Thank you for the kind offer.
Yes, the article is very superficial, but I am still struggling to see where
there is a need to assume particles in a test of the hypotheses that could be
made about the observations in it.
Can't this all be done with field equations or some other approach? Or am I not
understanding some form of "duality" that makes it irrelevant whether one uses
"fields" or particles?
It sort of is all done with field equations, the particles are just implied. But let me start from the beginning:
The most well tested field equation we have for gravity are the Einstein field equation. Those relate mass and energy to time and space. For laboratory tests these work very well: There are experiments with very high precision that test this equation and so far, all these tests were positive. They also work very well for describing "small" scale astronomical systems, like our solar system.
The problem occurs when you look at large scale astronomical systems like galaxies or the entire visible universe. If you put in all mass and energy that you know about from observations, you will notice some discrepancies of how the universe should behave to how it does behave.
There are two possible solutions to this:
The first one is to modify the field equations in a way that all works out with just visible mass and energy. This is what these alternative theories of gravity try to do. This isn't easy, because the new equations need to fit all observations that we have made. It is easy to introduce a term that works for a specific galaxy, but it has to work for all galaxies without becoming overly complicated. The problem is, you will never run out of possible ways to modify the equations, so "We need to modify the equation" is not a proper scientific hypothesis. To transform the idea into a specific hypothesis, you need to make assumptions how the equations should look like and then you can derive predictions from that which you can compare to observations.
The second possibility is to keep the equation as it is and fill in mass and energy that we cannot see (yet). The "missing" mass and energy are called dark matter and dark energy. Now, the field equation doesn't care much what you put in there - it doesn't have to be particles. For example, dark matter could consist of ordinary matter clumped together in many small black holes, which we would have no way of detecting. Again, you will never run out of possibilities of invisible stuff to put in, so "There is dark matter" isn't much of a scientific hypothesis, either. Again you need to make assumptions about the properties of the dark matter, so that you can derive predictions about the behavior of the universe.
Theoretical physicists concerned with gravity need to publish something, so they will keep up making new hypotheses about dark matter or the lack thereof. Over time, certain theories and classes of theories will become favorites, because they explain observations best, are not overly complex, or have the best justifications for the assumption they make. Of course, these criteria are quite soft, so there will be at least some gut feeling involved in this. To progress beyond these gut feelings, you need to find differences between predictions that these theories make and then compare these to observation to see which theory matches the observations best. To do that you need to commit to the assumptions of these theories long enough to calculate the consequences. So you go: If the equations would look like this, this galaxy should behave like that. Or if the dark matter distribution would look like this, the galaxy should behave in that different way. That is what they are trying to do in the paper this story is about.
The most successful model of the universe and dark matter is ΛCDM. To avoid the "it could be anything up to invisible pink unicorns"-problem, it makes certain assumptions of how dark matter behaves. It doesn't explicitly require dark matter to be some kind of particle, but it is difficult to see how these properties could be realized without some kind of particle or particle-like object. So it is implied that dark matter is composed of some kind of new particles. However, if you could propose some kind of dark matter which doesn't require new particles but still fulfills the assumptions of ΛCDM, this would not invalidate the theory. At them moment, I am not sure, how much this has been tried and how successful such approaches could be.
Since ΛCDM is so popular, the best way to make an alternative theory more popular is to show that it is better than ΛCDM. If it's worse, why should anyone go for it? This is why these studies usually compare with ΛCDM. If you could find an observation that ΛCDM could not explain, this will not prove that dark matter doesn't exist or isn't some kind of particle, but the next best theory might be quite a bit worse at explaining everything so that more people switch to alternative theories instead.
The field-particle duality isn't really relevant here, since the Einstein field equation is not a quantum field theory. So you cannot really work with the duality here. However, since QFT is the best theory we have to describe particles, any new particle is assumed to have a corresponding quantum field. This doesn't affect the gravity calculations, but if you search for a specific particle with an associated field, you usually need to involve the field-particle duality to calculate how that hypothetical particle would interact with whatever experiment you want to perform.