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We think that the mass boundary between neutron stars and stellar mass black holes is around three solar masses.

The maximum mass of the neutron stars now is two solar masses and we may find a 2.6 solar mass neutron star in the future. That is based on the equation of state.

The stellar mass black holes may have different origins. Even the smallest stellar mass black hole originates in SN explosion, we do not know the details of the explosion.

Why do not we think a two solar mass BH, or even a smaller BH, could exist?

Why do we think that the companion of Taylor-Hulse binary is a neutron star instead of a BH?

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I think you made a lot of confusion. Please, quote all the references for your statements, and ask just one question per time: your two questions are very different. – Py-ser May 30 '14 at 4:36
There are obvious connections. An expert will think it is one unified question, the mass range of NSs and BHs. – questionhang May 30 '14 at 6:37
Or maybe an expert is saying (among other things) they are not, as you like! – Py-ser May 30 '14 at 6:41
Relax, OK? This is not the first time that what you said has nothing to do with my question. If you do not know this field well, just ignore my question. If I am lucky, I can meet a professional people here. – questionhang May 30 '14 at 6:49
Don't need relaxing, I was just suggesting. If you think my comment is inappropriate just ignore it, don't start useless discussions. We are not on a let's-chat-forum, neither I am 15yo anymore. I just care about the physics behind the question. – Py-ser May 30 '14 at 7:08

As you said in your question we only know about Black Holes creating in Super Novae, which lead to relatively big BH. The reason they can't be smaller in these cases it that they are created when the core of the star reaches a critical mass that overflows both the electron and the neutron degeneracy. Lower masses can be hold by those degenerate mass states and won't collapse any further.

Theoretically BH with masses less than the solar one could exists if there was some method capable to compress it's mass reaching radii smaller than their Schwarzschild radius.

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The mass of a black hole is dependent on the formation mechanism, which as far as we are aware is predominantly as a supernovae remnant. The Tolman–Oppenheimer–Volkoff limit provides the theoretical lower mass bound of a black hole formed from a collapsing star; this is between 1.5-3.0 solar masses depending on the interpretation.

You must also remember that the mass of a Black Hole is not static; due to Hawking radiation and absorbtion from the Cosmic Microwave Background it can grow or shrink depending on its size. The following wiki quote sums it up well

A stellar black hole of one solar mass has a Hawking temperature of about 100 nanokelvins. This is far less than the 2.7 K temperature of the cosmic microwave background radiation. Stellar-mass or larger black holes receive more mass from the cosmic microwave background than they emit through Hawking radiation and thus will grow instead of shrink. To have a Hawking temperature larger than 2.7 K (and be able to evaporate), a black hole needs to have less mass than the Moon. Such a black hole would have a diameter of less than a tenth of a millimeter.[87]

If a single solar mass black hole did form through some exotic process it would grow through absorbtion of the CMB.

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