In the aftermath of our paper (with Maxim Pospelov) "Hunting for topological dark matter with atomic clocks" having been published, there were quite a number of e-mails with questions about our proposal. There was even an offer for a free-of-charge use of a powerful computational cluster (thank you!). I apologize for not answering all e-mails individually - just not enough time.  One of my friends has also sent me a link to this reddit thread - there is a genuine interest to the details of the proposal. This post is intended to answer some of these questions.

First of all see the previous post that outlines the basic idea of the search.

Topological dark matter: 
There are two components that go into dark-matter model building: (i) what the dark matter objects are and (ii) how these objects interact non-gravitationally with us (baryonic or ordinary matter). I emphasize the word non-gravitationally, as the gravitational interaction is a must due to multiple observations of gravitational interactions between dark and ordinary matter (and consistency with general relativity).

Additional model constraints come from various observations and cosmological simulations. Still the allowed parameter space is enormous: even if one were to assume that the dark matter objects are made out of elementary particles, the allowed masses span 50 orders (!) of magnitude. This is just a testament to the current state of confusion in modern physics and cosmology.  The field is ripe for discoveries.

First of all I admit that our model (due to Maxim Pospelov) is speculative, but it is as good as any model out there. WIMPs and axions have additional attractive features as they also might solve other outstanding problems in physics (for example, strong-CP problem in physics can be solved with axions).

So what is the model? (here you might get lost, just read on). For experts, technical discussion can be found in the extensive supplementary material to our paper.

Well, you start with a quantum field and this field has some self-interaction built in. The interaction is such that it allows for several identical minima. For example, the same value of potential minima could be reached at two distinct values of the field +A and -A. Now when the Universe expands it cools down and the field has to settle at the minima of the potential. The field is torn apart by which value to chose - the choice of +A or -A are equivalent. So in some regions of space it picks +A and in the other regions it picks -A. This is called "spontaneous symmetry breaking".

Nature does not like discontinuities and you have to smoothly connect  +A and  -A domains. This transition region is the topological defect or cosmic wall. The thickness of the wall is given by the particle Compton wavelength = h/(m c), where m is the particle mass, h is the Plank constant and c is the speed of light.

This example is overly-simplistic but it demonstrates the idea of how topological defects are formed as the Universe cools down: in fact, for a dark-matter model you would like to have the field to be zero everywhere except inside the defects (see the supplement). All the energy (or mass) is stored in topological defects.

Depending on the field's degrees of freedom (scalar vs vector fields) and the self-interaction potential  one may form defects of various geometry: monopoles, strings or domain walls. Especially interesting is the case of monopoles (spherically-symmetric objects) as the gravitationally-interacting gas of monopoles mimics dark matter. The size of the defect is a free parameter - we do not have constraints on how large it could be. GPS would be sensitive to Earth-sized monopoles (huge Compton wavelength translating into particle mass ~10^-14 eV).

Here is a real-life example of spontaneous symmetry breaking and topological defects (due to Rafael Lang, the interview to appear in Sensing Our Planet magazine)

“There’s a wedding and a hundred people are sitting at this big round table. Somebody starts eating the salad. They pick up the fork on their left, so the person next to them has to pick up the fork on the left. Now the bride also starts eating, picking up the fork on the right, so everybody around the bride picks up the right fork. At some point in between this poor guy will be sitting with no fork; on his other side will be someone with two forks. Those two guys are called a topological defect. There’s nothing special going on around the left, the right, but where those two guys are sitting, there’s a disruption of the forks.”

Ok so we are done with choosing dark-matter objects. Now the second ingredient is the non-gravitational interaction between dark matter objects and us. Here you do need to pick one that is "reasonable" (e.g., Lorentz-invariant)  and is sufficiently weak that it went unnoticeable in dedicated experiments and observations. The interaction that we picked is of this kind. Effectively when the defect overlaps with us, it pulls on the particle (electron, proton, neutron, etc) masses and forces acting between the particles. Mind you this pull is really weak, otherwise we would have noticed it. However, there are ultra-sensitive devices, like atomic clocks (see this post) that may be sensitive to such pulls. You might ask - why it might have gone unnoticed before in atomic clocks -  some of the reasons are purely psychological and are related to how an experimentalist discerns signal from noisy background (see this post.)