QCD Matter in Neutron Star Environments

November 9, 2017 @ NTU

QCD Matter in


Neutron Star Environments

Kenji Fukushima

The University of Tokyo

Workshop of Recent Developments in QCD and QFT

1

Neutron Star (NS) Constraint(s)

November 9, 2017 @ NTU

Neutron Star Constraint

3

common feature of models that include the appearance of ‘exotic’

hadronic matter such as hyperons^4,5or kaon condensates³at densities of a few times the nuclear saturation density (ns), for example models GS1 and GM3 in Fig. 3. Almost all such EOSs are ruled out by our results. Our mass measurement does not rule out condensed quark matter as a component of the neutron star interior^6,21, but it strongly constrains quark matter model parameters¹². For the range of allowed EOS lines presented in Fig. 3, typical values for the physical parameters of J1614-2230 are a central baryon density of between 2nsand 5nsand a radius of between 11 and 15 km, which is only 2–3 times the Schwarzschild radius for a 1.97M_[ star. It has been proposed that the Tolman VII EOS-independent analytic solution of Einstein’s equations marks an upper limit on the ultimate density of observable cold matter²². If this argument is correct, it follows that our mass mea- surement sets an upper limit on this maximum density of (3.74 6 0.15) 3 10¹⁵g cm²³, or ,10ns.

Evolutionary models resulting in companion masses .0.4M_[gen- erally predict that the neutron star accretes only a few hundredths of a solar mass of material, and result in a mildly recycled pulsar²³, that is one with a spin period .8 ms. A few models resulting in orbital para- meters similar to those of J1614-2230^23,24predict that the neutron star could accrete up to 0.2M_[, which is still significantly less than the

>0.6M[needed to bring a neutron star formed at 1.4M_[ up to the observed mass of J1614-2230. A possible explanation is that some neutron stars are formed massive (,1.9M_[). Alternatively, the trans- fer of mass from the companion may be more efficient than current models predict. This suggests that systems with shorter initial orbital periods and lower companion masses—those that produce the vast majority of the fully recycled millisecond pulsar population²³—may experience even greater amounts of mass transfer. In either case, our mass measurement for J1614-2230 suggests that many other milli- second pulsars may also have masses much greater than 1.4M_[.

Received 7 July; accepted 1 September 2010.

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The´or. 44, 263–292 (1986).

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(in the press).

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58, 661–710 (1985).

17. O¨zel, F. Soft equations of state for neutron-star matter ruled out by EXO 0748 - 676. Nature 441, 1115–1117 (2006).

18. Ransom, S. M. et al. Twenty-one millisecond pulsars in Terzan 5 using the Green Bank Telescope. Science 307, 892–896 (2005).

19. Freire, P. C. C. et al. Eight new millisecond pulsars in NGC 6440 and NGC 6441.

Astrophys. J. 675, 670–682 (2008).

20. Freire, P. C. C., Wolszczan, A., van den Berg, M. & Hessels, J. W. T. A massive neutron star in the globular cluster M5. Astrophys. J. 679, 1433–1442 (2008).

21. Alford,M.etal.Astrophysics:quarkmatterincompactstars?Nature445,E7–E8(2007).

22. Lattimer, J. M. & Prakash, M. Ultimate energy density of observable cold baryonic matter. Phys. Rev. Lett. 94, 111101 (2005).

23. Podsiadlowski, P., Rappaport, S. & Pfahl, E. D. Evolutionary sequences for low- and intermediate-mass X-ray binaries. Astrophys. J. 565, 1107–1133 (2002).

24. Podsiadlowski, P. & Rappaport, S. Cygnus X-2: the descendant of an intermediate- mass X-Ray binary. Astrophys. J. 529, 946–951 (2000).

25. Hotan, A. W., van Straten, W. & Manchester, R. N. PSRCHIVE and PSRFITS: an open approach to radio pulsar data storage and analysis. Publ. Astron. Soc. Aust. 21, 302–309 (2004).

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0207156æ (2002).

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Supplementary Information is linked to the online version of the paper at www.nature.com/nature.

Acknowledgements P.B.D. is a Jansky Fellow of the National Radio Astronomy Observatory. J.W.T.H. is a Veni Fellow of The Netherlands Organisation for Scientific Research. We thank J. Lattimer for providing the EOS data plotted in Fig. 3, and P. Freire, F. O¨zel and D. Psaltis for discussions. The National Radio Astronomy Observatory is a facility of the US National Science Foundation, operated under cooperative agreement by Associated Universities, Inc.

Author Contributions All authors contributed to collecting data, discussed the results and edited the manuscript. In addition, P.B.D. developed the MCMC code, reduced and analysed data, and wrote the manuscript. T.P. wrote the observing proposal and created Fig. 3. J.W.T.H. originally discovered the pulsar. M.S.E.R. initiated the survey that found the pulsar. S.M.R. initiated the high-precision timing proposal.

Author Information Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests.

Readers are welcome to comment on the online version of this article at www.nature.com/nature. Correspondence and requests for materials should be addressed to P.B.D. (pdemores@nrao.edu).

0.07 8 9 10 11

Radius (km)

12 13 14 15

0.5 1.0 1.5 2.0

AP4

J1903+0327 J1909-3744

systems Double neutron s

Double neutron star sysy

J1614-2230

AP3 ENG

MPA1

GM3 GS1

PAL6

FSU SQM3 SQM1

PAL1

MS0

MS2 MS1

2.5 GR

Causality

Rotation P < ∞

Mass (M()

Figure 3|Neutron star mass–radius diagram. The plot shows non-rotating mass versus physical radius for several typical EOSs²⁷: blue, nucleons; pink, nucleons plus exotic matter; green, strange quark matter. The horizontal bands show the observational constraint from our J1614-2230 mass measurement of (1.97 6 0.04)M_[, similar measurements for two other millisecond pulsars^8,28 and the range of observed masses for double neutron star binaries². Any EOS line that does not intersect the J1614-2230 band is ruled out by this

measurement. In particular, most EOS curves involving exotic matter, such as kaon condensates or hyperons, tend to predict maximum masses well below 2.0M_[and are therefore ruled out. Including the effect of neutron star rotation increases the maximum possible mass for each EOS. For a 3.15-ms spin period, this is a=2% correction²⁹and does not significantly alter our conclusions. The grey regions show parameter space that is ruled out by other theoretical or observational constraints². GR, general relativity; P, spin period.

LETTER RESEARCH

2 8 O C T O B E R 2 0 1 0 | V O L 4 6 7 | N A T U R E | 1 0 8 3 Macmillan Publishers Limited. All rights reserved

©2010 parameters, with MCMC error estimates, are given in Table 1. Owing to the high significance of this detection, our MCMC procedure and a standard x²fit produce similar uncertainties.

From the detected Shapiro delay, we measure a companion mass of (0.500 60.006)M_[, which implies that the companion is a helium–

carbon–oxygen white dwarf¹⁶. The Shapiro delay also shows the binary

system to be remarkably edge-on, with an inclination of 89.17u 6 0.02u.

This is the most inclined pulsar binary system known at present. The amplitude and sharpness of the Shapiro delay increase rapidly with increasing binary inclination and the overall scaling of the signal is linearly proportional to the mass of the companion star. Thus, the unique combination of the high orbital inclination and massive white dwarf companion in J1614-2230 cause a Shapiro delay amplitude orders of magnitude larger than for most other millisecond pulsars.

In addition, the excellent timing precision achievable from the pulsar with the GBT and GUPPI provide a very high signal-to-noise ratio measurement of both Shapiro delay parameters within a single orbit.

The standard Keplerian orbital parameters, combined with the known companion mass and orbital inclination, fully describe the dynamics of a

‘clean’ binary system—one comprising two stable compact objects—

under general relativity and therefore also determine the pulsar’s mass.

We measure a pulsar mass of (1.97 6 0.04)M_[, which is by far the high- est precisely measured neutron star mass determined to date. In contrast with X-ray-based mass/radius measurements¹⁷, the Shapiro delay pro- vides no information about the neutron star’s radius. However, unlike the X-ray methods, our result is nearly model independent, as it depends only on general relativity being an adequate description of gravity.

In addition, unlike statistical pulsar mass determinations based on measurement of the advance of periastron^18–20, pure Shapiro delay mass measurements involve no assumptions about classical contributions to periastron advance or the distribution of orbital inclinations.

The mass measurement alone of a 1.97M_[neutron star signifi- cantly constrains the nuclear matter equation of state (EOS), as shown in Fig. 3. Any proposed EOS whose mass–radius track does not inter- sect the J1614-2230 mass line is ruled out by this measurement. The EOSs that produce the lowest maximum masses tend to be those which predict significant softening past a certain central density. This is a a

b

c –40 –30 –20 –10 0 10 20 30

–40 –30 –20 –10 0 10 20 30

Timing residual (μs)

–40 –30 –20 –10 0 10 20 30

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Orbital phase (turns)

Figure 1|Shapiro delay measurement for PSR J1614-2230. Timing residual—the excess delay not accounted for by the timing model—as a function of the pulsar’s orbital phase. a, Full magnitude of the Shapiro delay when all other model parameters are fixed at their best-fit values.

The solid line shows the functional form of the Shapiro delay, and the red points are the 1,752 timing measurements in our GBT–GUPPI data set.

The diagrams inset in this panel show top-down schematics of the binary system at orbital phases of 0.25, 0.5 and 0.75 turns (from left to right). The neutron star is shown in red, the white dwarf companion in blue and the emitted radio beam, pointing towards Earth, in yellow. At orbital phase of 0.25 turns, the Earth–pulsar line of sight passes nearest to the companion (,240,000 km), producing the sharp peak in pulse delay. We found no evidence for any kind of pulse intensity variations, as from an eclipse, near conjunction.

b, Best-fit residuals obtained using an orbital model that does not account for general-relativistic effects.

In this case, some of the Shapiro delay signal is absorbed by covariant non-relativistic model parameters. That these residuals deviate significantly from a random, Gaussian distribution of zero mean shows that the Shapiro delay must be included to model the pulse arrival times properly, especially at conjunction. In addition to the red GBT–GUPPI points, the 454 grey points show the previous ‘long-term’ data set. The drastic improvement in data quality is apparent. c, Post-fit residuals for the fully relativistic timing model (including Shapiro delay), which have a root mean squared residual of 1.1 ms and a reduced x²value of 1.4 with 2,165 degrees of freedom. Error bars, 1s.

89.1 89.12 89.14 89.16 89.18 89.2 89.22 89.24

a b

0.48 0.49 0.5 0.51 0.52

Inclination angle, i (°)

Companion mass, M₂ (M₍)

1.8 1.85 1.9 1.95 2 2.05 2.1 2.15

Probability density

Pulsar mass (M₍)

Figure 2|Results of the MCMC error analysis. a, Grey-scale image shows the two-dimensional posterior probability density function (PDF) in the M2–i plane, computed from a histogram of MCMC trial values. The ellipses show 1s and 3s contours based on a Gaussian approximation to the MCMC results.

b, PDF for pulsar mass derived from the MCMC trials. The vertical lines show the 1s and 3s limits on the pulsar mass. In both cases, the results are very well described by normal distributions owing to the extremely high signal-to-noise ratio of our Shapiro delay detection. Unlike secular orbital effects (for example precession of periastron), the Shapiro delay does not accumulate over time, so the measurement uncertainty scales simply as T^21/2, where T is the total observing time. Therefore, we are unlikely to see a significant improvement on these results with currently available telescopes and instrumentation.

RESEARCH LETTER

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Macmillan Publishers Limited. All rights reserved

©2010

Demorest et al. (2010) Precise determination of


NS mass using Shapiro delay 1.928(17) M

sun

(slightly changed in 2016)

(J1614-2230)

2.01(4) M

sun

(PSRJ0348+0432)

Antoniadis et al. (2013)

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Neutron Star Constraint

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LETTER

doi:10.1038/nature09466

A two-solar-mass neutron star measured using Shapiro delay

P. B. Demorest¹, T. Pennucci², S. M. Ransom¹, M. S. E. Roberts³ & J. W. T. Hessels^4,5

Neutron stars are composed of the densest form of matter known to exist in our Universe, the composition and properties of which are still theoretically uncertain. Measurements of the masses or radii of these objects can strongly constrain the neutron star matter equation of state and rule out theoretical models of their composi- tion^1,2. The observed range of neutron star masses, however, has hitherto been too narrow to rule out many predictions of ‘exotic’

non-nucleonic components^3–6. The Shapiro delay is a general-relat- ivistic increase in light travel time through the curved space-time near a massive body⁷. For highly inclined (nearly edge-on) binary millisecond radio pulsar systems, this effect allows us to infer the masses of both the neutron star and its binary companion to high precision^8,9. Here we present radio timing observations of the binary millisecond pulsar J1614-2230^10,11 that show a strong Shapiro delay signature. We calculate the pulsar mass to be (1.97 6 0.04)M_[, which rules out almost all currently proposed^2–5 hyperon or boson con- densate equations of state (M_[, solar mass). Quark matter can sup- port a star this massive only if the quarks are strongly interacting and are therefore not ‘free’ quarks¹².

In March 2010, we performed a dense set of observations of J1614- 2230 with the National Radio Astronomy Observatory Green Bank Telescope (GBT), timed to follow the system through one complete 8.7-d orbit with special attention paid to the orbital conjunction, where the Shapiro delay signal is strongest. These data were taken with the newly built Green Bank Ultimate Pulsar Processing Instrument (GUPPI).

GUPPI coherently removes interstellar dispersive smearing from the pulsar signal and integrates the data modulo the current apparent pulse period, producing a set of average pulse profiles, or flux-versus-rota- tional-phase light curves. From these, we determined pulse times of arrival using standard procedures, with a typical uncertainty of ,1 ms.

We used the measured arrival times to determine key physical para- meters of the neutron star and its binary system by fitting them to a comprehensive timing model that accounts for every rotation of the neutron star over the time spanned by the fit. The model predicts at what times pulses should arrive at Earth, taking into account pulsar rotation and spin-down, astrometric terms (sky position and proper motion), binary orbital parameters, time-variable interstellar disper- sion and general-relativistic effects such as the Shapiro delay (Table 1).

We compared the observed arrival times with the model predictions, and obtained best-fit parameters by x² minimization, using the TEMPO2 software package¹³. We also obtained consistent results using the original TEMPO package. The post-fit residuals, that is, the differences between the observed and the model-predicted pulse arrival times, effectively measure how well the timing model describes the data, and are shown in Fig. 1. We included both a previously recorded long-term data set and our new GUPPI data in a single fit.

The long-term data determine model parameters (for example spin- down rate and astrometry) with characteristic timescales longer than a few weeks, whereas the new data best constrain parameters on timescales of the orbital period or less. Additional discussion of the

long-term data set, parameter covariance and dispersion measure vari- ation can be found in Supplementary Information.

As shown in Fig. 1, the Shapiro delay was detected in our data with extremely high significance, and must be included to model the arrival times of the radio pulses correctly. However, estimating parameter values and uncertainties can be difficult owing to the high covariance between many orbital timing model terms¹⁴. Furthermore, the x²surfaces for the Shapiro-derived companion mass (M₂) and inclination angle (i) are often significantly curved or otherwise non-Gaussian¹⁵. To obtain robust error estimates, we used a Markov chain Monte Carlo (MCMC) approach to explore the post-fit x²space and derive posterior probability distributions for all timing model parameters (Fig. 2). Our final results for the model

1National Radio Astronomy Observatory, 520 Edgemont Road, Charlottesville, Virginia 22093, USA.²Astronomy Department, University of Virginia, Charlottesville, Virginia 22094-4325, USA.³Eureka Scientific, Inc., Oakland, California 94602, USA.⁴Netherlands Institute for Radio Astronomy (ASTRON), Postbus 2, 7990 AA Dwingeloo, The Netherlands.⁵Astronomical Institute ‘‘Anton Pannekoek’’, University of Amsterdam, 1098 SJ Amsterdam, The Netherlands.

Table 1 | Physical parameters for PSR J1614-2230

Parameter Value

Ecliptic longitude (l) 245.78827556(5)u

Ecliptic latitude (b) 21.256744(2)u

Proper motion in l 9.79(7) mas yr²¹

Proper motion in b 230(3) mas yr²¹

Parallax 0.5(6) mas

Pulsar spin period 3.1508076534271(6) ms

Period derivative 9.6216(9) 3 10²²¹s s²¹

Reference epoch (MJD) 53,600

Dispersion measure* 34.4865 pc cm²³

Orbital period 8.6866194196(2) d

Projected semimajor axis 11.2911975(2) light s First Laplace parameter (esin v) 1.1(3) 3 10²⁷ Second Laplace parameter (ecos v) 21.29(3) 3 10²⁶

Companion mass 0.500(6)M_[

Sine of inclination angle 0.999894(5)

Epoch of ascending node (MJD) 52,331.1701098(3)

Span of timing data (MJD) 52,469–55,330

Number of TOAs{ 2,206 (454, 1,752)

Root mean squared TOA residual 1.1 ms

Right ascension (J2000) 16 h 14 min 36.5051(5) s

Declination (J2000) 222u 309 31.081(7)99

Orbital eccentricity (e) 1.30(4) 3 10²⁶

Inclination angle 89.17(2)u

Pulsar mass 1.97(4)M_[

Dispersion-derived distance{ 1.2 kpc

Parallax distance .0.9 kpc

Surface magnetic field 1.8 3 10⁸G

Characteristic age 5.2 Gyr

Spin-down luminosity 1.2 3 10³⁴erg s²¹

Average flux density* at 1.4 GHz 1.2 mJy

Spectral index, 1.1–1.9 GHz 21.9(1)

Rotation measure 228.0(3) rad m²²

Timing model parameters (top), quantities derived from timing model parameter values (middle) and radio spectral and interstellar medium properties (bottom). Values in parentheses represent the 1s uncertainty in the final digit, as determined by MCMC error analysis. The fit included both ‘long-term’ data spanning seven years and new GBT–GUPPI data spanning three months. The new data were observed using an 800-MHz-wide band centred at a radio frequency of 1.5 GHz. The raw profiles were polarization- and flux-calibrated and averaged into 100-MHz, 7.5-min intervals using the PSRCHIVE software

package²⁵, from which pulse times of arrival (TOAs) were determined. MJD, modified Julian date.

* These quantities vary stochastically on>1-d timescales. Values presented here are the averages for our GUPPI data set.

{ Shown in parentheses are separate values for the long-term (first) and new (second) data sets.

{ Calculated using the NE2001 pulsar distance model²⁶.

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Neutron Star Constraint

5

Equation of State (unknown)

M-R Relation (observed) Pressure : p

Mass density : r p = p(⇢)

NS mass : M NS radius : R

Tolman-Oppenheimer-
 -Volkoff (TOV) Eqs

Mathematically one-to-one correspondence

gravity pressure diff

(Energy density : " = ⇢c

²

)

M = M (⇢ _max )

R = R(⇢ _max )

Neutron Star Constraint

Old Picture

µ

P Nuclear EoS

Quark EoS

r P

Soft Stiff

It is hard to see if there is a 1st-order transition or not
 from the M-R relation, but
 a flat behavior can be


reconstructed mathematically

November 9, 2017 @ NTU

Neutron Star Constraint

7

Lindblom (1992)

Some simple test cases : useful for a 1st-order transition?

Test data set by hand

Solve TOV

Reconstructed

Yes, it is useful, in principle

Thanks to Y. Fujimoto

Neutron Star Constraint

IF there is a 1st-order phase transition with large density gap
 (i.e. strong 1st-order) at small densities,

EoS cannot be stiff enough to support massive NS

Remember: the slope is bounded by causality, and cannot
 exceed the speed of light.

Strong 1st-order transition excluded, which means…

November 9, 2017 @ NTU

Neutron Star Constraint

9

Alford et al. (2015)

Constraining and applying a generic high-density equation of state

Mark G. Alford¹, G. F. Burgio², S. Han (È )¹, G. Taranto^2,3, and D. Zappalà²

1Physics Department, Washington University, Saint Louis, Missouri 63130, USA

2 INFN Sezione di Catania, Via Santa Sofia 64, 95123 Catania, Italy and

3Dipartimento di Fisica e Astronomia, Universitá di Catania, Via Santa Sofia 64, 95123 Catania, Italy (Dated: 1 Oct 2015)

We discuss the “constant speed of sound” (CSS) parametrization of the equation of state of high-density mat- ter and its application to the field correlator method (FCM) model of quark matter. We show how observational constraints on the maximum mass and typical radius of neutron stars are expressed as constraints on the CSS parameters. We find that the observation of a 2M star already severely constrains the CSS parameters, and is particularly difficult to accommodate if the squared speed of sound in the high-density phase is assumed to be around 1/3 or less.

We show that the FCM equation of state can be accurately represented by the CSS parametrization, which assumes a sharp transition to a high-density phase with density-independent speed of sound. We display the mapping between the FCM and CSS parameters, and see that FCM only allows equations of state in a restricted subspace of the CSS parameters.

PACS numbers: 25.75.Nq, 26.60.-c, 97.60.Jd

I. INTRODUCTION

There are many models of matter at density significantly above nuclear saturation density, each with their own param- eters. In studying the equation of state (EoS) of matter in this regime it is therefore useful to have a general parametrization of the EoS which can be used as a generic language for relat- ing different models to each other and for expressing experi- mental constraints in model-independent terms. In this work we use the previously proposed “constant speed of sound”

(CSS) parametrization [1–3] (for applications, see, e.g., [4]).

We show how mass and radius observations can be expressed as constraints on the CSS parameters. Here we analyze a spe- cific example, where the high-density matter is quark matter described by a model based on the field correlator method (Sec. IV), showing how its parameters can be mapped on to the CSS parameter space, and how it is constrained by cur- rently available observations of neutron stars.

The CSS parametrization is applicable to high-density equations of state for which (a) there is a sharp interface be- tween nuclear matter and a high-density phase which we will call quark matter, even when (as in Sec. II) we do not make any assumptions about its physical nature; and (b) the speed of sound in the high-density matter is pressure-independent for pressures ranging from the first-order transition pressure up to the maximum central pressure of neutron stars. One can then write the high-density EoS in terms of three parameters:

the pressure ptrans of the transition, the discontinuity in energy density De at the transition, and the speed of sound c_QM in the high-density phase. For a given nuclear matter EoS e_NM(p), the full CSS EoS is then

e(p) = ⇢ e_NM(p) p < p_trans e_NM(p_trans) +De + c_QM² (p p_trans) p > p_trans The CSS form can be viewed as the lowest-order terms of(1) a Taylor expansion of the high-density EoS about the tran- sition pressure. Following Ref. [1], we express the three

parameters in dimensionless form, as p_trans/etrans, De/etrans

(equal to l 1 in the notation of Ref. [5]) and c²_QM, where e_trans ⌘ eNM(p_trans).

The assumption of a sharp interface will be valid if, for ex- ample, there is a first-order phase transition between nuclear and quark matter, and the surface tension of the interface is high enough to ensure that the transition occurs at a sharp in- terface (Maxwell construction) not via a mixed phase (Gibbs construction). Given the uncertainties in the value of the sur- face tension [6–8], this is a possible scenario. One can also formulate generic equations of state that model interfaces that are smeared out by mixing or percolation [9–11].

The assumption of a density-independent speed of sound is valid for a large class of models of quark matter. The CSS parametrization is an almost exact fit to some Nambu–Jona- Lasinio models [2, 12–14]. The perturbative quark matter EoS [15] also has roughly density-independent c²_QM, with a value around 0.2 to 0.3 (we use units where ¯h = c = 1), above the transition from nuclear matter (see Fig. 9 of Ref. [16]).

In the quartic polynomial parametrization [17], varying the coefficient a₂ between ±(150MeV)², and the coefficient a₄ between 0.6 and 1, and keeping n_trans/n₀ above 1.5 (n₀ ⌘ 0.16fm ³ is the nuclear saturation density), one finds that c²_QM is always between 0.3 and 0.36. It is noticeable that mod- els based on relativistic quarks tend to have c²_QM ⇡ close to 1/3, which is the value for systems with conformal symmetry, and it has been conjectured that there is a fundamental bound c²_QM < 1/3 [18], although some models violate that bound, e.g. [19, 20] or [14] (parametrized in [2]).

In Sec. II we show how the CSS parametrization is con- strained by observables such as the maximum mass M_max, the radius of a maximum-mass star, and the radius R_1.4 of a star of mass 1.4M . In Secs. III–IV we describe a specific model, based on a Brueckner-Hartree-Fock (BHF) calculation of the nuclear matter EoS and the field correlator method (FCM) for the quark matter EoS. We show how the parameters of this model map on to part of the CSS parameter space, and how the observational constraints apply to the FCM model param-

arXiv:1501.07902v3 [nucl-th] 29 Sep 2015

Parameters (choices) : Nuclear EoS, c

³ QM

, De, p

^trans

FIG. 2: (Color online). Contour plots showing the maximum hybrid star mass as a function of the CSS parameters of the high-density EoS.

Each panel shows the dependence on the CSS parameters ptrans/etransandDe/etrans. The left plots are for c²_QM=1/3, and the right plots are for c²_QM=1. The top row is for a DHBF (stiff) nuclear matter EoS, and the bottom row is for a BHF (soft) nuclear matter EoS. The grey shaded region is excluded by the measurement of a 2M star. The hatched band at low density (where ntrans<n0) is excluded because bulk nuclear matter would be metastable. The hatched band at high density is excluded because the transition pressure is above the central pressure of the heaviest stable hadronic star.

lows a wider range of CSS parameters to be compatible with the 2M measurement.

In Fig. 2 the dot-dashed (red) contours are for hybrid stars on a connected branch, while the dashed (blue) contours are for disconnected branches. As discussed in Ref. [1], when crossing the near-horizontal boundary from region C to B the connected hybrid branch splits into a smaller connected branch and a disconnected branch, so the maximum mass of the connected branch smoothly becomes the maximum mass of the disconnected branch. Therefore the red contour in the C region smoothly becomes a blue contour in the B and D regions. When crossing the near-vertical boundary from re- gion C to B a new disconnected branch forms, so the con- nected branch (red dot-dashed) contour crosses this boundary

smoothly.

In each panel of Fig. 2, the physically relevant allowed re- gion is the white unshaded region. The grey shaded region is excluded by the existence of a 2M star. We see that increas- ing the stiffness of the hadronic EoS or of the quark matter EoS (by increasing c²_QM) shrinks the excluded region.

For both the hadronic EoSs that we study, the CSS param- eters are significantly constrained. From the two left panels of Fig. 2 one can see that if, as predicted by many models, c²_QM. 1/3, then we are limited to two regions of parame- ter space, corresponding to a lowpressure transition or a high pressure transition. In the low-transition-pressure region the transition occurs at a fairly low density ntrans. 2n0, and a connected hybrid branch is possible. In the high-transition-

Okay if…

QM only at very high density 1st trans. at very high density 1st trans. very weak

NM EoS very stiff etc, etc

Looks generic, but


a bit misleading to say…

Neutron Star Constraint

Caveats

Based on the old picture of 1st-order transition to QM

Is there any reason to require 1st-order transition? NO!

Based on the extrapolation of NM EoS to high densities

0 1 2 3 4 5 6

0 0.1 0.2 0.3 0.4 0.5

Baryon Density [n0]

Pressure [GeV/fm³]

SLy

(APR+crust)

Can it be extrapolatable?

NO!

apart from an infamous
 causality problem…

⇠ 2M

November 9, 2017 @ NTU

Neutron Star Constraint

11

IF nucleons are surrounded by interaction clouds of pions,
 such clouds undergo a classical percolation transition at

1.4 n 0

Percolation transition
 allows for mobility


enhancement of quarks?

Quantum fluctuations
 (Anderson localization) induce “confinement”

(quantum percolation)

(Picture of H. Satz)

Neutron Star Constraint

One may think that the constraint may be strong for light NS BUT…

R is fixed by TOV with p(R)=0 and interestingly…

dp/dr(r = R) = 0

d ² p/dr ² (r = R) / M ² /R ²

If M is small or R is large,
 uncertainty becomes huge.

People do not care assuming that NS mass > 1.2 M

sun

Very uncertain


“by definition”

November 9, 2017 @ NTU

Neutron Star Constraint

13

Here, NS-NS merger will not be discussed, but
 another constraint is already available:

2 deformability between polytropes and “realistic” EOS.

In this paper, we calculate the deformability for realistic EOS, and show that a tidal signature is actually only marginally detectable with Advanced LIGO.

In Sec. II we describe how the Love number and tidal deformability can be calculated for tabulated EOS. We use the equations for k

₂

developed in [15], which arise from a linear perturbation of the Oppenheimer-Volko↵

(OV) equations of hydrostatic equilibrium. In Sec. III we then calculate k

₂

and as a function of mass for several EOS commonly found in the literature. We find that, in contrast to the Love number, the tidal deformability has a wide range of values, spanning roughly an order of magnitude over the observed mass range of neutron stars in binary systems.

As discussed above, the direct practical importance of the stars’ tidal deformability for gravitational wave ob- servations of NS binary inspirals is that it encodes the EOS influence on the waveform’s phase evolution during the early portion of the signal, where it is accurately mod- eled by post-Newtonian (PN) methods. In this regime, the influence of tidal e↵ects is only a small correction to the point-mass dynamics. However, when the signal is integrated against a theoretical waveform template over many cycles, even a small contribution to the phase evo- lution can be detected and could give information about the NS structure.

Following [11], we calculate in Sec. IV the measurabil- ity of the tidal deformability for a wide range of equal- and unequal- mass binaries, covering the entire expected range of NS masses and EOS, and with proposed detector sensitivity curves for second- and third- generation detec- tors. We show that the measurability of is quite sensi- tive to the total mass of the system, with very low-mass neutron stars contributing significant phase corrections that are optimistically detectable in Advanced LIGO, while larger-mass neutron stars are more difficult to dis- tinguish from the k

₂

= 0 case of black holes [16, 17]. In third-generation detectors, however, the tenfold increase in sensitivity allows a finer discrimination between equa- tions of state leading to potential measurability of a large portion of proposed EOSs over most of the expected neu- tron star mass range.

We conclude by briefly considering how the errors could be improved with a more careful analysis of the detectors and extension of the understanding of EOS ef- fects to higher frequencies.

Finally, in the Appendix we compute the leading or- der EOS-dependent corrections to our model of the tidal e↵ect and derive explicit expressions for the resulting cor- rections to the waveform’s phase evolution, extending the analysis of Ref. [11]. Estimates of the size of the phase corrections show that the main source of error are post- 1 Newtonian corrections to the Newtonian tidal e↵ect itself, which are approximately twice as large as other, EOS-dependent corrections at a frequency of 450 Hz.

Since these point-particle corrections do not depend on unknown NS physics, they can easily be incorporated into

the analysis. A derivation of the explicit post-Newtonian correction terms is the subject of Ref. [18].

Conventions: We set G = c = 1.

II. CALCULATION OF THE LOVE NUMBER AND TIDAL DEFORMABILITY

As in [11] and [15], we consider a static, spherically symmetric star, placed in a static external quadrupolar tidal field E

^ij

. To linear order, we define the tidal de- formability relating the star’s induced quadrupole mo- ment Q

_ij

to the external tidal field,

Q

_ij

= E

^ij

. (1)

The coefficient is related to the l = 2 dimensionless tidal Love number k

₂

by

k

₂

= 3

2 R

⁵

. (2)

The star’s quadrupole moment Q

_ij

and the external tidal field E

^ij

are defined to be coefficients in an asymp- totic expansion of the total metric at large distances r from the star. This expansion includes, for the met- ric component g

_tt

in asymptotically Cartesian, mass- centered coordinates, the standard gravitational poten- tial m/r, plus two leading order terms arising from the perturbation, one describing an external tidal field grow- ing with r

²

and one describing the resulting tidal distor- tion decaying with r

³

:

(1 + g

_tt

)

2 = m

r

3Q

_ij

2r

³

n

ⁱ

n

^j

+ . . . + E

^ij

2 r

²

n

ⁱ

n

^j

+ . . . , (3) where n

ⁱ

= x

ⁱ

/r and Q

_ij

and E

ij

are both symmetric and traceless. The relative size of these multipole components of the perturbed spacetime gives the constant relating the quadrupole deformation to the external tidal field as in Eq. (1).

To compute the metric (3), we use the method dis- cussed in [15]. We consider the problem of a linear static perturbation expanded in spherical harmonics following [19]. Without loss of generality we can set the azimuthal number m = 0, as the tidal deformation will be axisym- metric around the line connecting the two stars which we take as the axis for the spherical harmonic decompo- sition. Since we will be interested in applications to the early stage of binary inspirals, we will also specialize to the leading order for tidal e↵ects, l = 2.

Introducing a linear l = 2 perturbation onto the spher- ically symmetric star results in a static (zero-frequency), even-parity perturbation of the metric, which in the Regge-Wheeler gauge [20] can be simplified [15] to give

ds

²

= e

^{2 (r)}

[1 + H(r)Y

₂₀

(✓, ')] dt

²

+e

^2⇤(r)

[1 H(r)Y

₂₀

(✓, ')] dr

²

+r

²

[1 K(r)Y

₂₀

(✓, ')] d✓

²

+ sin

²

✓d'

²

, (4) 2

deformability between polytropes and “realistic” EOS.

In this paper, we calculate the deformability for realistic EOS, and show that a tidal signature is actually only marginally detectable with Advanced LIGO.

In Sec. II we describe how the Love number and tidal deformability can be calculated for tabulated EOS. We use the equations for k

₂

developed in [15], which arise from a linear perturbation of the Oppenheimer-Volko↵

(OV) equations of hydrostatic equilibrium. In Sec. III we then calculate k

₂

and as a function of mass for several EOS commonly found in the literature. We find that, in contrast to the Love number, the tidal deformability has a wide range of values, spanning roughly an order of magnitude over the observed mass range of neutron stars in binary systems.

As discussed above, the direct practical importance of the stars’ tidal deformability for gravitational wave ob- servations of NS binary inspirals is that it encodes the EOS influence on the waveform’s phase evolution during the early portion of the signal, where it is accurately mod- eled by post-Newtonian (PN) methods. In this regime, the influence of tidal e↵ects is only a small correction to the point-mass dynamics. However, when the signal is integrated against a theoretical waveform template over many cycles, even a small contribution to the phase evo- lution can be detected and could give information about the NS structure.

Following [11], we calculate in Sec. IV the measurabil- ity of the tidal deformability for a wide range of equal- and unequal- mass binaries, covering the entire expected range of NS masses and EOS, and with proposed detector sensitivity curves for second- and third- generation detec- tors. We show that the measurability of is quite sensi- tive to the total mass of the system, with very low-mass neutron stars contributing significant phase corrections that are optimistically detectable in Advanced LIGO, while larger-mass neutron stars are more difficult to dis- tinguish from the k

₂

= 0 case of black holes [16, 17]. In third-generation detectors, however, the tenfold increase in sensitivity allows a finer discrimination between equa- tions of state leading to potential measurability of a large portion of proposed EOSs over most of the expected neu- tron star mass range.

We conclude by briefly considering how the errors could be improved with a more careful analysis of the detectors and extension of the understanding of EOS ef- fects to higher frequencies.

Finally, in the Appendix we compute the leading or- der EOS-dependent corrections to our model of the tidal e↵ect and derive explicit expressions for the resulting cor- rections to the waveform’s phase evolution, extending the analysis of Ref. [11]. Estimates of the size of the phase corrections show that the main source of error are post- 1 Newtonian corrections to the Newtonian tidal e↵ect itself, which are approximately twice as large as other, EOS-dependent corrections at a frequency of 450 Hz.

Since these point-particle corrections do not depend on unknown NS physics, they can easily be incorporated into

the analysis. A derivation of the explicit post-Newtonian correction terms is the subject of Ref. [18].

Conventions: We set G = c = 1.

II. CALCULATION OF THE LOVE NUMBER

AND TIDAL DEFORMABILITY

As in [11] and [15], we consider a static, spherically symmetric star, placed in a static external quadrupolar tidal field E

^ij

. To linear order, we define the tidal de- formability relating the star’s induced quadrupole mo- ment Q

_ij

to the external tidal field,

Q

_ij

= E

^ij

. (1)

The coefficient is related to the l = 2 dimensionless tidal Love number k

₂

by

k

₂

= 3

2 R

⁵

. (2)

The star’s quadrupole moment Q

_ij

and the external tidal field E

^ij

are defined to be coefficients in an asymp- totic expansion of the total metric at large distances r from the star. This expansion includes, for the met- ric component g

_tt

in asymptotically Cartesian, mass- centered coordinates, the standard gravitational poten- tial m/r, plus two leading order terms arising from the perturbation, one describing an external tidal field grow- ing with r

²

and one describing the resulting tidal distor- tion decaying with r

³

:

(1 + g

_tt

)

2 = m

r

3Q

_ij

2r

³

n

ⁱ

n

^j

+ . . . + E

^ij

2 r

²

n

ⁱ

n

^j

+ . . . , (3) where n

ⁱ

= x

ⁱ

/r and Q

_ij

and E

^ij

are both symmetric and traceless. The relative size of these multipole components of the perturbed spacetime gives the constant relating the quadrupole deformation to the external tidal field as in Eq. (1).

To compute the metric (3), we use the method dis- cussed in [15]. We consider the problem of a linear static perturbation expanded in spherical harmonics following [19]. Without loss of generality we can set the azimuthal number m = 0, as the tidal deformation will be axisym- metric around the line connecting the two stars which we take as the axis for the spherical harmonic decompo- sition. Since we will be interested in applications to the early stage of binary inspirals, we will also specialize to the leading order for tidal e↵ects, l = 2.

Introducing a linear l = 2 perturbation onto the spher- ically symmetric star results in a static (zero-frequency), even-parity perturbation of the metric, which in the Regge-Wheeler gauge [20] can be simplified [15] to give

ds

²

= e

^{2 (r)}

[1 + H(r)Y

₂₀

(✓, ')] dt

²

+e

^2⇤(r)

[1 H(r)Y

₂₀

(✓, ')] dr

²

+r

²

[1 K(r)Y

₂₀

(✓, ')] d✓

²

+ sin

²

✓d'

²

,

(4) (tidal deformability) ⇤(1.4M ) < 800

= quadrupole moment external tidal field

Hinderer et al. (2009)

(~ Love number)

Often divided by M

⁵

to make it dimensionless → L

See: Annala-Gorda-Kurkela-Vuorinen (2017)

CERN-TH-2017-236, HIP-2017-30/TH

Gravitational-wave constraints on the neutron-star-matter Equation of State

Eemeli Annala,¹ Tyler Gorda,¹ Aleksi Kurkela,² and Aleksi Vuorinen¹

1Department of Physics and Helsinki Institute of Physics, P.O. Box 64, FI-00014 University of Helsinki, Finland

2Theoretical Physics Department, CERN, Geneva, Switzerland and Faculty of Science and Technology, University of Stavanger, 4036 Stavanger, Norway The LIGO/Virgo detection of gravitational waves originating from a neutron-star merger, GW170817, has recently provided new stringent limits on the tidal deformabilities of the stars involved in the collision. Combining this measurement with the existence of two-solar-mass stars, we generate the most generic family of neutron-star-matter Equations of State (EoSs) that inter- polate between state-of-the-art theoretical results at low and high baryon density. Comparing to results from similar calculations prior to the tidal deformability constraint, we witness a dramatic reduction in the family of allowed EoSs. Based on our analysis, we conclude that the maximal radius of a 1.4-solar-mass is 13.4 km, and that smallest allowed tidal deformability of a similar mass star is ⇤(1.4M ) = 224.

I. INTRODUCTION

The collective properties of the strongly-interacting dense matter found inside neutron stars (NS) are noto- riously difficult to predict [1, 2]. While the Sign Prob- lem prevents lattice Monte-Carlo simulations at nonzero chemical potentials [3], nuclear-theory tools such as Chi- ral E↵ective Theory (CET) are limited to sub-saturation densities [4] and perturbative QCD (pQCD) becomes re- liable only at much higher densities [5]. No controlled, first-principles calculations are applicable at densities en- countered inside the stellar cores.

Despite the above difficulties, it is possible to obtain robust information on the properties of neutron-star mat- ter at core densities. In particular, the requirement that the Equation of State (EoS) must reach its known low- and high-density limits while behaving in a thermody- namically consistent fashion in between poses a strong constraint on its form. This was demonstrated, e.g., in [6, 7], where a family of EoSs was constructed that in- terpolate between a CET EoS below saturation density and a pQCD result at high densities. This family quan- tifies the purely theoretical uncertainty on the EoS at intermediate densities, but the quantity can be further constrained using observational information about the macroscopic properties of NSs.

The first significant constraint for the EoS comes from the observation of two-solar-mass (2M ) stars [8, 9], im- plying that the corresponding mass-radius curve has to be able to support massive enough stars, Mmax> 2M . This requires that the EoS be sti↵ enough, which in com- bination with the fact that the high-density EoS is rather soft (with c²_s. 1/3, where csis the speed of sound) lim- its the possible behavior of the quantity at intermediate densities. In particular, it was shown in [6, 7] that — upon imposing the 2M constraint — the current un- certainty in the EoS when expressed in the form p(µB), where p is the pressure and µB is the baryon chemical potential, is±40% at worst.

On 16 October 2017, the LIGO and Virgo collabo-

8 9 10 11 12 13 14 15

R [km]

0.5 1 1.5 2 2.5 3

M [Msolar]

Λ<400 Λ<800 Λ>800

M_max<2M_sol

FIG. 1: The mass-radius clouds corresponding to our EoSs.

The cyan area corresponds to EoSs that cannot support a 2M star, while the rest denote EoSs that fulfill this re- quirement and in addition have ⇤(1.4M ) < 400 (green), 400 < ⇤(1.4M ) < 800 (violet), or ⇤(1.4M ) > 800 (red), so that the red region is excluded by the LIGO/Virgo measure- ment at 90% credence. This color coding is used in all of our figures. The dotted black lines denote the result that would have been obtained with bitropic interpolation only.

rations reported the first event, GW170817, where a gravitational-wave (GW) signal was observed from a merger of two compact stars [10]. Remarkably, this very first set of GW data has already o↵ered a second con- straint for the behavior of NS matter. The inspiral phase of a NS-NS merger creates extremely strong tidal grav- itational fields that deform the multipolar structure of the stars, which in turn leaves a detectable imprint on the observed gravitational waveform of the merger. This e↵ect can be quantified in terms of the so-called tidal de- formabilities ⇤ of the stars [11, 12], which measure the degree to which their shape and structure is modified during the inspiral. Assuming a low-spin prior for both stars involved in the merger (for details, see [10]), LIGO

arXiv:1711.02644v1 [astro-ph.HE] 7 Nov 2017

What is Known from Theory ?

November 9, 2017 @ NTU

What is known from theory?

15 Chemical Potential μ

Nuclear Superfluid

B

Almost nothing…

Fukushima-Sasaki (2013)

What is known from theory?

Most important lesson from high-T low-r QCD matter

QCD transition from hadronic to quark-gluon matter
 is a continuous crossover with an overlapping region
 (dual region) of hadrons and of quarks and gluons

HRG Lattice

Quark Matter 2014 (Fukushima)

pQCD

November 9, 2017 @ NTU

What is known from theory?

17

A hint to understand a crossover

N N

¼

Baryon int. at large N

c

⇠ O(N ^c )

Pressure of large-N

c

NM
 scales as ~ O(N

c

) as if
 it were QM.

Quark d.o.f. perceived
 through interactions even
 in baryonic matter

Quarkyonic Matter

McLerran-Pisarski (2007)

NM and QM indistinguishable !?

Hidaka, Kojo, etc…

What is known from theory?

Another hint to understand a crossover

Chiral symmetry more broken at higher density

h¯qqi 6= 0

Nuclear Matter hNNi 6= 0

Quark Matter hq

^R

q

_R

i 6= 0 hq

^L

q

_L

i 6= 0

breaks SU (N

_f

)

_R

breaks SU (N

_f

)

_L

Vectorial rotation can be canceled by color rot.

SU (N

_f

)

_R

⇥ SU(N

^f

)

_L

⇥ U(1)

^V

! SU(N

^f

)

_V

Color superconducting QM has the same symmetry as NM

Schaefer-Wilczek (1998)

NM and QM indistinguishable indeed

November 9, 2017 @ NTU

What is known from theory?

19

All excitations must be continuously connected…

Hyper Nuclear Matter CFL

BEC of colorless H BEC of colored qq BCS

pion

phason

small

qq qq_

_ _qq

qq

q_ _ _qq

qq qqq qqq

qq qq

qq qq qq

none (apart from U^A(1) breaking)

q q

q_

q q

q q

q q µ

Fukushima (2003)

q

q q q q q q

qq q

qq

Alford-Baym-Fukushima-
 -Hatsuda-Tachibana (2017)

What is known from theory?

µ P

NM EoS

(So far best) Bottom-up Approach

QM EoS

Smooth Interpolation Little ambiguity

Well constrained
 but still ambiguous

Masuda, Hatsuda, Takatsuka, Kojo, Baym, …

NJL

QCD