A neutron star is the dead core of a massive star, crushed to the size of a city. A teaspoon of its material weighs as much as a mountain. Dark matter particles fly through such a dense medium almost unimpeded, but occasionally they do hit neutrons and get stuck, like bullets bogging down in deep sand. Once settled in the center, they deposit energy, and the star warms up slightly.
If dark matter can interact with itself, the process changes. Imagine a crowd squeezing through a narrow corridor: when everyone moves alone, passage takes time; but if they also grab each other's hands, motion slows and more people pile up in the corridor. Similarly, inside the star, self-interactions help particles get captured and transfer heat faster. As a result, the star's surface becomes warmer than it would be without such an effect.
Such cold stars are rare, but they make ideal targets. If the James Webb Space Telescope or the future Thirty Meter Telescope spots such an object, it would be a strong argument for self-interacting dark matter. Constraints on the strength of this interaction would be hundreds of times tighter than those from the famous Bullet Cluster, where galactic clusters collided. A neutron star becomes a natural laboratory where invisible matter betrays itself through heat.
A neutron star the size of Manhattan can be hotter than the Sun in its early years, but it cools over millions of years; finding a star that has cooled to a thousand kelvins is like discovering a barely warm ember in a cosmic bonfire.
In Robert Wilson's novel Spin, neutron stars serve as beacons for an alien network — much like here, except instead of a signal, we catch the heat of dark matter.