Dark matter in the Solar System. III. The distribution function of WIMPs at the Earth from gravitational capture

In this last paper in a series of three on weakly interacting massive particle (WIMP) dark matter in the Solar System, we focus on WIMPs bound to the system by gravitationally scattering off of planets. We present simulations of WIMP orbits in a toy solar system consisting of only the Sun and Jupiter. As previous work suggested, we find that the density of gravitationally captured WIMPs at the Earth is small and largely insensitive to the details of elastic scattering in the Sun. However, we find that the density of gravitationally captured WIMPs may be affected by external Galactic gravitational fields. If such fields are unimportant, the density of gravitationally captured WIMPs at the Earth should be similar to the maximum density of WIMPs captured in the Solar System by elastic scattering in the Sun. Using standard assumptions about the halo WIMP distribution function, we find that the gravitationally captured WIMPs contribute negligibly to direct detection event rates. While these WIMPs do dominate the annihilation rate of WIMPs in the Earth, the resulting event rate in neutrino telescopes is too low to be observed in next-generation neutrino telescopes.

Figure 1

The total distribution function from the gravitational capture simulations compared against several theoretical distribution functions.

Figure 2

The initial angular momentum distributions for all bound WIMPs (upper two curves), and the distribution of the initial angular momenta for bound WIMPs that contribute to the WIMP DF at the Earth (lower two curves). The distributions are divided by whether the WIMPs were initially moving prograde or retrograde with respect to Jupiter.

Figure 3

Lifetime distributions for initially prograde or retrograde WIMPs. The solid lines show the lifetime distributions for WIMPs with ${\widehat{J}}^2<0.5$, and dashed curves mark those with ${\widehat{J}}^2>0.5$.

Figure 4

Distribution of times at which WIMPs cross the Earth’s orbit.

Figure 5

The distribution of total optical depth per particle of particles that enter the Sun. The solid line indicates the distribution for the Regular run simulation, while the dashed line indicates that of the High Perihelion run.

Figure 6

The DF of bound particles from the gravitational capture simulations. The squares mark the DF of all gravitationally bound WIMPs. The circles indicate the DF of particles before they are lost to the Solar System by Galactic tides. The triangles indicate the DF of particles that never enter the Sun.

Figure 7

Comparison of the geocentric bound distribution functions for $m_{\chi }=500\mathrm{AMU}$ and ${\sigma }_p^{\mathrm{SI}}={10}^{-43}{\mathrm{cm}}^2$. The dot-dashed line indicates the phase space density of unbound orbits. The squares show the results from the gravitational capture simulations, the circles are the Large Mass DF (${\sigma }_p^{\mathrm{SD}}=0$), and the solid magenta line indicates the estimated maximum DF resulting from spin-dependent scattering in the Sun (${\sigma }_p^{\mathrm{SD}}={10}^{-36}{\mathrm{cm}}^2$).

Figure 8

The differential direct detection signal from the halo, the gravitationally bound WIMP population, and the detailed balance estimate for the toy model solar system. The shaded region indicates the XENON10 analysis region [6], and the vertical dashed line indicates the lower limit to the CDMS analysis window (which extends to $Q=100\mathrm{keV}$) [8].

Figure 9

The maximum contribution to the differential direct detection rate for $m_{\chi }=500\mathrm{AMU}$ and ${\sigma }_p^{SI}={10}^{-43}{\mathrm{cm}}^2$. The upper lines represent the spin-independent event rate of halo WIMPs. The lower solid lines show the event rate of gravitationally bound WIMPs; the lower dashed lines also include WIMPs bound to the Solar System by solar capture assuming ${\sigma }_p^{\mathrm{SD}}={10}^{-36}{\mathrm{cm}}^2$).

Figure 10

Capture rate of WIMPs in the Earth as a function of WIMP mass for ${\sigma }_p^{\mathrm{SI}}={10}^{-43}{\mathrm{cm}}^2$.

Figure 11

Muon event rates from (a) halo WIMPs unbound to the Solar System, (b) halo and gravitationally bound WIMPs, and (c) halo and all bound WIMPs. Open circles mark MSSM models for which ${\sigma }_p^{\mathrm{SI}}$ is above the 2006 CDMS limit [48], filled triangles mark those with limits between that limit and the current best limits on ${\sigma }_p^{\mathrm{SI}}$ (set by XENON10 for $m_{\chi }<40\mathrm{GeV}$ [6] and CDMS for $m_{\chi }>40\mathrm{GeV}$ [8]), and filled squares denote models consistent with the best limits on elastic scattering cross sections. The solid line is an optimistic detection threshold for the IceCube experiment (25 and references therein).