Neutrinos were actually an early candidate for dark matter. The numbers didn't work out, but it's an example of something that would interact weakly.
And sure, dark matter is falsifiable. If observations are made where gravity appears to deviate from what's visible and can't be explained by adding invisible mass, that would falsify it. Or if someone finally comes up with a change to the known laws of physics which explains the observations better than dark matter does, that would do it.
Remember, nothing is truly "directly observed." All we can do to observe stuff is to see how it affects other stuff. Whether it's with photons or with gravity, it's all ultimately indirect.
Whats to stop us from "adding epicycles" when that happens?
Two dark matter particles headed directly at each other would pass through each other and be only slightly deflected by their gravitational interaction.
Based on observed dimming events while monitoring many stars, we have ruled out brown dwarfs as more than 20% of the total. Other possibilities still remain, but none have shown evidence of being sufficient to explain dark matter. However we have not ruled out the possibility of it being something that we haven't thought of yet.
The spacetime curvature, mass/energy, and gravity have quite a few interesting interactions as in gravity as a force, vs gravity as a feature of spacetime, vs potential other forms of gravity. We aren't even sure at this point if a curved spacetime without mass/energy that would generate that curve, would actually produce gravity or not since we aren't able to observe this nor do we have a complete theory of gravity.
A hypothesis that there's local unexplained curvature...Well that's really what dark matter is all about. What's curving it?
(for posterity) My understanding is that spacetime being flat is akin to it being the surface of the universe's curves. Explain, expand, or correct me if that's wrong; I'm certainly no physicist.
Except space isn't perfectly smooth. It does curve, and that curvature is gravity. That's what general relativity is all about. If you want to postulate some way for the universe to curve absent observable mass, go ahead: you're doing nothing but restating the dark matter problem in a (minimally) different way.
The de Sitter vacuum is an exact solution to the Einstein Field Equations of General Relativity that does exactly that.
Our expanding universe will very likely closely approximate de Sitter space in the extreme far future.
There are many other vacuum solutions; it's normal in General Relativity to think of curvature as being a background laid down by hand (ideally approximating nature) and then perturbing that background with matter or probing it with test particles.
> Except space isn't perfectly smooth.
General Relativity is defined on a smooth (infinitely differentiable) manifold.
This is very important! https://en.wikipedia.org/wiki/Smoothness
There can be arbitrary curvature in the manifold, however, and unfortunately with the Einstein Field Equations you can pretty easily produce manifolds with curvature singularities, and those are manifestly not smooth (and that is one driver for research aiming to extend General Relativity).
But to the first order, your experience of gravitation near the surface of the Earth emerges from the very weak manifold curvature the planet and its microscopic components and their motions source. The nearest strong curvature is a couple of kiloparsecs away.
But then again there are multiple contenders for dark matter.
Correct me if I'm wrong, which is quite likely, but isn't it one of the qualities ascribed to dark matter that it doesn't interact with 'ordinary' matter or photons in any way other than gravity, and therefore can't be directly observed?
Or to turn the question around, is the dark matter hypothesis actually falsifiable?