Effects of Onset of Phase Transition on Binary Neutron Star Mergers

Quantum Chromodynamics predicts phase transition from hadronic matter to quark matter at high density, which is highly probable in astrophysical systems like binary neutron star mergers. To explore the critical density where such phase transition can occur, we performed numerical relativity simulations of binary neutron star mergers with various masses (equal and unequal binaries). We aim to understand the effect of the onset of phase transition on the merger dynamics and gravitational wave spectra. We generated a set of equations of states by agnostically changing the onset of phase transition, having the hadronic matter part and quark matter part fixed. This particular arrangement of the equation of states explores the scenario of mergers where mixed phases of matter are achieved before or during the merger. Under these circumstances, if the matter properties with hadronic and quark degrees differ significantly, it is reflected in the stability of the final merger product for the intermediate mass binary. We performed a case study on mixed species merger, where one of the binary companions is hybrid star. If quark matter appears at low densities, we observe significant change in post-merger gravitational wave analysis in terms of higher peak frequencies and post-merger frequencies in power spectral density. We report indications expressed as spikes in phase difference plots at merger time for mixed mergers. We found that the expression of phase transition in post-merger gravitational wave signals is more significant for unequal mass binary than for equal mass binary having the same total baryonic mass.


INTRODUCTION
The advancement of detection capabilities of astrophysical detectors and robust high-resolution numerical simulations of a large number of complicated astrophysical systems has given us the hope to resolve many astrophysics problems, which has intrigued the scientific community for decades.One of the most sought-after questions is about the constituent particles and the fundamental nature of the force that governs matter at high density.Quantum chromodynamics (QCD) predicts a transition (mostly believed as a first order) from hadronic matter (HM) to quark matter (QM) at high density (Shuryak 1980).Although the prediction was made a few decades ago, theoretical or experimental evidence regarding this hypothesis is yet to be found.In fact, constructing an experimental setup to probe such high matter has proved challenging over the years.On the other hand, ab-initio calculations (like lattice QCD (de Forcrand 2010)) have failed due to the famous sign problem (Goy et al. 2017), and the perturbative QCD calculations (Lattimer & Prakash 2004) give reliable results only at asymptotically large densities.However, the phase transition (PT) is expected to occur at much lower densities.
Earth-based experiments and theoretical calculations fail in this regard; however, neutron star (NS) astrophysics comes to the forefront.NSs are very dense objects with a mass of few solar masses (most NS mass lies between 1.2 ∼ 1.8  ⊙ (Valentim et al. 2011)) and a radius of ∼ 10 km.Therefore, at the centre of such stars, ★ E-mail: shamims@iiserb.ac.in (SH) † E-mail: mallick@iiserb.ac.in (RM) the density is expected to reach a few times that of nuclear density, which is considered an ideal condition where the PT can take place.There had been several works which made an attempt to understand the PT process and its feasibility in isolated NSs (Glendenning 1992;Alford 2001;Alford et al. 2005;Beni´c, Sanjin et al. 2015;Annala et al. 2020;Olinto 1987;Bhattacharyya et al. 2006Bhattacharyya et al. , 2007;;Drago et al. 2007;Prasad & Mallick 2018, 2020;Baldo et al. 2003;Blaschke et al. 2009;Dexheimer & Schramm 2010;Orsaria et al. 2014;Ferreira et al. 2020;Han & Prakash 2020;Prasad & Mallick 2022;Mallick et al. 2021).However, verifying such inferences observationally is challenging, as the interior of the NS is not directly visible to us.The detectors can only observe the surface and various emissions from the surface.Presently, the mass of the NS can be measured with high accuracy (Cromartie et al. 2020), and its radius with improved accuracy up to a few hundreds of meters (Miller et al. 2019;Riley et al. 2019).Earlier, this was insufficient to constrain the properties of matter (known as the equation of state (EoS)); however, the discovery of a few massive NSs in the last decade has changed the picture entirely.
The discovery of heavy pulsars (highly rotating NSs) like PSR J0348+0432 (2.01  ⊙ ) (Antoniadis et al. 2013), and PSR J0740+6620 (2.08  ⊙ ) (Fonseca et al. 2021), has ruled out soft EoSs which are unable to produce massive NSs.Furthermore, the recent observation of the pulsar PSR J0740+6620 by NICER has given a lower bound on the radius of the pulsar to be  ≳ 11 km (Miller et al. 2021;Riley et al. 2021).Such observations have put constraints on the EoS of NS to a great degree.While such observations are gaining momentum, additional constraints came from the observa- ρ/ρ0 4.9 tion of binary NS merger (BNSM) GW170817 (Abbott et al. 2017;Radice et al. 2018;Raithel 2019;Bauswein et al. 2017;Paschalidis et al. 2018;Li et al. 2018;Kashyap et al. 2022).The gravitational wave (GW) observation of the merger has put a severe constrain of the tidal deformability of the stars (Hinderer et al. 2010;Most et al. 2018;Annala et al. 2018;Zhang et al. 2018;Nandi & Pal 2021).The present bound on the tidal deformability is Λ ≤ 720.The deformability of the star is directly proportional to the star compactness, which in turn depends on the EoS.Combining all these observations has narrowed down the EoS to a great extent.
It is an accepted fact that at low density, the matter is in the hadronic state.However, the exact density from which QM starts to appear (the deconfinement transition happens) is not known.In this paper, we are simulating BNSM systems (with different mass mergers) to understand the effects of the onset of PT on merger dynamics.We will use a hadronic EoS and the MIT bag model (Chodos et al. 1974) quark EoS to describe the HM and QM, respectively.These are combined (bridged by Gibbs construction (Benvenuto & Lugones 1999)) to form a hybrid EoS, which is mimicked using piece-wise polytrope to include in simulations.For our set of EoSs, we keep our HM part and QM part of the EoS fixed and only change the onset of PT agnostically.In all the previous works, the QM part differs after the onset point changes, which may happen when an underlying formalism is considered (to include microphysics) to calculate the EoS.Since we used an agnostic approach, we have restricted any such change in our study.From the onset of PT, we smoothly match to the common mixed phase region using the piece-wise polytrope method.Having the HM part and QM part fixed for all the EoSs, we ensure that the difference in BNSM systems appears only due to changes in the onset point.We have studied the possible effects in GW emission spectra and the final merger state (and their lifetimes) depending on the various onset points of PT.It is important to note that this particular choice of the EoSs explores the scenario of mergers where mixed phases of matter are achieved before or during the merger.Secondly, post-merger dynamics are highly affected by other factors like magnetic fields and neutrino cooling (Palenzuela et al. 2015;Anderson et al. 2008), which are not considered in this work.
Alongside the set of equal-mass mergers performed, we also reported a case study on a particular unequal-mass merger, which is an NS-HS (neutron star -hybrid star) merger.Here, only one of the companions in the NS pair has a quark seed inside it (an HS); the other companion has no quark seed (an NS).Such mixed systems have been acknowledged in previous work (Bauswein & Blacker 2020) but require an in-depth study.In this case study, the initial binary configuration is chosen such that the lighter companion is identical for all EoSs.However, the heavier companion has a different interior structure for each EoS.Hence, this case probes the scenario where the effects in the GW emission spectra and the resulting post-merger state are due to the differences in interior initial state properties of the heavier companion only.We have studied how the onset of PT affects these mixed systems.We report the indications from the GW analysis for such a scenario, which are expressed as the spikes in phase difference plots during the merger time.If such an observation is made in the near future, this particular merger will make greater clarity on the understanding of this topic.
The paper is arranged as follows: In Section 2, we explain the formalism and framework used for performing the simulations of the binary merger system and methods used for extracting and postprocessing GW signatures.In Section 3, we present our results of different equal and unequal mass binary cases (both pre-merger and post-merger simulations).Finally, in Section 4, we summarise our results and build conclusions from them.

Equation of State
We used DD-ME2 EoS (Lalazissis et al. 2005) and BPS EoS (Baym et al. 1971;Negele & Vautherin 1973) to describe the HM and crust respectively.We constructed hybrid EoSs where QM is described using the MIT bag model (Chodos et al. 1974), setting the bag constant  = 150, and parameter  4 = 0.60.The Gibbs construction is used for a smooth PT of HM to QM (Benvenuto & Lugones 1999), which creates a mixed-phase region inside the star where HM and QM coexist.At the point where the mixed-phase starts to appear, we identify that position as the onset point of PT.We varied the onset points of the mixed-phase in each hybrid EoS by stitching it up at different densities.All the density values are presented in terms of nuclear saturation density ( 0 = 2.51 × 10 14 g/cm 3 (Glendenning 1997)).We constructed three hybrid EoSs -EoS Hyb-A (onset point-2.170 ), EoS Hyb-B (onset point-2.360 ) and EoS Hyb-C (onset point-2.570 ).All the hybrid EoSs have a common point at 6.11 0 for the transition to pure QM.We mimic the EoSs using piece-wise polytrope fitting (Rezzolla & Zanotti 2013;Read et al. 2009), plotted in Fig. 1 (left).All the EoSs satisfy the following astrophysical constraints -mass of J0348+0432 (2.01 ± 0.04  ⊙ ) (Antoniadis et al. 2013), mass of J0740+6620 (2.08 ± 0.07  ⊙ ) (Fonseca et al. 2021;Cromartie et al. 2020), radius measurements by NICER (Riley et al. 2019;Miller et al. 2019;Riley et al. 2021;Miller et al. 2021).The maximum masses are constrained using J0740+6620 (2.08 ± 0.07  ⊙ ) for all the EoSs, that is, to be greater than 2.01  ⊙ .Secondly, we use the upper bound on binary tidal deformability ( λ < 720) by GW170817 (Abbott et al. 2019).The M-R curves are plotted in Fig 2, with markers on mass-radius of 1.1 and 1.4  ⊙ NSs, and the maximum mass generated by these EoSs.The tidal deformability Λ are -679, 671, 679, 679 for EoSs -Had, Hyb-A, Hyb-B, Hyb-C, respectively.
These EoSs are supplemented by an ideal-fluid thermal component (Janka et al. 1993), which accounts for shock heating in the system that dissipates the kinetic energy into the internal energy.The thermal component Γ th is set to 1.8 (Takami et al. 2015).More details on the formalism of EoS are available in Appendix D. The constant thermal coefficient does not exactly reproduce the effects of temperature at a high-density regime given by the finite-temperature EoSs (Bauswein et al. 2010b;Raithel et al. 2021;Figura et al. 2020).Secondly, the onset of phase transition could also be affected by the rise in local temperatures.These effects are not considered in the scope of this work.A framework is yet to be found which can create a thermal counterpart from a cold part of the EoS to form a finite-temperature EoS.All available finite-temperature EoSs use a predefined cold part of their own.Hence, for a study like this, where agnostic modifications in the cold EoS are required, it is difficult to use finite-temperature EoS, and hence, the use of a supplemented thermal part becomes necessary.

Computational Setup
For the evolution of the BNSM systems, we use the Einstein Toolkit (Löffler et al. 2012;Thornburg 2003;Diener et al. 2007;Dreyer et al. 2003) with McLachlan (Reisswig et al. 2011;Brown et al. 2009) (implementation of spacetime evolution) and Illinois-GRMHD (Etienne et al. 2015;Del Zanna, L. et al. 2003;Noble et al. 2006) (GRMHD solver).The initial data are generated using Lorene (Gourgoulhon et al. 2001).The grid and iteration parameters for our initial configurations were set identical to the version available at the Subversion repository server of the Gravitational Physics Group at the Parma University (De Pietri et al. 2016).We have set the initial physical separation between the stars to be 40 km with the irrotationality of the fluid flow.More details on the numerical setup and uncertainties due to the choice of grid size are available in Appendix B and Appendix C, respectively.We extract gravitational waveforms from our simulations by calculating the Weyl scalar (particularly Ψ 4 ) using the Newman-Penrose formalism (Newman & Penrose 1963).We analysed the dominant mode  =  = 2 of ℎ strain at 100 Mpc.Now, ℎ 22 can be given as , where |ℎ 22 | is the GW amplitude, and  is the phase of the complex waveform.We have set the merger time at the point where |ℎ 22 | is maximum.The instantaneous frequency is calculated by  GW = (1/2)(/).We used Kuibit (Bozzola 2021) for handling GW data.We calculated the Power Spectral Density (PSD) of the GW amplitude ( h), given as, 2 h2 = | h+ | 2 + | h× | 2 where h+,× (  ) are the Fourier transforms of ℎ +,× ().More details on the formalism of GW analysis are available in Appendix E.
The simulations carried out for this work used the coarsest grid size of Δ = 25  ⊙ (with 7 refinement levels), having the finest resolution resolving the star to be Δ = 0.39  ⊙ (∼ 576 m).The total extent of the 3D domain is ∼ 3 × 1000  ⊙ (∼ 3 × 1470 km).This grid setup is considerably low with respect to standard high-resolution simulations carried out in numerical relativity.As discussed in Appendix C, the qualitative analysis holds true for respective finer grid simulations.However, the quantitative figures are expected to change if finer grids are used with this framework.Hence, they must be used only as an assistance to the qualitative inferences of this article.

RESULTS
Using the EoS classifications, we simulated three equal mass binary merger cases (1.2v1.2, 1.4v1.4,1.6v1.6)and one unequal mass binary merger case (1.2v1.6).In some cases, we identify the hypermassive remnant core to be unstable due to high fluctuations of the maximum density in the core.Here, we reported the Δ, which is the local measure of the difference between the highest and lowest values of maximum density in ∼ 1 ms.For the scenarios where a hypermaissive remnant core collapses into a Black Hole (BH), it is not straightforward to mark the time stamp when the BH is formed.Hence, calculating the collapse times is non-trivial.In this article, we identify the onset of collapse when the maximum density in the simulation suddenly peaks (30 − 100 0 within ∼ 0.5 ms) and mark the collapse time when the maximum density crosses 30 0 .These markers are highly sensitive to additional effects and, hence, must be used for qualitative understanding of the merger dynamics specific to the framework (including grid resolution and EoS set) of this article only.The initial compact stars, which have central densities in the HM region, are identified as NS.If the central density (or maximum) is in the mixed-phase region, we interpret that the star endures a quark seed containing the mixed-phase matter and is identified as an HS.We observe that a lower onset of PT in an EoS leads to a larger quark seed.Similarly, the classifications -supramassive NS (SMNS), supramassive HS (SMHS), hypermassive NS (HMNS) and hypermassive HS (HMHS) are drawn for the identification of final remnant compact stars (Sarin & Lasky 2021).

Equal Mass Binary Simulations
Merger-1.2v1.2 -in Fig. 1 (centre), for all EoS cases, we measure the central densities of the initial compact objects to be 2.17 0 approximately, which is in the pure HM regime.Hence, these initial compact objects are NSs during the inspiral phase.The central densities of NS pair constructed using EoS Hyb-A sit at the onset point of PT.In Fig. 3, for the EoS Hyb-A case, we observe that the mixedphase appears for a short period due to the spiking maximum density at the merger time.The resulting compact object is an SMHS, as it carries a core ( max = 3.18 0 ) with mixed-phase ( = 0.23).As the onset point of PT is set up at higher densities for EoSs -Hyb-B and Hyb-C, the simulations introduce mixed-phase in these systems at subsequent times.At the end of the post-merger simulation, the final states are SMHSs with maximum densities of 3.03 0 ( = 0.17) and 2.91 0 ( = 0.09) respectively.In all the hybrid EoS cases, the simulations could not reach densities that favour the appearance of pure QM.Secondly, EoS Had, being purely hadronic, gives rise to an SMNS with a maximum density of 2.81 0 .GW analysis for this case is reported in Appendix A.
Merger-1.4v1.4 -in Fig. 1 (centre), we observe that the compact objects in initial binary configurations made using EoSs -Had, Hyb-B and Hyb-C are NSs.The NS pair constructed by Hyb-B has cores with central densities sitting at the onset of PT.EoS Hyb-A has constructed an initial HS binary configuration with comparatively higher central densities for the same initial masses.The cores of these HSs are just inside the mixed-phase regime.In Fig. 3, for all the hybrid EoS cases, we observe that the post-merger forms an HMHS, which leads to collapse into a Black Hole (BH).For EoSs -Hyb-A, Hyb-B and Hyb-C, the collapse times are ∼12 ms, ∼13 ms and ∼15 ms respectively.It infers that the earlier the mixed-phase appears in the system, the faster the HMHS collapses.Hence, the onset point of PT has an important role in defining the collapse times of post-merger remnants.EoS Had gives rise to an HMNS, and no collapse scenario is observed up to 45 ms of post-merger evolution.Since the description of mixed-phase in hybrid EoSs is softer than EoS Had, the systems using hybrid EoSs allow the core of HMHS to crunch more matter.It leads to a quicker increment of core densities, resulting in faster collapse to BH.We infer with certainty that, for this particular case of equal mass binary, hybrid EoSs favour core collapse in contrast to hadronic EoS, which forms a stable HMNS.In a realistic scenario of such mergers, a core collapse can confirm the presence of QM in hypermassive stars and extracting the collapse time can optimise the onset point of PT in our EoS models.
Merger-1.6v1.6 -in Fig. 1 (centre), we observe that the EoSs -Had and Hyb-C, form binary NS configurations.The cores of NS pair constructed using Hyb-C are at the onset of PT.The EoSs -Hyb-A and Hyb-B, form binary HS configurations.In Fig. 3, we observe that all the EoS cases lead to BH collapse scenario.However, the onset of collapse for EoS Had case is very far with respect to hybrid EoS cases.For the EoSs -Hyb-A, Hyb-B and Hyb-C, the rapidly differentially-rotating structure collapses to BH in about 2 ms.In realistic scenarios for high mass binaries, we see that collapse times become an essential parameter to understand the appearance of QM in these systems.

Unequal Mass Binary Simulation
The unequal mass binary (1.2v1.6)case has an interesting contrast to previous equal binary cases.In equal mass mergers, each EoS was constructing compact binaries with distinct interior structures, but the companions in each initial configuration were identical.In the unequal case, all the EoSs have constructed a 1.2  ⊙ (lighter) companion, which is identical for all the initial configurations, but  the 1.6  ⊙ (heavier) companion has a different interior structure in each initial configuration.So this case probes the scenario when particularly the heavier companion's interior varies, but the lighter companions remain the same.So, the effects and differences in the GW emission spectra and the resulting state are due to the differences in the initial state properties of the heavier companion.Secondly, an equal mass merger scenario can only allow the mergers of the same species (NS-NS or HS-HS) depending on the combination of initial mass and EoS used for constructing initial data.In this case, our mass choices are such that, by choosing the description of EoS smartly, the combination can allow us to form mixed species (NS-HS) mergers.Such a merger is possible if the lighter companion is purely hadronic and the heavier mass companion has a pre-established mixed-phase core.
Table 1.Table of different mass quantities at initial-() configuration (coalescence state) and final-(  ) states (after 36.34 ms of post-merger simulation) for 1.2v1.6 merger case.Column-2 reports the  max (in terms of  0 ) at the final state.Column-3 reports the initial ADM masses ( ADM ) for all the EoS cases.Column-4 and column-5 report the initial and final baryonic masses ( b ) of the configurations, respectively.The final  b is the total baryonic mass calculated inside a 45 km radius sphere.Column-6 reports the quark masses ( q ) at the final state. q are calculated from uniformly rotating equilibrium configurations of NS (setting the central density and baryonic mass of this NS as  max (  ) as  b (  ) respectively) using the RNS code (Stergioulas & Friedman 1995;Nozawa, T. et al. 1998).Coulmn-7 reports the quark mass fraction () at the final state, which is calculated as  q / b at the final state.Column-8 and column-9 report gravitational and baryonic mass at the Keplerian configuration for each EoS, respectively, constructed at  max (  ).Column-10 and column-11 report the threshold mass ( thres ) (Bauswein et al. 2020) for prompt collapse for each EoS at mass ratio  = 1 and  = 0.7 respectively.

EoS
max ( 0 )  In Fig. 1 (left), we observe that the 1.2  ⊙ companion (left) constructed by all the EoS are NSs with central density 2.18 0 approximately.The central density of NS formed using EoS Hyb-A is at the onset of PT.Fig. 4 confirms that none of these NSs have quark seeds inside them.The 1.6  ⊙ companion (right) constructed by EoSs -Had and Hyb-C, are NSs with central densities 2.60 0 and 2.61 0 respectively, where NS core by Hyb-C is at the onset of PT.The 1.6  ⊙ companion formed by EoSs -Hyb-A and Hyb-B, are HSs, with central densities 2.82 0 ( = 0.16) and 2.69 0 ( = 0.08) respectively.Fig. 4 indicates the same by showing hot spots in the core of these HSs in quark fraction distribution plots.The initial configuration constructed by EoS Hyb-A forms an HMHS, which leads to core collapse with a collapse time of ∼37 ms (Fig. 3).Simulations with EoSs -Hyb-B and Hyb-C, led to the formation of HMHSs with unsteady cores having maximum densities of 4.4 0 ( = 0.41, Δ = 0.16 0 ) and 4.06 0 ( = 0.33, Δ = 0.14 0 ) respectively.The simulation with EoS Had evolves to form an HMNS with a relatively stable core with maximum density 3.3 0 , having negligible fluctuations in the central density.The density, quark fraction, and temperature distributions for the evolved configurations for 1.2v1.6 mergers cases (at ∼ 36 ms) are reported in Fig. 5. Table 1 reports the  max at ∼ 36 ms, its initial ADM mass and baryonic mass, final baryonic mass, final quark mass and final quark mass fraction.These quantities are compared with the Keplerian mass and threshold mass for prompt collapse, which are characteristic quantities for each EoS.Such unequal mass mergers make unique observatories for a clear understanding of the onset of PT to mixed phases.Finding such realistic scenarios leading to core collapse can confirm the presence of mixed phases in the systems.Upon being able to reproduce it in simulations, one can further constrain the EoSs, emphasising the onset of PT.We compared the GW signal properties of each hybrid EoS with EoS Had for 1.2v1.6 merger scenario in Fig. 6.A short yet clear contrast between the GW signals is observed during the merger time (especially for Had v Hyb-A case), which is supported by the spike in Δ at the merger time.The GW signals deviate at a later stage, as also verified by the increasing Δ.All the inspiral stage systems have mixed-phase seed inside 1.6  ⊙ companion, resulting in the contrast in GW signature.The only exception is the EoS Hyb-C case, where 1.6  ⊙ NS has no mixed-phase seed (Fig. 4).The fact that its central density is at the onset of PT, it still creates enough mixedphase the density increase at the time of merging, resulting in a small spike in phase difference.It carries to the consequence that the magnitude of the spike depends on the amount of mixed-phase present at the initial configuration.From the spectrogram, we infer that the peak frequency for EoS Had is at ∼ 4.1 kHz, which is lower than all the hybrid EoSs.The GW signal amplitude flattens after 20 ms, making the  GW plot noisy in that time regime.For hybrid EoSs -Hyb-A, Hyb-B and Hyb-C, the peak frequency are ∼4.3,∼4.2 and ∼4.1 kHz respectively.The peak frequencies are higher for the EoS, which seeds more mixed-phase in the initial configuration.In PSD analysis, we compared the frequencies with the highest power.We found that HMHSs formed in the hybrid EoS cases have higher frequencies than the HMNS in EoS Had case.Particularly, the HMHS formed in the EoS Hyb-A case has the highest frequency of 2.4 kHz with respect to EoS Had with frequency 2.3 kHz.It demonstrates that a lower onset point of PT in a hybrid EoS leads to a higher rotational frequency of the hypermassive remnant object.

CONCLUSIONS
We simulated equal and unequal mass BNSM systems and observed the change in dynamics when the star is entirely made up of hadrons as to the case when the QM core can appear after the PT.The hybrid EoS has the hadronic degrees of freedom at low density, the mixed-phase region at intermediate density, and pure QM at very high density.The density at which quarks first appear (or the starting point of the mixed-phase region) is the onset point of PT, which is not known priory.To understand the effect of the onset point of PT on the numerical relativity simulations of BNSM systems, we used agnostic approach to construct a set of hybrid EoSs such that the onset of PT is varied, keeping HM and QM parts fixed for all the EoSs.This ensures that the difference in dynamics of BNSM systems appears only due to changes in the onset point.This set of EoSs examines a category of mergers where mixed phases of matter can be achieved before or during the merger.
The initial and final state data of all the simulations are reported in Table 2.When small mass equal binaries merge, the resultant compact object (Supramassive NS/HS) attains a stable configuration and does not collapse to a BH.In the case of intermediate-mass mergers, if the onset of PT is low, then initial stars may have a quark seed.We observed that the early onset of PT triggered early core collapse into BH.The delay in collapse times hints at the stiffness of the EoS.However, intermediate-mass equal binary constructed with hadronic EoS form an HMNS with a stable core.In the case of heavymass mergers, all the mergers collapsed into BH.For the hybrid EoSs, the collapse times did not differ significantly with the change in onset points of PT in the EoS.However, the merger constructed using purely hadronic EoS had notably delayed collapse into BH with respect to hybrid EoS cases.
We reported a case study on a particular unequal-mass merger, an NS-HS merger.These systems are known as mixed merger systems.In this case study, the initial binary configuration is chosen such that the lighter companion is identical for all EoSs.However, the heavier companion has a different interior structure for each EoS.This case probes the scenario where the effects in the GW emission spectra and the resulting post-merger state are due to the differences in interior initial state properties of the heavier companion only.We have studied how the onset of PT affects these mixed systems.We report the indications from the GW analysis for such a scenario, which are expressed as the spikes in phase difference plots during the merger time.Early onset of PT leads to stronger spikes at merger time.If the onset point of PT is at comparatively low density, then the signature of the GW signal is distinct with respect to the GW signal from the hadronic EoS merger, leading to significantly larger phase differences at post-merger.For such a case, we observed higher peak frequencies at the merger time in the GW spectogram.For the same, we observed larger values in PSD frequency with the highest power, indicating that an early onset of PT leads to a higher rotational frequency of the hypermassive remnant.Both the amplitude and frequency of the said post-merger signal are well within the next generation of GW detectors (Punturo et al. 2010).
The qualitative aspects of the simulation can be improved and made more robust by checking results with an exhaustive set of EoSs and using finer grid resolutions for the numerical relativity simulations (later improves quantitative aspects as well).Since post-merger dynamics are highly affected by phenomena like magnetic fields and neutrino cooling, it is vital to incorporate them into these studies as well.We find that observation of an unequal binary is extremely important to probe into estimating the onset of PT, as the GW signal clearly expresses the change in the onset of PT than in the case of equal mergers.We have seen from the final state of the BNSM that the information about the stiffness of the EoS can be gauged if we can make several observations of such kind.The critical aspects of matter properties, like the degrees of freedom at high density and the information about the onset point, can be extracted when future detectors are available with sensitivity to probe the post-merger phase of the BNSM.

APPENDIX A: GRAVITATIONAL WAVE ANALYSIS FOR 1.2V1.2 MERGER
We compared the GW signal properties of each hybrid EoS case with the EoS Had case for 1.2v1.2merger scenario in Fig. A1.Once the mixed-phase appears in the system, the GW signals start to deviate in each case.The divergence can be seen on the overlap and confirmed by the rising phase difference.Since the instantaneous frequencies are not well defined during the merger,  GW can peak up or down at the merger time.The second frequency peak is almost equal for all the EoSs and is marked around 2.7 kHz.As the GW signals deviate, the hybrid EoS frequencies also start to deviate from EoS Had accordingly.In the case of EoSs Hyb-A and Hyb-B, the simulations tend to gain higher frequencies at post-merger spectra and start to become noisy after ∼ 25 ms, that is when the GW signal amplitudes start to decrease.
In PSD plots, we marked the frequencies with the highest power for each EoS, selectively the one corresponding to the  =  = 2 fundamental mode of the respective SMNS/SMHSs.They are equal to twice the rotational frequency of the bar deformations of the SMNS/SMHSs.A detailed discussion about the spectral properties of the hypermassive remnants in numerical relativity simulations can be found in Ref. Rezzolla & Takami (2016).The frequencies are equal for all the cases marked at 2.24 kHz except EoS Hyb-A, which has a higher frequency marked at 2.27 kHz.It indicates that an early appearance of mixed-phase leads to a higher rotational frequency of SMHS.

APPENDIX B: NUMERICAL SETUP
We used the "Turing" ET_2021_05 version of the Einstein Toolkit.It implements the Baumgarte-Shapiro-Shibata-Nakamura-Oohara-Kojima (BSSNOK) formulation via McLachlan, where    is conformally transformed as, where, Φ is the logarithmic conformal factor and γ  is the conformal metric (constrained by det γ  = 1).These are the new variables alongside the trace of extrinsic curvature , the conformal trace free extrinsic curvature Ã  and the conformal connection functions Γ , which are defined as, to be evolved using the fourth-order finite-differencing method.The gauge functions are determined using 1 + log slicing (for lapse function) and Γ-driver shift (for shift vectors) condition (Alcubierre et al. 2003).During the evolution, a Sommerfeld-type radiative boundary condition (Alcubierre et al. 2000) is applied to all the components of the evolved BSSNOK variables, and to discard the high-frequency noise, a fifth order Kreiss-Oliger dissipation term is added using the module-Dissipation.We used the IllinoisGRMHD code (Etienne et al. 2015;Del Zanna, L. et al. 2003) for solving the GRMHD equations, which is an open-source code developed by the Illinois Numerical Relativity (ILNR) group and available with the Einstein Toolkit.The GRMHD equations are defined in a conservative form, and the flux terms are calculated using the second-order finite-volume high-resolution shock capturing (HRSC) scheme (Rezzolla & Zanotti 2013).It ensures that Rankine-Hugoniot shock jump conditions are satisfied.A third-order accurate piece-wise parabolic method (PPM) (Colella & Woodward 1984) is used for the reconstruction step.The standard Harten-Lax-van Leer-Einfeldt (HLLE) approximate Riemann solver (Harten et al. 1983;Einfeldt 1988) is applied.The method of lines module-MoL takes the time derivatives of the evolved GRMHD variables and integrates them forward in time using the Runge-Kutta fourth-order (RK4) scheme (Runge 1895;Kutta 1901).A two-dimensional Newton-Raphson solver is employed to compute the primitive variables from the conservative variables (Noble et al. 2006(Noble et al. , 2009)).where square braces are the antisymmetric part.This null tetrad is constructed from an orthonormal regular tetrad on each hypersurface based on the spherical coordinate system (bar indicates the complex conjugate).In 3+1 decomposition, by defining a polar orthogonal basis for 3D spatial hypersurface   r ,   θ ,   φ and a normal timelike vector on the hypersurface   t , null tetrad can be constructed as, (E3) The complex variable  4 provides a measure of outgoing radiation and can be related to the complex GW strain ℎ by its second-order time differentiation (Teukolsky 1973), where ℎ + and ℎ × are the polarisation modes.Now, Ψ 4 can be can be decomposed using spin-weighted spherical harmonics (Goldberg et al. 1967)  The Ψ 4 is provided by the modules-WeylScal4 and Multipole, as a complex grid function at different coordinate radius surfaces away from the source.
This paper has been typeset from a T E X/L A T E X file prepared by the author.

Figure 1 .
Figure 1.[Coloured Online] Left: Pressure-density plot of all the EoSs with their respective quark fractions.The inset zooms on the region where EoS splittings are clearly visible, with the tick marks set as per the separation points.Centre: Central (maximum) densities of the initial-() (final-(  )) stars for all the equal mass binary.Right: Central (maximum) densities of the initial-() (final-(  )) stars for all the unequal mass binary.The central densities of 1.2  ⊙ companion and 1.6  ⊙ companion are plotted separately.All the final merger remnant maximum densities are observed after 45 ms post-merger evolution.The marker sizes are adjusted to make all the overlapped markers visible.Differently shaded regions in the background inform the transitions in phases of matter (HM, mixed-phase, pure QM).Light-blue( ) shaded region is the common density range where all the EoSs have pure HM.Light-yellow( ) shaded region(and  beyond)  shows the presence of mixed-phase in EoS Hyb-A.For EoS Hyb-B, mixed-phase starts to appear in light-green( ) shaded region.EoS Hyb-C shows mixed-phase in light-red( ) shaded region.The light-gray( ) shaded region represents the pure QM regime for all the hybrid EoSs.

Figure 2 .
Figure 2. M-R curves for all the EoSs.The dashed lines on the y-axis mark the maximum mass of respective EoSs.The circular markers on the curves define the point after which the heavier NSs will have quark seed.The black dashed lines mark the mass-radius of 1.1  ⊙ and 1.4  ⊙ NSs formed by all the EoSs.

Figure 3 .
Figure 3. [Coloured Online] Evolution of  max plotted with mixed-phase and pure QM regions of the EoSs in the background.Collapse times are marked for the cases where compact objects collapsed into BHs.

Figure 6 .
Figure 6.[Coloured Online] GW emission properties all the EoSs in 1.2v1.6 merger case.Left panel reports the ℎ 22 + polarisation of the GW signal extracted at 100 Mpc.The phase difference Δ between the two signals is plotted in each case.The inset highlights the spike in Δ near the merger time.Right panel reports the spectrogram of the GW signal of the hybrid EoS case with the instantaneous frequencies  GW of both the signals, marked with peak frequencies.PSDs of both signals are plotted on the right of the spectrogram, marked with the highest power frequencies.

Figure A1 .
Figure A1.[Coloured Online] GW emission properties comparison for all the EoSs in 1.2v1.2merger case.Left panel reports the ℎ 22 + polarisation of the GW signal extracted at 100 Mpc for all the hybrid EoSs, overlapped on the GW signal for EoS Had.The phase difference Δ between the two signals is plotted in each case.The background colours on the plot indicate the presence of pure HM( ) matter and the appearance of mixed-phase (respective colours in Fig. 1) in the simulations.Right panel reports the spectrogram of the GW signal extracted from simulations of the hybrid EoS case with the instantaneous frequencies  GW of both the signals marked with peak frequencies.PSDs of both signals are plotted on the right of the spectrogram, marked with the highest power frequencies.

Table 2 .
Type of compact objects at initial-() configuration (coalescence state) and final-(  ) states (after 45 ms of post-merger simulation).Respective  max (in terms of  0 ) are mentioned in brackets with quark fraction in subscript (if HS or HMHS).If the final state is BH, then collapse time (in ms) is mentioned in brackets.For an equal binary case, only one of the star's properties is mentioned because the other companion is identical.