Can the Boltzmann equation accurately predict the relic density of dark matter?

[shahbaznihal]

Hi,

My question is: Whats the whole point of solving the Boltzmann equation for relic density of dark matter? I mean, we already know the current density of dark matter in the Universe so that end is fixed by experiment. If we have a model of dark matter evolution then it has to comply to the observed density of dark matter today. Correct? Now given that, do we use the observed density to "fix" the cross-section? Or is it something I am entirely missing?

 

[marcus]

That seems like a good question. "what's the whole point?"
In your post you hint at a possible answer. I think my reply here is just to confirm that I think your hinted explanation is right. If I understand your post, then I would guess that your idea is correct.

But I will spell all that out in detail in case anyone else is curious.
For a basic discussion see page 15 of http://arxiv.org/pdf/1006.2483v2.pdf
(a 2010 paper so viewpoint may be a little out of date)
or the Caltech website HTML version section 7.3
http://ned.ipac.caltech.edu/level5/March10/Garrett/Garrett_contents.html
http://ned.ipac.caltech.edu/level5/March10/Garrett/Garrett7.html#7.3
" Particle Production and Relic Density: The Boltzmann Equation"

I think the good thing about calculating relic densities is you have to first specify things like the mass of the particle and the reactions (at high temperature and density) you expect it to undergo.
And then you have to solve to find when the "freeze-out" occurs for that type of particle.
And then you solve for the density of that particle at "freeze-out".

So it is a way of investigating what reasonable assumptions about the particle could be. What type of particle, or what characteristics, lead to something that has the required abundance?

And then if you have an idea of what type of particle might be the DM particle, then you can look for it. Design detectors and experiments to try to see it.

So to me it makes sense to do that kind of Boltzmann equation analysis. As you say we know the overall density of DM, observationally, so the analysis is not supposed to discover that. The analysis is supposed to give hints as to what type of particle to look for.

BTW for what it's worth here is a particle physicist's plot of the half-dozen or so proposed identities of the DM particle
http://resonaances.blogspot.com/2014/03/weekend-plot-all-of-dark-matter.html
More actually, he shows 10 of them. It serves mainly as a way of listing them, so as to keep them in mind: there are enough ideas floating around that it's hard to visualize the field of candidates.

 

[shahbaznihal]

Thanks for the references. So it is about the mass and cross-section of the DM particle. So its like you derive the relationship between mass, cross-section and relic density from the Boltzmann equation and find out which mass and cross-section complies with your model given a fixed relic density. Correct?

 

[marcus]

Thanks for the references. So it is about the mass and cross-section of the DM particle. So its like you derive the relationship between mass, cross-section and relic density from the Boltzmann equation and find out which mass and cross-section complies with your model given a fixed relic density. Correct?

I can continue to respond in a provisional way, but we really must hope that someone more expert than I is interested and will notice this thread.

You want to understand how the relic density info is USED to restrict the range of parameter space of likely prospects, and to identify likely prospects (for the DM particle).

One way to learn is simply to click on one of the links and see for yourself the useful role that is being played by the relic density calculation. for example one of the links is called "Boehmot" and you get this on page 3:
==quote http://arxiv.org/pdf/astro-ph/0309686v3.pdf ==
V. RELIC DENSITY AND ASTROPHYSICAL CONSTRAINTS
A. Relic Density
We will now attempt to address the question of: is the range of annihilation cross sections found in section IV consistent with constraints from relic density and other measurements?…
==end quote==

The point is there has been observed a 511 keV gamma ray line in radiation coming from the central bulge of the galaxy where you might expect a concentration of DM. This likely comes from positrons annihilating with electrons in the environment. There are various possible explanations for what is making the positrons, one explanation is the mutual annihilation of two DM particles. So in section IV they get a formula for what the annihilation cross section would have to be (the density of DM being known and a mass being assumed).

then they have to check if the number density (based on the known DM density and the mass) is consistent with calculated relic density. I think that is what section V is about. BTW I think T_F stands for the "freeze-out temperature" which you would need to know if using the Boltzmann equation.

Basically you want to understand how relic density info is USED in evaluating possible signals from DM and there is an example to study. You can look on page 3 of that paper and see for yourself (better, I think, than I can paraphrase or describe in general.)

And that is only one case. Relic density info may be used in several different logical ways to cross check hypotheses about DM particles and the sources of various signals.

There was another recent very interesting signal detected recently, a 3.6 keV line. If you click on "Bulbulon" you can read about that. I don't know if relic density entered the analysis there, but it could well have. The idea there is that the DM particle (or at least one of them, if there are several) could be a "sterile neutrino".

 

[Chalnoth]

Thanks for the references. So it is about the mass and cross-section of the DM particle. So its like you derive the relationship between mass, cross-section and relic density from the Boltzmann equation and find out which mass and cross-section complies with your model given a fixed relic density. Correct?

Well, sort of. That's three unknowns and only two "equations" (the Boltzmann equation and the observation of the dark matter density). So you can't quite nail down the cross-section and mass. But you can narrow down the possible values of cross-section and mass that a dark matter particle might have.

Unfortunately, you can also come up with alternative models for dark matter that have either entirely different production mechanisms (e.g. axions), or models with multiple types of dark matter particle.

Thus, in and of itself, the calculation of the relic density doesn't get you a whole lot. But it does serve as a data point that any future tests of dark matter must pass.

 

[shahbaznihal]

Well, sort of. That's three unknowns and only two "equations" (the Boltzmann equation and the observation of the dark matter density). So you can't quite nail down the cross-section and mass. But you can narrow down the possible values of cross-section and mass that a dark matter particle might have.

Yeah, that's what I meant. You get a range of mass and cross-section consistent with the given evolution model and the relic density today.

Thus, in and of itself, the calculation of the relic density doesn't get you a whole lot. But it does serve as a data point that any future tests of dark matter must pass.

True, like Marcus mentioned.