Determining the neutron star (NS) maximum mass is a one of the most fascinating, unresolved issues in modern astrophysics. The answer is intimately related to identifying the correct equation of state (EoS) that descibes matter at supranuclear densities. Currently the highest observed NS mass is $2.14^{+0.10}_{-0.09}\ M_\odot$. In principle, the upper limit allowing only for causality and a matching density to a well-understood EoS somewhere around nuclear density, can be as high as $4.8\ M_\odot$, while recent studies based on the detection of the gravitational wave (GW) signal GW170817 place it around $\sim 2.2-2.3 M_\odot$. All these studies adopt a number of underlying assumptions whose validity will require new observations to be verified or modified accordingly. Observationally, merging binary black holes (BHBHs), black hole-neutron stars (BHNSs) or binary neutron stars (NSNSs) whose companions have masses that fall into the mass gap range $(3-5\ M_\odot)$ are hard to distinguish. The identification of a compact object becomes even more challenging when one includes exotic solutions, such as quark stars, boson stars, etc.

The parameter that encodes how much mass a compact star can hold in a certain volume is the compactness, defined as the dimensionless ratio $C = \frac{GM}{Rc^2}$. Here $M$ is the Arnowitt-Desser-Misner (ADM) mass, and $R$ the areal (Schwarzschild) radius of an isolated, nonrotating star with the same rest mass. Our sun has $C=2\times 10^{-6}$, a small number indicative of its nonrelativistic nature, while the upper limit, $C=1/2$, is set by a Schwarzschild BH. Typical NSs have compactions around $\sim 0.1-0.2$ with the precise number determined by the as yet unknown EoS. An extreme case is the incompressible fluid limit that yields $C=4/9=0.4\bar{4}$, the so called Buchdahl limit. This limit is unrealistic since it predicts an infinite sound speed. If one satisfies the causality criterion for the sound speed (i.e. $c_s\leq c$) then the upper limit for compactness drops to $C_{\rm max}=0.355$.

Compact binary systems provide some of the best laboratories to test the predictions of general relativity, as well as to probe possible deviations from its description of strong gravity. Despite the large progress that has been achieved in numerical relativity we are still lacking theoretical simulations that involve extremely compact NSs in binaries.

The purpose of this work is to quantify the difference between a BHBH and an NSNS system when the total ADM mass falls inside the mass gap and to provide useful GW diagnostics that may distinguish them. First we construct the most massive NSNSs in quasicircular orbit with the highest compactness to date using the initial data solver COCAL. The system has ADM mass $M=7.90 M_\odot$ and each star a compactness of $C=0.336$. This value (which is even higher than the maximum posible compactness that can be achieved by solitonic boson stars) is only slightly smaller than the limiting compactness $C_{\rm max}=0.355$ set by causality. Second, using the Illinois GRMHD code we evolve this NSNS system and perform a detailed comparison of the gravitational waveforms with a BHBH system having the same initial ADM mass.

We find that an NSNS system having the above compactness inspirals very similarly to the BHBH system and merges $\textit{without essentially any tidal disruption}$. We conjecture that finding to be true irrespective of the EoS for this level of compaction. The merged NSNS remnant collapses to a BH even before a common core forms. The GW phase difference at the peak GW amplitude of the NSNS system is $\sim 4$ rads with respect to the BHBH binary inside the band $[0.6,1]\ {\rm KHz}$. This phase difference corresponds to $\sim 20\%$ of the accumulated phase difference during the last $\sim 1.7$ orbits (corresponding to an initial separation of $\sim 81$ km) and $\textit{can}$ be measured by the aLIGO/Virgo network, although uncertanties in the orbital parameters will likely prevent distinguishing such NSNS binaries from BHBHs. Moreover, the postmerger remnants have ringdown waveforms that $\textit{cannot}$ be distinguished from the Kerr BH ringdown within the accuracy of our simulations.