The Reason We Haven’t Directly Detected Dark Matter

In the standard cosmology, dark energy is \Lambda, the cosmological constant.

The cosmological constant is taken to be a geometrical phenomenon rather than some dynamical field.

We can within the limits of observational accuracy use a "swiss-cheese" cosmology model. We start with a Friedmann-Lemaître-Walker-Robertson (FLRW) background, which is an expanding spacetime with a uniform dust that dilutes away uniformly with expansion. The expanding spacetime's metric is Robertson-Walker, which is an exact solution of the Einstein Field Equations (EFEs) of General Relativity. The dust particles are galaxy clusters, which are gravitationally bound, and are manifestly not expanding in the same way, so they cannot have the same metric. Around each particle, we cut out a "hole" in the background and replace it with a collapsing spacetime metric, like Schwarzschild or Lemaître-Tolman, which are two other exact solutions of the EFEs.

We can use Israel junctions to stitch together a pair of metrics like Robertson-Walker and Lemaître-Tolman, and while it's annoying procedurally, it produces good results. Indeed, a simpler case is the Einstein-Strauss swiss cheese, which served as a practical cosmological model until the late 1980s, when the evidence began piling up for the presence of a small positive cosmological constant.

In swiss cheese models, the cosmological constant vanishes in the inner metric (the "holes") and is only non-zero in the outer metric in which the holes are embedded. Since dark energy is simply the representation of the cosmological constant given a particular slicing of the universe into things-in-space+time rather than spacetime-filling fields, this means that mathematically there is no dark energy in galaxy clusters and other gravitationally collapsing "holes" in the otherwise expanding universe.

This seems shocking ("why isn't there dark energy everywhere?" seems to demand a mechanism rather than just a statement of geometry) but it's testable, and so far there is no evidence for the metric expansion of space within the solar system, or within galaxy clusters.

If we find out that space does expand near and in galaxies, then we have a variety of ways to go beyond the swiss-cheese approach, and we might explore a couple of them anyway since we now have powerful computers and do not have to lean on exact or analytical solutions of the Einstein Field Equations. This is the research field of "inhomogeneous cosmology".

Alternatively, we can stop making holes and instead treat the cosmological constant as a (location-dependent) dynamical field that is weaker near matter except in the early universe (when matter is all squashed close together). Various proposals along those lines like "quintessence" ("quint" because such a field automatically produces a fifth force) have been written about. Evidence strongly constrains a fifth fundamental force of nature, however, so this approach seems much more speculative than either simply accepting that the cosmological constant is geometry and the universe is swiss-chese-like geometrically, or pursuing a much more complicated "real" metric rather than starting with a simple, exact, analytical metric and perturbing it where that's important (e.g. because of how the matter in large structures might be laid out in ways that are hard to be considered pointlike or axisymmetric when viewed from large distances).

> gravity well

Not a useful concept and definitely not an object in General Relativity.

> anti-gravity

Kinda, but it's important to understand what that means. When gravitation squashes matter into a denser shape you can think of it as creating pressure on and in the matter. The metric expansion of space is not preventing the gravitational collapse of galaxy clusters: they're still shrinking and the pressure inside them is still increasing. Galaxy clusters are essentially bubbles floating in a sea that is getting larger around them. By contrast, the pressure in the "sea" is reducing over time, proportional to the value of the cosmological constant. Since we can treat pressure as a component of the stress-energy tensor, the "matter" or "sources" side of the EFEs, we can treat increasing pressure inside collapsing stars as a source of gravitation (an IMPORTANT source when stars collapse into white dwars, neutron stars or black holes -- pressure dominates the other masses and energies as a gravitational source in those cases). Likewise the increasingly negative pressure in the regions outside galaxies source can be treated as a source of gravitation, but really this is just a special way of looking at the fact that galaxy clusters are separating from one another without any motion-distortion (shear, for example) being evident in our images of the galaxy clusters at increasing distances. There is obvious shear within collapsing galaxy clusters and their internal components -- galaxies and objects near the centres of clusters are more stretched radially than those further from the centres of clusters, and the shearing strength depends on the overall mass of the cluster. There is no mass-dependency on cosmological redshift from receding galaxy clusters; individual galaxy clusters are not stretched towards us at different redshifts.

> interference pattern between dark-energy and gravity

Well, between dark-energy and collapsing matter, yes. The closest concept to your idea of an interference pattern is the presence of holes in the swiss cheese. If superclusters are gravitationally bound and form long filament structures that collapse collectively (rather than there being a line-up of individually roughly-spherically-collapsing galaxy clusters, with the individual clusters not moving towards each other over time) then the geometry would not be quite so swiss-cheese like, or at least not everywhere.

You are free to do handstands and other contortions to describe this in terms of waves-and-interference. You'd probably use perturbation theory, where you throw away the holes and complicate the background metric or the matter fields. When you do that you're engaging in the study of inhomogeneous cosmology or quintessence-like dynamical dark energy. Those are decent search-engine terms if you want to do a quick survey of those fields of research. They aren't popular because the standard cosmology with swiss cheese matches observations to extremely high precision while being much easier to work with than the other two approaches.

> ... at the galactic boundary ...

The Israel junction is described in Chapter 21 of one of the standard textbooks, _Gravitation_ by Misner, Thorne & Wheeler ("MTW"). In brief, there is an infinitesimally thin shell drawn as a boundary around the collapsing spacetime arranged carefully so that an internal time coordinate matches the external time coordinate, which is the scale factor (or lookback time) in the standard model of cosmology. There's a mathematical matching of values of the electromagnetic fields and other fields of the standard model of particle physics on either side of that thin shell.

In reality, the boundary around real galaxy clusters is not that sharp; the edge is a fuzzy end to the sparse gas and dust one finds at the outer limits of galaxy clusters' gravitational influence, so it ends kinda like Earth's atmosphere. It's mostly gone at 100km up, but not enough that satellites and spacecraft much higher don't have to deal with tiny drag from stray molecules and atoms. But in practice, above 100km the residue of atmosphere doesn't enter into equations, and in practice far from the centres of galaxy clusters the residue of gas doesn't enter into equations either.

But if you had much much much more powerful computers and software than we have today, you could in principle do numerical relativity that accounts for all that, and would be "fuzzing out" the Israel junction procedure most likely. (Or, again, you could go right in and study and account for inhomogeneities right down to photons travelling between galaxy clusters, wheeeee! But where do you cut it off? A stroke of lightning on Earth around eight million years ago flashed some light in a direction that caused a tiny fraction of the flash to exit our galaxy cluster. Should we subtract that out from our galaxy cluster's position in the expanding universe? We're also near edge than the centre of our cluster, so the direction of the flash is relevant to how much of it exited, and when it exited. And so on and so on and so on. At some point, the light in question is still contributing to the gravitational collapse of our galaxy cluster; at another point, it's removed some of the galaxy cluster's stress energy from the region in which everything in it is gravitationally bound. We just choose an arbitrary point and say "there's the crossing-over". (We also would ignore the flash because it is such a tiny fraction of the total stress-energy of the galaxy cluster).

As with almost all physical models at some point you have to say "I can only be so precise in modelling and in matching the model to nature" and hope that precision keeps improving over time.

We can detect dark matter via its mass, that's why we know it exists. We can see it speeding up galactic rotations and acting to gravitationally lens more distant galaxies and so forth. But we have yet to detect it on the small scale, especially at the individual particle level.

Keep in mind that one of the key differences between dark matter and atomic matter within a galaxy, for example, is that dark matter has a nearly uniform density over very large volumes (many light years across) whereas atomic matter has enormous density variations, with huge expanses of near vacuum punctuated by ultra dense stars, planets, neutron stars, etc. Within our own Solar System the amount of dark matter inside a spherical volume that would extend out to Neptune's orbit is only as much as a comparatively small asteroid. As you scale out to larger and larger scales the fact that the density of dark matter is relentless causes the mass enclosed inside a volume to start to match (and ultimately exceed at the largest scales) the mass of stars, nebulae, planets, etc.

Your experiment has several fundamental flaws. Firstly, dark matter is weakly interacting (even with itself) which is why it doesn't "clump" the way atomic matter does. The only "clumping" of dark matter happens on galactic scales, because the typical speeds of dark matter particles are in the range of orbital speeds around galaxies. Our measurements of the dark matter mass in galaxies don't have nearly enough precision to be able to determine the relationship you posit with much certainty.

As to your own experimental design, using large spheres of matter, it's completely non-workable. As mentioned above, dark matter particles have velocities in the range of orbital speeds around the galaxy, which is 100s of km/s. So most of the dark matter particles in the vicinity of Earth are going to simply pass through your spheres without stopping, and certainly without increasing their mass.

Additionally, you seem to be confused about the scales of densities here. Dark matter is distributed in a somewhat uniform density on interstellar scales (there are density gradients across the galaxy however). Near the Sun the density of dark matter is about 0.0025 solar masses per cubic light-year, or about 5e-20 kg per cubic meter. So, there really is not a lot of dark matter passing through objects around Earth, mass wise. There's only maybe 6 kg within the entire volume of the Earth at any given time, for example.

What do you think of the possibility of 'dark sector' interactions?[1]

The idea that dark matter might consist of a class of self-interacting particles, and that we might be embedded in a universe full of hidden phenomena as rich as the ordinary-matter phenomena that are visible to us (e.g. dark 'planets', dark 'stars', dark 'galaxies', or something very different) was always intriguing to me, but it seems that a consensus is emerging, based on observations of large-scale distribution, that dark matter is dominated by a single type of particle incapable of self-interaction.

Is it still possible that some fraction of the dark matter in the universe is self-interacting, capable of 'clumping' and exhibiting physics similar to ordinary condensed matter, or are all the indications now pointing strongly towards a single non self-interacting particle?

[1] https://en.wikipedia.org/wiki/Dark_photon

In short: "because science", quite literally.

Astronomers are scientists, they don't make shit up just out of boredom, they only believe theories when they've withstood rigorous testing through observational evidence.

It's important to understand the history of dark matter / "missing matter". It's a multi-decade history that originally started out with a small amount of intriguing but seemingly persistent observational data that couldn't be explained easily (namely that when you measure how much galaxies weigh by looking at how fast the stars are orbiting that figure differs a lot from the weight you get by calculating up the contribution from all the stars and gas and dust and whatnot that we can see (the "light" matter)). In response a huge number of different "theories" (more properly hypothesis) were brought up to try to explain the mystery, while new ways of studying the problem were thought up as well.

And over the many years after the initial evidence came to light (in the '70s) considerably more evidence came to light from a wide diversity of observations. I won't list them here but I'll point out that the wikipedia page on dark matter has a good run down. Anyway, the fascinating twist to the story here is that as this evidence came to light it started eliminating various hypotheses about this missing matter until finally only one was left: the current theory of dark matter. It's not just a crazy idea, it's a crazy idea that fits all of the evidence when nothing else did.

So don't look at the WIMP dark matter theory as though it's just some half-baked idea, it's a hard fought veteran of numerous campaigns to kill it, but it just keeps going because by all the evidence it seems to be the only theory that explains reality.

As to specifically the "modified gravity" theories that compete with dark matter, they have a hard time explaining several observed phenomena, especially things like the famed "Bullet Cluster". There are a few examples where we can observe collisions between galaxy clusters and through different techniques we can map out the distribution of stars and of gas and of mass in the collision. What we observe in the Bullet Cluster as well as some others is that the gas is in a completely different place than the stars (because stars in galaxies mostly pass through one another whereas gas clouds squish together) and the center of mass of the stars of the galaxies is separated from the center of the non-visible mass of the galaxies. This absolutely cannot be explained by any of the modified gravity theories.