Reduced-order surrogate models for scalar-tensor gravity in the strong fieldregime and applications to binary pulsars and GW170817

Junjie Zhao ,1 Lijing Shao ,2,3,* Zhoujian Cao,4 and Bo-Qiang Ma1,5,6,†1School of Physics and State Key Laboratory of Nuclear Physics and Technology,

Peking University, Beijing 100871, China2Kavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China3Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, D-53121 Bonn, Germany

4Department of Astronomy, Beijing Normal University, Beijing 100875, China5Collaborative Innovation Center of Quantum Matter, Beijing, China

6Center for High Energy Physics, Peking University, Beijing 100871, China

(Received 1 July 2019; published 19 September 2019)

We investigate the scalar-tensor gravity of Damour and Esposito-Farese (DEF), which predicts nontrivialphenomena in the nonperturbative strong-field regime for neutron stars (NSs). Instead of solving themodified Tolman-Oppenheimer-Volkoff equations, we construct reduced-order surrogate models, coded inthe pySTGROM package, to predict the relations of a NS radius, mass, and effective scalar coupling to itscentral density. Our models are accurate at ∼1% level and speed up large-scale calculations by 2 ordersof magnitude. As an application, we use pySTGROM and Markov-chain Monte Carlo techniques to constrainparameters in the DEF theory, with five well-timed binary pulsars, the binary NS (BNS) inspiralGW170817, and a hypothetical BNS inspiral in the Advanced LIGO and next-generation GW detectors.In the future, as more binary pulsars and BNS mergers are detected, our surrogate models will be helpfulin constraining strong-field gravity with essential speed and accuracy.

DOI: 10.1103/PhysRevD.100.064034

I. INTRODUCTION

The theory of general relativity (GR), proposed byAlbert Einstein in 1915, postulates that gravity is mediatedonly by a long-range, spin-2 tensor field, gμν [1]. For morethan a century, tests of GR have never stopped. The bulkof accurate experimental tests from, (i) the Solar System[2], (ii) the high-precision timing of binary pulsars [3–5],and (iii) the gravitational-wave (GW) observations ofcoalescing binary black holes (BBHs) [6–8] and binaryneutron stars (BNSs) [9–11], have all been proven to be inline with GR.However, there are various motivations to look for

theories beyond GR [12]. As one of the most naturalalternatives, the scalar-tensor gravity, in addition to gμν,adds a long-range, spin-0 scalar field, φ. This theory wasconceived originally by Scherrer [13] and Jordan [14].Viewpoints similar to the one in the scalar-tensor gravityhave also been sketched in the Kaluza–Klein theory, thestring theory, and the brane theory [15]. The extra scalardegree of freedom is potentially related to the dark energy,the inflation, and a possible unified theory of quantumgravity.

From the 1960s to the present time, a healthy version ofscalar-tensor gravity has played the most influential role(see Refs. [12,15] for reviews). We now call it the Jordan-Fierz-Brans-Dicke (JFBD) theory [14,16–18]. Inspired byMach’s principle, the gravitational constant G is promotedto a time-varying dynamical field in the JFBD theory [18].With the additional scalar field coupled nonminimally tothe Einstein-Hilbert Lagrangian, the JFBD theory leads to aviolation of the strong equivalence principle (SEP) [5,19].In this paper, we focus on a class of special mono-scalar-tensor gravity, formulated by Damour and Esposito-Farese(DEF) [20–22]. Relative to the JFBD theory, the DEFtheory extends the conformal coupling function by includ-ing a quadratic term, which dictates the way matter couplesto the scalar field. Within a certain parameter space, itsignificantly modifies the level where the SEP is violatedfor strongly self-gravitating NSs [21].The theory of scalar-tensor gravity has been extensively

investigated in the weak field, mainly from experiments inthe Solar System [2]. The most stringent constraint comesfrom the Cassini probe [23]. In the parametrized post-Newtonian (PPN) framework [2], it is verified to a highprecision ∼10−5 that the DEF theory is very close to GR inthe weak-field regime [24]. In the nonperturbative strong-field regime, Damour and Esposito-Farese [21] noticeda sudden strong activation of the scalar field for NSs.

*lshao@pku.edu.cn†mabq@pku.edu.cn

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This kind of intriguing feature, where the DEF theorycan be reduced to GR in the weak field while beingsignificantly different in the strong field, has causedenormous interest [25–28]. The process that causes sig-nificant differences from GR is referred to as strong-fieldscalarization. It has been comprehensively investigatedfor decades in terms of spontaneous scalarization [21,22,28–31], as well as induced and dynamical scalarizations[27,32–35].Different kinds of scalarization phenomena are defined

in the following ways.(1) Spontaneous scalarization occurs in an isolated and

compact star whose compactness exceeds a criticalvalue [21,29]. This phenomenon can be regarded asa phase transition [22,35].

(2) Induced and dynamical scalarizations are discoveredin numerical-relativity (NR) simulations of BNSsin the DEF theory. They hasten the plunge andmerger phases relative to GR [32]. The inducedscalarization occurs in those binary systems whereone component of the binary has already sponta-neously scalarized while the other has not in theearly inspiral [32].

(3) Dynamical scalarization corresponds to the binarysystems, where none of the binary components canspontaneously scalarize in isolation, but they getscalarized when the gravitational binding energy ofthe orbit exceeds a critical value [32].

In this paper, we pay particular attention to the strong-field region, where the spontaneous-scalarization phenom-ena are significant. Compared with an unscalarized system,the scalarized binary system brings the following manifestchanges: (i) an additional gravitational binding energy forthe orbit, (ii) an enhancement in the decay rate of thebinary’s orbital period, and the energy flux by an extradipolar radiation [24]. As we know, dipolar radiationcorresponds to the −1 post-Newtonian (PN) correction.1

It enters at a lower order relative to the typical quadrupolarradiation in GR. This means that, the smaller the relativespeed, the greater the relative radiating effect of the dipolarradiation.With the dominant radiating component being the

dipolar emission at early time, a binary system emits extraenergy in addition to GR. In certain binary pulsar systems,it is a powerful means to probe the strength of dipolarcontribution that can be caused by the spontaneous scala-rization [4,22]. With a small characteristic speed ∼10−3c,the gravitational radiation of binary pulsar systems could bedominated by the dipolar emission.

A pulsar emits ratio signals like a lighthouse. In a binarypulsar system, the spin period of the recycled pulsar isusually extremely stable. For several years to decades,those periodic signals have been continuously monitoredby large radio telescopes on the Earth. It makes the pulsar aclock that can rival the best clocks for precision funda-mental physics [36,37]. The accurate measurement tech-nique, the so-called pulsar timing, models the times ofarrival (TOAs) of pulses emitted from the pulsar anddetermines timing parameters to a high precision [3,36].In order to accommodate alternative gravity theories,

the parametrized post-Keplerian (PPK) formalism wasdeveloped as a generic pulsar timing model [38]. A setof theory-independent Keplerian and post-Keplerian timingparameters are determined with high precision in a fit of thetiming model to the TOAs [24]. We can obtain extremelyprecise physical parameters to describe those systems.They can be used to place constraints on alternative gravitytheories [3–5]. Binary pulsars are currently one of the bestavailable strong-field test beds for testing gravity [24].Recently, GWs have started to compensate with binary

pulsars in probing the strong-field gravity. The first GWevent of coalescing BNSs, GW170817, was detected by theLIGO/Virgo Collaboration in August 2017 [9]. GW170817provides a powerful laboratory in the highly dynamicalstrong field. The spacetime of BNSs is strongly curved andhighly dynamical in the vicinity of NSs in the late inspiral.If the DEF theory correctly describes the gravity, the GWphase evolution of BNSs is modified. For now, limited bythe sensitivity of the LIGO/Virgo detectors below tens ofHz, the precision to constrain the dipolar radiation fromthe short duration of GW170817 is still less than binarypulsars [27].Observations of BNSs at lower frequency are beneficial

in the dipolar-radiation test. To increase the detectorsensitivity, LIGO/Virgo have once more upgraded theirequipment, and recently started the observing run 3 (O3) onApril 1, 2019. Meanwhile, as the first kilometer-scaleunderground GW detector, the Kamioka GravitationalWave Detector (KAGRA) [39,40] is likely to join O3before the end of 2019 as well [41]. The next-generationground-based GW detectors, such as the Cosmic Explorer(CE) led by the United States, and the Einstein Telescope(ET) led by Europe [42], will further improve the sensitivityin the future. In particular, they extend the sensitivity bandsto be below 10 Hz. At the time of the third-generationdetectors, we expect to discover more BNS merger eventswith higher sensitivities and larger signal-to-noise ratios(SNRs). These events are to put more stringent limits onalternative gravity theories, with the DEF theory being animportant example.In deriving constraints on the scalar-tensor gravity,

the structure of NSs needs to be solved [21,22]. Thus,the equation of state (EOS) of NS matters plays a role. TheEOS is used to infer a NS radius, mass, and the scalar

1We refer nPN to Oðv2n=c2nÞ corrections with respect to theNewtonian order, where v is the characteristic relative speed inthe binary. Here, we follow the convention in the GW phase,where the quadrupolar radiation reaction is denoted as 0PN.Therefore, the dipolar radiation is at −1 PN order.

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charge, by integrating the modified Tolman-Oppenheimer-Volkoff (TOV) equations [21,22]. There are still largeuncertainties in the NS EOS. In this work we choose nineEOSs that are all consistent with the maximummass of NSslarger that 2 M⊙. Since more observations are being madefor pulsars at radio and x-ray wavelengths, and BNSs withan increasing statistics, the uncertainty in the nuclear EOSis to be reduced in the near future.Given an EOS, with inputs from binary pulsars and the

recent BNS observation, we carry out advanced algorithmsto constrain the DEF theory in a statistically sound way. Inparticular, we employ Bayesian methods through Markov-chain Monte Carlo (MCMC) simulations. Those simula-tions update the posterior probability distribution ofparameters in the DEF theory by evaluating the likelihoodfunction millions of times. Every step requires the corre-sponding NS properties (including mass, radius, and scalarcharge), which are derived from the modified TOV equa-tions iteratively. Being computationally intensive, suchstudies have been carried out already in Refs. [27,28].Being statistically sound, MCMC simulations lead to a

large number of iterative calculations however. Here webuild a new model to reduce the computational burden.Instead of solving the modified TOV equations iterativelyand repeatedly, we construct reduced-order surrogate mod-els (ROMs) to predict NS properties. There are twoparameters characterizing the DEF theory, α0 and β0(see the next section). We explore a sufficiently largeparameter space for them in the regime of strong-fieldscalarization. With our surrogate models, the process ofobtaining NS properties is no longer an iterative integra-tion, but a linear algebraic operation. It costs a fixed amountof time, much shorter than that in the previous method. Inpractice, for a given DEF theory, we use the central matterdensity ρc of a NS to predict its radius R, mass mA, and theeffective scalar coupling αA. Our models are numericallyaccurate at ∼1% level for αA, and better than 0.01% for mAand R. They accelerate the processes of parameter estima-tion significantly everywhere in the parameter space weexplore. With the speedup of those models, one canperform MCMC simulations much more efficiently yetstill accurately. According to our performance test, theyspeed up calculations at least 100 times, compared with theprevious method in Ref. [27]. Various practical exampleswith binary pulsars and BNS events are demonstrated inthis paper.The organization of the paper is as follows. In Sec. II,

we briefly review the nonperturbative spontaneous-scalarization phenomena for isolated NSs. The additionaldipolar radiation and the modification of mass-radiusrelations for different EOSs in the scalar-tensor gravitywill be discussed. Section III analyzes the difficulties insolving the modified TOV equations with large-scalecalculations. We develop a better numerical method, andcode it streamlinedly in the pySTGROM package. We make it

public for an easy use for the community.2 In Sec. IV, withthe speedup from pySTGROM, we stringently constrain theDEF theory by combining the dipolar-radiation limits fromobservations of five NS-white dwarf (WD) systems, andthat of GW170817, which also includes a modified mass-radius relation. Our results are in good agreement with thatfrom Shao et al. [27]. We also forecast constraints involv-ing a hypothetical BNS event, to be detected by theAdvanced LIGO at its design sensitivity [43], and next-generation ground-based GW detectors. Finally, the mainconclusions and discussions are given in Sec. V.

II. SPONTANEOUS SCALARIZATIONIN THE DEF GRAVITY

In our study, we concentrate on the DEF theory. It isdefined by the following action in the Einstein frame[21,22]:

S ¼ c4

16πG�

Zd4xc

ffiffiffiffiffiffiffiffi−g�

p ½R� − 2gμν� ∂μφ∂νφ − VðφÞ�

þ Sm½ψm;A2ðφÞg�μν�: ð1Þ

In Eq. (1), g� ≡ det g�μν denotes the determinant of theEinstein metric g�μν, R� is the Ricci curvature scalar of g�μν,G� is the bare gravitational coupling constant, φ is thedynamical scalar field that is added to GR, ψm describesany matter fields, and AðφÞ is the conformal coupling factorthat determines how φ couples to ψm in the Einstein frame.The potential of the scalar field VðφÞ can be neglected

for a slowly varying φ compared with the typical scale ofthe system. Therefore, in our study we set VðφÞ ¼ 0.Alternatively, Refs. [44,45] considered the effects of amassive scalar field, via VðφÞ ≈ 2mφφ

2. The field equa-tions of the DEF theory can be derived by varying theaction (1). They are [21,22],

R�μν ¼ 2∂μφ∂νφþ 8πG�

c4

�T�μν −

1

2T�g�μν

�; ð2Þ

□g�φ ¼ −4πG�c4

αðφÞT�; ð3Þ

where Tμν� ≡ 2cð−g�Þ−1=2δSm=δg�μν denotes the matter

stress-energy tensor and,

αðφÞ≡ ∂ lnAðφÞ∂φ : ð4Þ

As Eq. (3) shows, αðφÞ measures the field-dependentcoupling strength between the scalar field φ and thetrace of the energy-momentum tensor of matter fields,T� ≡ g�μνT

μν� .

2https://github.com/BenjaminDbb/pySTGROM

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In the JFBD theory, lnAðφÞ was chosen to be linear in φ,i.e., lnAðφÞ ¼ α0φ. It is extended to the polynomial formup to the quadratic order in the DEF theory [21],

lnAðφÞ ¼ 1

2β0φ

2; ð5Þ

with αðφÞ≡ ∂ lnAðφÞ=∂φ ¼ β0φ. We denote α0 ¼ β0φ0

with φ0 being the asymptotic (cosmological background)scalar field value of φ at infinity. Therefore, there are onlytwo extra parameters, α0 and β0 (or equivalently, φ0 andβ0), to describe a DEF theory uniquely. In GR, wehave α0 ¼ β0 ¼ 0.Within the PPN framework in the weak field, Solar-

System experiments can be used to narrow down theparameter space in the DEF theory. However, generallyonly the α0 or the combination β0α20 can be constrained (seeRefs. [2,24]).For NSs, the aforementioned nonperturbative scalariza-

tion phenomena happen when [21,32],

β0 ≡ ∂2 lnAðφÞ∂φ2

����φ¼φ0

≲ −4: ð6Þ

From Fig. 1 in Ref. [27], it is clear that with certainconditions a negative β0 can trigger an instability in thescalar field [35]. It describes the strength of nonperturbativephenomena. The more negative for β0 from the criticalvalue −4.0, the more manifest the spontaneous scalariza-tion in the strong-field regime.In the strong field, the effective scalar coupling for

a NS A,

αA ≡ ∂ lnmAðφÞ∂φ

����φ¼φ0

; ð7Þ

measures the “sensitivity” of the coupling between the NSmass and variations in the background scalar field φ0. Itappears directly in the Keplerian binding energy betweentwo stars, A and B,

V int ¼ −G�mAmB

rABð1þ αAαBÞ: ð8Þ

Besides the gravitational attraction in GR, the effectivescalar coupling αA brings an additional scalar interaction.It also affects the strength of the orbital period decay ofbinaries [22].In the following, we investigate the dipolar contribution

to the orbital decay from the scalar field, _Pdipoleb , and the

quadrupolar contribution from the tensor field, _Pquadb . They

read [21,46]

_Pdipoleb ¼ −

2πG�c3

gðeÞ�2π

Pb

�mpmc

mp þmcðαp − αcÞ2; ð9Þ

_Pquadb ¼ −

192πG5=3�

5c5fðeÞ

�2π

Pb

�5=3 mpmc

ðmp þmcÞ1=3; ð10Þ

where Pb is the orbital period, p and c denote the pulsarand its companion, respectively. Subdominant contribu-tions to _Pquad

b , that are totally negligible to our study, can befound in Eq. (6.52d) in Ref. [20]. In the above equations,gðeÞ and fðeÞ are functions of the orbital eccentricity e,

gðeÞ≡ ð1 − e2Þ−5=2�1þ e2

2

�; ð11Þ

fðeÞ≡ ð1 − e2Þ−7=2�1þ 73

24e2 þ 37

96e4�: ð12Þ

In Eqs. (9) and (10), GN ¼ G�ð1þ α20Þ denotes theNewtonian gravitational coupling in the weak field [24].The effective scalar coupling equals to zero for BHs

due to the no-hair theorem [6,47], and approaches to α0 forweak-field WDs in the DEF theory. Therefore, theinfluences from dipolar radiation on the orbital perioddecay occur dominantly in NS-WD and NS-BH binaries, aswell as in asymmetric NS-NS binaries.3 Since the dipolarcontribution relates to −1 PN contribution in the GWphase, instructively, a precise bound on the gravitationaldipole emission can be derived with the early inspiral stageof relevant GW events.For a comprehensive study on NS spontaneous scalari-

zation in the strong field, we here turn our attention to thenuclear EOS and the other NS properties. Usually, the NSmatter can be treated as a perfect fluid. In GR, given anEOS, we can solve the classical TOVequations [49,50] forNSs. The NS radius R and mass mA can be derived whenthe central matter density ρc, and the EOS are given. In theDEF theory, we involve the additional scalar field φ.Correspondingly, the modified TOV equations [Eq. (7) inRef. [21] and Eqs. (3.6a) to (3.6f) in Ref. [22]] should beused. In the Jordan frame,4 the physical quantities R, mA,and αA can be obtained via integrating from ρc and thevalue of the scalar field at the center of a NS, φc.We show an example of mass-radius relation for NSs

in the DEF theory with log10jα0j ¼ −5.0 and β0 ¼ −4.5 in

3In the BH-BH systems, scalar charges of binaries both equalto zero. They hardly contribute to the orbital decay. But in themost general scalar-tensor theory, the Horndeski gravity [48],such systems are able to have scalar hairs which, depending onthe details of the theory, might be the case only for certain massranges.

4The Jordan frame, also known as the physical frame,equivalents to the Einstein frame by a conformal transformationwith redefinitions of the metric and the scalar field [51,52].

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Fig. 1 [53]. Two massive pulsars, PSRs J0740þ 6620 [54]and J0348þ 0432 [26], show that the maximum mass ofNSs is most likely above 2 M⊙. Therefore, we choose theEOSs that satisfy this condition in our study. In total, weuse nine EOSs. From Fig. 1 we can see that,(1) the mass-radius relation in the DEF theory is very

close to that of GR in the majority of mass range;(2) there are “bumps” that correspond to the sponta-

neous scalarization phenomena and, when comparedwith GR, the bump for each EOS predicts a largerradius for a same NS mass;

(3) in GR, the EOSs SLy4 and H4 are getting more andmore in tension with pulsar mass measurementsbecause their maximum masses hardly reach the 1-σlower limit of PSR J0740þ 6620 [54].

It is worthy to note that, the EOSs H4 and PAL1predict relatively larger radii for NSs. They are startingto be disfavored by the LIGO/Virgo’s observation ofGW170817 [55,56].

III. THE REDUCED-ORDER MODEL

As discussed in the previous section, the initial values forthe TOV integrator include the NS center matter density ρc,and the center value of the scalar field φc. The macroscopicquantities of NSs, namely, R, mA, and αA can be obtainedfrom integrating the modified TOV equations. From theviewpoint of the DEF gravity, instead of φc, it is moreconvenient to have the scalar field at infinity φ0 as an initialvalue. There is correspondence between φ0 and φc. Thus, to

obtain a desired φ0 we need to find the value of φc. To solvesuch a boundary value problem, the shooting method turnsout to be a practical way [27]. Nevertheless, sometimes itleads to a large number of iterations, especially aroundthe region of spontaneous scalarization. Therefore, it isnot efficient to perform large-scale calculations, such as theparameter estimation with the MCMC approach, based onthe shooting method. It is helpful to find a faster and moreefficient, but still accurate, surrogate algorithm for a betterperformance in solving the NS structure in the DEF theory.Before discussing the surrogate algorithm for the DEF

theory, we turn to GW data analysis for an idea. A similarissue, which is equally computationally challenging anddifficult, arises when considering GW data analysis [57].In the case of compact binary coalescences, the matched-filtering technique is used. It involves matching the dataagainst a set of template waveforms (see, e.g., Ref. [58]), toinfer a potential astrophysical signal. The analysis couldlead to a large computational cost. Therefore, effectiveapproaches, the surrogate models, were built.Distinct examples are developed by Pürrer [59,60] and

Field et al. [61,62]. This kind of model is constructed by aset of highly accurate sparse waveforms. It brings a newwaveform model to provide fast and accurately compressedapproximations. These accurate surrogate models are builtwithout sacrificing the underlying accuracy for data analy-sis. Recently, several such methods have been proposed inthe literature. Based on the singular value decomposition(SVD) method, time-domain inspiral surrogate waveformsare generated in Ref. [59]. Another popular constructionmethod is the greedy reduced basis method, which isusually combined with the empirical interpolation method(EIM) [63,64]. It has been applied to GW waveforms [61]and BH ringdown [65].Inspired by the above optimization for large-scale

calculations in the GW data analysis, we adopt the greedyreduced basis method to the NS structure in the DEF theory.In our surrogate models, for each EOS, the NS propertiesare accurately and rapidly inferred with given NS centralmatter density ρc. In the following subsections, we brieflyintroduce the processes of general ROM, and describe indetail how to specialize this model to the DEF theory.Finally, we assess our ROMs’ accuracy, and find that thefinal results are consistent with our expectation. Our ROMswill be a helpful tool in generating NS properties withessential speed and accuracy for future studies.

A. A brief technical introduction to ROM

The theoretical aspects and the processes of buildingthe ROM have been discussed extensively in Fig. 1 andAppendixes A–D in Ref. [61]. We here give a briefoverview.Let us use the GW data analysis as a prototype, and

denote a curve hðt; λÞ, representing a waveform withparameters λ (for GW waveforms, parameters in λ include

FIG. 1. The mass-radius relations of NSs for different EOSs.We adopt nine EOSs whose corresponding maximum masses forNSs are larger than 2 M⊙. The relations of mass and radius arederived from GR (dotted lines), and from the DEF theory withlog10jα0j ¼ −5.0 and β0 ¼ −4.5 (solid lines). The precise massconstraints (indicated with their 1-σ uncertainties) from PSRsJ0740þ 6620 [54] and J0348þ 0432 [26] are depicted in gray.In the figure, the curves from GR and the DEF theory overlaplargely, except the bumps that result from the nonperturbativespontaneous scalarization. In the region of bumps, the DEFtheory shows a larger radius for a same NS mass relative to GR.

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the masses, the spins, and so on). In order to generate aROM for hðt; λÞ, we need the data of hðtÞ, produced with aset of given parameters fλig on a grid. We denote thetraining space V ≡ fhðt; λiÞg. With the reduced basis (RB)method, we select a certain number of bases derived fromthe training space V. Those bases are regarded as a set ofrepresentative bases for the remaining waveforms. Insteadof choosing m existing fλig and their corresponding spacefhðt; λiÞg, we seek a more complete set of bases that are asindependent as possible from each other.The greedy algorithm is currently one of the methods to

select m orthonormal RBs (see Ref. [66] and Appendix.A in Ref. [61] for details). The corresponding chosen spaceis RV ¼ feigmi¼1. Actually, the process of generatingorthonormal RBs is mentioned as iterated Gram-Schmidtorthogonalization algorithm with greedy selection [67–70].First, a seed, which can be any waveform in V, is chosen asthe starting RB (i ¼ 0). Then, we define the maximumprojection error,

σi ≡maxh∈V

jjhð·; λÞ − Pihð·; λÞjj2; ð13Þ

and a user-specified tolerable error bound Σ to infer thenext RB iteratively in the way that is explained here. InEq. (13),Pi describes the projection of hðt; λÞ onto the spanof the first i RBs. The waveform corresponding to themaximum σi is chosen as the next (iþ 1)th RB, eiþ1ðtÞ,after orthogonalized by the iterated Gram-Schmidt algo-rithm. In practice, Σ is set as a threshold to terminateiterations of the greedy selection. When σm−1 ≲ Σ, thedesired orthonormal m bases are complete.With RBs obtained above, every waveform in the

training space is well approximated by an expansion ofthe form

hðt; λÞ ≈Xmi¼1

ciðλÞeiðtÞ ≈Xmi¼1

hhð·; λÞ; eið·ÞieiðtÞ: ð14Þ

In Eq. (14), ciðλÞ is the corresponding coefficient of the ithRB. It is calculated by a special “inner product” in the spaceRV. We collect such coefficients derived from V as thegreedy data for the ROM. After that, we perform the EIMalgorithm (see Appendix B in Ref. [61] for details) toidentifym time samples, which we call the empirical nodes.An interpolation can be built to accurately reconstruct anyfiducial waveform by the RBs. Finally, at each empiricalnode, we perform a fit to the parameter space fλig andfinish the construction of the ROM.5 For convenience, thefitted data are saved in a model file.Now, we assume that a waveform, parametrized by

an unknown set of parameters λ⋆, is required in the

calculation. The parameters λ⋆ are different from λi, butwithin the boundary ranges of fλig. We can use the ROMand the fitted data on each empirical node to get a completewaveform as a function of t, hðt; λ⋆Þ. The value of hðt⋆; λ⋆Þis derived for a given t⋆ in a fast and accurate manner.

B. Constructing ROMs for the DEF gravity

In this subsection, we specialize the greedy reducedbasis method to build the ROMs for the DEF theory. Thedetailed approach is explained according to the stepsoutlined above. Because the idea of building ROMs is“borrowed” from the GW waveform studies, we use theword “waveform” to represent the desired functional forms.The parameters λ are specialized as ðα0; β0Þ.Ideally, we want to get the NS properties R and αA as a

function of the mass parameter mA. But in some parameterspace pathological behaviors might happen, while simplyintegrating the modified TOV equations. We use the DEFtheory with log10jα0j ¼ −5.3 and β0 ¼ −4.8 for the EOSAP4 as an example. We illustrate the different relations ofNS effective scalar coupling αA, radius R, and center matterdensity ρc in Fig. 2. In the left panel, the effective scalarcoupling can obtain a value significantly larger than 0.1within the NS mass interval mA ∈ ½1.4 M⊙; 2.0 M⊙�. Thisstrength of spontaneous scalarization in general decreasesmonotonically and rapidly when mA ≳ 2.0 M⊙. But, themultivalued relations between αA and mA arise whenβ0 ≲ −4.6, namely, multiple values for αA are possiblefor a given mA. The curve in the middle panel shows themass-radius relation where the bump happens (see Fig. 1for a similar behavior). There exists the multivaluedrelation as well. In the bump region, an mA can correspondto multiple values of R within the interval mA ∈ ½2.0 M⊙;2.05 M⊙�, as indicated in gray. Similarly, in the right panel,the mass of NS has a sudden drop when ρc exceeds a certainvalue in the gray band.In Fig. 2, in the interval mA ∈ ½2.0 M⊙; 2.05 M⊙�, a

fixed NS mass mA can correspond to multiple values for αAand R. The existences of those phenomena have also beenconfirmed in the other EOSs, when β0 is below a criticalvalue. For the EOS AP4, the critical value approximates to−4.6. Combining this observation and the relation of mAand ρc in the right panel of Fig. 2, we have become fullyaware where the pathologies come from. It is due to theexcessive density at the center (about ≳1.5 × 1015 g cm−3for the EOS AP4), which causes the NS to further collapseinto a BH. In order to avoid dealing with those multivaluedrelations in the ROMs, we promote the implicit parameterρc, to an independent variable which corresponds to the“time” t in Sec. III A. Therefore, we finally decide toconstruct three independent ROMs for the DEF theory,mAðρcÞ, RðρcÞ, and αAðρcÞ.6 Those relations are all single

5Particularly, a two-dimensional 5th spline interpolating fit isused in our study. A different two-dimensional fitting methodworks without practical difference.

6Actually, we use ln jαAj instead of αA in the ROM for a betterperformance.

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valued. The NS properties, ðR;mA; αAÞ, are expected to bederived with a given parameter ρc injectively in the ROMs.For different EOSs, the ranges of ρc are different. We

choose the range of ρc to have mA ∈ ð1 M⊙; mmaxA Þ where

the maximum NS mass mmaxA is EOS dependent. The

spacing Δρc is selected according to the steepness of thescalarization in different regions where we choose morepoints in rapidly changing parameter space. After buildingthe ROMs, we can have results for all ρc in its range directlyfrom our ROMs.In a short summary, we build three ROMs, mAðρcÞ,

RðρcÞ, and αAðρcÞ. Though with one more implicit param-eter ρc. This choice has avoided the pathologies if we hadbuilt two ROMs, RðmAÞ and αAðmAÞ.For the sake of completeness, we should generate all

training waveforms of interest at locations lying in theparameter space grid,

V ≡A × B: ð15Þ

In Eq. (15), A and B are one-dimensional sets coveringthe desired parameter ranges. They are representing theparameters in the DEF theory, α0 and β0, respectively. Thespacing in the parameters of log10jα0j and −β0 is chosen inan uneven way, shown in Fig. 3. We concentrate moresamples near the rapidly changing domain in the greedydata for each empirical node. Particularly, we choose A tocover the range of log10jα0j ∈ ½−5.3;−2.5� with 42 nodes.For β0, 101 nodes are chosen to cover the interval−β0 ∈ ½4.0; 4.8�. Totally, we have involved 42 × 101 ¼4242 sparse waveforms to form the training space Vfor constructing the ROMs. For the boundary values inbuilding the ROMs, the particular value α0 ≈ 10−2.5

approximately equals to the upper limit given by theCassini spacecraft [23,24], and the critical value β0 ≲−4.0 indicates places where the spontaneous scalarization

FIG. 3. An uneven grid in the parameter space ðlog10jα0j;−β0Þis used in building the ROMs of the DEF theory. We generate aset of 42 × 101 ¼ 4242 waveforms as the training data.

FIG. 2. Pathological phenomena occur when integrating the modified TOV equations. The curves represent different relationsbetween NS properties. The calculation assumes the DEF parameters, log10jα0j ¼ −5.3 and β0 ¼ −4.8. We use the EOS AP4. All graybands denote the mass range mA ∈ ½2.0 M⊙; 2.05 M⊙� for a NS. The left panel shows the relation between log10 jαAj and mA. Notably,the multivalued relation between αA and mA arises in the gray band. The middle panel shows a similar pathology for building the ROM.In the right panel, an excessive center matter density (about ρc ≳ 1.5 × 1015 g cm−3) of the NS leads to a collapse, and form a BH ratherthan a NS. These pathologies are caused by the gravitational instability when ρc exceeds a critical value.

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happens in the DEF theory [21,32]. These choice will alsoserve as our priors in the parameter estimation that we willdiscuss in Sec. IV.The greedy selection of RBs from the training space V is

iteratively performed to generate the next RB until thedesired accuracy is achieved, namely, σi ≲ Σ. In practice, Σis set to terminate the iteration of the greedy selection.In our ROMs, we choose Σ ¼ 10−7 for R and mA, andΣ ¼ 10−5 for αA.In the iterative process, the relative projection errors,

σi ≡ σi=σ0, are recorded for convenience. We illustrate σias a function of the basis size in Fig. 4. In the figure, themost notable features are the decreasing speed for σi withan increasing basis size. As the basis size increases, the σi’sof R and mA quickly arrive at the level ≲10−8 within thebasis size of 20. Particularly, the ROM of mAðρcÞ has thefastest decline. In contrast, σi falls off smoothly to Σ ∼ 10−5

for about 150 steps in building the ROM of αA. This meansthat, more RBs are needed to ensure the accuracy of αA.Actually, considering the tolerable error involved by theshooting method when integrating the modified TOVequations, which is about ∼1%, we find that the precisionloss of the above processes in constructing ROMs almostnegligible. This conclusion is verified when assessing theaccuracy of the ROMs in Sec. III C.Finally, we follow the steps in Sec. III A to obtain RBs

with the EIM algorithm, and build the entire ROMs. ThoseROMs can generate NS properties R,mA, and αA efficientlyand rapidly as a function of ρc. We tested the ROMs withrandomly generated parameters, and found that the timefor one solve of the modified TOV equations improvesfrom ∼300 ms to ∼1 ms on our Intel Xeon E5computers. We implement those three ROMs for the

DEF theory in the pySTGROM package. As is shown withapplications in Sec. IV, our ROMs can be performed at least2 orders of magnitude faster than the shooting method.

C. Assessing the ROMs of the DEF gravity

After the ROMs of the DEF gravity are constructed, weneed to assess their accuracy. We define,

EðXÞ ¼���� XROM − XmTOV

XROM þ XmTOV

����; X ∈ fmA; R; αAg; ð16Þ

that measures the fractional accuracy of the ROMs. InEq. (16), EðXÞ denotes the relative error with respect to thetrue value. The values we predict by the ROMs are denotedas XROM; the values generated by the shooting algorithm(with a relative tolerable error ∼1%) are denoted as XmTOV.We randomly generate 5 × 105 sets of parameters for α0,

β0, and ρc in the valid ranges of the ROMs. We thengenerate the corresponding values for XROM and XmTOVfrom the ROMs and the shooting algorithm, respectively.EðXÞ is calculated according to Eq. (16). The distributionsof EðXÞ are illustrated in Fig. 5. As the figure shows, therelative errors of R and mA are ≲10−5. In contrast, EðαAÞ isapproximately 3 orders of magnitude larger. The upperlimit of EðαAÞ is less than the tolerable error, 1%, of theshooting method. Therefore, these results are consistentwith the relative maximum projection error discussed inFig. 4. Notice that, the parameters used in making Fig. 5 arerandomly generated, not necessarily on the grid in Fig. 3.

FIG. 4. The relative maximum projection error, σi, in buildingthe ROMs for the EOS AP4. For the ROMs of R and mA, we setΣ ¼ 10−7, and for the ROM of αA, we set Σ ¼ 10−5. As theorthonormal basis size increases in the greedy selection, the σi’sof R and mA decrease rapidly to ∼10−8 with a basis size of 20. Incontrast, for αA, the error falls off slowly and smoothly to the levelof 10−7 for a few hundreds of basis size.

FIG. 5. The kernel density estimation (KDE) distributions ofthe relative error EðXÞ, where X ∈ fmA; R; αAg. The dashed linecorresponds to the relative tolerable error (≲1%) involved inintegrating the modified TOV equations in the shooting method.As the illustration shows, the relative errors of R and mA aresmall, ≲10−5. In contrast, the upper limit of EðαAÞ is less than 1%as expected, but worse than those of R and mA. The results areconsistent with Fig. 4.

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Therefore, the results in Fig. 5 have included all errors forour ROMs, including the interpolating errors. Although theROM of αA leads to a larger error relative to those of Rand mA, the error can be neglected when compared withthe error from the shooting method. It has no noticeableinfluence in the parameter estimation that we are to discussin Sec. IV.

IV. CONSTRAINTS FROM BINARY PULSARSAND GRAVITATIONAL WAVES

In this section, we apply our ROMs to various scenarios,and discuss the improvement in deriving NS properties. Inthese applications, we use the MCMC technique [27,71,72]to constrain the DEF theory.

A. The setup

As mentioned before, two independent parameters, α0and β0, are needed to characterize the DEF theory. Oneof them, α0, has been well constrained by the Cassinimission [23], that measures the Shapiro time delay in theweak-field regime. It gives a limit, jα0j < 3.4 × 10−3 at68% confidence level (C.L.) [24]. In the nonperturbativestrong field, the DEF theory can be significantly differentfrom GR. The constraints in the strong field were studied inRefs. [4,25–28].NSs, whose gravitational binding energy can be as large

as ∼20% of their rest-mass energy, are powerful objectsto test the strong-field gravity. We concentrate on theobservations relevant to NSs. We consider (i) systems ofbinary pulsars in the quasistationary regime, and (ii) GWsfrom BNSs in the highly dynamical regime. With theconstruction of ROMs in Sec. III, we now have a numeri-cally faster way to derive NS properties. The current andfuture constraints for the DEF theory are investigated in thefollowing.

Some stringent limits via the MCMC approach to thespontaneous scalarization of the DEF theory have beenobtained in Ref. [27]. Five asymmetric NS-WD binarieswere used. They are PSRs J0348þ 0432 [26], J1012þ5307 [73–75], J1738þ 0333 [25], J1909 − 3744 [75–77],and J2222 − 0137 [78]. Relevant parameters for thesebinary pulsars are listed in Table I. The WD companioncorresponds to a weakly self-gravitating object. It has a tinyeffective scalar coupling αc ≈ α0. The small effective scalarcoupling of the WD would lead to a large dipole term,∝ ðαp − αcÞ2, in the parameter space we are investigating[see Eq. (9)].Pulsar parameters and orbit parameters are measured

by the TOAs of pulses, including Keplerian and post-Keplerian parameters. Some parameters, such as the timederivative of the orbital period _Pb are functions of themasses of the pulsar and its companion [38]. The chosensystems are all well timed. The uncertainty in _Pb can bevery small for the pulsars we consider (see Table I). Suchprecise measurements place strong bounds on variousalternative theories of gravity [4].On the other hand, a completely new era for testing

highly dynamical strong field with NSs has begun withGW170817 [9]. It offers an extraordinary opportunity,completely different from previously detected BBH mergerevents. From the phase of GW170817, one can derive theNS properties, for instance, the mass and radius of each NS.The LIGO/Virgo Collaboration used EOS-insensitive rela-tions to derive those properties in Ref. [56]. The results areshown in Table II, and they are included to constrain theDEF theory in the MCMC approach.7

TABLE I. Binary parameters of the five NS-WD systems (PSRs J0348þ 0432 [26], J1012þ 5307 [73–75], J1738þ 0333 [25],J1909 − 3744 [75–77] and J2222 − 0137 [78]) that we use to constrain the DEF theory. The observed time derivatives of the orbitperiod, _Pobs

b , will be corrected with the latest Galactic model in Ref. [79]. The intrinsic derivatives of the orbital period, _Pintb , are obtained

from _Pobsb , by subtracting the other kinematic effects, such as the acceleration effect [80] and the “Shklovskii” effect [81]. For PSRs

J0348þ 0432, J1012þ 5307, and J1738þ 0333, the pulsar masses, mobsp , are not derived from ratio observations. They are obtained

from the companion masses,mobsc , and the mass ratios, q, without assuming the validity of GR. Similar consideration was made for PSRs

J1909 − 3744 and J2222 − 0137 [27]. The standard 1-σ errors in the least significant digit(s) are shown in parentheses.

PulsarJ0348þ 0432

[26]J1012þ 5307

[73–75]J1738þ 0333

[25]J1909 − 3744

[75–77]J2222 − 0137

[78]

Orbital period, Pb (d) 0.102424062722(7) 0.60467271355(3) 0.3547907398724(13) 1.533449474406(13) 2.44576454(18)Eccentricity, e 2.6ð9Þ × 10−6 1.2ð3Þ × 10−6 3.4ð11Þ × 10−7 1.14ð10Þ × 10−7 0.00038096(4)Observed _Pb, _Pobs

b (fs s−1) −273ð45Þ −50ð14Þ −17.0ð31Þ −503ð6Þ 200(90)Intrinsic _Pb, _Pint

b (fs s−1) −274ð45Þ −5ð9Þ −27.72ð64Þ −6ð15Þ −60ð90ÞMass ratio, q≡mp=mc 11.70(13) 10.5(5) 8.1(2) … …

Pulsar mass, mobsp (M⊙) … … … 1.48(3) 1.76(6)

Companion mass, mobsc (M⊙) 0.1715þ0.0045

−0.0030 0.174(7) 0.1817þ0.0073−0.0054 0.208(2) 1.293(25)

7Different from the LIGO/Virgo Collaboration, De et al. [82]used another EOS-insensitive relation. They obtained similarresults.

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In binary pulsars, timing parameters are measured byradio techniques. Some of them have achieved extremelyhigh precision from decades of observation. In contrast, theNS properties derived from GW170817, have a relativelypoor precision. But, the radii of NSs were inferred fromGW170817 [9,56]. This measurement is unique to con-strain the alternative theories of gravity.In addition to the systems mentioned above, we expect

that, more BNSs are to be detected with the future GWdetectors. KAGRA, operated in Japan, has two 3 kmorthonormal arms to form an underground GW interfer-ometer. It has a similar designed sensitivity as that of theAdvanced LIGO [41]. Moreover, the next-generationground-based detectors are expected to detect more GWevents with larger SNRs, due to their even higher sensi-tivities. CE [85] is an L-shaped 40 km interferometer(compared to 4 km for LIGO, and 3 km for Virgo andKAGRA). It is to be built on the experience and success ofcurrent detectors. It is roughly 10 times more sensitive thanthe Advanced LIGO. Another proposed next-generationGW detector, ET [42], is a 10 km interferometer. It issupported by the European Commission. There are threearms forming an equilateral triangle, located undergroundto reduce seismic noises. It is also 10 times more sensitivethan the Advanced LIGO. Particularly, in the frequencyband below ∼10 Hz, ET can achieve a higher sensitivitythan CE [85]. We want to find out how the DEF theory willbe constrained with those future GW detectors.Shibata et al. [33] discovered the fact that, depending on

the EOS one can still have strong scalarization in a massrange that is not yet constrained by pulsar experiments.After accounting for new pulsar tests, Shao et al. [27]showed that there is a so-called “scalarization window” atmp ∼ 1.7 M⊙. This is the place where there could still havea large deviation from GR, given all the current constraints.If the future GW detectors can observe asymmetric BNSswith masses around 1.6–1.7 M⊙, this window could beclosed. In this paper, we assume that a GW event froman inspiraling BNS with masses, ð1.65 M⊙; 1.22 M⊙Þ,is detected at luminosity distance DL ¼ 200 Mpc. ThisBNS system has similar NS masses as those of PSRJ1913þ 1102 [86]. We denote this hypothetical BNS asBNS@200Mpc. Then, we perform MCMC simulations toinvestigate this hypothetical BNS. In the foreseeable future,

BNS coalescence will be observed in large numbers.Correspondingly, a tight constraint on parameters of theDEF theory, α0 and β0, can be obtained.Some properties of NSs are governed by the EOS. NS

EOS describes the relation between pressure and density ofNS matters. It is involved in the modified TOV equationintegration. However, because of our lack of knowledgeabout the inner structure of NSs, the EOS is still uncertain.Following our previous discussion, there exists a lowerlimit for the maximum mass of NSs, Mmax ≳ 2 M⊙. NineEOSs, AP3, AP4, ENG, H4, MPA1, PAL1, SLy4,WFF1, and WFF2, are adopted in this study (see Ref. [87]for a review). We have illustrated the mass-radius relationsof NSs for these EOSs in Fig. 1. They are all consistent withthe above maximum-mass limit. Moreover, for our studies,we believe that they are sufficient to investigate the EOS-dependent properties with spontaneous scalarization [33].

B. The MCMC framework

Combining the observations of five pulsars, GW170817,and BNS@200Mpc, we perform MCMC simulations witheach of these EOSs. MCMC techniques update posteriordistributions of underlying parameters. After convergence,those distributions will be consistent with astrophysicalobservations by evaluating the likelihood function. We usethe PYTHON implementation of an affine-invariant MCMCensemble sampler, EMCEE.8 We use the ROMs, that is builtin Sec. III and coded in the pySTGROM package, to speed uptheMCMC calculations for parameter estimation within theBayesian framework [27,88].In the Bayesian inference, given priors, the posterior

distribution of ðα0; β0Þ can be inferred with data D and ahypothesis H. We use the formula of Bayes’ theorem,

Pðα0; β0jD;H; IÞ

¼Z

PðDjα0; β0;Ξ;H; IÞPðα0; β0;ΞjH; IÞPðDjH; IÞ dΞ; ð17Þ

where Pðα0; β0jD;H; IÞ is an updated (marginalized)posterior distribution of ðα0; β0Þ, PðDjα0; β0;Ξ;H; IÞ≡L is the likelihood function, Pðα0; β0;ΞjH; IÞ is the prioron parameters ðα0; β0;ΞÞ, and PðDjH; IÞ is the modelevidence. In Eq. (17), the hypothesisH represents the DEFtheory, I denotes all other relevant knowledge, Ξ collec-tively denotes all other unknown parameters in additionto ðα0; β0Þ.For our studies of the DEF theory, the priors of ðα0; β0Þ

are chosen carefully to cover the interesting region wherethe spontaneous scalarization occurs. In order to speed upthe MCMC simulations, the values of ðlog10jα0j; β0Þ arerestricted to the same rectangle region as in our ROMs thatare built in Sec. III B. We pick a uniform prior on log10 jα0j

TABLE II. Properties of GW170817 from the LIGO/VirgoCollaboration [9,11,55,56]. The primary mass m1 and secondarymass m2 are determined with the low-spin prior assumption [56].The radii of NSs, R1 and R2, are derived with the EOS-insensitiverelations [83,84].

GW170817[9]

mobs1 ðM⊙Þ[56]

mobs2 ðM⊙Þ[56]

Robs1 ðkmÞ[55]

Robs2 ðkmÞ[55]

90% C.L. (1.36,1.60) (1.16,1.36) (9.1,12.8) (9.2,12.8)68% C.L. (1.41,1.55) (1.20,1.32) (9.7,11.9) (9.6,11.8)

8https://github.com/dfm/emcee

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in the range ½−5.3;−2.5�. The prior on β0 is chosen to beuniform in the interval β0 ∈ ½−4.8;−4.0�.The parameters α0 and β0 can be constrained by

evaluating the likelihood function. Because of their obser-vational characteristics, the five pulsars in Table I,GW170817, and BNS@200Mpc contribute differently tothe likelihood function. Their corresponding log-likelihoodfunctions are given separately as follows.For binary pulsars, we have the log-likelihood function

lnLPSR ¼ −1

2

XNPSR

i¼1

��_Pthb − _Pint

b

σ _Pintb

�2

þ�mp −mobs

p

σmobsp

�2

þ�mc −mobs

c

σmobsc

�2�i

ð18Þ

for NPSR binary pulsar systems (we have NPSR ¼ 5 in thisstudy). We have assumed that observations with differentbinary pulsars are independent. For each pulsar, the intrinsicorbital decay, _Pint

b , the pulsar mass,mobsp , and the companion

mass, mobsc are given in Table I when applicable. In some

cases, we have the mass ratio q≡mp=mc and mobsc instead;

it is easy to obtain mobsp . The 1-σ uncertainty, σX in Eq. (18),

is the observational uncertainty for the quantity X, whereX ∈ f _Pint

b ; mobsp ;mobs

c g. The predicted orbital decay _Pthb

from the DEF theory equals to _Pdipoleb þ _Pquad

b [seeEqs. (9) and (10)]. The quantities, _Pth

b and mp, are implicitlydependent on the parameter set, ðα0; β0;ΞÞ. During theMCMC simulations, they are derived from those parameters.The companion mass mc is picked randomly within its 1-σuncertainty.For the full calculation, it is worth noting that, the orbital

period Pb and the orbital eccentricity e should also beincluded in Eq. (18). Those quantities have been deter-mined very well from the observations. We adopt theircentral values directly for simplifying the MCMC proc-esses. It is verified that the uncertainties of them have littleeffect on our final limits of ðα0; β0Þ.In general, the log-likelihood function for NGW GWs can

be expressed as

lnLGW ¼ −1

2

XNGW

i¼1

Xj¼1;2

��mj −mobs

j

σmobsj

�2

þ�Rj − Robs

j

σRobsj

�2�i

þXNGW

i¼1

Diðjα1 − α2j; jΔαjupperÞ: ð19Þ

Here, the properties of BNSs, such as the massesmj (j ¼ 1, 2) the radii Rj are given in Table II. The 1-σuncertainties of those properties σmobs

jand σRobs

jare given

at the 68% C.L. The function for the dipole radiationDiðjα1 − α2j; jΔαjupperÞ describes the contribution of the ithGW’s dipolar radiation to lnLGW. It returns 0, when the

effective scalar couplings of the BNS, α1 and α2, satisfyjα1 − α2j ¼ jΔαj ≤ jΔαjupper, otherwise it returns −∞.Here jΔαjupper is the upper limit of the absolute differencebetween α1 and α2 from observations. It is derived fromthe relation B ¼ 5jΔαj2=96 [89], where B is a theory-dependent parameter regulating the strength of thedipolar radiation in scalar-tensor theories. The presence ofdipole radiation in GW170817 is constrained to be B ≤1.2 × 10−5 [11]. Therefore, the upper bound of jΔαj,jΔαjupper ≃ 0.015, is obtained for GW170817. Notice thatwe are using NS masses and radii derived from GR.Because here we are constraining the non-GR effects,we feel our approach sufficient. However, at the stage ofdiscovering non-GR effects, we need a fully consistentapproach that uses NS masses and radii derived from thescalar-tensor gravity, instead of GR.In addition to GW170817, we introduce a hypothetical

GW from an inspiraling BNS, BNS@200Mpc, for fore-casting future constraints. It is meaningful to investigate theimprovement with future GW detectors. The log-likelihoodfunction of N⋆

GW hypothetical GWs is chosen as

lnLBNS ¼ −1

2

XN⋆GW

i¼1

��α1 − α2σðjΔαjÞ

�2�i: ð20Þ

In Eq. (20), σðjΔαjÞ is the expected 1-σ uncertainty of jΔαj.It is obtained from the approach of the Fisher informationmatrix. We will introduce this approach briefly and performthe MCMC calculations with the log-likelihood functionlnLBNS in Sec. IV D. Compared with Eq. (19), we neglectthe contributions from masses and radii, because, giventhe starting frequency of CE and ET, the dipolar radiationcontribution is more constraining.In a short summary, the likelihood function we use in

Eq. (17) is the sum of Eqs. (18) to (20) that have includedall contributions from binary pulsars, GW170817, andfuture BNSs.Now, we will explain how to employ the MCMC

technique to get the posteriors from the priors on ðα0; β0Þand the log-likelihood functions. Combining the observa-tions of binary pulsars and GWs, we have N ¼ NPSR þ2ðNGW þ N⋆

GWÞ NSs to constrain the ðα0; β0Þ parameterspace. To fully describe the contribution of the gravitationaldipolar radiation of these systems, N þ 2 free parameters,denoted collectively as θ, are required. The parameters, θ,

include α0, β0, and ρðiÞc ði ¼ 1;…; NÞ [27]. The initial central

matter densities, ρðiÞc , are needed in the Jordan frame. Theyare fed to our ROMs to derive the NS properties. Initiallythey will be sampled around their GR values, but they areallowed to explore a sufficiently large range in the MCMCprocess. In principle, if we were using the shooting method,

the initial central values of the scalar field φðiÞc are needed as

well. In our ROMs, they are no longer involved. Those NS

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properties can be obtained uniquely with α0, β0, and ρðiÞc .Then, those derived properties are passed to the log-like-lihood functions, Eqs. (18) to (20), to evaluate the posteriors.The constraints from the same five binary pulsars have

been investigated in great detail in Ref. [27]. In this paper,in order to verify the validity of our newly built ROMs,we will reproduce their result. In addition to the fivepulsars, we also use GW170817 and a hypotheticalBNS, BNS@200Mpc. The BNS@200Mpc is assumed tobe detected by the Advanced LIGO at its designedsensitivity, and next-generation detectors, CE and ET.Now, we can use those observations to constrain theDEF theory, and obtain the bounds in the ðα0; β0Þ param-eter space. Furthermore, we also quantify how much thetests will be improved with those next-generation detectors.For each EOS, we perform six separate MCMC runs

with different combinations of observations. These sixscenarios are shown in Table III. Here the investigationsof the DEF theory are divided into two catalogs.

(i) Scenarios (i)–(iii) correspond to the present obser-vations that contain five pulsars and GW170817. Wewill compare the constraining results of five pulsarswith those of GW170817. The corresponding con-straints from binary pulsars and GW170817 aregiven in Sec. IV C.

(ii) Scenarios (iv)–(vi) involve five pulsars and a hypo-thetical BNS@200Mpc to be observed by differentGW detectors. We expect to obtain tighter con-straints compared with present observations. Corre-sponding constraints from binary pulsars and ahypothetical BNS are investigated in Sec. IV D.

In the following subsections, we use the EOS AP4 as anexample. At the end of this section, a total of 54 MCMCruns (6 scenarios × 9EOSs) are discussed.For each run, we produce 2.6 millions of samples in total

using multiple chains ð26 chains × 105 samples for eachchain). According to the guides in Refs. [71,72], the firsthalf chain samples of these 54 runs are discarded as theBURN-IN phase. Then, we “thin” remaining samples, with athinning factor of 10, to reduce the correlation of adjacentpoints. Finally, there are 1.3 × 105 “thinned” samples

remaining for each scenario. In our studies, it is verifiedthat, those thinned samples have passed the Gelman–Rubinconvergence diagnostic very well [90]. It indicates thatall parameters in θ have lost the memory of their initialvalues, such that they can be used to infer the parametersincluding ðα0; β0Þ.

C. Constraints from binary pulsars and GW170817

In this subsection, we investigate three scenarios with realobservations, GW170817, PSRs, and PSRs+GW170817.We use the EOS AP4 as an example. Following our previousdiscussion, we distribute initial values of log10 jα0j and −β0uniformly in the rectangle region of the parameter space.Then, after the MCMC simulations, we obtain the posteriordistributions, from where we can get upper limits of theoryparameters, α0 and β0.First, we investigate the scenario GW170817. The

properties of GW170817 are given in Table II. They areused to perform the MCMC simulations. The marginalizedtwo-dimensional distribution in the parameter space ofðlog10 jα0j;−β0Þ is illustrated in Fig. 6. It is evident that,GW170817 almost has no constraint on α0. Actually, it iswithin our expectation. Because of its short duration,compared with binary pulsar observations, the parametersare derived with lower precision. Current data are notaccurate enough to give tight constraints. Nevertheless, theupper limit of −β0 is constrained to be ≲4.5 at 90% C.L.It is consistent with the argument that β0 plays the majorrole in controlling the strength of scalarization. Therefore,β0 is constrained.We use five binary pulsars for the scenario PSRs.

Different from the illustration in Fig. 6, the results ofthe scenario PSRs show very good constraints on theparameters of the DEF theory in Fig. 7. They are consistentwith results in Ref. [27] (in particular, we compared ourresults with Table II of Ref. [27]). In Fig. 7, the posteriorsamples are gathered in the lower left corner in themarginalized two-dimensional distribution. The corre-sponding parameters log10jα0j and −β0 are constrained.Especially, the parameter α0 is constrained tightly in thisscenario within our MCMC setting. The upper limit of it

TABLE III. The different scenarios used in the paper to constrain the DEF theory. Different combinations of five binary pulsars (seeTable I), GW170817 (see Table II), and a hypothetical BNS BNS@200Mpc, are investigated. Their corresponding log-likelihoodfunctions, lnLPSR, lnLGW, and lnLBNS, are expressed in Eqs. (18) to (20). The scenarios (i) to (iii) correspond to real observations,while the scenarios (iv) to (vi) involve a hypothetical BNS@200Mpc to be observed by the Advanced LIGO, CE, and ET. Notice that thescenario (ii) was investigated in great detail in Ref. [27], while the other five scenarios include new extensions.

Scenario Log-likelihood function

(i) GW170817 lnLGW GW170817 only(ii) PSRs lnLPSR Five pulsars(iii) PSRs+GW170817 lnLPSR þ lnLGW Combining five pulsars and GW170817(iv) PSRs+aLIGO lnLPSR þ lnLBNS;aLIGO Combining five pulsars and a BNS@200Mpc observed by the Advanced LIGO(v) PSRs+CE lnLPSR þ lnLBNS;CE Combining five pulsars and a BNS@200Mpc observed by CE(vi) PSRs+ET lnLPSR þ lnLBNS;ET Combining five pulsars and a BNS@200Mpc observed by ET

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achieves the level of ≲10−3.7 at 90% C.L. Compared withthe constraint in Fig. 6, it dictates that binary pulsarobservations can lead to better constraints on α0 thanGW170817. Moreover, the bound on the parameter β0becomes tighter as well. The right and upper parts of theparameter space have no support from MCMC samples.It is worth noting that the nonsmoothness of the one-

dimensional marginalized distributions, e.g., in Fig. 7, iscaused by the statistical fluctuations in the MCMC sim-ulations. It has no statistical bearing, and will disappear ifwe have infinite samples. As we noted before, our sampleshave passed the Gelman-Rubin test for convergence, thusour limits on α0 and β0 are reliable.In Fig. 8, we illustrate the results for the scenario PSRs

+GW170817. It involves the combination of the observa-tions of five pulsars and GW170817. Basically, its result isconsistent with the conclusion in Fig. 7. It is verified againthat the observation of GW170817 has little effect inconstraining the DEF theory. In contrast to GW170817,the contemporary observation of radio pulsars gives us aquite powerful tool in probing the strong-field regime forgravity.

D. Constraints from binary pulsarsand a hypothetical BNS

In the previous subsection, we find that, compared withbinary pulsars, GW170817 has little effect in constrainingthe DEF theory. It is interesting to investigate whether futureground-based GW detectors can surpass binary pulsarobservations. We investigate the other three scenarios inTable III, namely, PSRs+aLIGO, PSRs+CE, and PSRs+ET. We assume that the hypothetical event BNS@200Mpc

FIG. 6. The marginalized two-dimensional distribution in theparameter space of ðlog10jα0j;−β0Þ in the scenario GW170817,for the EOS AP4. The marginalized one-dimensional KDEdistribution of log10 jα0j is illustrated in the upper panel, whilethat of −β0 is illustrated in the right panel. The upper limits ofparameters are shown in the red dashed lines at 68% C.L. and inthe green dashed lines at 90% C.L. The “sharp dives” in themarginalized one-dimensional distributions near the boundaryare the “boundary effects” of the KDE process; same for theother figures that use KDE.

FIG. 7. Same as Fig. 6, for the scenario PSRs. FIG. 8. Same as Fig. 6, for the scenario PSRs+GW170817.

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has component masses (1.65 M⊙, 1.22 M⊙). One of themasses is chosen to be close to the scalarization window [27]where, given the binary pulsar observations, there could stillexist a large deviation from GR.We investigate the power spectral density (PSD) SnðfÞ

in GW detectors [85]. The SNR of a GW event isρ ¼ ðhðfÞjhðfÞÞ1=2, where hðfÞ is a Fourier-domain wave-form, and the inner product is defined as [91],

ðh1ðfÞjh2ðfÞÞ≡ 2

Zfmax

fmin

h�1ðfÞh2ðfÞ þ h1ðfÞh�2ðfÞSnðfÞ

df:

ð21Þ

Here, the initial frequency fmin is chosen as the startingfrequency, fin, for a GW detector, the final frequency is setto be twice of the frequency of the innermost stable circularorbit (see Ref. [27]). Those starting frequencies and SNRsare listed in Table IV.In Eq. (20), the expected 1-σ uncertainties of jΔαj,

denoted as σðjΔαjÞ, are needed for different GW detectors.The technique of Fisher information matrix is used tocalculate them. The Fisher information matrix is a measureof an experiment’s resolving power for the waveformparameters, collectively denoted as ξ. It is defined as [91]

I ij ¼�∂h∂ξi

���� ∂h∂ξj�: ð22Þ

See Appendix A in Ref. [27] for more details.For a parameter ξi, a lower bound on its variance

expected from an experiment can be placed with theinequality of the Cramer-Rao bound [92,93] σðξiÞ ≥ffiffiffiffiffiffiffiffiffiffiffiffiffiffiðI−1Þii

p. A lower bound on jΔαj can be got from the

diagonal component of the inverse Fisher matrix. Thecorresponding 1-σ uncertainties of jΔαj for the GWdetectors are given in Table IV. The limits from thenext-generation detectors will be better than that of theAdvanced LIGO, due to better low-frequency sensitivities.In the following, we give the results for the scenarios

PSRs+aLIGO, PSRs+CE, and PSRs+ET.For the scenario PSRs+aLIGO, in addition to the five

binary pulsars, we use a hypothetical BNS@200Mpc,

assumed to be observed by the Advanced LIGO. Weillustrate the result of this scenario in Fig. 9. Comparedwith the scenario PSRs+GW170817 in Fig. 8, it is evidentthat the constraints on DEF theory are not significantlyimproved.The scenarios, PSRs+CE and PSRs+ET for the EOS

AP4, are illustrated, respectively, in Figs. 10 and 11. Theparameter space of the DEF theory is highly constrainedwith those next-generation detectors. We can obtain thetightest constraints on the parameters, α0 and β0, in our

TABLE IV. The first column lists three ground-based GWdetectors. The second column gives their frequencies at low end.The third and fourth columns are respectively the expected SNRsand the expected 1-σ uncertainties in the difference of theeffective scalar couplings from the Fisher-matrix analysis [27],for a hypothetical event, BNS@200Mpc.

Detector fin (Hz) ρ σðjΔαjÞaLIGO 10 10.6 8.8 × 10−3

CE 5 450 7.9 × 10−4

ET 1 153 4.0 × 10−4

FIG. 9. Same as Fig. 6, for the scenario PSRs+aLIGO.

FIG. 10. Same as Fig. 6, for the scenario PSRs+CE.

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studies. Here, the upper limit of jα0j achieves the level of≲10−4.0 at 90% C.L., given our priors. The parameter −β0can be constrained to ≲4.3 at 90% C.L. Because of its low-frequency sensitivity, ET can track a much longer time of aBNS inspiral than the Advanced LIGO and CE. It willconstrain the DEF theory better.In our study, we investigate all the scenarios mentioned

above for nine EOSs. The upper limits of the parametersjα0j, −β0 at 90% C.L. are illustrated in Figs. 12 and 13,respectively. Figures 15 and 16 in the Appendix showthe corresponding marginalized one-dimensional KDEdistributions.We collect our marginalized limits in Figs. 12 and 13.

In Fig. 12, for nine EOSs we adopted, the scenarioGW170817 shows that jα0j cannot be well constrained.With the five binary pulsars, the constraints are improvedenormously down to the level of jα0j≲ 2 × 10−4. It isevident that the scenarios PSRs, PSRs+GW170817, andPSRs+aLIGO give similar upper bounds on jα0j. Itdictates that the constraints from GW170817 are weakerthan those from binary pulsars. The constraints on jα0j areexpected to be greatly enhanced with the next-generationdetectors, CE and ET. Their upper bounds on jα0j canachieve the level of ≲1 × 10−4. It is worthy to note that theconstraints on jα0j given in Fig. 12 are heavily influencedby our priors (especially the one on β0). A different priorwill give a different limit, but the relative strength of theseobservational scenarios is fixed.In Fig. 13, the bounds on β0 from GW170817 are

hardly constrained for most of the EOSs. Only for theEOS WFF1, β0 can be limited to a meaningful level, −β0 ≲4.35 at 90% C.L. In contrast, for the scenarios, PSRs,

FIG. 11. Same as Fig. 6, for the scenario PSRs+ET.

FIG. 12. 90% C.L. upper bounds on the parameter jα0j for nineEOSs and six scenarios in our studies. Their marginalized one-dimensional distributions are shown in Figs. 15 and 16. Noticethat the limit on jα0j is influenced by our priors (see text).

FIG. 13. Same as Fig. 12 for the parameter −β0. In order toexpress those constraints on −β0 more clearly around −β0 ∼ 4.3,a special logarithmic axis is adopted in this figure.

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PSRs+GW170817, and PSRs+aLIGO, the limits areimproved up to the level of β0 ≈ −4.30 for most of theEOSs (except for H4). When considering the scenariosPSRs+CE and PSRs+ET, we find that the bounds on β0have almost no improvement for all EOSs, except for AP4,SLy4, WFF1, and WFF2. The results of β0 are differentfrom those of α0. Particularly, the parameter β0 plays themajor role in controlling the strength of scalarization. Wecould not expect to obtain a tighter constraint on β0 onlyby improving the precision of observations. The results inFig. 13 are consistent with Fig. 7 in Ref. [27]. In particular,for the EOSs AP4, SLy4, WFF1, and WFF2, CE andET have the potential to improve the current limit frombinary pulsars.We investigate the improvement from the scenario PSRs

+ET, with respect to the scenario PSRs+GW170817. Theupper bounds on the NS effective scalar coupling αA, as afunction of the NS massmA, are illustrated in Fig. 14. Here,the upper bounds at 90% C.L. on theory parameters (seeFigs. 12 and 13) are used for those curves. The area belowthe curve for each EOS corresponds to the unconstrainedregion for a NS. It is evident that the allowed maximumeffective scalar couplings are constrained strongly for allEOSs except for the EOS H4. The narrow range of the NSmass that corresponds to the scalarization peak, is aroundmA ∼ 1.9 M⊙ for the EOS H4. All NS masses from binarypulsars in Table I, GW170817, and BNS@200Mpc arenot in this region. It is also the reason why the constraintsfor the EOS H4 in Fig. 13 are hardly improved, with thenext-generation detectors.Combining the results in Figs. 13 and 14, we can

understand some results qualitatively. In the previousdiscussion, the upper bounds on β0 at 90% C.L. can beplaced at the level of ∼ − 4.35 for the EOS WFF1. It is the

tightest constraint in the scenario GW170817. Actually, itcan be understood in Fig. 14. For the specific range of NSmasses, the spontaneous scalarization is significant. For theEOS WFF1, the corresponding NS mass is around thevalue, mA ≈ 1.4–1.7 M⊙. In contrast, the other EOSs allowNSs to scalarize strongly when mA ≳ 1.7 M⊙. One of theBNS masses derived from GW170817 is in the rangeð1.36 M⊙; 1.60 M⊙Þ. It is within the scalarization massrange for the EOS WFF1. Therefore, solely withGW170817, we can constrain β0 for this EOS.In a short summary, the constraints on jα0j improve with

the precision of the observations. But, for β0, not only theprecision of the observations, but also the choice of theEOS can influence the limit. Different EOSs allow NSs toscalarize at different NS masses. We can use the observa-tions, binary pulsars, and BNSs, to constrain the DEF theorywith different EOSs, if suitable systems are observed.

V. CONCLUSION

In this paper, we studied the DEF theory in the non-perturbative strong field. In the parameter space thatwe investigate, NSs could develop a phenomenon calledspontaneous scalarization. Instead of solving the modifiedTOV equations for NSs (as was done in an earlier study[27]), we have built efficient ROMs, implemented in thepySTGROM package, to predict NS properties. The code ismade public for the community use. With the speedup ofROMs, MCMC calculations were carried out to constrainthe parameter space, ðα0; β0Þ. Comparisons between vari-ous scenarios are discussed in detail. We summarize themain points in the following.(1) To speed up large-scale calculations, pySTGROM

is constructed for studying the DEF theory.

FIG. 14. The 90% C.L. upper bounds on the NS effective scalar coupling, αA, from the scenarios PSRs+GW170817 (left) andPSRs+ET (right).

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Considering six scenarios that include currentlyavailable as well as future projected observations,we have tested our ROMs in practice. It turns outthat, our ROMs are performed at least 2 orders ofmagnitude faster than the previous method. In theforeseeable future, we wish our ROMs to be used inrelevant calculations for the DEF theory.

(2) For a fixed EOS, a NS is allowed to scalarizestrongly in a specific mass range of the NS. Abinary pulsar and/or a BNS with a particular NSmass in this mass region may be observed. They canbe used to constrain the DEF theory stringently forthis EOS.

(3) Using the uncertainty in jΔαj from the Fisher-matrixcalculation [27,94], the tightest constraints comefrom the scenario PSRs+ET in our studies. Givenour specific priors, we can bound the parameters tobe jα0j ≲ 10−4.0 and −β0 ≲ 4.3 at 90% C.L. Becauseof its low-frequency sensitivity, ET for sure willprovide us with significant improvement over cur-rent constraints on the dipole radiation.

(4) The spontaneous scalarization is mainly controlledby the β0 parameter in the DEF theory. It is likelythat, the constraints on the parameter jα0j can beconstrained more tightly with the improvement inthe observational precision. Different from jα0j, theconstraints on the scalarization parameter β0 arerelated to the property of EOS, in addition to theprecision of the observations.

The current and projected bounds were obtained inSecs. IV C and IV D, respectively. It indicates that the next-generation GW detectors, especially the ET, have thepotential to further improve current limits, set by the binarypulsars and GW170817. Those current limits are to beimproved over time, especially if suitable systems fillingthe scalarization windows are discovered [27]. Some binarypulsar systems, like PSRs J1012þ 5307 [73] and J1913þ1102 [86], still have large uncertainties in their masses. Ifthe masses are constrained to be around the scalarizationwindow, they may eliminate the possibility of a strongscalarization below 2 M⊙. The new large radio telescopes,

such as the Five-hundred-meter Aperture Spherical radioTelescope (FAST) in China [95–97] and the SquareKilometre Array (SKA) in Australia and South Africa[98–100], can help to improve the timing precision. Inaddition, they may discover new systems that meet therequirements. On the other hand, O3 of LIGO/Virgodetectors began in April 2019. It has been discovering ahandful of GW candidates.9 By the end of the observingruns, some events may happen to be suitable systems toclose the scalarization windows. Asymmetric BNSs (withαA ≠ αB) as well as NS-BHs (with αNS ≠ αBH ¼ 0) wouldgive asymmetric systems and therefore could be particu-larly interesting. Furthermore, the next-generation ground-based GW detectors are expected to observe many moresystems in the future, and they can be used to studyalternative gravity theories in a more precise way. OurROMs are built to meet the requirements of new observa-tions to constrain the DEF theory in an efficient yetaccurate way.

ACKNOWLEDGMENTS

We are grateful to Bin Hu and Michael Kramer forcomments.We thankNorbertWex for stimulatingdiscussionsand carefully reading the manuscript. This work was sup-ported by the Young Elite Scientists Sponsorship Programby the China Association for Science and Technology(2018QNRC001), and was partially supported by theNational Natural Science Foundation of China (11721303,11475006, 11690023, 11622546), the Strategic PriorityResearch Program of the Chinese Academy of Sciencesthrough the Grant No. XDB23010200, the EuropeanResearch Council (ERC) for the ERC Synergy GrantBlackHoleCam under Contract No. 610058, and the High-performance Computing Platform of Peking University.Z. C. was supported by the “Fundamental Research Fundsfor the Central Universities.”

9GW candidates are collectively shown in the gravitational-wave candidate event database (GraceDB) by the LIGO/VirgoCollaboration in the link https://gracedb.ligo.org/latest.

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APPENDIX: THE MCMC RESULTS FOR NINE EOSs

The marginalized one-dimensional KDE distributions of the parameters log10 jα0j and −β0 for the EOSs in the set {AP3,AP4, ENG, H4, MPA1, PAL1, SLy4, WFF1, WFF2}, are illustrated in Figs. 15 and 16. Their corresponding upperbounds at 90% C.L. are shown with dashed lines.

(a) (b)

(c) (d)

(e) (f)

FIG. 15. The marginalized one-dimensional KDE distributions for the MCMC posteriors for the parameters log10 jα0j and −β0. Theyare shown for nine EOSs and the scenarios GW170817 (top), PSRs (middle), and PSRs+GW170817 (bottom). The distributions oflog10 jα0j are shown in the left; the posterior distributions of −β0 are shown in the right. Their corresponding upper bounds at 90% C.L.are illustrated with dashed lines.

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(a) (b)

(c) (d)

(e) (f)

FIG. 16. Same as Fig. 15, for the scenarios, PSRs+aLIGO (top), PSRs+CE (middle), and PSRs+ET (bottom).

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