Our Galaxy resides in the center of a vast "Halo" of Dark Matter (DM). This concentration produces, in many viable particle physics models, an indirect Weakly Interacting Massive Particle (WIMP) annihilation signal that peaks in the Fermi-LAT's energy range. Our knowledge of the diffuse background is essential to placing reasonable limits on the DM mass and cross-section. We incorporate a systematic variation of the GALPROP galactic diffuse background model, constrained by current cosmic-ray measurements, into a profile likelihood analysis and present preliminary upper limits on the DM annihilation cross-section using the Fermi-LAT data.
The Milky Way rests at the center of a massive, O(10 12 ) M [Diemand et al. (2007)], Halo of Dark Matter (DM). We have a good sense of the large-scale mass distribution of this Halo from N-body simulations, and can approximate that with a simple density profile function. For example, here we use an Einasto profile, where ρ(r) ∝ exp (-Ar α ). If this Halo is comprised of Weakly Interacting Massive Particles (WIMPs) capable of annihilation or decay into Standard Model (SM) particles, the Fermi-LAT provides a unique all-sky energy window to search for gamma-ray signal resulting from that process.
To capture the uncertainty in our DM signal, we consider WIMPs of mass 25 to 500 GeV annihilating to a variety of SM channels, including b b, t t, τ + τ -, and µ + µ -. And while N-body simulations are powerful, their inability to resolve small scale structure leaves another uncertainty in the “boost” that comes from extrapolating their mass functions down to the free-streaming cutoff. We therefore use three different mass function exponents (1.8,1.9,2.0) to cover the reasonable [Diemand et al. (2008)] range of possibilities.
While channels like b b deliver their gamma-ray signal mostly through π 0 decay, others (e.g.
µ + µ -) require an extra step in calculating the predicted Fermi-LAT signal. This is the propagation and interaction of the final SM decay products. This calculation is performed with the same methodology as the rest of our cosmic-ray (CR) backgrounds discussed in the next section.
Since we expect the Halo signal to be large-scale and slowly varying, masking out regions of the sky containing point sources greatly simplifies our analysis without much affecting our sample size. We mask the entire galactic plane to ±10 • , and Fermi sources with an energy-dependent sized mask corresponding to the LAT PSF.
After masking, we contend only with diffuse backgrounds to the Halo signal: extragalactic, instrumental, point source residuals, and galactic diffuse. The first two enter the fit as completely isotropic signals with fixed energy spectra [Abdo et al. (2010)]. Point source residuals, products of the extended tails of the PSF that reach outside our masking scheme, are modeled using Fermi-LAT software1 tuned to the Fermi First Source Catalogue [Abdo et al. (2010)] measured source parameters.
Originating from interactions (bremsstraulung, inverse compton, and π 0 decay) of CR with the gas, light, dust, and magnetic fields in the Milky Way, the galactic diffuse is by far the most difficult background to model. GALPROP [Strong et al. (2000)] is a code designed to self-consistently solve the transportation and interaction of CR within the galaxy from source injection to arrival at Earth radius. We assess the quality of a particular realization of GALPROP’s parameter space by comparing with available CR measurements. This is advantageous in the respect that it makes our model independent of the gamma rays we wish to probe for DM signals.
The large number and uncertainty of parameters in the GALPROP framework, however, requires us to make a thorough investigation of that space to quantify our systematics. we fit with models generated within the parameter space in Table 1. Each model’s validity is quantified by a χ 2 fit to 10 Be/ 9 Be, B/C, and proton from the HEAO-3, IMP, ATIC-2, CREAM, ACE, ISOMAX, AMS01, CAPRICE, and BESS experiments. The explicit form is,
where D and T are model and data respectively, ∆φ 2 comes from uncertainty in the solar modulation (taken to be ±100MV), and the sum is over all points and experiments.
One uncertainty obviously missing from the parameter scan is the normalization of the primary electron sources. The sensitivity of a local measurement to nearby sources (or lack of) makes this a particularly difficult value to set a parameter range for. Running each model with and without primary electrons and finding the difference isolates their contribution so that we can let it float freely in our fits to the data.
After binning into 12 annular and 80 logarithmic energy bins, we compare our model with the Fermi-LAT gamma-ray data and find the Poisson Likelihood, L [Mattox et al. (1996)]. For each DM channel and mass we then produce a Maximum Likelihood,
by selecting the best-fitting linear parameters, α (EGB, instrumental, and primary electron normalizations), for each DM normalization, θ DM . The j represents the diffuse model and the product is over each bin, i.
Finally, we convert L into a Test Statistic for every GALPROP model that includes the CR-χ 2 (Eq. 2.1) information, This leaves us with a family of curves (Fig. 1), the minimum contour of which comprises the Profile Likelihood. The contour should behave as a χ 2 with one degree of freedom, and we set our 95% confidence upper limit on θ DM to where it rises above the minimum by 3.84.
To be confident with the limit it is necessary to have sufficiently populated the model space such that the minimum is the true minimum,
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