DARK MATTER LINE EMISSION CONSTRAINTS FROM NuSTAR OBSERVATIONS OF THE BULLET CLUSTER

We fit the spectra of both observations and both detectors simultaneously using the Xspec spectral fitting package (Arnaud 1996) and explored two different approaches to background modeling and subtraction. The spectra were binned to contain at least three counts per bin, and we assume the background to be Poisson distributed and consequently use Cash-statistics (C) to optimize the parameter values (Cash 1979). The Cash-statistics is similar to ${\chi }^{2}$ but generally regarded as more suitable for analyzing spectra with a few counts per channel (Nousek & Shue 1989; Arzner et al. 2007) and less biased than ${\chi }^{2}$ (Leccardi & Molendi 2007). The magnitude of C depends on the number of bins included in the fit and the values of the data themselves, and consequently one cannot analytically assign a goodness-of-fit measure to a given value of C. However, for large number counts ($\gt 5$) the C-distribution is similar to the ${\chi }^{2}$-statistics and ${\rm{\Delta }}C$ can be used instead of ${\rm{\Delta }}{\chi }^{2}$ for model comparison tests.

In the first method of background treatment we simply subtract the spectrum of the dark matter offset region from the spectrum of the source region. Since the spectra are almost identical we can fit any residual with a single power law (the statistics for each region are given in Table 1). Subsequently, we added a Gaussian to represent a single emission line at a fixed energy and flux to the best-fit model above, and searched for the improvement of C as a function of line energy and intensity, allowing for simultaneous variation of the power-law parameters and the Gaussian normalization. We consider line intensities from 0 to ${10}^{-5}\;\mathrm{photons}\;{{\rm{cm}}}^{-2}\;{{\rm{s}}}^{-1}$ and line energies of 3–80 keV in steps of ${\rm{\Delta }}E=0.1\;\mathrm{keV}$. We assume that the line is narrow compared to the detector resolution, fixing the intrinsic line width at $0.001\;\mathrm{keV}$ and noting that as long as the assumed width is less than $\sim 0.03\;\mathrm{keV}$, our results do not change.

The reduction of the C parameter by the extra line is shown as a function of line energy in the upper panel of Figure 3. At most energies, the additional Gaussian does not lead to any significant improvement ($| {\rm{\Delta }}C| \lt 9$ for 1105–3821 degrees of freedom), and instead we constrain the flux by increasing the Gaussian normalization and refitting all other parameters until ${\rm{\Delta }}C=C-{C}_{\mathrm{base}}=2.71$, corresponding to the one-sided 95% confidence level marginalized over the power-law normalizations. These flux levels are shown in Figure 3 for the Peak and Left regions as well as for the combined analysis.

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In the second approach we model the background instead of subtracting it. In Wik et al. (2014) the background emission in the Bullet Cluster was thoroughly investigated and we use their results as a baseline model for the background, and check if there is room for any line emission above this model. The model consists of four components: (i) the aperture background (smooth gradient across the detector with a normalization uncertainty of 10%); (ii) a focused cosmic ray background from unresolved sources (smooth with a normalization uncertainty of 10%); (iii) instrumental continuum (we use a 10% normalization uncertainty even though the systematic uncertainty is probably much smaller); and (iv) the thermal solar continuum and instrumental lines from reflections (smooth component with a normalization uncertainty of 10% plus known detector emission lines). Additionally we fit a line-free plasma model (apec; Smith et al. 2001) to account for any gas emission from the cluster. The redshift and abundances of the plasma model are kept fixed at z = 0.296 and A = 0.2, respectively (consistent with the value found in Wik et al. 2014), while the temperature is required to be the same for both detectors and both observations, but with individual normalizations.

The $3-150\;\mathrm{keV}$ energy interval is well fitted by the background model with the statistics given in Table 1. As before, for each energy between 3 and $80\;\mathrm{keV}$ in steps of ${\rm{\Delta }}{E}_{\gamma }=0.1\;\mathrm{keV}$ we add a Gaussian and determine the best-fit normalization considering line intensities from $-{10}^{-5}$ to ${10}^{-5}\;\mathrm{photons}\;{{\rm{cm}}}^{-2}\;{{\rm{s}}}^{-1}$. The line intensities are allowed to be negative to account for overestimation of the background. The normalization is then increased until ${\rm{\Delta }}{C}^{2}=2.71$. We derive upper flux limits from the Gaussian alone as well as from the base model plus the Gaussian integrated over the FWHM of the NuSTAR spectral resolution crudely approximated by (Harrison et al. 2013)

$$\begin{eqnarray}&&{\rm{\Delta }}{E}_{\mathrm{FWHM}}=0.01{E}_{\gamma }+0.3\;\mathrm{keV}.\end{eqnarray} \tag{ 1 }$$

The addition of the Gaussian only improves the model by $| {\rm{\Delta }}C| \gt 9$ around 50 and $84\;\mathrm{keV}$ where ${\rm{\Delta }}C\approx -12$, reflected in weaker constraints at those energies. While $| {\rm{\Delta }}| C\gt 9$ would appear to imply a significant detection, we need to take the look-elsewhere effect into account (Gross & Vitells 2010). The energy of the line is unknown and by scanning over energy we perform a number of independent searches that increase the chance of seeing statistical outliers. Consequently the probability of detection is degraded by the number of independent attempts (given the spectra resolution of NuSTAR we search of the order of 150 independent energies).

Line-like features may arise from fluorescent and activation-induced instrumental lines if imperfectly modeled, or due to statistical/systematic fluctuations between target and background regions (see appendix of Wik et al. 2014). These lines are strongest between 20 and 30 keV, and may explain the region of larger C-values at those energies.

The flux limits from the two approaches for the background subtraction are compared in Figure 3. The solid lines show the results of modeling the background of the nearby dark matter offset region, and the dotted lines show the results of subtracting the nearby dark matter offset region. The results from the individual regions are consistent with each other, and the constraints tighten by combining the regions. The background modeling generally provides slightly stronger constraints, but with gaps where the preferred flux is negative. The results are very similar, and we consider the background treatment to be robust. We also derive flux limits from fitting the two regions simultaneously, taking their mass difference and consequently expected signal strength into account.

While the background modeling provides slightly stronger constraints, there is a risk of spurious line detections from small residuals due to a gain drift between the actual observation and the time of collection of data from which the background model was constructed. This is unlikely to be the case for direct background subtraction.

The weak lensing shear maps¹² from Clowe et al. (2006) can be integrated to provide the masses within the three regions for a fiducial cosmology of ${{\rm{\Omega }}}_{{\rm{m}}}=0.3$, ${{\rm{\Omega }}}_{{\rm{\Lambda }}}=0.7$, ${H}_{0}=70\;{\rm{km}}\;{{\rm{s}}}^{-1}\;{{\rm{Mpc}}}^{-1}$. The map contours are shown in Figure 1 and the obtained region masses are given in Table 1.

Paraficz et al. (2012) presented a mass map of the Bullet Cluster based on strong lensing rather than weak lensing. This map provides region masses that are almost twice as big as for the weak lensing map. This indicates an uncertainty on the mass estimates of the order of 50%. In the remaining analysis and plots we conservatively use the values from the map by Clowe et al. (2006) and assume that the entire mass is made up of dark matter since the observed regions have been chosen to minimize the gas presence, so for the Peak region ${M}_{\mathrm{gas}}\approx 0.1{M}_{\mathrm{tot}}$.

Assuming all of the dark matter to be Majorana-type sterile neutrinos, we can interpret the flux constraints from Figure 3 in terms of the sterile neutrino parameters of mass, m_s, and mixing angle, ${\mathrm{sin}}^{2}(2\theta )$, where the latter describes the mixing probability with the lightest of the active neutrinos. The constraints are converted as (Riemer-Sørensen et al. 2006; Boyarsky et al. 2007b):

$$\begin{eqnarray}&&{\mathrm{sin}}^{2}(2\theta )\\ &&\quad \leqslant {10}^{18}(\displaystyle \frac{{F}_{\mathrm{obs}}}{\;{\rm{erg}}\;{{\rm{cm}}}^{-2}\;{{\rm{s}}}^{-1}})(\displaystyle \frac{{m}_{{\rm{s}}}}{\;\mathrm{keV}})[\displaystyle \frac{({M}_{\mathrm{fov}}/\;{M}_{\odot })}{{({D}_{{\rm{L}}}/\;{\rm{Mpc}})}^{2}}]\end{eqnarray} \tag{ 2 }$$

where F_obs is the observed flux limit, M_fov is the total dark matter mass within the field of view, and D_L is the luminosity distance. Equation (2) is only concerned with the decay of the sterile neutrinos and applies regardless of how they were formed. Theoretically, the sterile neutrino can have any mass, but as explained in Section 1 observations restrict the range to approximately $1-100\;\mathrm{keV}$.

As illustrated in Figure 4, very large mixing angles will lead to overproduction of dark matter and are consequently ruled out. Similarly, the resonant production mechanisms require a primordial lepton asymmetry (Boyarsky et al. 2008b), which may affect Big Bang nucleosynthesis and cosmic element abundances. This gives a lower limit on the mixing angle. As mentioned in Section 1, the mass range is limited from below by the Tremaine–Gunn bound (Tremaine & Gunn 1979; Boyarsky et al. 2008b) and from above by line emission searches. The NuSTAR Bullet Cluster constraints from the Peak region alone are weaker than previously existing constraints from the diffuse cosmic background, but by combining the Peak and Left regions we get similar constraints. This provides an important independent cross-check of the many assumptions about, e.g., source distribution, that go into the diffuse background constraints.

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In Figure 5 we present the constraints on generic dark matter decays leading to photon emission for two photons per decay. For one-photon interactions, the constraints are weaker by a factor of two. For particles of Majorana-type where the particles are their own anti-particles, the interaction probability doubles and the constraints are strengthened by a factor of two.

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Again, the NuSTAR constraints do not significantly improve on existing constraints, but as a pointed observation with robust background treatment they provide an important cross-check on previous analysis of diffuse emission observed with HEAO, Fermi, INTEGRAL, COMPTEL, and EGRET (Boyarsky et al. 2006a, 2008b; Yüksel & Kistler 2008; Yüksel et al. 2008; Ng et al. 2015).

Unfortunately, the claims of possible line detections (Loewenstein & Kusenko 2010; Boyarsky et al. 2014; Bulbul et al. 2014) all lie below the lower sensitivity cutoff for NuSTAR of $3.89\;\mathrm{keV}$ introduced by the redshift of the Bullet Cluster.

The Milky Way halo has provided strong constraints on line emission in the X-ray range (Riemer-Sørensen et al. 2006; Riemer-Sørensen 2014). The advantage of local constraints is that the dark matter source is nearby and there exists a wealth of observations, but the disadvantage is the number of sources of non-thermal "background" radiation and the uncertainty of the inner mass profile. The background radiation issue can be reduced significantly by point source removal if one has sufficient spatial resolution. This is now becoming possible with NuSTAR, and will be investigated further in future work. The profile uncertainty problem can be mitigated somewhat by excluding the center of the halo from the analysis.