Characterizing the nature of the unresolved point sources in the Galactic Center: An assessment of systematic uncertainties

Laura J. Chang, Siddharth Mishra-Sharma, Mariangela Lisanti, Malte Buschmann, Nicholas L. Rodd, and Benjamin R. Safdi

Phys. Rev. D 101, 023014 – Published 27 January 2020

Abstract

The Galactic Center excess (GCE) of GeV gamma rays can be explained as a signal of annihilating dark matter or of emission from unresolved astrophysical sources, such as millisecond pulsars. Evidence for the latter is provided by a statistical procedure—referred to as non-Poissonian template fitting (NPTF)—that distinguishes the smooth distribution of photons expected for dark matter annihilation from a “clumpy” photon distribution expected for point sources. In this paper, we perform an extensive study of the NPTF on simulated data, exploring its ability to recover the flux and luminosity function of unresolved sources at the Galactic Center. When astrophysical background emission is perfectly modeled, we find that the NPTF successfully distinguishes between the dark matter and point source hypotheses when either component makes up the entirety of the GCE. When the GCE is a mixture of dark matter and point sources, the NPTF may fail to reconstruct the correct contribution of each component. These results are related to the fact that in the ultrafaint limit, a population of unresolved point sources is exactly degenerate with Poissonian emission. We further study the impact of mismodeling the Galactic diffuse backgrounds, finding that while a dark matter signal could be attributed to point sources in some outlying cases for the scenarios we consider, the significance of a true point source signal remains robust. Our work enables us to comment on a recent study by Leane and Slatyer (2019) that questions prior NPTF conclusions because the method does not recover an artificial dark matter signal injected on actual Fermi data. We demonstrate that the failure of the NPTF to extract an artificial dark matter signal can be natural when point sources are present in the data—with the effect further exacerbated by the presence of diffuse mismodeling—and does not on its own invalidate the conclusions of the NPTF analysis in the Inner Galaxy.

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Received 24 September 2019

DOI:https://doi.org/10.1103/PhysRevD.101.023014

Physics Subject Headings (PhySH)

Dark matter Gamma ray astronomy Particle astrophysics

Milky Way

Particles & FieldsGravitation, Cosmology & Astrophysics

Authors & Affiliations

Laura J. Chang ¹, Siddharth Mishra-Sharma², Mariangela Lisanti¹, Malte Buschmann³, Nicholas L. Rodd^4,5, and Benjamin R. Safdi³

¹Department of Physics, Princeton University, Princeton, New Jersey 08544, USA
²Center for Cosmology and Particle Physics, Department of Physics, New York University, New York, New York 10003, USA
³Leinweber Center for Theoretical Physics, Department of Physics, University of Michigan, Ann Arbor, Michigan 48109, USA
⁴Berkeley Center for Theoretical Physics, University of California, Berkeley, California 94720, USA
⁵Theoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

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Vol. 101, Iss. 2 — 15 January 2020

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Figure 1
Recovery of two benchmark source populations, both normalized to account for 100% of the GCE flux. The black points correspond to the source-count distribution of the simulated sources with a hard (left) and soft (right) flux distribution. At this stage, the simulated maps contain no contributions other than from the PSs, and the sources themselves are not PSF smoothed. We run the NPTF on these maps using three templates: (i) NFW PS, (ii) NFW DM, and (iii) Galactic diffuse emission. The solid red line indicates the best-fit source-count function for the PSs, with the bands corresponding to the 68 and 95% confidence interval. We assume that all sources above the 3FGL threshold of $F \sim 4 - 5 \times 10^{- 10}$ counts ${cm}^{- 2} s^{- 1}$ would be resolved and thus choose, by design, source-count functions that are suppressed above this threshold flux. The vertical gray dotted line indicates the flux corresponding to $\sim 1 ph$ . Above this flux, the NPTF successfully recovers the source-count distributions in both cases. Below $\sim 1 ph$ , the uncertainty on the recovered source-count function increases dramatically for the soft population, as this is the regime where the statistical method loses sensitivity. In both cases, the total flux of the PSs recovered by the NPTF matches the true value. We also show for reference the millisecond pulsar (MSP) source-count distribution derived using the luminosity function for disk MSPs obtained in Ref. [31] and assuming the energy spectrum found in Ref. [30], normalized to account for 100% of the GCE flux (gray dash-dotted line). The soft source-count distribution is a reasonable approximation to the MSP source-count distribution, while the hard source-count distribution significantly underpredicts the number of dim PSs.
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Figure 2
Differential source-count distributions (top panels) and cumulative flux distributions, integrated above a given threshold flux (bottom panels), shown for simulations with PSs following a soft source-count distribution, accounting for 100% of the GCE (no DM contribution). We run the NPTF on these maps using three templates: (i) NFW PS, (ii) NFW DM, and (iii) Galactic diffuse emission. Black points indicate the true simulated distributions, while solid lines indicate the median best-fit distributions recovered over 100 Monte Carlo realizations of the simulated data maps. The bands show the median 68 and 95% confidence bands over these 100 iterations. In order, the results are shown without including PSF effects in the simulation or NPTF analysis and without Galactic diffuse emission in the simulated data (left column); without PSF effects but including Galactic diffuse emission (middle column); and finally, accounting for PSF effects while also including Galactic diffuse emission (right column). In the left column, the dotted lines designate the photon counts associated with a given flux; in the middle and right columns, the lines designate the approximate significance of a source with flux $F$ . Note that the 3FGL threshold corresponds roughly to $\sim 5 σ$ sources. We observe that PS recovery at low fluxes gets progressively worse as Galactic diffuse emission is added into the simulated data maps and PSF effects are included. The individual flux posteriors for a random subset of 50 of the 100 realizations summarized in the right panel are provided in Fig. 11. All subsequent figures in this paper include Galactic diffuse emission in the simulated data maps, as well as PSF effects in both the simulated data and the NPTF analysis.
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Figure 3
Comparison of the flux posterior (relative to the true injected flux) for the Galactic diffuse, NFW DM, and NFW PS components. The last column shows the Bayes factors (BFs), characterizing the statistical preference of a model of the data that includes NFW PSs over a model that does not include them, for each realization shown. These results pertain to maps where 100% of the GCE is accounted for by DM and there are no PSs present in the simulated data. We run the NPTF on these maps using three templates: (i) NFW PS, (ii) NFW DM, and (iii) Galactic diffuse emission. Each row in the figure represents a different Monte Carlo iteration of the map. In cases where the flux distributions are recovered exactly, the posteriors are centered at zero. Importantly, we see that when there are no unresolved PSs in the map, the NPTF analysis does not erroneously attribute the DM to PSs. For this example, the simulated map is made using the p6v11 model, and the same template is used in the analysis—this represents the case where there is no diffuse mismodeling. As expected, the $\ln (BF)$ is negative for the majority of realizations, pointing to a preference for a model without PSs. These are a random subset of 50 of the 100 iterations shown in the left half of Fig. 7.
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Figure 4
Comparison of the flux posterior (relative to the true injected flux) for the Galactic diffuse, NFW DM, and NFW PS components, as well as the total flux, varying the relative fraction of the GCE accounted for by DM and PS. These results pertain specifically to the soft source-count function. We consider the four cases where the GCE is 100% PS (first column), 25% DM and 75% PS (second column), 50% DM and 50% PS (third column), and 75% DM and 25% PS (fourth column). We run the NPTF on these maps using three templates: (i) NFW PS, (ii) NFW DM, and (iii) Galactic diffuse emission. In each panel, the solid line represents the median and the shaded region spans the minimum and maximum values in a given flux bin, across 100 Monte Carlo iterations. In each case where there are contributions from both DM and PSs, there is a probability that the DM signal is absorbed by the PS template, as evidenced by the peaks at “ $- True$ NFW DM” in the second row of each of the right three panels; the probability of the PS signal being absorbed by the NFW DM template increases with increased DM fraction, as evidenced by the increasingly large peak at “ $- True$ NFW PS” in the third row of each of the right three panels. The corresponding plot for the hard source-count distribution is provided in the Appendix as Fig. 18.
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Figure 5
Same as Fig. 4, but with mock data constructed by injecting an additional GCE-strength DM signal on top of an existing GCE signal. Each panel shows the median (solid lines) and minimum/maximum (shaded regions) value in a given flux bin over 100 Monte Carlo iterations. The simulated data maps consist of 10 distinct “base” maps in which the GCE is entirely accounted for by DM (left two columns) or PSs (right two columns), onto which the additional DM signal is injected. The injected DM signal is Poisson fluctuated to generate ten realizations for each base case. All of the simulated data maps are made using the p6v11 diffuse model; to explore the effects of diffuse mismodeling, the second and fourth columns are analyzed using a Galactic diffuse template that is based on Model F from Ref. [14]. Importantly, when injecting the additional DM signal onto an existing GCE-strength DM signal, the analysis is well behaved and there is no significant misattribution of flux between the DM and PS components, even in the presence of diffuse mismodeling. When injecting the additional signal onto an existing PS signal, there is confusion between the two components and the flux posteriors for the DM and PS templates become bimodal. Notably, in the absence of diffuse mismodeling, the amount by which the median PS flux posterior is shifted negative matches the true total flux contributed by ultrafaint sources below the $\sim 1 σ$ significance threshold. In the presence of diffuse mismodeling, the DM signal is typically absorbed by the PS template, as evidenced by the median DM posterior peaking at “ $- True$ NFW DM” in the fourth column. These results clearly demonstrate that an artificial DM signal injected on the data may fail to be recovered by the NPTF if point sources are already present, and especially when the diffuse emission is mismodeled, thereby providing a simple possible explanation for the results presented in Ref. [44]. The corresponding plot for the hard source-count distribution is provided in Fig. 19.
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Figure 6
Same as the rightmost panel of Fig. 2, but analyzed using Model F rather than the p6v11 model, which was used to generate the simulated data maps. Results are shown for the soft PSs (left panels) as well as the hard PSs (right panels). Even at the $\sim 95 %$ level over 100 Monte Carlo iterations, the recovered source-count function fails to capture the low-flux sources for the soft PSs. The cumulative flux distribution for the soft PSs shows that in the presence of diffuse mismodeling, the recovered flux is consistent with the true injected flux down to $\sim 2 \times 10^{- 10}$ counts ${cm}^{- 2} s^{- 1}$ , but is in excess for fluxes between $\sim 1 - 2 \times 10^{- 10}$ counts ${cm}^{- 2} s^{- 1}$ . In the case of the hard source-count distribution, the recovered function reliably captures the input at the $\sim 95 %$ level. The individual flux posteriors for a random subset of 50 of the 100 runs shown in the left panels are provided in Fig. 12. For comparison, we also provide (in Fig. 13) the flux posteriors for the case where the GCE consists of 100% DM and there is diffuse mismodeling.
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Figure 7
Bayes factors (BFs) characterizing the statistical preference for a model of the data that includes NFW PSs over a model that does not include them. The mock data consist of the GCE, accounted for by either 100% NFW DM (left) or by 100% NFW PS following the soft source-count distribution (right), and diffuse emission modeled by p6v11. Results are shown for 100 Monte Carlo iterations. We run the NPTF on these maps using three templates: (i) NFW PS, (ii) NFW DM, and (iii) Galactic diffuse emission. The BFs recovered using the p6v11 model for the diffuse template are shown along the horizontal axes, while the BFs recovered using the Model F template are shown along the vertical axes. (Note the difference in scale for the axes between the left and right halves of the figure.) Even in the presence of diffuse mismodeling, the NPTF robustly picks up evidence for PSs when they constitute 100% of the GCE. When the GCE is 100% DM, the evidence for PSs is always negligible in the p6v11 case; in the Model F case, while there is a strong peak around $\ln (BF) \sim 0$ , there are 35 iterations in which the diffuse mismodeling leads to residuals that are picked up as PSs with $5 < \ln (BF) ≲ 13$ . The significance of these detections is still smaller than the BF range when true PSs are actually present. We emphasize that while the relative values of the BFs are useful in comparing the different tests studied here, their overall scale should not be compared to any results on Fermi data as these maps are not intended to closely model an actual data realization.
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Figure 8
Same as Fig. 7, but where the mock data consist of, in addition to the DM(PS) signal: p6v11 diffuse emission, emission from the Fermi bubbles [60], isotropic emission, and emission from 3FGL sources [48]. The left(right) panel shows results where the GCE is 100% DM(soft PSs). The NPTF analysis now also includes templates to model the isotropic, 3FGL, and bubbles emission. Compared to what we see in Fig. 7, the overall scale of the BFs is lower. Additionally, in the 100% DM case with diffuse mismodeling, the distribution is much more strongly peaked near a low value of $\ln (BF) \sim 1$ , and the number of iterations for which residuals are picked up as PSs with $5 < \ln (BF) ≲ 13$ is reduced to 7 (compared to 35 cases in Fig. 7). We emphasize again that while the relative values of the BFs are useful in comparing the different tests studied here, their overall scale should not be compared to any results on Fermi data as the simple setup is not intended to accurately model the analysis on real data.
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Figure 9
Differential source-count distributions in the presence of diffuse mismodeling. The simulated data consist of the GCE, which is entirely accounted for by DM, and Galactic diffuse emission (corresponding to the left panel of Fig. 7). We run the NPTF on these maps using three templates: (i) NFW PS, (ii) NFW DM, and (iii) Galactic diffuse emission. The simulated maps are generated using the p6v11 model and analyzed using Model F. We separately show the results for iterations with $\ln (BF) < 5$ (left panel) and iterations with $\ln (BF) > 5$ (right panel). In each case, the solid green line is the median best-fit distribution, and the bands denote the median $68 / 95 %$ confidence intervals, recovered over 100 Monte Carlo realizations. In the cases with $\ln (BF) < 5$ , the typical best-fit source-count function is suppressed relative to the examples we have considered thus far and does not look like the source-count function recovered in the NPTF analysis on Fermi data [32]; in the outlying cases with $\ln (BF) > 5$ , which correspond to the tail of the distribution shown in the leftmost panel of Fig. 7, the median recovered source-count function resembles the hard PS population in the left panel of Fig. 1. Note that when adding in additional Poissonian contributions to the mock data, as in Fig. 8, the number of instances where $\ln (BF) > 5$ is significantly reduced.
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Figure 10
Differential source-count distributions recovered when the simulated data consist of the GCE, comprises entirely of soft PSs, with an additional injected GCE-strength DM signal. We run the NPTF on these maps using three templates: (i) NFW PS, (ii) NFW DM, and (iii) Galactic diffuse emission. The p6v11 model is used to generate the Galactic diffuse emission in the simulated data, and the analysis is performed using the “correct” diffuse model (p6v11, left panel) or the “incorrect” diffuse model (Model F, right panel). In each case, the solid green line is the median best-fit distribution, and the bands denote the median $68 / 95 %$ confidence intervals, recovered over 100 Monte Carlo realizations. The left(right) panel corresponds to the third(fourth) column of Fig. 5. These results can be compared with the cases without the additional DM injection, shown in the top right panel of Fig. 2 for p6v11 and the top left panel of Fig. 6 for Model F. In the analysis with diffuse model p6v11, a significant portion of the PS flux is typically absorbed by the DM template (correspondingly, the source-count distribution is suppressed in the $\sim 5 \times 10^{- 11} - 2 \times 10^{- 10}$ counts ${cm}^{- 2} s^{- 1}$ flux range, relative to Fig. 2, top right panel). This is similar to the observed misattribution of injected DM flux to PSs in Ref. [44]. On the other hand, when the data are analyzed with the diffuse Model F, the injected DM is consistently absorbed into the PS model (correspondingly, the source-count distribution captures excess flux in the $\sim 10^{- 11} - 10^{- 10}$ counts ${cm}^{- 2} s^{- 1}$ range, relative to Fig. 6, top left panel).
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Figure 11
Comparison of the flux posterior (relative to the true injected value) for the Galactic diffuse, NFW DM, and NFW PS components. The last column shows the Bayes factors (BFs), characterizing the statistical preference of a model of the data that includes NFW PSs over a model that does not include them. These results pertain to maps where 100% of the GCE is accounted for by soft PSs (corresponding to the rightmost panel of Fig. 2). We run the NPTF on these maps using three templates: (i) NFW PS, (ii) NFW DM, and (iii) Galactic diffuse emission (p6v11 model). Each row in the figure represents a different Monte Carlo iteration of the map. These are a random subset of 50 of the 100 iterations shown in the left half of Fig. 7. In cases where the PS flux is underestimated, it is typically picked up by the NFW DM template. Even when the PS flux is underestimated, decisive evidence for a PS population is still seen based on the Bayes factors. We emphasize that the overall scale of the BFs should not be compared to any results on Fermi data as these maps are not intended to closely model an actual data realization.
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Figure 12
Same as Fig. 11, but this time analyzed using Model F as the Galactic diffuse emission model. This corresponds to the case where the diffuse emission is mismodeled. Decisive evidence for a PS population is seen for each realization, even with the diffuse mismodeling. We emphasize that the overall scale of the BFs should not be compared to any results on Fermi data as these maps are not intended to closely model an actual data realization.
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Figure 13
Same as Fig. 3, but this time analyzed using Model F as the Galactic diffuse emission model (note the different scale in the rightmost panel from Fig. 3). Misattribution of the DM flux to PSs is significantly exacerbated in the presence of diffuse mismodeling, and evidence for a PS population can be erroneously inferred for a subset of the realizations due to residuals from the mismodeling. However, the misattribution is ameliorated when more DM signal is injected into the mock data, as demonstrated in the second column of Fig. 5. We emphasize that while the relative values of the BFs are useful in comparing the different scenarios and realizations studied here, their overall scale should not be compared to any results on Fermi data as these maps are not intended to closely model an actual data realization.
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Figure 14
Same as the right two columns of Fig. 5, but with even brighter DM signals injected on top of an existing GCE-strength soft PS signal. The simulated data maps consist of ten distinct “base” maps in which the GCE is entirely accounted for by soft PSs, onto which an additional 200% GCE flux (left two columns) or 300% GCE flux (right two columns) DM signal is injected. The injected DM signal is Poisson fluctuated to generate ten realizations for each base case. In the absence of diffuse mismodeling (first and third columns), the injection of increasingly bright DM signals reduces the bimodality of the DM and PS posterior flux distributions (see the third column of Fig. 5 for comparison). On average, the injected DM signal is fully recovered, and the DM template additionally picks up the flux contribution from ultrafaint PSs below the $1 σ$ significance threshold. In the presence of diffuse mismodeling, the injection of brighter and brighter DM signals mitigates the absorption of the DM signal by the PS template. Note the different scale of the horizontal axes in this case compared to Fig. 5.
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Figure 15
An example triangle plot for an NPTF scan that passes the convergence criteria on $S_{b, 1}$ described in Sec. 2a: the posterior distribution for $S_{b, 1}$ is nicely converged within the prior range. The simulated data in this particular instance consist of 100% soft PSs and diffuse emission. The templates used in this scan are: (i) NFW PS, (ii) NFW DM, and (iii) Galactic diffuse emission (p6v11). Where applicable, the true simulated values are indicated on the one-dimensional posterior distributions by thick solid red lines.
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Figure 16
Same as Fig. 15, but where the simulated data consist of 100% DM and diffuse emission. In this case, the posterior distribution for $S_{b, 1}$ is unconstrained over the prior range. Where applicable, the true simulated values are indicated on the one-dimensional posterior distributions by thick solid red lines.
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Figure 17
An example triangle plot for an NPTF scan that fails the convergence criteria on $S_{b, 1}$ described in Sec. 2a: the posterior distribution for $S_{b, 1}$ is pushed against the lower prior edge. The simulated data in this particular instance consist of 100% DM and diffuse emission, and there is diffuse mismodeling present in the scan. The templates used in the scan are (i) NFW PS, (ii) NFW DM, and (iii) Galactic diffuse emission (Model F). Where applicable, the true simulated values are indicated on the one-dimensional posterior distributions by thick solid red lines—the “true” diffuse model normalization here corresponds to the norm that yields equivalent flux to the true simulated (p6v11) flux. Scans like this are discarded from all results presented in this paper. Additionally, we note that implementing a lower flux cutoff in the source-count function (detailed in Appendix pp3) drastically reduces the number of such scans.
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Figure 18
Same as Fig. 4, except for the case of the hard PS source-count distribution. In this case, the PSs never get fully absorbed by the DM template, as is evidenced by the absence of a strong peak at “ $- True$ NFW PS” in the NFW PS posterior flux distributions. However, there is still a probability that the DM signal gets absorbed by the PS template.
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Figure 19
Same as Fig. 5, but for the case of the hard PS source-count distribution. Compared to case of the soft source-count distribution, there is less spread in the NFW DM and NFW PS flux posteriors. In particular, the PS flux is never absorbed by the DM template in these cases, while the DM template still may be absorbed by the PS template. The injected DM flux is consistently misatrributed to PSs in the presence of diffuse mismodeling (right column).
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Figure 20
Residual photon counts in our ROI from fitting the data with a model that includes (i) the Galactic diffuse emission, (ii) the Fermi bubbles [60], (iii) isotropic emission, and (iv) resolved Fermi 3FGL PSs [48]. We show the residual sky maps for the cases where the Galactic diffuse emission is described by the p6v11 model (left panel) or Model F (middle panel). We also show the difference between the best-fit sky maps obtained using each of the two models. For presentation purposes, we have smoothed each map by a Gaussian with $σ = 1 °$ .
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Figure 21
Histogram of residual photon counts in our ROI. The green dashed line indicates the case where the Fermi data are analyzed using the p6v11 diffuse model (this corresponds to the left panel of Fig. 20). The gray dotted line indicates the case where the Fermi data are analyzed using Model F (this corresponds to the middle panel of Fig. 20). For comparison, we generate 100 Monte Carlo (MC) data maps consisting of the following components (best fit from the Fermi data): (i) p6v11 Galactic diffuse emission, (ii) the Fermi bubbles [60], (iii) isotropic emission, and (iv) resolved Fermi 3FGL PSs [48], which we analyze using Model F and Poissonian components (ii)–(iv). The median result is shown by the solid blue line, while the shaded blue bands span the minimum and maximum values in each bin over the 100 simulated maps. The three cases are roughly comparable, although on average, there tend to be fewer residuals with large magnitudes when fitting the p6v11 simulated data with Model F than when fitting the data with either model.
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Figure 22
Differential source-count distributions (top panels) and cumulative flux distributions, integrated above a given threshold (bottom panels), shown for simulations with PSs with a soft source-count distribution, accounting for 100% of the GCE (no DM contribution). The diffuse emission is not mismodeled in this case. The solid lines indicate the median best-fit distributions recovered over 100 Monte Carlo realizations of the simulated data maps. The bands show the median 68 and 95% confidence bands over these 100 iterations. We show a copy of the right column of Fig. 2 (left) and the corresponding result when a flux cutoff is added to the parametrization of the source-count function in the NPTF analysis (right). The addition of the cutoff does not strongly affect the recovered source-count distribution. However, a slight discrepancy (at the 68% level) is introduced between the recovered and true cumulative flux distributions near the cutoff.
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Figure 23
Same as Fig. 22, except for the case where the diffuse emission is mismodeled. We show a copy of the left column of Fig. 6 (left) and the corresponding result when a flux cutoff is added to the parametrization of the source-count function in the NPTF analysis (right). Here, we find that the cutoff helps to reduce the peak in the recovered source-count distribution near $\sim 2 \times 10^{- 10}$ counts ${cm}^{- 2} s^{- 1}$ .
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Figure 24
Same as Fig. 22, except for the case where the GCE consists entirely of DM.
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Figure 25
Same as Fig. 23, except for the case where the GCE consists entirely of DM.
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Figure 26
The Bayes factors in preference for PSs when the GCE consists entirely of DM in the absence (left panel) or presence (right panel) of diffuse mismodeling. In each case, solid(hatch)-shaded histogram corresponds to recovered source-count distributions without(with) a flux cutoff. In the p6v11 analyses (left panel), the addition of the flux cutoff shifts the distribution of BFs to lower values while leaving the shape of the distribution unchanged. On the other hand, in the Model F analyses (right panel), the cutoff reduces the number of iterations with $\ln (BF) ≳ 5$ while increasing the number of iterations with $1 ≲ \ln (BF) ≲ 5$ .
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Figure 27
Posterior flux distributions for the case where the simulated data are made up of the GCE, comprising soft PSs, with an additional GCE-strength DM signal injected on top. The first and third columns are duplicates of the third and fourth columns of Fig. 5, respectively, and are provided here for comparison. The second and fourth columns show the corresponding results when a flux cutoff is applied to the source-count distribution in the NPTF analysis. In the absence of diffuse mismodeling (first and second columns), the cutoff mitigates the bimodality of the posterior flux distributions. In the presence of diffuse mismodeling (third and fourth columns), the cutoff reduces—on average—the probability that the injected DM signal is absorbed by the PS template. This is demonstrated by reduced peak in the median DM posterior at “ $- True$ NFW DM” in going from the third column to the fourth column.
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