Gravitational wave in f(R) gravity: possible signature of sub- and super-Chandrasekhar limiting mass white dwarfs
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Gravitational wave in f ( R ) gravity: possible signature of sub- and super-Chandrasekhar limiting masswhite dwarfs Surajit Kalita and Banibrata Mukhopadhyay Department of Physics, Indian Institute of Science, Bangalore 560012, India (Received XXX; Revised YYY; Accepted ZZZ)
Submitted to ApJABSTRACTAfter the prediction of many sub- and super-Chandrasekhar (at least a dozen for the latter) limitingmass white dwarfs, hence apparently peculiar class of white dwarfs, from the observations of luminosityof type Ia supernovae, researchers have proposed various models to explain these two classes of whitedwarfs separately. We earlier showed that these two peculiar classes of white dwarfs, along with theregular white dwarfs, can be explained by a single form of the f ( R ) gravity, whose effect is significantonly in the high-density regime, and it almost vanishes in the low-density regime. However, since thereis no direct detection of such white dwarfs, it is difficult to single out one specific theory from thezoo of modified theories of gravity. We discuss the possibility of direct detection of such white dwarfsin gravitational wave astronomy. It is well-known that in f ( R ) gravity, more than two polarizationmodes are present. We estimate the amplitudes of all the relevant modes for the peculiar as well asthe regular white dwarfs. We further discuss the possibility of their detections through future-basedgravitational wave detectors, such as LISA, ALIA, DECIGO, BBO, or Einstein Telescope, and therebyput constraints or rule out various modified theories of gravity. This exploration links the theory withpossible observations through gravitational wave in f ( R ) gravity. Keywords:
White dwarf stars (1799) — Gravitational waves (678) — Scalar-tensor-vector gravity(1428) — Chandrasekhar limit (221) — Rotation powered pulsars (1408) — Stellar mag-netic fields (1610) — Stellar surfaces (1632) INTRODUCTIONWhite dwarfs (WDs) are the end-state of stars withmass (cid:46) M (cid:12) . A WD attains its stable equilibriumconfiguration by balancing the outward force due tothe degenerate electron gas with the inward force ofgravity. If the WD has a binary companion, it pullsout matter from the companion, and as a result, themass of WD increases. Once the mass of the WDreaches the Chandrasekhar mass-limit (Chandrasekhar1931) ( ∼ . M (cid:12) for carbon-oxygen non-rotating non-magnetized WDs), the pressure balance no longer sus-tains, and the WD bursts out to produce a type Ia super-nova (SNIa) (Choudhuri 2010). The similarity in peak [email protected]@iisc.ac.in luminosities of SNeIa is used as one of the standardcandles to estimate the luminosity distances for vari-ous astronomical and cosmological objects (Lieb & Yau1987; Nomoto et al. 1997). However, recent discoveriesof various under- and over-luminous SNeIa question thecomplete validity of considering luminosities of SNeIa asstandard candle. SNeIa such as SN 1991bg (Filippenkoet al. 1992; Mazzali et al. 1997), SN 1997cn (Turattoet al. 1998), SN 1998de (Modjaz et al. 2001), SN 1999by(Garnavich et al. 2004), and SN 2005bl (Taubenbergeret al. 2008) were discovered with extremely low lumi-nosities, which were produced from WDs with Ni masscontent as low as ∼ . M (cid:12) (Stritzinger et al. 2006). Onthe other hand, a different class of SNeIa, such as, SN2003fg (Howell et al. 2006), SN 2006gz (Hicken et al.2007), SN 2009dc (Yamanaka et al. 2009; Tanaka et al.2010; Silverman et al. 2011; Taubenberger et al. 2011;Kamiya et al. 2012), SN 2007if (Scalzo et al. 2010; Yuan a r X i v : . [ a s t r o - ph . H E ] J a n Kalita & Mukhopadhyay et al. 2010; Scalzo et al. 2012), SN 2013cv (Cao et al.2016), any many more was discovered with an excessluminosity, with the observed mass of Ni as high as ∼ . M (cid:12) (Kamiya et al. 2012), violating the Khokhlovpure detonation limit (Khokhlov et al. 1993). It was in-ferred that these under-luminous SNeIa were producedfrom WDs with mass ∼ . M (cid:12) (Mazzali et al. 1997;Turatto et al. 1998), while the same for over-luminousSNeIa could be ∼ . M (cid:12) (Scalzo et al. 2010; Kamiyaet al. 2012). Hence. these progenitor WDs of peculiarSNeIa violate the Chandrasekhar mass-limit: the under-luminous SNeIa were produced from sub-Chandrasekharlimiting mass WDs (WDs burst before reaching the mass ∼ . M (cid:12) ), and the over-luminous SNeIa were producedfrom super-Chandrasekhar limiting mass WDs (WDsburst well above the mass ∼ . M (cid:12) ). These new mass-limits are important as they may lead to modifying thestandard candle.Various groups around the world have proposed differ-ent models to explain the formation of these two pecu-liar classes of SNeIa. Sub-Chandrasekhar limiting massWDs were believed to be formed by merging two sub-Chandrasekhar mass WDs (double degenerate scenario)leading to another sub-Chandrasekhar mass WD, ex-ploding due to accretion of a helium layer (Pakmor et al.2010; Hillebrandt & Niemeyer 2000). On the other hand,the super-Chandrasekhar WDs were often explained byincorporating different physics, such as a double degen-erate scenario (Hicken et al. 2007), presence of magneticfields (Das & Mukhopadhyay 2013, 2014), presence ofa differential rotation (Hachisu et al. 2012), presence ofcharge in the WDs (Liu et al. 2014), ungravity effect(Bertolami & Mariji 2016), lepton number violation inmagnetized white dwarfs (Belyaev et al. 2015), general-ized Heisenberg uncertainty principle (Ong 2018), andmany more. However, none of these theories can self-consistently explain both the peculiar classes of WDs.Moreover, each of these has some caveats or incomplete-ness, mostly based on the stability (Komatsu et al. 1989;Braithwaite 2009). Furthermore, numerical simulationsshowed that a merger of two massive WDs could neverlead to the mass as high as 2 . M (cid:12) due to the off-centerignition, and formation of a neutron star rather thanan (over-luminous) SNIa (Saio & Nomoto 2004; Mar-tin et al. 2006). Hence, all the conventional picturesfailed to explain the inferred masses of both the sub- andsuper-Chandrasekhar progenitor WDs and also both theclasses of progenitor WDs simultaneously by invokingthe same physics. Moreover, each of the theories canexplain only one regime of SNIa but it seems more likelythat the nature would prefer only one scenario/physicsto exhibit the same class of supernovae. Whether it be an under- or over-luminous SNeIa, other physics such asthe presence of Si etc. remains the same. Therefore, weseem to require just one theory to explain all the SNeIa.Einstein’s theory of general relativity (GR) is un-doubtedly the most beautiful theory to explain the the-ory of gravity. It can easily explain a large numberof phenomena where the Newtonian gravity falls short,such as the deflection of light in strong gravity, gener-ation of the gravitational wave (GW) in 3 + 1 dimen-sion, perihelion precession of Mercury’s orbit, gravita-tional redshift of light, to mention a few. It is well-known that in the asymptotically flat limit where thetypical velocity is much small compared to the speedof light, GR reduces to the Newtonian theory of grav-ity (Ryder 2009). According to the Newtonian theory,Chandrasekhar mass-limit for WDs is achieved only atzero radius with infinite density, whereas GR can con-sistently explain this for a WD with a finite radius anda finite density (Padmanabhan 2001). Nevertheless, fol-lowing several recent observations in cosmology (Joyceet al. 2016; Casas et al. 2017) and in the high-densityregions of the universe, such as at the vicinity of com-pact objects (Held et al. 2019; Banerjee et al. 2020a,b;Moffat 2020), it seems that GR may not be the ultimatetheory of gravity. Starobinsky (1980) first used one ofthe modified theories of gravity, namely R -gravity with R being the scalar curvature, to explain the cosmologyof the very early universe. Eventually, researchers haveproposed a large number of modifications to GR, e.g.,various f ( R ) gravity models, to elaborate the physicsof the different astronomical systems, such as the mas-sive neutron stars (Astashenok et al. 2013, 2014), ac-cretion disk around the compact object (Multam¨aki &Vilja 2006; Pun et al. 2008; P´erez et al. 2013; Kalita &Mukhopadhyay 2019a) and many more. Our group hasalso shown that using the suitable forms of f ( R ) gravity,we can unify the physics of all WDs, including those pos-sessing sub- and super-Chandrasekhar limiting masses(Das & Mukhopadhyay 2015; Kalita & Mukhopadhyay2018). We showed that by fixing the parameters in aviable f ( R ) gravity model such that it satisfies the so-lar system test (Guo 2014), one can obtain the sub-Chandrasekhar limiting mass WDs at a relatively lowdensity and super-Chandrasekhar WDs at high den-sity (Kalita & Mukhopadhyay 2018). Of course, themass–radius relation alters from Chandrasekhar’s origi-nal mass–radius relation depending on the form of f ( R )gravity model. Nevertheless, from the recent detectionof GW through LIGO/Virgo detectors, researchers haveput constraints on the f ( R ) gravity theory (Jana & Mo-hanty 2019; Vainio & Vilja 2017). ravitational wave in f ( R ) gravity f ( R ) gravity is a better bet to explain and unifyall the WDs, including the peculiar ones, to study its va-lidity from observation is extremely necessary. Due tothe failure of their direct detections in electromagneticsurveys, GW astronomy seems to be the prominent al-ternate to detect the peculiar WDs directly. In this way,one can estimate both the mass and the size of the ob-jects, thereby ruling out or putting constraints on thevarious models of f ( R ) gravity. Moreover, there is ahuge debate on the existence of modifications to GR.Hence, if the futuristic GW detectors such as LISA,ALIA, DECIGO, BBO, or Einstein Telescope can de-tect such WDs, it will also be a simple verification forthe existence of the modified theories of gravity. We, inthis article, present various mechanisms that can pro-duce GWs from f ( R ) gravity induced WDs, and discusshow to rule out/single out various theories from suchobservations.The article is organized as follows. In §
2, we discussthe properties of gravitational radiation in f ( R ) gravity.In §
3, we discuss the generation of GW from f ( R ) grav-ity induced WDs through various mechanisms such asthe presence of roughness at the surface of WDs, or thepresence of magnetic fields and rotation in the WDs. In §
4, we discuss the amplitude and luminosity of the gravi-tational radiation emitted from these isolated WDs, andwhether the futuristic detector can detect them or not.In §
5, we discuss the various results and their physicalinterpretations. In this section, we mainly discuss how to extract information about the WDs from GW de-tection, and thereby to put constraints on the modifiedtheories of gravity, before we conclude in § GRAVITATIONAL WAVE IN f ( R ) GRAVITYAssuming the metric signature to be ( − , + , + , +) infour dimensions, the action in f ( R ) gravity (modifiedEinstein-Hilbert action) is given by (De Felice & Tsu-jikawa 2010; Will 2014; Nojiri et al. 2017) S f ( R ) = (cid:90) (cid:20) c πG f ( R ) + L M (cid:21) √− g d x , (1)where c is the speed of light, G Newton’s gravitationalconstant, L M the Lagrangian of the matter field and g =det( g µν ) the determinant of the spacetime metric g µν .Varying this action with respect to g µν , with appropriateboundary conditions, we obtain the modified Einsteinequation in f ( R ) gravity, given by F ( R ) R µν − g µν f ( R ) − ( ∇ µ ∇ ν − g µν (cid:50) ) F ( R ) = κT µν , (2)where R µν is the Ricci tensor, T µν the matter stress-energy tensor, F ( R ) = d f ( R )/d R , κ = 8 πG/c , (cid:50) thed’Alembertian operator given by (cid:50) = − ∂ t /c + ∇ with ∂ t being the temporal partial derivative and ∇ the 3-dimensional Laplacian. For f ( R ) = R , it is obvious thatEquation (2) will reduce to the field equation in GR(Ryder 2009). The trace of Equation (2) is given by RF ( R ) − f ( R ) + 3 (cid:50) F ( R ) = κg µν T µν = κT. (3)Since we plan to explore f ( R ) gravity models, which canexplain both the sub- and super-Chandrasekhar limitingmass WDs together, the first higher-order correction toGR, i.e., f ( R ) = R + αR seems to suffice for this pur-pose. However, in this model, one needs to vary themodel parameter α to obtain both the regimes of theWDs (Das & Mukhopadhyay 2015) and, hence, this isprobably not the best model for this purpose. There-fore, we need to consider the next higher order correctionterms, i.e., f ( R ) = R + αR { − γR + O ( R ) } (Kalita& Mukhopadhyay 2018), to remove the deficiency of theprevious model. In this model, one does not need to varythe parameters α and γ present in the model. Ratherone needs to fix them from the Gravity Probe B ex-periment and, then, just changing the central density,one can obtain both the sub- and super-Chandrasekharlimiting mass WDs. Kalita & Mukhopadhyay (2018)provided a detailed analysis of this considering varioushigher order corrections to GR and establishing thatthey still pass the solar system test.Since we are interested in the most common sce-nario where GW propagates in vacuum, i.e. in the flat Kalita & Mukhopadhyay
Minkowski space-time, we need to linearize both g µν aswell as R . The perturbed forms for g µν and R are givenby g µν = η µν + h µν , (4) R = R + R , (5)with | h µν | (cid:28) | η µν | , where η µν is the backgroundMinkowski metric, R the unperturbed backgroundscalar curvature, and h µν and R are respectivelythe tensor and scalar perturbations. Of course, forMinkowski vacuum background, R = 0 and T = 0,where T being the trace of the background stress-energytensor. Now perturbing the equations (2) and (3), andsubstituting the above relations, we obtain the linearizedfield equations, given by (Capozziello et al. 2008; Lianget al. 2017) (cid:50) ¯ h µν = − πGc T µν (6) (cid:50) h f − m h f = 8 πG F ( R ) c T, (7)where ¯ h µν = h µν − ( h/ − h f ) η µν with h = η µν h µν and h f = F (cid:48) ( R ) R /F ( R ). Here, m is the effective massassociated with the scalar degree of freedom in f ( R )gravity (Sotiriou & Faraoni 2010; Prasia & Kuriakose2014; Sbis`a et al. 2019), given by m = 13 (cid:26) F ( R ) F (cid:48) ( R ) − R (cid:27) , (8)where F (cid:48) = d F /d R . Of course, in Minkowski vacuumbackground, m = 1 / { F (0) /F (cid:48) (0) } . It is evident that m depends on the background density, which is knownas the chameleon mechanism in f ( R ) gravity (Liu et al.2018; Burrage & Sakstein 2018). In Minkowski vacuumbackground, for f ( R ) = R + αR (1 − γR ) with F ( R ) =1 + αR (2 − γR ), the equations (6) and (7) reduce to (cid:50) ¯ h µν = − πGc T µν (9) (cid:0) (cid:50) − m (cid:1) R = 4 πG αc T, (10)with m = 1 / α . When the GW propagates in thevacuum, these equations reduce to (Liang et al. 2017;Capozziello et al. 2008) (cid:50) ¯ h µν = 0 , (cid:0) (cid:50) − m (cid:1) R = 0 . (11)The first equation is similar to the equation obtained inGR, which means ¯ h µν satisfies the transverse-traceless(TT) gauge condition. This leads to the fact that thereis a presence of only two propagating degrees of free-dom/polarization for ¯ h µν (namely ¯ h + and ¯ h × ). In GR, since α →
0, or equivalently m → ∞ , only the ten-sor equation gives the propagating polarization modes.The scalar mode, being infinitely massive in GR, can nolonger propagate and, hence, the corresponding scalarequation serves as a constraint equation for the tensormodes . On the other hand, in f ( R ) gravity, since R (cid:54) = 0(even in vacuum, due to the presence of R ), there is apresence of an extra propagating scalar degree of polar-ization, also known as the breathing mode. Hence thenumber of polarizations in f ( R ) gravity turns out to be3, unlike the case for GR where it is 2 (Kausar et al.2016; Gong & Hou 2018). For a plane wave traveling in z − direction, the solutions of the above wave Equations(11), are given by¯ h µν ( z, t ) = ˆ h µν exp [ i ( ωz/c − ωt )] , (12) R ( z, t ) = ˆ R exp (cid:104) i (cid:16)(cid:112) ˜ ω − m c z/c − ˜ ωt (cid:17)(cid:105) , (13)where ω is the frequency of the tensorial modes and˜ ω is that of the scalar mode. It is evident that thetensorial modes, being massless, propagate at a speed v t = c , whereas the massive scalar mode propagateswith a group velocity v s = c √ ˜ ω − m c / ˜ ω < c (Yanget al. 2011).Our aim is to calculate the strength of GW gener-ated from f ( R ) gravity induced WDs. Hence, we needto solve Equations (9) and (10). Since these are inho-mogeneous differential equations, we use the method ofGreen’s function. The Green’s function for the operator( (cid:50) − m ) or ( − ∂ t /c + ∇ − m ) is given by (Berry &Gair 2011; Dass & Liberati 2019) G m ( x, x (cid:48) ) = (cid:90) d p (2 π ) exp [ ip. ( x − x (cid:48) )] m + p , (14)where p ≡ ( ω/c, k ) with k being the wavenumber, suchthat p = − ω /c + k . For a spherically symmetric sys-tem with x ≡ ( ct, r = | x − x (cid:48) | , θ, φ ), where x (cid:48) and x arerespectively source and observer (detector) positions, itreduces to G m ( x, x (cid:48) ) = (cid:82) d ω πc exp [ − iω ( t − t (cid:48) )] πr exp (cid:20) i (cid:113) ω c − m r (cid:21) if ω > m c (cid:82) d ω πc exp [ − iω ( t − t (cid:48) )] πr exp (cid:20) − (cid:113) m − ω c r (cid:21) if ω < m c . (15) We can also investigate the other extreme regime, i.e., m → α → ∞ . From Gravity Probe B experiment, the bound on α is | α | (cid:46) × cm (N¨af & Jetzer 2010). It is evident that α → ∞ naturally violates this bound, and hence, this limit is unphysicalin the present context. ravitational wave in f ( R ) gravity (cid:50) operator is given by (Berry & Gair 2011) G ( x, x (cid:48) ) = δ ( t − t (cid:48) − r/c )4 πrc . (16)Therefore, from Equation (9), the solution for ¯ h µν isgiven by ¯ h µν = − Gc (cid:90) d x (cid:48) T µν ( t − r, x (cid:48) ) r . (17)Since ¯ h µν follows TT gauge condition like GR, only thespace part of the ¯ h µν contributes. Assuming the detec-tor to be far from the source such that x (cid:29) x (cid:48) , the aboveequation reduces to the following form (Ryder 2009)¯ h ij = − Gc r ¨ Q ij , (18)where Q ij is the quadrupolar moment of the system with i, j = 1 , ,
3. Moreover, from Equation (10), the solutionfor R is given by R = 4 πG αc (cid:90) d x (cid:48) G m ( x, x (cid:48) ) T ( x (cid:48) ) . (19)The stress-energy tensor for a perfect fluid is given by T µν = ( ρc + P ) u µ u ν + P g µν , (20)where P is the pressure, ρ the density and u µ the four-velocity of the fluid. The trace of T µν is given by T = − ρc + 3 P . For the case of WD, the equation of state(EoS), known as Chandrasekhar EoS, is governed by thedegenerate electron gas. It is given by (Chandrasekhar1935) P = m e c π (cid:126) (cid:20) x F (2 x F − (cid:113) x F + 1 + 3 sinh − x F (cid:21) ,ρ = µ e m H ( m e c ) π (cid:126) x F , (21)where x F = p F /m e c , p F is the Fermi momentum, m e the mass of an electron, (cid:126) the reduced Planck’s con-stant, µ e the mean molecular weight per electron and m H the mass of a hydrogen atom. For our work, wechoose µ e = 2 indicating the carbon-oxygen WDs. Itis evident, from this EoS, that ρc (cid:29) P and, hence, T ≈ − ρc . Moreover, from the Gravity Probe B ex-periment,the bound on α is | α | (cid:46) × cm (N¨af &Jetzer 2010). We already showed in our previous workthat α = 3 × cm is enough to probe both sub-and super-Chandrasekhar limiting mass WDs simulta-neously (Kalita & Mukhopadhyay 2018). For this valueof α , in vacuum background, m = 1 / α ≈ . × − cm − and, hence, the cut-off frequency turns out to be ω c = mc ≈ . − . The rotation period Ω rot ∼ ω of a WD is always (cid:46)
10 rad s − . Hence, in Equation(15), ω (cid:28) m c = ω c is satisfied, and the Green’s func-tion reduces to (Stabile 2010) G m ( x, x (cid:48) ) = δ ( t − t (cid:48) )4 πrc e − mr . (22)Therefore, using Equation (19), for a WD, the solutionfor R is given by R = G αc (cid:90) d x (cid:48) ρ ( x (cid:48) ) | x − x (cid:48) | e − m | x − x (cid:48) | . (23)The typical distance of a WD from the earth is ∼ ≈ . × cm), which means x (cid:29) x (cid:48) . Hence,for a WD with mass M , R reduces to R ∝ GM αc r e − mr . (24)For the chosen values of m and r , mr ≈ . × , whichmeans the scalar mode’s amplitude is exponentially sup-pressed enormously (Katsuragawa et al. 2019), and thedetectors cannot detect them. Hence, in the rest of thearticle, we discuss only the tensorial modes ¯ h ij givenby Equation (18), and for convenience, we remove ‘bar’from h hereinafter. MECHANISMS FOR GENERATING GW FROMAN ISOLATED WDIn this section, we discuss a couple of mechanisms thatcan generate gravitational radiation from an isolatedWD. From Equation (18), it is evident that a systemcan emit gravitational radiation if and only if the systempossesses a time-varying quadrupolar moment such that¨ Q ij (cid:54) = 0. Hence, neither a spherically symmetric systemnor an axially symmetric system can radiate GW, anda tri-axial system is required. In a tri-axial system, themoment of inertia is different along all the three spa-tially perpendicular axes. There are mainly two waysby which a WD can be tri-axial (Shapiro & Teukolsky1983). First, the rotating WD already possesses rough-ness at its surface, may be due to the presence of moun-tains and holes (craters). The second possibility beingthat the WD possesses the magnetic field and rotation,with a non-zero angle between their respective axes. Wenow discuss each of these possibilities one by one.3.1. GW due to roughness of the surface
If a WD possesses asymmetry of matter at its sur-face, the moments of inertia of the WD are differentalong all directions, making the system a tri-axial one.If such a WD rotates with angular velocity Ω rot , it can
Kalita & Mukhopadhyay emit effective gravitational radiation continuously. Sup-pose the moments of inertia for such a system be I , I and I along x − , y − , z − axes respectively such that I < I < I . Thereby, using Equation (18), the twotensorial polarizations of GW, at time t , are given by(Zimmermann 1980; Van Den Broeck 2005; Maggiore2008) h + = A + , cos (2Ω rot t ) + A + , cos [(Ω rot + Ω p ) t ]+ A + , cos [2(Ω rot + Ω p ) t ] ,h × = A × , sin (2Ω rot t ) + A × , sin [(Ω rot + Ω p ) t ]+ A × , sin [2(Ω rot + Ω p ) t ] , (25)where A + , = ( h / (cid:0) i (cid:1) ,A + , = 2 h (cid:48) ( I a/I b ) sin i cos i,A + , = 2 h (cid:48) ( I a/I b ) (cid:0) i (cid:1) ,A × , = h cos i,A × , = 2 h (cid:48) ( I a/I b ) sin i,A × , = 4 h (cid:48) ( I a/I b ) cos i, (26)with i being the inclination angle between the rotationaxis of the WD and the detector’s line of sight, and h = − G Ω I rc (cid:18) I − I I (cid:19) ,h (cid:48) = − G (Ω rot + Ω p ) I rc (cid:18) − I + I I (cid:19) . (27)Here a and b are given by a = (cid:115) EI − ˜ M I ( I − I ) , b = (cid:115) ˜ M − EI I ( I − I ) , (28)where ˜ E = (cid:0) I Ω + I Ω + I Ω (cid:1) / M = I Ω + I Ω + I Ω , with Ω , Ω , Ω are the components of ini-tial angular velocity along x − , y − , z − axes respectively(Landau & Lifshitz 1982). The precession frequency isgiven by Ω p = πb K ( ˜ m ) (cid:20) ( I − I )( I − I ) I I (cid:21) / , (29)where K ( ˜ m ) is the complete elliptic integral of the firstkind, with the ellipticity parameter ˜ m given by˜ m = ( I − I ) I a ( I − I ) I b . (30)The rotation frequency is given byΩ rot = ˜ MI + 2 bK ( ˜ m ) (cid:20) ( I − I )( I − I ) I I (cid:21) / × ∞ (cid:88) n =1 q n − q n sinh (2 πnc ) − Ω p , (31) where q = exp {− πK (1 − ˜ m ) /K ( ˜ m ) } , and c satisfiesthe following equationsn [2 ic K ( ˜ m ) , ˜ m ] = i I bI a . (32)It is evident from the set of Equations (25) that for atri-axial system, GW is associated with three frequen-cies, which implies that one should expect three distinctlines in the spectrum. In reality, not only three in thespectrum, but lines with higher frequencies, arising fromhigher order terms, may also be present, whose, however,the intensity is suppressed (Maggiore 2008).3.2. GW due to breaking of axial symmetry throughrotation
One more possibility of generating GW from an iso-lated WD is by breaking the pre-existence axial symme-try through rotation. A WD can be axially symmetric ifit possesses a magnetic field (either toroidal or poloidalor any other suitable mixed field configuration). Now,if the WD rotates with a misalignment between its ro-tation and magnetic axes (similar configuration like aneutron star pulsar), it can emit gravitational radiationcontinuously. For such an object, with χ being the an-gle between the magnetic field and rotation axes, usingEquation (18), the two tensorial polarizations of GW aregiven by (Zimmermann & Szedenits 1979; Bonazzola &Gourgoulhon 1996) h + = ˜ A + , cos (Ω rot t ) − ˜ A + , cos (2Ω rot t ) ,h × = ˜ A × , sin (Ω rot t ) − ˜ A × , sin (2Ω rot t ) , (33)where ˜ A + , = ˜ h sin 2 χ sin i cos i, ˜ A + , = 2˜ h sin χ (1 + cos i ) , ˜ A × , = ˜ h sin 2 χ sin i, ˜ A × , = 4˜ h sin χ cos i, (34)with ˜ h = Gc Ω ( I − I ) r . (35)Hence, in this configuration, continuous GW is emittedat two frequencies viz. Ω rot and 2Ω rot . STRENGTH OF GW EMITTED FROM ANISOLATED WDIn this section, we discuss the strength of GW emit-ted from an f ( R ) gravity induced isolated WD. We ear-lier showed that in the presence of f ( R ) gravity with f ( R ) = R + αR (1 − γR ), where α = 3 × cm and γ = 4 × cm , it is possible to obtain sub-Chandrasekhar limiting mass WDs as well as super-Chandrasekhar WDs, along with the regular WDs just ravitational wave in f ( R ) gravity R ( k m ) A BC D f ( R ) gravityGR0.5 1.0 1.5 2.0 2.5 3.0 M ( M fl )10 ρ c ( g c m − ) A BC D
Figure 1.
The variation of radius and central density withrespect to the mass of the WD. varying the central density ρ c of the WDs (Kalita &Mukhopadhyay 2018). We also showed that, with thechosen values of the parameters, this model is valid interms of the solar system test (Guo 2014). For demon-stration, we recall the key results from that work, de-picted in Figure 1, which shows the variation of radius R and ρ c with respect to M . WDs following GR areshown in green dashed line and the red solid line corre-sponds to the f ( R ) gravity induced WDs. Indeed, thereare some WDs, e.g., EG 50, GD 140, J2056-3014, etc.(Provencal et al. 1998; Lopes de Oliveira et al. 2020),which do not follow the standard Chandrasekhar mass–radius relation. Moreover, it is to be noted that weadopt the perturbative calculations and consider the ex-terior solution of the WD to be the Schwarzschild solu-tion while obtaining the mass–radius curve in f ( R ) grav-ity. As a result, R is asymptotically flat outside the WD(Ganguly et al. 2014; Capozziello et al. 2016). More-over, in perturbative analysis, a WD is unstable underthe radial perturbation if it falls in the branch where ∂M/∂ρ c < ∂M/∂ρ c > ∂M/∂ρ c > ρ c , the effect of modified gravity isnegligible, as both the curves overlap with each other(mostly in the branch AB). As ρ c increases, the mass– radius curve reaches a maximum at the mass ∼ M (cid:12) and ρ c ∼ . × g cm − (point B). Beyond this ρ c ,the curve turns back which violates the positivity con-dition and, hence, BC is an unstable branch. Therefore,point B corresponds to the sub-Chandrasekhar limit-ing mass WD. Further, reaching a minimum value, thecurve again turns back from the point C, and quicklyenters in the super-Chandrasekhar WD regime follow-ing ∂M/∂ρ c >
0. Since the branch CD is stable, thesuper-Chandrasekhar WDs are stable under radial per-turbation. The maximum ρ c is chosen in such a waythat it does not violate any of the known physics for COWDs, such as neutron drip (Shapiro & Teukolsky 1983),pycno-nuclear reactions and inverse beta decay (Otonielet al. 2019), etc. The empirical relation between R and M , in the branch AB is approximately R ∝ M − / , inthe branch BC is R ∝ M , and that of in the branchCD is R ∝ M − / . In this way, this form of f ( R ) grav-ity, with the chosen parameters, can explain both thesub- and super-Chandrasekhar limiting mass WDs justby varying the ρ c . It is however evident that there is nosuper-Chandrasekhar mass-limit in this model. Such amass-limit is possible if we consider higher-order correc-tions (at least 16 th order) to the Starobinsky model asdiscussed earlier (Kalita & Mukhopadhyay 2018). Thebeauty of this model is that at the low-density limit, theextra terms of modified gravity have a negligible effect,and GR is enough to describe the underlying physics.In the case of a star, its density is relatively small com-pared to the WD’s density and, hence, even if the same α and γ remain intact, their effect is not prominent inthe star phase. It brings in significant effect only whenthe star becomes a WD depending on the density. Now,based on this mass–radius relation, we discuss the cor-responding strength of GW separately for each of thepossibilities, mentioned in the previous section.4.1. Presence of roughness at WD’s surface
We have already mentioned that if a pre-existed tri-axial WD rotates, it can produce continuous gravita-tional radiation. One possibility for a WD of being tri-axial is due to the asymmetry of matter present at itssurface. One can imagine this configuration as a sim-ilar structure of the earth or moon, where there aremountains and craters (holes) at the surface. We assumethat there are excess ‘mountains’ along the x − axis and‘holes’ along the z − axis of a WD, as shown in Figure 2.Such a configuration guarantees a tri-axial system with I < I < I . For simplicity, we assume Ω = Ω = Ω .Let us now check the maximum possible height of amountain on a WD’s surface. Assuming the mountains Kalita & Mukhopadhyay xz mountainsholes Figure 2.
Exaggerated figure of the presence of mountainsand holes at the surface of a WD. to be generated due to the shear in the outer envelopeof WDs (same as the case for the earth where thereare mountains at its crust), the maximum height of amountain is given by (Sedrakian et al. 2005) H = Sρg , (36)where ρ is the average density of the mountain, g theacceleration due to gravity at the surface of the WD,and S the shear modulus which is given by (Mott &Jones 1958; Baym & Pines 1971) S = 0 . Z e n / e , (37)with e being the charge of an electron, Z the atomicnumber, and n e the electron number density in themountain. Substituting this expression, the maximumheight of the mountain reduces to H = 3 . × Z g ρ / cm . (38)Assuming the C-O WDs with the surface mostly con-sisting of He, we choose Z = 2. Moreover, assuming themountains’ average density to be the density just be-low the surface of the WDs, i.e. ρ ≈ − ρ c , we obtain H for various WDs with different values of ρ c , which isshown in Figure 3. From the empirical relations men-tioned in the previous section, we also obtain the samebetween H and M : in the branch AB, H ∝ M − / , inCD, H ∝ M − / , while in BC, H is nearly constant.It is evident from Figure 3 that H (cid:28) R . However,if there is a series of mountains of similar heights on M ( M fl )10 -2 -1 H ( k m ) A BC D
Figure 3.
The maximum height of a mountain present ona WD as a function of mass. the x − direction and big holes on the z − direction, asshown in Figure 2, effective radii of the WDs alter. Theeffective radii in x − , y − and z − directions become R + H , R and R − H respectively, resulting in a tri-axialsystem. Hence, the moments of inertia along differentdirections are given by I = M (cid:104) R + ( R − H ) (cid:105) ,I = M (cid:104) ( R + H ) + ( R − H ) (cid:105) ,I = M (cid:104) R + ( R + H ) (cid:105) . (39)Moreover, we know that rotation can also increase themass and radius of a WD. Hence, we assume the angularfrequency to be 1 / / (cid:112) GM/ R , such that it doesnot affect the mass and size of the WD. Using the setof Equations (26), we obtain the dimensionless strainamplitude of GW, A (e.g., A + , , A × , , etc.), emittedfrom WDs with rough surface, corresponding to all threefrequencies of the spectrum. Moreover, we know thatthe integrated signal to noise ratio (SNR) increases ifwe observe the source for a long period of time T . Therelation between SNR and T is given by (Maggiore 2008;Thrane & Romano 2013; Sieniawska & Bejger 2019)SNR = 1 √ (cid:18) T S ( ν ) (cid:19) / A, (40)where S ( ν ) is the power spectral density (PSD) ata frequency ν . We show PSD of various detectorsas a function of frequency (Sathyaprakash & Schutz http://gwplotter.com/ ravitational wave in f ( R ) gravity − − − − − ν (Hz)10 − − − − − − p S ( ν ) & p T / A ( H z − / ) ETLISA ALIA BBO DECIGO
Massive binariesGalactic binariesExtrememassratioinspirals Compact binary inspirals i = 0 ◦ A + , A + , A + , A × , A × , A × , (a) − − − − − ν (Hz)10 − − − − − − p S ( ν ) & p T / A ( H z − / ) ETLISA ALIA BBO DECIGO
Massive binariesGalactic binariesExtrememassratioinspirals Compact binary inspirals i = 30 ◦ A + , A + , A + , A × , A × , A × , (b) − − − − − ν (Hz)10 − − − − − − p S ( ν ) & p T / A ( H z − / ) ETLISA ALIA BBO DECIGO
Massive binariesGalactic binariesExtrememassratioinspirals Compact binary inspirals i = 60 ◦ A + , A + , A + , A × , A × , A × , (c) − − − − − ν (Hz)10 − − − − − − p S ( ν ) & p T / A ( H z − / ) ETLISA ALIA BBO DECIGO
Massive binariesGalactic binariesExtrememassratioinspirals Compact binary inspirals i = 90 ◦ A + , A + , A + , A × , A × , A × , (d) Figure 4. (cid:112) T / A for f ( R ) gravity induced WDs with rough surfaces for different i over 5 s integration time along with variousdetectors’ PSD. (cid:112) T / A for f ( R ) gravity induced WDs with ρ c =10 , , , , , . × g cm − over T = 5s assuming r = 100 pc. It is evident that many ofthese dense WDs will readily, or at most in a few sec-onds, be detected by DECIGO and BBO with SNR (cid:38) ≈ T ∼
102 mins. However, to detect these WDs by ALIAor LISA with the same SNR, the integration time turnsout to be
T ∼
T ∼ (cid:112) T / A for different integration times for these WDs. Itis evident from this figure that SNR increases as the inte-gration time increases, allowing possibility of detectingthese WDs even by the Einstein Telescope and ALIA.Furthermore, due to the emission of gravitational radi-ation, it is associated with the quadrupolar luminosity, given by (Zimmermann 1980) L GW = − d E d t = − I Ω rot ˙Ω rot ≈ G c b ( I − I ) + 2 G c a b (cid:18) I − I + I (cid:19) . (41)Since there is no magnetic field in this WD configura-tion, there will be no associated electromagnetic coun-terpart. In other words, these WDs do not emit anydipole radiation. Nevertheless, due to the emission ofGW radiation, the WD starts spinning down, i.e., Ω rot decreases with time. After a certain period of time(characteristic timescale, P , of a WD pulsar), it will loseall its rotational energy and can no longer radiate anygravitational radiation. Using the expression for L GW ,we obtain P ≈ I c G Ω Y ( I − I ) + XY (2 I − I − I ) , (42)with X = 1 + I ( I − I ) I ( I − I ) and Y = 1 + I ( I − I ) I ( I − I ) . Kalita & Mukhopadhyay
Figure 6 shows the variation of L GW and P with respectto M of WD. The empirical relations for L GW and P invarious branches are given in Table 1. It is evident thatthe life-time of massive WD pulsars is shorter than thatof the lighter WDs.4.2. Presence of magnetic field in WD
As mentioned in the previous section, if a magnetizedWD rotates with a misalignment between its magneticfield and rotation axes (similar configuration of a pul-sar), it can emit continuous GW. We already provided adetailed discussion on GW emitted from WDs with dif-ferent magnetic field geometries and strengths in GR(Kalita & Mukhopadhyay 2019b; Kalita et al. 2020).Figure 7 shows an illustrative diagram of a magnetizedWD where the magnetic field is along z (cid:48) − axis and rota-tion is along the z − axis, with χ being the angle betweenthese two axes. We calculate the amplitude of GW us-ing the set of Equations (34) assuming the difference inradii of the WD between those along x − and z − axesbe 0 . (cid:15) = | I − I | /I ≈ × − , due to thepresence of a very weak magnetic field and slow rota-tion. The choice of weak fields and slow rotation assuresthat the underlying WD mass–radius solutions do notpractically differ from the solutions based on the f ( R )gravity without magnetic fields and rotation. In future,we plan to check rigorously by solving the set of equa-tions, if indeed such (cid:15) is possible in the presence of weakmagnetic fields and rotation keeping the mass and ra-dius practically intact. As we will show below, however,the chosen (cid:15) appears to be minimally required value tohave any appreciable effect. Nevertheless, there are ex-amples of weakly magnetized WD pulsars, which canbe explained even in GR framework, e.g. AE Aquarii(Bookbinder & Lamb 1987), AR Scorpii (Marsh et al.2016), where magnetic fields hardly affect their mass–radius relations. Figure 8 shows PSD as a functionof frequency for various detectors along with (cid:112) T / A over 5 s integration time for various f ( R ) gravity in-duced WD pulsars with different i assuming χ = 90 ◦ and r = 100 pc. It is evident that while DECIGO andBBO can immediately detect such weakly magnetizedsuper-Chandrasekhar WDs, the Einstein Telescope candetect them in T ∼ ≈ T ∼
T ∼ . Hence, it is also possible to detect Note that even if the threshold SNR for detection increasesslightly (say 5 to 20), many of these sources can still be detectedin a few seconds to a few days of integration time depending onthe type of the detectors. such weakly magnetized WDs using ALIA, whereas forLISA, it is quite impossible. Figure 5(b) depicts (cid:112) T / A for these WDs with different integration times to showthat SNR increases if the integration time increases sothat various detectors can detect them eventually. Forsuch a system, the GW luminosity is given by (Zimmer-mann & Szedenits 1979) L GW ≈ G c ( I − I ) Ω sin χ (cid:0) χ (cid:1) . (43)It is expected that a source can emit electromagneticradiation in the presence of a magnetic field, and it is thedipole radiation in the case of a WD pulsar. However,because of the presence of a weak magnetic field, thedipole radiation emitted from such a WD is minimal,and the corresponding dipole luminosity is negligible ascompared to L GW . Hence the spin-down timescale ismostly governed by L GW , given by (Kalita et al. 2020) P ≈ (cid:32) I c G ( I − I ) Ω (cid:33) χ (cid:0) χ (cid:1) . (44)Figure 9 shows the variation of L GW and P with respectto M for various WDs with χ = 90 ◦ . The maximum L GW in the case of a WD is ∼ erg s − . The em-pirical relations of L GW and P , in various branches, aresame as the previous case provided in Table 1. It is alsoclear from the figure that the massive WD pulsars areshort-lived as compared to the lighter ones. DISCUSSIONFrom Figures 4 and 8, we observe that the GW fre-quency of isolated WDs could be much larger as com-pared to that of galactic binaries. This is because, in thecase of isolated WDs, the spin frequency is responsiblefor the GW generation, whereas, in the case of binaries,their orbital periods are essential. Therefore, the confu-sion noise of the binaries does not affect the detection ofthe isolated WDs. However, the GW frequency of someother sources, such as massive binaries and compact bi-nary inspirals, is similar to that of isolated WDs. Hence,using specific templates for binaries, the objects can bedistinguished from each other. Of course, the frequencyrange of isolated WDs is such that neither the nano-hertz detectors, such as IPTA, EPTA, and NANOGrav,nor the other currently operating ground-based detec-tors such as LIGO, VIRGO and KAGRA, can detectthem.From Figure 1, it is evident that the GR dominatedWDs are considerably big as compared to the f ( R )gravity dominated ones, particularly at higher centraldensities, provided they possess the same central den-sity. Moreover, using the equations (27) and (35), we ravitational wave in f ( R ) gravity − − − − − ν (Hz)10 − − − − − − p S ( ν ) & p T / A ( H z − / ) ETLISA ALIABBO DECIGO
Massive binariesGalactic binariesExtrememassratioinspirals Compact binary inspirals i = 30 ◦ T = 5 s T = 2 hrs T = 3 months (a) WDs with surface roughness. − − − − − ν (Hz)10 − − − − − − p S ( ν ) & p T / A ( H z − / ) ETLISA ALIABBO DECIGO
Massive binariesGalactic binariesExtrememassratioinspirals Compact binary inspirals i = 30 ◦ T = 5 s T = 7 mins T = 1 week (b) WDs with magnetic field. Figure 5. (cid:112) T / A for f ( R ) gravity induced WDs with i = 30 ◦ and different integration time along with various detectors’PSD. Table 1.
The empirical relations of various quantities with respect to mass of the WDs.Quantity AB branch BC branch CD branchRadius
R ∝ M − / R ∝ M R ∝ M − / Maximum height H ∝ M − / H ≈ constant H ∝ M − / Luminosity L GW ∝ M L GW ≈ constant L GW ∝ M / Timescale P ∝ M − / P ∝ M P ∝ M − L G W ( e r g s − ) A BC D0.5 1.0 1.5 2.0 2.5 3.0 M ( M fl )10 P ( y r) A BC D
Figure 6.
The variation of L GW and P with respect to themass of WD with mountains and holes. have h ∝ I with I being the typical moment of iner-tia of the body. As a result, if we compare two WDswith the same ellipticity and angular frequency, the GRdominated WD emits stronger gravitational radiation.Since this paper is dedicated to studying the effect of f ( R ) gravity, we do not explicitly calculate h in thecase of GR, like we have calculated it in detail in ourearlier papers (Kalita & Mukhopadhyay 2019b, 2020; xz x z Ω rot χ χ Figure 7.
Cartoon diagram of a magnetized WD with mag-netic field is along z (cid:48) − axis and rotation is along z − axis. Kalita et al. 2020). Moreover, we target to explorethe detectability of those WDs which exhibit sub- andsuper-Chandrasekhar limiting mass WDs in the same M − R relation, which GR based theory cannot, andit has many consequences outlined in the Introduction.However, it is to be noted that f ( R ) gravity dominatedWDs, being smaller in size, can rotate much faster withfrequency (cid:38) Kalita & Mukhopadhyay − − − − − ν (Hz)10 − − − − − − p S ( ν ) & p T / A ( H z − / ) ETLISA ALIA BBO DECIGO
Massive binariesGalactic binariesExtrememassratioinspirals Compact binary inspirals i = 0 ◦ ˜ A + , ˜ A + , ˜ A × , ˜ A × , (a) − − − − − ν (Hz)10 − − − − − − p S ( ν ) & p T / A ( H z − / ) ETLISA ALIA BBO DECIGO
Massive binariesGalactic binariesExtrememassratioinspirals Compact binary inspirals i = 30 ◦ ˜ A + , ˜ A + , ˜ A × , ˜ A × , (b) − − − − − ν (Hz)10 − − − − − − p S ( ν ) & p T / A ( H z − / ) ETLISA ALIA BBO DECIGO
Massive binariesGalactic binariesExtrememassratioinspirals Compact binary inspirals i = 60 ◦ ˜ A + , ˜ A + , ˜ A × , ˜ A × , (c) − − − − − ν (Hz)10 − − − − − − p S ( ν ) & p T / A ( H z − / ) ETLISA ALIA BBO DECIGO
Massive binariesGalactic binariesExtrememassratioinspirals Compact binary inspirals i = 90 ◦ ˜ A + , ˜ A + , ˜ A × , ˜ A × , (d) Figure 8. (cid:112) T / A for f ( R ) gravity induced weakly magnetized WD pulsars for different i over 5 s integration time along withvarious detectors’ PSD. L G W ( e r g s − ) A BC D0.5 1.0 1.5 2.0 2.5 3.0 M ( M fl )10 P ( y r) A BC D
Figure 9.
Same as Figure 6 except that the WDs havemagnetic field instead of mountains. a regular WD governed by GR. In this paper, we haveshown that in the presence of small deformation, such asthe presence of a rough surface or magnetic fields, these WDs can emit intense gravitational radiation, which canlater possibly be detected by BBO, DECIGO, ALIA,and Einstein Telescope.The birth rate of He-dominated WDs is ∼ . × − pc − yr − (Guseinov et al. 1983), which means within100 pc radius, only one WD is formed in approxi-mately 10 yrs. Hence, continuous GW from some mas-sive WDs, which have radiation timescales (life span) ∼ − yrs, can be detected, as shown in the Figures 4and 8. If the advanced futuristic detectors, such as DE-CIGO, BBO, or Einstein Telescope detect the isolatedWDs, one can quickly check whether the physics is gov-erned by GR, or f ( R ) gravity, or any other modifiedtheory of gravity as follows. Once the GW detectors de-tect such a WD, we have the information of h and Ω rot .Consequently, if the distance to the source r is knownby some other method, then by using the Equations (27)and (35), one can estimate the ellipticity and, thereby,can predict the mass and size of the WD. This will be adirect detection of the WD with low thermal luminosity(usually a super-Chandrasekhar WD which is smaller insize). In this way, we obtain the exact mass–radius rela-tion of the WD. Since different theories provide differentmass–radius relations of the WD, by obtaining the ex- ravitational wave in f ( R ) gravity f ( R ) gravity model but with different sets of model pa-rameters. Comparing these results with those shown inFigure 1 reveals that the range of radius for stable WDsdepends on the chosen model parameters for the sameranges of mass and central density. Direct detection ofthe WDs will provide valuable information to identifythe correct radius range and hence mass–radius relationof the WDs. R ( k m ) ( α , γ )( α , γ ) M ( M fl )10 ρ c ( g c m − ) Figure 10. M −R and M − ρ c relations for WDs for f ( R ) = R + αR (1 − γR ) model. The values of α and γ in the units ofcm are ( α , γ ) = (10 , ) and ( α , γ ) = (3 × , × ). 6. CONCLUSIONSIn this paper, we have established a link between the-ory and possible GW observations of the WDs in a f ( R )gravity. Various researchers have already proposed hun- dreds of modified theories of gravity, including many f ( R ) gravity theories, and each one of them possessesits own peculiarity. However, because of the lack of ad-vanced observations near extremely high gravity regime,nobody, so far, can rule out most of the models to singleout one specific theory of gravity. In this paper, we con-sider one valid class of f ( R ) gravity model from the so-lar system constraints, which can explain both the sub-and super-Chandrasekhar WDs, along with the normalWDs, depending only on their central densities keepingthe parameters of the model fixed. However, from thepoint of observation, the primary difficulty is that we donot know the size of the peculiar WDs and, hence, theexact mass–radius curve is still unknown. Thereafter, wecalculate the strength of GW emitted from these WDs,assuming they slowly rotate with little deformation dueto some other factors, such as the presence of rough-ness of the envelope, or the presence of a weak magneticfield. If the advanced futuristic GW detectors, such asALIA, DECIGO, BBO, or Einstein Telescope can de-tect these WDs, one can estimate the ellipticity of theWDs and, thereby, put bounds on the WD’s size. Thiswill restrict the mass–radius relation of the WD, whichcan rule out various modified theories, and we will beinching towards the ultimate theory of gravity.ACKNOWLEDGMENTSThe authors would like to thank Clifford M. Will ofthe University of Florida for his insightful comments onthe effect of modified gravity in tri-axial systems. S. K.would also like to thank Khun Sang Phukon of Nikhef,Amsterdam, for the useful discussions on integratedSNR in the case of various detectors. Finally, thanks aredue to the anonymous referee for a thorough reading themanuscript and comments that have helped to improvethe presentation of the work, particularly comments onthe mass–radius relations of various models. B.M. ac-knowledges a partial support by a project of Departmentof Science and Technology (DST), India, with Grant No.DSTO/PPH/BMP/1946 (EMR/2017/001226).REFERENCES Astashenok, A. V., Capozziello, S., & Odintsov, S. D. 2013,JCAP, 12, 040—. 2014, PhRvD, 89, 103509Banerjee, I., Chakraborty, S., & SenGupta, S. 2020a,PhRvD, 101, 041301Banerjee, I., Sau, S., & SenGupta, S. 2020b, PhRvD, 101,104057 Baym, G., & Pines, D. 1971, Annals of Physics, 66, 816Belyaev, V. B., Ricci, P., ˇSimkovic, F., et al. 2015, NuclearPhysics A, 937, 17Berry, C. P. L., & Gair, J. R. 2011, PhRvD, 83, 104022Bertolami, O., & Mariji, H. 2016, PhRvD, 93, 104046Bhattacharya, M., Mukhopadhyay, B., & Mukerjee, S.2018, MNRAS, 477, 2705 Kalita & Mukhopadhyay
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