A joint analysis of Planck and BICEP2 B modes including dust polarization uncertainty

TL;DR

This paper tackles the challenge of constraining primordial gravitational waves via B-mode polarization in the presence of polarized dust foregrounds. It jointly analyzes Planck and BICEP2 data under a three-component model (lensing, gravity waves, and dust) with a conservative dust power-spectrum prior $\\Delta_{BB,{ m dust},l}^2 \\propto l^{-0.42}$ and a flexible dust amplitude, using importance sampling to propagate dust uncertainties into the tensor-to-scalar ratio $r$. The main findings are upper limits $r<0.11$ (95% c.l.) without the Planck 353 GHz dust constraint and $r<0.09$ when that Planck constraint is included, disfavoring large-$r$ inflation models (e.g., $r>0.14$ at ~99.5% c.l.). The analysis also shows that current multi-frequency data do not decisively discriminate dust from no-dust scenarios due to sampling variance, and it highlights the crucial need for accurate dust polarization maps processed through the BICEP2 pipeline to sharpen future constraints or enable a potential detection around $r\,\sim\,0.1$.

Abstract

We analyze BICEP2 and Planck data using a model that includes CMB lensing, gravity waves, and polarized dust. Planck dust polarization maps have highlighted the difficulty of estimating the dust polarization in low intensity regions, suggesting that the polarization fractions have considerable uncertainties and may be significantly higher than previous predictions. In this paper, we start by assuming nothing about the dust polarization except for the power spectrum shape, which we take to be $C_{l}^{BB} \propto l^{-2.42}$. The resulting joint BICEP2+Planck analysis favors solutions without gravity waves, and the upper limit on the tensor-to-scalar ratio is $r<0.11$, a slight improvement relative to the Planck analysis alone which gives $r<0.13$ (95% c.l.). The estimated amplitude of the dust polarization power spectrum agrees with expectations for this field based on both HI column density and Planck polarization measurements at 353 GHz in the BICEP2 field. Including the latter constraint in our analysis improves the limit further to $r < 0.09$, placing strong constraints on inflation (e.g., models with $r>0.14$ are excluded with 99.5% confidence). We address the cross-correlation analysis of BICEP2 at 150 GHz with BICEP1 at 100 GHz as a test of foreground contamination. We find that the null hypothesis of dust and lensing with $r=0$ gives $Δχ^2<2$ relative to the hypothesis of no dust, so the frequency analysis does not strongly favor either model over the other. We also discuss how more accurate dust polarization maps may improve our constraints. If the dust polarization is measured perfectly, the limit can reach $r<0.05$, but this degrades quickly to almost no improvement if the dust calibration error is 20% or larger or if the dust maps are not processed through the BICEP2 pipeline, inducing sampling variance noise. (Abridged.)

Figures (4)

• Figure 1: Left: Joint constraints (68% and 95% c.l.) on $r$ and the amplitude of the dust polarization spectrum at $l=100$ from Planck+WP+BICEP2, assuming a flat prior on the dust amplitude (blue contours) or including the constraint on the dust polarization power in the BICEP2 field estimated by extrapolating Planck data at 353 GHz to 150 GHz (red bands and purple contours). Right: Constraints from the same combinations of data in the $r$--$n_s$ plane (blue and purple contours), compared with constraints from Planck+WP alone (yellow contours). The thick solid line shows the relation between $n_s$ and $r$ predicted by inflation models with $\phi^2$ potentials and the number of $e$-folds varying from 50 to 65; the dotted line shows the same relation for linear potentials.
• Figure 2: Marginalized constraints on $r$ from Planck+WP (dashed curve), Planck+WP+BICEP2 with free dust polarization amplitude or including the constraint from Planck 353 GHz data (solid and dotted curves), and the BICEP2 likelihood alone (dot-dashed curve).
• Figure 3: $B$ mode spectrum predictions compared with BICEP2 data (black points with error bars, showing measurement uncertainty only). Each model curve shows the expected signal in the 9 BICEP2 bandpowers. The dashed curve is the sum of a gravity wave component with $r=0.2$ and the lensing spectrum (dotted curve); the solid curve assumes $r=0$ and adds to the lensing spectrum a dust polarization spectrum $\Delta_{BB,{\rm dust},l}^2 = (0.01\,\mu{\rm K}^2)\, (l/100)^{-0.42}$. Error bars on the dust model spectrum indicate approximate sampling variance uncertainties; although not shown here, sampling variance on the $r=0.2$ model is comparable in magnitude.
• Figure 4: Forecasts for marginalized constraints on $r$ from Planck+WP+BICEP2 assuming lensing plus a gravity wave component with either $r=0$ (left) or $r=0.1$ (right), with the remainder of the signal measured by BICEP2 coming from polarized dust. For the black curves, this polarized dust contribution is assumed to be known within the BICEP2 field perfectly (solid curve), with 10% accuracy (dot-dashed), or with 20% accuracy (dotted). The solid blue curve shows the marginalized-dust constraint from Figure \ref{['fig_1d']} again for comparison.