Line-of-sight extrapolation noise in dust polarization

The B-modes of polarization at frequencies ranging from 50–1000 GHz are produced by Galactic dust, lensing of primordial E-modes in the cosmic microwave background (CMB) by intervening large scale structure, and possibly by primordial B-modes in the CMB imprinted by gravitational waves produced during inflation. The conventional method used to separate the dust component of the signal is to assume that the signal at high frequencies (e.g. 350 GHz) is due solely to dust and then extrapolate the signal down to a lower frequency (e.g. 150 GHz) using the measured scaling of the polarized dust signal amplitude with frequency. For typical Galactic thermal dust temperatures of $\sim 20\mathrm{K}$, these frequencies are not fully in the Rayleigh-Jeans limit. Therefore, deviations in the dust cloud temperatures from cloud to cloud will lead to different scaling factors for clouds of different temperatures. Hence, when multiple clouds of different temperatures and polarization angles contribute to the integrated line-of-sight polarization signal, the relative contribution of individual clouds to the integrated signal can change between frequencies. This can cause the integrated signal to be decorrelated in both amplitude and direction when extrapolating in frequency. Here we carry out a Monte Carlo analysis on the impact of this line-of-sight extrapolation noise on a greybody dust model consistent with Planck and Pan-STARRS observations, enabling us to quantify its effect. Using results from the Planck experiment, we find that this effect is small, more than an order of magnitude smaller than the current uncertainties. However, line-of-sight extrapolation noise may be a significant source of uncertainty in future low-noise primordial B-mode experiments. Scaling from Planck results, we find that accounting for this uncertainty becomes potentially important when experiments are sensitive to primordial B-mode signals with amplitude $r\lesssim 0.0015$ in the greybody dust models considered in this paper.

Figure 1

Ratio of the blackbody spectral radiance (as a function of frequency) to the blackbody spectral radiance at 350 GHz for three different dust temperatures. The Rayleigh-Jeans law is plotted for comparison. In the limit of a large temperature ($T\gg 30\mathrm{K}$), the plotted ratio is independent of temperature, approaching the Rayleigh-Jeans scaling of ${\nu }^2$. But at typical diffuse dust cloud temperatures of $\sim 20\mathrm{K}$, the scaling of the blackbody spectral radiance function with frequency has a significant temperature dependence. For example, note the difference between this ratio at $\nu =150\mathrm{GHz}$ when the temperature is 10 K as opposed to 30 K.

Figure 2

Histogram of the quantity ${\log }_{10}(N_{d,i}/N_{d,0})$ from Pan-STARRS 1 and 2MASS photometry data, where $N_{d,0}$ is the median cloud column density defined for each line of sight. The best fit Gaussian, with a mean of 0 and standard deviation of 0.42, is overplotted in blue. Dashed vertical lines indicate values of ${\log }_{10}(N_{d,i}/N_{d,0})$ for the three-cloud model by Vergely et al. [25]. Further discussion on the data analysis is provided in Sec. 3a1.

Figure 3

Top: Integrated reddening map at 1 kpc from Pan-STARRS 1 photometry. The black contour line indicates the 30° region used in our analysis. Bottom: Reddening as a function of distance for 100 lines-of-sight near the North Galactic Pole out to 1 kpc. The white contour line in (a) indicates where these sight lines are located.

Figure 4

Cartesian projections of stimulated maps of the Stokes Q parameter made using fiducial model 2. Maps are centered on the North Galactic Pole (see Sec. 3a2 for details). Left: map of true Q Stokes parameters at 150 GHz. Center: map of predicted Stokes Q parameters at 150 GHz, obtained from extrapolating Stokes Q map at 350 GHz down to 150 GHz using Eq. (5). Both maps are unnormalized. Right: map of the ratio of the true Stokes Q parameters to the estimated Stokes Q parameter at 150 GHz.

Figure 5

Full projected and marginal distributions of line-of-sight extrapolation noise quantities. In the marginal distribution plots, the median (50th percentile) value and 68th percentile limits are plotted as dashed lines, whose values are stated above each plot. The different 2D projections of the Monte Carlo samples are also directly plotted, with denser regions binned. The contour lines in each 2D projection correspond to the 0.5, 1, 1.5 and $2\sigma$ confidence intervals (the $0.5\sigma$ line is obscured in some of the plots.) There appears to be a slight correlation between $p_{\text{True}}/p_{\text{predicted}}$ and $Q_{\text{True}}/Q_{\text{predicted}}$. This is expected, since $p$ has dependencies on the Stokes $Q$ parameter [Eq. (11)]. We do not observe a correlation between ${\alpha }_{\text{True}}-{\alpha }_{\text{predicted}}$ and any of the other observables, however.

Figure 6

A comparison of the marginal distributions of line-of-sight extrapolation noise observables from the two fiducial models, with corresponding input parameters specified in Table 1. As in Fig. 5, the dashed lines corresponds to the 68th percentile confidence intervals. Both models produce very similar distributions, with model 2 producing samples with slightly wider confidence intervals than model 1. The exact values of the confidence intervals are given in both Table 2 and Fig. 5.

Figure 7

The line-of-sight extrapolation noise $\mathrm{\Delta }\chi =({\chi }_{84\%}-{\chi }_{16\%})/2$, for the three line-of-sight extrapolation noise parameters. The left column corresponds to model 1 for various values of the mean number of cloud, while the right column corresponds to model 2 at various cumulative distance bins. Stars indicate values for the fiducial models we describe in Sec. 4a.

Figure 8

Extrapolation noise as a function of various input distributions using model 1, with stars indicating fiducial values. The line-of-sight extrapolation noise has no dependence on the mean values of ${\log }_{10}(N_{d,i}/N_{d,0})$ or $\alpha$ in the distribution, and so those dependencies are not plotted. In the rightmost column, we investigate the possible effect of correlations in the polarization angles of clouds along a line-of-sight on the extrapolation noise by changing the polarization angle distribution from the fiducial choice of a uniform random distribution to a Gaussian distribution around an arbitrary mean angle, and plot the extrapolation noise as a function of the standard deviation of the polarization angle distribution.

Figure 9

A comparison of the marginal distributions of line-of-sight extrapolation noise observables from (1) the fiducial model 2 (green), where every cloud draws a temperature from the same temperature distribution, and (2) a model in which the mean temperature of the clouds decrease with distance from the Galactic disk (yellow). Dashed lines indicate 68% confidence intervals for each model. The model used to produce the latter is described in Sec. 4c2.

Figure 10

Unnormalized marginal distribution of line-of-sight extrapolation noise parameters at various $N_{\mathrm{side}}$ angular resolution using model 2 and fiducial distributions (Table 1). Dashed lines of the same color indicate 68% confidence intervals for each of the four distributions. For each $N_{\mathrm{side}}$ value, the dust reddening map was downsampled from a native resolution near the Galactic Pole of $N_{\mathrm{side}}=256$ down to its target resolution by averaging the reddening of all the native resolution pixels in each target resolution pixel. Other input parameters were drawn from the same fiducial distribution.

Figure 11

Normalized histograms of the three extrapolation error parameters for various degrees of smoothing at angular resolutions of 3.4’ ($N_{\mathrm{side}}=1024$) and when degraded to a pixel angular resolution of 13.7’ ($N_{\mathrm{side}}=128$), the angular resolution of the pixel used in our fiducial analysis as well as the Planck analysis [15]. Vertical dashed lines indicate 68% confidence intervals of histograms of the same color.