# What is the upper-limit on intrinsic heating due to dark matter?

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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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