# Compact Star of Holographic Nuclear Matter and GW170817

###### Abstract

We use a holographic model of quantum chromodynamics to extract the equation of state (EoS) for the cold nuclear matter of moderate baryon density. This model is based on the Sakai-Sugimoto model in the deconfined Witten’s geometry with the additional point-like D4-brane instanton configuration as the holographic baryons. Our EoS takes the following doubly-polytropic form: with a tunable parameter of order , where and are the energy density and pressure, respectively. The sound speed satisfies the causality constraint and breaks the sound barrier. We solve the Tolman-Oppenheimer-Volkoff equations for the compact stars. We reach the reasonable compactness for the proper choices of . Based on these configurations we further calculate the tidal deformability of the single and binary stars. We find our results agree with the inferred values of LIGO/Virgo data analysis for GW170817.

## I Introduction

Tremendous gravity can transform the ordinary matter in a compact star into exotic nuclear matter such as neutron liquid or quark-gluon plasma, which are hard to produce on earth and whose properties remain to be clarified after decades of studies Lattimer:2004pg (); Baym:2017whm (); Baym:2019yyo (). By the same token, the gravitational tidal force acting on the nuclear matter of a compact star can cause shape deformation, which can reveal nuclear matter’s hydrodynamical properties such as equation of state (EoS). A novel way of observing the tidal deformation is to detect the gravitational wave emitted during the binary merger of compact stars. An recent example is LIGO/Virgo’s GW170817 TheLIGOScientific:2017qsa (); Abbott:2018wiz (); Abbott:2018exr (), which has inspired closer examination of the EoS Tews:2018chv (); De:2018uhw (); Malik:2018zcf (); Zhang:2018bwq (); Zhao:2018nyf (); Lau:2018mae (); Dudi:2018jzn (); Christian:2018jyd (); Han:2018mtj (); Torres-Rivas:2018svp (); Liuti:2018ccr (); Choi:2018zbi (); Carson:2018xri (); Ayriyan:2018blj (); Coughlin:2019kqf (); Bauswein:2019ybt (); Tews:2019cap (); Kumar:2019xgp (). One shall expect to observe more such kind of binary mergers in the coming future to clarify the nature of nuclear matter inferred from the mass, radii and tidal deformability of the compact stars. To detect gravitational waves and perform parameter estimation of the sources by the method of matched filter TheLIGOScientific:2016qqj (), one needs to prepare template banks of waveform produced from specific gravity theory and EoS of nuclear matters. Especially, one needs the theoretical understanding of EoS in order to discern the modified gravity’s effect from the hydrodynamical one in the waveforms.

However, it is notoriously difficult to derive the EoS of exotic nuclear matters at moderate baryon density, i.e., about a few of the saturation density of nuclei, either from the first principle such as lattice quantum chromodynamics (QCD) by suffering a sign problem at finite chemical potential deForcrand:2010ys (); Cristoforetti:2012su (), or from perturbative QCD and chiral perturbation theory due to a sizable coupling at moderate densities Kraemmer:2003gd (); Kurkela:2016was (); Vuorinen:2018qzx (). Therefore, most of EoS currently used for nuclear matter in compact stars are phenomenological. It is important to derive EoS systematically based on some physical principle, and the holographic QCD is suitable for such a purpose.

Holographic QCD is a mean-field theory for QCD in terms of the dual bulk gravity dynamics based on the spirit of AdS/CFT correspondence Maldacena:1997re (), and it has been adopted to address many QCD problems with success, e.g., clarify the hydrodynamical nature of quark-gluon plasma Kovtun:2004de (); Liu:2006ug (); Herzog:2006gh (); Gubser:2006bz (); Yee:2009vw (); Gynther:2010ed () realized in the experiments of heavy ion collisions. Among many holographic QCD models, the Sakai-Sugimoto (SS) model Sakai:2004cn (); Sakai:2005yt (), where the mesons are introduced by the -branes (or called meson-brane) in the background of Witten’s geometry Witten:1998zw (), is the best model so far with very few free parameters. Unlike the other holographic QCD models Kruczenski:2003be (); Kruczenski:2003uq (); Erlich:2005qh (), the SS model has spontaneous supersymmetry breaking, and realizes the quark confinement and chiral symmetry breaking in a natural and geometric manner. Moreover, the effective field theory derived from SS model takes the form of the chiral effective Lagrangian, and yields the well-fitted meson and hadron spectra and the decay amplitudes Sakai:2004cn (); Sakai:2005yt (); Hata:2007mb ().

Our goal in this paper is to extract EoS of holographic nuclear matters from SS model, and use it to study the properties of the associated compact stars. We start with the SS model in the deconfined Witten’s geometry, in which both the broken and unbroken phases of chiral symmetry can be realized. This is suitable for the consideration of QCD at finite baryon density because the chiral symmetry is expected to be restored at high enough baryon density. The baryons are introduced as the D4-brane instanton Witten:1998xy (); Hata:2007mb (). Here we will only consider the point-like instanton configuration Bergman:2007wp () which should be good enough approximation for the case of moderate baryon density. Our EoS has only one tunable parameter, and by the proper choice we find the mass, radius and tidal deformability of the compact stars are in excellent agreement with the inferred values from the data analysis of GW170817 TheLIGOScientific:2017qsa (); Abbott:2018wiz (); Abbott:2018exr (). Compact stars with EoS from more general instanton configurations with finite size and multi-layers Ghoroku:2012am (); Preis:2016fsp (); Preis:2016gvf (); Li:2015uea () will be considered in the future works.

## Ii Holographic nuclear matters

The meson-brane action of the SS model consists of the non-Abelian Dirac-Born-Infeld (DBI) action and the Chern-Simons (CS) action. For our study of holographic uniform nuclear matter, the total action is formally given by Sakai:2004cn (); Sakai:2005yt (); Hata:2007mb ()

(1) |

where is the radial coordinate of Witten’s geometry, and and denote spatial volume and temperature of the dual QCD, respectively. Besides, is the profile of meson-brane wrapping around the Kaluza-Klein circle of Witten’s geometry; is the 0-th component of the part of the flavor gauge field, whose boundary value is holographic dual to the baryon chemical potential, i.e., ; and is the -brane instanton configuration of part of the flavor gauge field with moduli ’s. We will set one of ’s, say which is the location of the -brane instanton on the meson-branes. We can scale the variables so that and are dimensionless, and all the dimension is encoded in the constant , which is given by

(2) |

where is the color number, the ’t Hooft coupling and is the mass gap of the holographic QCD.

There are two scenarios of studying the holographic baryonic matters in SS model. The first one is to consider the antipodal separation of and in the confined Witten’s geometry which ensures the chiral symmetry is always broken. The antipodal separation means with

(3) |

where is the asymptotic separation of and . This scenario is used to describe QCD states in the mesonic vacuum and yields well-fitted decay amplitudes, meson and baryon spectra Sakai:2004cn (); Sakai:2005yt (); Hata:2007mb (), and serves as a reference for our study based on the second scenario. One can fit the parameters in (2) by comparing the predicted values of physical pion decay constant and the rho meson mass with the experimental data, so that the results are

(4) |

Thus, which will be adopted later on.

The second scenario is to consider the decompactified limit of and , i.e., , in the deconfined Witten’s geometry, which is a black hole with with a blacken factor and is dual to QCD at temperature . In this geometry, the chiral symmetry can be restored at high enough temperature or baryon density. This is similar to the key feature of Nambu-Jona-Lasinio (NJL) model. This scenario is thus the holographic version of NJL model Antonyan:2006vw (); Davis:2007ka (); Li:2015uea () and is suitable for the purpose of this work to study QCD at moderate baryon density, which is the arena for the nuclear matter inside the neutron stars.

Given a -brane instanton configuration, one can solve the equations of motion for and to obtain the on-shell meson-brane action, and then interpret it as the (dimensionless) grand canonical potential density given the baryon density , i.e.,

(5) |

Here the dimensionless baryon density is related to the -brane instanton number, and appears as the overall coefficient of the CS term, i.e.,

(6) |

with the radial instanton profile normalized by . For a given set of and , the thermal equilibrium configuration can then be obtained by minimizing w.r.t. and ’s, i.e.,

(7) |

This gives and for the thermal equilibrium configuration at temperature and chemical potential so that one can derive the other thermal dynamical quantities such as the pressure , the entropy density , and the energy density . Note also . From these one can extract the EoS of the nuclear matter ^{1}^{1}1We also need to make sure the phase of nuclear matter is dominated over the mesonic phase, i.e., , and quark phase, i.e., with mesonic branes ending on horizon by comparing their free energies. Besides, we do not include the vacuum energy in our EoS, for such possibility see Csaki:2018fls ()..

Here and in the following, we will adopt the scalings as done in Li:2015uea () so that , and are dimensionless partners of the dimensionful , and , respectively. After restoring the dimension and -dependence, we have the dimensional quantities, i.e.,

(8) |

In this framework, the resultant EoS depends on the choice of the -brane instanton configuration and the form of the since there is no consensus form of non-Abelian DBI action. In this work, we choose the so-called point-like -brane instanton configuration proposed by Bergman:2007wp (), for which in (6) and

(9) |

The first term is the DBI action without instanton, and the second term is the source Lagrangian of D4-brane instanton located at . The point-like instanton configuration will not restore chiral symmetry at high baryon density Bergman:2007wp (); Li:2015uea (), but should capture the feature of nuclear matter inside neutron stars. In Ghoroku:2012am (); Li:2015uea () more general instanton configurations and DBI actions are proposed in order to recover the chiral symmetry restoration through a hadron-quark phase transition at high enough baryon density. These configurations could be relevant when holographically considering the twin stars Gerlach:1968zz (); Kampfer:1981yr (); Heiselberg:1992dx (); Oestgaard:1994gy (); Schertler:1999xn (); Steiner:2000bi (); Ranea-Sandoval:2015ldr (); Gomes:2018bpw (); Montana:2018bkb (), hybrid star of mixing both hadrons and quarks.

## Iii Equation of state

To extract the EoS of the holographic nuclear matter, we need to numerically solve the integral equations (3) and (7). Using the solutions we can evaluate , and as the functions of and , from which we then numerically fit the EoS by the piecewise (multi)-polytropic form. We find that for small value of , i.e., , the EoS can be fitted well by a doubly-polytropic function

(10) |

where and depend only on the temperature . It turns out that this regime of is the one for producing reasonable features of neutron stars, such as mass, core pressure and energy density. On the other hand, for the higher , the EoS can be fitted by piecewise single-polytopic form: with and .

Based on our numerical results, we find that the values of and in (10) are

(11) |

for , and

(12) |

for , i.e., for . Note that this temperature is considerably high from typical astrophysical point of view for a neutron star, however, we see that its effect to EoS is quite small. Thus, we will neglect the temperature effect in the following discussions. Moreover, the sound speed squared derived from our EoS (10) and (11) satisfies the causality constraint, i.e., and also break the sound barrier, i.e., . Thus, our EoS is stiff enough to support more massive neutron stars. This is in contrast to the holographic neutron star model based on D3/D7-branes proposed in Hoyos:2016zke (); Hoyos:2016cob (); Annala:2017tqz (), where they need the additional inputs outside their model to break the sound barrier.

We can also fit the relation between and , and the result for in the regime is

(13) |

If we adopt (4) with , then from (8) it gives where is the saturation density of nuclei. The baryon density inside a typical neutron star is about to Lattimer:2004pg (). Thus, this requires .

As we will use EoS to solve the Tolman-Oppenheimer-Volkoff (TOV) equations to get the mass-radius relation and furthermore to calculate the tidal deformability of neutron stars, it is better to write the EoS again in terms of dimensionless pressure and energy density which are however with respect to astrophysical units: , and where is the Newton constant and is the solar mass. Thus, we can turn (10) into

(14) |

with . Note that (14) is a one-parameter family of EoS parameterized by . In the following we will adopt (4) with so that , which is equivalently to be parametrized by .

Next, we need to estimate the order of magnitude of (or ) if adopt (4) with to fix the EoS for the use of solving TOV equations and the tidal deformability. The typical core pressure of neutron star is about to . Note that . Since is about , thus we should have about for the core pressure to be . Moreover, from (8) we have . Combing with the fact that (or ), we have so that the requirement for the neutron star’s baryon is also satisfied. Indeed, we will see that our EoS (14) with do yield the reasonable mass-radius relation and the tidal deformability fitted well with the observational data of GW170817 TheLIGOScientific:2017qsa (); Abbott:2018wiz ().

## Iv The holographic stars

Based on EoS (14), we solve TOV equations for different values of with the prescribed order of magnitude around , and then yield the mass-radius ( vs ) relation of the holographic stars, etc. In Fig. 1 we show (a) the mass-radius relation and (b) the relation between the core pressure and the mass ( vs ), for ten values of equally ranging from to , which are labelled from to , respectively. We see that the maximal mass can reach more than for . In (a) of Fig. 1 the lowest maximal mass is about for , and we expect this value will be lower if one further increases . We choose because it is still larger than the upper bound shown in the data analysis of GW170817 TheLIGOScientific:2017qsa (); Abbott:2018wiz (). Moreover, we can also infer that the compactness increase as increases. In (b) of Fig. 1 we see that the core pressure is about as reasoning above, and it also implies that the baryon density should be around a few .

We see that our EoS satisfies the causality constraint, breaks the sound barrier, and can be stiff enough by tuning to support the star with mass larger than than as required by astrophysical observations Lattimer:2004pg (). However, we can constrain further by also evaluating the tidal deformability and compare with the inference values from observation data of GW170817 TheLIGOScientific:2017qsa (); Abbott:2018wiz (). The tidal deformability characterizes how the shape of the star is deformed by the external gravitational field, and is defined as the dimensionless coefficient in the following linear response relation

(15) |

where is the mass of the star, is the induced quadrupole moment, and is the external gravitational tidal field strength. Given the EoS and a neutron star configuration, we can follow the perturbative method of Hinderer:2007mb () to calculate the tidal deformability. For our EoS (14) and the star configurations shown in Fig. 1, the relation of tidal deformability and mass ( vs ) is shown in Fig. 2. We see that increases as decreases, this implies that it is easier to deform for less compact star as intuitively expected.

We see from Fig. 2 that covers a very large range. However, the data analysis of GW170817 TheLIGOScientific:2017qsa (); Abbott:2018wiz () shows that the tidal deformability is moderately constrained. As GW170817 is a system of binary neutron stars, the data is fitted for the following combined quantity

(16) |

where are the masses of two neutron stars with , and are their associated tidal deformabilities. The analysis of GW170817 in TheLIGOScientific:2017qsa (); Abbott:2018wiz () yields an estimate on . This will serve as a further constraint on of our EoS.

To obtain for our EoS and star configurations, we first fit the curve of vs for the low-spin prior in Fig. 5 of Abbott:2018wiz () with , and then plug into (16) to get vs . The result is shown in Fig. 3. We see that the ’s in Fig. 2 are too large for lower so that we present only the ones closer to the estimate of TheLIGOScientific:2017qsa (); Abbott:2018wiz (). From Fig. 3 we see that only the ones , and are consistent with the observation. However, for these two cases or the ones of higher where our predicted radii shown in Fig. 1 are also consistent with the inferred values of LIGO/Virgo observation of GW170817, i.e., and for Abbott:2018exr (), the maximal mass is less than , which is less than .

## V Conclusion

From our study, we see that the tidal deformability gives a more stringent constraint on our 1-parameter family of EoS, which requires should be larger than . This however rule out the possibility of having stars. Despite that, our EoS enjoy all the other nice features of a neutron star’s EoS, i.e., simple polytropic, satisfying the causality constraints, breaking sound barrier, and the most importantly, deriving almost from first principle for strongly interacting QCD and in excellent agreement with observation of GW170817. It is reasonable to conjecture that the more massive stars are hybrid twin stars Gerlach:1968zz (); Kampfer:1981yr (); Heiselberg:1992dx (); Oestgaard:1994gy (); Schertler:1999xn (); Steiner:2000bi (); Ranea-Sandoval:2015ldr (); Gomes:2018bpw (); Montana:2018bkb () with a more dense quark star core surrounded by the shell of holographic nuclear matter dictated by our EoS. One can generate the EoS for the holographic dense quark matter by a more general D4-brane instanton configurations, such as the instanton gas ones proposed in Ghoroku:2012am (); Li:2015uea (). We are currently working on this and will report in the near future.

FLL is supported by Taiwan Ministry of Science and Technology (MoST) through Grant No. 103-2112-M-003-001-MY3. LWL is supported by Academia Sinica Career Development Award Program through Grant No. AS-CDA-105-M06. KZ(Hong Zhang) thanks Yutaka Matsuo for useful advice and is supported by MoST through Grant No. 107-2811-M-003-511. We thank Alessandro Parisi, Meng-Ru Wu for helpful discussions. We also thank NCTS for partial financial support.

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