Properties of scalar wave emission in a scalar-tensor theory with kinetic screening Masaru Shibata12and Dina Traykova1 1Max Planck Institute for Gravitational Physics Albert Einstein Institute

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Properties of scalar wave emission in a scalar-tensor theory with kinetic screening

Masaru Shibata1,2∗and Dina Traykova1†

1Max Planck Institute for Gravitational Physics (Albert Einstein Institute),

Am Mühlenberg 1, Potsdam-Golm 14476, Germany and

2Center for Gravitational Physics and Quantum-Information,

Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan

We study numerically the scalar wave emission by a non-spherical oscillation of neutron stars in a scalar-

tensor theory of gravity with kinetic screening, considering both the monopole and quadrupole mode emission.

In agreement with previous results in the literature, we ﬁnd that the monopole is always suppressed by the

screening effect, regardless of the size of the screening radius, rsc. For the quadrupole mode, however, our

analysis shows that the suppression only occurs for screening radius larger than the wavelength of scalar waves,

λwave, but not for rsc <λwave. This demonstrates that to fully understand the nature of this theory, it is necessary

to study other more complex systems, such as neutron star binaries, considering a wide range of rsc values.

I. INTRODUCTION

The ample evidence for the current accelerated expansion

of the Universe has hinted at the existence of some new

physics at cosmological scales [1–8]. One of the simplest

modiﬁcations to general relativity (GR), which can provide

a possible explanation of this phenomenon, is the so-called

scalar-tensor theories, where an additional scalar degree of

freedom is minimally (e.g. quintessence [9–12]; see also

Refs. [13,14] for reviews) or non-minimally coupled to the

gravitational metric (see Refs. [15–18] for a review on scalar-

tensor gravity). On cosmological scales, it is possible to mea-

sure and constrain physical parameters that capture this novel

behaviour [19–23], showing that modiﬁcations to GR that can

account for the observed accelerated expansion of the Uni-

verse on these scales with the dark sector whose density is of

the order of the critical density, ρc. This means that we can

expect similar deviations on small scales too. However, So-

lar System [24,25] and binary pulsar [26–30] tests show no

violations of the predictions of GR there. In addition, radio

observations of pulsars (neutron stars) accompanying white

dwarfs constrain the emissivity of scalar-type gravitational

waves (hereafter referred to simply as scalar waves), and thus,

the parameter space for some scalar-tensor theories has been

signiﬁcantly limited [30–32]. More recently, consistency with

GR has also been shown by null tests with gravitational-wave

observations [33–37].

One possible solution to this problem is employing an ap-

propriate screening mechanism, by which the effects of the

scalar ﬁeld are suppressed on local scales so that GR phenom-

ena can be reproduced, while on cosmological scales, mod-

iﬁcations to GR remain appreciable. Some well-studied ex-

amples of this behaviour are the chameleon [38], symmetron

[39], and Vainshtein [40–42] screening (see also Refs. [43–45]

for reviews). Even though screening effects have been studied

extensively in a range of simpliﬁed scenarios, such as weak-

gravity and spherical symmetry approximations (see, e.g.,

Refs. [46–56]), they are not so well-understood in strongly

∗masaru.shibata@aei.mpg.de

†dina.traykova@aei.mpg.de

self-gravitating and dynamical environments, such as the dy-

namical neutron star spacetime. For example, the emission

mechanism of scalar waves has not been yet well-understood.

In order to fully characterise the screening effect in dynami-

cal spacetimes, for which no linearization or symmetry of the

system can be employed, numerical relativity (NR), by which

the solution of the fully non-linear systems can be obtained, is

needed.

NR simulations of compact objects in scalar-tensor theories

with a kinetic screening effect have been performed in a few

recent studies [57–62], some of which report a non-trivial na-

ture of the scalar-wave emission. In particular, in Ref. [59],

the authors ﬁnd that the quadrupole scalar wave emission may

not be screened in the case of a binary neutron star inspiral.

This study focuses on the cases with a small screening radius

(.140 km), which is smaller than the wavelength of gravita-

tional and scalar waves. We argue here that, in a such setting,

the screening effect may not be signiﬁcant and one could ex-

pect different behaviour when larger screening radii, which

are more realistic, are considered.

In this paper, we study numerically the emission of scalar

waves from non-spherically oscillating neutron stars in the

same scalar-tensor theory employed in Ref. [59]. It has been

shown in Ref. [63] that scalar waves in a scalar-tensor theory

of gravity can be detected by interferometers in the same way

as gravitational waves. Their analysis, done in the framework

of the Brans-Dicke theory shows that, for a simple Michel-

son interferometer, the antenna sensitivity pattern depends

strongly on the frequency of the scalar gravitational waves,

with essentially the same features as those of the tesnosr mode

of GWs. Thus showing that as long as the dependence of the

antenna sensitivity pattern on the wave length of scalar waves

is taken into account in the same way as for the tensor modes,

scalar waves would be detectable in the case of a scalar-tensor

theory. Therefore in this work we treat both scalar and tensor

modes as gravitational waves.

Our NR simulation is performed in the Jordan frame in con-

trast to previous works [58,59], which employ the Einstein

frame instead. Doing this has three advantages: (i) the equa-

tions for hydrodynamics are not changed and have a conser-

vative form, same as in GR; (ii) the gravitational and scalar

waves are extracted independently from the spacetime metric

and the scalar ﬁeld, respectively; and (iii) unlike the Einstein

arXiv:2210.12139v2 [gr-qc] 21 Feb 2023

frame case, the Jordan frame metric couples universally to the

matter ﬁelds and so observables can also be computed in the

same way as one does in GR. We will show that if the screen-

ing radius is larger than the wavelength of scalar waves, the

screening effect on the scalar waves (i.e., the suppression of

the scalar wave emission) is always signiﬁcant irrespective of

the multipoles considered.

The paper is organised as follows. In Sec. II we summarise

the basic equations that we employ. Section III presents a

formulation for computing equilibrium and quasi-equilibrium

states, necessary for the initial conditions in NR simulation.

Section IV presents numerical solutions of 1.4Mspherical

neutron stars and summarises the properties of a neutron star

spacetime in the presence of the kinetic screening. In Sec. V

we explore the non-spherical oscillation of neutron stars ob-

tained in Sec. IV, in particular focusing on the generation and

propagation of quadrupole scalar waves. Finally, we discuss

our results and summarise our conclusions in Sec. VI.

Throughout this paper, we use the units of c=1=¯

where cand ¯

hdenote the speed of light and the reduced

Planck constant, respectively. In these units the Planck length,

`p:=G1/2=1.616 ×10−33 cm and the Planck mass, Mp:=

G−1/2=2.176 ×10−5g. The subscripts aand bdenote the

spacetime tensor components, and i,j, and kdenote the spa-

tial components.

II. BASIC EQUATIONS

In this work we consider a scalar-tensor theory with kinetic

screening, in which the action in the so-called Jordan frame is

given by [24,64–66],

S=1

16πGZd4x√−gφR+3

2+ˆ

α2

sgab ∇aφ∇bφ

φ2

+Smatter(χmatter,gab).(1)

The corresponding action in the Einstein frame can be found

in, e.g., Refs. [59,67]. Here Rand ∇aare the Ricci scalar and

covariant derivative associated with the spacetime metric gab,

φ(>0)is the gravitational scalar ﬁeld and ˆ

Kis a function of

the canonical kinetic term of the scalar ﬁeld, X.Smatter is the

action of the perfect ﬂuid, with χmatter representing the matter

ﬁelds. The kinetic term of the scalar ﬁeld is deﬁned as,

X=¯gab ¯

∇a¯

ϕ¯

∇b¯

ϕ=φ−1gab∇a¯

ϕ∇b¯

ϕ,(2)

where ¯gab is the spacetime metric in the Einstein frame, ¯

∇a

is its covariant derivative, ¯

ϕ=lnφ/p16πGα2

s, and αsis a

coupling constant. Following Ref. [59], we consider the case,

K(X) = −1

2+γ1

4Λ4X−γ2

8Λ8X2··· ,(3)

where Λis the strong-coupling scale (i.e., λ:=Λ−1deter-

mines the length scale of screening), and γ1and γ2are con-

stants of order unity. Here we choose γ1=0 and γ2=1 as

it has been shown (see Refs. [68,69]) that this is a necessary

condition for having a well-posed initial value formulation, as

well as screening static solutions. Screening is expected to

occur in the strong ﬁeld zone, where X>Λ4is satisﬁed. We

suppose that φ→1 (i.e., ¯

ϕ→0 and X→0) for r→∞.

For γ1=γ2=0 this theory is equivalent to the Fierz-Jordan-

Brans-Dicke (FJBD) theory [64–66], with Brans-Dicke pa-

rameter of the form,

ω(X):=−3

2−ˆ

K(X)

α2

,(4)

with X=0, so that ω(X) = −1

23−αs−2.

Then the basic equations for the geometry, scalar ﬁeld, en-

ergy momentum tensor, Tab, and rest-mass continuity are as

follows,

Gab =8πGφ−1Tab

−3

2+ˆ

α2

sφ−2(∇aφ)∇bφ−1

2gab(∇cφ)∇cφ

−X

α2

sφ2

∂ˆ

∂X∇aφ∇bφ+φ−1(∇a∇bφ−gab2gφ),(5)

∇a(F∇aφ) = 8πGα2

sT,(6)

∇aTa

b=0,(7)

∇a(ρua) = 0,(8)

where Gab is the Einstein tensor associated with gab,T=Ta

uais the ﬂuid four velocity, ρis the rest-mass density, and

F:=−2∂(Xˆ

∂X=1−γ1

Λ4+3γ2

Λ8+··· .(9)

To derive Eq. (6), we used the trace of Eq. (5),

−R=8πGφ−1T+3

2+ˆ

α2

s−X

α2

∂ˆ

∂X(∇aφ)∇aφ

φ2

−3

φ2gφ,(10)

where 2g=∇a∇a.

For Tab, we consider the stress-energy tensor for a perfect

ﬂuid,

Tab = (ρ+ρε +P)uaub+Pgab,(11)

where εand Pare the speciﬁc internal energy and pressure

of the ﬂuid. In the Jordan frame the ﬂuid matter is coupled

only to the gravitational ﬁeld, as seen in Eq. (7). Hence, the

equations for the perfect ﬂuid are the same as those in GR in

this frame.

The basic equations in the 3+1 formulation for the grav-

itational ﬁeld are derived simply by contracting nanb,naγb

and γa

iγb

jwith Eq. (5). Here, γab =gab +nanbdenotes the

spatial metric, and nais the unit normal to the spatial hyper-

surfaces. The 3+1 form of the scalar ﬁeld equation is derived

from Eq. (6) by deﬁning Π:=−na∇aφor ˆ

Π:=−Fna∇aφ.

The evolution of the scalar ﬁeld and its conjugate momen-

tum have the following form,

(∂t−βk∂k)φ=−αΠ,(12)

(∂t−βk∂k)ˆ

Π=−DiαFDiφ+αKˆ

+8πGαα2

sT,(13)

where Diis the covariant derivative with respect to γi j . In

terms of Πand φ,Xcan be written as,

X=1

16πGα2

sφ3h(Dkφ)Dkφ−Π2i.(14)

From these one can also obtain an algebraic equation for X,

f(X):=X−1

16πGα2

sφ3(Dkφ)Dkφ−ˆ

Π2

F(X)2=0.(15)

For a detailed description of the 3+1 equations of the system,

we refer the reader to Appendix A.

The evolution equations for the gravitational ﬁelds

are solved numerically in the Baumgarte-Shapiro-Shibata-

Nakamura (BSSN) formalism [70,71] with the moving-

puncture gauge [72,73], as done in Ref. [74]. In particu-

lar, we evolve the conformal factor W:=ψ−2(with ψ:=

(detγi j)1/12), the conformal metric ˜

γi j :=ψ−4γi j , the trace part

of the extrinsic curvature K, the conformally weighted trace-

free part of the extrinsic curvature ˜

Ai j :=ψ−4(Ki j −Kγi j/3)

(with Ki j – the extrinsic curvature), and the auxiliary variable

Γi:=−∂j˜

γi j. Introducing the auxiliary variable Biand a pa-

rameter ηs, which is typically set to be ∼M−1,Mbeing the

total mass of the system1, we employ the moving-puncture

gauge in the form of [75],

(∂t−βj∂j)α=−2αK,(16)

(∂t−βj∂j)βi= (3/4)Bi,(17)

(∂t−βj∂j)Bi= (∂t−βj∂j)˜

Γi−ηsBi,(18)

where αand βiare the lapse function and shift vector, respec-

tively.

The spatial derivative is evaluated by a fourth-order cen-

tral ﬁnite difference scheme, except for the advection terms,

which are evaluated by a fourth-order non-centred ﬁnite dif-

ference. For the time evolution, we employ a fourth-order

Runge-Kutta method (see Ref. [76]). We use the same scheme

for the evolution of the scalar ﬁeld as for the tensor, because

the structure of the equations is essentially the same.

To solve the hydrodynamics equations, we evolve ρ∗:=

ραutW−3, ˆui:=hui, and e∗:=hαut−P/(ραut), with hbe-

ing the speciﬁc enthalpy. The advection terms are handled

with a high-resolution shock capturing scheme of a third-order

piecewise parabolic interpolation for the cell reconstruction.

For the equation of state (EOS), we decompose the pressure

and the speciﬁc internal energy into cold and thermal parts as,

P=P

cold +P

th ,ε=εcold +εth .(19)

Here, P

cold and εcold are functions of ρ, and their forms are

determined by nuclear-theory-based zero-temperature EOSs.

Speciﬁcally, the cold part of both variables are determined

using the piecewise polytropic version (see, e.g., Ref. [77])

1We note that the total mass includes a contribution both from the ADM

mass and the scalar charge (see the tensor mass in Sec. III).

of the APR4 EOS [78], for which the maximum mass of the

neutron stars in GR is ≈2.2M.

Then the thermal part of the speciﬁc internal energy is de-

ﬁned from εas εth :=ε−εcold. Because εth vanishes in

the absence of shock heating, εth is regarded as the ﬁnite-

temperature part (and thus, this part is minor in the present

study). The thermal pressure is determined by a Γ-law EOS,

th = (Γth −1)ρεth ,(20)

and we choose Γth equal to 1.8, following Refs. [74,77].

III. FORMULATION FOR INITIAL CONDITIONS

Here we outline the formulation for computing quasi-

equilibrium conﬁgurations for a binary in a circular orbit with

angular velocity Ωfollowing Refs. [79–81]. This description

is also valid for computing static spherical stars with Ω=0.

To derive quasi-equilibrium conﬁgurations, for simplicity,

we assume the conformal ﬂatness of the three metric, such

that

γi j =ψ4fi j ,(21)

where fi j is the ﬂat spatial metric, and employ the confor-

mal thin-sandwich prescription. We also impose the maxi-

mal slicing K=0. For integrating the hydrodynamics equa-

tions, we assume the presence of a helical Killing vector,

ξa= (∂t+Ω∂ϕ)a. For the ﬂuid part, the basic equations in

the Jordan frame are the same as those in GR. Thus, assuming

that the velocity ﬁeld is irrotational, the ﬁrst integral of the

hydrodynamics equations is readily determined in the same

manner as those in GR [82,83].

The basic equations for the tensor ﬁeld are obtained from

the Hamiltonian and momentum constraints, together with the

evolution equation for K(see Appendix A) under the maximal

slicing condition, K=0=∂tK. Except for the modiﬁcations

introduced by the presence of the scalar ﬁeld, φ, the equations

are again the same as in GR. The Hamiltonian and momentum

constraints are written as,

(0)

∆ψ=−2πGφ−1ρhψ5−1

Ai j ˜

Ai jψ5

−ψ5

8ω

φ2Π2+ (Diφ)Diφ+2φ−1DiDiφ

−2Π2

α2

sφ2X∂ˆ

∂X,(22)

and

(0)

Di(ψ6˜

Aij) = ψ6"8πGφ−1Jj+ω−X

α2

∂ˆ

∂Xφ−2Π

(0)

Djφ

+φ−1(

(0)

DjΠ−˜

Aij

(0)

Diφ)#,(23)

respectively. Here

(0)

∆and

(0)

Diare the Laplacian and covari-

ant derivatives with respect to fi j,ρh:=Tabnanb, and Ji:=

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Propertiesofscalarwaveemissioninascalar-tensortheorywithkineticscreeningMasaruShibata1;2andDinaTraykova11MaxPlanckInstituteforGravitationalPhysics(AlbertEinsteinInstitute),AmMühlenberg1,Potsdam-Golm14476,Germanyand2CenterforGravitationalPhysicsandQuantum-Information,YukawaInstituteforTheoreticalPh...

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Properties of scalar wave emission in a scalar-tensor theory with kinetic screening Masaru Shibata12and Dina Traykova1 1Max Planck Institute for Gravitational Physics Albert Einstein Institute

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