A Measurement of the CMB E-mode Angular Power Spectrum at Subdegree Scales from 670 Square Degrees of POLARBEAR Data

Measurements of the cosmic microwave background (CMB) provide the foundation for our current understanding of cosmology. However, temperature measurements are now largely sample-variance-limited (Planck Collaboration et al. 2020a) out to small angular scales where extragalactic foregrounds become significant (Dunkley et al. 2013; Das et al. 2014; George et al. 2015). As a result, the focus of recent experiments has shifted to measuring the polarization of the CMB. CMB polarization anisotropies encode comparable amounts of information per angular multipole to the temperature anisotropy (Galli et al. 2014). Additionally, the relatively small polarization fraction of extragalactic sources (Seiffert et al. 2007; Battye et al. 2011; Gupta et al. 2019) means that measurements can be extended to smaller angular scales before becoming foreground-dominated.

The polarization patterns in the CMB are commonly separated into curl-free modes (E-modes) and gradient-free modes (B-modes). This division is made because density fluctuations will produce E-modes, but not B-modes, at first order. B-modes are instead produced by gravitational waves and gravitational lensing (Kamionkowski et al. 1997; Seljak & Zaldarriaga 1997).

E-mode anisotropy was first detected by DASI in 2002 (Kovac et al. 2002). Since then, the field has moved from detecting power to high signal-to-noise ratio measurements of the power spectrum by a number of experiments (Louis et al. 2017; BICEP2 Collaboration et al. 2018; Henning et al. 2018; Planck Collaboration et al. 2020a). To date, these E-mode measurements have supported the ΛCDM cosmological model. Due to the lower levels of polarized foregrounds, polarization measurements have the potential to surpass the amount of information that can be extracted from the CMB temperature anisotropy, and thus improve our ability to constrain cosmological models (Galli et al. 2014; Louis et al. 2017). Measuring CMB polarization can also help disentangle effects that are degenerate in the temperature data.

In this paper, we report a measurement of the E-mode auto-power spectrum (EE) in the angular multipole range, $500\leqslant {\ell }\lt 3000$, using new data collected between July 2014 and December 2016 from the Polarbear experiment. The expanded Polarbear survey covers 670 ${\deg }^{2}$ of sky at 150 GHz, a twenty-fivefold increase in area over the initial deep but small surveys by Polarbear (Polarbear Collaboration et al. 2017). The survey region overlaps the SPTpol and BICEP2/Keck Array surveys, and the new Polarbear bandpowers provide an independent measurement of the E-mode power spectrum on small angular scales. A measurement of the B-mode power spectrum on large angular scales on this field was presented by the Polarbear Collaboration et al. (2020, hereafter PB20), which overlaps this work in the narrow range of angular scales $500\leqslant {\ell }\leqslant 600$. We combine the Polarbear bandpowers with other recent CMB power spectrum measurements (Story et al. 2013; Louis et al. 2017; Planck Collaboration et al. 2020a) as well as CMB lensing power spectrum measurements (Wu et al. 2019; Planck Collaboration et al. 2020a), baryon acoustic oscillation (BAO) results (Beutler et al. 2011; Ross et al. 2015; Alam et al. 2017) and Hubble constant measurements (Riess et al. 2019) to study the implications for cosmology. This is the first time the cosmological implications of this combined data set have been presented.

This paper is organized as follows. In Section 2, we give a brief overview of the Polarbear instrument and the 670 ${\deg }^{2}$ survey. We continue to describe the low-level data processing and mapmaking in Section 3. The power spectrum analysis is outlined in Section 4. We test for systematic errors in Section 5. In Section 6, we present the measurements of the E-mode power spectra. Subsequently, we study the cosmological implications in Section 7. We conclude in Section 8.

Polarbear is a receiver with 1274 cryogenically cooled, transition-edge-sensor (TES) bolometers and a continuously rotating half-wave plate mounted on the 2.5 m aperture Huan Tran Telescope at the James Ax Observatory on the Atacama plateau in Chile. The elevation (5190m) and the low Precipitable Water Vapor (PWV) of the Atacama plateau make the site one of the best in the world for microwave observations. Information on the instrument and telescope can be found in Arnold et al. (2012), Kermish et al. (2012), and Takakura et al. (2017).

This work uses data taken with Polarbear on a 670 ${\deg }^{2}$ field in three observing seasons from July 2014 to December 2016. The field is centered at (RA, Dec)=($+{0}^{{\rm{h}}}{12}^{{\rm{m}}}{0}^{{\rm{s}}},-59^\circ {18}^{{\prime} }$), and largely overlaps the survey fields of BICEP2/Keck Array (BICEP2 Collaboration et al. 2018) and SPTpol (Henning et al. 2018). The data are taken in 1 hour blocks by scanning back and forth at a constant velocity ($0\buildrel{\circ}\over{.} 4\,{{\rm{s}}}^{-1}$) and constant elevation as the sky rotates past. After every four hours, the telescope is adjusted to track the field and the bolometers are retuned. We only use the Polarbear polarization data in this work, as the temperature noise level is substantially higher due to atmospheric noise. More information on the scan strategy can be found in PB20.

In this section, we review the data selection and filtering of the time-ordered data (TOD). We then briefly describe the mapmaking process, and the determination of the beam function and absolute calibration. These steps closely follow the treatment in PB20, and we refer the reader to that work for more details while highlighting any differences from that work below.

3.1. Data Selection and Filtering of the Time-ordered Data

3.2. Mapmaking

3.3. Noise

3.4. Beams and Calibration

The power spectrum is measured using a pseudo-${C}_{{\ell }}$ cross-spectrum method (Hivon et al. 2002; Tristram et al. 2005). The Polarbear implementation of this method has been previously described by Polarbear Collaboration (2014), and as "Pipeline A" by Polarbear Collaboration et al. (2017) and PB20. In this section we outline the basic method while highlighting any changes from PB20. We express the bandpowers in terms of ${D}_{{\ell }}\equiv {\ell }({\ell }+1){C}_{{\ell }}/(2\pi )$ unless otherwise noted.

Pseudo-${C}_{{\ell }}$ methods are based on measuring the biased power spectrum, or pseudo-${C}_{{\ell }}$, from the fast Fourier transform (FFT) of an apodized map (or a spherical harmonic transform in curved sky), and then correcting these pseudo-${C}_{{\ell }}$'s for the finite sky coverage, beams and filtering to recover the true spectrum on the sky. Cross-spectrum methods iterate on this approach by replacing auto-spectra with cross-spectra between maps with independent noise properties to avoid any noise bias.

The binned pseudo-${C}_{{\ell }}$'s can be written as

$$\begin{eqnarray}&&\begin{array}{l}{\tilde{D}}_{b}^{{EE}}=\displaystyle \frac{1}{{\displaystyle \sum }_{{i}^{{\prime} }\ne {j}^{{\prime} }}{w}_{E,{i}^{{\prime} }}{w}_{E,{j}^{{\prime} }}}\\ \times \ \displaystyle \sum _{i\ne j}{w}_{E,i}{w}_{E,j}\displaystyle \sum _{{\boldsymbol{k}}\in b}\displaystyle \frac{k(k+1)}{2\pi }{\tilde{{\boldsymbol{m}}}}_{{\boldsymbol{k}}}^{E,i}{\tilde{{\boldsymbol{m}}}}_{{\boldsymbol{k}}}^{* E,j}.\end{array}\end{eqnarray} \tag{ 1 }$$

Here w is a weight factor, and the indices i and j specify different bundle maps. The ℓ-bin is denoted by b, the angular wavevector by ${\boldsymbol{k}}$, and the Fourier transform of an apodized bundle map by ${\tilde{{\boldsymbol{m}}}}_{{\boldsymbol{k}}}$.

The true on-sky power spectrum is related to the binned pseudo-${C}_{{\ell }}$'s by

$$\begin{eqnarray}&&{D}_{b}={{\boldsymbol{K}}}_{{bb}^{\prime} }^{-1}{\tilde{D}}_{b^{\prime} }\end{eqnarray} \tag{ 2 }$$

where the matrix ${{\boldsymbol{K}}}_{{bb}^{\prime} }$ is known as the kernel matrix and defined by

$$\begin{eqnarray}&&{{\boldsymbol{K}}}_{{{bb}}^{{\prime} }}=\displaystyle \sum _{{\ell }{{\ell }}^{{\prime} }}{{\boldsymbol{P}}}_{b{\ell }}{{\boldsymbol{M}}}_{{\ell }{{\ell }}^{{\prime} }}{F}_{{{\ell }}^{{\prime} }}{B}_{{{\ell }}^{{\prime} }}^{2}{{\boldsymbol{Q}}}_{{{\ell }}^{{\prime} }{b}^{{\prime} }}.\end{eqnarray} \tag{ 3 }$$

Here, ${\boldsymbol{P}}$ and ${\boldsymbol{Q}}$ are binning and interpolation operators. The mode-coupling matrix ${{\boldsymbol{M}}}_{{\ell }{{\ell }}^{{\prime} }}$ accounts for the finite frequency resolution in the FFT of a finite area of sky. The beam function of the instrument is represented by ${B}_{{{\ell }}^{{\prime} }}$ (see Section 3.4), while the transfer function ${F}_{{{\ell }}^{{\prime} }}$ accounts for the effects of filtering at the TOD and map levels.

We will discuss these factors in more detail in the following subsections.

4.1. Apodization Mask and Mode-coupling Matrix

4.2. Simulations

4.3. Filter Transfer Function and Bandpower Window Functions

The transfer function, F_ℓ, is calculated by comparing the E-mode power spectrum of these simulations to the original input power spectrum as described in Hivon et al. (2002). The transfer function is shown in Figure 1.

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We also report the bandpower window functions necessary to compare the binned spectra to a theory curve. In the pseudo-${C}_{{\ell }}$ formalism, these bandpower window functions, ${{\boldsymbol{w}}}_{b{\ell }}$, can be expressed as

$$\begin{eqnarray}&&{{\boldsymbol{w}}}_{b{\ell }}=\displaystyle \sum _{{b}^{{\prime} }{{\ell }}^{{\prime} }}{{\boldsymbol{K}}}_{{{bb}}^{{\prime} }}^{-1}{{\boldsymbol{P}}}_{{b}^{{\prime} }{{\ell }}^{{\prime} }}{{\boldsymbol{M}}}_{{{\ell }}^{{\prime} }{\ell }}{F}_{{\ell }}{B}_{{\ell }}^{2}.\end{eqnarray} \tag{ 4 }$$

The bandpower window functions are applied to an assumed theory spectrum, ${{\boldsymbol{C}}}_{{\ell }}^{\mathrm{theory}}$, to get the binned expectation bandpowers,

$$\begin{eqnarray}&&{{\boldsymbol{C}}}_{b}^{\mathrm{theory}}={{\boldsymbol{w}}}_{b{\ell }}{{\boldsymbol{C}}}_{{\ell }}^{\mathrm{theory}},\end{eqnarray} \tag{ 5 }$$

for comparison with the measured bandpowers.

We test the stability of the transfer function and bandpower window functions by running smaller numbers of simulations with different input cosmologies, and testing if the average resulting bandpowers for each simulated set match the expected bandpowers for the product of the bandpower window functions with the assumed cosmological model. We find agreement in all tests, validating the power spectrum pipeline.

4.4. Bandpower Covariance

We also need to estimate the uncertainty on the measured bandpowers. The total uncertainties will include sample and noise variance as well as the beam and calibration uncertainties. To allow the simulations to be run before settling on the final absolute calibration, we calculate the sample and noise variance separately before combining the two estimates.

We use the 192 mock-observed noiseless CMB maps from Section 4.2 to estimate the covariance matrix due to sample variance. We use the calibrated, sign-flip noise maps from Section 3.2 to estimate the noise variance, while cross-checking the results with the simulated noise maps. For both the sample and noise variance, we estimate the covariance matrix at an initial binning of ${\rm{\Delta }}{\ell }=50$, and condition this matrix following Henning et al. (2018) to reduce the impact of uncertainties in the covariance estimate. Specifically, we require the correlation matrix to be a symmetric Toeplitz matrix. Given the expected correlation length, we also zero out the correlation for ${\rm{\Delta }}{\ell }\gt 150$. The observed correlation at these ${\rm{\Delta }}{\ell }{\rm{s}}$ is consistent with zero (although the uncertainty is large). We then rebin this estimate of the sample variance into the final bandpower binning.

Beam and calibration uncertainties are dealt with separately. We handle the calibration uncertainty by adding a calibration factor to the cosmological analysis with a prior set by the expected 2% calibration uncertainty. The beam uncertainty is propagated into a beam correlation matrix, ${\rho }_{{bb}^{\prime} }$. At each step in the chain, this beam correlation matrix is combined with the binned theory spectrum D_b and added to the sample and noise covariance matrix to yield the total covariance at that step:

$$\begin{eqnarray}&&{C}_{{bb}^{\prime} }^{\mathrm{tot}}={C}_{{bb}^{\prime} }^{{\rm{S}}+{\rm{N}}}+{\rho }_{{bb}^{\prime} }{D}_{b}{D}_{b^{\prime} }\end{eqnarray} \tag{ 6 }$$

We test the data for unknown systematics using null tests. Each null test splits the data set in approximately half, with the splits chosen to be sensitive to likely sources of systematic bias. The difference between the two halves removes nearly all true sky signal, thus suppressing the sample variance and allowing a more sensitive test for systematics. As will be described in more detail below, the null test suite shows no evidence for systematics in the data.

We have also run a suite of simulations for expected sources of systematic errors (information on the simulation procedure can be found in PB20). At ${\ell }\gt 1050$, the most significant systematics are related to detector cross-talk, pointing, and the half-wave plate; the estimated systematic uncertainty is less than 0.25 the statistical uncertainty in all bins (Polarbear Collaboration 2020, in preparation). Given that the systematic uncertainties are small compared to the statistical uncertainties on the E-mode bandpowers, we choose to neglect the systematic errors in this work.

We run a suite of 19 null tests to search for potential bias in our data set. Our framework has been previously used by the Polarbear Collaboration (2014, 2017, PB20), which is based on the formalism developed originally by the quiet Collaboration (Bischoff 2010). The binned null spectrum, ${\hat{C}}_{b}^{\mathrm{null}}$, is constructed as

$$\begin{eqnarray}&&{\hat{C}}_{b}^{\mathrm{null}}={\hat{C}}_{b}^{A}+{\hat{C}}_{b}^{B}-2{\hat{C}}_{b}^{{AB}},\end{eqnarray} \tag{ 7 }$$

where ${\hat{C}}_{b}^{A/B}$ are the spectra calculated following Section 4 for each half of the data split, and ${\hat{C}}_{b}^{{AB}}$ is the cross-spectrum between the two halves (all after correcting for the appropriate filter transfer functions and mode-coupling matrices). For each null test, the data from each half is rebundled to maximize the overlapping area. The binning in ℓ used in these null tests is the same one used in Table 1.

Table 1.  Bandpowers

Multipole Range	${{\ell }}_{\mathrm{eff}}$	DbEE $(\mu {{\rm{K}}}^{2})$	$\sigma ({D}_{b}^{{EE}})$ $(\mu {{\rm{K}}}^{2})$
$500\leqslant {\ell }\lt 550$	525.4	7.36	0.64
$550\leqslant {\ell }\lt 600$	575.3	11.33	0.87
$600\leqslant {\ell }\lt 650$	625.3	26.96	1.68
$650\leqslant {\ell }\lt 700$	675.2	37.72	2.19
$700\leqslant {\ell }\lt 750$	725.4	36.51	2.13
$750\leqslant {\ell }\lt 800$	775.2	21.98	1.46
$800\leqslant {\ell }\lt 850$	825.3	12.20	1.09
$850\leqslant {\ell }\lt 900$	875.2	19.79	1.45
$900\leqslant {\ell }\lt 950$	925.3	28.79	1.87
$950\leqslant {\ell }\lt 1000$	975.2	40.78	2.43
$1000\leqslant {\ell }\lt 1050$	1025.3	34.95	2.25
$1050\leqslant {\ell }\lt 1100$	1075.2	26.92	2.00
$1100\leqslant {\ell }\lt 1150$	1125.2	16.67	1.72
$1150\leqslant {\ell }\lt 1200$	1175.1	10.89	1.65
$1200\leqslant {\ell }\lt 1250$	1225.2	23.02	2.16
$1250\leqslant {\ell }\lt 1300$	1275.1	31.09	2.55
$1300\leqslant {\ell }\lt 1400$	1350.6	30.71	1.93
$1400\leqslant {\ell }\lt 1500$	1450.6	15.88	1.84
$1500\leqslant {\ell }\lt 1600$	1550.6	15.56	2.11
$1600\leqslant {\ell }\lt 1700$	1650.5	18.98	2.54
$1700\leqslant {\ell }\lt 1800$	1750.5	7.74	2.65
$1800\leqslant {\ell }\lt 2000$	1901.8	11.06	2.40
$2000\leqslant {\ell }\lt 2200$	2101.6	6.62	3.27
$2200\leqslant {\ell }\lt 2500$	2353.2	5.08	4.08
$2500\leqslant {\ell }\lt 3000$	2757.6	3.62	6.47

Note. The angular multipole range, E-mode bandpowers, and uncertainties for the Polarbear survey. The uncertainties are taken from the square root of the diagonal of the covariance matrix, and do not include beam or calibration errors.

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Most of these null tests have been previously described by PB20, but five are added to test specific potential concerns for the E-mode measurement. The new tests include: (1) a second test on Sun contamination, splitting the data by the distance to the Sun; (2) a test splitting the data based on the observed level of temperature-to-polarization leakage in each CES; (3 and 4) two tests of HWP contamination by splitting the data on the level of either the $2f$ or $4f$ line amplitude in each CES; and (5) a random split of the bolometers to test the quality of the noise model. A short description of all 19 tests can be found in Appendix A. We estimate the uncertainty on each null test bin, $\sigma ({\hat{C}}_{b}^{\mathrm{null}})$, by looking at the standard deviation of a suite of 48 simulated null spectra. We then define the statistic

$$\begin{eqnarray}&&{\left({\chi }_{b}^{\mathrm{null}}\right)}^{2}\equiv {\left(\displaystyle \frac{{\hat{C}}_{b}^{\mathrm{null}}}{\sigma ({\hat{C}}_{b}^{\mathrm{null}})}\right)}^{2}.\end{eqnarray} \tag{ 8 }$$

We compare the values of ${({\chi }_{b}^{\mathrm{null}})}^{2}$ from the real data to simulations to calculate the PTE for each test. Summing across all tests and all bins, we find the PTE for the total ${\chi }^{2}$ to be 67.9%. The data thus show no evidence for systematic biases.

We also test that the set of PTEs is consistent with a uniform distribution as expected. Specifically, we perform a Kolmogorov–Smirnov (KS) test on the three sets of PTEs of the ${\chi }_{\mathrm{null}}^{2}$ values by test, by bin, and overall. All three distributions are consistent with uniform distributions (PTE = 0.45, 0.30, 0.13), showing no evidence for a bias.

The E-mode bandpowers measured by applying the analysis method of Section 4 to the Polarbear 670 ${\deg }^{2}$ survey are shown in Figure 2 and tabulated in Table 1. E-mode power is detected at very high significance, with zero E-mode power excluded at 61σ. The Polarbear bandpowers are consistent with the ΛCDM model; the Polarbear data have a PTE of 0.38 relative to the best-fit ΛCDM model for the Polarbear and Planck (Planck Collaboration et al. 2020a) data sets. The Polarbear bandpowers trace out the third through seventh acoustic peaks in the E-mode spectrum, and extend to ${\ell }=3000$ well into the Silk damping tail (Silk 1968).

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We show the current state of E-mode power spectrum measurements in Figure 3. In this figure, we compile the bandpowers of this work with other recent E-mode measurements (Louis et al. 2017; BICEP2 Collaboration et al. 2018; Henning et al. 2018; Planck Collaboration et al. 2020a). The observed E-mode spectra agree well, enhancing our confidence in the E-mode measurements.

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We now turn to the cosmological implications of the Polarbear E-mode power spectrum along with other recent cosmological observations. We look at parameter constraints for the standard, six-parameter, ΛCDM cosmological model. We also look at two one-parameter extensions to ΛCDM, ${N}_{\mathrm{eff}}$, or ${Y}_{\mathrm{He}}$. These extensions are constrained primarily by the Silk damping scale in temperature data. Finally, we consider the two-parameter extension of ${Y}_{\mathrm{He}}$+${N}_{\mathrm{eff}}$, of interest as a test of big bang nucleosynthesis (BBN).

7.1. Methodology

We derive parameter constraints using the 2019 version of the Markov Chain Monte Carlo (MCMC) package CosmoMC (Lewis & Bridle 2002). We have extended CosmoMC to include the Polarbear bandpowers in a manner similar to the public likelihood for Henning et al. (2018). The Polarbear likelihood code and associated data are available on the LAMBDA website.

In addition to the usual cosmological parameters in CosmoMC, we have added four nuisance parameters specific to the Polarbear data, most with an informative prior. The first parameter is the calibration factor (in power) for the Polarbear E-mode power spectrum. We set a prior on this factor based on the expected 2% uncertainty in the absolute calibration. The other three parameters relate to on-sky signals. First, we have one term to describe the polarized Poisson-distributed point source power in the field after masking. This power scales with ℓ as ${D}_{{\ell }}\propto {{\ell }}^{2}$, and we report the power at ${\ell }=3000$, D₃₀₀₀^PS. We use a weakly informative prior on the point source power that is uniform for ${D}_{3000}^{{PS}}\in [0,10]\,$ $\mu {{\rm{K}}}^{2}$. Second, we have one parameter to describe polarized Galactic dust, which we model as having a power spectrum

$$\begin{eqnarray}&&{D}_{{\ell }}^{\mathrm{dust}}={D}_{80}^{\mathrm{dust}}{\left(\displaystyle \frac{{\ell }}{80}\right)}^{{\alpha }_{\mathrm{dust}}}.\end{eqnarray} \tag{ 9 }$$

Given that the Polarbear bandpowers are at much higher angular multipoles, we apply strong priors from BICEP2 Collaboration et al. (2018). Specifically, we fix ${\alpha }_{\mathrm{dust}}=-0.58$ and apply a Gaussian prior that ${D}_{80}^{\mathrm{dust}}$ is drawn from $N(0.0188,{0.0042}^{2})\mu {{\rm{K}}}^{2}$. The results are insensitive to this term; we have run one chain for ΛCDM+${N}_{\mathrm{eff}}$ with the dust power zeroed and have seen no shifts larger than 0.1σ. Finally, we allow for "super-sample lensing" variance (Manzotti et al. 2014). This is parameterized by the mean lensing convergence across the field, κ, to which we apply a Gaussian prior centered at zero with a 1σ width of 0.001.³⁴

7.2. Data Sets

7.3. Constraints on the ΛCDM Model

As has been previously noted, the Planck 2018 CMB power spectrum data alone do an excellent job of constraining all six parameters in the standard ΛCDM model. We report the median parameter values and 68% confidence intervals in Table 2. While the optical depth is relatively uncertain with a 14% error bar, the other five parameters are measured with percent-level precision. Adding the ground-based CMB power spectrum measurements to the set reduces the allowed parameter volume by a factor of 2.0, or roughly a 7% reduction in parameter uncertainties. These small-scale power spectrum measurements do not help recover the optical depth, however, as that parameter constraint depends on the reionization bump at ${\ell }\lt 10$. Adding the CMB lensing, BAO, and ${H}_{0}$ data yields a further reduction in uncertainties of order 10%. Notably, lensing provides another avenue to measure the amplitude of fluctuations, A_s, which partially breaks the ${A}_{s}{e}^{-2\tau }$ degeneracy in the power spectrum alone and helps the determination of A_s and τ individually.

Table 2.  ΛCDM Parameter Constraints

	Planck	CMBselect	CMBext
${{\rm{\Omega }}}_{b}{h}^{2}$	0.02237 ± 0.00015	0.02230 ± 0.00014	0.02234 ± 0.00015
${{\rm{\Omega }}}_{c}{h}^{2}$	0.1200 ± 0.0014	0.1203 ± 0.0013	0.1198 ± 0.0011
$100{\theta }_{{MC}}$	1.04089 ± 0.00032	1.04102 ± 0.00030	1.04104 ± 0.00030
τ	0.0548 ± 0.0079	0.0527 ± 0.0070	0.0518 ± 0.0065
$\mathrm{ln}({10}^{10}{A}_{s})$	3.044 ± 0.016	3.043 ± 0.015	3.040 ± 0.014
ns	0.9650 ± 0.0045	0.9636 ± 0.0042	0.9647 ± 0.0038
H0 (km s−1 Mpc−1)	67.42 ± 0.69	67.20 ± 0.57	67.41 ± 0.51

Note. The median values and 68% confidence intervals for the six ΛCDM parameters for the Planck, CMBselect, and CMBext data sets. Adding the ground-based CMB power spectrum measurements to Planck reduces the parameter uncertainties by ∼7% (except for the amplitude terms A_s and τ which depend on the low $-{\ell }$ polarization bump). Adding the BAO, ${H}_{0}$ and CMB lensing data further reduces uncertainties by an additional ∼10%.

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There has been much discussion recently about the degree of tension between local determinations of the Hubble constant, ${H}_{0}$, and the values inferred from the Planck data (e.g., Riess et al. 2019; Wong et al. 2020). Adding the ground-based CMB E-mode bandpowers to the Planck data supports the current tension on the Hubble constant by reducing the uncertainty by a factor of 1.2 to

$$\begin{eqnarray}&&{H}_{0}=67\mathrm{.}20\pm 0\mathrm{.}57\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}.\end{eqnarray} \tag{ 10 }$$

As a side note, we see a similar level of improvement when adding only the Polarbear bandpowers to Planck (a factor of 1.13 reduction in uncertainty on ${H}_{0}$) or adding only the other ground-based CMB bandpowers (a factor of 1.19). The tension with the local determination by Riess et al. (2019) of ${H}_{0}\,=74.03\pm 1.42\,\mathrm{km}\,{{\rm{s}}}^{-1}\,{\mathrm{Mpc}}^{-1}$ remains essentially unchanged from 4.3 to 4.5σ.

7.4. Constraints on the Primordial Helium Abundance

We also look at the inferred primordial helium fraction, which can be viewed as a test for new particles or physics during the epoch of big bang nucleosynthesis (BBN). The expected helium fraction under BBN consistency in the ΛCDM model for the CMBselect data set is extremely tightly constrained at ${Y}_{\mathrm{He}}=0\mathrm{.}246696\pm 0\mathrm{.}000062$. Relaxing BBN consistency substantially weakens what we can infer about the helium fraction. However, the CMB anisotropies have some sensitivity to the helium fraction as it changes the number of free electrons preset at recombination. Higher helium fractions lead to fewer free electrons, a longer photon mean free path and thus more Silk damping. We show the posterior probability distribution function for the primordial helium fraction for three data sets in Figure 4. Using Planck alone, we find ${Y}_{\mathrm{He}}=0\mathrm{.}239\pm 0\mathrm{.}013$. Adding the other CMB measurements improves this slightly to

$$\begin{eqnarray}&&{Y}_{\mathrm{He}}=0\mathrm{.}248\pm 0\mathrm{.}012,\end{eqnarray} \tag{ 11 }$$

in excellent agreement with the expectation from BBN. There is no further improvement from adding the other cosmological data.

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7.5. Constraints on the Number of Relativistic Species

The energy density of relativistic particles in the early universe is proportional to ${N}_{\mathrm{eff}}$, the effective number of relativistic species. The standard model of particle physics predicts that ${N}_{\mathrm{eff}}=3.046$ for the three neutrino species plus a small correction from positron annihilation (Mangano et al. 2005). The preferred ${N}_{\mathrm{eff}}$ from the CMBselect data set is within $0.4\sigma$ of this prediction

$$\begin{eqnarray*}&&{N}_{\mathrm{eff}}=2\mathrm{.}94\pm 0\mathrm{.}16.\end{eqnarray*}$$

Adding the nonCMB data (i.e., the CMBext data set) slightly reduces the preferred value of ${N}_{\mathrm{eff}}$, but it remains within $0.9\sigma$ of the expectation:

$$\begin{eqnarray*}&&{N}_{\mathrm{eff}}=2\mathrm{.}90\pm 0\mathrm{.}18.\end{eqnarray*}$$

When ${N}_{\mathrm{eff}}$ is allowed to vary, as shown in Figure 5 we see that it correlates strongly with ${{\rm{\Omega }}}_{c}{h}^{2}$ and n_s. As discussed by Hou et al. (2013), increasing the matter density as ${N}_{\mathrm{eff}}$ increases avoids shifting the redshift of matter-radiation equality. A side effect is that the CMB constraint on the Hubble constant significantly weakens: the uncertainty on the Hubble constant nearly triples from ±0.57 to ±1.5 km s⁻¹ Mpc⁻¹ (the central value changes by less than $0.4\sigma$ of the weakened constraint).

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7.6. Constraints on the ΛCDM+ ${Y}_{\mathrm{He}}$ + ${N}_{\mathrm{eff}}$ Model

We now consider the results when allowing both ${N}_{\mathrm{eff}}$ and ${Y}_{\mathrm{He}}$ to vary, as both parameters affect the damping tail. As with the other extensions to ΛCDM considered, freeing these two parameters does not significantly improve the quality of the fit (${\rm{\Delta }}{\chi }^{2}\simeq -1.2$ for two new parameters for the CMBext data set). The resulting parameter posteriors are shown in Figure 6. We find the following for the CMBselect data set:

$$\begin{eqnarray*}\begin{array}{rcl}{N}_{\mathrm{eff}} & = & 2\mathrm{.}70\pm 0\mathrm{.}26,\\ {Y}_{\mathrm{He}} & = & 0\mathrm{.}262\pm 0\mathrm{.}015.\end{array}\end{eqnarray*}$$

The CMBext data set prefers essentially the same values as well:

$$\begin{eqnarray*}\begin{array}{rcl}{N}_{\mathrm{eff}} & = & 2\mathrm{.}65\pm 0\mathrm{.}26,\\ {Y}_{\mathrm{He}} & = & 0\mathrm{.}263\pm 0\mathrm{.}014.\end{array}\end{eqnarray*}$$

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Adding the ground-based CMB measurements of the damping tail to the Planck bandpowers pushes along the ${N}_{\mathrm{eff}}$/${Y}_{\mathrm{He}}$ degeneracy toward higher values of ${Y}_{\mathrm{He}}$, and lower values of ${N}_{\mathrm{eff}}$. However, Figure 6 shows that the $2\sigma$ parameter ellipses still contain the ΛCDM values, and as mentioned above, the quality of the fit does not substantially improve. Table 3 summarizes the median and 68% confidence intervals for the parameters in the ΛCDM case and extensions, for the CMBext data set.

Table 3.  Parameter Constraints for the CMBext Data Set

	ΛCDM	ΛCDM+${N}_{\mathrm{eff}}$	ΛCDM +${Y}_{\mathrm{He}}$	ΛCDM+${Y}_{\mathrm{He}}$+${N}_{\mathrm{eff}}$
${{\rm{\Omega }}}_{b}{h}^{2}$	0.02234 ± 0.00015	0.02220 ± 0.00022	0.02236 ± 0:00019	0.02221 ± 0.00020
${{\rm{\Omega }}}_{c}{h}^{2}$	0.1198 ± 0.0011	0.1177 ± 0.0027	0.1197 ± 0.0011	0.1138 ± 0.0040
$100{\theta }_{{MC}}$	1.04104 ± 0.00030	1.04125 ± 0:00040	1.04104 ± 0.00050	1.0424 ± 0.0011
τ	0.0518 ± 0.0065	0.0511 ± 0:0075	0.0511 ± 0.0074	0.0509 ± 0.0073
$\mathrm{ln}({10}^{10}{A}_{s})$	3.040 ± 0.014	3.031 ± 0.016	3.039 ± 0.015	3.028 ± 0.016
ns	0.9647 ± 0.0038	0.9592 ± 0.0075	0.9648 ± 0.0066	0.9580 ± 0.0078
${N}_{\mathrm{eff}}$	⋯	2.90 ± 0.18	⋯	2.65 ± 0.26
${Y}_{\mathrm{He}}$	⋯	⋯	0.246 ± 0.011	0.263 ± 0.014
H0 (km s−1 Mpc−1)	67.41 ± 0.51	66.5 ± 1.4	67.49 ± 0.59	65.1 ± 1.6

Note. The median parameter values and 68% confidence intervals with the CMBext data set for the ΛCDM model as well as the three extensions to the ΛCDM model considered in this work. In the last row, we also show the constraints on a derived parameter, the Hubble constant, H₀, which has received attention recently due to tensions in the preferred value between different experiments.

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We have presented a measurement of the CMB E-mode power spectrum on angular multipoles $500\leqslant {\ell }\leqslant 3000$ from 670 ${\deg }^{2}$ surveyed with the Polarbear instrument. E-mode polarization is detected at high significance across the third through the seventh acoustic peaks of the E-mode power spectrum. We find no evidence for significant systematic biases in the null suite data. The Polarbear E-mode bandpowers provide an independent confirmation of the observed CMB E-mode power spectrum at intermediate-to-small angular scales.

We combine the Polarbear E-mode bandpowers with other recent CMB measurements (Louis et al. 2017; Henning et al. 2018; Planck Collaboration et al. 2020a) to explore the current state of CMB cosmological constraints. Adding the ground-based CMB bandpowers does not reduce the Hubble constant tension between the Planck-inferred value and direct local measurements (4.3σ vs 4.5σ). We find no significant preference in the data for any of the extensions considered: ${N}_{\mathrm{eff}}$, ${Y}_{\mathrm{He}}$, ${N}_{\mathrm{eff}}$+${Y}_{\mathrm{He}}$.

For the ΛCDM+${Y}_{\mathrm{He}}$ model extension, adding the ground-based CMB power spectrum measurements brings the helium abundance toward the BBN expectation of 0.2467. With Planck-only, the data prefer ${Y}_{\mathrm{He}}=0\mathrm{.}239\pm 0\mathrm{.}013$, shifting to ${Y}_{\mathrm{He}}=0\mathrm{.}248\pm 0\mathrm{.}012$ when the other power spectrum measurements are added. As expected, the nonCMB-power-spectrum data does little for ${Y}_{\mathrm{He}}$.

We also look at varying the effective number of relativistic species, ${N}_{\mathrm{eff}}$. We find only minor improvements and shifts from adding data beyond the Planck bandpowers. For the combined CMBext data set, the data favor ${N}_{\mathrm{eff}}=2\mathrm{.}90\pm 0\mathrm{.}18$, which is within 1σ of the expected value of 3.046.

Finally, we allow both ${Y}_{\mathrm{He}}$ and ${N}_{\mathrm{eff}}$ to vary to study the degeneracies between the two. Here, the full CMB data set slightly pulls ${Y}_{\mathrm{He}}$ upward and ${N}_{\mathrm{eff}}$ downward relative to the Planck constraints and expected values. We find for CMBext, ${N}_{\mathrm{eff}}=2\mathrm{.}65\pm 0\mathrm{.}26$ and ${Y}_{\mathrm{He}}=0\mathrm{.}263\pm 0\mathrm{.}014$. However, the actual improvement in the quality of fit from adding these two parameters is small (${\rm{\Delta }}{\chi }^{2}\simeq -1.2$), suggesting that these shifts are not significant.

While the Polarbear survey has finished, its successor, the Simons Array, had first light in 2019. The complete Simons Array will have three telescopes with a total of about 20 times more detectors than Polarbear and will survey a large fraction of the Southern sky (Suzuki et al. 2016; Hasegawa et al. 2018). The Simons Array will also extend the survey area to the North at the equator, facilitating studies of cross-correlations with experiments at other wavelengths. The E-mode power spectrum measurement from the Simons Array survey will dramatically improve upon current E-mode constraints and enable new tests of cosmology.

The Polarbear project is funded by the National Science Foundation under grants AST-0618398 and AST-1212230. The analysis presented here was also supported by Moore Foundation grant No. 4633, the Simons Foundation grant No. 034079, and the Templeton Foundation grant No. 58724. The James Ax Observatory operates in the Parque Astronómico Atacama in Northern Chile under the auspices of the Comisión Nacional de Investigación Científica y Tecnológica de Chile (CONICYT).

The Melbourne group acknowledges support from the University of Melbourne and an Australian Research Council's Future Fellowship (FT150100074). A.K. acknowledges the support by JSPS Leading Initiative for Excellent Young Researchers (LEADER) and by the JSPS KAKENHI grants No. JP16K21744 and 18H05539. C.B., N.K., and D.P. acknowledge support from the ASI-COSMOS Network (cosmosnet.it) and from the INDARK INFN Initiative (web.infn.it/CSN4/IS/Linea5/InDark). G.F. acknowledges the support of the European Research Council under the European Union's Seventh Framework Programme (FP/2007-2013)/ERC grant Agreement No. [616170] and of the UK STFC grant ST/P000525/1. H.N. acknowledges JSPS KAKENHI grant JP26800125. M.A. acknowledges support from CONICYT UC Berkeley-Chile Seed Grant (CLAS fund) Number 77047, Fondecyt project 1130777 and 1171811, DFI postgraduate scholarship program and DFI Postgraduate Competitive Fund for Support in the Attendance to Scientific Events. M.D. acknowledges funding from the Natural Sciences and Engineering Research Council of Canada and Canadian Institute for Advanced Research. M.H. acknowledges the support from the JSPS KAKENHI grants No. JP26220709 and JP15H05891. N.K. acknowledges the support from JSPS Core-to-Core Program, A. Advanced Research Networks. O.T. acknowledges the SPIRITS grant in the Kyoto University, and JSPS KAKENHI JP26105519. S.T. was supported by Grant-in-Aid for JSPS Research Fellow JP14J01662 and JP18J02133. Y.C. acknowledges the support from the JSPS KAKENHI grants No. 18K13558, 18H04347, and 19H00674. The APC group acknowledges the travel support from the Labex Univearths grant. This work was supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan. This research used resources of the Central Computing System, owned and operated by the Computing Research Center at KEK. Support from the Ax Center for Experimental Cosmology at UC San Diego is gratefully acknowledged. Work at LBNL is supported in part by the U.S. Department of Energy, Office of Science, Office of High Energy Physics, under contract No. DE-AC02-05CH11231. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under contract No. DE-AC02-05CH11231.

As mentioned in Section 5, we used 19 null tests to search for hidden systematics. Of these tests, 14 were previously used in PB20 and are listed here for the readers' convenience. The PTEs for each multipole range and each null test are shown in Table 4.

• 1.  
  "First half versus second half": the data set is split into two equal-weight halves chronologically to probe for time-dependent changes in the instrument, such as drifting calibration.
• 2.  
  "Middle versus rising and setting": the three different CES types are split in middle range elevation scans versus rising plus setting scans to detect, for example, elevation-dependent miscalibration or residual ground synchronous signal.
• 3.  
  "Left-going versus right-going subscans": the data set is split in half according to the direction of motion of the telescope to test for, for example, microphonic or magnetic pickup in the data.
• 4.  
  "High gain versus low gain observations": the data set is split into observations with above and below average mean detector gain coefficients to search for problems with the gain calibration.
• 5.  
  "High PWV versus low PWV": the data set is split by PWV as measured by the nearby apex radiometer to check for loading- or weather-dependent effects.
• 6.  
  "Mean temperature to polarization leakage by channel": split the data set into detectors that see small and large temperature leakage coefficients to test the subtraction and search for residual contamination.
• 7.  
  "$2f$ amplitude by channel", "$4f$ amplitude by channel": split the data by HWP signal amplitude to check for problems removing the HWP structure or systematic contamination coupling into the data through these terms.
• 8.  
  "Q versus U pixels": each detector wafer is fabricated with two sets of polarization angles. We split the data into the two pixel types to check for problems in the device fabrication.
• 9.  
  "Sun above or below the horizon", "Moon above or below the horizon": we split observations based on whether or not the Sun or moon is up to check for residual sidelobe contamination.
• 10.  
  "Top half versus bottom half", "left half versus right half": we split detectors by the boresight axis of the telescope to check for optical distortion and problems due to far sidelobes.
• 11.  
  "Top versus bottom bolometers": with a continuous HWP each bolometer TOD independently measures Q and U. We explicitly separate detector pairs to check for temperature aliasing or device mismatch.

The other five splits are:

• 1.  
  "Mean temperature to polarization leakage by CES": split the data set into CESs that see small and large temperature leakage coefficients to test the subtraction and search for residual contamination.
• 2.  
  "$2f$ amplitude by CES", "$4f$ amplitude by CES": split the data by HWP signal amplitude for CES to check for problems removing the HWP structure or systematic contamination coupling into the data through these terms.
• 3.  
  "Low distance or high distance from Sun": we split observations based on the distance to the Sun to check for residual sidelobe contamination.
• 4.  
  "Random splits of bolometers": we randomly split bolometers into two halves to check the noise model.

Table 4.  Null Test PTE Values

ℓbin Summed Over Null Tests	Total EE ${\chi }^{2}$ PTE	Null Test Summed Over ℓbins	Total EE ${\chi }^{2}$ PTE
$500\leqslant {\ell }\lt 550$	34.7%	First half vs. second half	90.1%
$550\leqslant {\ell }\lt 600$	8.9%	Middle vs. rising and setting	88.0%
$600\leqslant {\ell }\lt 650$	96.8%	Left-going vs. right-going subscans	39.5%
$650\leqslant {\ell }\lt 700$	52.2%	High gain vs. low gain CESs	58.0%
$700\leqslant {\ell }\lt 750$	90.9%	High PWV vs. low PWV	43.6%
$750\leqslant {\ell }\lt 800$	47.1%	Mean temperature leakage by bolometer	11.4%
$800\leqslant {\ell }\lt 850$	82.3%	Mean temperature leakage by CES	84.5%
$850\leqslant {\ell }\lt 900$	96.8%	$2f$ amplitude by bolometer	71.7 %
$900\leqslant {\ell }\lt 950$	39.2%	$4f$ amplitude by bolometer	9.2%
$950\leqslant {\ell }\lt 1000$	25.8%	$2f$ amplitude by CES	47.5 %
$1000\leqslant {\ell }\lt 1050$	90.1%	$4f$ amplitude by CES	24.4%
$1050\leqslant {\ell }\lt 1100$	44.8%	Q vs. U pixels	89.4%
$1100\leqslant {\ell }\lt 1150$	79.6%	Low or high distance from Sun	96.7%
$1150\leqslant {\ell }\lt 1200$	52.5%	Sun above or below the horizon	65.8%
$1200\leqslant {\ell }\lt 1250$	69.3%	Moon above or below the horizon	82.1%
$1250\leqslant {\ell }\lt 1300$	72.0%	Top half vs. bottom half	69.9%
$1300\leqslant {\ell }\lt 1400$	95.4%	Left half vs. right half	88.8%
$1400\leqslant {\ell }\lt 1500$	2.3%	Top vs. bottom bolometers	96.9%
$1500\leqslant {\ell }\lt 1600$	96.5%	Random splits of bolometers	24.8%
$1600\leqslant {\ell }\lt 1700$	18.8%
$1700\leqslant {\ell }\lt 1800$	48.5%
$1800\leqslant {\ell }\lt 2000$	47.6%
$2000\leqslant {\ell }\lt 2200$	87.0%
$2200\leqslant {\ell }\lt 2500$	66.9%
$2500\leqslant {\ell }\lt 3000$	80.2%

Note. The PTE values for the total ${\chi }^{2}$ of the null tests when summed over all tests for a single ℓbin, and and when summed over all bins for a single null test. We see no evidence for a statistically significant excess in the null tests.

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We show parameter constraints for the Planck + Polarbear data alone in Table 5. For the ΛCDM, adding Polarbear data alone reduces the uncertainty of H₀ by 13%, which is substantial compared to the overall reduction by the CMBselect data set. Polarbear slightly improves the constraint of other parameters. For ΛCDM+${N}_{\mathrm{eff}}$, the Planck+Polarbear data set, and CMBext have comparable central values and uncertainties for ${N}_{\mathrm{eff}}$ and H₀. Polarbear does not alter the constraint for the ΛCDM +${Y}_{\mathrm{He}}$ and ΛCDM+${Y}_{\mathrm{He}}$+${N}_{\mathrm{eff}}$ models.

Table 5.  Parameter Constraints for the Planck + Polarbear

	ΛCDM	ΛCDM+${N}_{\mathrm{eff}}$	ΛCDM +${Y}_{\mathrm{He}}$	ΛCDM+${Y}_{\mathrm{He}}$+${N}_{\mathrm{eff}}$
${{\rm{\Omega }}}_{b}{h}^{2}$	0.02236 ± 0.00015	0.02224 ± 0.00022	0.02226 ± 0.00021	0.02222 ± 0.00022
${{\rm{\Omega }}}_{c}{h}^{2}$	0.1201 ± 0.0014	0.1183 ± 0.0030	0.1203 ± 0.0014	${0.1175}_{-0.0051}^{+0.0043}$
$100{\theta }_{{MC}}$	1.04086 ± 0.00031	1.04110 ± 0.00044	1.04057 ± 0.00057	1.0413 ± 0.0012
τ	${0.0547}_{-0.0082}^{+0.0070}$	0.0536 ± 0.0078	0.0539 ± 0.0077	0.0525 ± 0.0078
$\mathrm{ln}({10}^{10}{A}_{s})$	${3.045}_{-0.017}^{+0.015}$	3.038 ± 0.019	3.042 ± 0.016	3.034 ± 0.019
ns	0.9648 ± 0.0043	0.9596 ± 0.0087	0.9609 ± 0.0074	0.9583 ± 0.0089
${N}_{\mathrm{eff}}$	⋯	2.91 ± 0.19	⋯	${2.86}_{-0.33}^{+0.28}$
${Y}_{\mathrm{He}}$	⋯	⋯	0.237 ± 0.013	0.246 ± 0.018
H0 (km s−1 Mpc−1)	67.30 ± 0.60	66.4 ± 1.4	67.05 ± 0.71	${66.0}_{-1.9}^{+1.7}$

Note. The median parameter values and 68% confidence intervals with the Planck+Polarbear data set for the ΛCDM model as well as the three extensions to the ΛCDM model considered in this work. In the last row, we also show the constraints on a derived parameter, the Hubble constant, H₀. For the ΛCDM, adding Polarbear data alone reduces the uncertainty of H₀ by 13% and improves the constraint of other parameters slightly. For ΛCDM+${N}_{\mathrm{eff}}$, Polarbear improves the constraint of ${N}_{\mathrm{eff}}$ and H₀ in similar orders as with CMBext. Adding Polarbear alone does not alter the constraint for the ΛCDM +${Y}_{\mathrm{He}}$ and ΛCDM+${Y}_{\mathrm{He}}$+${N}_{\mathrm{eff}}$ models.

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