Direct detection of dark matter bound to the Earth

We study the properties and direct detection prospects of an as of yet neglected population of dark matter (DM) particles moving in orbits gravitationally bound to the Earth. This DM population is expected to form via scattering by nuclei in the Earth’s interior. We compute fluxes and nuclear recoil energy spectra expected at direct detection experiments for the new DM population considering detectors with and without directional sensitivity, and different types of target materials and DM-nucleon interactions. DM particles bound to the Earth manifest as a prominent rise in the low-energy part of the observed nuclear recoil energy spectrum. Ultra-low threshold energies of about 1 eV are needed to resolve this effect. Its shape is independent of the DM-nucleus scattering cross-section normalization.

Figure 1

$K_A$ as a function of the DM particle mass $m_{\chi }$ for two elements in the Earth, namely Oxygen and Iron. $K_A$ is proportional to the probability of scattering towards a bound orbit of given ellipticity $e$ and perihelion $r_m$. In the figure we vary $e$ as reported in the legends, and fix $r_m$ to $R_{\oplus }/2$. The left panel refers to the interaction ${\mathcal{O}}_1$ with $c_1^0=2/m_V^2$ and $c_1^1=0$, whereas the right panel to the interaction ${\mathcal{O}}_{11}$ with $c_{11}^0=2/m_V^2$ and $c_{11}^1=0$. The parameter $m_V=246.2\mathrm{GeV}$ corresponds to the electroweak scale.

Figure 2

Accumulation time $T$ as a function of the perihelion $r_m$ for two reference values of the DM particle mass $m_{\chi }$ and of the ellipticity $e$. The left panel refers to the interaction ${\mathcal{O}}_1$ with $c_1^0=2/m_V^2$ and $c_1^1=0$, whereas the right panel to the interaction ${\mathcal{O}}_{11}$ with $c_{11}^0=2/m_V^2$ and $c_{11}^1=0$. Coupling constants are expressed in terms of the electroweak scale $m_V=246.2\mathrm{GeV}$. The accumulation time for the operator ${\mathcal{O}}_{11}$ is significantly larger than that of ${\mathcal{O}}_1$. Overall, $T\times K_A$ for $O_{11}$ is larger than the corresponding of $O_1$.

Figure 3

Rate of nuclear recoil events $dR/d{E}_R$ as a function of $E_R$. We assume dark matter-nucleon interactions of type ${\mathcal{O}}_1$ and $c_1^0=2\epsilon /m_V^2$, $c_1^1=c_4^0=c_4^1=c_{11}^0=c_{11}^1=0$ ($m_V=246.2\mathrm{GeV}$), with $\epsilon =1$ in the left panel, and $\epsilon$ equal to the value in the legends in the right panel (this allows us to consider different target materials in the same figure). The left panel reports results obtained for three different values of the DM particle mass, and assuming Germanium as a target material. In the right panel we fix $m_{\chi }=50\mathrm{GeV}$, and consider different target materials for $d{\sigma }_N/d{E}_R$, namely, Xenon, Germanium, Sodium, Iodine and Fluorine. In both panels, solid lines correspond to the total rates, including the contribution from halo and bound DM particles. Dashed lines represent the contribution to $dR/d{E}_R$ from halo DM particles. Vertical lines show illustrative energy thresholds of running or proposed dark matter direct detection experiments.

Figure 4

Same as for Fig. 3 but now for the interaction ${\mathcal{O}}_{11}$ and with $\epsilon =2$ for Xenon and 1 otherwise (information omitted in the legends for simplicity).

Figure 5

Left: Ratio of Eqs. (32) and (34) as a function of $m_{\chi }$ for three different dark matter-nucleon interactions. From the top to the bottom in the legend: ${\mathcal{O}}_1$, a modified version of ${\mathcal{O}}_1$ obtained by replacing $c_1^0$ with $c_1^0/v$ (i.e. resonant scattering), ${\mathcal{O}}_4$ and ${\mathcal{O}}_{11}$. In all cases we set $c_1^1$, $c_4^1$ and $c_{11}^1$ to zero, and assume Germanium as a target material. In the figure, we introduce the symbol $d\sigma /dE$ to characterize the scaling of $d{\sigma }_A/d{E}_R$ and $d{\sigma }_N/d{E}_R$ as a function of the dark matter-nucleus relative velocity $v$ and of the momentum transferred $q$. Right: Contribution of ${}^{16}\mathrm{O}$, ${}^{28}\mathrm{Si}$, ${}^{24}\mathrm{Mg}$, ${}^{56}\mathrm{Fe}$, ${}^{40}\mathrm{Ca}$, ${}^{23}\mathrm{Na}$, ${}^{32}\mathrm{S}$, ${}^{58}\mathrm{Ni}$, and ${}^{27}\mathrm{Al}$ to the ratio of Eqs. (32) and (34) as a function of the DM mass $m_{\chi }$. We assume ${\mathcal{O}}_1$ as dark matter-nucleon interaction, $c_1^0=2/m_V^2$ ($m_V=246.2\mathrm{GeV}$) and $c_1^1=0$. In both panels we assume a nuclear recoil energy of 1 eV in the evaluation of the scattering rates.

Figure 6

Left: Ratio of the double differential rates in Eqs. (52) and (56) as a function of the DM particle mass. We assume ${\mathcal{O}}_1$ as dark matter-nucleon interaction. The solid blue line refers to a hypothetical detector composed of Fluorine, whereas the dashed red line corresponds to a second hypothetical detector which uses ${}^3\mathrm{He}$ as a target material. Right: Same ratio as in the left panel now evaluated for the operators ${\mathcal{O}}_1$ (blue solid line) and ${\mathcal{O}}_{11}$ (red dashed line) for comparison. In both cases we assume Fluorine as a target material. In the two panels we assume a nuclear recoil energy of 1 eV.