Effect of Gravitational Focusing on Annual Modulation in Dark-Matter Direct-Detection Experiments

The scattering rate in dark-matter direct-detection experiments should modulate annually due to Earth’s orbit around the Sun. The rate is typically thought to be extremized around June 1, when the relative velocity of Earth with respect to the dark-matter wind is maximal. We point out that gravitational focusing can alter this modulation phase. Unbound dark-matter particles are focused by the Sun’s gravitational potential, affecting their phase-space density in the lab frame. Gravitational focusing can result in a significant overall shift in the annual-modulation phase, which is most relevant for dark matter with low scattering speeds. The induced phase shift for light $O(10)\mathrm{GeV}$ dark matter may also be significant, depending on the threshold energy of the experiment.

Figure 1

A schematic illustration of the effect of gravitational focusing on unbound DM particles. The phase-space density of DM at Earth is greater around March 1 than around September 1 due to this effect.

Figure 2

For the standard halo model, the phase-space density $\rho f_v{|}_t$ (dashed black curve) is maximal at $300\mathrm{km}/\mathrm{s}$; however, the difference in the phase-space density between Spring and Fall (dotted red curve) is maximal at $\sim 200\mathrm{km}/\mathrm{s}$. Note that these two plots are normalized separately. Gravitational focusing is most significant at low Earth-frame speeds $v$, as indicated by the plot of the modulation fraction (inset).

Figure 3

Left: The time $t_0$ when the differential rate is maximized changes as a function of $v_{\min }$. Without GF, this time is around June 1 for $v_{\min }>200\mathrm{km}/\mathrm{s}$ and about half a year earlier for smaller $v_{\min }$. With GF, $t_0$ is still approximately June 1 at high $v_{\min }$, but as $v_{\min }$ decreases, GF becomes more significant, and $t_0$ ultimately asymptotes to a value $\approx 21$ days later than that expected with no GF. The dot-dashed brown line marks March 1, which is the time when GF is maximal. The orange region roughly accounts for the astrophysical error in $t_0$ by varying $v_0$ from 180 to $260\mathrm{km}/\mathrm{s}$. Right: The time ${\overline{t}}_0$ that maximizes the binned rate as a function of the minimal bin energy $E_{\min }$ for DM masses 8, 15, and 50 GeV (thick lines). The thin lines show the corresponding phases when GF is neglected. We assume $1{\mathrm{keV}}_{\mathrm{nr}}$ energy bins at a germanium detector; note that the shapes of these curves are highly sensitive to the bin size and target nucleus.