First, I don’t know what dark matter is, or even if it is, and while they might have ideas, neither does anyone else know. However, the popular press tells us that there is at least five times more of this mysterious stuff in the Universe than ordinary matter and we cannot see it. As an aside, it is not “dark”; rather it is transparent, like perfect glass. The reason is light does not interact with it, nevertheless we also know that there are good reasons for thinking that something is there because assuming our physics are correct, certain things should happen, and they do not happen as calculated. The following is a very oversimplified attempt at explaining the problem.
All mass exerts a force on other mass called gravity. Newton produced laws on how objects move according to forces, and he outlined an equation for how gravity operates. If we think about energy, follow Aristotle as he considered throwing a stone into the air. First we give the stone kinetic energy (that is the energy of motion) but as it goes up, it slows down, stops, and then falls back down. So what happened to the original energy? Aristotle simply said it passed away, but we now say it got converted to potential energy. That permits us to say that the energy always stayed the same. Note we can never see potential energy; we say it is there because in makes the conservation of energy work. The potential energy for a mass munder the gravitational effect of a mass Mis given by V = GmM/r. Gis the gravitational constant and ris the distance between them.
When we have three bodies, we cannot solve the equations of motion, so we have a problem. However, the French mathematician Lagrange showed that any such system has a function that we call a Lagrangian, in his honour, and this states that the difference between the total kinetic and potential energies equals this term. Further, provided we know the basic function for the potential energy, we can derive the virial theorem from this Lagrangian, and for gravitational interactions, the average kinetic energy has to be half the magnitude of the potential energy.
So, to the problem. As the potential energy drops off with distance from the centre of mass, so must the kinetic energy, which means that velocity of a body orbiting a central mass must slow down as the distance from the centre increases. In our solar system Jupiter travels much more slowly than Earth, and Neptune is far slower still. However, when measurements of the velocity of stars moving in galaxies were made, there was a huge surprise: the stars moving around the galaxy have an unexpected velocity distribution, being slowest near the centre of the galaxy, then speeding up and becoming constant in the outer regions. Sometimes the outer parts are not quite constant, and a plot of speed vs distance from the centre rises, then instead of flattening, has wiggles. Thus they have far too much velocity in the outer regions of the galactic disk. Then it was found that galaxies in clusters had too much kinetic energy for any reasonable account of the gravitational potential energy. There are other reasons why things could be considered to have gone wrong, for example, gravitational lensing with which we can discover new planets, and there is a problem with the cosmic microwave background, but I shall stick mainly with galactic motion.
The obvious answer to this problem is that the equation for the potential is wrong, but where? There are three possibilities. First, we add a term Xto the right hand side, then try to work out what Xis. Xwill include the next two alternatives, plus anything else, but since it is essentially empirical at this stage, I shall ignore it in its own right. The second is to say that the inverse dependency on ris wrong, which is effectively saying we need to modify our law of gravity. The problem for this is that Newton’s gravity works very well right out to the outer extensions of the solar system. The third possibility is there is more mass there than we expect, and it is distributed as a halo around the galactic centre. None of these are very attractive, but the third option does save the problem of why gravity does not vary from Newtonian law in our solar system (apart from Mercury). We call this additional mass dark matter.
If we consider modified Newtonian gravity (MOND), this starts with the proposition that with a certain acceleration, the force takes the form where the radial dependency on the potential contained a further term that was proportional to the distance rthen it reached a maximum. MOND has the advantage that it predicts naturally the form to the velocity distribution and its seeming constancy between galaxies. It also provides a relationship for the observed mass and the rate of rotation of a galaxy, and this appears to hold. Further, MOND predicts that for a star, when its acceleration reaches a certain level, the dynamics revert to Newtonian, and this has been observed. Dark matter has a problem with this. On the other hand, something like MOND has real trouble trying to explain the wiggly structure of velocity distributions in certain galaxies, it does not explain the dynamics of galaxy clusters, it has been claimed it offers a poor fit for velocities in globular clusters, the predicted rotations of galaxies are good, but they require different values of what should be constant, and it does not apply well to colliding galaxies. Of course we can modify gravity in other ways, but however we do it, it is difficult to fit it with General Relativity without a number of ad hocadditions, and there is no real theoretical reason for the extra terms required to make it work. General Relativity is based on ten equations, and to modify it, you need ten new terms to be self-consistent; the advantage of dark matter is you only need 1.
The theory that the changes are due to dark matter has to assume that each galaxy has to incorporate dark matter roughly proportional to its mass, and possibly has to do that by chance. That is probably it biggest weakness, but it has the benefit that it assumes all our physics are more or less right, and what has gone wrong is there is a whole lot of matter we cannot see. It predicts the way the stars rotate around the galaxy, but that is circular reasoning because it was designed to do that. It naturally predicts that not all galaxies rotate the same way, and it permits the “squiggles” in the orbital speed distribution, again because in each case you assume the right amount of dark matter is in the right place. However, for a given galaxy, you can use the same dark matter distribution to determine motion of galaxy clusters, the gas temperature and densities within clusters, and gravitational lensing, and these are all in accord with the assumed amount of dark matter. The very small anisotropy of the cosmic microwave background also fits in very well with the dark matter hypothesis, and not with modified gravity.
Dark matter has some properties that limit what it could be. We cannot see it, so it cannot interact with electromagnetic radiation, at least to any significant extent. Since it does not radiate energy, it cannot “cool” itself, therefore it does not collapse to the centre of a galaxy. We can also put constraints on the mass of the dark matter particle (assuming it exists) from other parts of physics, by how it has to behave. There is some danger in this because we are assuming the dark matter actually follows those relationships, and we cannot know that. However, with that kept in mind, the usual conclusions are that it must not collide frequently, and it should have a mass larger than about 1 keV. That is not a huge constraint, as the electron has a mass of a little over 0.5 MeV, but it says the dark matter cannot simply be neutrinos. There is a similar upper limit in that because the way gravitational lensing works, it cannot really be a collection of brown dwarfs. As can be seen, so far there are no real constraints on the mass of the dark matter constituent particles.
So what is the explanation? I don’t know. Both propositions have troubles, and strong points. The simplest means of going forward would be to detect and characterize dark matter, but unfortunately our inability to do this does not mean that there is no dark matter; merely that we did not detect it with that technique. The problem in detecting it is that it does not do anything, other than interact gravitationally. In principle we might detect it when it collides with something, as we would see an effect on the something. That is how we detect neutrinos, and in principle you might think dark matter would be easier because it has a considerably higher mass. Unfortunately, that is wrong, because the neutrino usually travels at near light speed; if dark matter were much larger, but much slower, it would be equally difficult to detect, if not more so. So, for now nobody knows.
Just to finish, a long shot guess. In the late 20th century, a German physicist B Heim came up with a theory of elementary particles. This is largely ignored in favour of the standard model, but Heim’s theory produces a number of equations that are surprisingly good at calculating the masses and lifetimes of elementary particles, both of which are seemingly outside the scope of the standard model. One oddity of his results is he predicts a “neutral electron” with a mass slightly greater than the electron and with an infinite lifetime. If matter and antimatter originally annihilated and left a slight preponderance of matter, and if this neutral electron is its own antiparticle, then it would survive, and although it is very light, there would be enough of it to explain why its total mass now is so much greater than matter. In short Heim predicted a particle that is exactly like dark matter. Was he right? Who knows? Maybe this problem will be solved very soon, but for now it is a mystery.