Terrestrial density of strongly-coupled relics

The simplest cosmologies motivate the consideration of dark matter subcomponents that interact significantly with normal matter. Moreover, such strongly coupled relics may have evaded detection to date if upon encountering the Earth they rapidly thermalize down to terrestrial temperatures, $T_{\oplus }\sim 300\mathrm{K}\sim 25\mathrm{meV}$, well below the thresholds of most existing dark matter detectors. This shedding of kinetic energy implies a drastic enhancement to the local density, motivating the consideration of alternative detection techniques sensitive to a large density of slowly moving dark matter particles. In this work, we provide a rigorous semianalytic derivation of the terrestrial overdensities of strongly coupled relics, with a particular focus on millicharged particles (MCPs). We go beyond previous studies by incorporating improved estimates of the MCP-atomic scattering cross section, new contributions to the terrestrial density of sub-GeV relics that are independent of the Earth’s gravitational field, and local modifications that can arise due to the cryogenic environments of precision sensors. We also generalize our analysis in order to estimate the terrestrial density of thermalized MCPs that are produced from the collisions of high-energy cosmic rays and become bound by the Earth’s electric field.

Figure 1

Cartoon depiction of a millicharged relic particle $\chi$ entering the Earth’s atmosphere with an initially virialized effective temperature $T_{\mathrm{vir}}\sim {10}^{-6}{m}_{\chi }$. Before reaching an underground detector, $\chi$ sheds its kinetic energy by scattering with nuclei in the Earth’s atmosphere and crust, thermalizing down to terrestrial temperatures $T_{\oplus }\sim 300\mathrm{K}$. Since the interaction rate is strongly peaked towards smaller momentum transfer, thermalization significantly shortens the mean free path of $\chi$. After thermalizing, $\chi$ diffuses throughout the Earth and may become gravitationally bound depending on its mass.

Figure 2

Schematic cartoon outlining the journey of a millicharged relic particle $\chi$ thermalizing in the Earth’s environment. A virialized MCP $\chi$ enters the Earth’s atmosphere from the left, traveling to the point $x_{\mathrm{FTS}}$ until it suddenly thermalizes with terrestrial matter [orange (dark gray) dot]. At terrestrial temperatures, the mean free path of $\chi$ is greatly reduced as it diffuses throughout the Earth. Once an outwardly traveling thermalized MCP passes a surface located at $x_{\mathrm{LSS}}$ [yellow (light gray) dot], it is able to free stream out of the atmosphere or remains gravitationally or electrically bound and begins the entire process again, depending on the MCP mass and charge. At positions to the left of $x_{\mathrm{FTS}}$, the flow of MCPs is governed by both the Galactic flux $j_{\mathrm{vir}}$ as well as the thermal diffusion flux $j_{\chi }$. The boundary condition involving $j_{\chi }$ and $j_{\mathrm{vir}}$ imposed at $x_{\mathrm{FTS}}$ and $x_{\mathrm{LSS}}$ is discussed in Sec. 7. In particular, at positions to the right of $x_{\mathrm{FTS}}$, the flow is solely governed by the thermal diffusion flux $j_{\chi }$.

Figure 3

Thermally averaged transfer cross section $⟨{\sigma }_T⟩$ for MCP scattering off of room-temperature nitrogen (averaged between positively and negatively charged MCPs) as a function of the MCP mass $m_{\chi }$ for various values of the effective charge $q_{\chi }$. The solid lines take the MCP phase space to track that of the virialized Galactic population, corresponding to MCPs that have not yet thermalized with the Earth, whereas the dotted lines assume that MCPs have equilibrated to room temperature $T_{\oplus }\simeq 300\mathrm{K}$.

Figure 4

Left: MCP-nuclear momentum drag length [see Eq. (5.8)] [i.e., the distance traversed by an MCP before exchanging an $\mathcal{O}(1)$ fraction of its momentum due to scattering with terrestrial nuclei] as a function of the MCP mass at a depth of 1 km underground and for various choices of the MCP coupling. The solid lines correspond to an MCP phase space that tracks the virialized Galactic DM distribution, whereas the dotted lines assume that the MCPs have thermalized to terrestrial temperatures. Right: position $x_{\mathrm{FTS}}$ at which a virialized MCP from the Galactic halo thermalizes with the Earth (solid lines) and the position $x_{\mathrm{LSS}}$ at which an outwardly traveling MCP that is thermalized with the Earth free streams out of the atmosphere (dotted lines) as a function of mass for various choices of the MCP coupling. These positions are shown as the corresponding height with respect to the Earth’s surface.

Figure 5

Gravitationally bound hydrostatic overdensity (solid lines) of MCP relics as a function of the MCP mass, evaluated at various depths underground (from darkest to lightest blue: 6 Mm, 4 Mm, 2 Mm, and 1 km underground). The MCP coupling is assumed to be large enough for MCPs to rapidly thermalize in the Earth’s environment and the temperature at which these particles evaporate from the Earth is fixed to $T_{\oplus }=300\mathrm{K}$. Also shown as the dashed line is an example of the electrically bound hydrostatic population, which tracks the gravitationally bound density at large masses.

Figure 6

Traffic jam (solid colored lines) and bound hydrostatic (dashed gray line) overdensity for positively charged effective MCP relics, evaluated at a depth of 1 km, as a function of mass and for various choices of the charge $q_{\chi }$. For the hydrostatic density, we assume a temperature at the last scattering surface of $T_{\oplus }=300\mathrm{K}$.

Figure 7

Terrestrial overdensity of MCP relics evaluated at a depth of 1 km underground, incorporating both the traffic jam and hydrostatic populations, as a function of the MCP mass $m_{\chi }$ and coupling $q_{\chi }$. The shaded green regions correspond to different values of the overdensity $n_{\chi }/n_{\mathrm{vir}}$, as labeled by the white text. In the top-left and top-right panels, these overdensities are evaluated for positively and negatively charged effective MCPs, respectively, corresponding to models in which the terrestrial electric field does not bind such particles to the Earth (see Sec. 4). The bottom panel presents our results for positively charged pure MCPs where the terrestrial electric field binds these particles to the Earth (we do not show results for negatively charged pure MCPs, since upon thermalization they are ejected from the lower atmosphere by the Earth’s electric field). In this panel, the incoming flux of MCPs is significantly affected by the Earth’s magnetic field above the pink dotted line. Also shown in gray are existing constraints on MCPs from accelerator probes and neutrino experiments [47, 48, 49, 50, 51] as well as observations of SN1987A [52]. Above the orange and red dashed lines, Galactic MCPs rapidly thermalize before traveling a distance of 1 m or $R_{\oplus }/10$ in the Earth’s crust, respectively. Thus, the sensitivity of underground direct-detection experiments is exponentially suppressed above the orange line, while below the red line MCPs do not rapidly thermalize in the Earth. For couplings above the dashed cyan line in the top-right panel, negatively charged MCPs efficiently form bound states with iron nuclei. Thus, in this region of parameter space, the formation of $\chi \text{−}N$ bound states should drastically modify our estimates.

Figure 8

In shaded blue with corresponding white labels we show the enhancement to the local traffic jam density in a 1-m-sized 100 mK cryogenic environment, as compared to room temperature, as a function of the MCP mass $m_{\chi }$ and coupling $q_{\chi }$. These overdensities are evaluated for positively charged effective MCPs scattering off of oxygen nuclei $N$. Also shown in gray are existing constraints on MCPs from accelerator probes and neutrino experiments [47, 48, 49, 50, 51] as well as observations of SN1987A [52]. Above the red dashed line, Galactic MCPs rapidly thermalize before traveling a distance of $R_{\oplus }/10$ in the Earth’s crust. In the region labeled $T_{\chi }\simeq 300\mathrm{K}$, room-temperature MCPs do not efficiently thermalize with the cryogenic region within a distance of 1 m. For MCP masses close to $m_N\sim 15\mathrm{GeV}$ or $m_{\mathrm{diff}}\equiv 100\mathrm{mK}/(g\times 1\mathrm{m})\sim 80\mathrm{GeV}$, our analytic approximations are not expected to hold.

Figure 9

Analogous to Fig. 7, we show the total thermalized density of MCPs evaluated at a depth of 1 km underground, but now shown for MCPs produced by the collisions of high-energy cosmic rays. In the left panel, we assume that the terrestrial electric field does not couple to MCPs (as in models where the coupling is generated by a kinetically mixed light dark photon), resulting in a greatly diminished terrestrial density. In this case, we average the momentum drag rate ${\mathrm{\Gamma }}_p$ for positively and negatively charged MCPs before evaluating the terrestrial density. In the right panel, we instead assume that MCPs efficiently couple to the Earth’s electric field and thus can remain electrically bound to the Earth after thermalizing to terrestrial temperatures. Since in this case negatively charged MCPs are expelled from the atmosphere, this panel only shows the density of positively charged MCPs.