## Wednesday, July 10, 2013

### Planck's summer: particle size matters!

second post in the "Planck's summer"-series by Youness:

Let us assume that dark matter is made of approximately collisionless particles evolving under the laws of gravity (just one type, for simplicity). Is there a minimum mass $m_\text{min}$ these particles need to have in order to fit observational constraints? Conversely, is there an upper limit $m_\text{max}$?

1. Pretty question!
My own reply to both parts of it is: yes. There is a minimum and a maximum. But I won't (presently!) tell why - in order to give others the chance to independently think and argue.
Uli Bastian

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1. Idea for a lower bound:

Very light particles will be hot dark matter. For a collisionless fluid there is something analoguous to a Jeans length, called the free streaming length (http://ned.ipac.caltech.edu/level5/March02/Bertschinger/Bert3.html). It depends on the particle mass. Fluctuations below approximately the free streaming length are damped out. This must be reflected in the Power Spectrum and (if we have only one type of dark matter) rule out masses that are too small.

3. I had the same idea at first Angelos, but isn't that only true for thermally produced dark matter? If the particles are produced by other means, their mass could be arbitrarily small.
So I would guess there is no way of telling in a model-independent way. Under additional assumptions (i.e. thermal production) it's possible to give constraints via the free streaming length. But axions or sterile neutrinos could basically have masses as low as they want.

4. Angelos'argument is fine - to my opinion - because I think that *completely* non-interacting dark-matter particles would be a really extreme assumption. If we want to avoid an extreme violation of time-reversal invariance, the creation process of the particles has an annihilation counterpart. If we go back far enough in time (and density!), this should at some time lead to a strong coupling. And a strong coupling entails thermalisation.
If I'm wrong, I would be curious to learn why.
Uli Bastian

5. Now for the upper mass limit: If the particles were *very* heavy, they would create a graininess of the gravitational field in galaxies. This in turn would perturb dynamically "cold" systems, like thin galactic disks, open star clusters and the like. In other words, very heavy particles would puff up galactic disks to large scale heights. As far as I remember, the critical mass at which the disk of our galaxy would become puffed up is of the order of 100 000 solar masses.
Uli Bastian

6. Youness's answer: There is no lower limit, except for the trivial condition
$m_\text{min} > 0$ (massless particles would necessarily be ultra-relativistic
and would not clump). Astronomical and cosmological observations indicate that
dark matter is cold. This can be inferred from the simulation of structure
formation, but an even stronger argument is related to the CMB. The CMB
temperature fluctuations are of order $10^{-5}$; from $z \approx 1000$ to $z = 0$, density perturbations would have grown by a factor of $10^3$ (since
$\delta_m \propto a$ in matter domination). We would thus only observe small
linear fluctuations of order $10^{-2}$, no galaxies, clusters and the like!
Cold dark matter is a way out: its density perturbations had already grown
while the baryons were prevented from clumping by the photons.

'Cold' dark matter, however, does not imply a large mass, it's just about the
comparison between the rest mass and the kinetic energy (or temperature). If
dark matter was in thermal equilibrium with the primordial plasma at some
point and then just cooled due to the expansion, we indeed get lower limits on
the mass because dark matter needs to be nonrelativistic early on (before the
onset of matter domination). If there is a process reducing the kinetic energy
of dark matter, such a limit does not apply and dark matter particles can be
light. A famous example is axion dark matter (sub-eV particles).

An upper mass limit may be related to discreteness effects like gravitational
distortions in the Solar System by single dark matter particles. But this
would probably give only very large upper limits. Note that particle masses
above the Planck mass mean that dark matter consists of black holes. This is
because for a particle with the Planck mass, the Compton wavelength just
equals the Schwarzschild radius. The requirement of these black holes to
survive until present is $m_\text{ini} > 10^{15}$g (for lower masses, the
black holes would have evaporated by Hawking radiation within the age of the
Universe).