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A black hole doesn't necessarily need to form from a star; theoretically, it could form from any extremely dense object. In fact, many astronomers differentiate certain black holes, like supermassive ones, from stellar ones (ones that form from stars).

However, could the same apply for neutron stars? Neutron stars only form because of the intense gravity during a star's collapse: electron capture is forced to happen, and the majority of the star becomes neutrons. Could this potentially happen to non-stellar objects, if the gravity forces electron capture?

If so, why don't we see as many of these "neutron objects" as we do non-stellar black holes?

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There can be no such thing as a "supermassive neutron star". The theoretical upper mass limit for a neutron star is somewhere between 2.2 and 3 solar masses. Any more massive and they inevitably form black holes. So I am not clear what kind of "neutron objects" you were thinking of?

Nor is it clear what you mean by "non-stellar" objects that will have the densities required to make neutron-degenerate matter? There aren't any apart from (I) the cores of massive stars at the ends of their lives. (II) Massive white dwarfs if they accrete matter over and above their Chandrasekhar limit. Furthermore, there is a minimum mass for a neutron star. Although the observed lower limit to those seen in nature is (so far) around $1.15 M_{\odot}$, there is a theoretical lower limit of about $0.1$ to $0.2M_{\odot}$ set by the stability of the neutron against decay in self-gravitating material ( See https://physics.stackexchange.com/questions/143166/what-is-the-theoretical-lower-mass-limit-for-a-gravitationally-stable-neutron-st/143174#143174 ).

It is true if you could arrange to compress any matter to densities above $\sim 10^{15}$ kg/m$^3$ it would form neutron-degenerate material. But this requires (as far as we know) the conditions I listed above.

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A neutron star will form if you have roughly 1.4-3 solar masses of matter which is not producing enough energy to hold itself up through radiation pressure. So in principle you could assemble 2 solar masses of iron and it would collapse under its own gravity and form a neutron star.

However, the vast majority of the universe is mixed hydrogen and helium, and if you assemble 2 solar masses of those materials, fusion will start well before it collapses and you get a normal star. So it seems likely that every natural neutron star in the real universe will have been through a nuclear burning phase, and so be a "stellar" neutron star.

It is just conceivable that a density fluctation in the very early universe might have been just the right size that the matter there cooled to neutronium rather than just a denser cloud of normal matter, but it would have been surrounded by fairly dense normal matter, so would probably grow into a black hole.

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As others have pointed out, the reason why we don't see non-stellar neutron stars is that the pressures needed to form them are usually only found in stars. Lower pressures don't form neutron degenerate matter and higher pressures form black holes.

I think part of your question may be whether or not smaller quantities of neutron-degenerate matter, which would be under lower constant pressure, are stable. I think the answer is either "no" or "not very much lower pressure", since even the crusts of neutron stars are believed to have separate nuclei: https://link.springer.com/article/10.12942/lrr-2008-10 . In small quantities, neutron-degenerate-matter would probably explode with extreme force, as if were just a ridiculously superheavy and super-neutron-rich atom: https://physics.stackexchange.com/questions/10052/what-would-happen-to-a-teaspoon-of-neutron-star-material-if-released-on-earth

However, some theories suggest that this would not be true for denser types of quark-gluon matter that are likely to be found inside massive "neutron stars". The concept of "strange matter" made of up, down, and strange quarks is well known, and it is has been famously predicted by some to be stable at room temperature, after it has been formed, perhaps even converting normal matter it touches into strange matter (or, alternatively, maybe not). "Strangelets", or tiny pieces of strange matter, are one candidate for what dark matter could be, and it has even been suggested that they might hit Earth about once a year and explain some weird craters: https://arxiv.org/ftp/arxiv/papers/2007/2007.04826.pdf

Similarly, it has also been suggested that very heavy atomic nuclei (A>about 300), may collapse into a sea of up and down quarks called "up-down quark matter" or udQM, which might actually be more stable than "strange matter" (uds-matter). This has been suggested to create a "continent of stability", where nuclei this big are actually stable, unlike smaller superheavy nuclei, and has been suggested as an alternative to strangelets as a dark matter candidate: https://en.wikipedia.org/wiki/Continent_of_stability

Needless to say, both of the types of QCD-matter are extremely theoretical because A) only supernova-like pressures can create them, even if they do turn out to be stable or metastable at low pressure, B) our methods of making superheavy nuclei with particle accelerators have not progressed that far yet, C) the proper quantum-chromodynamics calculations are very difficult to do, and really good calculations of this sort have only been done for small nuclei, so the math these predictions (including the one about neutron-degenerate matter) are based on is all approximations that probably introduce significant errors.

It is possible that further investigations of neutron-star collisions will reveal more about the nature of their interiors, and whether these collisions might even be able to liberate some ultradense matter without it decaying into normal-sized nuclei. (If it doesn't, that doesn't mean this ultradense matter couldn't be stable, since it could just be the energy of the collision that destroyed it, and even if it does, such a source is not likely to be a major source of dark matter, because that would result in the amount of dark matter increasing as the universe aged, which I think goes against observation.)

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