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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 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.

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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