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Cold dark matter is thought to fill our galactic neighborhood with a density $\rho$ of about 0.3 GeV/cm${}^3$ and with a velocity $v$ of roughly 200 to 300 km/s. (The velocity dispersion is much debated.) For a given dark matter mass $m$ and nucleon scattering cross section $\sigma$, this will lead to a constant collision rate of roughly

$r \sim \rho v \sigma / m$

for every nucleon in normal matter. The kinetic energy transferred to the nucleon (which is essentially at rest) will be roughly

$\Delta E \sim 2 v^2 \frac{M m^2}{(m+M)^2}$,

where $M \approx 1$ amu $\approx 1$ GeV/c${}^2$ is the mass of a nucleon. The limits for light ($m \ll M$) and heavy ($m \gg M$) dark matter are

$\Delta E_\mathrm{light} \sim 2 v^2 \frac{m^2}{M}$ and $\Delta E_\mathrm{heavy} \sim 2 v^2 M$.

This leads to an apparent intrinsic heat production in normal matter

$\tilde{P} \sim r \Delta E / M$,

which is measured in W/kg. The limits are

$\tilde{P}_\mathrm{light} \sim 2 \rho v^3 \sigma m / M^2$ and $\tilde{P}_\mathrm{heavy} \sim 2 \rho v^3 \sigma / m$.

What existing experiment or observation sets the upper limit on $\tilde{P}$?

(Note that $\tilde{P}$ is only sensibly defined on samples large enough to hold onto the recoiling nucleon. For tiny numbers of atoms--e.g. laser trap experiments--the chance of any of the atoms colliding with dark matter is very small, and those that do will simply leave the experiment.)

The best direct limit I could find looking around the literature comes from dilution refrigerators. The NAUTILUS collaboration (resonant-mass gravitational wave antenna) cooled a 2350 kg aluminum bar down to 0.1 K and estimated that the bar provided a load of no more than 10 $\mu$W to the refrigerator. Likewise, the (state-of-the-art?) Triton dilution refrigerators from Oxford Instruments can cool a volume of (240 mm)${}^3$ (which presumably could be filled with lead for a mass of about 150 kg) down to ~8mK. Extrapolating the cooling power curve just a bit, I estimated it handled about $10^{-7}$ W at that temperature.

In both cases, it looked like the direct limit on intrinsic heating is roughly $\tilde{P} < 10^{-9}$W/kg.

However, it looks like it's also possible to use the Earth's heat budget to set a better limit. Apparently, the Earth produces about 44 TW of power, of which about 20 TW is unexplained. Dividing this by the mass of the Earth, $6 \times 10^{24}$ kg, limits the intrinsic heating to $\tilde{P} < 3 \times 10^{-12}$W/kg.

Is this Earth-heat budget argument correct? Is there a better limit elsewhere?

To give an example, the CDMS collaboration searches for (heavy) dark matter in the range 1 to 10${}^3$ GeV/c${}^2$ with sensitivities to cross sections greater than 10${}^{-43}$ to 10${}^{-40}$ cm${}^2$ (depending on mass). A 100 GeV dark matter candidate with a cross-section of 10${}^{-43}$ cm${}^2$ would be expected to generate $\tilde{P} \sim 10^{-27}$ W/kg, which is much too small to be observed.

On the other hand, a 100 MeV dark matter particle with a cross-section of $10^{-27}$ cm${}^2$ (which, although not nearly as theoretically motivated as heavier WIMPs, is not excluded by direct detection experiments) would be expected to generate $\tilde{P} \sim 10^{-10}$ W/kg. This would have shown up in measurements of the Earth's heat production.

EDIT: So it looks like I completely neglected the effects of coherent scattering, which has the potential to change some of these numbers by 1 to 2 orders of magnitude. Once I learn more about this, I will update the question.

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Great Question! Starting a bounty on it! –  FrankH Oct 22 '12 at 20:21
Unfortunately still no answers. Perhaps this preprint on astroseismology might tangentially spark ideas. It just came out today, and is very preliminary, but the basic idea is DM may cool cores of stars via scattering, and such cooling may alter the internal structure enough to detectably change the pulsation spectrum. –  Chris White Dec 17 '12 at 0:10
I can't offer much --- but I did recently hear a talk about DM direct detection (regarding: arxiv.org/abs/1211.1377.pdf), and this question was asked. The rough answer was that heating can't provide as good constraints on cross-sections as lower-limits from WMAP signatures, and upper-limits from line-surveys. –  zhermes Dec 26 '12 at 21:48
Concerning the geological heating limit: potassium-40 decays are not included in the neutrino measurements (due to threshold effects) and contribute to the otherwise unexplained 20 TW. See arxiv.org/abs/1003.0284 and nature.com/ngeo/journal/vaop/ncurrent/full/ngeo1205.html for descriptions of the LOS geo-neutrino measurements. –  dmckee Mar 14 '13 at 20:41
The Mack et al. article the question cites about Earth's heat budget say "Obviously, the possibility to make direct heat flow measurements under the surface is unique to Earth", however, two Apollo missions actually did drill into the Moon and determine heat flux. lpi.usra.edu/lunar/missions/apollo/apollo_17/experiments/hf –  DavePhD Feb 28 at 14:09
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protected by Qmechanic Feb 27 '13 at 17:06

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