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Dark matter appears to dominate the matter component of the universe as compared to luminous, or baryonic, matter. Though it does not interact electromagnetically (it doesn't absorb, scatter, or emit photons), there is an ever-increasing mountain of evidence for its existence through it's gravitational interactions with stars, galaxies, and clusters, as well as it's influence on objects behind it through what's known as gravitational lensing.

My question is, what are the most promising particle candidates of dark matter, and which experiments currently exist (or will exist) to attempt to answer this question?

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When you say "particle" candidates, I assume you're excluding MACHOs and RAMBOs. MACHOs are "dark" objects at the stellar scale like black holes, neutron stars, brown dwarfs, etc. RAMBOs are clusters of similar dark objects. MACHOs and RAMBOs are made of primarily baryonic matter (everyday stuff like protons and neutrons — electrons are not baryons but they can be a part of this too).

This leaves us with the WIMPs and possible other particle candidates. I don't know the details of any experiments looking for these offhand (I'll see if I can find it out and update the answer in a few days), but the major particle candidates are:

  • Neutrinos: It's highly unlikely that these make up any significant chunk of the DM in our universe, but they deserve a mention.
  • Sterile neutrinos (I'm using the term in the specific sense here, not the general sense): This type of neutrino is right handed and only interacts gravitationally (note that there can be right handed active neutrinos, too). There is some preliminary evidence suggesting that these might exist.
  • Axions: These are hypothetical, but if they exist (and make up the majority of DM), we kill two birds with one stone by both explaining DM and resolving the lack f symmetry breaking in the strong interaction.
  • Gravitinos and the Lightest Supersymmetric Partner: Both of these require supersymmetry to hold, but they are major DM candidates.
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    $\begingroup$ Dark matter almost certainly does not interact in any way, shape, or form electromagnetically. For a while people thought that MACHOs were potentially the cause (whether they're brown dwarves or stellar mass black holes), but this has been ruled out through microlensing studies of the Mikly Way. However, there is a significant change that dark matter does interact via the weak force (which is why various types of neutrinos have been proposed - right-handed neutrinos, sterile neutrinos, etc..). $\endgroup$
    – astromax
    Commented Sep 27, 2013 at 15:53
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    $\begingroup$ Also, can you be more specific when you say "Neutrinos". There are a number of possibilities here. $\endgroup$
    – astromax
    Commented Sep 27, 2013 at 15:57
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    $\begingroup$ @astromax Aren't MACHOs still thought of as part of the DM in the universe? I know that they're a minority, but IIRC they still are considered DM. You're right about the neutrino bit, I planned on adding a bit but forgot. Doing so now. $\endgroup$
    – Manishearth
    Commented Sep 27, 2013 at 18:49
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    $\begingroup$ Though MACHOs are tough to detect (since they're very faint; e.g. - brown dwarves, neutron stars, etc..), they are certainly made of baryonic content. Some work has been done to estimate the amount of mass due to objects like these, and it still doesn't even come close to adding up. Black holes, though nobody know what happens inside of the event horizon, also do not add up to what is needed to describe what galaxy/clusters dynamics and mass reconstructions from lensing, x-ray, and S-Z measurements tell us is there. So far as we know, gravity is the only confirmed interaction. $\endgroup$
    – astromax
    Commented Sep 27, 2013 at 18:59
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    $\begingroup$ Well - things that are composed of composite particles with charge (protons/electrons, or quarks), radiate photons. Accelerating/decelerating charged particles produce radiation, so no, there is still a fundamental difference between the two. $\endgroup$
    – astromax
    Commented Sep 27, 2013 at 20:36

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