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

The next generation of dark matter (DM) direct detection experiments and neutrino telescopes will probe large swaths of dark matter parameter space. In order to interpret the signals in these experiments, it is necessary to have good models of both the halo DM streaming through the Solar System and the population of DM bound to the Solar System. In this paper, the first in a series of three on DM in the Solar System, we present simulations of orbits of DM bound to the Solar System by solar capture in a toy solar system consisting of only the Sun and Jupiter, assuming that DM consists of a single species of weakly interacting massive particle (WIMP). We describe how the size of the bound WIMP population depends on the WIMP mass $m_{\chi }$, spin-independent cross section ${\sigma }_p^{\mathrm{SI}}$, and spin-dependent cross section ${\sigma }_p^{\mathrm{SD}}$. Using a standard description of the Galactic DM halo, we find that the maximum enhancement to the direct detection event rate, consistent with current experimental constraints on the WIMP-nucleon cross section, is $<1\%$ relative to the event rate from halo WIMPs, while the event rate from neutrinos from WIMP annihilation in the center of the Earth is unlikely to meet the threshold of next-generation, ${\mathrm{km}}^3$-sized (IceCube, KM3NeT) neutrino telescopes.

Figure 1

Jacobi constant errors as a function of distance from the primary for a trajectory with $a=1.73\mathrm{AU}$, followed for $2\times {10}^4$ Kepler periods. This trajectory was integrated with 500 steps/orbit. Errors are calculated at perihelion and aphelion. Points to the left of the vertical line lie within the volume of the Sun; however, we used a point-mass Sun for this integration.

Figure 2

Errors in the Jacobi constant as a function of eccentricity and semimajor axis. Each point shows the maximum error for 10 trajectories initialized with the same eccentricity but with random initial orientation, and followed for $2\times {10}^4$ Kepler orbits. Open points denote those trajectories for which the semimajor axis $a=a_{\mathrm{J}}/3=1.73\mathrm{AU}$; closed points refer to trajectories with $a=a_{\mathrm{J}}/6=0.87\mathrm{AU}$. Circles mark trajectories with initial eccentricity $e_i=0.9999$, squares denote those with $e_i=0.999$, diamonds indicate those with $e_i=0.99$, and triangles those with $e_i=0.9$.

Figure 3

Error in the Jacobi constant as a function of time for several particles. The Jacobi constant is recorded at aphelion at ${10}^5\mathrm{yr}$ intervals. Top left: A particle with $a=1.54\mathrm{AU}$. This particle repeatedly goes through the Sun (about ${10}^7$ times), but never goes through the bubble around Jupiter. It is integrated with $h=6\times {10}^{-5}{R}_{\odot }^{-1}\mathrm{yr}$, which corresponds to $\approx 100\mathrm{steps}/\mathrm{orbit}$. Top right: A particle that gets stuck near a Sun-skimming $2:1$ resonance with Jupiter. This particle repeatedly goes through the Jupiter bubble. It is integrated with $h=2\times {10}^{-5}{R}_{\odot }^{-1}\mathrm{yr}$, or $\approx 350\mathrm{steps}/\mathrm{orbit}$. Bottom left: A particle gets stuck near a $3:2$ resonance with Jupiter. This orbit was integrated with $h=1.5\times {10}^{-5}{R}_{\odot }^{-1}\mathrm{yr}$, or $\approx 650\mathrm{steps}/\mathrm{orbit}$. Bottom right: This particle repeatedly crosses $r_c$, the transition radius between barycentric and heliocentric coordinates (dashed line marks $r_c/2$, the crossing semimajor axis for an orbit with $e\sim 1$) and has its last aphelion before ejection from the solar system at $t=1.6\times {10}^6\mathrm{years}$. It is integrated with $h=2\times {10}^{-6}{R}_{\odot }^{-1}\mathrm{yr}$, or $9\times {10}^3\mathrm{steps}/\mathrm{orbit}$.

Figure 4

Points in the ${\sigma }_p^{\mathrm{SI}}-m_{\chi }$ parameter space used for the solar capture simulations, plotted along with exclusion curves from recent experiments. This plot was generated with http://dendera.berkeley.edu/plotter/entryform.html.

Figure 5

Flowchart for the simulation algorithm for the solar capture experiments.

Figure 6

Geocentric distribution functions from the simulations. (a) Results from all simulations. (b) The CDMS distribution function relative to theoretical distribution functions for unbound WIMPs.

Figure 7

Distribution functions divided by $n_{\chi }$ in the $v-\cos \theta$ plane (integrated over $\varphi$) for both (a) heliocentric and (b) geocentric frames. These DFs come from the CDMS simulation, and the units are $(\mathrm{km}{\mathrm{s}}^{-1}{)}^{-3}$.

Figure 8

Locations of various types of orbits in the (a) $\varphi =0$ and (b) $\varphi =\pi /2$ slices of heliocentric velocity-space, and (c) $\varphi =0$ and (d) $\varphi =\pi /2$ slices of geocentric velocity-space.

Figure 9

Particle lifetime distributions for the DAMA (solid), CDMS (dot-dashes), Medium Mass (short dashes), and Large Mass (long dashes) simulations.

Figure 10

Lifetime distributions as a function of initial semimajor axis.

Figure 11

Growth of the distribution functions as a function of time for the (a) DAMA and (b) CDMS simulations.

Figure 12

Percentages of particles in each initial perihelion bin. Poisson errors smaller than points.

Figure 13

In each plot, the red solid line denotes all species in the Sun, and the dotted blue line represents hydrogen. (a): The capture rate $\dot{N}$ of WIMPs by the Sun for ${\sigma }_p^{\mathrm{SI}}={10}^{-43}{\mathrm{cm}}^2$, divided by the halo number density of WIMPs. The short solid black line gives the slope $\dot{N}/n_{\chi }\propto m_{\chi }^{-1}$, the limiting slope for $m_{\chi }\gg m_A$ for a nuclear species $A$. (b): The capture rate ${\dot{N}}_{\oplus }$ to Earth-crossing orbits divided by the halo WIMP number density.

Figure 14

DFs for the three simulations with ${\sigma }_p^{\mathrm{SI}}={10}^{-43}{\mathrm{cm}}^2$ scaled by ${\dot{N}}_{\oplus }$.

Figure 15

The estimated maximum DF consistent with current exclusion limits if spin-independent scattering dominates in the Sun, for a WIMP mass $m_{\chi }=2\mathrm{TeV}$ and WIMP-proton cross section ${\sigma }_p^{\mathrm{SI}}={10}^{-42}{\mathrm{cm}}^2$.

Figure 16

Predicted geocentric DF if ${\sigma }_p^{\mathrm{SD}}={10}^{-36}{\mathrm{cm}}^2$, assuming $f(v)\propto {\sigma }_p^{\mathrm{SI},\mathrm{SD}}$ for Jupiter-crossing orbits. This prediction is based on the output of the DAMA simulation.

Figure 17

Estimated geocentric DFs for ${\sigma }_p^{\mathrm{SD}}=1.3\times {10}^{-39}$, ${10}^{-38}$, ${10}^{-37}$, and ${10}^{-36}{\mathrm{cm}}^2$. The estimated DF for ${\sigma }_p^{\mathrm{SD}}=1.3\times {10}^{-39}{\mathrm{cm}}^2$ is based on the DAMA simulation result, since the optical depth of the Sun for ${\sigma }_p^{\mathrm{SI}}={10}^{-41}{\mathrm{cm}}^2$ is approximately the same as ${\sigma }_p^{\mathrm{SD}}=1.3\times {10}^{-39}{\mathrm{cm}}^2$.

Figure 18

The differential direct detection rate for $m_{\chi }=500\mathrm{AMU}$ and ${\sigma }_p^{\mathrm{SI}}={10}^{-43}{\mathrm{cm}}^2$ assuming the DF is dominated by spin-dependent scattering in the Sun with ${\sigma }_p^{\mathrm{SD}}={10}^{-36}{\mathrm{cm}}^2$. The shaded region indicates the XENON10 analysis region [32], and the vertical dashed line indicates the lower limit to the CDMS analysis window (which extends to $Q=100\mathrm{keV}$) [64].

Figure 19

Capture rate of WIMPs in the Earth as a function of WIMP mass for ${\sigma }_p^{\mathrm{SI}}={10}^{-43}{\mathrm{cm}}^2$. All capture rates include the capture of unbound halo WIMPs as well as the capture of bound WIMPs.

Figure 20

Muon event rates from halo WIMPs unbound to the Solar System. Open circles mark MSSM models for which ${\sigma }_p^{\mathrm{SI}}$ is above the 2006 CDMS limit [85], filled triangles mark those with limits between that limit and the current best limits on ${\sigma }_p^{\mathrm{SI}}$ (set by XENON10 for $m_{\chi }<40\mathrm{GeV}$ [32] and CDMS for $m_{\chi }>40\mathrm{GeV}$ [64]), 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(36, and references therein).

Figure 21

Muon event rates including bound WIMPs. Symbols mark the same models as in Fig. 20.