Cold dark matter is thought to fill our galactic neighborhood with a density ρ of about 0.3 GeV/cm3 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 σ, this will lead to a constant collision rate of roughly
r∼ρvσ/m
for every nucleon in normal matter. The kinetic energy transferred to the nucleon (which is essentially at rest) will be roughly
ΔE∼2v2Mm2(m+M)2,
where M≈1 amu ≈1 GeV/c2 is the mass of a nucleon. The limits for light (m≪M) and heavy (m≫M) dark matter are
ΔElight∼2v2m2M and ΔEheavy∼2v2M.
This leads to an apparent intrinsic heat production in normal matter
˜P∼rΔE/M,
which is measured in W/kg. The limits are
˜Plight∼2ρv3σm/M2 and ˜Pheavy∼2ρv3σ/m.
What existing experiment or observation sets the upper limit on ˜P?
(Note that ˜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 μW 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 ˜P<10−9W/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, 6×1024 kg, limits the intrinsic heating to ˜P<3×10−12W/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 103 GeV/c2 with sensitivities to cross sections greater than 10−43 to 10−40 cm2 (depending on mass). A 100 GeV dark matter candidate with a cross-section of 10−43 cm2 would be expected to generate ˜P∼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 cm2 (which, although not nearly as theoretically motivated as heavier WIMPs, is not excluded by direct detection experiments) would be expected to generate ˜P∼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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