract by forming cross-spectra between or with the input . Putting it all together, the unbiased MV and POL lensing potentials are constructed as,
where for the MV estimator and for the POL estimator.
3.2. Input CMB map filtering
We filter our input maps with an inverse-variance (C) filter, which is derived such that the variance of a lensing field reconstructed by a quadratic estimator is minimized. The filter is constructed identically as in S15: we assume the data maps to be composed of three components: sky signal, “sky noise”, and pixel domain noise. The sky signal and noise are modeled in the Fourier domain and they include the CMB, astrophysical foregrounds, and atmospheric noise. The pixel domain noise is modeled as white, uncorrelated, and spatially non-varying inside the mask. Concretely, we solve for the inverse-variance filtered Fourier modes in the following expression using conjugate-gradient-descent:
Here describes the sky signal and noise components, is the inverse of the map noise variance and is zero for masked pixels, applies the filter transfer function and inverse-Fourier transforms the map from pixel space to Fourier space. additionlly transforms to for polarization maps, and denotes the input pixel-space maps.
This filter approaches the simple form in regions far away from the mask boundary. To see that, one can absorb the pixel-domain noise into the term (which is valid under our model of the data for pixels far from the mask boundary); in the limit where , the diagonal form is exact.
The inputs to this filter are as follows. For the sky signal component of , we use the same lensed CMB and astrophysical foreground spectra used to generate simulations (Section 2.3). For the sky noise component, we take the averaged power spectra from the temperature and polarization noise realizations and from them subtract noise floors of 7 K–arcmin to form . We set 7 K–arcmin as the white noise level of the pixel-domain noise of the , , and maps. The filter transfer function includes time-stream filtering, beam, and pixel-window function. We approximate the time-stream filtering transfer function using simplified simulations that capture the lost modes along the axis due to our scan strategy. We measure the SPT beam using the response of the detectors to Venus, as done in H18. In addition, we fit the measured beam with a function of form to obtain a smooth profile at . We incorporate the pixel-window function for the map pixel size of . Our input CMB map range for this analysis is set by the timestream -space high-pass filter and set by concerns of foreground contamination (Section 5.2).
3.3. Estimating the lensing potential power spectrum
Upon obtaining the unbiased estimate of the lensing potential , we calculate the raw power spectrum of the lensing potential by forming cross-spectra of and where :
where is the average value of the fourth power of the mask.
This power spectrum is biased, and we correct four sources of biases in this analysis. Two of them, the and biases, arise from spurious correlations of the input fields. The third source of bias comes from foregrounds, . Finally, we correct for a multiplicative bias due to higher-order coupling of the source mask . Therefore, the unbiased lensing spectrum is estimated as
Figure 1 shows , , and for the MV reconstruction. and are similar to those in S15 because the CMB map noise levels for these two works are about the same. We describe how each bias term is estimated in the following paragraphs.
The lensing spectrum is a 4-point function, or trispectrum, of the observed fields and thus contains, in addition to the connected term caused by lensing, a disconnected term from correlations of Gaussian fields (). Secondary contractions of the trispectrum give rise to connected terms that also bias the lensing spectrum (). It is called because it is first order in (Kesden et al., 2003; Hanson et al., 2011). We estimate the power of the and terms using simulations and subtract them from the raw lensing spectrum.
We estimate the disconnected term using a realization-dependent method (, Namikawa et al. 2013). The standard is estimated by computing the lensing spectra with two input maps that form from different simulation realizations. Since input maps from different realizations do not share the lensing potential, the resultant power in the lensing spectra comes from spurious correlations of the maps. With the realization-dependent method, we use a combination of simulation and data maps. Specifically, on top of the mix of simulation realizations, we also construct lensing potentials using data in one of the input maps. Accounting for all the different combinations, we have:
where is expressed as to more clearly indicate the sources of the input maps. Here and denote simulation skies from different realizations, and denotes the data map. The realization-dependent method produces a better estimate of the disconnected term because it reduces the bias from the mismatch between the fiducial cosmology used for generating the simulations and that in the data. In addition, this method reduces the covariance between the lensing potential bandpowers.
The second bias term comes from spurious correlations between the CMB and the lensing potential and is proportional to . We estimate it using pairs of simulated skies that have different realizations of unlensed CMB lensed with the same :
where the subscript denotes CMB maps lensed by the same realization.
We account for biases to the lensing spectrum due to foregrounds in the temperature map. As studied in van Engelen et al. (2014), both tSZ and CIB have a trispectrum, which leads to a response in the lensing power spectrum. This bias enters the lensing spectrum through the 4-point function of the temperature map, thus modifying the T spectrum and the MV specturm. Additionally, since both tSZ and CIB trace the same large-scale structure as the lensing field, the non-Gaussianities of both fields can mimic lensing and couple through the estimator in a coherent way that correlates with , forming a non-zero bispectrum of the matter density field (denoted by tSZ- and CIB-). This effect biases the MV lensing spectrum and spectra from pairs of estimators of the form , where . The level of foreground bias is scale-dependent. It is negative and close to flat at 2% for for the total bias contributed from tSZ and CIB through their trispectra and correlations with . As increases, the magnitude of this bias decreases and reaches a null at about of 2000. We subtract the relevant terms from the T and MV spectra by using the bias estimates from van Engelen et al. (2014). Specifically, we subtract the total foreground bias coming from the tSZ trispectra, CIB trispectra, tSZ-, and CIB- from the T lensing spectrum. For the MV spectrum, we subtract the total foreground bias that enters through , and tSZ- and CIB- biases that enter through with . We compute the bias fraction by forming a weighted average of the foreground biases to and . We use as weights the fractional contribution of the estimators to the MV estimator. The size of this bias in the MV spectrum is shown in Figure 1.
MC bias describes the difference between the recovered amplitude in simulations and the input spectrum. It is typically found to be small (e.g. Sherwin et al., 2017; Planck Collaboration et al., 2018b). In S15, we found that the mean MV amplitude from simulations was 3% below unity and treated this discrepancy as a source of systematic uncertainty. In this work, we find that the main source of our MC bias comes from the inclusion of the point source and cluster mask. When analyzing the set of simulations with the source mask removed (while keeping the boundary mask), we are able to recover to within 1 of the standard error of 400 sky realizations () for the multipole range we report in this work. Furthermore, comparing the mean recovered spectrum of the MV, POL, and T estimators with and without the source mask applied to the input maps, we observe a relatively constant multiplicative offset in the range of . We therefore conclude that the main source of our MC bias is due to higher-order coupling generated from the presence of source masks in the map that is not accounted for by . We construct a multiplicative correction for this MC bias as an inverse of the mean of the simulation lensing amplitudes estimated from :
To check the stability of this estimate, we vary the maximum range used between of 446 and 813 and find the resultant simulation from all pairs of estimators to be consistent with unity to within 2 of their standard errors. The correction from this step is about 5%. Specifically, for the MV, POL, and T are = 1.05, = 1.05, and = 1.07. For , the mean of the simulation spectra with the source mask removed is below the input spectrum. Therefore, this MC bias correction is not applicable for multipoles below 100, and we do not report results below of 100.
In constructing our simulations and applying the quadratic estimator to recover the lensing potential, we have assumed to be Gaussian. However, nonlinear structure growth and post-Born lensing would introduce non-Gaussianities to (Böhm et al., 2016; Pratten & Lewis, 2016). These non-Gaussianities produce the so-called bias, as studied in Böhm et al. (2016, 2018); Beck et al. (2018). For the range considered, the size of this bias is 0.5% for the temperature reconstruction and is negligible for the reconstruction for input CMB maps similar in noise levels and multipole range to those in this work (Böhm et al., 2018; Beck et al., 2018). These are small compared to our other sources of uncertainties and we neglect them in our results.
We report our results in binned bandpowers. First, we derive the per-bin amplitude as the ratio of the unbiased lensing spectrum to the input theory spectrum:
with the subscript denoting a binned quantity. is a weighted average of the inputs for inside the boundaries of the bin, with the weights designed to maximize signal-to-noise: . We obtain the variance from the corresponding set of simulation cross-spectra.
We report the bandpowers in lensing convergence () instead of the lensing potential . The convergence field is of the divergence of the deflection field, which is the gradient of the lensing potential (Lewis & Challinor, 2006):
In Fourier space, they are related by . The reported bandpowers are derived as the product of the data amplitude and the input theory spectrum at bin center ,
The overall lensing amplitude for each estimator is calculated identically as the per-bin amplitude in Eq. (11) using the whole reported range.
Differences between the fiducial cosmology and the cosmology of the SPTpol patch would produce different measured lensing amplitudes. The cosmology dependence enters through the bias. In this work, since we choose a fiducial cosmology that is consistent with data, we expect the difference in the bias to be small. To test this, we sample the lensing amplitude given the fiducial cosmology with and without corrections to . We find the difference in the lensing amplitude for the MV estimator to be 0.007 (0.1 ). We therefore neglect this correction in this work.
In the following, we discuss our main results: the lensing convergence map, the lensing spectrum, and the lensing amplitude measurement. We compare our MV map to S15’s. We discuss the relative weights of the T and the POL results, compare the sizes of the systematic uncertainties to the statistical uncertainties, and put our measurement in the context of other lensing measurements.
In Figure 2, we show the MV, T, and POL lensing convergence. The maps are smoothed by a 1-degree FWHM Gaussian to highlight the higher signal-to-noise modes at the larger scales. At the angular scales shown, the POL reconstruction has higher signal-to-noise (lower ) than the T reconstruction. Therefore, while the T and the POL map fluctuations both trace those in the MV map in broad strokes, we see that the POL modes traces the MV modes more faithfully. The lensing modes in the MV map are reconstructed with S/N better than unity for . It is the largest lensing map reconstructed with this S/N level from the CMB to date.
In Figure 3, we compare the MV map from S15 and the one from this work over the same region of the sky. We observe that the two convergence maps show nearly identical degree-scale structure. The S15 data were taken before the SPTpol 500 deg survey data (see Section 2.1), and thus constitute an independent dataset from that used in this work. In addition, because the two analyses have similar per-mode reconstruction noise, the visual agreement of the modes between the two is a useful consistency check.
We present the lensing power spectrum measurement in logarithmically spaced bins in the range . We list the MV lensing bandpowers and their uncertainties in Table 1. The lower bound of the range is chosen by the region of validity of the MC bias correction. The upper bound is set by computing the uncertainties on the lensing amplitudes as we include higher multipoles and seeing no gain in signal-to-noise of the amplitude going beyond of 2000. In Figure 4, we show the bandpowers from the MV spectrum, the POL spectrum, and the T spectrum. We see that the errorbars of the POL spectrum are smaller than those of the T spectrum at , and vice versa on smaller angular scales. This is consistent with the of the T reconstruction at small angular scales being lower than that of the POL reconstruction – the small angular scale modes are better reconstructed by the T estimator.
We measure the overall lensing amplitude for each estimator and find:
We derive the statistical uncertainties from the standard deviations of the lensing amplitudes from simulations. We detail the sources of systematic uncertainties in Section 5.2 and break them down in Table 3. Considering statistical uncertainties alone, we measure the lensing amplitude with uncertainty using the MV estimator and with uncertainty with the POL estimator. For the T estimator, we measure the lensing amplitude with uncertainty. Having chosen the same cuts in multipole space for both the input temperature and polarization maps, this shows that the signal-to-noise per mode in the input polarization maps are now high enough that the POL estimators give more stringent measurements of the lensing amplitude than the T estimator. In future analyses, the T lensing spectrum is sample-variance limited and cannot be improved by lowering the temperature map noise levels. Instead, it can be improved by including information from higher multipoles and/or more sky area. However, lowering the noise levels of the polarization maps can still improve the lensing measurement from polarization estimators. Specifically, unlike the temperature estimator, the of the estimator is not limited by unlensed power in the map, because there is little unlensed mode power to contribute to in the multipole range important for lensing reconstruction. In addition to surpassing the measurement uncertainty of the T lensing amplitude, considering statistical uncertainties alone, our POL lensing amplitude is the most precise amplitude measurement () using polarization data alone to date.
The systematic uncertainties for the MV and the POL estimators are 40% of their respective statistical uncertainties, whereas the systematic uncertainty is subdominant for the T estimator compared to its statistical uncertainty. For both the MV and the POL estimators, the systematic uncertainty budget is dominated by the uncertainty (Section 5.2). Including the systematic uncertainties in the MV amplitude measurement, we measure with uncertainty.
We detect lensing at very high significance. From reconstructing using 400 unlensed simulations, the standard deviation of is 0.024. The observed amplitude of would thus correspond to a fluctuation.
Compared to other ground-based measurements, our result has the tightest constraint on the lensing amplitude. In Figure 5, we show our lensing power spectrum measurement against previous measurements. Our measurement is consistent with the measurement by Omori et al. (2017). In that work, they reconstruct lensing using a combined temperature map from SPT-SZ and Planck over the common 2500 deg of sky. They measure the lensing amplitude to be relative to the best-fit CDM model to the Planck 2015 plikHM_TT_lowTEB_lensing dataset (same as the fiducial cosmology used in this work). The most recent lensing analysis of all-sky Planck data found the best-fit lensing amplitude to be against the Planck 2018 TTTEEE_lowE_lensing cosmology (Planck Collaboration et al., 2018b). To compare our measurement to this model, we refit our minimum-variance bandpowers and get , consistent with Planck’s lensing measurement.
5. Null Tests, Consistency Checks, and Systematic Uncertainties
In this section, we summarize the null and consistency tests we have performed on the data and account for the systematic uncertainties in our lensing amplitude measurements. We report test results from the MV, POL, and T estimators.
5.1. Null Tests and Consistency Checks
We quantify the results of our tests using summary statistics comparing data bandpower differences to simulation bandpower-difference distributions, where the differences are taken between bandpowers obtained from the baseline analysis pipeline and from a pipeline with one analysis change. We plot these bandpowers in Figure 6 to provide an absolute sense of how much the bandpowers and their errorbars change given the various analysis variations.
Quantitatively, we calculate the of the data-difference spectrum against the mean of the simulation-difference spectrum using the variance of the simulation-difference spectra (), as the difference bandpowers are largely uncorrelated:
The probability-to-exceed (PTE) is calculated from a distribution with 10 degrees of freedom, as we have 10 bandpowers. For the curl test, comes from the distribution of the simulation curl spectra. Figure 7 provides a visual summary of these tests. It shows the data difference bandpowers for each test for the three estimators. The errorbars are generated from the distributions of simulation difference bandpowers. The data points and the errorbars from each estimator in each bin are scaled by the 1 lensing spectrum uncertainties from that estimator in that bin. We list the and PTE values for each test and each estimator in Table 2.
We compare the differences in overall lensing amplitude similarly.
We calculate the difference in the lensing amplitudes
between the baseline and the alternate analysis choice setup: .
We form the by comparing the data difference amplitude against the distribution
of the difference amplitudes in simulations.
Varying and :
We vary the multipole range of the
input CMB maps used for lensing reconstruction to check
(1) consistency of bandpowers as we include more or fewer CMB modes,
(2) impact of foregrounds as we increase the maximum multipole.
As we increase the maximum multipole used in the input temperature map,
the contamination of the CMB by foregrounds like tSZ and CIB increases.
As discussed in Section 3.3, both of these inputs can bias the lensing spectrum.
On the low side, we remove modes because the combination of our
observing strategy and time-stream filtering removes
modes approximately along the axis.
We run two cases for the cut: and
From Table 2, we see that the differences of the cases
are consistent with the expected simulation distribution.
We can in principle set the in the baseline analysis to be 50.
However, since the applied timestream filtering removes modes,
and the number of modes available between of 50 and 100 is small
compared to the number of modes at the high- end,
we do not expect there to be significant improvement in signal-to-noise and therefore
choose to set for the baseline analysis.
For the tests, we vary the cut between
and in steps of 500.
We observe from Table 2 that moving between of 3000 and
2500 and between of 3000 and 3500, the data bandpower differences
are consistent with bandpower differences from simulations for all three sets of estimators.
We set the baseline to be 3000 to keep the systematic uncertainty due
to the subtraction of foreground biases subdominant to the statistical uncertainty of the measurements.
For future analyses, estimators designed to reduce foreground
biases (Osborne et al., 2014; Schaan & Ferraro, 2018; Madhavacheril & Hill, 2018)
can be employed, potentially sacrificing some statistical power for reduced systematic uncertainty.
We test for the effects of using a tophat function
for the boundary and source mask by
redoing the entire analysis using a cosine taper instead.
The radius of the cosine taper is 10 for the boundary mask
and 5 for the sources.
The data difference is typical given the simulation-difference distribution for
all three estimators.
Curl test: The deflection field that remaps the primordial CMB anisotropies can be generically decomposed into a gradient and a curl component:
where is a 90 rotation operator and is a
psuedo-scalar field that sources the curl component (see e.g. Hirata & Seljak, 2003; Namikawa et al., 2012).
In the Born approximation, the lensing potential only sources the gradient component
of the deflection field.
At current noise levels and barring unknown physics, the curl component is expected
to be consistent with zero.
Therefore, estimating the curl of the deflection field serves as a test of
the existence of any components in the data generated from non-Gaussian
secondary effects or foregrounds (Cooray et al., 2005).
Analogous to estimating the lensing field, for the curl component,
we estimate the psuedo-scalar field .
The curl estimator is orthogonal to the lensing (gradient) estimator and
has the same form as the lensing estimator Eq. (1).
The weights of the curl estimator are designed to have response to the curl
instead of the gradient.
We implement the weights as presented in Namikawa et al. (2012).
The curl spectrum is derived analogous to the lensing spectrum with two differences:
(1) the theory input is set to a flat spectrum , where
it is used for uniformly weighting the modes when binning and as a reference
spectrum for the amplitude calculation;
(2) no response correction from simulations is applied to the psuedo-scalar field as the
expected signal is zero.
The bottom panel of Figure 7 shows the power spectra for the MV, POL,
and T curl estimators and their uncertainties scaled by the respective uncertainties
of their lensing spectra.
The bandpower PTEs and the amplitude PTEs for the three estimators are listed in Table 2.
They are all consistent with zero.
The bandpower PTE and the amplitude PTE are 0.50 and 0.20 for the MV curl estimator.
We thus see no evidence of contamination to our lensing estimate from non-Gaussian secondaries.
Consistency across estimators: In Figure 8, we show the reconstructed lensing spectrum from all 15 pairs of the 5 estimators , and the MV spectrum. We test for consistency of the lensing spectra from the 15 pairs of estimators with the best fit from the MV estimator by calculating and PTE. We calculate the of the 150 bandpowers against the binned theory spectrum scaled by the best-fit MV amplitude:
where is the 150 data bandpowers, is 15 copies of the scaled binned theory spectrum, and denotes the covariance matrix. We construct the covariance matrix of the 150 bandpowers of 15 estimators from 400 simulations. We set the off-diagonal terms of each sub-block to zero as we expect there to be little to no correlation across different bins between estimators. For the main diagonal, the first and second bins are correlated at the 10% level for 12 of the estimators and we keep them while setting the rest of the covariance elements to zero. The PTE compared to the set of 400 sets of simulated bandpowers is 0.44. Comparing instead the best-fit amplitudes of the 15 pairs of estimators to the MV best-fit amplitude, we note that the EBTB and EBEB pairs are low by 2.4 and 2.1 respectively, while the rest of the pairs are within 2 of the MV’s amplitude. With the number of tests we have performed, it is not unusual to see 2 outliers.
5.2. Systematic Uncertainty
In this section, we summarize the sources of systematic uncertainty and our
accounting of them in the lensing amplitude measurement.
We quantify the uncertainties in the lensing amplitude measurement due to uncertainties
in the beam measurement, temperature and polarization calibrations,
TP leakage correction, global polarization angle rotation,
and the applied foreground template.
We address the potential impact of non-Gaussian polarized foregrounds.
The sources of systematic uncertainty and their respective impact on the lensing amplitude
measurements are listed in Table 3.
Beam uncertainty: We take the beam measurement uncertainty
derived from the beam covariance matrix in H18 and convolve (1+ )
with the input maps while keeping all the simulation maps the same.
We analyze the data maps as if their beam were the mean measured beam
as opposed to the modified .
The difference in the lensing amplitude between this analysis and the baseline analysis
quantifies the effect of an underestimation of the beam profile by 1 of the
beam measurement uncertainty across the entire angular multipole range.
We find , , and
The systematic uncertainties induced by the beam uncertainty are small (0.1 )
compared to the respective statistical uncertainties
of the measured lensing amplitudes in T, POL, and MV.
Temperature and Polarization calibrations: We apply the temperature and polarization calibration factors derived in H18 to our data maps as described in Section 2.2. From the posterior distributions of H18, we obtain the uncertainties of the factor and the factor which are 0.3% and 0.6% respectively. The and are applied to the raw temperature and polarization maps through and . Keeping the simulated maps fixed to the baseline analysis, we scale the data maps by ) for the temperature map and (1+)(1+) for the polarization maps and we recalculate the data lensing amplitudes***In S15, we accounted for the systematic uncertainties from calibration uncertainty as 4 and 4. This underestimates the systematic uncertainty because the object that gets scaled by and is instead of . This means that the calibration offset applies to the noise biases terms as well as the signal term and can be a factor of a couple larger than itself. . The two pieces in the amplitude calculation that change because of the modified data maps are and . To illustrate how they contribute to the overall change in the lensing amplitude, we consider here the temperature-only estimator. , which contains four copies of the data map, would shift by . For , the four terms where two of the four input maps are replaced by data maps get a multiplicative correction of . The overall shift in is therefore:
with denoting simulation realizations and denoting data.
The polarization-only case has a similar form as the temperature-only case
The difference in the resultant measured amplitudes due to an offset in the
calibration factors depends on the relative amplitudes of
and at each multipole .
Therefore, we quantify the difference by running the baseline analysis
with the temperature and polarization calibration of the data maps shifted by 1 .
We find that , , and
The shifts in the lensing amplitudes are dominated by the polarization calibration uncertainty.
Furthermore, the calibration systematic uncertainty is
almost half of the statistical uncertainty of the polarization-only amplitude.
Since the signal-to-noise of future lensing measurements will be driven by the polarization estimators,
we will need more precise polarization calibration in order for the measurements to remain dominated by
This can be achieved by cross-calibrating with deeper CMB maps or over larger areas of sky,
assuming an external CMB map exists with more accurate polarization calibration than Planck.
Relatedly, lensing measures mode-coupling
which in principle can be extracted irrespective of the input maps calibration.
An example of circumventing the systematic uncertainty contribution caused by
calibration uncertainties in the input maps is to use the measured power spectra
in one of the weight functions in the analytic response in Eq. (2).
In this approach, the response moves together with the calibration of the input maps
and therefore eliminates this systematic uncertainty.
We estimate the bias to the lensing amplitude measurement caused by misestimating
the TP leakage factors.
Similar to quantifying the systematic offsets by the beam uncertainty and the / uncertainties, here
we modify the data polarization maps by over-subtracting a -scaled
copy of the temperature map by 1 in and ,
while keeping the rest of the analysis the same.
We find the change in and to be negligible, at
0.001 and 0.002 respectively.
Global polarization angle rotation:
There is a 6% uncertainty in the global rotation angle measured through the minimization
of the and spectra discussed in Section 2.2.
How much would an offset in the polarization angle rotation bias the lensing amplitude
We run the baseline analysis with the data polarization maps rotated an extra 6% on top
of the measured angle from the minimization procedure.
We find that and change by less than 0.01 of their
respective statistical uncertainties.
As discussed in Section 3.3, we subtract templates of expected foreground biases
from the T and MV lensing power spectra
given models of CIB and tSZ from van Engelen et al. (2014).
The templates are taken from the mean of the range of models.
We estimate the systematic uncertainty from the template subtraction step
by measuring the lensing amplitude difference using the maximum and minimum of
template values allowed by the model space.
We find the lensing amplitudes to shift by and
to shift by , both less than 0.1 of their statistical
For the temperature estimator, this source of bias is 1% of the lensing amplitude
and is of the same magnitude as the bias from the temperature calibration.
For polarization, we expect the only significant source of extragalactic foregrounds
to come from the unclustered point-source component,
because SZ and the clustered CIB are negligibly polarized.
The SZ effects are expected to be polarized at the
less than 1% level for the cluster masking threshold of this work (Birkinshaw, 1999; Carlstrom et al., 2002).
The clustered CIB component is a modulation of the mean power
from all sources (Scott & White, 1999) and is thus effectively unpolarized.
For the unclustered point sources, we measure the mean squared polarization
fraction to be
(3% polarized) at 150 GHz
for sources above 6 mJy in the SPTpol field (Gupta et al. in prep.).
Assuming the polarization fraction is constant with flux density,
the extragalactic foregrounds are thus on average significantly less polarized than the CMB.
Since the level of bias introduced from temperature foregrounds is at the
percent level, we conclude that
the bias from polarized sources is negligible for this work.