Accounting for Flux Boosting

MAP CLEANING

3.1 Point Source Removal

3.1.1 Accounting for Flux Boosting

The effect of flux boosting arises in the regime where the differential source counts 𝑑𝑁/𝑑𝑆 are a steep function of flux density. Because of the steeply rising 𝑑𝑁/𝑑𝑆, a source with an observed flux density 𝑆_𝜈 is more likely to be a dimmer source that has fluctuated upward in flux due to noise than a brighter source that has fluctuated downward. In effect, the posterior distribution of true flux for a given observed flux 𝑆_𝜈 is biased to lower fluxes than 𝑆_𝜈, or, more simply put, observed fluxes are biased high on average. Failing to account for flux boosting

1This convention for the threshold is not equal to the signal-to-noise ratio (SNR) of the detected source when the PSF is multiple pixels wide. We apply an aggressive detection threshold of 1𝜎_pixel and rely on a manual cut at the end to discard sources with SNR<4.

2We use a value of 0.4 in this work.

30 leads to the subtraction of too large a flux on a source-by-source basis, which causes negative pointlike artifacts to appear in the source-subtracted map. StarFinder does not perform a correction (“deboosting”) for this effect on the flux densities of the sources it detects, so we manually implemented this functionality into our pipeline.

We considered two methods for accounting for flux boosting. The first of these is the Bayesian formalism of Crawford et al. (2010), which gives the posterior probability distribution of the flux density of the brightest source within a pixel 𝑆_max given the detected flux density of the source 𝑆_𝑝,𝑚 and the prior probability distribution𝑃(𝑆_max)derived from the source counts. For the sake of computational tractability, our implementation of the formalism makes the approximation (also described in Crawford et al. (2010)) that the likelihood 𝑃(𝑆_𝑝,𝑚|𝑆_max) is Gaussian, which greatly simplifies the calculation at the cost of some accuracy. From the posterior probability density function (PDF) 𝑃(𝑆_max|𝑆_p,m), this method chooses a single representative value of𝑆_maxto be the true flux density for CIB reconstruction;

in practice, we use the median value of the posterior as a representative.

The second method we considered is to calibrate an empirical relation between the known true flux density of a source 𝑆_true and the flux density of the detection 𝑆_detected. There is some ambiguity inherent in the definition of 𝑆_true, as a detected source does not necessarily correspond exactly to a single true source in the CIB realization, either in position or in flux density, due to the contributions of nearby sources. Based on our testing, we have found that the most useful way to define 𝑆_truegiven a detection is as the amplitude of a fit of the PSF to the noiseless input CIB map at the detected source position. Furthermore, we remove all contributions to 𝑆_true from sources above a 4𝜎 detection threshold that are outside StarFinder’s minimum source separation of 1 FWHM. The calibration is performed on a noisy map of the full field of the SIDES catalog; it consists of fitting a curve to 𝑆_true vs.

𝑆_detected (Figure 3.2). We chose the form of a piecewise linear function for this fitting curve as a compromise between the robustness of the fitting behavior and the flexibility to deviate from linearity. Finally, the true flux density of an unknown source can be estimated by evaluating the piecewise calibration function at𝑆_detected.

Figures of Merit

In tuning the parameters of the source removal algorithm, we have used several figures of merit (FoM) to evaluate its performance. The simplest of these is the

Figure 3.2: Empirical calibration of StarFinder’s flux estimate for all sources de- tected in a large (2 deg²) map. For each detected source (blue), StarFinder’s estimate 𝑆_detectedis compared to a fit to the true flux density𝑆_truewithin a beam of the detected position in the noiseless map. The red curve shows the piecewise linear fit to the calibration data, while the black line shows 𝑆_true = 𝑆_detected for comparison. (The vertical black lines indicate boundaries of the piecewise linear fitting regions.) It can be seen that, on average,𝑆_detectedonly exceeds 𝑆_truein dim sources.

residual map-space RMS, calculated as the RMS of the difference between the input map of source fluctuations 𝑀_in and the map of reconstructed sources 𝑀_out, where both maps have been convolved with the beam and 𝑀_in has been calculated before the addition of instrument noise:

FoM=RMS(𝑀_in− 𝑀_out). (3.1)

A more informative figure of merit is the histogram of all the pixels of the residual map𝑀_in−𝑀_out, which can be used to see deviations in the residual from a Gaussian noise distribution. Another option is to consider the angular power spectrum of residual map pixels. We use these FoM to compare the two techniques for boost corrections, as shown in Figure 3.3.

The postage stamps stacked on source positions (Figure 3.3a) are the most transparent way to visualize how each method affects the residual in the map.

Applying no correction results in a clear over-subtraction on average, while there is apparently a trade-off between the two correction methods. The empirical method

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(a)

(b) (c)

Figure 3.3: Demonstrations of the performance of the boosting correction algorithm in the case of a 30m telescope with confusion-limited noise at 270 GHz. a Postage stamps of residual CIB maps (𝑀_in − 𝑀_out) stacked on detected source positions, where 𝑀_outis uncorrected, corrected empirically, and corrected with the Bayesian method of Crawford et al. (2010), respectively. The postage stamps are normalized relative to the peak of the stack on the true CIB map. b Residual CIB power spectra within postage stamps stacked on detected source positions, where the wavenumber 𝑢 is defined in the flat sky approximation such that when 𝑢 has units of rad⁻¹, 2𝜋𝑢 = ℓ, where ℓ is the angular multipole. Power spectra are given for both the residual CIB maps—with the Crawford et al. (2010) boost correction (blue), the empirical correction (orange), and no correction (red)—and the true CIB map 𝑀_in (green). c Histograms of flux density values in each pixel within a postage stamp of the source in the true CIB map (blue), the uncorrected residual map (orange), and the empirically corrected residual map (green). Compare the performance to the 4𝜎 detection threshold (dashed line). The map pixels have been rebinned by a factor of 5 to match the FWHM of the beam.

reduces the over-subtraction but suffers from an annular feature in the stack at a radius of slightly less than one FWHM of the beam (in this case, 10⁰⁰). This suggests that the source positions are being slightly misestimated due to other nearby dim sources. It is interesting to note that the annulus does not disappear in the case where the sources in the map follow a uniform distribution in space (unlike SIDES, where clustering is taken into account); hence, it seems that source clustering does not cause the feature. The Crawford et al. (2010) method, on the other hand, lacks the annular feature but tends to under-subtract sources, likely because it yields the posterior PDF of only the brightest source within a beam without including contributions from dimmer sources.

These characteristics can also be seen in the postage stamps’ power spectra (Figure 3.3b): the Crawford et al. (2010) technique performs better at small angular scales, while the empirical correction performs better on scales of the beam size and larger. Since the SZ signal we seek occurs mostly at angular scales larger than that of the beam, the SZ signal recovery is presumably less sensitive to sub-beam-scale residuals; hence, we choose the empirical correction technique to apply to the main analysis.

The histograms of Figure 3.3c illustrate how the algorithms’ performance affects the entire recovered map. The histogram bins count residuals per beam by binning the FoM map (Equation 3.1) into beam-sized pixels and only considers pixels located at detected source positions. This procedure roughly corresponds to counting the residual per source detection while avoiding the ambiguity in pairing detected sources with one or more sources in the ground truth catalog. As these histograms illustrate, any of the source subtraction methods (with or without boost correction) modify the map histogram by removing the long positive tail of bright sources and leaving a distribution resembling Gaussian noise. The effect of the boost corrections is to reduce the negative tail where oversubtraction is occurring. The Crawford et al.

(2010) is arguably more effective in this regard, although perhaps at the expense of undersubtraction for some sources.