Constraints on f(R) theories of gravity from GW170817
aa r X i v : . [ g r- q c ] M a r Constraints on f ( R ) theories of gravity from GW170817 Soumya Jana a and Subhendra Mohanty b Theoretical Physics Division, Physical Research Laboratory, Ahmedabad 380009, India
Abstract
A novel constraint on f ( R ) theories of gravity is obtained from the gravitational wave signalemitted from the binary neutron star merger event GW170817. The f ( R ) theories possess an addi-tional massive scalar degree of freedom apart from the massless spin-2 modes. The correspondingscalar field contributes an additional attractive, short-ranged “fifth” force affecting the gravita-tional wave radiation process. We realize that chameleon screening is necessary to conform withthe observation. A model independent bound | f ′ ( R ) − | < × − has been obtained, where theprime denotes the derivative with respect R and R is the curvature of our Universe at present.Though we use the nonrelativistic approximations and obtain an order of magnitude estimate ofthe bound, it comes from direct observations of gravitational waves and thus it is worth noting.This bound is stronger/equivalent compared to some earlier other bounds such as from the Cassinimission in the Solar-System, Supernova monopole radiation, the observed CMB spectrum, galaxycluster density profile, etc., although it is weaker than best current constraints ( | f ′ ( R ) − | . − )from cosmology. Using the bound obtained, we also constrain the parameter space in the f ( R )theories of dark energy like Hu-Sawicki, Starobinsky, and Tsujikawa models. a [email protected] b [email protected] . INTRODUCTION The recent detection of gravitational waves (GW) by the LIGO collaboration [1–5] pro-vides an unprecedented opportunity to test the theories of gravity beyond GR in the extremestellar environment or strong-field regime per se. Previously, no significant deviation fromGR was found in vacuum or in the weak-field regime through several precision tests [6].Recently, some model independent constraints on deviations from GR have been studiedbased on various GW generation and propagation mechanisms in the observed GW signalsfrom compact black hole binaries [7, 8]. More recently, constraints on a number of theoriesbeyond GR have been obtained from the constraints on the speed of gravitational waves[9, 10].There are several unsolved puzzles in GR, such as resolving the singularities (in black holesand the big bang singularity in cosmology), understanding the dark matter and dark energy,etc. which motivate many researchers to pursue modified gravity theories in the classicaldomain which deviate from GR in ultraviolet and/or infrared energy scales. The simplestand well studied modification is the f ( R ) theory of gravity which is a generalization of theEinstein-Hilbert action by replacing the Ricci scalar ( R ) by a function f ( R ) (see [11, 12]and the references therein for a review). Such theories have some important cosmologicalimplications. For example, Starobinsky [13] gave the first successful f ( R ) = R + αR ( α >
0) model of cosmic inflation, which can account for the early inflationary era withoutany inflationary scalar field. The observed cosmic acceleration (at present) can arise in some f ( R ) theories of gravity without requiring the cosmological constant and the dark energyi.e. a new exotic form of matter. Initial form of the models proposed for this purpose was f ( R ) = R − α/R n ( α > n >
0) [14]. However, this model suffers from various instabilityproblems [15] mainly due to the fact that f ,RR = ∂ f /∂R is negative in this model. Also,it does not satisfy the local gravity constraints [16]. Initially, some viable f ( R ) models wereproposed by Nojiri and Odintsov [17] for resolving these problems. Later, Hu and Sawicki [18]designed a class of models which avoid the instability problems and do satisfy cosmologicaland Solar-System constraints under certain limits of parameter space. Other such viable f ( R ) models were proposed by Starobinsky [19] and Tsujikawa [20]. There are also otherviable models [21] which unify the inflationary paradigm and the late time acceleration alongwith the satisfaction of local tests. Modification at the large scale dynamics in these f ( R )2odels leaves several interesting observational signatures such as the modification to thespectra of galaxy clustering, CMB, weak lensing, etc. [22]. For astrophysical and otherworks in f ( R ) gravity, see [23].An important feature of f ( R ) gravity is that it carries a massive scalar degree of freedomapart from the usual massless spin-2 tensor modes [24]. It can be shown that f ( R ) gravity isdynamically equivalent to Einstein gravity minimally coupled to a scalar field in the Einsteinframe [11]. The scalar field is associated with a nontrivial potential that depends upon theform of the f ( R ) model and couples to matter through the trace of the energy-momentumtensor. In the nonrelativistic limit, the scalar field sources a (finite-range) fifth force whichis added to the usual Newtonian force. The role of this extra scalar field in gravitationalradiation and weak-field metric for simple sources was studied in [25] using the linearizedform of f ( R ) gravity. Their results are among those ones which clearly show the needfor some screening mechanism to suppress this fifth force at the astronomical scales. Insome f ( R ) theories, the fifth force can be screened only at the galactic or Solar-Systemscales through the chameleon mechanism [18, 26, 27]. This mechanism facilitates the abovementioned viable models to conform the local gravity constraints as well as the modifieddynamics at the large scale. Recently, in Refs. [28, 29], the authors have discussed howsuch screening mechanisms in scalar-tensor theories affect the gravitational radiation fromcompact binary systems.Constraints on such f ( R ) theories were obtained by several authors in Solar-System tests[18], and cosmology [30–36] using various observations such as galaxy cluster profiles [31],cluster abundances [32, 33], CMB [34], redshift-space distortions [35], etc. For astrophysicaltests based on the studies of stellar structure, distance measurements, galaxy rotation curves,etc. see Refs. [37–39]. On the other hand, binary systems of compact objects are excellentlaboratory to probe the gravity in the strong field regime. Recently, the authors of Ref. [40]obtained the constraints from the study of orbital period decay of quasicircular neutron star-white dwarf (NS-WD) binary systems using the observational data of PSR J0348 +0432 andPSR J1738 + 0333 [41]. In Ref. [29], the authors compute the waveforms of gravitational-waves (GWs) emitted by such inspiral compact binaries such as neutron star-black hole(NS-BH) and use it to constrain screened modified gravity including the f ( R ) theories.Also, there are some other constraints [42] from the stochastic background of gravitationalwaves. 3n this paper, we constrain independently the f ( R ) gravity (with chameleon mechanism)from the observed GW signals at the LIGO-VIRGO detectors. Static black holes in f ( R ) andother scalar-tensor theories do not have scalar hair [43] and, therefore, are identical as in GR.Although the additional scalar fields are excited in dynamical situations (such as the late-inspiral and merger stage of the binary black hole (BBH) coalescence, or ringdown of singleblack holes [44–46]), the early stages of BBH inspirals in these theories are indistinguishablefrom GR [47]. Therefore, GWs from the inspirals of BBHs [2–4] are not as useful as othercompact binaries (such as NS-BH, BNS, NS-WD, etc.) to constrain f ( R ) gravity. Theauthors of [48] have studied the possibilities to use the future observations of gravitationalradiation from the binary neutron star mergers (BNS) as the probe of f ( R ) gravity. However,they do not consider the chameleon mechanism. Our study is aimed at a phenomenologicalinsight of the observed GW170817 [5] from a BNS merger. In Sec. II, we discuss theGW radiation from the coalescence of binaries in the presence of additional short-rangedscalar force and use GW170817 to constrain it. We use this result in Sec. III to show thatchameleon screening is must in f ( R ) gravity. Then, in Sec. IV, we obtain constraints ongeneral f ( R ) theories which accommodate chameleon mechanism and apply it on the specificdark-energy models such as the Hu-Sawicki, Starobinsky, and Tsujikawa models. Finally,we summarize our results in Sec. V. II. COALESCENCE OF BINARIES AND GW RADIATION FOR THE NEWTONIAN-YUKAWA POTENTIAL
Consider a binary system of two compact objects with masses m and m moving aroundeach other. Let us assume the presence of a short-ranged Yukawa-type modification to thegravitational potential (originated from some scalar field) in addition to the Newtonian term.For nonrelativistic and quasicircular motion of the system, the effective Lagrangian becomes L = 12 µ (cid:16) ˙ r + r ˙ θ (cid:17) + Gm m r + αq q r e − m φ r , (1)where µ = m m m + m is the reduced mass, r is relative separation between the compact objects, α is the coupling constant of the scalar interaction, q and q are the scalar charges, and m φ defines the length scale for which the modification in the potential is important. Theeffect of such an additional term in the gravitational potential will be observed in the LIGO-4IRGO detection window of gravitational waves, if m − φ > O (10) km. When the distancebetween the binary compact objects such as NS-NS is large ( r ≫ m − φ ) the modificationin the gravitational interaction can be neglected. However, when they are close enough( r . m − φ ) the gravitational “fifth” force is switched on. Note that m − φ cannot be verysmall as in that case the “fifth” force becomes relevant when the two NS are very close, in aregime where relativistic corrections are important and the tidal effects may dominate overthe scalar force. However, m − φ should be at least much greater than the impact parameterof the BNS collision, which is of the order of O (10) km. On the other hand, m − φ cannot betoo large also, as in that case the “fifth” force will be turned on during the whole detectionof the GW signal and, hence, this extra force cannot be distinguished from the Newtonianforce. The typical binary separation when the signal enters LIGO-VIRGO window is up to O (1000) km. Therefore, we assume the mass range 10 km << m − φ . r ≪ m − φ , when the signal enters LIGO-VIRGO detectors. Then the modifiedKepler’s law becomes ω = G ( m + m ) r (1 + ˜ α ) (2)where ˜ α = αq q Gm m . As the inspiraling binary radiate gravitational waves, the orbital energy( E ) of the binary system decreases, where E = − GM µ r (1 + ˜ α ) = − µv . (3)In the above equation, M = m + m and v = ωr .5he luminosity of GW emitted is related to the quadrupole moment of the binary massand is given by, L GW = 32 G c µ r ω . (4)Using Eq. (2), we get L GW = 325 c G η (cid:16) vc (cid:17) α ) , (5)where η = µM is called the symmetric mass ratio.In general, scalar dipole radiation also contributes to the total energy loss. However, itvanishes if the scalar-charge to mass ratio of the compact objects are same (i.e. q m = q m )[49, 52]. This is true also for the scalar field originated from f ( R ) gravity, where, for thebinaries consisting of the same type of compact objects (such as NS-NS mergers), the scalardipole radiation vanishes [48]. Thus L GW = − dEdt . Using Eqs. (3) and (5) we get ddt (cid:16) vc (cid:17) = 32 η c GM (cid:16) vc (cid:17) α ) (6)The angular frequency ( ω gw ) of the gravitational wave radiation is directly related to theorbital angular frequency ( ω ) of the binary source such that ω gw = 2 ω . As the orbit decays,the frequency as well as the amplitude of the gravitational wave sweeps upward. This isknown as a chirp and such an inspiral wave form is known as chirp wave form. Using πf gw = ω = v GM (1+˜ α ) in Eq. (6), we get df gw dt = 965 π / G ˆ M c c ! / f / gw , (7)where f gw is the frequency of the emitted gravitational waves. ˆ M c is the modified chirpmass given by ˆ M c = ( m m ) / ( m + m ) / (1 + ˜ α ) / = M c (1 + ˜ α ) / . (8)For ˜ α = 0 we get back the standard chirp mass ( M c ). The expression for GW amplituderemains unchanged as in GR, [49] A gw = 4 Gc D L µω r , (9)where D L is the luminosity distance of the source from the detector.Below, we describe how the extra “fifth” force can be probed from just the chirp masswithout going into the full waveform analysis:6 i) km << m − φ . km: For this mass range of the Yukawa potential, the “fifth”force is switched off during the early binary inspirals of the GW signal and the relevant chirpmass is given by that in GR (i.e. M c ). Where as, for later inspiral stages when the “fifth”force is switched on, the chirp mass gets modified and is given by ˆ M c = M c (1 + ˜ α ) / . So,one can express the modified chirp mass in a compact notation as,ˆ M c = M c ( r > m − φ ) , M c (1 + ˜ α ) / ( r < m − φ ) . (10)Consequently, a signature of “fifth force” in GW signal is that the entire gravitationalwaveform cannot be fitted with a single standard template with a unique chirp mass. Thentwo templates with different masses m E and m L are required to fit the early wave form andlate waveform, respectively. The value of ˜ α can be obtained from the difference between m E and m L . However, if ˜ α is sufficiently small, a single chirp mass may be used for fitting thewhole waveform. Then, an estimation of upper bound on the size of ˜ α can be done fromthe uncertainty in the observed chirp mass (∆ M c , obs ) such that ∆ M c = | ˆ M c − M c | < ∆ M c , obs . This is the case for GW170817, where the observed chirp mass is M c,obs =1 . +0 . − . M ⊙ [5]. From Eq. (8) we get ∆ M c M c = ˆ M c −M c M c ≈ ˜ α and using it we obtain anestimation on the upper bound ˜ α < . α , one can constrain thetheories of gravity where such a gravitational short-ranged “fifth” force appears. (ii) m − φ > km: For this mass range, although the “fifth force” is switched on for thewhole LIGO-Virgo detection band, one can still distinguish the two forces (pure Newtonianand Newtonian-Yukawa) from total mass estimation of the binary system. Note that in GRthe total mass ( M = m + m ) is estimated by using the explicit formula of chirp mass,i.e. M c = (( M − m ) m ) / /M / . Given an observed value of chirp mass, M is minimizedw.r.t. m . This gives an estimation of M as well as the component masses m and m . Asin the case of m − φ & α from the comparison of the estimated total mass and an independent measurement of it(if possible) from the observations (such as GRB etc.) other than GW.However, in our case, we consider 10 km << m − φ . α . 7 II. REVIEW OF f ( R ) THEORIES OF GRAVITY AND THEIR NEWTONIANLIMIT
Scalar-tensor theories of gravity can be possible origin of the additional short-ranged“fifth” force in the Newtonian limit. Massive scalar mode appears in addition to the masslessspin-2 graviton modes in such theories [54]. This massive scalar mode coupled with mattercan generate Yukawa-type potential and, consequently, the “fifth” force at the nonrelativisticlimit. We consider metric f ( R ) theories of gravity, which falls under this class, in our study.In this section, we review the well-known properties of f ( R ) theories of gravity, in particularthe Newtonian limit and the chameleon screening, which we use in the next section. f ( R )theories of gravity are given by the gravitational action in the Jordan frame S J = 116 πG Z d x √− gf ( R ) + S M [ g, Ψ] , (11)where g µν and R are metric tensor components and Ricci scalar in Jordan frame, and Ψ isthe matter field. The field equations are given as f ′ ( R ) R µν − f ( R ) g µν − ∇ µ ∇ ν f ′ ( R ) + g µν ✷ f ′ ( R ) = 8 πGT µν . (12)and trace of the above equation is3 ✷ f ′ ( R ) + f ′ ( R ) R − f ( R ) = 8 πGT. (13)In the Einstein frame, f ( R ) theory can be written down in the form of a scalar-tensorgravity [11] S E = Z d x p − ˜ g " ˜ R πG − ∂ µ φ∂ µ φ − V ( φ ) + S M [ A ( φ )˜ g µν , Ψ] , (14)where the Jordan frame metric is related to Einstein frame metric as g µν = A ( φ )˜ g µν . Theconformal factor A ( φ ) is directly related to f ′ ( R ) = dfdR as A = f ′ ( R ) − . Here, the scalarfield φ is defined as φ = − r πG ln f ′ ( R ) . (15)Then A ( φ ) becomes A ( φ ) = e √ πG φ , (16)and the potential V ( φ ) is V ( φ ) = Rf ′ ( R ) − f ( R )16 πGf ′ ( R ) . (17)8owever, particles follow the geodesics of Jordan frame metric ( g µν ). In the nonrelativisticlimit, it turn out to be [27] d x i dt = − ∂ i Φ N − β ( φ ) M pl ∂ i φ, (18)where Φ N is the Newtonian potential. Thus the “fifth” force is a = − β ( φ ) M pl ∂ i φ, (19)where β ( φ ) = M pl d ln Adφ . (20)Note that M − pl = 8 πG . For f ( R ) theories of gravity, β ( φ ) = 1 / √ φ is ✷ φ = dV ( φ ) dφ − β ( φ ) M pl ˜ T , (21)where ˜ T = ˜ g µν ˜ T µν . ˜ T µν = √− ˜ g ∂ ( √− ˜ gL M ) ∂ ˜ g µν is the stress-energy tensor defined in the Einsteinframe. However, it is not conserved ˜ ∇ µ ˜ T µν = 0. The stress-energy tensor defined in theJordan frame is T µν = √− g ∂ ( √− gL M ) ∂g µν . Actually, the stress-energy tensor in Jordan frameis physically relevant and also conserved, i.e. ∇ µ T µν = 0. The definitions of stress-energytensor in Jordan and Einstein frames are related as T µν = A − ˜ T µν . In the nonrelativisticlimit, T = − ρ ≈ − ˜ ρ = ˜ T . Then Eq. (21) becomes ∇ φ = dV ( φ ) dφ + β ( φ ) ρM pl = dV eff dφ , (22)where the effective potential V eff = V ( φ ) + ρ ln A ( φ ) . (23)The scalar field φ settle down at the minimum of effective potential ( V eff ( φ )) instead of theactual potential ( V ( φ )). The minimum of the effective potential depends upon the density ρ of matter distribution. Consider a spherical object of mass m and radius r ◦ embeddedin the medium of background density ρ . This could represent a star inside a galaxy or agalaxy/dark matter halo/cluster embedded in the cosmological background, in which case ρ is the mean cosmic density. Then the effective potential has minimum at φ = φ min ( ρ ).Far away from the object φ ( r ) → φ . The object of mass m act as the source of perturbationin the uniform background scalar field φ , such that φ = φ + δφ . Then Eq. (22) becomes ∇ δφ − m φ ( φ ) δφ = βM pl δρ ( r ) , (24)9here m φ ( φ ) = V ′′ eff ( φ ) and δρ ( r ) is the mass density profile of the spherical object.Outside the source, the solution for δφ looks like δφ = β πM pl f ( m, r ◦ ) r e − m φ r , (25)where the constant f ( m, r ◦ ) depends upon the structure of the spherical object. For a pointmass (i.e. r ◦ = 0), f ( m, r ◦ ) = m , and assuming m φ r << a = − Gm r , (26)and the total gravitational acceleration (Eq. (18)) in the nonrelativistic limit becomes a r = − Gmr (cid:18) (cid:19) . (27)This is true irrespective of any model of f ( R ) gravity. Thus for point mass in f ( R ) theoriesof gravity and at distances m φ r <<
1, the nonrelativistic gravitational force deviates largelyfrom the Newtonian force up to a factor of 4 /
3; i.e. ˜ α ≈ . f ( R ) theories [29, 43] and, hence, the “fifth” force is absent there (i.e. ˜ α =0). Although BH-BH mergers are dynamical phenomena, still they are not very useful toconstrain f ( R ) gravity as the early stages (motion through 2.5 post-Newtonian order [47])of the binary inspirals are indistinguishable from GR. Therefore, in our case, we considerthe Neutron stars which have finite size. The above mentioned large contribution from the“fifth” force can be suppressed in some f ( R ) theories through the chameleon screening. A. Chameleon screening and thin shell effect
For the models of f ( R ) theories of gravity which admit chameleon screening mechanism(see [27] for review), gravitational “fifth” force is suppressed at small scale such as solarsystems, while strong modification in gravity appears at the cosmological scales. In suchmodels, the form of V ( φ ) becomes such that the effective mass of the scalar field m φ becomesheavier in high density ( ρ ) region and lighter in the low density region. The first exampleof such a model was that of Hu and Sawicki [18]. Other notable examples are Starobinsky[19] and Tsujikawa [20] dark energy models. Chameleon screening is also applicable to finite10ize compact objects such as neutron stars. Therefore, using BNS mergers, we can constrainsuch f ( R ) theories of gravity.In such theories, the field can reach a minimum of the effective potential ( V ′ eff ( φ s ) = 0)also at the centre of the spherical object (neutron star) and remain there ( φ = φ s ) upto some radius r s , at which it enters in the second regime and begins to roll towards itsasymptotic value φ (see Fig. 1(a)). Therefore, there is no “fifth” force interior to r s calledas the screening radius. Then Eq. (24) becomes ∇ δφ = βM pl δρ ( r ) , r s ≤ r ≪ m − φ , , r < r s , (28) (a)For r ≤ r s , φ = φ s and for r → ∞ , φ → φ .Ref. [37] (b)Thin shell effect. Ref. [27]. FIG. 1. Chameleon screening.
After integrating Eq. (28) we get dφdr = β ( m ( r ) − m ( r s ))4 πM pl r , (29)outside the screening radius, where m ( r ) = R r πr ′ δρ ( r ′ ) dr ′ . Then the “fifth” force(Eq. (19)), outside the screening radius, becomes a = − Gm ( r )3 r (cid:18) − m ( r s ) m ( r ) (cid:19) = a N (cid:18) − m ( r s ) m ( r ) (cid:19) . (30)If r s ≪ r ◦ , the “fifth” force is of the order of the Newtonian gravitational force ( a /a N ≈ / r s ≈ r ◦ and a /a N ≪
1. In this case, the fifth-force only receives contributions from the11ass in a thin shell outside the screening radius (see Fig. 1(b)). This phenomenon is calledas the thin-shell effect [27]. Assuming φ s ≈ N as [37] φ ( r ) ≈ βM pl h Φ N ( r ) − Φ N ( r s ) + r s Φ ′ N ( r s ) (cid:16) r − r s (cid:17)i , r ≥ r s , , r < r s . (31)The screening distance r s is related to the background field φ through the following equation χ ≡ φ β M pl = − Φ N ( r s ) − r s Φ ′ N ( r s ) . (32)From Eq. (15), we note that φ depends on the model of f ( R ) gravity as | f ′ ( R ) − | = r φ M pl . (33)Thus the information about the screening distance from the observations can be used toconstrain different f ( R ) theories. IV. CONSTRAINTS ON f ( R ) THEORIES FROM GW170817
From Eqs. (18) and (30), the total gravitational acceleration outside a neutron star ofmass m becomes a r = − Gmr (cid:20) (cid:18) − m ( r s ) m (cid:19)(cid:21) . (34)Using this result for a BNS system of masses m and m , we obtain the effective gravitationalpotential energy of the binary system (in the nonrelativistic limit) V grav = − i = j X i,j = { , } Gm i m j r ij (cid:20) (cid:18) − m ( r s,j ) m j (cid:19)(cid:21) . (35)Note that r = r = r (the binary separation). Then the effective force acting on thereduced mass µ becomes F r = − µ ∂∂r V grav ( r ) = − Gm m r (cid:20) (cid:18) − (cid:18) m ( r s, ) m + m ( r s, ) m (cid:19)(cid:19)(cid:21) . (36)Therefore, ˜ α in Eq. (8) becomes˜ α = 13 (cid:20) − (cid:18) m ( r s, ) m + m ( r s, ) m (cid:19)(cid:21) . (37)12or neutron stars, we assume that the mass density inside the star is almost constant.Therefore, m ( r s ) /m = r s /r ◦ . Further, we assume that the neutron stars for GW170817 arealmost similar (i.e. m ≈ m and r ◦ , ≈ r ◦ , = r ◦ ) and hence, r s, ≈ r s, = r s . Then˜ α ≈ (cid:18) − r s r ◦ (cid:19) (38)Since ˜ α < .
013 from the observations of GW170817, we get r s > . r ◦ using Eq. (38). Thetypical neutron star radius is r ◦ ∼
15 km. This result reveals that neutron stars are differentfrom the main sequence stars where a substantial part of the interior can be unscreened suchthat r s ≈ . r ◦ [37].Next we note that the background field φ is same for both the neutron stars, i.e. χ = φ βM pl = − Φ N ( r s, ) − r s, Φ ′ N ( r s, ) = − Φ N ( r s, ) − r s, Φ ′ N ( r s, ) ≈ − Φ N ( r s ) − r s Φ ′ N ( r s ) . (39)We assume the Newtonian potential for each of the neutron star of masses m ≈ m = m ,Φ N ( r ) ≈ Gm r ◦ (cid:0) r − r ◦ (cid:1) . (40)Then using Eq. (39), we get χ ≈ Gm c r ◦ (cid:18) − r s r ◦ (cid:19) (41)where we divided r.h.s. by c to get the match the dimension and get the correct number.The total mass of BNS merger (GW170817) is M = m + m = 2 . +0 . − . M ⊙ ( M ⊙ is the massof the Sun.). Hence, we assume m ≈ . M ⊙ . Then, χ < × − . Using the estimated χ in Eq. (33) we get | f ′ ( R ) − | < × − . (42)Note that this is still an model independent result, provided the model allows the chameleonscreening. This result is consistent with above mentioned difference between the neutronstars and the main sequence stars. For the Sun (an example of a main sequence star), weget | f ′ ( R ) − | ≈ × − using r s ≈ . r ◦ , m = M ⊙ = 2 × kg (Solar mass), and r ◦ = R ⊙ = 7 × m (Solar radius).Also, we note that above analysis and the result is correct when 10 km << m − φ . << λ c . . × − eV . E φ << . × − eV . Here,13e emphasize on the fact that the energy scale mentioned here is not related to the boundon the graviton mass [53] which was used in [55, 56]. In f ( R ) gravity, graviton is masslessas the spin-2 modes are massless and the mass of the scalar mode ( m φ ) signifies only therange of the scalar force and dispersion in the associated scalar wave [54]. Using Eqs. (15),(17), and (23) we get V ′ eff ( φ ) = βM pl ( Rf ′ ( R ) − f ( R )) f ′ ( R ) + βρM pl , (43) m φ = V ′′ eff ( φ ) = 13 (cid:20) Rf ( R ) + 1 f ′′ ( R ) − f ( R ) f ′ ( R ) (cid:21) . (44)At the background scalar field ( φ ), V ′ eff ( φ ) = 0, which leads to m φ ( φ ) = 13 (cid:20) f ′′ ( R ) − R f ′ ( R ) − πGρ (cid:21) . (45)From Eq. (42), we can safely use f ′ ( R ) ≈ R ≈ πGρ .Then Eq. (45) becomes m φ ( φ ) ≈ f ′′ ( R ) − πGρ . (46)Considering the cosmological background and using the above said assumption on mass ofthe scalar field (10 km << m − φ . . × m << f ′′ ( R ) . . × m . (47)Using the bound on f ′ ( R ) (42) and assumption on f ′′ ( R ) (47), we next constrain Hu-Sawicki, Starobinsky, and Tsujikawa dark energy models. A. Hu-Sawicki model
The Hu-Sawicki dark-energy model is given by f ( R ) = R − µR ( R/R ) n b ( R/R ) n + 1 , (48)where n ≥ µ, b > n , µ , and b are dimensionlessquantities.At present, the Universe is mostly dominated by dark energy. So, we work in the constantcurvature (de Sitter) cosmological background. Then, from Eq. (13), we get f ′ ( R ) R − f ( R ) ≈ , (49)14here R = 4Λ and Λ is the cosmological constant. Using the Hu-Sawicki model (Eq. (48))in Eq. (49), we get [55] b ± = − µ ± p µ ( µ − n ) . (50)Note that µ > n . From Eq. (33), we have | f ′ ( R ) − | = 2 nµ (1 + b ± ) < × − . (51)Assuming n/µ <<
1, the above inequality can not be satisfied for b − . Therefore the allowedroot is b + . Then we obtain nµ < × − . (52)On the other hand we have f ′′ ( R ) = 1 R (cid:20) n µ + 2 nµ ( b + + 1) − n µ ( b + + 1) (cid:21) (53) ≈ ( n + n/ µR , n/µ << . (54)Then using Eq. (47) and Λ ≈ . × − m − , we obtain1 . × − << ( n + n/ µ . . × − . (55)Thus, from Eqs. (52) and (55), we get for n = 1, 10 > µ > , and, for n = 2,3 . × > µ > . × .Relating the galactic density ( ρ gal = 10 − g cm − for the Milky Way) to the cosmologicaldensity we find | f ′ ( R gal ) − | ≈ (cid:18) πρ gal G c Λ (cid:19) − n − | f ′ ( R ) − | , (56)where we used b + ≈ µ >> R gal > R , R gal ≈ πρ gal G/c , and R ≈
4Λ (Λ = 1 . × − m − ). Using n = 1 and Eq. (42) we get at the galactic scale, | f ′ ( R gal ) − | < × − . (57)Above bound on f ′ ( R gal ) is stronger than the bound from Cassini test where | f ′ ( R gal ) − | < × − [18]. 15 . Starobinsky model The Starobinsky dark-energy model [19] is given by f ( R ) = R + λ " R (cid:18) R R (cid:19) − n − , (58)where n ≥ λ >
0. For this model | f ′ ( R ) − | = 2 − n nλ < × − . (59)So λ < × n n × − . On the other hand, using Eq. (47), we have f ′′ ( R ) = n λR n . (60)Using Eq. (47) we obtain 1 . × − << n λ n . . × − . For n = 1, we have 3 × − <λ < × − and, for n = 2, we get 1 . × − < λ < . × − . C. Tsujikawa model
Another such dark energy model is given by [20] f ( R ) = R − νR tanh (cid:18) RR (cid:19) . (61)For this model | f ′ ( R ) − | = 0 . × ν, (62)and f ′′ ( R ) = 0 . × νR . (63)Then we obtain 2 . × − < ν < . × − . V. CONCLUSIONS
In this paper we constrain f ( R ) theories of gravity from recently detected gravitationalwaves at LIGO-VIRGO detectors. We use the observation of GW170817, the first GW signalfrom a binary neutron star merger.In f ( R ) gravity, an extra massive scalar mode appears apart from the massless spin-2modes. This extra scalar mode affects the GW generation in two ways. One is that an16ttractive short ranged “fifth” force adds up to the usual Newtonian gravitational forcebetween two compact objects. The other effect is that the scalar dipole radiation carriesaway some part of the total mechanical energy of the binary system. However, for the BNSmerger, the scalar dipole radiation is negligible as the scalar charge to mass ratio ( q/m ) forboth the objects are same. We assumed that the range of the scalar force is smaller thanthe binary separation when the GW signal enters in the LIGO-VIRGO detection window,such that the scalar force is switched on only for the late binary inspirals. As a result, theeffective chirp mass behaves differently for early and late binary inspirals. Then, from theuncertainty in the observed chirp mass for GW170817, we obtained an estimation of upperbound on the strength of the scalar force ( ˜ α < . f ( R ) theories of gravity will contribute by a largefactor ( ˜ α = 1 / f ( R ) models such as the Hu-Sawicki model admit thechameleon screening which can suppress the effect of the scalar field considerably to conformwith the observations. Due to the chameleon mechanism, the compact objects like stars areself-screened such that only a shell of its interior contributes to the scalar force, which iscalled as the thin shell effect. The observation from GW170817 reveals that most part of theinterior of the neutron stars are screened ( r s > . r ◦ ). This results in a model independentbound on f ( R ) theories of gravity such that | f ′ ( R ) − | < × − where the R is curvatureof the cosmological background spacetime at present. Our assumption on the range of scalarforce translates into the relation 3 . × m << f ′′ ( R ) . . × m . We appliedthese two results in the Hu-Sawicki, Starobinsky, and Tsujikawa models to constrain theparameter space.In the Table I, we compare the constraint on | f ′ ( R ) − | that we obtained with otherbounds available in the literature. We note that, although we have obtained an order ofmagnitude estimate of the bound, it is better than the bounds from Cassini test, Supernovamonopole radiation, and also is as good as the bounds from the study of galaxy cluster den-sity profiles and CMB spectrum. However, this bound is weaker than the bounds obtainedfrom cluster abundances, strong gravitational lensing, redshift-space distortions, distanceindicators in dwarf galaxies, etc. Our present work is based on the analysis in the nonrela-tivistic/Newtonian limit. However, through simple analysis we highlighted some importantnew features such as:(i) even direct observation of chirp mass of compact binaries can be used to constrain f ( R )17 ABLE I. Comparison of the bounds on f ′ ( R ) from different observationsObservations | f ′ ( R ) − | constraints Ref.Solar-System bounds (Cassini mission) . a [18, 27]Supernova monopole radiation < − [58]Cluster density profiles (Max-BCG) < . × − [31]CMB spectrum < − [34] GW170817 (GW from BNS merger) < × − our current work Cluster abundances < . × − [32, 33]CMB + BAO + σ − Ω m relationship b < . × − [59]Strong gravitational lensing (SLACS) < . × − [57]Redshift-space distortions < . × − [35]Distance indicators in dwarf galaxies < × − [38] a This is obtained for the Hu-Sawicki model with n = 1, when translated from the bound | f ′ ( R gal ) − | . × − at the galactic scale [27]. b Taking into account cluster number counts (PSZ catalog) and weak-lensing tomography measurements(CFHTLens). This analysis assumes the Hu-Sawicki model. gravity, without going into detail analysis of the GW waveform,(ii) chameleon screening mechanism is inevitable in f ( R ) theories of gravity in order to con-front with the GW observation from compact binaries,which are worth noting. We intend to study the post-Newtonian phases, in future, whichmay improve the bound we obtained. Also, future observations of the GWs from otherBNS mergers will put tighter constraints on theories of f ( R ) gravity and other scalar-tensorgravity with Chameleon mechanism. ACKNOWLEDGMENTS
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