Clumpiness of Dark Matter and Positron Annihilation Signal: Computing the odds of the Galactic Lottery

Abstract

The small-scale distribution of dark matter in Galactic halos is poorly known. Several studies suggest that it could be very clumpy, which turns out to be of paramount importance when investigating the annihilation signal from exotic particles (e.g. supersymmetric or Kaluza-Klein). In this paper we focus on the annihilation signal in positrons. We estimate the associated uncertainty, due to the fact that we do not know exactly how the clumps are distributed in the Galactic halo. To this aim, we perform a statistical study based on analytical computations, as well as numerical simulations. In particular, we study the average and variance of the annihilation signal over many Galactic halos having the same statistical properties. We find that the so-called boost factor used by many authors should be handled with care, as i) it depends on energy and ii) it may be different for positrons, antiprotons and gamma rays, a fact which has not received any attention before. As an illustration, we use our results to discuss the positron spectrum measurements by the HEAT experiment.

Figures (11)

• Figure 1: The effective boost factor $B_\text{eff}$ is featured as a function of the positron energy $E$ in the case of a 100 GeV line. A fraction $f = 0.2$ of the DM distribution is in the form of substructures whose individual boost factor $B_{c}$ -- relative to the solar neighbourhood density -- has been varied from 3 to 100. An isothermal halo -- panel a -- and a NFW profile -- panel b -- are considered. They illustrate the influence of the central profile index. The increase of $B_\text{eff}$ is noticeable especially around $E \sim 10$ GeV.
• Figure 2: The relative variance ${\sigma_{r}}/{\langle \phi_{r} \rangle}$ of the random component of the positron flux at the Earth -- solid lines -- and its hard-sphere approximation -- long-dashed curves -- are featured as a function of the positron energy $E$ for three different values of the clump mass $M_{c}$. The injected positron energy $E_{S}$ has been set equal to 100 GeV. A NFW profile with typical scale 25 kpc has been assumed. At fixed clump mass, the variance increases with $E$ and matches its hard-sphere approximation above $\sim$ 40 GeV. As the number of clumps is decreased, the curves are shifted upwards by a factor of $1/\sqrt{N_H} \propto \sqrt{M_c}$. The relative variance ${\sigma_{B}}/{B_\text{eff}}$ of the boost factor is also displayed -- short-dashed curve. In the limit where the clump boost factor $B_{c}$ is large -- a value of 100 has been assumed here -- ${\sigma_{B}}/{B_\text{eff}}$ and ${\sigma_{r}}/{\langle \phi_{r} \rangle}$ are approximately equal.
• Figure 3: The effective boost factor $B_\text{eff}$ -- black line -- is plotted as a function of the positron energy $E$ for an injected energy $E_{S} = 100$ GeV. The 1-$\sigma$ range of its fluctuations extends from $B_{\rm min} = B_\text{eff} - {\sigma_{B}}$ up to $B_\text{max} = B_\text{eff} + {\sigma_{B}}$. At fixed clump mass, that range opens up as $E$ approaches the injected energy $E_{S} = 100$ GeV. It also widens significantly at fixed positron energy $E$ when the number of clumps is decreased.
• Figure 4: The density of probability $\mathcal{P} \! \left( \Phi \right)$ is plotted as a function of the reduced flux $\Phi = \varphi / \varphi_\text{max}$ which a single clump generates. A NFW halo has been assumed with a scale radius of 25 kpc. The domain $\mathcal{D}_{H}$ over which the probability is normalized to unity is the Milky Way DM halo up to a radius of 20 kpc. The injection energy is $E_{S} = 100$ GeV. The smaller the positron energy $E$ at the Earth, the larger the probability density for a non-vanishing flux. The fully numerical calculations -- solid curves -- are compared to the infinite 3D approximation (\ref{['analytic_mathcal_P_Phi']}) that corresponds to the long-dashed lines.
• Figure 5: The density of probability $\mathcal{P} \! \left( \Phi \right)$ as well as $\Phi \, \mathcal{P} \! \left( \Phi \right)$ and $\Phi^{2} \, \mathcal{P} \! \left( \Phi \right)$ are featured as a function of the reduced flux $\Phi = \varphi / \varphi_\text{max}$ for a positron energy at the Earth of 50 GeV.