What's Left From The Junk Heap of Failed Dark Matter Models?

Simple Dark Matter Models Have Failed

Lots of phenomena in astronomy are assumed to be the product of dark matter not made out of the same kind of particles that are found in ordinary matter that interact with ordinary matter predominantly via gravity alone.  For example, dark matter is invoked to explain why galaxies rotate at speeds inconsistent with their visible mass.

Very simple collisionless warm dark matter (keV scale) and cold dark matter (many GeV scale) dark matter models are sufficient to explain the dark matter phenomena observed at the cosmology level.  Thermal relic "hot dark matter" with mass in the neutrino range has long been ruled out because this would create a universe that has far less large scale structure than is observed.  But, warm dark matter and cold dark matter models have historically been less accurate at describing phenomena at the scale of an individual galaxy and cold dark matter models predict that there would be far more satellite galaxies to galaxies like the Milky Way than are observed.

A pure collisionless cold dark matter model leads to a cuspy NFW dark matter halo shape (named with the initial of the three scientists to formulate it) which is inconsistent with observational evidence.

Much lighter dark matter particles called warm dark matter, solve the satellite galaxy problem and fit a variety of other observational constraints in a narrow mass window around 2-3 keV, and also produce a less severely cuspy dark matter halo shape.  But, improved observations of the movements of visible matter in galaxies used to more precisely infer the shape of the dark matter halos that would be necessary to produce the observed movements of visible matter still rule out even simple collisionless warm dark matter models.  These models still produce halos that are too cuspy.

But, there is basically no form of dark matter that is capable of being consistent with observational evidence and is a thermal relic that can fit the bill of cold dark matter for the highly successful lamda CDM cosmology model which is not also accompanied by a new force that acts between particles of dark matter and is mediated by an MeV scale mass force carrying boson that creates a Yukawa potential between dark matter particles.

Thus, without a major overhaul of the foundations of cosmology, both a new fundamental particle and a new force with its own particle are necessary, at a minimum.  A complete formulation of physics in a particle model, if it is possible at all, would also require a hypothetical graviton tensor boson, and possible a dark energy scalar particle and an inflaton scalar particle as well - although it is possible that the Higgs field, dark energy phenomena, and cosmological inflation (or at least a couple of them) could have a common scalar particle source.

Finally, A Dark Matter Halo Model That Works

On Friday, I reviewed a new paper by Toky Randriamampandry and Claude Carignan, entitled "Galaxy Mass Models: MOND versus Dark Matter Halos" which recapped two important developments in dark matter research and put it to the test by consistently fitting galactic rotation curves better than MOND models, which modify the laws of gravity in weak gravitational fields to make them weaken more slowly than they do in Newtonian gravity and General Relativity.

This wasn't an entirely fair comparison. The dark matter halo model using two parameters, while the MOND model used just one. The test assumed an empirically fixed mass to light ratio which is not part of the MOND paradigm for which M/L equivalent is an output rather than an input of the model (indeed, predictions regarding mass to light ratio equivalents were key valdations of the early MOND models). And, the galaxies examined in which there were significant discrepancies from the MOND prediction were dwarf galaxies with a very spatially extended region in which there was a transition from Newtonian to non-Newtonian behavior with respect to visible mass. This raises the possibility that the MOND interpolation function, an ad hoc formula for transitioning between the Newtonian and non-Newtonian MOND regime, rather than the core MOND concept may have been at fault for the poor fits. In most galaxies, the transition from the MOND to non-MOND regime is sufficiently abrupt that rotation curve fits are indifferent to the exact form of the interpolation function. But, in these kinds of dwarf galaxies, where the transition is much more gradual, the interpolation function, which there has been little empirical work to refine because rotation curve fits aren't influenced much by it in other kinds of galaxies, is much more important. So, a poor MOND fit could be due to problems with a part of the theory that even its supporters acknowledge needs more work.

But, what are the two important developments?

First, there is a dark matter halo model that can produce consistently accurate rotation curve. In a nut shell this is a pseudo-isothermal dark matter halo model in which core radius and central halo density, are not independent of each other and the surface density of the halo has a constant value which is calculated as the product of these two parameters and is approximately 120 Mp/c2 (two previous papers estimated the value of this constant at 100 and 150, respectively).

We actually know, from precision observations of the rotation curve of the Milky Way galaxy that this spherically symmetric model isn't correct. Instead, the major axis of the dark matter halo which coincides with the galaxy's axis of rotation is a bit longer than the radial axis of the dark matter halo that is parallel to the spiral arms of a spiral galaxy. But, the deformation of the halo shape from purely spherical to an ellipsoid that is modestly deformed relative to a spherical shape, is pretty modest (e.g. the ratio of the major axis to the minor axis of the halo is probably less than 2.5-1 although it is more than 1-1).

Of course, this isn't a complete dark matter model. It doesn't tell you what kind of dark matter particles and dark matter particle interactions are necessary to produce that kind of halo. But, it provides a mathematically well defined and empirically validated intermediate target for people trying to determine what kind of dark matter particle models could produce such halos. And, of course, any particle model that can produce the right halo in N-body simulations also needs to be able to fit other constraints from scales larger than individual galaxies, satellite galaxy formation rate limitations, and particle physics. For example, the mass density of dark matter in the universe needs to be a bit more than five times that of ordinary baryonic matter, it must have neutral electric charge, and it must not appear in W or Z boson decays at energies possible to generate at the LHC so far.

Second, it is possible to reduce this dark matter halo model to a single parameter without undue loss of accuracy. One of the basic consequences of the fact that MOND can make a reasonable fit to the rotation curve of almost any galaxy with a single parameter is that any dark matter halo model must also be able to do so.

Trying To Fit The Right Halo To Specific Kinds Of Dark Model Particles

Finding a particle with the right mass and interactions to produce this kind of halo is another problem.

No Known Forces Give Rise To Cold Dark Matter-Ordinary Matter Interactions

We know that no particles that could produce the right kind of dark matter halo are produced in the decays of W and Z bosons, ruling out, for example, any neutrino-like particle with a mass of 45 GeV or less.  In other words, no light dark matter candidate can be "weakly interacting".

We also know that direct detection of dark matter experiments such as XENON rule out essentially all of the cold dark matter mass parameter space (below 10 GeV to several hundreds of GeVs with the exclusion most definitive at 50 GeV) through cross-sections of interaction on the order of 10^-43 to 10^-45 which is a far weaker cross section of interaction than the neutrino has via the weak force.  If dark matter does interact via the weak force, it differs from all other weakly interacting particles which has much higher weak force cross sections of interactions and integer or simply fraction of integer weak force charges.

XENON also places strong limits on interactions between ordinary photons and "dark photons".

We know that dark matter has zero net electric charge (it dark matter is composite, its components could have electric charge) and is not produced or inferred in any strong force interactions observed to date in collider experiments.

Taken together these facts rule out any kind of interaction between between cold dark matter and ordinary matter via any recognizable version of the three Standard Model forces (electromagnetic, weak and strong).

Of course, by hypothesis, dark matter and ordinary matter interact via gravity just like any other massive particles.

Thus, interactions between dark matter and ordinary matter other than via gravity are strongly disfavored.

Any non-gravitational interactions between ordinary matter and dark matter must be very, very weak and involve some kind of new force, or must involve radically new manifestations of an existing force.  For example, they might be transmitted via some kind of composite particle by analogy to the nuclear binding force in atoms carried mostly by pions that is derivative of, but much weaker in this manifestation than, the strong force that binds quarks within hadrons.

There are not currently any models for such forces that produce the observed dark matter halos of which I am aware.  The trick is to figure out how to make the interaction with ordinary matter weak enough to make it "nearly collisionless" as it must be to fit cosmology models, while strong enough to in some manner track the distribution of ordinary matter to a greater extent than gravity requires it to match that distribution.  Theoretical interest has focused on interactions between the sectors via the Higgs boson and interactions between "dark photons" and ordinary Standard Model photons, and on very slight weak force interactions that are somehow suppressed when they involve dark matter.

Completely Collisionless Dark Matter Is Inconsistent With Observational Evidence 

We know that purely collisionless dark matter (i.e. dark matter that interacts with other dark matter only via the gravitational force), that has a particular mass anywhere from the keV range to the TeV+ range produces cuspy halos inconsistent with observational evidence.

We know that multiple kinds of collisionless dark matter simultaneously present in the universe at the same time produce worse fits to the data than single variety of collisionless dark matter models.

Collisionless bosonic dark matter, as well as fermionic collisionless dark matter, is likewise excluded over a wide range of parameters.

Thus, the purest form of sterile neutrino, with a particular mass and no non-gravitational interactions at all, is ruled out by observational evidence from the shape of dark matter halos.

Simple Self-Interacting Dark Matter Models Still Fail

We know that self-interactions between dark matter particles with each other with cross-sections of interaction on the order of 10^-23 to 10^-24 greatly improve the fit to the halo models observed (self-interactions on the order of 10^-22 or more, or of 10^25 or more, clearly do produce the observed halos). Notably, this cross section of self-interaction is fairly similar to the cross-section of interaction of ordinary matter (e.g. helium atoms) with each other.  So, if dark matter halos are explained by self-interaction, the strength of that self-interaction ought to be on the same order of magnitude as electromagnetic interactions.

But, our observations and simulations are now sufficiently precise that we can determine that ultimately, a simple constant coupling constant between dark matter particles or velocity dependent coupling constant between dark matter particles fails to fit the observed dark matter halos.

Generically, these models generate shallow spherically symmetric halos which are inconsistent with the comparatively dense and ellipsoidal halos that are observed.

Sophisticated Self-Interacting Dark Matter Models Might Work

Next generation self-interacting dark matter models look at more a general Yukawa potential generated by dark matter to dark matter forces with massive force carriers (often called "dark photons") that have masses which empirically need to be on the order of 1 MeV to 100 MeV (i.e. between the mass of an electron and a muon, but less than the lightest hadron, the pion, which has a mass on the order of 135-140 MeV) to produce dark halos that are a better fit to the dark matter halos that are observed.  Sean Carroll was a co-author on one of the early dark photon papers in 2008.

There have been some efforts to constrain these models directly in the case where photons and dark photons are allowed to interact to some extent (more recently here).  But, since these experimental searches depend upon dark photons decaying into particles outside the dark sector, and generally, decays of force carrying particles only involve particles that the force carrying particle couples to, and since photons are only known to couple to electric charge and nothing else, the prospects of this experimental search producing anything other than a null result are dim.  Indirect searches based upon precision measurements of the trajectories of visible matter and understandings of how invisible ordinary matter are distributed around that visible matter, that are used to infer dark matter distributions in a wide variety of systems observed by astronomers, are likely to be more fruitful in pinning down the properties of dark photons than direct detection experiments.

This force could be scalar (like Newtonian gravity or electromagnetism without polarization), pseudoscalar (like pions) or vector (like photons, gluons, W bosons and Z bosons).  This force could have a single repulsive charge (like imaginary number valued mass in Newtonian gravity, or like a universe made entirely of electrons without any protons), or could have positive and negative charges akin to the electric charge.  Often it is modeled as a second U(1) group interaction, like the U(1) electromagnetic force, with a massive boson much like the weak force but lighter than the W and Z bosons.  In contrast, the strong force is SU(3) and the weak force is SU(2) (strictly speaking the unified electroweak force is SU(2) * U(1), but not necessarily neatly separated into weak and electromagnetic components).

Still, the bottom line, is that to explain dark matter, one needs at least a new dark matter fermion and a new massive boson carrying a new force.

The apparent relationship between the known particle masses and the Higgs vacuum expectation value in which the sum of the square of the mass of each particle that obtains its mass via the Higgs boson equals the square of the Higgs vev disfavors new heavy particles that gain their mass via the Higgs mechanism.  But, these measurements are not so precise that they could disfavor new light particles that gain their mass via the Higgs mechanism.  Current experimental uncertainties in this equivalence could accommodate both a new massive boson of 100 MeV and a new massive fundamental fermion of up to about 3 GeV, so both particles could couple to the Higgs boson and obtain their mass entirely from interactions with it, even though they don't couple to the other Standard Model forces, consistent with current knowledge from colliders (which cannot yet quantify if there are any "missing" Higgs boson decays from new long lived electrically neutral particles).

Creating and Annihilating Dark Matter: Thermal Relics v. Other Sources

One of the things we know about dark matter is its overall abundance and expected mass density in the vicinity of Earth per volume of empty space.  Any model that calls for too much dark matter to be created or for too little dark matter to be created is disfavored.

The generally very successful six parameter lamda CDM cosmology model, which accurately describes a wide variety of properties of the observed universe such as its cosmic background radiation and the relative frequency of stars at different redshifts, assumes that dark matter is nearly collisionless and has an origin as a thermal relic consistent with a mass of ca. 2 keV or more (i.e. warm dark matter or cold dark matter).

The assumption that dark matter is largely a thermal relic (i.e. that the number of dark matter particles "freeze out" at a certain point when the temperature of the universe becomes cool enough) tightly relates the mass of a dark matter particle to its average velocity and kinetic energy, and places a floor on dark matter mass.

In models where dark matter has some self-interaction via a massive boson creating a Yukawa potential, thermal relic dark matter abundance and velocity is still a function of dark matter particle properties.  But, in that case, it is a product of the coupling constant of the self-interaction force, the mass of the force carrying boson and the mass of the dark matter particle itself, rather than merely the mass of the dark matter particle.  This creates a three dimensional parameter space window for combinations of these parameters in the model.

Once can create a dark matter model where dark matter abundance is not a function of the moment at which it freezes out as a thermal relic, and doing so greatly loosens this constraint on the parameter space for dark matter particles.

For example, axion dark matter, which has a mass so low that in most cases it would be "hot dark matter" rather than "warm dark matter" or "cold dark matter" if it was a thermal relic, might be possible to reconcile with observation if it was created non-thermally, for example, via strong force interactions that create and destroy it at a rate consistent with the current observed abundance of dark matter in the universe.

But, any dark matter model with a dark matter candidate that is not a thermal relic makes it necessary to radically rethink and overhaul the lamda CDM cosmology model from first principles and reconcile the revised cosmology model to what is observed.  Given how many pieces of the puzzle of the universe's cosmology come together nicely if it is possible merely to find a suitable thermal relic dark matter candidate that can reproduce the galaxy scale properties of dark matter phenomena, as well as the large scale structure dark matter phenomena that are observed, thermal relic dark matter candidates are particularly attractive.

Analysis

The strong possibility that dark matter is entirely sterile with respect to Standard Model particles and interactions (apart, perhaps from the Higgs boson), dramatically limits the usefulness of direct dark matter detection experiments.  The gravitational impact of individual dark matter particles is simply far to slight to be detected experimentally at any time in the foreseeable future.  Direct dark matter detection experiments are likely to do nothing more than to rule out all forms of dark matter that have meaningful non-gravitational interactions with ordinary matter.

But, the prospects for inferring the properties of one or more dark matter particles from their aggregate distribution as inferred from their collective gravitational effects are looking good.  The most simple models of the dark sector don't work, but a straightforward two particle model with one fermion and one massive boson whose properties are quite tightly bound by observational evidence may suffice to do the job of explaining all of our observations.  If this were accomplished, almost all of the "what" questions in physics would be answered, even if some of the "why" questions remained open, and there might be a few loose ends with observable consequences only capable of being seen in extremely esoteric circumstances (e.g. are there unstable, second and third generation dark matter fermions akin to muons and tau leptons).

Footnote

I am writing most of this from memory and will try to track down references to specific points later, probably as unacknowledged added links to this post.