Smallfield $$r$$-equivalent Maps

— Kenny Lau

Introduction

The posting continues Clem, Walt and Sergi's work on small sky patch analysis. Questions I want to answer here are, how strong is the dust foreground generated by PySM in the context of $$r$$? Does it match the reality? In order to get a clear conclusion, I made $$r$$-equivalent maps from PySM and Planck 353GHz full-sky maps, and put them together for a comparison.

PySM Simulation and Planck Data

PySM (arXiv:1608.02841) is a publicly available numerical code used for the simulation of Galactic emission in intensity and polarization at microwave frequencies - Clem has a CMB-S4 posting checking dust decorrelation in these models. In this analysis there are 3 PySM simulation maps involved.

• d1_150: 150GHz simulation map generated by PySM d1 model (as in arXiv:1608.02841) with tophat bandpass.
• d4_150: 150GHz simulation map generated by PySM d4 model (as in arXiv:1608.02841) with tophat bandpass.
• d7_150: 150GHz simulation map generated by PySM d7 model (Hensley/Draine model) with tophat bandpass.

On the data side there are 5 Planck maps available.

• Pl353 Det Set Split: Planck 353GHz detector-set maps (nside = 2048). The detectors at this frequency are divided into two groups.
• Pl353 Year Split: Planck 353GHz yearly maps (nside = 2048). The data from odd/even years (year1+3/year2+4) of observations are used for the two maps.
• Pl353 Half-ring Split: Planck 353GHz half-ring maps (nside = 2048). The data from each pointing period is divided in halves.
• Pl353 Half-mission Split: Planck 353GHz half-mission maps (nside = 2048). The data from the first/second half-mission are used for the two maps.
• PIP XXX: a redigitalized $$r$$-equivalent map directly passed from figure 8 of PIP XXX paper (arXiv:1409.5738). According to the description of PIP XXX it was derived from Pl353 det set split maps.

Computation

This analysis is computed by numpy, scipy and healpy - I made Python code on top of Walt's groundwork. Here is the recipe (same as in PIP XXX) of how I compute an $$r$$-equivalent map from a full-sky input map.

• Work in Galactic coordinates ($$l$$, $$b$$).
• Downgrade the input map to nside = 512 whenever it is larger than that.
• Define the resolution of the output map by a value of nside. In this analysis nside = 8 is used as in PIP XXX.
• On a single pixel of the output map
• Create a mask centering on this pixel - the mask chosen is an 11.3° radius circular disk (i.e. 400 sq. deg., ~1% of sky) tapered with a FWHM=2° Gaussian as in PIP XXX. It yields $$f_{\text{sky}}^{\text{eff}} = 0.0080$$ as stated in PIP XXX.
• Subtract mean T/Q/U (of the area covered by the mask) from the input map. Then apply the mask to it.
• Run anafast on this masked input map to get $${{C_l}^{\prime}}^{TT,EE,BB}$$
• Planck data maps: for each input map, as it composes map A and map B coming from the split of data, there are three ways to make $${C_l}^{XX}$$: $$C_l^{AA}, C_l^{BB}, C_l^{AB}$$. In this analysis $$C_l^{AB}$$ is chosen to remove noise bias appearing in auto spectrum.
• PySM input maps: as simulation input maps should have no noise, proceed to use the auto spectrum.
• Multiply factor $$l(l+1)/2\pi$$ to $$C_l^{\prime}$$ to get $$D_l^{\prime}$$.

(Note: an old version of this recipe stated that $$C_l^{\prime}$$ is first binned before it is transformed to $$D_l^{\prime}$$. It was incorrect. The figures shown in this posting have always been produced by the results coming from binning of $$D_l^{\prime}$$.)
• $${D_l}^{\prime}$$ is binned using top-hat binning on the intervals defined between $$l = 40,70,110,160,220,290,370$$. This also yields bin centers $$l_c = 55, 90, 135, 190, 255, 330$$.
• $$D_l^\prime$$ is corrected for partial sky patch size (i.e. divided by the square of the integral of the apodization mask.) in order to scale it back to $$D_l$$. No E/B purification.
• Removal of $$\Lambda\text{CDM}$$ contribution
• Planck data maps: $$D_l^{XX} \rightarrow D_l^{XX} - {D_l}_{\Lambda\text{CDM}}^{XX}$$, where $$XX = TT, EE$$. As in PIP XXX only T and E are removed - the argument is that CMB B-mode power is negligible with respect to the dust polarization at 353GHz even at $$l=500$$.
• PySM input maps: it seems like the input maps contain no CMB contribution at all, so this step is skipped.
• Error in $$D_l$$
• Planck data maps: calculate $$\Delta D_l$$ by Xpol formula. The details are well-documented in equation (29) - (32) of Tristram et al., arXiv:astro-ph/0405575. Sample variacne is quadratically subtracted from it to make $$\Delta D_l$$ entirely due to instrument noise.
• PySM input maps: simulation has no noise, so there is no $$D_l$$ error.
• Fit $$(l_c, {D_{l_c}}^{BB}, \Delta {D_{l_c}}^{BB})$$ to the model function $${D_l}^{BB} = A^{BB}\big(\frac{l}{80}\big)^{\alpha^{BB}+2}$$ where $$\alpha^{BB} = -2.42$$ to get amplitude $$A^{BB}_\text{fit}$$ and the corresponding $$\Delta A^{BB}_\text{fit}$$.
• Frequency scaling
• Planck data maps: scale $$A^{BB}_\text{fit}$$ obtained from 353GHz to 150GHz by using factor $$0.0395^2$$ as in PIP XXX. $$\Delta A^{BB}_\text{fit}$$ is scaled as well with uncertainty in the scaling factor added in quadrature.
• PySM input maps: the input maps are already at 150GHz - no scaling is needed.
• As PIP XXX states $$A^{BB} = 6.71 \times 10^{-2} \mu\text{K}^{2}_{\text{CMB}}$$ when $$r=1$$ and $$l=80$$ (at 150GHz), and as this amplitude increases linearly with $$r$$, calculate the equivalent $$r_d$$ by $$r_d = A^{BB}_\text{fit} \, / \, (6.71 \times 10^{-2} \mu\text{K}^{2}_{\text{CMB}})$$. So this is the $$r_d$$ value of a single pixel. A corresponding $$\sigma(r_d)$$ can be derived as well from $$\Delta A^{BB}_\text{fit}$$.
• Loop over all pixels to get an equivalent $$r_d$$ and $$\sigma(r_d)$$ value on all of them.

Comparison

Figure 4.1 presents the smallfield $$|r|$$-equivalent maps in Galactic coordinates and the perspective of orthographic projection. I cut the highly saturated rim of each map to show $$|b| > 35^\circ$$ only. The graticule used in these plots is defined by $$(\Delta l, \Delta b) = (45^\circ, 20^\circ)$$. Patches with negative $$r$$ are marked by white crosses while the center of BICEP field ($$l=316.1^\circ,b=-58.3^\circ$$) is marked by a circular dot with color showing the magnitude coming from the measurement of BK14 - $$A_\text{dust@353GHz} = 4.300^{+1.200}_{-1.100} \mu\text{K}^2$$ or $$r_d = 0.238^{+ 0.066}_{-0.061}$$.

Note: The conversion is done by exploiting the modified blackbody law with $$\beta_d = 1.59$$ and $$T_d = 19.6\text{K}$$. We can scale $$A_\text{dust@353GHz}$$ to $$A_\text{dust@150GHz}$$ by multiplying the factor $$((150\text{GHz})^{\beta_d}B(\nu = 150\text{GHz}, T = T_d) \big/ (353\text{GHz})^{\beta_d}B(\nu = 353\text{GHz}, T = T_d))^2 = (0.0609)^2$$.

It is clear that d4_150 is an overestimate. d1_150 and d7_150 is closer to the reality but they still have higher $$r$$. Moreover PySM models in general make the sky look more even - they cannot reproduce the spatial variation.

(Note: Pl353 Det Set Split is supposed to be the same as PIP XXX but they don't match. Walt has found this before and I again get the same conclusion.)

Updated on 20180503: I reran the analysis with the mask area increased to 1000 sq. deg. (~2.4% of sky). Now there is an extra option in all figures showing r/σ(r) from a larger area. Also the circular dot now has a color reflecting $$A_\text{dust}$$ value from BK14.

• All negative r values disappear; 4 data set split r/σ(r) maps basically converge - det set split still shows a smaller $$r$$ but they in general agree with each other. This is due to the increase in S/N ratio. Moreover adjacent patches become more correlated as common area is larger.
• Both PySM and Planck maps yield higher r. However the conclusion that PySM models overestimate $$r$$ still holds.
• In the 400 sq. deg. case, $$A_\text{dust}$$ from BK's measurement is basically the same as Planck's estimate; in the 1000 sq. deg. case, it is a bit smaller. $$|r|$$ and $$\sigma(r)$$ maps from PySM simulations and Planck data. The northern (southern) Galactic hemishpere is shown on the left (right).

In figure 4.2 I make histograms comparing Planck data vs. PySM d1/d4/d7 models. As before I drop all data in $$|b| \leq 35^\circ$$, since the emission directly coming from Galactic plane is out of our intereset. One can see Planck data cluster around smaller $$r$$. Histogram of Planck data vs. PySM simulations. Only patches with $$|b| > 35^\circ$$ are included in this statistics.

I also make the following graphs in order to show how $$r_d$$ varies over Galactic latitude $$b$$. In figure 4.3, "r_d vs. b" presents every $$r_d$$ value at a particular $$b$$, while "median(r_d) vs. b" takes median over Galactic longitude $$l$$. The vertical dashed black line indicates $$b$$ of BICEP field center and the black dot stands for BK14 measurement. Statistics of Planck data vs. PySM simulations. Note that there are no PIP XXX data in the range $$|b| \leq 35^\circ$$.

Products

Here are the nside = 8 HEALPix $$r$$-equivalent maps. In each FITS file it contains 4 columns: $$\big( r_d, \sigma(r_d), r_E, \sigma(r_E) \big)$$. The latter two columns contain $$r$$ and $$\sigma(r)$$ derived from E-modes - under the assumption $$\alpha_{EE} = \alpha_{BB}$$, I used $$C_l^{EE}$$ in fitting to get amplitude $$A^{EE}_{\text{fit}}$$, and calculated $$A^{BB}_{\text{fit}}$$ by the relation $$A_{EE}/A_{BB} = 2$$. They are relatively experimental so I did not show them in figure 4.1 .

• PySM $$r$$-equivalent maps:
• Planck $$r$$-equivalent maps:

And here are the nside = 8 HEALPix $$A^{BB}_\text{fit}$$-equivalent maps at 353GHz. They are in the unit of $$\mu\text{K}^{2}_{\text{CMB}}$$.

• Planck $$A^{BB}_\text{fit}$$-equivalent maps:

Related postings and papers

1. 2017 June 27: Checking dust decorrelation in models d1/d4/d7 and hipdt (Clem P.)
2. The Python Sky Model: software for simulating the Galactic microwave sky (arXiv:1608.02841)
3. Planck intermediate results. XXX. The angular power spectrum of polarized dust emission at intermediate and high Galactic latitudes (arXiv:1409.5738)
4. Xspect, estimation of the angular power spectrum by computing cross-power spectra with analytical error bars (arXiv:1409.5738)