LIGO has detected several NSNS and NSBH merger events. However, it’s difficult to tell their identities directly unless the neutron stars are very light in an NSNS merger (such as GW170817) or the black hole is small and spinning very fast in an NSBH merger. If the binary neutron stars are too massive as is the case of GW190425, they will collapse into a black hole almost immediately upon contact, leaving no accretion disk or hypermassive neutron star intermediate. Similarly, if the black hole in an NSBH merger is not small or fast spinning, the neutron star will plunge into the black hole directly instead of being torn apart. In both cases the gravitational wave and EM signals are not much different from BHBH mergers of identical masses and the identity of companions were inferred based on the theoretical upper bound of neutron star masses (the TOV limit). So what “smoking guns” in the GW signals can be useful in telling the identity of a compact object whose mass is close to the TOV limit? If we can tell their identities directly from their GW signatures, we can put a tighter constraint on the upper bound of neutron star masses and the lower bound of black hole masses, and find out whether there is an overlap between them.
Are there any subtle differences in the gravitational waves emitted from NSNS, NSBH and BHBH of identical masses
astrophysicsblack-holesgravitational-wavesneutron-stars
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We think that most neutron stars are produced in the cores of massive stars and result from the collapse of a core that is already at a mass of $\sim 1.1-1.2 M_{\odot}$ and so as a result there is a minimum observed mass for neutron stars of about $1.2M_{\odot}$ (see for example Ozel et al. 2012). Update - the smallest, precisely measured mass for a neutron star is now $1.174 \pm 0.004 M_{\odot}$ - Martinez et al. (2015).
The same paper also shows that there appears to be a gap between the maximum masses of neutron stars and the minimum mass of black holes.
You are correct that current thinking is that the lower limit on observed neutron star and black hole masses is as a result of the formation process rather than any physical limit (e.g. Belczynski et al. 2012 [thanks Kyle]).
Theoretically a stable neutron star could exist with a much lower mass, if one could work out a way of forming it (perhaps in a close binary neutron star where one component loses mass to the other prior to a merger?). If one just assumes that you could somehow evolve material at a gradually increasing density in some quasi-static way so that it reaches a nuclear statistical equilibrium at each point, then one can use the equation of state of such material to find the range of densities where $\partial M/\partial \rho$ is positive. This is a necessary (though not entirely sufficient) condition for stability and would be complicated by rotation, so let's ignore that.
The zero-temperature "Harrison-Wheeler" equation of state (ideal electron/neutron degeneracy pressure, plus nuclear statistical equilibrium) gives a minimum stable mass of 0.19$M_{\odot}$, a minimum central density of $2.5\times10^{16}$ kg/m$^3$ and a radius of 250 km. (Colpi et al. 1993). However, the same paper shows that this is dependent on the details of the adopted equation of state. The Baym-Pethick-Sutherland EOS gives them a minimum mass of 0.09$M_{\odot}$ and central density of $1.5\times10^{17}$ kg/m$^3$. Both of these calculations ignore General Relativity.
More modern calculations (incorporating GR, e.g. Bordbar & Hayti 2006) get a minimum mass of 0.1$M_{\odot}$ and claim this is insensitive to the particular EOS. This is supported by Potekhin et al. (2013), who find $0.087 < M_{\rm min}/M_{\odot} < 0.093$ for EOSs with a range of "hardness". On the other hand Belvedere et al. (2014) find $M_{\rm min}=0.18M_{\odot}$ with an even harder EOS.
A paper by Burgio & Schulze (2010) shows that the corresponding minimum mass for hot material with trapped neutrinos in the centre of a supernova is more like 1$M_{\odot}$. So this is the key point - although low mass neutron stars could exist, it is impossible to produce them in the cores of supernovae.
Edit: I thought I'd add a brief qualitative reason why lower mass neutron stars can't exist. The root cause is that for a star supported by a polytropic equation of state $P \propto \rho^{\alpha}$, it is well known that the binding energy is only negative, $\partial M/\partial \rho>0$ and the star stable, if $\alpha>4/3$. This is modified a bit for GR - very roughly $\alpha > 4/3 + 2.25GM/Rc^2$. At densities of $\sim 10^{17}$ kg/m$^3$ the star can be supported by non-relativistic neutron degeneracy pressure with $\alpha \sim 5/3$. Lower mass neutron stars will have larger radii ($R \propto M^{-1/3}$), but if densities drop too low, then it is energetically favorable for protons and neutrons to combine into neutron-rich nuclei; removing free neutrons, reducing $\alpha$ and producing relativistic free electrons through beta-decay. Eventually the equation of state becomes dominated by the free electrons with $\alpha=4/3$, further softened by inverse beta-decay, and stability becomes impossible.
The two black holes observed by LIGO were around 30 solar masses each - they were formed from stellar sources - that is, a supernova or similar event. They are not the same "kind" of black holes which are found in the middle of galaxies.
(sidenote: The fact that they are 30 solar masses is actually interesting. In this paper they discuss how the environment had to be a little bit special for these black holes to form).
In regards to the condition of "truth", it conforms to established scientific norms. For instance, the detector has been very well-modeled and every reasonable error has been accounted for, so we have very good reason to believe that the signal is real (to say nothing about the fact that it was observed in TWO detectors, one in Louisiana and one in Washington, and the signals are nearly identical). To determine the details of the merger, people have been working very hard over the past decade to develop a library of signals, for a variety of objects (neutron stars and black holes) and a wide variety of parameters (masses and orbital parameters). So they determined the characteristics of the merger by comparison with those models.
Of course, we aren't in a spaceship floating over this merger viewing it with our own eyes. But on the basis of the scientific method (hypothesis testing and independent verification), this establishes the existence of gravitational waves.
(for the full paper talking about the observation)
EDIT: I'm going to try to tackle your clarifying questions.
This one is slightly tricky, since all (extra-solar) astronomy is indirect in this way - we only observe the cosmos via the light we receive from it. For example, the existence of the star Polaris is indirect, and depends on the assumption that stars produce light (which is on very solid footing, obviously). Some examples that might be closer to what you're thinking of - Dark matter is only detected via it's gravitational influence (never directly), but most people consider it to be a real phenomena. Pulsars being associated to neutron stars is mostly theoretical - although we can associate them to SNR sometimes. And actually, the vast majority of extrasolar planets are detected indirectly, via the Doppler shift or transit methods.
I think the answer is "no". You would have to explore each one individually, since the argument in each case is rather unique. I once listened to an interesting podcast about how astronomy is observational, not experimental. I think it's here. I think the best you can do is list evidence for discovery and let the community decide. This is not a unique problem, BTW - no one has ever seen a Higgs particle, in the traditional sense - we inferred it's existence at a level sufficient for the scientific community.
LIGO releases it's data to the public at proscribed times. Here's a list of projects using LIGO data. I don't think I see specifically what you are interested in ("We checked LIGO, it's right!"), but this list is only the past few months.
Best Answer
There are a few different layers to the logic for how this can be done. Here I will focus on gravitational-wave observations of binary systems, but there is also relevant information that can be extracted from electromagnetic observations of binaries (kilonovae) from observations of individual neutron stars (either electromagnetically using, eg, NICER) or, in principle, from gravitational waves emitted from "neutron star mountains").
Gravitational-wave observations of a single binary
Inspiral phase
The main feature that distinguish binary black holes (BBHs), from binaries with matter (binary neutron stars (BNSs) and neutron-star--black-holes (NSBHs)) is the tidal deformability parameter $\Lambda$. This parameter is related to the Love numbers of neutron stars in the system (the Love number of black holes is zero, modulo some subtleties not directly relevant for the gravitational waveform). The tidal deformability changes the phasing of the binary system relative to a BBH starting at the 5-th post-Newtonian level. A measurement of the tidal deformability therefore lets one rule out BBHs, and gives information about the NS equation of state. Finally, there are non-linear tides, although these are difficult to measure [2].
[1] https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.121.161101
[2] https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.061104
Merger and postmerger phase
As you pointed out, for NSBHs with a near-equal mass ratio (the details of how "near-equal" depends on the equation of state), the neutron star will get torn apart by the black hole. This tidal disruption cuts off the amplitude before the normal merger phase of the waveform. However, the disruption happens at a high frequency, so is hard to measure with current detectors.
Additionally, as you pointed out, if a hyper-massive neutron star forms in the aftermath of a BNS merger, there are characteristic post-merger signals that differ from the BBH ringdown frequency. In principle, there is a lot of information contained in the post-merger since the system is highly excited. However in practice, the modeling of this phase is extremely difficult, and the post-merger happens at such high frequencies that it is essentially impossible for current detectors to directly probe this phase [3,4]. There are proposals for future detectors that would be optimized for high frequencies to measure the post-merger spectrum [5].
[3] https://iopscience.iop.org/article/10.3847/2041-8213/aa9a35
[4] https://iopscience.iop.org/article/10.3847/1538-4357/ab0f3d
[5] https://journals.aps.org/prd/abstract/10.1103/PhysRevD.99.102004
Analyses of populations of gravitational-wave sources
The summary of the previous section is essentially that it's possible in principle to rule out a BBH hypothesis using gravitational-wave observations if the tidal deformability parameter can be measured, but to really distinguish BNS and NSBH requires some observation of the merger and post-merger phase, which is incredibly difficult with current observatories (but maybe can be done in the future).
However, there is more information that can be obtained by considering a collection of observations as a population. There are a few ways this can be done.
[6] https://science.sciencemag.org/content/370/6523/1450
[7] https://iopscience.iop.org/article/10.3847/2041-8213/ab960f