High-energy astrophysics suffers from an energy problem. The total integrated luminosity observed in the universe cannot be completely accounted for by existing theoretical models. In almost all astrophysical explosive events that generate 1053 ergs or more in kinetic energy and radiation, the engine remains elusive. For example, the energies observed in core-collapse supernovae [1] or gamma-ray bursts cannot be reproduced consistently with computer simulations [2]. Specifically, in the case of core-collapse supernovae, computer simulations cannot form robust explosions from first principles for all the relevant progenitor masses [1]. In the case of even more energetic phenomena, such as superluminous supernova, that have kinetic energies of around 1052 ergs, the engine remains even more elusive. Similar issues appear with gamma-ray bursts, which suffer from related energy budget problems. Recently, the associated gamma-ray burst observation of the gravitational wave [3] of a neutron star merger showed the same energy budget problems, where the observed luminosity was much milder than for other known GRBs. These anomalies suggest the need for a novel source that can “balance” this budget problem and can be accounted for by the physics required to fix it.

There are various observational phenomena that indicate the explosion of a neutron star. For example, most models that seek to explain the large luminosities and kinetic energy of super-luminous supernovae do so by using a “point source” that injects energy into an envelope of (1–20) M⊙, whether this point source is a core-collapse supernova, or a magnetar [4]. However, in the case of transforming a core-collapse supernova’s energy into luminosity by shocking it with a 1 M⊙ envelope or “wall”, it is necessary to explain the source of the envelope itself, which is a non-trivial problem. In the case of a magnetar, it is necessary to assume almost 100 percent efficiency of conversion between the rotational energy and the luminosity/kinetic energy of the envelope [4]. Furthermore, it is necessary to explain the source of the large magnetic field.

If (u,d,s) quark-matter is the most stable form of matter in the universe, then it follows that neutron stars may decay into more stable quark stars through an exothermic process. According to the BWT hypothesis, the reason why hadronic matter does not spontaneously decay into (u,d,s) matter is that there is an intermediate higher energy state of (u,d) matter. To diminish this energy barrier, there need to be sufficient s-quarks available to trigger the combustion process. Another way of stating this, is that s-quarks act as catalysts that lower the free energy barrier, allowing hadronic matter to decay into a lower state of (u,d,s) matter. 

Since this hypothesis was proposed, many interesting scenarios have been postulated in both astrophysics and particle physics. For example, the existence of pure strange quark stars, and fragments of (u,d,s) matter, called strangelets, have been suggested. Beyond the existence of macroscopic objects, such as strange quark stars, another interesting consequence of the BWTH is the release of large amounts of energy when hadronic matter converts to (u,d,s) matter. Assuming a bag constant of B=145 MeV, using the above model, the energy per baryon becomes ∼840 MeV which is roughly 100 MeV less than for ordinary hadronic matter (∼930 MeV) [9]. This implies that a conversion from hadronic to (u,d,s) matter should release about 100 MeV per converted baryon. Assuming a neutron star has about 1057 baryons, conversion of every baryon into (u,d,s) matter would generate ∼1053 ergs in total energy. While this is of the same order of magnitude for typical explosive events in astrophysics, such as core-collapse supernovae, the energy is hardly harnessed since it is emitted as neutrinos. The advent of the quark-nova allowed novel channels which would convert this energy to photon fireballs and to the kinetic energy of the quark-nova ejecta which can be easily harnessed with revolutionary consequences for high-energy astrophysics. 

Glendenning [11] pointed out that, in complex systems of more than one conserved charge, the system does not need to be locally electrically neutral, only globally so. This allows for a complex mixed phase to exist during the transition of nuclear to quark matter where various charges, including baryon number, electric charge, and quark flavors, are conserved. This led to a rich literature exploring the hadron-to-quark matter transition which can be divided into three main streams including smooth (i.e., cross-over), Gibbs (i.e., soft) and Maxwell (i.e., sharp first-order) transitions. The nature of the transition depends strongly on the EoS of the hadronic and quark matter. In the Maxwell construction, the nuclear-quark phase transition is first-order (e.g., [12,13,14,15] and references therein) and the imposition of local charge neutrality would lead to a sharp interface (because of the high surface tension) with a width in the order of femtometers (for details see [11,16,17,18]). This is in contrast to a Gibbs construction where there is a mixed region where hadron matter and quark matter coexist [19,20,21,22,23,24]. In the case of a smooth cross-over, interpolation procedures are used to connect the two phases (e.g., [25,26] and references therein). The Gibbs construction also appeals to a smooth transition into the mixed phase but the fraction of each phase is determined self-consistently and is independent of the interpolation method adopted. 

The Gibbs potential is typically chosen to model most phase transitions since the timescales are usually large enough that the sound waves flatten any pressure spatial gradient across the interface. The Gibbs potential is generally deployed in many studies of phases of matter inside compact stars, since the objects of study are in a steady state, sufficient time has passed so that the phases are in mechanical equilibrium, and the variables that are being studied, such as the radius and mass, are steady-state, time-independent values. Yet, not all phase transitions are in mechanical equilibrium. If the timescales are short enough so that sound waves have not flattened the pressure gradients, then the Gibbs potential becomes inaccurate. The correct choice for the thermodynamic potential to represent the free energy depends on which thermodynamic quantities are approximated as constant when a system changes its thermodynamic state. If it is assumed that the pressure P and the temperature T remain constant through the change (i.e., dP=dT=0), then the decrease in free energy dG≤ 0 is equivalent to the second law; that is, the increase in entropy dS≥ 0. In the case of the Helmholtz energy, dF≤ 0 is equivalent to dS≥ 0 if dV=0 (where V is the volume) and dT=0. This difference between the Gibbs and Helmholtz potential is crucial in the context of hadron-quark phase transitions (see discussion in [10]).

The mechanism of the quark-nova is intimately linked to the strong force which governs the interaction between quarks and also gives rise to the nuclear force. In astrophysics, quantum chromodynamics (QCD) becomes relevant in the context of compact objects. This is because the cores of compact objects are so dense that they become thermodynamically ideal sites for the phase-transition of hadronic to quark matter.

Quark deconfinement appears at extremely high temperatures or densities. This is due to the property of asymptotic freedom where the high momentum exchange between quarks weakens the attractive interaction between them. So, for the “quarks” to be released/deconfined, they need to collide with extreme momenta. Since temperature is a measure of kinetic energies, high temperatures are a way to trigger this deconfinement. In the case of high densities, fermions, such as quarks, are compressed into having high Fermi energies, triggering high momentum exchange.

In Earth-based experiments, particle accelerators tap into the high temperature regime by triggering very high energy collisions. However quark deconfinement in compact stars cannot be probed through experiments, since deconfinement appears at low temperatures but high (∼1015 g cm−3) densities. This give rise to the need to use compact star observations to probe the QCD phase diagram. The existence of exotic particles in the core of compact stars is, therefore, an ideal laboratory for the study of exotic particles. Given the high Fermi energies, and, therefore, high momentum exchanges in the cores of compact stars, nucleation of quark matter inside them can be expected.

The BWTH hypothesis referred to previously states that matter with the lowest binding energy could be (u,d,s) quark matter. The main reason for this is that the existence of a third degree of freedom in the form of s-quarks in general lowers the Fermi energy of the matter. In the MIT bag model, a simple approximation is that quark matter is in the form of a Fermi gas with a constant B that acts as the confinement pressure. A range of bag constants can be found where (u,d,s) quark matter is lower than the hadronic binding energy of ∼930 MeV, but, at the same time, where (u,d) matter has a higher binding energy than hadronic matter. This hypothesis therefore implies that macroscopic objects made of (u,d,s) matter are thermodynamically plausible.

The conversion of hadronic to quark matter could occur in the following way: Once two-flavoured quark matter is nucleated in the core of neutron stars, the weak interaction can turn some of the d quarks into s-quarks, lowering the Fermi energy of the quark matter. Because, at this point, the free energy of (u,d,s) matter is lower than the free energy of hadrons, the hadrons accreted by the quark core would find it energetically favourable to deconfine into lower energy quark matter. Eventually, the quark core would grow, engulfing the whole compact star, turning it into a pure (u,d,s) star. There are alternate scenarios for conversion of a whole compact star to a (u,d,s) star including, for example, through “seeding” of cosmic strangelets (e.g., [30]), or dark-matter annihilation in neutron stars heating up parcels of neutron star matter making conditions favorable for the creation of quark bubbles [31].

Although the 1053 ergs of energy release predicted by energetics compares favorably to explosive events such as supernovae, whether this energy is released explosively or in a slow simmer is not defined. Since the 1980s, different groups have sought to elucidate the phenomenology of this energy release. Olinto [30] pioneered a hydrodynamic formalism for exploring the conversion of hadronic to (u,d,s) quark matter as a hydrodynamic combustion process. The conversion was modeled as a “combustion front” that “burns” hadronic fuel into (u,d,s) ash. However, the exact equations that govern this reaction zone, the reaction-diffusion-advection equations, cannot be solved in analytic form since they are non-linear. Olinto therefore needed to linearize the equations and impose mechanical equilibrium and derive semi-analytic, steady-state solutions. The study yielded timescales of conversion of minutes to days for the whole compact star, which would imply a slow simmer, since the timescale of a supernova explosion is about one second.

Another pioneering paper was published by Benvenuto et al. [32] in the late 1980s. In contrast to Olinto, the authors assumed an initial shock and solved the relativistic jump conditions. The model proposed leads to a steady-state solution but without the mechanical equilibrium assumed by Olinto. The approach yields a supersonic detonation, which Benvenuto et al. argue can provide enough kinetic energy to make a core-collapse supernova explode. Although this explosive solution contrasts with Olinto’s much slower timescales, the reason is that a shock is assumed on an a priori basis, with arguments presented that the initial deconfined bubble of quark matter creates a sharp pressure discontinuity. Drago et al. [33] also solved the jump conditions without assuming mechanical equilibrium, but in their case they found that the combustion takes the form of subsonic deflagration.

All the literature on hadron-quark combustion before the 2010s can be roughly categorized as following either a mechanical equilibrium approach [30] or a jump condition approach [33], and has always assumed a steady state. Because of the variety of the assumptions made, such as whether a pressure equilibrium is assumed or not, or whether a shock is hypothesized as an initial condition or not, the timescales of conversion predicted for the compact star have varied by various orders of magnitude, from milliseconds to days.

Previous literature on this topic has reported very different results on the transition speed and energy as a consequence of incorrectly assuming premixed combustion (see discussion on this in [34]). However, the (u,d)-to-(u,d,s) combustion is of the non-premixed type, a distinction that is critically important. In a hadron-quark combustion flame, the thermal conductivity plays a negligible role, since the activation occurs through the s-quark fraction, because it is ultimately the quantity of s-quarks in the quark phase that determines whether the quark matter has a lower free energy than the hadronic matter. A minimal amount of s-quarks in the NS core is sufficient to create “oxidation” (to represent it in chemical activation terminology). In traditional pre-mixed combustion, the oxidant must be mixed with the fuel so that, once the activation temperature is achieved, the fuel is burned. Some fuels come premixed with the oxidant, and, therefore, the combustion is fundamentally mediated by thermal conductivity.

A hydrodynamic combustion code (the Burn-UD code; [35,36]) was developed by the Quark-Nova group to model in detail the non-premixed phase transition of hadronic to quark matter. The Burn-UD code allows the adoption of the Helmholtz, instead of the Gibbs, potential and self-consistently couples the thermodynamics to the hydrodynamics, which is of crucial importance. It can be shown that this coupling allows for a more rigorous capture of the propagation of the burning front with implications for the energetics in the case of a burning neutron star. For example, if the propagation of the burning front is too slow, the energy released is not efficiently transformed into kinetic energy, with the energy simply leaking out slowly as transparent neutrinos. The Burn-UD code consistently calculates how the weak interaction gives rise to particle and temperature spatial gradients that, in turn, trigger pressure gradients. A pressure gradient acts as a source of momentum density in the fluid, transforming some of the energy released into mechanical energy. The inclusion of the Helmholtz thermodynamic potential was found to lead to much larger neutrino luminosities (about two orders of magnitude larger than for the Gibbs potential) and larger burning speeds. Furthermore, the Helmholtz approach offers advantages numerically since it can borrow from the Gibbs construction, which avoids sharp density gradients in numerical experiments by employing a mixed phase (see Section 2.2 in [10]).

The Burn-UD code models the flame micro-physics for different equations of state (EOS) on both sides of the interface, i.e., for both the ash (up-down-strange quark phase) and the fuel (up-down quark phase). It also allows the user to explore strange-quark seeding produced by different processes. It is an advection-reaction-diffusion code which is essential for a proper treatment of the micro-physics of a burning front. Furthermore, having a precise understanding of the phase transition dynamics for different EOSs further aids in constraining the nature of the non-perturbative regimes of QCD in general (see Section 4.1). The Burn-UD code has evolved into a platform/software which can be used and shared by the QCD community exploring the phases of quark matter and by astrophysicists working on compact stars. The code provides a unique physical window to diagnose whether the combustion process will simmer quietly and slowly, lead to a transition from deflagration to detonation, or entail a (quark) core-collapse explosion.

Niebergal et al. [37], for the first time, in 2010 published a study that numerically solved the reaction-diffusion-advection equations for hadron-quark combustion. This study combined transport, chemical, and entropic processes into a numerical simulation. Not only was the burning velocity much faster than many of the previous estimates, but Niebergal et al. sugested that leptons may trigger feedback that can accelerate the burning front into supersonic detonation or quench it. Their argument was based on solving the jump conditions and parameterizing the cooling behind the front.

Later, Ouyed et al. [38] solved the reaction-diffusion-advection equations and coupled them to neutrino transport using a flux-limited diffusion scheme, and added an electron EOS and a hadronic matter (HM) EOS. Ouyed et al. confirmed numerically that leptons can trigger extreme feedbacks, with the burning halting completely for certain choices of the initial conditions. Ouyed et al.’s study was important in that it showed that, due to non-linear couplings between lepton physics and hydrodynamics, the simulation was extremely sensitive to the details of neutrino transport. This indicated that the system is genuinely a non-linear, dynamic process, and that simplifying it by imposing mechanical equilibrium or steady-state conditions was extremely inaccurate.

There are multiple ways that a quark-nova could be triggered. There are two “mechanisms” for initiating the combustion of a neutron star into a quark star. One mechanism relates to the core of a neutron star in some way reaching sufficiently high densities that favour the deconfinement of quark matter. These nucleated (u,d) quark bubbles, in turn, would beta equilibrate into (u,d,s) matter, and then, in accordance with the BWTH, grow, engulfing the whole neutron star [39]. However, the density at which quark matter deconfines is very uncertain. It could be that most neutron stars achieve the deconfinement density and, therefore, turn into quark stars. However if the deconfinement density is higher than that of the average core of a neutron star, then sufficiently high density could be achieved through other processes, such as accretion, fall-back from supernovae, or spin-down evolution, leading to a two-family compact star scenario, where quark stars and neutron stars coexist [40].

The other mechanism for triggering a quark-nova could be through “seeding”. According to the BWTH, a strangelet that interacts with hadronic matter can convert the latter, provided that there is not an electrostatic barrier preventing the interaction. Since neutrons do not have a charge, neutron stars are an ideal site for strangelet contamination. The source of strangelets can be arbitrary; for example, cosmic strangelets can be released by a (u,d,s) star merger, or stranglets may be formed through the annihilation of dark matter in the core of neutron stars.

In summary, the quark-nova hypothesis (and the underlying microphysics) implies that the traditional picture of stellar evolution is not the whole story. There is the possibility that the neutron star would experience further collapse into a (u,d,s) star. Such a phenomenon would have similar dynamics and energetics as a core-collapse supernova, with approximately ∼1053 ergs released of both chemical and gravitational binding energy. Such an addition to the stellar evolution picture has immense phenomenological consequences. This hypothesis essentially argues that the neutron star “explodes” (i.e., the neutron-star-to-quark-star combustion is explosive). The neutron-rich ejecta released from the outer layers of the exploding neutron star also constitute a very suitable site for nucleosynthesis and r-process elements, since there is a very low proton fraction.

In the quark-nova investigations, it was found that a very natural way of triggering the detonation of a neutron star is via a “quark core-collapse” where the neutron star core simply collapses into a more compact, (u,d,s) configuration, releasing massive amounts of energy. This relies on the crucial coupling between the conversion front (the micro-physics; see [10]) and the dynamics it induces at the scale of the star (∼106 cm; the macro-physics), as discussed in the next section.