# Prompt Black Hole Formation in Binary Neutron Star Mergers

Christian Ecker<EMAIL_ADDRESS>Institut für Theoretische Physik,
Goethe Universität, Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany
Konrad Topolski<EMAIL_ADDRESS>Institut für Theoretische
Physik, Goethe Universität, Max-von-Laue-Str. 1, 60438 Frankfurt am Main,
Germany  Matti Järvinen<EMAIL_ADDRESS>Asia Pacific Center for
Theoretical Physics, Pohang, 37673, Korea  Department of Physics, Pohang
University of Science and Technology, Pohang, 37673, Korea  Alina Stehr
<EMAIL_ADDRESS>Institut für Theoretische Physik, Goethe
Universität, Max-von-Laue-Str. 1, 60438 Frankfurt am Main, Germany

###### Abstract

We carry out an in-depth analysis of the prompt-collapse behaviour of binary
neutron star (BNS) mergers. To this end, we perform more than $80$ general
relativistic BNS merger simulations using a family of realistic Equations of
State (EOS) with different stiffness, which feature a first order
deconfinement phase transition between hadronic and quark matter. From these
simulations we infer the critical binary mass $M_{\rm crit}$ that separates
the prompt from the non-prompt collapse regime. We show that the critical mass
increases with the stiffness of the EOS and obeys a tight quasi-universal
relation, $M_{\rm crit}/M_{\rm TOV}\approx 1.41\pm 0.06$, which links it to
the maximum mass $M_{\rm TOV}$ of static neutron stars, and therefore provides
a straightforward estimate for the total binary mass beyond which prompt
collapse becomes inevitable. In addition, we introduce a novel gauge
independent definition for a one-parameter family of threshold masses in terms
of curvature invariants of the Riemann tensor which characterizes the
development toward a more rapid collapse with increasing binary mass. Using
these diagnostics, we find that the amount of matter remaining outside the
black hole sharply drops in supercritical mass mergers compared to subcritical
ones and is further reduced in mergers where the black hole collapse is
induced by the formation of a quark matter core. This implies that $M_{\rm
crit}$, particularly for merger remnants featuring quark matter cores, imposes
a strict upper limit on the emission of any detectable electromagnetic
counterpart in BNS mergers.

neutron stars, equation of state, binary neutron star mergers, black holes

††preprint: APCTP Pre2024 - 006

##  I Introduction

Neutron stars have become leading laboratories to study dense matter in the
course of the past decade. As a key component in this development, multi-
messenger signals from binary neutron star (BNS) mergers include information
about the properties of strongly interacting matter at densities not
accessible to terrestrial experiments. The detection of gravitational waves
(GW) from the GW170817 event [1] already provided constraints on the neutron
star Equation of State (EOS) [2] that complement those from direct mass [3, 4,
5, 6] and radius measurements [7, 8, 9, 10, 11] as well as from microphysics
calculations in nuclear theory [12, 13, 14, 15] and perturbative Quantum
Chromodynamics (QCD) [16, 17, 18, 19]. Neutron stars with masses close to the
maximal limiting value set by collapse into a black hole are particularly
interesting for EOS inference, since their cores reach densities multiple
times larger than the nuclear saturation density $n_{s}=0.16/{\rm fm}^{3}$.
Our theoretical understanding of QCD, the fundamental theory for matter, at
these densities is very limited, which makes observations of highly massive
neutron stars and their mergers currently the only reliable source of
information about their material properties.

There exists a simple and well defined notion for the limiting mass of
isolated static ($M_{\rm TOV}$) and rotating stars ($M_{\rm max}$), beyond
which they become unstable against gravitational attraction and collapse into
a black hole, in terms of the so-called turning-point criterion [20]. While
the values of $M_{\rm TOV}$ and $M_{\rm max}$ individually depend strongly on
the EOS, in case of uniformly rotating stars they were shown to satisfy the
quasi-universal (EOS independent) relation $M_{\rm max}\approx 1.2~{}M_{\rm
TOV}$ [21, 22, 23]. In contrast to isolated neutron stars, which are well
approximated by a stationary and axially symmetric spacetime, the notion of a
limiting mass is more intricate for dynamic systems like BNS mergers, because
they evolve non-linearly in time and possess essentially no symmetries.
Crucially, the time-dependent differential angular velocity profile of the
merger remnant influences the stability and changes the simplified picture
observed for isolated, uniformly rotating stars, so that simple recipes like
the turning-point criterion do not apply. Depending on the total mass and its
division into its binary constituents, a BNS merger can either result in the
prompt formation of a black hole or a (meta-)stable neutron star merger
remnant, which may collapse into a black hole at later times. These two
scenarios feature different patterns of GW emission and electromagnetic (EM)
radiation [24].

Significant effort has been put in the field towards investigating the
collapse of isolated neutron stars [25, 26, 27] and black hole formation in
neutron star head-on collisions [28, 29]. More recently the mass-threshold to
prompt black hole formation has also been investigated in numerical
simulations of BNS mergers starting from more realistic inspiral initial
conditions [30, 31, 32, 33, 34, 35] rather than head-on conditions. Such
threshold mass analysis, in combination with observational data from the
GW170817 event, lead to estimated lower bounds of neutron star radii [36, 31]
that agree surprisingly well with those from entirely different theoretical
approaches [37]. The aforementioned studies provided important insights about
the collapse behaviour of BNS mergers, but also rely on various simplifying
assumptions to estimate the threshold mass. These assumptions include for
example an approximate description for the gravitational field [36], simple
gauge dependent criteria based on a Newtonian notion of free-fall [31] or the
use of EOS models that are in tension with some of the recent neutron star
measurements that constrain the EOS at the densities and temperatures reached
right before and during the collapse.

In this article we address these limitations by introducing the notion of the
critical mass $M_{\rm crit}$ that separates the prompt from the non-prompt
collapse regime, as well as by suggesting a gauge independent definition for a
one parameter family of threshold masses $M^{(p)}_{\rm th}$ in terms of
curvature invariants of the Riemann tensor which does not rely on a Newtonian
free-fall criterion. We deduce the critical mass from a delay in the black
hole formation time that appears generically in a small mass range close to
the prompt collapse regime, while the threshold masses of promptness $p$ are
inferred by demanding positivity of the $p$th time derivative of the spatial
maximum of the Kretschmann scalar in the prompt collapse regime, therefore
characterizing the competition between the gravitational pull and internal
pressure of the hypermassive neutron star formed in the merger. We demonstrate
the robustness and analyse the consequences of these definitions with a set of
more than $80$ individual large-scale general relativistic hydrodynamic
simulations of BNS mergers with different total masses populating the prompt
and non-prompt collapse regime. We find that especially the critical mass
$M_{\rm crit}$ plays an important role, since it sets a strict upper bound
beyond which essentially all matter ends up inside the black hole during the
merger and where the possibility of any subsequent EM emission is strongly
suppressed.

To account for uncertainties in the matter description we employ three
different variants (soft, intermediate, stiff) of the holographic V-QCD hybrid
EOS model [38]. All models in this article are by construction consistent with
nuclear theory and perturbative QCD at low and high densities, respectively,
and in-between faithfully cover the central region of the EOS-uncertainty band
of Ref. [39] which has been obtained by combining recent constraints from
neutron star mass-radius measurements and GW data. A distinguished
characteristic of the V-QCD framework is that it is non-perturbative and
provides a unified description of both the nuclear and quark matter phases,
and therefore also the (first order) phase transition between these phases.
This feature allows us to track the amount of quark matter formed and study
its impact on the lifetime of the post-merger remnant and the various limiting
masses introduced in this article.

To determine the impact of the phase transition on our results we construct
and simulate for comparison also a model without quark matter phase that is
otherwise identical to the intermediate variant of the V-QCD EOSs. As we will
show, near critical-mass mergers produce remnants that have the highest
fraction of quark matter inside their cores. For mergers with binary mass
slightly above the critical value, this in turn leads to a quicker collapse
with approximately one order of magnitude less residual matter outside the
black hole and therefore a stronger suppression of EM emission in the model
with quark matter than in the model without quark matter. We argue that this
suppression-mechanism may be useful to infer the existence of quark matter in
neutron stars from future multi-messenger observations of BNS mergers.
Furthermore, we find that short-lived hypermassive neutron stars formed in
binaries with total mass slightly below the critical value can experience
periodic episodes of quark matter formation in their cores for several
milliseconds before they collapse into a black hole.

The 3D-rendering in Fig. 1 summarises the main ingredients of our analysis.
Shown in orange are the iso-contours of the Kretschmann scalar, in green those
of baryon number densities close to $n_{s}$ and in blue those of the volume
fraction of deconfined quark matter in the core region, as they manifest in a
post-merger remnant close to the critical mass only moments before it
collapses into a black hole.

Figure 1: Iso-contours of the Kretschmann scalar (orange), the baryon number
density enclosing values larger than the nuclear saturation density (green)
and the fraction of quark matter (blue), for a remnant formed in a binary
neutron star merger moments before collapse into a black hole. Light and dark
colours indicate small and large values of these quantities, respectively
(only results for negative values of $y$ are shown for clarity).

The rest of this article is organized as follows. Section II summarises the
properties of the V-QCD hybrid EOS models used. In Sec. III we review the
relevant definitions and properties of the curvature invariants used in our
analysis, give definitions for $M_{\rm crit}$ and $M_{\rm th}^{(p)}$ and
explain how to compute them in BNS merger simulations. In Sec. IV we present
and interpret our results and end with a summary and conclusions in Sec. V.
For completeness, we provide four appendices with additional details on the
EOS construction (App. A), a number of simulation results that are not shown
in the main text (App. B), numeric checks with the Reissner-Nordström geometry
of the curvature invariants implemented in our numeric code (App. C) as well
as an comparison of our simulation results with the free-fall criterion (App.
D).

Throughout this article we use natural units where $c=\hbar=k_{B}=G_{N}=1$.

##  II Equation of State

Currently, calculations from-first-principles of even the most basic
thermodynamic quantities as the phase diagram and the EOS at finite
temperature and baryon density directly in QCD are illusive because of the
non-perturbatively large coupling strength as well as the infamous fermionic
sign-problem in lattice QCD. Traditional approaches to tackle this problem
include effective field theory models (see, e.g., [40, 14, 41]), which however
need to be extrapolated to densities way beyond their range of applicability
[42], or generic (model independent) parametrizations of the EOS that are
tuned to satisfy theoretical and observational constraints [37, 43, 39, 44],
but do not contain any information about the particle composition. A promising
alternative are non-perturbative models for QCD that are based on the
holographic gauge/gravity duality and where it is feasible to rigorously
compute the phase structure and the EOS of theories that closely mimic the
expected behaviour of QCD [45, 38, 46, 47, 48, 49, 50].

Figure 2:  Top-Left: Phase boundaries between hadronic, mixed and quark matter
together with the critical end points of the soft (red), intermediate (green)
and stiff (blue) VQCD variants in the energy density and temperature plane.
Bottom-Left: Cold, beta-equilibrium slices of VQCD together with a variant
(dashed, labeled “no QM”) of the intermediate model with suppressed first
order phase transition. Blue and green areas denote the uncertainty bands from
nuclear theory [12] and perturbative QCD [51], respectively. The solid black
and grey lines enclose the $100\%$ and $95\%$ confidence intervals constructed
from a large ensemble of generic EOS models interpolating between the colored
bands [39], respectively. Right: Corresponding Mass-Radius relations where
coloured stars mark (half) the critical mass and ellipses indicate the radius
measurements of J0030+0451 [7, 8] (blue) and J0740+6620 [9, 10] (orange) by
the NICER experiment, while colored bands are direct mass measurements of
J0348+0432 [3] (green) and J0952-0607 [6] (pink).

Formation of quark matter during and after the merger of two neutron stars is
expected to affect significantly the properties of the resulting merger
remnant, because quark matter may have drastically different compressibility
properties compared to nuclear matter. In particular, the presence of quark
matter is expected to impact the limiting mass at which a BNS merger collapses
promptly into a black hole as well as modify the observable GW longevity and
spectrum [52, 53, 54, 55, 56, 57, 58, 59, 60] and influence the conditions
dictating the emergence of an EM signal [61]. In order to study this effect
reliably, we opt to use a set of three EOS variants (soft, intermediate, and
stiff) constructed in [38], which contain both nuclear and quark matter
phases. In this setup, based on the gauge/gravity duality, both the EOS of the
nuclear and the quark matter phase arises from the same model (V-QCD, [62]),
making it possible to obtain sound predictions for the location and the
strength of the nuclear to quark matter phase transition. This is demonstrated
in the top left panel of Fig. 2. At low temperatures the phase transition is
relatively strong with latent heat $\sim 10^{3}$ MeV/fm3 [63, 64, 38]. With
increasing temperature, the phase transition becomes weaker, ending in a
critical point above $T=100$ MeV. The model for the quark matter phase follows
the construction in [65, 66], with the details of the five-dimensional gravity
action pinned down through fit to lattice data in [63]. The model for the
nuclear matter phase, within the same V-QCD setup, was established in [67] by
using the approach of [68].

The three EOS variants at zero temperature are shown in Fig. 2 (bottom left
panel) as the colored solid curves. We point out that they compare well with
the probability distribution of EOSs obtained by sampling a large set of
interpolations between the low and high density regimes [39], where the EOS is
known to a good precision. In addition, the intermediate variant without a
quark matter phase (see App. A) is shown as the dashed green curve. A
characteristic feature of the EOS in the regime of the phase transitions is
that the nuclear matter EOS is rather stiff, with the speed of sound squared
clearly above the value $1/3$ of conformal theories, whereas the quark matter
EOS is soft, with the speed of sound below the conformal value. Therefore
there is a strong first order phase transition from nuclear to quark matter
inside a neutron star which tends to destabilize the star. This holds true
both for static and rotating stars [69, 22] as well as the hypermassive
neutron stars formed in neutron star mergers [57].

The agreement of the EOSs at zero temperature is illustrated through
comparison with the measurements by the NICER collaboration [7, 8, 9, 10] and
direct mass measurements [3, 6] on the right panel of Fig. 2. The agreement is
generically good [69, 64, 70]. The $M_{\mathrm{TOV}}$ for the soft EOS, set by
the kink of the mass-radius curve where a quark core develops, is nevertheless
somewhat lower when compared to the recent observations of neutron star masses
[6, 11]. Therefore this variant is slightly disfavoured.

##  III Characterizing Prompt Collapse

There have been several attempts to determine the mass-threshold to prompt
black hole formation in BNS mergers [30, 31, 32, 33, 34, 35]. Here we
introduce a novel approach, which uses curvature scalars built from the
Riemann tensor $R^{a}_{\;\;bcd}$ to locate the borderline between prompt and
non-prompt black hole formation and to formulate a gauge independent
definition for a one-parameter family of threshold masses. There exists an
infinite number of curvature scalars that can be formed by contracting tensor
products of an arbitrary number of Riemann tensors. For simplicity, our focus
in this manuscript will be on the three quadratic polynomial invariants that
are built by contracting Riemann and epsilon tensors. These three quantities
are the principal invariants of the Riemann tensor on four dimensional
Lorenzian manifolds, known as the Kretschmann scalar $K_{1}$, the Chern-
Pontryagin scalar $K_{2}$, and the Euler scalar $K_{3}$ and are defined as
follows [71]

$\displaystyle K_{1}$ $\displaystyle:=$ $\displaystyle R_{abcd}\,R^{abcd}\,,$
(1) $\displaystyle K_{2}$ $\displaystyle:=$
$\displaystyle{{}^{\star}R}_{abcd}\,R^{abcd}\,,$ (2) $\displaystyle K_{3}$
$\displaystyle:=$ $\displaystyle{{}^{\star}R}^{\star}_{abcd}\,R^{abcd}\,,$ (3)

where the left and right duals of a generic four index tensor $A_{abcd}$ are
defined by the following contractions with the epsilon tensor 111Note, that it
is possible to define $K_{2}$ using the right dual instead of the left dual,
but this difference is irrelevant for our purposes.

$\displaystyle{{}^{\star}A}_{abcd}$ $\displaystyle:=$
$\displaystyle\epsilon_{abef}{A^{ef}}_{cd}\,,$ (4) $\displaystyle
A^{\star}_{abcd}$ $\displaystyle:=$
$\displaystyle{A_{ab}}^{ef}\epsilon_{efcd}\,.$ (5)

It is useful to express these contractions in terms of the Weyl tensor, i.e.,
the trace free part of the Riemann tensor

$\displaystyle
C_{abcd}=R_{abcd}-\left(g_{a[c}R_{d]b}-g_{b[c}R_{d]a}\right)+\frac{1}{3}R~{}g_{a[c}g_{d]b}\,,$
(6)

which has only two linear independent principle invariants
$I_{1}=C_{abcd}\,C^{abcd}$ and $I_{2}={{}^{\star}C}_{abcd}\,C^{abcd}$, because
${{}^{\star}C^{\star}}_{abcd}\,C^{abcd}=-I_{1}$. The principle invariants of
the Riemann tensor can then be written as

$\displaystyle K_{1}$ $\displaystyle=$ $\displaystyle
I_{1}+2R_{ab}R^{ab}-\frac{1}{3}R^{2}\,,$ (7) $\displaystyle K_{2}$
$\displaystyle=$ $\displaystyle I_{2}\,,$ (8) $\displaystyle K_{3}$
$\displaystyle=$ $\displaystyle-I_{1}+2R_{ab}R^{ab}-\frac{2}{3}R^{2}\,.$ (9)

As it turns out, the numerical values of $K_{1},K_{3}$ in our simulations are
dominated by their trace-free contributions $\propto I_{1}$ resulting, except
for an overall sign, in very similar structures and numeric values of these
quantities. Both $K_{1}$ and $K_{3}$ attain their finite extremal values
outside the black hole horizon in the orbital plane $z=0$, which makes them
viable probes to analyse the collapse behaviour of BNS mergers. The scalar
$K_{2}$ has negative parity $K_{2}(z)=-K_{2}(-z)$, which makes $K_{2}=0$ by
definition in the orbital plane $z=0$ and therefore not suitable as a probe
for black hole formation in the centre of the merger (see App. B). In order to
simplify the discussion we report henceforth only results for $K_{1}$ and note
that it can be replaced by $-K_{3}$ without loss of generality in the
following.

For the numeric evaluation it turns out to be advantageous to use Einstein
equations to replace the explicit Ricci tensor and Ricci scalar contributions
in $K_{1}$ with the corresponding expressions of the energy momentum tensor
$T^{ab}$, which leads to

$\displaystyle
K_{1}=I_{1}+128\pi^{2}\left(T_{ab}T^{ab}-\frac{1}{6}T^{2}\right)\,,$ (10)

where $T=g_{ab}T^{ab}$ denotes the trace of the energy momentum tensor.

We proceed now to outline the procedure of threshold mass determination. As
will be shown shortly, the spatial maximum of $K_{1}$ has an interesting and
generic structure when plotted as a function of the total binary mass $M_{\rm
total}$ and the post-merger time $t-t_{\rm merge}$, where the merger time
$t_{\rm merge}$ is set by the global maximum of the GW strain amplitude. For
small values of $M_{\rm total}$ the function $K_{1}$ is non-monotonic in
$t-t_{\rm merge}$, but rises monotonically for sufficiently large values of
$M_{\rm total}$. These two regions are identified as the non-prompt collapse
regime and the prompt collapse regime, respectively. However, as it turns out,
in order to precisely define the critical mass $M_{\mathrm{crit}}$ separating
the two regions it is simpler to use the black hole formation time $t_{\rm
crit}(M_{\rm total})$ instead of the properties of $K_{1}$. That is, the
critical mass turns out to be clearly distinguished by a sharp local maximum
in this critical time, where a trapped surface (apparent horizon) is found in
the simulation:

$\left\\{M_{\rm crit}={\rm min}(M_{\rm total}):\frac{dt_{\rm crit}}{dM_{\rm
total}}<0\,\,\forall\,\,t>t_{\rm merge}\right\\}\,.$ (11)

While this definition does not depend on $K_{1}$ explicitly, the local maximum
of $t_{\rm crit}$ also implies a sharp feature in the behavior of $K_{1}$,
which grows rapidly when the black hole is formed, as we will demonstrate in
the following section. The specific value of $M_{\rm crit}$ depends on the EOS
222We expect $M_{\rm crit}$ to also depend in a non-trivial way on the mass
ratio, but leave an investigation of this dependence for future work., but the
overall behaviour of $K_{1}(t-t_{\rm merge})$ and the sharp delay in $t_{\rm
crit}(M_{\rm total})$ are generic and robust features that are EOS-
independent. Fine-tuning the total binary mass $M_{\rm total}$ to $M_{\rm
crit}$ establishes a subtle dynamic balance between the gravitational
attraction and the material pressure in the post-merger remnant which leads to
a significant delay of black hole formation.

In order to determine when gravitational pull dominates over residual
repulsive forces in neutron star matter we introduce, in addition, a one-
parameter family of threshold masses $M_{\rm th}^{(p)}$ in terms of $K_{1}$
via the following min-max criterion 333Strictly speaking, the criterion is not
entirely covariant because of the coordinate time derivatives, but can
probably be phrased in an entirely covariant way by replacing the time
coordinate $t$ by the parameter of any non-spacelike geodesic that passes
through the point $p$ where $K_{1}(p)={\rm max}(K_{1}(x))$

$\left\\{M_{\rm th}^{(p)}={\rm min}(M_{\rm total}):\frac{d^{p}}{dt^{p}}{\rm
max}(K_{1})\geq 0\,\,\forall\,\,t>t_{\rm merge}\right\\}\,.$ (12)

The p-th derivative characterises the promptness of the collapse, or in other
words, the competition between the gravitational and repulsive forces in
neutron star matter as well as the overall collision dynamics after the
merger. In principle, this definition incorporates arbitrarily high
promptness, i.e., all values $p\geq 1$. For practical purposes, however, we
restrict our analysis to $p=1,2$, as the determination of higher derivatives
of $K_{1}$ becomes increasingly difficult given the finite temporal resolution
of the simulations. The critical and threshold masses satisfy $M_{\rm
crit}<M_{\rm th}^{(1)}<M_{\rm th}^{(2)}<\ldots$ as we shall demonstrate
explicitly below.

We have implemented $K_{1},K_{2}$ and $K_{3}$ by extending the Einstein
Toolkit thorn WeylScalar4 [75] using the Mathematica-based [76] computer-
algebra package KRANK [77] for code generation. To construct the binary
neutron star initial data we use the Frankfurt University Kadath code (FUKA)
[78] which is based on the KADATH spectral solver library [79]. For simplicity
we consider here only non-spinning, equal mass binaries with an initial
separation of $42\rm km$ and leave a more challenging exploration including
unequal masses and/or spins of the binary components to future work. We have
checked the independence of our simulation results on the choice of $42\rm km$
initial separation by comparing to selected cases with $40\rm km$ and $45\rm
km$ and found no noticeable difference. Furthermore, we neglect magnetic
fields in our simulations, because the merger and collapse dynamics is
predominantly determined by binary parameters and the properties of the
neutron star fluid and less by the magnetic field.

To perform the binary evolution we use the Einstein-Toolkit [75]
infrastructure and its fixed-mesh box-in-box refinement framework Carpet [80].
All simulations in this article are performed with six refinement levels,
finest grid-spacing of $295\rm m$ and total domain size of $(3025{\rm
km})^{3}$ while imposing reflection symmetry orthogonal to the orbital plane.
Opting for such intermediately fine resolution allows us to perform $\approx
20$ individual simulations per EOS which we found necessary to resolve
sufficiently well the mass dependence of $K_{1}$ and other quantities for each
of the four different EOS models. As shown in [57], simulations with higher
resolution result in minor quantitative changes, but generic features are
expected to remain robust. For the spacetime evolution we solve the constraint
damping formulation of the Z4 system [81, 82] with the Antelope [83] code. We
use the Frankfurt-Illinois (FIL) [83] code, which is based on the
IllinoisGRMHD code [84], to evolve the hydrodynamic part. Finally, we locate
the black hole horizon by checking the geometry every $6.3\rm ns$ for trapped
surfaces with the Einstein Toolkit thorn AHFinderDirect, which we find more
than sufficient to resolve the delay in black hole formation close to $M_{\rm
crit}$.

##  IV Results

This section provides a comprehensive analysis of our simulations based on a
representative selection of simulation results while postponing a discussion
of the rest of our results for the EOS variants not shown here to the
Appendix. We start with analysing the generic structure of the curvature
invariant $K_{1}$ that emerges in BNS mergers.

Fig. 3 shows one of our main results, namely the dependence of
${\rm{max}}\;K_{1}$ on the total binary mass $M_{\rm total}$ when plotted as
function of $t-t_{\rm merge}$.

Figure 3: Maximum of $K_{1}$ as a function of $t-t_{\rm merge}$ and total
binary mass $M_{\rm total}$ of the intermediate V-QCD model. Prompt and non-
prompt collapse regions are colored in red and orange, respectively, and are
separated by the critical mass $M_{\rm crit}$ marked in violet. The orange and
brown lines mark the threshold masses of promptness $p=1$ and $p=2$,
respectively, and black regions (labeled “BH collapse”) indicate where a black
hole horizon is formed.

The 2D surface of ${\rm{max}}\;K_{1}$ has an interesting structure which
allows to rigorously separate merger remnants experiencing at least one bounce
and therefore a prolonged lifetime from those collapsing immediately after the
merger without any additional features. This clear qualitative difference
allows us to uniquely identify the critical boundary value $M_{\rm crit}$
(violet line) between the non-prompt (orange) and the prompt collapse regime
(red), while regions where a black hole is formed are marked in black. Also
shown are two curves in the prompt collapse region that correspond to the
threshold mass of promptness $p=1$ (orange line) and $p=2$ (brown line) as
defined by Eq. (12). The plot in Fig. 3 is based on $\approx 20$ individual
BNS merger simulations with different total binary masses $M_{\rm total}$
performed using the intermediate V-QCD EOS variant. Similar results for the
other models considered in this article are given in the Appendix, which show
that this structure is generic (EOS independent) and that only details, such
as the values of $M_{\rm crit}$, depend on the EOS.

Fig. 4 illustrates how we identify $M_{\rm crit}$ and $M_{\rm th}^{(p)}$ from
our mass scan of $K_{1}$.

Figure 4: Spatial maxima of $K_{1}$ (left) and their first (middle) and second
derivatives (right) as function of the post merger time $t-t_{\rm merge}$. The
central values of the various limiting masses, as determined by Eq. (11) for
$M_{\rm crit}$ in the leftmost panel and Eq. (12) for $M^{(p)}_{\rm th}$ in
the other two, are plotted as solid lines, whereas dotted and dashed lines
indicate their nearest neighbours that bracket the central values in our mass
scan. Finally, the region of negative values is indicated by grey areas.

The left, middle and right panels here show $K_{1}$, its first time
derivative, and its second time derivative, respectively, at fixed values of
the total mass as indicated in the plots. The solid lines mark the central
values for $M_{\rm crit}$ and $M_{\rm th}^{(p)}$, while dotted and dashed
lines represent their corresponding neighbours $M_{\rm crit}^{\pm}$ and
$M_{\rm th}^{(p)\pm}$ with slightly lower (-) and higher (+) total masses that
bracket the central values. The values of the critical and threshold masses
for all EOSs as computed via Eq. (11) and Eq. (12) are collected in Table 1,
where the $\pm$ uncertainties follow from the corresponding values for
$M^{\pm}_{\rm crit}$ and $M^{(p)\pm}_{\rm th}$ obtained from our mass scan.

As shown in the left panel, the solid curve for the central value of $M_{\rm
crit}$ diverges later, indicating a delay in black hole formation, than those
for subcritical (dotted) and supercritical (dashed) values $M^{\pm}_{\rm
crit}$ which provide estimates for the upper and lower bounds of the central
value. Analogously, the dotted curves in the middle and right panel for
$M^{(p)-}_{\rm th}$ reach negative values at some $t-t_{\rm merge}>0$, which
violates the criterion given in Eq. (12), and therefore set lower bounds for
$M^{(p)}_{\rm th}$, while the dashed curves for $M^{(p)+}_{\rm th}$ remain
strictly positive and provide upper bounds.

EOS | $M_{\rm TOV}/M_{\odot}$ | $C_{\rm TOV}$ | $\tilde{\Lambda}_{\rm GW}$ | $M_{\rm crit}/M_{\odot}$ | $M_{\rm th}^{(1)}/M_{\odot}$ | $M_{\rm th}^{(2)}/M_{\odot}$
---|---|---|---|---|---|---
soft | $2.01$ | $0.249$ | $537$ | $2.95_{-0.013}^{+0.002}$ | $2.963_{-0.006}^{+0.013}$ | $3.05_{-0.038}^{+0.075}$
interm. | $2.14$ | $0.267$ | $565$ | $3.038_{-0.003}^{+0.002}$ | $3.05_{-0.01}^{+0.05}$ | $3.15_{-0.05}^{+0.05}$
no QM | $2.15$ | $0.275$ | $565$ | $3.065_{-0.005}^{+0.005}$ | $3.075_{-0.005}^{+0.025}$ | $3.15_{-0.05}^{+0.05}$
stiff | $2.34$ | $0.293$ | $617$ | $3.19_{-0.005}^{+0.005}$ | $3.195_{-0.005}^{+0.005}$ | $3.25_{-0.025}^{+0.025}$
Table 1:  Static and binary neutron star properties of the four simulated EOS
models. The $\pm$ uncertainty values are estimated from the nearest (super-
and sub-critical) neighbours to the central values in our mass scan.

We find a maximum difference of $\approx 0.25~{}M_{\odot}$ in $M_{\rm crit}$
between the soft and the stiff EOS variants. From this we arrive at the
approximate inequality in equal-mass BNS mergers

$2.95M_{\odot}\lesssim M_{\rm crit}\lesssim 3.19M_{\odot}\,,$ (13)

while the corresponding limits for $M_{\rm th}^{(p)}$ are slightly higher

$\displaystyle 2.963M_{\odot}\lesssim$ $\displaystyle M_{\rm th}^{(1)}$
$\displaystyle\lesssim 3.195M_{\odot}\,,$ (14) $\displaystyle
3.05M_{\odot}\lesssim$ $\displaystyle M_{\rm th}^{(2)}$ $\displaystyle\lesssim
3.25M_{\odot}\,,$ (15)

with all of them consistent with the short compact binary gamma-ray burst
regime reported in [24]. Furthermore, because $M_{\rm crit}<M_{\rm
th}^{(1)}<M_{\rm th}^{(2)}<\ldots$ and since our lower bound for $M_{\rm
crit}=2.95M_{\odot}$ is significantly larger than the total mass $M_{\rm
total}^{\rm GW170817}=2.74^{+0.04}_{-0.01}M_{\odot}$ inferred from GW170817,
our results are consistent with a non-prompt collapse outcome of this BNS
merger as suggested by the observed EM emission [85, 86, 35].

The critical and threshold masses scale in the expected way with the stiffness
of the EOS in the sense that soft models with low TOV-mass, compactness and
stellar radii lead to smaller limiting masses than stiffer models, which
provide more internal pressure to prevent black hole formation at high
densities. This suggests a tight and unique quasi-universal relation between
the TOV-mass $M_{\rm TOV}$ of static neutron stars and the total BNS mass
beyond which a prompt collapse after merger is inevitable

$M_{\rm crit}/M_{\rm TOV}\approx 1.41\pm 0.06\,,$ (16)

where the lower and upper bounds in this relation are inferred from the stiff
and soft model, respectively, while the central value is approximated with the
corresponding arithmetic average. The uncertainty estimate in this relation is
therefore a consequence of selecting V-QCD EOSs that coincide well with the
boundaries of the $95\%$ confidence interval from generic EOS inference shown
in Fig.2. Because of this we expect that this estimate reflects well the
uncertainty for all possible consistent choices of the EOSs, while some
specific choices may lead to mass ratios slightly outside the error band.

Comparing the results for the intermediate EOS variants with and without quark
matter shows only a small decrease of the limiting masses due to the presence
of quark matter in the merger (see Table 1). This means the formation of quark
matter during and after the merger has almost negligible impact on the value
of the critical mass and the threshold masses, which we are nevertheless able
to accurately resolve in our simulations as demonstrated in App. B.

Let us next discuss the impact of quark matter on the GW signal. In Fig. 5 we
show in the left panel the GW strain (top), the quark volume fraction $Y_{\rm
quark}\in[0,1]$ averaged over a region inside the merger remnant $\langle
Y_{\rm quark}\rangle_{n_{b}\geq n_{s}}=\int_{n_{b}\geq
n_{s}}d^{3}x\sqrt{\gamma}~{}Y_{\rm quark}/\int_{n_{b}\geq
n_{s}}d^{3}x\sqrt{\gamma}$ with number density $n_{b}\geq n_{s}$ (bottom) and
in the right panel the corresponding power spectral density (PSD) together
with sensitivity curves for the advanced LIGO (aLIGO) detector and the
Einstein Telescope (ET). Shown are curves for a sub-critical mass $M_{\rm
total}=2.9M_{\odot}$ in orange, for the critical mass $M_{\rm crit}$ in violet
and for the threshold mass $M_{\rm th}^{(2)}$ of promptness $p=2$ in red, with
thick and thin lines corresponding to the model with and without quark matter
phase, respectively.

Figure 5: Left: On top we show the plus-polarization of the GW strain
extrapolated to the estimated luminosity distance of $40\,{\rm Mpc}$ of the
GW170817 event [1] and viewing angle $\theta=15^{\circ}$ determined from the
jet of GW170817 [87] (using $\phi=0^{\circ}$ without loss of generality).
Shown are results for binaries with sub-critical ($2.9M_{\odot}$), critical
($M_{\rm crit}$) and supercritical mass ($M_{\rm th}^{(2)}$), using orange,
violet, and red colors respectively (as in Fig. 3), with cases with and
without quark matter plotted with thick lines and thin lines, respectively. On
the bottom we show the corresponding quark matter fraction averaged over
densities $n_{b}>n_{s}$ inside the merger remnant. Right: Corresponding PSD
with the dominant $f_{2}$ mode marked by a dashed line whenever it is
identifiable, together with sensitivity curves of the advanced LIGO (aLIGO)
detector and the Einstein Telescope (ET).

As expected, larger masses lead to a clear truncation of the waveform due to
the earlier black hole formation. The post-merger signal for $M_{\rm
th}^{(2)}$ is almost entirely dominated by the exponential ring-down of the
black hole, while the critical ($M_{\rm total}=M_{\rm crit}$) and the
subcritical ($M_{\rm total}=2.9M_{\odot}$) binaries survive for approximately
$2\rm ms$ and $8\rm ms$ after the merger, respectively.

As shown in the bottom panel, there is a significant amount of quark matter
formed during the various post merger phases. This demonstrates that models
like V-QCD that do not allow quark matter cores inside static neutron stars,
because their strong phase transition destabilises the stars, can still lead
to extended periods on the order of $10\rm ms$ during which a significant
amount of quark matter is present inside a post-merger remnant. The spectral
properties of the emitted GW signal during this period are potentially
accessible with existing detectors. Furthermore, what all cases have in common
is the formation of a pure quark matter core right before the collapse. In
mergers with slightly sub-critical mass $M_{\rm total}=2.9M_{\odot}\lesssim
M_{\rm crit}$ a curious phenomenon can be observed. Namely, an approximately
$10\rm ms$ long stage is present, with periodic reoccurrence of quark matter
inside the core of the merger remnant before it ultimately collapses into a
black hole. As shown in the Appendix, this stage appears generically for
$M_{\rm total}\lesssim M_{\rm crit}$ in all our simulations with EOS models
that include quark matter. Nevertheless, the impact of quark matter on the GW
spectrum, as shown in the power spectral density (PSD) in the right panel of
the plot, depends most strongly on the total binary mass. Here we see that
only the critical mass mergers (violet curves) are impacted by the presence of
quark matter while sub- and supercritical masses result in essentially no
noticeable difference in the GW spectrum. However, because of the short
duration ($\approx 2\rm ms$), accurately accessing this information presents
an extreme challenge to current GW detectors, but may be accessible to those
of the next generation such as the Einstein Telescope.

As argued above, the impact of quark matter in the vicinity of the prompt
collapse regime is rather subtle and it is currently not clear if it will be
possible in the near future to infer the formation of quark matter in BNS
mergers just by analysing their GW signal. For lack of clear GW signatures, it
is essential to harvest the potential of other information channels in
identifying quark matter formation, such as EM counterparts of merger events.
The prospects of observing an EM signal are crucially determined by the amount
of mass left outside of the black hole. Fig. 6 shows how the residual amount
of matter available for EM emission after the merger depends on the total mass
of the binary.

Figure 6: Post-collapse distribution and abundance of baryonic matter in the
sub-critical (top left), critical (top right), and super-critical (bottom)
regimes as obtained $10\rm ms$ after the merger. Notably, for $M_{\rm
total}\geq M_{\rm crit}$, the highest baryon densities within the region of
$\approx 30\rm km$ radius around the central black hole, drop by several
orders of magnitude.

Shown here is the number density in the orbital plane outside the black hole
horizon, indicated by the black region in the centre, $10\rm ms$ after the
merger for binaries with sub-critical mass $M_{\rm total}=2.9M_{\odot}$ (top
left), critical mass $M_{\rm total}=M_{\rm crit}$ (top right) and for
supercritical masses $M_{\rm th}^{(1)}$ (bottom left) and $M_{\rm th}^{(2)}$
(bottom right). There is a clearly visible drop in the number density outside
the horizon when crossing the critical mass bound. In order to quantify this
drop, we evaluate the total remnant baryonic mass outside of the apparent
horizon in all our simulations by evaluating the integral

$M_{b}=\int_{r>r_{\rm AH}}W\rho\sqrt{\gamma}d^{3}x$ (17)

at $10\rm ms$ after the collapse within a sphere of radius $\sim 74\rm km$,
where $W$ stands for the Lorentz factor, $\rho$ is the matter density, and
$\sqrt{\gamma}$ the square root of the determinant of the spatial metric
$\gamma_{ij}$.

We present the results of this analysis in the top panel of Fig. 7. The
critical mass of the binary separates those BNS systems that result in small
tori masses $5\times 10^{-3}-10^{-2}M_{\odot}$ from the supercritical binaries
which leave only negligible amount of mass outside the apparent horizon,
$M_{b}<10^{-3}M_{\odot}$. Moreover, the resulting torus masses decrease by a
factor of a few with the increasing stiffness of the EOS. One can check that
our findings are also consistent with the results for mass ejection in other
studies on equal-mass [88] and unequal-mass (see, e.g., [89]) binary mergers.
Because the collapse can occur in different phases of the oscillating
hypermassive neutron star (HMNS), small variations of the tori masses are
present on both sides of the critical line.

Importantly, we find that the drop of the torus mass at criticality is much
less pronounced and smaller in magnitude for the intermediate model where the
quark matter phase is neglected (light green data points and curves in the top
panel of Fig. 7). Since the nuclear to quark matter transition in our models
is relatively strong, we expect that a similar reduction is found for EOS
models that feature a weaker nuclear to quark matter transition. We stress
however that a clear signal is visible even in the case where the phase
transition is removed altogether.

For all the EOS employed in this study, dimensionless spins of the formed
black holes of at least $\chi_{\rm BH}\geq 0.65$ are measured. Assuming a
large scale, ordered magnetic field, subcritical binaries should be capable of
generating a short gamma-ray burst (sGRB) by launching a relativistic jet via
the Blandford-Znajek mechanism (see [24] and references therein).

Furthermore, binaries closer to the critical mass (i.e., as $M_{\rm total}$
approaches $M_{\rm crit}$ from below) result in higher spins $\chi_{\rm
BH}\sim 0.8$, which leads to a more efficient energy extraction via the
Blandford-Znajek process [90]. On the other hand, the tori masses for
supercritical binaries are too small to give rise to a detectable EM emission
driven by the mass accretion onto the black hole [91] and have also not been
seen in the BNS event GW190425 [92] with a high total mass of $3.4M_{\odot}$.
Formation of a detectable kilonova is also disfavoured, because in prompt-
collapse scenarios and short-lived HMNS only negligible ejecta masses are
present [93, 94]. We conclude that the only EM counterpart for equal-mass
binaries with $M_{\rm total}\gtrsim M_{\rm crit}$ should be seen in the late
inspiral or at the moment of merger, for example in the form of fast radio
bursts [95, 96], quasi-periodic X-ray bursts in magnetars [97] and precursor
flares [98, 99].

Even though our simulations were carried out without magnetic fields, we
expect our predictions regarding the disc masses in the prompt vs. non-prompt
regimes to remain unchanged for realistic magnetic field strengths $B\lesssim
10^{13-14}\rm G$. The reason for this is that the decisive mechanisms
preventing the prompt-collapse of a post-merger HMNS are hydrodynamical in
nature, i.e., dominated by the rotational energy and the pressure support of
the fluid and less by the magnetic field.

Finally, we show in the bottom panel of Fig. 7 the persistence of the quark
matter, which is a diagnostic we introduce to simultaneously quantify the
temporal extent and the abundance of the quark phase after the merger.

Figure 7: Top: Remnant baryon mass $M_{\rm rem}$ outside the horizon at $10\rm
ms$ after black hole formation as a function of the total mass for different
EOS models as indicated in the legend. Dashed lines mark the critical masses
and the grey band indicates the region dominated by the density of the
atmosphere in the simulation $\rho_{\rm atm}=10^{-14}M_{\odot}^{-2}$ as
estimated by integrating $\int d^{3}x~{}\rho_{\rm atm}\approx 8.6\times
10^{-5}M_{\odot}$ over the entire simulation domain of size
$(2048M_{\odot})^{3}$. Bottom: Time integral over $3\rm ms$ after merger of
the volume averaged quark fraction inside regions with number density larger
than nuclear saturation density.

More specifically, we evaluate the integral of the volume-averaged quark
fraction $\int\langle Y_{\rm quark}\rangle_{n_{b}\geq n_{s}}dt$ over a $3\rm
ms$ long time interval after the merger. This auxiliary quantity may be used
to measure the overall contribution of the quark matter regions to the global
dynamics of the system. We find the largest amount of quark matter to be
formed in critical mass mergers. In addition, for each EOS a secondary peak at
$\sim 0.1M_{\odot}$ smaller total mass is present. In consequence, we expect
also more quark matter to be formed in binaries that are slightly lighter than
the critical mass because of the corresponding delay in collapse time.

##  V Conclusion

In this work we performed an extensive numerical analysis of the prompt
collapse behaviour of BNS mergers for a family of realistic EOS models that
satisfy known observational bounds for behaviour of QCD at the highest
densities realised inside neutron stars and their mergers. In particular, we
explored three EOS variants of different stiffness (from soft to stiff)
constructed with the holographic V-QCD framework which provides a non-
perturbative description of the phase transition between dense nuclear and
quark matter within the same approach. This framework also allows us to
suppress the quark matter phase and therefore isolate in a consistent way the
effect of the phase transition in our simulations.

As the main novel feature we introduce a set of criteria, based on gauge
independent curvature invariants of the Riemann tensor, to identify the
critical mass $M_{\rm crit}$ that separates the prompt from the non-prompt
collapse regime as well as a one-parameter family of threshold masses $M_{\rm
th}^{(p)}$ with promptness $p$ which further characterises the prompt collapse
regime of BNS mergers. Using these criteria we uncover a rather tight relation
$M_{\rm crit}/M_{\rm TOV}\approx 1.41\pm 0.06$ between the TOV-mass and the
critical binary mass beyond which any BNS system undergoes a prompt collapse
after merger.

Furthermore, we demonstrate that the critical mass $M_{\rm crit}$ plays a
crucial role in analysing the impact of quark matter in BNS mergers.
Importantly, we find that most quark matter is formed in critical mass mergers
and not in sub- or supercritical ones. This can be explained by the subtle
metastable equilibrium realised at the critical mass which results in a few
millisecond long prolongation of the remnant lifetime compared to non-critical
binaries. This leads to a clearly visible impact on the spectral properties of
the post-merger GW spectrum in simulations near the critical mass, which we
nonetheless expect to be challenging to isolate from future observations with
existing detectors because of the short duration of the signal.

As the most promising indicator for the presence of quark matter we identify
the residual mass outside the black hole formed after the merger. We find that
the amount of baryonic mass outside the horizon not only drops by several
orders of magnitude right at the critical mass, but also that this amount
depends on whether the collapse was induced by a quark matter or by a pure
nuclear matter core. The results of our simulations suggest that a black hole
collapse induced by a softer quark matter core cleans the black hole exterior
from residual matter more efficiently than in the case where the collapse
happens against the resistance of a stiffer nuclear matter core. In
consequence, we find approximately one order of magnitude less baryonic mass
outside the black hole produced in supercritical mass mergers with quark
matter compared to those formed without quark matter.

There exists a number of possible extensions of our work. A straightforward
generalisation involves extending the BNS parameter space by taking into
account different mass ratios and spins of the binary components similar to
what has been done in [34]. We have not attempted this yet because of the
considerable computational costs required, but plan to do so in future work.
Another interesting avenue to explore are dissipative effects due to viscosity
and their impact on the critical mass. Bulk viscosity in particular could be
studied using the recent developments in microphysics calculations of the
relevant transport properties in holographic and perturbative QCD [100].
Finally, it would be important to rigorously quantify the actual impact of
quark matter on the expected EM emission in the prompt collapse regime, which
could be done by adding magnetic fields together with a prescription for
neutrino transport and a nuclear reaction network to our simulations. Such a
study would clearly require significantly higher resolution than it has been
used in the present work in order to faithfully resolve the dynamics and
amplification mechanism of the magnetic field. In combination with the
necessary mass scan to identify the critical mass, however, such a study would
result in computational costs that are larger than we can currently afford,
but may be addressed in future.

###### Acknowledgements.

We thank Tyler Gorda, Carlo Musolino and Luciano Rezzolla for useful
discussions. C. E. acknowledges support by the Deutsche Forschungsgemeinschaft
(DFG, German Research Foundation) through the CRC-TR 211 “Strong-interaction
matter under extreme conditions”– project number 315477589 – TRR 211. C. E.
and K. T. acknowledge the support by the State of Hesse within the Research
Cluster ELEMENTS (Project ID 500/10.006). M. J. has been supported by an
appointment to the JRG Program at the APCTP through the Science and Technology
Promotion Fund and Lottery Fund of the Korean Government, by the Korean Local
Governments – Gyeongsangbuk-do Province and Pohang City – and by the National
Research Foundation of Korea (NRF) funded by the Korean government (MSIT)
(grant number 2021R1A2C1010834). Some calculations were performed on the local
ITP Supercomputing Clusters Iboga and Calea. The merger simulations were
performed on HPE Apollo HAWK at the High Performance Computing Center
Stuttgart (HLRS) under the grant BNSMIC. For our visualisations we made use of
the Kuibit library [101], as well as Mathematica [76] and Thermo Scientific™
Amira 3D visualization software.

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##  Appendix A Construction of the EOS variant without quark matter

In order to isolate the effects of quark matter formation during the merger on
the collapse behavior, we construct a hybrid V-QCD EOS where the quark matter
phase has been removed by hand. To accomplish this we start from the
intermediate EOS variant of [38]. Obtaining a consistent model is, however,
more involved than simply neglecting the quark matter phase, because the
nuclear matter component of this model is well-defined only up to about ten
saturation densities. This is because of the van der Waals setup that was used
to define the nuclear matter EOS at finite temperature: the excluded volume
correction in the setup immediately sets the maximum density to the inverse of
the value of the excluded volume per nucleon. Notice that the densities at
which this issue appears are so high that they are not expected to play a
major role in our simulations. That is, when densities exceed ten saturation
densities, this happens in a limited part of the system, and collapse to a
black hole is imminent, thus ending the HMNS stage which is the most
interesting stage for us. Nevertheless, we want to construct a fully
consistent extension of the EOS (rather than an ad-hoc extension) to densities
above the limiting value set by the van der Waals setup.

In order to remove this limitation, we modify the excluded volume correction
in the van der Waals setup such that it is smoothly turned off at extremely
high densities. Because the van der Waals model is only used to extrapolate
the V-QCD nuclear matter EOS from zero temperature to nonzero temperature, so
that the EOS at zero temperature (both at $\beta$-equilibrium and out-of-
equilibrium) remains unchanged. Therefore the modification solely affects the
temperature dependence of the nuclear matter EOS above five saturation
densities.

Let us then discuss the precise definition in detail. In [38] we defined the
excluded volume corrected pressure though the set of equations (see also [102,
103, 104])

$\displaystyle
p_{\mathrm{ex}}(T,\\{\mu_{k}\\})=p_{\mathrm{id}}(T,\\{\tilde{\mu}_{k}\\})\,,$
(18)
$\displaystyle\tilde{\mu}_{i}=\mu_{i}-v_{i}p_{\mathrm{ex}}(T,\\{\mu_{k}\\})\,,$
(19)

where $p_{\mathrm{ex}}$ is the corrected pressure, $p_{\mathrm{id}}$ is the
ideal gas pressure, and the indices $i,k$ run over the particles included in
the van der Waals model, i.e., the nucleons, the elecrons as well as their
antiparticles: $i,k\in\\{n,p,e,\bar{n},\bar{p},\bar{e}\\}$. We chose
$v_{p}=v_{\bar{p}}=v_{n}=v_{\bar{n}}=v_{0}$, $v_{e}=v_{\bar{e}}=0$, and set
$v_{0}=0.56\,\text{fm}^{3}$. However, without the need of changing the
framework in any other way, we may consider a generalized definition where we
replace Eq. (19) by

$\tilde{\mu}_{i}=\mu_{i}-F_{i}(p_{\mathrm{ex}}(T,\\{\mu_{k}\\}))\,,$ (20)

where $F_{i}(p)$ is an arbitrary function. A reasonable choice for the
nucleons employs the earlier definition at small densities, which means that
$F_{i}$ is linear in $p$, and turns off the correction smoothly at large
densities, which is achieved by making $F_{i}$ constant. We choose

$F_{i}(p)=v_{0}\,p\,\frac{1+\alpha\left(\frac{p}{p_{c}}\right)^{w}+\frac{p_{c}}{p}\left(\frac{p}{p_{c}}\right)^{2w}}{1+\left(\frac{p}{p_{c}}\right)^{2w}}\equiv
F(p)\ ,$ (21)

for the nucleons ($i\in\\{n,p,\bar{n},\bar{p}\\}$), where

$\alpha=-0.1\ ,\qquad w=15\ ,\qquad p_{c}=911~{}\textrm{MeV/fm}^{3}\,.$ (22)

For the electrons and positrons the function $F_{i}$ is set to zero.

The expression in Eq. (21) may appear overly complicated. Nevertheless, we
wished to choose a function that represents a rather rapid step between the
linear behavior at small densities and a constant one at high densities, which
can be expressed in terms of elementary functions, and for which the excluded
volume $F^{\prime}(p)$ decreases monotonically with increasing $p$. Using a
simpler Ansatz than that of Eq. (21) typically fails to satisfy some of these
properties.

No other changes are necessary in the construction of the EOS. The modified
correction implies that, for example, the corrected number densities are given
by

$\displaystyle n_{\mathrm{ex}}^{(i)}(T,\\{\mu_{k}\\})$ $\displaystyle=$
$\displaystyle\frac{\partial
p_{\mathrm{ex}}(T,\\{\mu_{k}\\})}{\partial\mu_{i}}$ $\displaystyle=$
$\displaystyle\frac{n_{\mathrm{id}}^{(i)}(T,\\{\tilde{\mu}_{k}\\})}{1+F^{\prime}(p_{\mathrm{ex}}(T,\\{\mu_{k}\\}))\,\sum_{l}n_{\mathrm{id}}^{(l)}(T,\\{\tilde{\mu}_{k}\\})}\,,\;\;$

where $n_{\mathrm{id}}^{(i)}=\partial
p_{\mathrm{id}}(T,\\{\tilde{\mu}_{k}\\})/\partial\mu_{i}$ and $i$ denotes all
the fermion species, while the index $l$ runs only over nucleons and
antinucleons.

Figure 8: Iso-contours of the $K_{2}$ scalar illustrating the negative parity
with respect to the equatorial plane. Positive and negative values are colored
with shades of red and blue respectively, with the intensity dictated by the
absolute value.

##  Appendix B Additional Simulation Results

This appendix is a collection of the simulation results which we have not
shown in the main text. We start with the 3D rendering in Fig. 8 which
illustrates the global structure of the Chern-Pontryagin scalar $K_{2}$, in
particular its negative parity with respect to the orbital plane. Here, the
positive values of $K_{2}$ are shown in red and the negative values in blue
colours, while values close to or identical to zero, present in the vicinity
of the orbital plane and far away from the centre, are left white. While
$K_{2}$ displays a non-trivial structure outside the orbital plane, the fact
that it vanishes in the centre clearly renders this quantity impractical to
characterise the collapse properties of a BNS merger.

Figure 9: Maximum of $K_{1}$ as a function of $t-t_{\rm merge}$ and total
binary mass $M_{\rm total}$ of the soft (left), intermediate without quark
matter (middle) and stiff (right) V-QCD model. Prompt and non-prompt collapse
regions are coloured in red and orange, respectively, and are separated by the
critical mass $M_{\rm crit}$ marked in violet, while the black regions
indicate where a black hole horizon is formed.

We next compare in Fig. 9 the maximal values of the Kretschmann scalar $K_{1}$
as a function of the total mass $M_{\rm total}$ and the post-merger time
$t-t_{\rm merge}$ for the soft, intermediate without quark matter and stiff
V-QCD model EOSs. This figure demonstrates the universality, i.e., EOS
independence of the overall structure of $K_{1}$. It also clearly shows the
trend of $M_{\rm crit}$ with respect to the EOS-stiffness: soft models result
in smaller values of $M_{\rm crit}$ than stiff models. Furthermore, a careful
comparison of the plot in the middle for the intermediate EOS without quark
matter to Fig. 3 (main text) for the corresponding EOS with quark matter shows
a small but clearly resolved positive shift of $\Delta M_{\rm crit}\approx
0.028M_{\odot}$ in the model without quark matter. While this small shift is
probably irrelevant for any observational purposes, it nevertheless
demonstrates the generic trend that quark matter tends to destabilise the
post-merger remnant, as indicated by the lower value of $M_{\rm crit}$.

Figure 10: Spatial maxima of $K_{1}$ (top) and their first (middle) and second
derivatives (bottom) as function of the post merger time $t-t_{\rm merge}$ of
the soft (left), intermediate without quark matter (center) and stiff model
(right). The central values of the various limiting masses, as determined by
Eq. (11) and Eq. (12), are plotted as solid lines, while dotted and dashed
lines indicate their nearest neighbours that bracket the central values in our
mass scan and the region of negative values is indicated by grey areas.

For completeness, we show in Fig. 10 the details of our limiting mass
inference for the other EOS models considered in this work. These figures
follow the same logic as Fig. 4 in the main text. Lastly, the largest time
delay in black hole formation at the critical mass takes place for the soft
EOS model, as can be seen from the plot in the top left corner.

Finally, we end this appendix with a discussion of the GW features and the
quark matter production for the soft and stiff EOS models shown in Fig. 11.

Figure 11: Left: In the top of each panel we show the plus-polarization of the
GW strain extrapolated to the estimated luminosity distance of $40\,{\rm Mpc}$
of the GW170817 event [1] and viewing angle $\theta=15^{\circ}$ determined
from the jet of GW170817 [87] (using $\phi=0^{\circ}$ without loss of
generality). Binaries with sub-critical, critical ($M_{\rm crit}$) and
supercritical mass ($M_{\rm th}^{(2)}$) are shown in different colors. In the
bottom panels we show the corresponding quark matter fraction averaged over
densities $n_{b}>n_{s}$ inside the merger remnant. The plots in the top row
correspond to the soft and those in the bottom row to the stiff EOS model.
Right: Corresponding PSD where we marked the dominant $f_{2}$ mode where it is
identifiable, together with sensitivity curves of the advanced LIGO (aLIGO)
detector and the Einstein Telescope (ET).

The results shown here are qualitatively similar to those in the corresponding
Fig. 5 in the main text, with one notable exception, namely that the stiff
model leads to the clearest periodic recurrence of quark matter in sub
critical mass mergers. As mentioned in our conclusion, it would be interesting
to study this curious recurrence pattern further by including bulk viscosity
for quark matter in further simulations, in particular for the stiff model
where this feature appears in a very clear form.

##  Appendix C Check of the Numeric Implementation

As a non-trivial check of the implementation of the curvature scalars we
compare the closed-form expressions for the Reissner-Nordström (RN) black hole
to the numerical results obtained with our numeric code. The line element of
the RN geometry can be written as follows

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle g_{ab}dx^{a}dx^{b}$
(24) $\displaystyle=$
$\displaystyle-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}d\Omega_{2}\,,$ $\displaystyle
f(r)$ $\displaystyle=$ $\displaystyle 1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\,,$
(25)

where the conserved charges $M$ and $Q$ parameterise the mass and the electric
charge of the black hole. The geometry has an inner and an outer horizon
located at $r_{BH}=M\pm\sqrt{M^{2}-Q^{2}}$. The case $|Q|=M>0$ corresponds to
the critical solution where both horizons coincide, while for $Q=0$ the
geometry reduces to the Schwarzschild solution with horizon radius $r_{BH}=2M$
and the second horizon coincides with the singularity at $r=0$. For our
comparison below we only consider the case $0<Q<M$ and the part of the
geometry beyond the outer horizon $r\geq M+\sqrt{M^{2}-Q^{2}}$.

The RN geometry is a solution to the Einstein equations with the following
energy momentum tensor

$\displaystyle T_{ab}$ $\displaystyle=$ $\displaystyle
F_{ac}F^{c}_{b}-\frac{1}{4}g_{ab}F^{2}\,,$ (26) $\displaystyle F_{ab}$
$\displaystyle=$ $\displaystyle\partial_{a}A_{b}-\partial_{b}A_{a}\,,$ (27)
$\displaystyle A_{a}$ $\displaystyle=$
$\displaystyle\left(\frac{Q}{r},0,0,0\right)\,.$ (28)

The energy momentum tensor is traceless $g_{ab}T^{ab}=0$, but the contraction
$T_{ab}T^{ab}=\frac{Q^{4}}{16\pi^{2}r^{8}}$ is non-trivial and rapidly
decreases with $r$. The three relevant curvature invariants evaluate for the
RN metric to

$\displaystyle K_{1}$ $\displaystyle=$
$\displaystyle\frac{8\left(6M^{2}r^{2}-12MQ^{2}r+7Q^{4}\right)}{r^{8}}\,,$
(29) $\displaystyle K_{2}$ $\displaystyle=$ $\displaystyle 0\,,$ (30)
$\displaystyle K_{3}$ $\displaystyle=$
$\displaystyle-\frac{8\left(6M^{2}r^{2}-12MQ^{2}r+5Q^{4}\right)}{r^{8}}\,,$
(31)

where $K_{2}=0$ follows from the spherical symmetry of the RN metric in
combination with negative parity of $K_{2}$.

In Fig. 12 we compare the analytic expressions of $K_{1}$ (left), $K_{2}$
(middle) and $T_{ab}T^{ab}$ (right) for $M=1$ and $Q=1/2$ to the numerically
evaluated solutions computed with the code which we also use in our binary
merger simulations.

Figure 12: Comparison between the exact (blue) the numerically obtained
(dashed red) solutions for $K_{1}$ (top), $K_{3}$ (middle) and $T_{ab}T^{ab}$
(bottom) for $M=1$ and $Q=1/2$. The gray-shaded area marks the region inside
the outer black hole horizon (black line) located at $r_{\rm BH}\approx
1.866~{}M$.

This comparison shows perfect agreement between the numeric and closed-form
solution and therefore provides a non-trivial consistency check of our
implementation.

##  Appendix D Free-Fall Criterion

The approach of [31] uses the lapse function $\alpha$ to determine the
threshold mass $M_{\rm th}$. The criterion that defines $M_{\rm th}$ requires
to determine the global minimum of the lapse function $\alpha(t)$ together
with the Newtonian free-fall time $\tau_{\rm
TOV}=\frac{\pi}{2}\sqrt{\frac{R_{\rm TOV}^{3}}{2\,M_{\rm TOV}}}$, where
$M_{\rm TOV}$ and $R_{\rm TOV}$ are the maximum mass and the corresponding
radius of the individual star. In this approach, the threshold mass $M_{\rm
th}$ is defined by the limit in which the binary collapses on the Newtonian
free-fall timescale:

$M/M_{\rm TOV}\to M_{\rm th}/M_{\rm TOV}\,\quad{\rm for}\,\quad t_{\rm
coll}\to\tau_{\rm TOV}\,,$ (32)

where the collapse time $t_{\rm coll}=t_{\rm BH}-\bar{t}_{\rm merge}$ is
defined as the difference between the time of black hole formation and the
merger. The merger and black hole formation times are in general ambiguous.
However, the following definitions in terms of the global minimum of the lapse
function were shown [31] to lead to robust predictions for approximately
symmetric binaries

$\displaystyle\bar{t}_{\rm merge}$ $\displaystyle:$ $\displaystyle\quad{\rm
min}(\alpha(t))=0.35\,,$ (33) $\displaystyle t_{\rm BH}$ $\displaystyle:$
$\displaystyle\quad{\rm min}(\alpha(t))=0.2\,,\,$ (34)

where the time $\bar{t}_{\rm merge}$ defined here via the lapse function
criterion should not be confused with the one determined by the global maximum
of the GW strain amplitude $t_{\rm merge}$. In practice, a series of different
binaries with $M_{\rm total}\approx M_{\rm th}$ is simulated and used to
extrapolate $M_{\rm total}$ to the point where $t_{\rm coll}/\tau_{\rm
TOV}=1$, assuming vanishing slope at this point.

In Fig. 13 we evaluate the free fall criterion for our family of V-QCD EOSs
and collect the relevant quantities in Tab. 2.

Figure 13:  Evaluation of the free-fall. Solid and dashed lines are the ratios of collapse and free fall time in the prompt and non-prompt regime, respectively. The solid black line indicates the free-fall limit of equal collapse and free-fall time. EOS | $\tfrac{M_{\rm TOV}}{M_{\odot}}$ | $\tfrac{R_{\rm TOV}}{\rm km}$ | $\tfrac{\tau_{\rm TOV}}{\rm\mu s}$ | $\tfrac{\bar{t}_{\rm merge}-t_{\rm merge}}{\rm\mu s}$ | $\tfrac{t_{BH}-t_{\rm merge}}{\rm\mu s}$ | $\tfrac{t_{\rm coll}}{\tau_{\rm TOV}}$
---|---|---|---|---|---|---
soft | $2.01$ | $12.41$ | $88.4$ | $324\,(252)$ | $795\,(471)$ | $5.33\,(2.47)$
interm. | $2.14$ | $12.50$ | $84.7$ | $296\,(220)$ | $768\,(436)$ | $5.34\,(2.44)$
no QM | $2.15$ | $12.50$ | $81.4$ | $272\,(219)$ | $760\,(450)$ | $5.99\,(2.83)$
stiff | $2.34$ | $12.64$ | $80.8$ | $245\,(217)$ | $857\,(482)$ | $7.57\,(3.28)$
Table 2:  Quantities necessary to evaluate the free-fall criterion. The values
listed are for $M_{\rm th}^{(1)}$ $(M_{\rm th}^{(2)})$ as evaluated from the
corresponding min-max criteria for $K_{1}$.

In our set of simulations we find no evidence that the slope of the collapse
time vanishes as the free-fall time scale is approached. This is because free-
fall is never realised in a collapsing fluid with finite internal pressure. We
therefore conclude that the threshold mass criterion defined in terms of
specific values of the gauge function $\alpha(t)$ such as Eqs.(33)-(34),
together with the Newtonian notion of the free-fall, does not lead to a well-
defined threshold mass, even though it may give rough approximations to the
threshold masses defined in this article.