RELEVANT EXAMPLES
5.1 Effects of Backgrounds and Noise Levels
We expect that the precision of the SZ constraints will be chiefly limited by two factors: the noise of the instrument, and the confusion noise. The noise level can be reduced by increasing the number of detectors (or increasing the integration time), while the confusion noise is fundamentally limited by the aperture diameter of the telescope. We list confusion limits for our choice of instrument diameters in Table 5.1, along with estimates of the integration times required to reach each confusion limit for a given diameter. Thus, we are interested in characterizing the effects of our assumptions about noise and the CIB on the resulting SZ constraints.
In this investigation, we explored four different scenarios, which we refer to with the numbers (1) through (4), or with the following shorthand:
1. “NoBkg”: confusion-limited instrument noise, no backgrounds;
2. “CIB”: confusion-limited instrument noise + CIB;
3. “LowNoise”: 10% of confusion-limited instrument noise, no backgrounds;
4. “LowNoiseCIB”: 10% of confusion-limited instrument noise + CIB.
Case (1) represents a roughly current-generation instrument, where the noise levels correspond to the confusion limits in the sense of Section 2.3.3, but it includes only instrument noise without astrophysical contaminants. Case (2) is a more realistic version of case (1) that includes the non-Gaussian CIB component. We were also
Frequency Band 5𝜎Confusion Limit (mJy) Required Integration Time (s)
(GHz) 10m 30m 50m 10m* 30m 50m
90 0.4 0.12 0.06 4262 684 355
150 1.1 0.24 0.09 1038 315 290
220 2.1 0.38 0.14 455 201 192
270 3.0 0.47 0.17 469 276 274
350 4.0 0.62 0.20 241 145 181
400 4.6 0.70 0.22 626 391 512
Table 5.1: Comparison of confusion limits for the instrument configurations. These values represent the 5𝜎detection threshold for CIB sources. These confusion limits were calculated using SIDES (Béthermin et al., 2017) with the formalism described in Zmuidzinas (2018, private communication). Confusion limits may be converted to map-space RMS values, with units of mJy/beam, by dividing by 5. We also list estimates of the integration times required to reach these confusion limits in practice, based on sensitivities at a high-altitude site such as the Chajnantor Plateau.
The integration times assume a focal plane with 0.5(𝐹/#)𝜆spacing at 350 and 400 GHz and (𝐹/#)𝜆 spacing at lower frequencies; appropriate penalty factors have been taken for the lower frequencies to ensure sufficiently dense sampling of the sky. *The integration times listed under 10m were in fact calculated for a diameter of 10.4 m.
interested in exploring a longer-term scenario in case (3), where the instruments are assumed to have noise levels well below the confusion limit; in particular, we assume an improvement by a factor of 10 relative to case (1). Case (4) is the same as case (3) but with a CIB component added. While cases (1) and (3) are less realistic than cases (2) and (4), they provide a baseline for the behavior of the parameter constraints. In addition, the factor-of-10 noise improvement of cases (3) and (4) may not be achievable in practice (a factor of 3 may be more realistic), but we include it to show the implications of a CIB-dominated scenario.
The mock observations to produce the constraints in this section are derived from a cluster model based on the cluster MCXC J1056.9-0337 (also known as MS 1054), which has a mass of 𝑀500 =8.5×1014𝑀· and a redshift𝑧 = 0.83. We provide a comprehensive set of corner plots for different cluster models in Appendix A.2.1. Additionally, we assume a fractional prior of 10% on the temperature𝑇 (see Section 4.1.2).
Figure 5.1 shows the constraints from all observational scenarios for this clus- ter model. Results for other cluster models are shown in Appendix A.2.1 and summarized in Section 5.3.
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(a) 10m Parameter Constraints
Figure 5.1: 1D histograms and 2D contours showing SZ parameter constraints with different levels of noise and CIB contamination. The 1D histograms comprise the best-fit parameter values for each noise realization, while the 2D contours are 68%
confidence regions based on curve fits to this distribution of optimal points for ease of visualization. We consider instruments with aperture diameters of 10m (Figure a), 30m (b), and 50m (c). For each aperture diameter, we show the following scenarios, corresponding to cases (1) through (4) in the text: instrument noise only, with noise at confusion limit (“NoBkg,” blue); instrument noise + CIB, with noise at confusion limit (“CIB,” orange); instrument noise only, with noise at 10% of the confusion limit (green); and instrument noise + CIB, with noise at 10% of the confusion limit (red). Ground-truth values are indicated with gray lines.
(b) 30m Parameter Constraints
In general, none of the observational scenarios 1-4 shows a large bias in the recovered parameters: all of the true parameter values (gray lines in Figure 5.1) fall within the 68% contours. This suggests that the source subtraction and fitting procedures are valid on at least a basic level.
We compare the constraints achieved by the different scenarios for each in- strument class below. Many of these constraints behave similarly for the different instrument classes; the main differences appear to be due to differences in the CIB removal.
5.1.1 10m Diameter
We first address the cases without CIB: cases (1) and (3). The constraints scale in a straightforward way with the noise level: reducing the noise by a factor of 10
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(c) 50m Parameter Constraints
simply shrinks the areas of the constraint regions while roughly maintaining their shapes. Accordingly, the velocity constraint improves with reduced noise, although the strength of this effect depends on the mass and redshift of the cluster as well as the telescope diameter; see Table 5.2 for precise 𝜎𝑣 ratios. Case (1) does not add significant constraining power on𝑇 beyond the 10% prior, suggesting that the data are insufficient to constrain the rSZ component. Case (3) also improves upon the rSZ constraint.
The inclusion of the CIB in case (2) significantly degrades the quality of the 𝑣𝑧 constraint relative to case (1). This degradation is likely due to the degeneracy between the CIB and the SZ signals. The level of this effect demonstrates the im- portance of carefully considering the CIB contamination even when the instrument
noise is confusion-limited.
It is surprising that the temperature constraint degrades in case (4) compared to case (2) considering the lower noise in case (4). Indeed, the𝑇 constraint in case (4) appears to be less precise than the 10% temperature prior. A likely explanation for this behavior is as follows. First, compared to the 30m and 50m cases, the 10m case lacks the resolving power to enable deep CIB cleaning towards the SZ peak. The residual CIB is degenerate with the SZ signals, so a given CIB realization can be fit with a𝑇 value that deviates from the prior. Finally, because of the low noise level, the uncertainty estimate from the least-squares fitter is interpreted as having greater precision than the 10%𝑇 prior, so the fitter converges to the deviant𝑇 value. We have compared the spread of𝑇 values for the realizations with the 𝜎𝑇 estimated by the fitter, and we have found that the empirical𝑇 spread is roughly 2 times as great, which is consistent with the above explanation. However, rigorously confirming the cause of this behavior is a subject for future investigation. In particular, it is not known with certainty whether the behavior is a fundamental limitation or a deficiency of the CIB removal algorithm.
Despite that case (4) yields a relatively poor𝑇 constraint, it still constrains 𝑣𝑧
with significantly better precision than in case (2). This may be because a typical dusty galaxy SED more closely resembles the rSZ spectrum than the kSZ spectrum.
In particular, the rSZ effect represents a larger fraction of the total SZ signal in the highest frequency bands, from 270 to 400 GHz, while the kSZ effect is fractionally strongest in the middle bands around 220 GHz; see Figure 3.1.
We note that the performance of case (4) is sensitive to the choice of the detection threshold supplied to StarFinder in the source subtraction algorithm. The StarFinder documentation indicates that the detection threshold should be set as the noise RMS in the map. We have found this choice to be valid for the confusion limited case (2), which was the context in which we developed the multiband detection and subtraction algorithm. However, when the instrument noise is reduced, the effective noise level becomes dominated by source confusion, and a threshold based on instrument noise alone becomes too aggressive, resulting in spurious detections.
We varied the value of the threshold in case (4), trying factors of 1, 3, 5, and 10 of the instrument noise. We found that the optimal threshold, as determined by the SZ constraints, varies as a function of telescope diameter. However, a factor of 5 is close to optimal for each case, so we use it in the reported constraints for case (4) for consistency. These results may not be fully general, and refining this threshold
64 is a subject for future work.
5.1.2 30m Diameter
As in the 10m case, the constraints from the noise-only cases (1) and (3) simply scale with the noise level. The constraints still degrade when the CIB is added, though the effect is less pronounced than in the 10m diameter case. Cases (2) and (4) improve more significantly due to the greater resolving power. Case (4) can now constrain𝑇 to a precision better than the prior’s, presumably due to the improved CIB removal.
5.1.3 50m Diameter
The 50m case is qualitatively similar to the 30m case. This suggests that a 30m telescope provides sufficient data to capture much of the behavior due to CIB. The degeneracy shape in case (2) is slightly modified, such that its𝑣𝑧 constraint is only slightly degraded relative to case (1). The improvement in the𝑇constraint is greater, which seems to be the primary benefit of the 50m telescope.