Framework for performance forecasting and optimization of CMB $B$-mode observations in the presence of astrophysical foregrounds

We present a formalism for performance forecasting and optimization of future cosmic microwave background (CMB) experiments. We implement it in the context of nearly full sky, multifrequency, $B$-mode polarization observations, incorporating statistical uncertainties due to the CMB sky statistics, instrumental noise, as well as the presence of the foreground signals. We model the effects of a subtraction of these using a parametric maximum likelihood technique and optimize the instrumental configuration with predefined or arbitrary observational frequency channels, constraining either a total number of detectors or a focal plane area. We showcase the proposed formalism by applying it to two cases of experimental setups based on the CMBpol and COrE mission concepts looked at as dedicated $B$-mode experiments. We find that, if the models of the foregrounds available at the time of the optimization are sufficiently precise, the procedure can help to either improve the potential scientific outcome of the experiment by a factor of a few, while allowing one to avoid excessive hardware complexity, or simplify the instrument design without compromising its science goals. However, our analysis also shows that even if the available foreground models are not considered to be sufficiently reliable, the proposed procedure can guide a design of more robust experimental setups. While better suited to cope with a plausible, greater complexity of the foregrounds than that foreseen by the models, these setups could ensure science results close to the best achievable, should the models be found to be correct.

Figure 1

1-$\sigma$ and 2-$\sigma$ contours in the ${\beta }_{\mathrm{dust}}\mathrm{\text{−}}{\beta }_{\mathrm{sync}}$ space of the spectral likelihood, $\mathcal{L}$, calculated using Eq. (3) for a random realization of the CMB and noise contributions, shaded areas, and compared against the Gaussian approximation with a dispersion as given by Eq. (5), solid lines. The former likelihood has been recentered at the true values of the parameters.

Figure 2

Schematic illustration of our optimization procedure in the case of an adjustable number of channels, a number of detectors per channel, and their central frequencies.

Figure 3

Three foreground masks as used in this work. Yellow (largest mask), dark red (large mask round the galactic bulge), and dark blue (narrowest mask around the galactic plane) mark sky areas excluded from the masks: Mask II, P06, and Mask I, respectively.

Figure 4

Pseudopower spectra of the foreground templates for the three different masks considered in this work and contrasted with the CMB $B$-mode power spectrum. For each mask the three lines show dust (solid line), synchrotron (dashed line), and their cross-correlation (dotted line). The foreground signals are computed at the 65 GHz. All the spectra used in this work are computed from HEALpix-pixelized maps with $\mathrm{nside}=512$.

Figure 5

Breakdown of the focal plane area between the frequency channels as originally proposed for the COrE, left, and CMBpol, right, satellites. In the case of COrE all the channels with frequencies larger than 250 GHz represent less than 10% of the total focal plane area.

Figure 6

Left: Optimized distributions of numbers of detectors per channel derived under the total focal plane area constraint for the COrE satellite, including only channels below 400 GHz. From top to bottom we show first the original distribution followed by the three optimized ones derived using FOM#1 to #3, respectively. Right: Corresponding power spectra of the residuals and the noise computed for the optimized configurations shown on the left and compared against the spectrum of the CMB $B$-modes with $r=0.001$.

Figure 7

As in Fig. 6 but imposing the constraint on the total number of detectors.

Figure 8

As in Fig. 6 but for the CMBpol satellite.

Figure 9

As in Fig. 7 but for the CMBpol satellite.

Figure 10

Left panel: Results of the FOM#1-based optimization derived in the case of the COrE experiment with a constraint on FOM#2 ($<{10}^{-4}$), and using the P06 mask and channels with frequencies below 400 GHz. Right panel: Comparison of the power spectra corresponding to the proposed and optimized versions of the COrE experiment as listed in Table  and visualized in the left panel. The spectra in blue (mid-level noise spectrum and highest residuals, these latter being depicted with dashed lines) correspond to the cases with the total area constraint. On the other hand, the spectra in magenta (lowest noise level, same residuals as previously) correspond to the cases with the detector number constraint. The foreground residual spectra in both of these cases overlap perfectly in the figure with the magenta curve being invisible.

Figure 11

Dependence of the values of FOM#1 (left), FOM#2 (middle), and FOM#3 (right), on a fractional change of a number of detectors in the hardware configuration as detailed in the fourth column of Table . The solid lines show cases with a number of detectors in only one selected channel being gradually decreased (left to right) and all the others being kept fixed at their optimized values. The circles show the case with a number of detectors in all channels decreasing by the same fraction simultaneously. The color schemes for the lines are the same in all the panels and described in the legend.

Figure 12

The worst values of each FOM, $\tilde{v}$, computed for each of the concentric hyperellipsoids, Eq. (20), defined by the threshold values, $v$, as shown on the horizontal axis. The dotted line shows $\tilde{v}=v$ case. Clearly, $\tilde{v}\simeq v$ in all shown cases, where the latter approximate equality holds to within 10%. The values of $\tilde{v}$ and $v$ given here are relative to the optimized values of the respective FOMs.

Figure 13

Summary of our robustness tests applied to the COrE configuration obtained via the optimization of FOM#1 with constraints of FOM#2 $\le {10}^{-4}$ and a fixed number of detectors. The lines of different colors correspond to different FOMs and different lines show: average (dotted), 95% confidence limit, (dot-dashed), and the worst value (solid).

Figure 14

Demonstration of the optimization results derived with respect to a variable number of channels, numbers of detectors per channel, and their central frequencies, while constraining the total number of detectors ($=6128$ as in the proposed COrE version). Upper left panel shows, from top to bottom, (1) the starting configuration with all the detectors evenly distributed among a fine-grid of channels; (2) a configuration after the first optimization of FOM#1 constrained to ensure that $\mathrm{FOM}\#2\le {10}^{-4}$; and (3) the reoptimization of configuration (2) restricted only to channels with a number of detectors larger than five and after adjacent channels merging and recentering as described in Sec. 4e. Right panel shows power spectra corresponding to these configurations contrasted against the expected CMB signals.