# Galactic Substructure and Energetic Neutrinos from the Sun and the Earth

###### Abstract

We consider the effects of Galactic substructure on energetic neutrinos from annihilation of weakly-interacting massive particles (WIMPs) that have been captured by the Sun and Earth. Substructure gives rise to a time-varying capture rate and thus to time variation in the annihilation rate and resulting energetic-neutrino flux. However, there may be a time lag between the capture and annihilation rates. The energetic-neutrino flux may then be determined by the density of dark matter in the Solar System’s past trajectory, rather than the local density. The signature of such an effect may be sought in the ratio of the direct- to indirect-detection rates.

###### pacs:

95.35.+d,98.35.-a,98.35.Pr,98.85.RyNumerous experimental probes have confirmed indirectly the presence of a yet unknown form of gravitationally-interacting matter in Galactic halos that contributes roughly 20% of the total cosmic energy density. It is generally assumed that “dark matter” is in the form of some yet undiscovered elementary particle. Among the plethora of proposed theoretical particle dark-matter candidates, weakly interacting massive particles (WIMPs) are favored because they provide, quite generally, the correct relic abundance and because they may be experimentally accessible in the near future. WIMPs arise naturally in supersymmetric extensions (SUSY) of the Standard Model Jungman:1995df as well as in models with Universal Extra Dimensions (UEDs) Hooper:2007qk .

The two principle avenues toward dark-matter detection are direct detection (DD) via observation of the recoil of a nucleus, when struck by a halo WIMP, in a low-background experiment Goodman:1984dc ; Griest:1988ma ; and neutrino indirect detection () via observation of energetic neutrinos from annihilation of WIMPs that have been captured in the Sun (and/or Earth) neutrinos ; Gould:1987ir .

The DD rate is proportional to the local dark-matter density. The rate is proportional to the rate at which WIMPs annihilate in the Sun, which in turn depends on an integral of the square of the dark-matter density over the volume of the Sun. However, the WIMPs depleted in the Sun by annihilation are replenished by the capture of new WIMPs. In most cases where the signal is large enough to be detectable, the timescale for equilibration of capture and annihilation is small compared with the age of the Solar System. The rate is then also determined by the local dark-matter density. Since the capture rate is controlled by the same elastic-scattering process that occurs in DD, the DD and rates are roughly proportional Kamionkowski:1994dp .

In this Letter we investigate the effects of Galactic substructure on this canonical scenario. Analytic arguments and numerical simulations suggest that realistic Galactic halos should have significant substructure, remnants of smaller halos produced in early stages of the structure-formation hierarchy (which may themselves house remnants of even smaller structures, and so on) substructure . Theoretical arguments suggest that the substructure may be scale invariant Kamionkowski:2008vw with subhalos extending all the way down to sub-Earth-mass scales earthmass . The local dark-matter density of different locations at similar Galactocentric radii in the Milky Way may thus differ by a few orders of magnitude. The analytic descriptions of substructure are rough, and the simulations are limited by finite resolution, and this motivates the pursuit of avenues toward empirically probing the existence of substructure.

The purpose of this Letter to show that measurements of the ratio of DD to rates can be used to test for Galactic substructure. If there is Galactic substructure, then the dark-matter density at the position of the Solar System may vary with time. There is a finite time lag between capture and annihilation, and so the current energetic-neutrino flux may be determined not by the local dark-matter density, but rather by the density of dark matter along the past trajectory of the Solar System. The ratio for the /DD rate may thus differ from the canonical prediction. A departure from the canonical ratio would thus, if observed, provide information about Galactic substructure. Since the equilibration timescale in the Earth is generally different from that in the Sun, additional information might be provided by observation of energetic neutrinos from WIMP annihilation in the Earth.

To illustrate, we suppose the WIMP has a scalar coupling to nuclei, but the formalism can be easily generalized to spin-dependent WIMPs. Then the DD rate for a WIMP of mass from a target nucleus of mass is Jungman:1995df ; Kamionkowski:1994dp ,

(1) | |||||

where is the local dark-matter density in units of 0.3 GeV cm, and (given in Ref. Griest:1988ma ) accounts for form-factor suppression. Here, is the cross section for WIMP-nucleon scattering in units of cm.

The flux of upward muons induced in a neutrino telescope by neutrinos from WIMP annihilation in the Sun is

(2) | |||||

while the corresponding flux from the Earth is obtained by replacing the prefactor of Eq. (2) by . The function varies over the range over the mass range for the Sun (with a slightly larger range for the Earth), while the function is in the range over the same mass range.

The factor in Eq. (2) quantifies the number of WIMPs in the Sun. Once WIMPs are captured in the Sun, they accumulate deep within the solar core, where they may annihilate to a variety of heavy Standard Model particles which then decay to produce high-energy neutrinos (which may escape the Sun). The time evolution of the number of WIMPs in the Sun, is governed by the differential equation,

(3) |

where is the capture rate of WIMPs by the Sun, and is twice (because each annihilation destroys two WIMPs) the effective annihilation rate. If both and are constant and the initial condition is , the solution to this equation is

(4) |

where

(5) |

and is the equilibration timecale. After a time , the number approaches , and the annihilation rate becomes equal to (one half) the capture rate, .

The capture rate and annihilation coefficient , and thus the equilibration timescale , are determined by the cross sections for WIMPs to annihilate and to scatter from nuclei. The equilibration timescale evaluates to

(6) | |||||

Here, is the annihilation cross section (times relative velocity in the limit ) in units of . The equilibration timescale for the Earth is obtained by replacing the prefactor of Eq. (6) by yr. Using the canonical numbers we have adopted, the equilibration timescales for the Sun and Earth are both small compared with the age of the Solar System, but the equilibration timescales may vary by several orders of magnitude over reasonable ranges of the WIMP parameter space (and even more if more exotic physics, like a Sommerfeld Hisano:2004ds or self-capture enhancement Zentner:2009is , is introduced). To illustrate, we show in Fig. 1 the equilibration timescales, for various DD and ID rates, for realistic supersymmetric dark-matter candidates (using DarkSUSY Gondolo:2004sc ).

Suppose now the WIMP model parameters are determined, e.g., from the LHC and/or by theoretical assumption/modeling. Then the unknowns in Eqs. (1) and (2) will be the halo density and the number of WIMPs in the Sun (or Earth). The measured DD rate will then provide the local halo density . Measurement of the rate will then determine (in both the Sun and the Earth).

For example, suppose the equilibration timescale is years in the Sun, and that the Solar System entered a region of density (where GeV cm is the smoothed local halo density) a time years ago, e.g., a halo (see Fig. 2). We would then see a boosted DD rate and a boosted energetic-neutrino flux from the Sun, but the energetic-neutrino flux from the Earth would be correspondingly weaker, since the equilibration time in the Earth is longer. Now, suppose that the Solar System exited this high-density region a million years ago. The DD rate would be at the canonical value, but the energetic-neutrino fluxes from the Sun/Earth would still be boosted. Finally, suppose that the Solar System exited the high-density region years ago. In that case, the DD rate and energetic-neutrino flux from the Sun would have the canonical values, but the neutrino flux from the Earth would still be boosted, as .

In reality, the capture rate in Eq. (3) is a function of time, and the equation for the number of WIMPs in the Sun or Earth can be integrated numerically to give the annihilation rate as a function of time. To illustrate, imagine that all of halo dark matter was distributed in objects of a single mass, , each with a density Koushiappas:2009du . The radius of these subhalos would then be . The transit time of the Solar System through such an object is . The mean-free time between encounters with such objects is . In this toy model, the dark-matter density (and hence the DD rate) is zero unless the Solar System is within a subhalo. Fig. 3 illustrates the effects from such a scenario for . For equilibration timescales which are of order , the energetic-neutrino signal is depleted completely prior to the next encounter, while for longer equilibration timescales, the net effect is an elevated signal at all times. For example, if , and , the signal from the Earth will be boosted relative to the signal from the Sun for most of the time.

Finally, substructure may also speed up the equilibration between capture and annihilation in cases where the smooth-halo equilibration timescale is larger than the age of the Solar System. Suppose the dark matter has a smooth component and some substructure down to very small scales, . The abundance of the smallest subhalos can be inferred by extrapolating the subhalo mass function measured in simulations at larger mass scales. If we take the density within these objects to be times the smooth value, the radius of these objects is ; the crossing time is 50 years; and the mean time between encounters is roughly a million years. For equilibration timescales less than the age of the Sun (), the signal will be roughly at the equilibrium value of the smooth component for most of the time. However, for long equilibration timescales (e.g. the Earth), the amount of depletion between interactions is negligible. This effect leads to a continuous build-up of WIMPs in the Earth, augmented by brief periods of an increased capture due to interactions with the subhalos. This results in an energetic-neutrino signal that today is higher than the signal that would be obtained from the smooth component. This can be understood as follows: For , the second term in Eq. (3) is small. While the cross section for the Solar System trajectory to intersect subhalos is , the capture rate while in them is , thus giving rise to a net increase in the capture rate . Fig. 4 shows the net effect of this speed-up.

In summary, we considered the effects of Galactic substructure on energetic neutrinos from WIMP annihilation in the Sun and the Earth. While DD experiments depend on the local dark-matter density and velocity distribution, the energetic-neutrino fluxes from the Sun and the Earth depend on the past trajectory of the Solar System through the clumpy Galactic halo. If experimental DD and signals are obtained before the dark-matter particle-physics parameters are known, then the potential for probing dark matter via the DD/ ratios will be compromised by the particle-physics uncertainties. If, however, the particle-physics parameters are known, then measurement of the DD rate and the rates from the Sun and Earth can be used to probe Galactic substructure at the solar radius, with the equilibration timescales setting roughly the mass scales that can be probed.

We acknowledge useful conversations with J. Beacom, I. Dell’Antonio, R. Gaitskell, A. Geringer-Sameth, G. Jungman, L. Strigari, and A. Zentner. SMK thanks the Caltech/JPL W. M. Keck Institute for Space Studies for hospitality during the preparation of this article. The work of MK was supported by DoE DE-FG03-92-ER40701 and the Gordon and Betty Moore Foundation.

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