No evidence for dust $B$-mode decorrelation in Planck data

Constraints on inflationary $B$ modes using cosmic microwave background polarization data commonly rely on either template cleaning or cross-spectra between maps at different frequencies to disentangle Galactic foregrounds from the cosmological signal. Assumptions about how the foregrounds scale with frequency are therefore crucial to interpreting the data. Recent results from the Planck satellite collaboration claim significant evidence for a decorrelation in the polarization signal of the spatial pattern of Galactic dust between 217 and 353 GHz. Such a decorrelation would suppress power in the cross-spectrum between high-frequency maps, where the dust is strong, and lower-frequency maps, where the sensitivity to cosmological $B$ modes is strongest. Alternatively, it would leave residuals in lower-frequency maps cleaned with a template derived from the higher-frequency maps. If not accounted for, both situations would result in an underestimate of the dust contribution and thus an upward bias on measurements of the tensor-to-scalar ratio, $r$. In this paper, we revisit this measurement and find that the no-decorrelation hypothesis cannot be excluded with the Planck data. There are three main reasons for this: (i) There is significant noise bias in cross-spectra between Planck data splits that needs to be accounted for. (ii) There is strong evidence for unknown instrumental systematics, the amplitude of which we estimate using alternative Planck data splits. (iii) There are significant correlations between measurements in different sky patches that need to be taken into account when assessing the statistical significance. Between $\ell =55–90$ and over 72% of the sky, the dust $BB$ correlation between 217 and 353 GHz is ${1.001}_{-.004/.000}^{+.004/.021}$ ($68\%\mathrm{stat}/\mathrm{syst}$.) and shows no significant trend with the sky fraction.

Figure 1

Half-mission cross-spectra on the PIPL LR regions in bins of $\mathrm{\Delta }\ell =10$. The points are the real data. Error bars are the standard deviation of the $\text{signal}+\text{noise}$ simulations. Solid lines are the mean of the simulations. Vertical dashed lines indicate the bin edges used in PIPL. Band powers for $\ell =20–50$ are plotted for completeness but are not used in PIPL.

Figure 2

Half-mission cross-spectra of the FFP8 Monte Carlo noise simulations in the LR63 region. The binning is the same as in Fig. 1. The solid line is the mean over FFP8 realizations, and the shaded region is the standard error, computed as the standard deviation over realizations divided by the square root of the number of realizations.

Figure 3

Correlation ratios ${\mathcal{R}}_{\ell }^{EE}$ (top panel) and ${\mathcal{R}}_{\ell }^{BB}$ (bottom panel) in the LR63 region. Correlation ratios calculated from the HM and DS splits are shown as the red diamonds and blue squares, respectively. The red and blue dots are the correlation ratios computed from the same spectra but without debiasing the expected noise correlation. The red x’s in the bottom panel are ${\mathcal{R}}_{\ell }^{BB}$ reported in the Appendix of PIPL and are directly comparable to the red dots. The dashed black line is the expected correlation ratio given the relative amplitudes of dust and CMB in the LR63 region. The model expectation values are shown as gray horizontal lines indicating the bin. The error bars are computed as the median absolute deviation of the $\text{signal}+\text{noise}$ simulations.

Figure 4

Noise debiased ${\mathcal{R}}_{\ell }^{BB}$ on all LR regions in bins of width $\mathrm{\Delta }\ell =10$. The HM and DS splits are shown as red and blue points, respectively. The error bars are computed as the standard deviation of the $\text{signal}+\text{noise}$ simulations. The dashed vertical gray lines indicate the PIPL bin edges. The dotted line is the model expectation value.

Figure 5

Noise debiased ${\mathcal{R}}_{\ell }^{BB}$ on the LR63 region. The gray points are the same HM and DS data points shown in Fig. 4. The red diamonds and blue squares use an alternative binning of $\mathrm{\Delta }\ell =35$ beginning from $\ell =20$. Error bars are the standard deviation of the $\text{signal}+\text{noise}$ sims. The dotted black line shows the ${\mathcal{R}}_{\ell }^{BB}$ model expectation for no dust decorrelation. The dashed gray/black lines show the decorrelation produced by PySM dust model $1/2$. PySM dust model 1 is indistinguishable from the no-decorrelation model.

Figure 6

Top panel: Model dust ${\mathcal{C}}_{\ell }^{BB}$ for 353 GHz and 217 GHz (dashed lines) in LR72, and excess power in the HM map difference null test relative to the FFP8 noise simulations (solid lines). Bottom panel: Predicted ${\mathcal{R}}_{\ell }^{BB}$ in LR72 with no decorrelation and no systematics (dotted black line) and in the presence of systematics given by the solid lines in the top panel (solid black line). The data points are the measured ${\mathcal{R}}_{\ell }^{BB}$ computed from the half-ring split for each detector set separately.

Figure 7

Expectation for ${\mathcal{R}}_{\ell }^{BB}$ in the presence of a correlated systematic of the same amplitude as the uncorrelated systematic shown in Fig. 6. The expectation without systematics is given by the corresponding dotted lines.

Figure 8

Left column: ${\mathcal{D}}_{\ell }^{BB}$ (top) and ${{\mathcal{R}}_{\ell }}^{BB}$ (bottom) on LR72 measured in bins of $\mathrm{\Delta }\ell =3$ using two independent cross-spectra of the four ${\mathrm{HM}}_i{\mathrm{DS}}_j$ data splits. The measurements from the independent cross-spectra are shown as x’s and squares, respectively. The error bars are computed as the standard deviation of 100 corresponding $\text{signal}+\text{noise}$ simulations with the signal realization held fixed. Right column: the differences, $\mathrm{\Delta }{\mathcal{D}}_{\ell }^{BB}$ (top three panels) and $\mathrm{\Delta }{\mathcal{R}}_{\ell }^{BB}$ (bottom panel), between the independently measured data points in the left column, which should be consistent with zero. The error bars are the standard deviation of the corresponding differences between simulation realizations.

Figure 9

Root-mean-square (rms) of the null test differences $\mathrm{\Delta }{\mathcal{D}}_{\ell }^{BB}$ and $\mathrm{\Delta }{\mathcal{R}}_{\ell }^{BB}$ shown in the right column of Fig. 8. The left three columns show $\mathrm{\Delta }{\mathcal{D}}_{\ell }^{BB}$ in $\mu {\mathrm{K}}^2$ for $353\times 353$, $217\times 353$, and $217\times 217$. The right column shows $\mathrm{\Delta }{\mathcal{R}}_{\ell }^{BB}$. Each row shows the rms computed in the indicated broad multipole bin. The $x$ axis of each panel indicates the fine binning of the raw spectra from which the rms values are calculated in the broad bins. The binning ranges from $\mathrm{\Delta }\ell =1$ to the full bin width. The dark and light gray regions enclose 68% and 95% of the corresponding $\text{signal}+\text{noise}$ simulations.

Figure 10

${\mathcal{R}}_{\ell }^{BB}$ in the 9 LR regions, plotted as a function of neutral hydrogen column density, $N_{\mathrm{HI}}$. Each panel is a different multipole bin. The red diamonds and blue squares show ${\mathcal{R}}_{\ell }^{BB}$ calculated from the HM and DS splits, respectively. The corresponding red and blue dots show ${\mathcal{R}}_{\ell }^{BB}$ without accounting for noise bias. The vertical bars indicate the regions enclosing 68% and 95% of the $\text{signal}+\text{noise}$ simulations (16–84 and 2.5–97.5 percentiles, respectively), and the dark gray horizontal lines show the median of the simulations. The red x’s in the bottom two panels show the corresponding PIPL points, taken from the Appendix, which should be the same as the “HM (no debias)” points. In each panel, there are four statistics listed: the $\chi$ and ${\chi }^2$ PTE for the HM and DS splits, calculated as the number of simulations having $\chi$ (${\chi }^2$) less (greater) than the observed value. A low/high $\chi$ PTE indicates data that are coherently low/high. A low/high ${\chi }^2$ PTE indicates data that have too much/little scatter. The PTEs do not account for systematic uncertainty. The cyan shaded region indicates the region between the pessimistic and optimistic estimates of the systematic bias on ${\mathcal{R}}_{\ell }^{BB}$, computed as the expectation for ${\mathcal{R}}_{\ell }^{BB}$ in the presence of a systematic upward bias on ${\mathcal{D}}_{\ell }^{BB}$ given by Eq. (6) and the $\mathrm{\Delta }\ell =3$ (pessimistic) and $\mathrm{\Delta }\ell =\text{full}\mathrm{width}/2$ (optimistic) data in Fig. 9.

Figure 11

${\mathcal{R}}_{\ell }^{BB}$ in the 9 LR regions plotted as a function of neutral hydrogen column density, $N_{\mathrm{HI}}$. The left and right columns are the HM and DS splits, respectively. The top and bottom rows are the $\ell =50–160$ bin and $\ell =55–90$ bin, respectively. The thick line shows the real data and is the same as the noise debiased points in the corresponding panels of Fig. 10. The thin lines show the first 50 $\text{signal}+\text{noise}$ realizations. The only component held fixed between LR regions in a single simulation realization is the $QU$ covariance generated noise map.

Figure 12

${\mathcal{R}}_{\ell }^{BB}$ correlation coefficient between multipole bins and LR regions.

Figure 13

Number of zero crossings of $[{\mathcal{R}}_{50-160}^{BB}(N_{\mathrm{HI}})-1]$ calculated from HM and DS splits (red and blue) and shown in the top left panel of Fig. 10. The vertical lines are the observed values, and the histograms are the distribution from the $\text{signal}+\text{noise}$ simulations. The dotted histogram is the expectation for nine uncorrelated random numbers distributed about 1 computed from the binomial distribution.