Impacts of dark matter on the curvature of the neutron star
H. C. Das, Ankit Kumar, Bharat Kumar, S. K. Biswal, S. K. Patra
MMNRAS , 000–000 (0000) Preprint 13 July 2020 Compiled using MNRAS L A TEX style file v3.0
Impacts of dark matter on the curvature of the neutron star
H. C. Das , (cid:63) , Ankit Kumar , † , Bharat Kumar , S. K. Biswal , S. K. Patra , Institute of Physics, Sachivalya Marg, Bhubaneswar-751005, India Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India Department of Physics & Astronomy, National Institute of Technology, Rourkela-769008, India Department of Astronomy, Xiamen University, Xiamen 361005, P. R. China
13 July 2020
ABSTRACT
The effects of dark matter (DM) on the curvatures of the neutron star (NS) are examined by using the stiff andsoft relativistic mean-field equation of states. The curvatures of the NSs are calculated with the variation of baryondensity. Also, it is found that the radial variation of the different curvatures significantly affected by the presence ofDM inside the NS. The effects of DM are less pronounced on the compactness of the maximum NS mass, but stillsignificant. The NS surface curvature is found to be more remarkable for the massive star. The binding energy of theNSs become positive with the increasing DM momentum.
Key words: dark matter– equation of state– stars: neutron– curvature
Neutron stars are the most known compact cosmological ob-jects without event horizons (Shapiro & Teukolsky 1983; Lat-timer & Prakash 2007) and its central density is 5–10 timesthe nuclear saturation density (Lattimer & Prakash 2004). Itis the best ground for testing many astrophysical observationsin the extreme gravitational field regime. However, a detailedstudy of the NS is not a settled issue yet due to its very com-plex structure. Its strong gravity may constraint or modifythe equation of general relativity (GR) (Psaltis 2008; Will2014). To explore the observables of the NS is quite trickydue to its uncertainty in the equation of state (EoS) at suprasaturation density, mainly the symmetry energy (Tsang et al.2009; Xu et al. 2010; Horowitz et al. 2014; Danielewicz & Lee2014; Baldo & Burgio 2016; Li et al. 2019).In August 2017, the LIGO and Virgo (Abbott et al. 2017)experiments found that the gravitational wave (GW) comesfrom the two NSs collision, and this unprecedented eventopened a new era of astronomy, which is the deepest mys-tery of the Universe. In their observation, they have takenthe Universe with no DM. We do not know the consequenceabout the DM on the GW due to its gravitationally attract-ing behaviour which bends the space-time curvature (Emspak2017). The DM percentage is more than 80 % (Joglekar et al.2019) in the Universe so that the compact object when rotatesaround the centre in the Galaxy; it will sweep through DMhalo and eventually capture some of the particles (Kouvaris2008). The captured DM particles may have effects on the ob-servational properties of NSs, which may use to constrain thenature of the DM, and also it curves the space-time which de- (cid:63) E-mail: [email protected] † [email protected] pends on the percentage of DM contained inside the NS. Thetwo things are capable of capturing the DM inside the NSs.First, its enormous gravitational force inside the NS. Second,the immense baryonic density inside the NS (Kouvaris 2008;G ˜Aijver et al. 2014).Because of the uncertain nature of the DM, different DMparticles like weakly interacting massive particle (WIMP),feebly interacting massive particle (FIMP) etc. are hypoth-esized. The WIMPs are the most abundant DM particles inthe early Universe due to its freeze-out mechanism (Kouvaris& Tinyakov 2011; Ruppin et al. 2014). They equilibrated withthe environments at freeze-out temperature and annihilatedto form different standard model particles and leptons. More-over, if the DM particles are annihilating, then it cools theold NSs (Kouvaris 2008; Ding et al. 2019; Bhat & Paul 2020).If the DM does not annihilate, then it interacts with baryonand also themselves, so that it affects the structure of the NS(De Lavallaz & Fairbairn 2010; Ciarcelluti & Sandin 2011).Once the DM contained inside the NS larger than the Chan-drasekhar limit, it may form a mini black hole in the core,which can destroy the NSs (Kouvaris 2008). This mechanismcan use to constrain the percentage of dark matter. Differ-ent approach has been used to calculated the NS propertieswith the inclusion of DM inside the NS (Sandin & Ciarcelluti2009; De Lavallaz & Fairbairn 2010; Kouvaris & Tinyakov2010; Ciarcelluti & Sandin 2011; Leung et al. 2011; Li et al.2012; Panotopoulos & Lopes 2017; Ellis et al. 2018; Bhat& Paul 2020; Ivanytskyi et al. 2019; Das et al. 2019; Qudduset al. 2020; Das et al. 2020). However, in the present scenario,we use the non-annihilating WIMP as DM particle inside theNS. The detailed discussions can be found in our previouscalculations (Das et al. 2020). As we know that the additionof DM softens the EoS, reduce mass-radius of the NS (Li et al. © a r X i v : . [ nu c l - t h ] J u l H. C. Das et al. r from anobject of mass M defined as η ≡ GMrc and its values lie inbetween 0 and 1 (Psaltis 2008; He et al. 2015). The value of η = corresponding to flat Minkowski space in special rela-tivity, while η = corresponds to the event horizon limit ofa black hole, i.e. strongest gravitational field. The strengthof the gravitational field is not due to the potential but dueto the curvature (Psaltis 2008). So the curvature is the cru-cial thing to quantify gravity and the amount of wrapped inspace-time. A massive body has larger curvature than thelighter one. Some experiments had already done to measurethe curvature of the space-time. Recently, the direct detectionof gravity-field curvature has done by using atom interferom-eters (Rosi et al. 2015).We describe the different curvature quantities briefly in theconcept of GR from the Ref. Carroll (2019). The Riemanntensor has twenty components in four dimensions. One cannotset all the components of the Riemann tensor to zero so thatit is an appropriate tensor to measure the curvature. Sincethe Kretschmann scalar is the square root of the Riemanntensor, it also has the same property as Riemann tensor. TheRicci tensor is the contraction of the Riemann tensor, and thetrace of the Ricci tensor is Ricci scalar or curvature scalar.One can say that the Ricci scalar and tensor contains all theinformation about the Riemann tensor leaving the tracelesspart of the Riemann tensor. If we remove all the contractionof Riemann tensor, then we have Weyl tensor. In the physicalsense, the Ricci tensor and Ricci scalar measures the volumet-ric change of a body in the presence of the tidal effect. While,the Weyl tensor is telling about the distortion of the shape ofthe body, not its volume. However, the Riemann tensor mea-sures both distortion of shape and volumetric change of thebody in the presence of tidal force. In Ref. Ek¸si et al. (2014)they have calculated the curvature of the NS, and they havenoticed that GR is not well tested in the whole star thanEoS and also found that the variation of Weyl tensor withthe radius follows the power law for the large part of thestar. Further, He et al. (2015) have taken both relativisticmean-field (RMF) and Skyrme-Hartree-Fock approaches andconcluded that the symmetry energy has significant effectson the curvature of the lower mass NS and lesser effects onthe massive NS. In the present work, we investigate the cur-vature of the NS in the presence of DM, with the differentLagrangian density of the models NL3 (Lalazissis et al. 1997),G3 (Kumar et al. 2017) and IOPB-I (Kumar et al. 2018).It is exciting to probe the deeper inside the strong fieldregime of the NS by the modern observational instruments,and we may hope that the measurement of mass and radiusof the NS more stringent constrain on the test of GR (Psaltis2008). Till now, the accurate measurement of M - R with min-imal uncertainty is not done due to to its unknown core. Thedensity in the core is around 15 times the density at thesurface (Ek¸si et al. 2014). It means that the large mass isconcentrated in the core. But the maximum mass and radiuscalculations are mainly from the unconstrained EoS regime.In contrary, the compactness and surface curvature are radi-ally increasing towards the surface of the NS, which may be the possibility to measure the maximum mass-radius that ismore prominent for the EoS rather than the gravity. If wecalculate the EoS in some way, but the application of GR toNS is difficult for all densities range. So that there will bethe degeneracy breaking between the nuclear EoS and grav-ity models, which is a longstanding problem (Harada 1998;Sotani 2014; Ek¸si et al. 2014). The detailed study requires forthe EoS in a better way at supra saturation density. Since theGR equations breakdown at the strong-field regime, so we caneither modify or extended up to some limit. In the Ref. Ek¸siet al. (2014), it is quantified that the unconstrained gravityof the NS in the framework of GR. Moreover, to quantify thedeviations come from GR in the strong-field regime, the de-tailed understanding required to study the properties of DMin the Universe (Psaltis 2008). Therefore, in the present cal-culations, we want to measure the curvature of the NS withthe addition of DM using RMF approach.In the present analysis, we take the RMF model to calculatethe EoS. The RMF model reproduces the experimental valuefor exotic and super-heavy nuclei (Rashdan 2001; Bhuyan& Patra 2012) very well and gives a good description of thefinite nuclei up to the β -stability line. The extended RMF (E-RMF) formalism (Kumar et al. 2017, 2018), which producedall the nuclear matter (NM) properties and satisfied the NSsobservables calculated till now. In Sec. 2, we calculate the EoSof NS with the addition of DM. The curvature calculations arepresented in 2.4. The detailed calculations are given in Ek¸siet al. (2014). In Sec. 3, we present the variation of curvatureswith the different quantities like baryon density, mass andradius with the addition of DM. The binding energy of theNS is also calculated with the inclusion of DM. Finally, wegive a brief conclusion on the curvatures of the NS in Sec. 4. In this section, we provide the formalism required to com-pute the curvature of the NS in the presence of DM. First, webriefly sketch the E-RMF model along with DM by present-ing model Lagrangian (Lalazissis et al. 1997; Kumar et al.2017, 2018). All the parameters used in the E-RMF modelwere fitted to reproduce the observables of finite nuclei andinfinite NM. The NS matter EoS is computed in the presenceof DM and hence on the NS properties to solve the Tolman-Openheimer-Volkoff (TOV) equations and its different curva-tures inside on it.
The RMF Lagrangian is built from the interaction of mesons-nucleons and their self ( σ , σ , σ , ω , ω , ρ and ρ ) andcross-couplings ( σ − ω , ω − ρ , σ − ω and σ − ρ ) up tofourth order. The RMF Lagrangian is discussed in these Refs.(Miller & Green 1972; Serot & Walecka 1986; Furnstahl et al.1987; Reinhard 1988; Ring 1996; Furnstahl et al. 1997; Kumar MNRAS , 000–000 (0000)
M effects on curvature of the NS et al. 2017, 2018). The RMF Lagrangian for NM system is L nucl . = (cid:213) α = p , n ¯ ψ α (cid:40) γ µ (cid:18) i ∂ µ − g ω ω µ − g ρ (cid:174) τ α · (cid:174) ρ µ (cid:19) − (cid:18) M nucl . − g σ σ − g δ (cid:174) τ α · (cid:174) δ (cid:19)(cid:41) ψ α + ∂ µ σ ∂ µ σ − m σ σ + ζ g ω ( ω µ ω µ ) − κ g σ m σ σ M nucl . − κ g σ m σ σ M nucl . + m ω ω µ ω µ − W µν W µν + η g σ σ M nucl . m ω ω µ ω µ + η g σ σ M nucl . m ω ω µ ω µ + η ρ m ρ M nucl . g σ σ (cid:18) (cid:174) ρ µ · (cid:174) ρ µ (cid:19) + m ρ (cid:18) (cid:174) ρ µ · (cid:174) ρ µ (cid:19) − (cid:174) R µν · (cid:174) R µν − Λ ω g ω g ρ (cid:0) ω µ ω µ (cid:1)(cid:0) (cid:174) ρ µ · (cid:174) ρ µ (cid:1) + ∂ µ (cid:174) δ ∂ µ (cid:174) δ − m δ (cid:174) δ , (1)where M nucl (=939 MeV) is the mass of the nucleon. m σ , m ω , m ρ and m δ are the masses and g σ , g ω , g ρ and g δ arethe coupling constants for the σ , ω , ρ and δ mesons respec-tively. κ (or κ ) and ζ are the self-interacting coupling con-stants of the σ and ω mesons respectively. η , η , η ρ and Λ ω are the coupling constants of non-linear cross-coupled terms.The quantities W µν and (cid:174) R µν being the field strength ten-sors for the ω and ρ mesons respectively, defined as W µν = ∂ µ ω ν − ∂ ν ω µ and (cid:174) R µν = ∂ µ (cid:174) ρ ν − ∂ ν (cid:174) ρ µ . The (cid:174) τ are the Paulimatrices and behave as the isospin operator. Parameters andsaturation properties for NL3 (Lalazissis et al. 1997), G3 (Ku-mar et al. 2017) and IOPB-I (Kumar et al. 2018) along withthe empirical/experimental values are given in Table 1.The meson fields for the NM system are calculated by solv-ing the mean-field equation of motions (Kumar et al. 2017,2018; Das et al. 2019) in a self-consistent way. The energydensity ( E nucl . ) and pressure ( P nucl . ) are calculated usingthe energy-momentum stress-tensor technique which is givenby (Walecka 1974; Glendenning 1997) E nucl . = γ ( π ) (cid:213) i = p , n ∫ k i d kE (cid:63) i ( k i ) + ρ b W + ρ R + m s Φ g s (cid:32) + κ Φ M nucl . + κ Φ M nucl . (cid:33) − ζ W g ω − m ω W g ω (cid:32) + η Φ M nucl . + η Φ M nucl . (cid:33) − Λ ω ( R × W ) − (cid:32) + η ρ Φ M nucl . (cid:33) m ρ g ρ R + m δ g δ D , (2) Table 1.
The parameter sets NL3 (Lalazissis et al. 1997), G3(Kumar et al. 2017) and IOPB-I (Kumar et al. 2018) are listed.All the coupling constants are dimensionless, except k which isin fm − . The NM parameters are given at saturation point forNL3, G3 and IOPB-I parameter sets in the lower panel. The em-pirical/experimental values for NM parameters are also given atthe saturation density. The references are [ a ] , [ b ] , [ c ] & [ d ] (Zylaet al. 2020), [ e ] & [ f ] (Bethe 1971), [ g ] (Garg & Col ˜Aˇs 2018), [ h ] & [ i ] (Danielewicz & Lee 2014), and [ j ] (Zimmerman et al. 2020).Parameter NL3 G3 IOPB-I Empirical/Expt. Value m σ / M nucl . [ a ] m ω / M nucl . [ b ] m ρ / M nucl . [ c ] m δ / M nucl . [ d ] g σ / π g ω / π g ρ / π g δ / π k k -5.688 1.694 -2.932 ζ η η η ρ Λ ω ρ ( f m − ) 0.148 0.148 0.149 0.148 – 0.185 [ e ] BE ( MeV ) -16.29 -16.02 -16.10 -15.00 – 17.00 [ f ] K ( MeV ) [ g ] J ( MeV ) [ h ] L ( MeV ) [ i ] K sym ( MeV ) [ j ] Q sym (MeV) 177.90 915.47 862.70 ———– P nucl . = γ ( π ) (cid:213) i = p , n ∫ k i d k k E (cid:63) i ( k i ) + ζ W g ω − m s Φ g s (cid:32) + κ Φ M nucl . + κ Φ M nucl . (cid:33) + m ω W g ω (cid:32) + η Φ M nucl . + η Φ M nucl . (cid:33) + Λ ω ( R × W ) + (cid:32) + η ρ Φ M nucl . (cid:33) m ρ g ρ R − m δ g δ D . (3)Where Φ , W , R and D are the redefined fields for σ , ω , ρ and δ mesons as Φ = g s σ , W = g ω ω , R = g ρ (cid:174) ρ and D = g δ δ respectively. The E (cid:63) i ( k i ) = (cid:113) k i + M (cid:63) i , where M (cid:63) i is theeffective mass and k i is the momentum of the nucleon and γ is the spin degeneracy factor which is equal to 2 for individualnucleons.Inside the NS, many particles like hyperons, nucleons andleptons are present. The neutron decays to proton, electronand anti-neutrino inside the NS (Glendenning 1997). Thisprocess is called as β –decay. To maintain the charge neu-trality condition, the inverse β –decay process occurred. The MNRAS000
The parameter sets NL3 (Lalazissis et al. 1997), G3(Kumar et al. 2017) and IOPB-I (Kumar et al. 2018) are listed.All the coupling constants are dimensionless, except k which isin fm − . The NM parameters are given at saturation point forNL3, G3 and IOPB-I parameter sets in the lower panel. The em-pirical/experimental values for NM parameters are also given atthe saturation density. The references are [ a ] , [ b ] , [ c ] & [ d ] (Zylaet al. 2020), [ e ] & [ f ] (Bethe 1971), [ g ] (Garg & Col ˜Aˇs 2018), [ h ] & [ i ] (Danielewicz & Lee 2014), and [ j ] (Zimmerman et al. 2020).Parameter NL3 G3 IOPB-I Empirical/Expt. Value m σ / M nucl . [ a ] m ω / M nucl . [ b ] m ρ / M nucl . [ c ] m δ / M nucl . [ d ] g σ / π g ω / π g ρ / π g δ / π k k -5.688 1.694 -2.932 ζ η η η ρ Λ ω ρ ( f m − ) 0.148 0.148 0.149 0.148 – 0.185 [ e ] BE ( MeV ) -16.29 -16.02 -16.10 -15.00 – 17.00 [ f ] K ( MeV ) [ g ] J ( MeV ) [ h ] L ( MeV ) [ i ] K sym ( MeV ) [ j ] Q sym (MeV) 177.90 915.47 862.70 ———– P nucl . = γ ( π ) (cid:213) i = p , n ∫ k i d k k E (cid:63) i ( k i ) + ζ W g ω − m s Φ g s (cid:32) + κ Φ M nucl . + κ Φ M nucl . (cid:33) + m ω W g ω (cid:32) + η Φ M nucl . + η Φ M nucl . (cid:33) + Λ ω ( R × W ) + (cid:32) + η ρ Φ M nucl . (cid:33) m ρ g ρ R − m δ g δ D . (3)Where Φ , W , R and D are the redefined fields for σ , ω , ρ and δ mesons as Φ = g s σ , W = g ω ω , R = g ρ (cid:174) ρ and D = g δ δ respectively. The E (cid:63) i ( k i ) = (cid:113) k i + M (cid:63) i , where M (cid:63) i is theeffective mass and k i is the momentum of the nucleon and γ is the spin degeneracy factor which is equal to 2 for individualnucleons.Inside the NS, many particles like hyperons, nucleons andleptons are present. The neutron decays to proton, electronand anti-neutrino inside the NS (Glendenning 1997). Thisprocess is called as β –decay. To maintain the charge neu-trality condition, the inverse β –decay process occurred. The MNRAS000 , 000–000 (0000)
H. C. Das et al. process can be expressed as n → p + e − + ¯ ν, p + e − → n + ν. (4)To maintain the stability of NSs, there must have both β -equilibrium and charge-neutrality conditions, which are ex-pressed in term of chemical potential µ n = µ p + µ e ,µ e = µ µ ,ρ p = ρ e + ρ µ . (5)Where µ n , µ p , µ e , and µ µ are the chemical potentials of neu-trons, protons, electrons, and muons, respectively. When thechemical potential of electron is equal to the muon rest mass,then muon appear inside the NS. The chemical potentials µ n , µ p , µ e , and µ µ are given by (Das et al. 2020) µ n , p = g ω ω ± g ρ ρ ∓ g δ δ + (cid:113) k n , p + ( M (cid:63) n , p ) , (6) µ e ,µ = (cid:113) k e ,µ + m e ,µ , (7)where M (cid:63) n and M (cid:63) p are the effective mass of neutron and pro-ton respectively. The particle fraction inside the NS is cal-culated by solving Eq. (5) using the Eqs. (6 – 7) for a givenbaryon density by self-consistently. The energy density andpressure of NS are given by, E NS = E nucl . + E l , and P NS = P nucl . + P l , (8)where, E l = (cid:213) l = e ,µ ( π ) ∫ k l d k (cid:113) k + m l , (9)and P l = (cid:213) l = e ,µ ( π ) ∫ k l d k k (cid:113) k + m l . (10)Where E l , P l and k l are the energy density, pressure andFermi momentum for leptons respectively. The Eq. (8) givesthe total energy, pressure of the NS. DM particles accreted inside the NS core due to its high grav-itational field (Goldman & Nussinov 1989; Kouvaris 2008; Xi-ang et al. 2014; Das et al. 2019) and the amount of accretedDM depends directly on its evolving life time. In this sce-nario, we consider the Neutralino (Martin 1998; Panotopou-los & Lopes 2017; Das et al. 2019, 2020) as a fermionic DMcandidate which interacts with nucleon via SM Higgs. Thedetailed formalism has been taken from our previous analysis(Das et al. 2020) and the total Lagrangian is written as: L tot . = L NS + ¯ χ (cid:2) i γ µ ∂ µ − M χ + y h (cid:3) χ + ∂ µ h ∂ µ h − M h h + f M nucl . v ¯ ϕ h ϕ, (11)where L NS is the NS Lagrangian and ϕ and χ are the nucle-onic and DM wave functions respectively. h is the Higgs field. All the parameters value like y ( = . ) , f ( = . ) and v ( = GeV) are given in (Das et al. 2020). From the Lagrangian inEq. (11), we get the total energy density ( E tot . ) and pressure( P tot . ) for NS with DM given as (Das et al. 2020) E tot . = E NS + ( π ) ∫ k DMf d k (cid:113) k + ( M (cid:63)χ ) + M h h , (12) P tot . = P NS + ( π ) ∫ k DMf d k k (cid:113) k + ( M (cid:63)χ ) − M h h , (13)where k DMf is the DM Fermi momentum. The M (cid:63) n , p and M (cid:63)χ are the effective masses of nucleon and DM, which are givenas M (cid:63) n , p = M nucl . + g σ σ ∓ g δ δ − f M nucl . v h , M (cid:63)χ = M χ − y h , (14)where the σ , δ and h are the field for σ , δ and Higgsrespectively. Here we calculate the NS observables like M and R etc. usingTOV equations. Hence, we take the EoSs of NS with DM andinput to the TOV equations (Tolman 1939; Oppenheimer &Volkoff 1939) are given as dP tot . ( r ) dr = − ( P tot . ( r ) + E tot . ( r ))( m ( r ) + π r P tot . ( r )) r ( r − m ( r )) , (15)and dm ( r ) dr = π r E tot . ( r ) , (16)where E tot . ( r ) and P tot . ( r ) are the total energy and pressuredensity as a function of radial distance. m ( r ) is the gravita-tional mass, and r is the radial parameter. These two coupledequations are solved to get the mass and radius of the NS atcertain central density. We adopt the mathematical form of different curvature quan-tities from Ek¸si et al. (2014), which measure the space-timecurvature for both inside and outside the compact objects.Several curvatures are Riemann tensor, Ricci tensor, Ricciscalar and Weyl tensor, which are given asThe Ricci scalar R( r ) = π (cid:20) E tot . ( r ) − P tot . ( r ) (cid:21) , (17)the full contraction of the Ricci tensor J ( r ) ≡ R µν R µν = ( π ) (cid:104) E tot . ( r ) + P tot . ( r ) (cid:105) , (18) MNRAS , 000–000 (0000)
M effects on curvature of the NS the Kretschmann scalar (full contraction of the Riemann ten-sor) K ( r ) = R µνρσ R µνρσ = ( π ) [ E tot . ( r ) + P tot . ( r ) + P tot . ( r )E tot . ( r )]− E tot . ( r ) m ( r ) r + m ( r ) r , (19)and the full contraction of the Weyl tensor W ( r ) ≡ C µνρσ C µνρσ = (cid:18) m ( r ) r − π E tot . ( r ) (cid:19) . (20)Where E tot . , P tot . , m ( r ) and r are the energy density, pressure,mass and radius of the NS respectively. At the surface m → M and r → R . The Ricci tensor and Ricci scalar vanishingoutside the NSs because all tensors depend on the E tot . ( r ) , P tot . ( r ) and m ( r ) , which are zero outside the star. There is anon-vanishing component of the Riemann tensor which doesnot vanish; R = − MR = − ξ , even in the outside of the star(Ek¸si et al. 2014; He et al. 2015). So the Riemann tensor isthe more relevant to measure the curvature of the NSs thanothers. In the present work, three sets of parameter are chosen,namely NL3 (Lalazissis et al. 1997), IOPB-I (Kumar et al.2018) and G3 (Kumar et al. 2017). The NL3 parameter setcorresponds to the standard RMF model, which contains non-linear interactions (self-interaction of the sigma meson). Ithas relatively high incompressibility in comparison to theother two parameter sets. The NM properties of all the three-parameter sets are given in Table 1. In the case of IOPB-I,there are two extra coupling parameters, Λ ω and ζ are takenon top of the NL3 parameter set. These two coupling param-eters play a vital role in both finite and infinite NM system(Singh et al. 2014; Biswal et al. 2015; Kumar et al. 2018). Theparameter Λ ω controls the symmetry energy (or neutron skinthickness of finite nuclei) as well as the maximum mass of theNS (Horowitz & Piekarewicz 2001). ζ , which is the couplingconstants for the self-interaction of the vector-meson, affectsthe EoS at higher density (Sugahara & Toki 1994; M ˜Aijller &Serot 1996). So it is imperative to include for the study of theNS physics. The interaction G3 corresponds to the completeE-RMF Lagrangian, as discussed in Sec. 2.1. One can notethat G3 parameter has extra six couplings constants ( η , η , η ρ , g δ , Λ ω and ζ ) in compare to the NL3 parameter set.All the couplings constants of the Lagrangian in Eq. (1) areobtained by fitting the several properties of finite nuclei andinfinite NM at saturation density. The NL3 being the stiffestEoS enrich with the higher NS mass (2.774 M (cid:12) ) in compari-son to other parameter sets. The maximum mass is given byIOPB-I (2.149 M (cid:12) ) is closer to the one suggested by the Cro-martie et al. (2019) along with the calculated values of canon-ical radius and tidal deformability are in accordance with theGW170817 (Abbott et al. 2017, 2018). We have given theresults with these parameter sets, for the comparative studyand better understanding of the parametric dependency ofthe dark matter effects on the curvatures of the neutron star. -3 -5 -3 P N S ( M e V / f m ) Outer - CrustInner - CrustEntire EOS -3 ε NS (MeV/fm ) -3 NL3 G3 IOPB-I C r u s t T r a n s i t i o n C r u s t - C o re T r a n s i t i o n Figure 1. (colour online) The EoS of the NS is shown for differentparameter sets for core part ( < ρ ), where ρ is the NM saturationdensity. The red and green shaded line represents the inner crust(3 × − – 8 × − fm − ) and outer crust (6 × − – 2.61 × − fm − )of Ref. (Sharma et al. 2015) respectively. EoS is the most vital equipment to understand the proper-ties of the NS. We use different EoS emerged from differentparameter sets like NL3 (Lalazissis et al. 1997), G3 (Kumaret al. 2017) and IOPB-I (Kumar et al. 2018) to explore thecurvature of NS. The calculated EoS in Eqs. (8) is only for thecore part of the NS and it is different for NL3, G3 and IOPB-Iparameter sets. However, for the crust part (both inner andouter crust), we adopt the Sharma et. al. EoS (Sharma et al.2015) which added to form unified EoS for the whole den-sity range. The entire EoS depicted in Fig. 1. Moreover, wehave found in our recent work Das et al. (2020) that the EoSbecomes softer with the addition of DM and reduces the M and R of the NS. The maximum masses, radii and its corre-sponding central densities are given in Table 2 for differentDM momenta. We calculate various curvatures like K , J , R and W of theNS in the presence of DM. NS has different density regionslike the inner core, inner crust and an outer crust (see Fig.1). The core part is the most crucial component of the star,which account the density in the range of 5-10 times of thenuclear saturation density (Lattimer & Prakash 2004). Here,we calculate the curvature of the NS with the variation ofthe baryon density with different DM momentum for differ-ent parameter sets, which is shown in Fig. 2. The curvaturesare increasing or decreasing with the increase of the baryondensity. It is observed that at lower density the quantity K and W give higher curvature than J , R . Near to the surface,the curvatures J , and R are almost vanished due to theirzero vacuum expectation value. The K and W approach eachother at the radius of the crust and have a local maximumin that region. Therefore, the crust region is the best site tomeasure the deviation of GR in the strong gravity regime, MNRAS000
M effects on curvature of the NS the Kretschmann scalar (full contraction of the Riemann ten-sor) K ( r ) = R µνρσ R µνρσ = ( π ) [ E tot . ( r ) + P tot . ( r ) + P tot . ( r )E tot . ( r )]− E tot . ( r ) m ( r ) r + m ( r ) r , (19)and the full contraction of the Weyl tensor W ( r ) ≡ C µνρσ C µνρσ = (cid:18) m ( r ) r − π E tot . ( r ) (cid:19) . (20)Where E tot . , P tot . , m ( r ) and r are the energy density, pressure,mass and radius of the NS respectively. At the surface m → M and r → R . The Ricci tensor and Ricci scalar vanishingoutside the NSs because all tensors depend on the E tot . ( r ) , P tot . ( r ) and m ( r ) , which are zero outside the star. There is anon-vanishing component of the Riemann tensor which doesnot vanish; R = − MR = − ξ , even in the outside of the star(Ek¸si et al. 2014; He et al. 2015). So the Riemann tensor isthe more relevant to measure the curvature of the NSs thanothers. In the present work, three sets of parameter are chosen,namely NL3 (Lalazissis et al. 1997), IOPB-I (Kumar et al.2018) and G3 (Kumar et al. 2017). The NL3 parameter setcorresponds to the standard RMF model, which contains non-linear interactions (self-interaction of the sigma meson). Ithas relatively high incompressibility in comparison to theother two parameter sets. The NM properties of all the three-parameter sets are given in Table 1. In the case of IOPB-I,there are two extra coupling parameters, Λ ω and ζ are takenon top of the NL3 parameter set. These two coupling param-eters play a vital role in both finite and infinite NM system(Singh et al. 2014; Biswal et al. 2015; Kumar et al. 2018). Theparameter Λ ω controls the symmetry energy (or neutron skinthickness of finite nuclei) as well as the maximum mass of theNS (Horowitz & Piekarewicz 2001). ζ , which is the couplingconstants for the self-interaction of the vector-meson, affectsthe EoS at higher density (Sugahara & Toki 1994; M ˜Aijller &Serot 1996). So it is imperative to include for the study of theNS physics. The interaction G3 corresponds to the completeE-RMF Lagrangian, as discussed in Sec. 2.1. One can notethat G3 parameter has extra six couplings constants ( η , η , η ρ , g δ , Λ ω and ζ ) in compare to the NL3 parameter set.All the couplings constants of the Lagrangian in Eq. (1) areobtained by fitting the several properties of finite nuclei andinfinite NM at saturation density. The NL3 being the stiffestEoS enrich with the higher NS mass (2.774 M (cid:12) ) in compari-son to other parameter sets. The maximum mass is given byIOPB-I (2.149 M (cid:12) ) is closer to the one suggested by the Cro-martie et al. (2019) along with the calculated values of canon-ical radius and tidal deformability are in accordance with theGW170817 (Abbott et al. 2017, 2018). We have given theresults with these parameter sets, for the comparative studyand better understanding of the parametric dependency ofthe dark matter effects on the curvatures of the neutron star. -3 -5 -3 P N S ( M e V / f m ) Outer - CrustInner - CrustEntire EOS -3 ε NS (MeV/fm ) -3 NL3 G3 IOPB-I C r u s t T r a n s i t i o n C r u s t - C o re T r a n s i t i o n Figure 1. (colour online) The EoS of the NS is shown for differentparameter sets for core part ( < ρ ), where ρ is the NM saturationdensity. The red and green shaded line represents the inner crust(3 × − – 8 × − fm − ) and outer crust (6 × − – 2.61 × − fm − )of Ref. (Sharma et al. 2015) respectively. EoS is the most vital equipment to understand the proper-ties of the NS. We use different EoS emerged from differentparameter sets like NL3 (Lalazissis et al. 1997), G3 (Kumaret al. 2017) and IOPB-I (Kumar et al. 2018) to explore thecurvature of NS. The calculated EoS in Eqs. (8) is only for thecore part of the NS and it is different for NL3, G3 and IOPB-Iparameter sets. However, for the crust part (both inner andouter crust), we adopt the Sharma et. al. EoS (Sharma et al.2015) which added to form unified EoS for the whole den-sity range. The entire EoS depicted in Fig. 1. Moreover, wehave found in our recent work Das et al. (2020) that the EoSbecomes softer with the addition of DM and reduces the M and R of the NS. The maximum masses, radii and its corre-sponding central densities are given in Table 2 for differentDM momenta. We calculate various curvatures like K , J , R and W of theNS in the presence of DM. NS has different density regionslike the inner core, inner crust and an outer crust (see Fig.1). The core part is the most crucial component of the star,which account the density in the range of 5-10 times of thenuclear saturation density (Lattimer & Prakash 2004). Here,we calculate the curvature of the NS with the variation ofthe baryon density with different DM momentum for differ-ent parameter sets, which is shown in Fig. 2. The curvaturesare increasing or decreasing with the increase of the baryondensity. It is observed that at lower density the quantity K and W give higher curvature than J , R . Near to the surface,the curvatures J , and R are almost vanished due to theirzero vacuum expectation value. The K and W approach eachother at the radius of the crust and have a local maximumin that region. Therefore, the crust region is the best site tomeasure the deviation of GR in the strong gravity regime, MNRAS000 , 000–000 (0000)
H. C. Das et al. C u r va t u re ( - c m - ) k fDM = 0.00 GeVk fDM = 0.02 GeVk fDM = 0.03 GeVk fDM = 0.04 GeV ρ ( x10 cm -3 ) NL3 G3 IOPB-I
Figure 2. (colour online) The variation of different curvatures K (red), J (green), R (blue) and W (magenta) with baryon densityfor NL3 (left), G3 (middle) and IOPB-I (right) in the presence ofDM for corresponding maximum mass. the activity of the pulsar glitches, and the modulation of thecooling of NS (Ek¸si et al. 2014).The radial variation of the curvatures is shown in Fig 3with the addition of DM. It is found that in the presence ofDM, the curvatures of the NS increase with the increasing of k DMf . All the curvatures are maximum at the centre of thestar except Weyl tensor. However, the Ricci scalar is negativewithin the star (for maximum mass), as shown in Fig. 3,whose magnitude is higher for stiffer EoS like NL3 than otherstwo G3 and IOPB-I. At the surface of the star, E tot . = , P tot . = , so K and W are equal to √ MR from Eqs. (19) and(20). Near the surface of the NS, the J and R are approachto zero. If we assume that the NS has uniform density, i.e. m = π r ρ , then the Eq. (20) is equal to zero. Therefore, the W tends to zero at the core. As we approach from outer crustto the surface, the density in this region is like diffuse state.So that W is maximum at the surface. Various curvaturesfollow the similar radial variation in the presence of the DM,but the magnitude of the curvature increases.The compactness parameter measures the degree of densityof the star. The NS has a larger mass and smaller radius ascompared to the Sun, so its compactness is times largerthan our Sun. Therefore, we study the radial variation of thecompactness of the NS in the presence of the DM, which isdepicted in Fig. 4. The compactness within the star increaseswith the increase of the DM momentum for different parame-ter sets, which is shown in Fig. 4. It has a larger value for themaximum mass NS in compare to the canonical star. Withthe increase of the DM percentage, the EoS becomes softer,which has less compactness as compared to the stiff EoS.We calculate the K (r) within the NS. To see the paramet-ric dependence of the curvature with different radius, we fixthe DM percentage (0.04 GeV), which is shown in Fig. 5 fordifferent masses of the star. If one see carefully, the radial vari-ation of the K (r) increases slowly with different mass star upto canonical mass; then the percentage is increasing for themaximum mass star. With the addition of DM, the change K (r) increases ≈
33 %, and this percentage is more for max-imum mass star. It is observed that the softer EoS, namelyG3, has larger curvature than stiffer like NL3 and IOPB-I k fDM = 0.04 GeVk fDM = 0.03 GeVk fDM = 0.02 GeVk fDM = 0.00 GeV C u r va t u re ( - c m - ) Radius (km)
NL3 1.4 M O G3 IOPB-I1.4 M O O .. . Max. Mass Max. MassMax. Mass Max. Mass
Figure 3. (colour online) The radial variation of all the curvatures K (red), J (green), R (blue) and W (magenta) for NL3 (left), G3(middle) and IOPB-I (right) in the presence of DM. Figure 4. (colour online) The radial variation of compactness ( η )for NL3 (left), G3 (middle) and IOPB-I (right) in the presence ofDM. both for the canonical and maximum mass star. Hence, weconclude that the effects of DM on the curvature of the NSchanges considerably not only on the K but also on the J , R , W .Here, we calculate the curvature at the surface of the NS,which is more prominent to quantify the space-time wrap inthe Universe. The variation of K (R)/ K (cid:12) with the mass ofthe NS is shown in Fig. 6. The curvature ( K (cid:12) ) and compact-ness ( η (cid:12) ) at the surface of the sun are . × − cm − and4.27 × − respectively (Ek¸si et al. 2014). In our calculations,it is found that the surface curvature for NS without DM are MNRAS , 000–000 (0000)
M effects on curvature of the NS Table 2.
The central density E c , mass M (M (cid:12) ), radius R, Surface curvature K(R) , binding energy B / M of the NS are tabulated with thevariation of k D Mf both for canonical (1.4 M (cid:12) ) and maximum mass star for NL3, G3 and IOPB-I parameter sets. k D Mf (GeV) Startype E c (MeV fm − ) M ( M (cid:12) ) R (km) K (R)( K (cid:12) ) B / M NL3 G3 IOPB-I NL3 G3 IOPB-I NL3 G3 IOPB-I NL3 G3 IOPB-I NL3 G3 IOPB-I0.00 Canonical 270 460 366 1.400 1.400 1.400 14.08 12.11 12.78 1.477 2.320 1.977 -0.084 -0.098 -0.092Maximum 870 1340 1100 2.774 1.997 2.149 13.16 10.78 11.76 3.584 4.695 3.894 -0.207 -0.162 -0.1650.02 Canonical 286 700 385 1.400 1.400 1.400 13.63 11.32 12.42 1.626 2.841 2.153 -0.038 -0.065 -0.057Maximum 890 1440 1120 2.734 1.543 2.118 12.91 10.25 11.54 3.741 4.218 4.061 -0.178 -0.071 -0.1390.03 Canonical 320 870 430 1.400 1.400 1.400 12.78 10.56 11.75 1.976 3.515 2.546 0.045 -0.011 0.016Maximum 940 1480 1190 2.646 1.491 2.050 12.39 9.81 11.06 4.097 4.651 4.492 -0.116 -0.024 -0.0090.04 Canonical 383 1470 518 1.400 1.400 1.400 11.60 9.18 10.76 2.638 6.365 3.317 0.016 0.055 0.105Maximum 1100 1500 1390 2.502 1.402 1.937 11.46 9.15 10.23 4.093 6.478 5.341 -0.023 -0.001 -0.002 O O O κ ( r ) ( - c m - ) O Radius (km) O O NL3
NL3+DM(0.04 GeV)
G3 IOPB-I
IOPB-I+DM(0.04 GeV)
G3+DM(0.04 GeV) . . ....
Figure 5. (colour online) The radial variation of K (r) withoutand with DM having momentum 0.04 GeV. The correspondingmaximum mass is shown bold line (without DM) and dashed-line(with DM). K (R)/ K (cid:12) ≈ and η / η (cid:12) ≈ re-spectively. The ratio K (R)/ K (cid:12) increases with the addition ofDM. Therefore, we conclude that less massive star is moresuitable for the detailed study of EoS. G3 is the softer EoSthan the IOPB-I, which gives larger surface curvature thanstiffer EoS both for the canonical and maximum mass star asgiven in Table 2. If the DM density is very high inside theNS, then the EoS becomes softer, which affects the curvaturessignificantly at the surface. The large curvature wrapped thespace-time more than the smaller one. Using the Shapiro de- κ ( R ) / κ O ( ) k fDM = 0.00 GeVk fDM = 0.02 GeVk fDM = 0.03 GeVk fDM = 0.04 GeV M/M O NL3 G3 IOPB-I . .
Figure 6. (colour online) The ratio of the surface curvature of NSand the Sun with the variation of NS mass with DM for NL3, G3and IOPB-I. lay measurement (Shapiro 1964), may one can find the NSusing modern technology. Recently, the Shapiro effect hasplayed an important role to test the GR in the strong-fieldregime of the NS using the binary pulsars (P˜A˝ussel 2019).
The gravitational binding energy (B) is defined as the massdifference between gravitational mass (M) and baryonic mass( M B ) of the NS; B = M − M B , where M is calculated as (Glen-denning 1997; He et al. 2015) M = c ∫ R dr π r E( r ) , (21)and M B = Nm b , where m b is the mass of baryons (931.5MeV/c ) and N is the number of baryons calculated by the MNRAS000
The gravitational binding energy (B) is defined as the massdifference between gravitational mass (M) and baryonic mass( M B ) of the NS; B = M − M B , where M is calculated as (Glen-denning 1997; He et al. 2015) M = c ∫ R dr π r E( r ) , (21)and M B = Nm b , where m b is the mass of baryons (931.5MeV/c ) and N is the number of baryons calculated by the MNRAS000 , 000–000 (0000)
H. C. Das et al. volume integration over the whole radius in the Schwarzchildlimit is N = ∫ R dr π r (cid:104) − Gm ( r ) rc (cid:105) − / . (22)The N is found to be ≈ same as given in the Ref. Glen-denning (1997). We divide the B with the NS mass to formfractional B , which is B / M for easier to compare with ourSun. The binding energy per particle of the symmetric NMis ≈ -16 MeV, i.e. it needs 16 MeV to make the system un-bound. For pure neutron matter (PNM) system, it is posi-tive (Serot & Walecka 1986). That means the PNM systemis already unstable. In the case of the NS, which consistsof ≈
90% of neutrons and ≈
10% of protons and leptons.As we know, the nuclear force is state-dependent and thenucleon-nucleon interaction either singlet-singlet or triplet-triplet, which are repulsive (Patra & Praharaj 1992; Satpathy& Patra 2004; Kaur et al. 2020). However, the singlet-tripletnucleon-nucleon interaction is attractive. Due to the excessnumber of neutrons, the repulsive part adds instability to NS.As a result, the NS becomes unbound, which give positivebinding of the system. However, its enormous gravitationalforce balanced the repulsive nuclear force. Thus for the wholeNS, the B is negative.With the addition of DM inside the NS, the B will goingtowards positive, that means it is going to be unstable. How-ever, the instability of NS depends on the DM percentage.The variation of B / M with the variation of k DMf depictedin Fig. 7 for different parameter sets. The numerical valuesgiven in Table 2. The careful inspection of the Table 2 showsthat up to the k DMf value 0.02 GeV the B of the canoni-cal and maximum mass neutron star both are negative. Thatindicates that both the canonical and maximum mass neu-tron star system are bound systems with this amount of DMinside the system. However, as we increase the DM momen-tum, the canonical neutron star system becomes unboundwith positive B . For example, with DM momentum 0.04 GeVthe binding energy for the canonical star becomes positivefor different parameter sets. But still, the maximum massneutron star shows a bound system with negative bindingenergy. From this, we conclude that one can constraint theDM percentage inside the NS. If the DM percentage is morethan canonical star, it forms a mini black hole at the core anddestroys the NS (Goldman & Nussinov 1989; De Lavallaz &Fairbairn 2010; Kouvaris & Tinyakov 2011; Kouvaris 2012).The cooling of NS is also faster with the increasing of DMmass (Ding et al. 2019; Bhat & Paul 2020). That means thepositive B may have a relation with the cooling properties ofthe NS or in other words; it may accelerate the urca process. In the present work, we study the impacts of dark matteron the curvatures of the NS. The calculations are done withsome well tested RMF parameters sets like NL3, the E-RMFparameter IOPB-I (two additional couplings to RMF) andG3 parameter (six extra couplings to RMF). The E-RMFformalism well suited to both finite nuclei properties as wellas NM in extreme conditions. The parameter sets G3 andIOPB-I established the recently measured the maximum massand the radius of the NS. The EoS of the NS is calculated -0.2-0.100.10.20.30.40.5 B / M k fDM = 0.00 GeVk fDM = 0.02 GeVk fDM = 0.03 GeVk fDM = 0.04 GeV M/M O NL3 G3 IOPB-I . Figure 7. (colour online) The variation of B / M with the M/ M (cid:12) of the NS with and without DM. by assuming that the DM particles present inside the NS.We find that the DM effects play a significant role in the NScurvatures, even in the E-RMF model, which yield a softerEoS.We calculate the various curvatures with the variationsof the baryon density, mass and radius of the NS in thepresence of the DM. The curvature increases or decreaseswith increase the baryon density. It is observed that at lowerdensity the quantity K and W gives more curvature than J , R . At the crust region, the curvatures J , and R almostvanish. The K and W approach each other within the crustand have a local maximum in that region. Moreover, theradial variation of K (r) increases with the increasing DMmomentum slowly up to canonical star and more for maxi-mum star. The percental change of K (r) with and withoutDM is approximately 33% for 1.4 M (cid:12) , and it increases forthe maximum mass star. From the surface curvature study,we conclude that the softer EoS gives large curvature thanthe stiffer one, which means the smaller massive star ismore suitable to study the EoS of the NS. The bindingenergy increases towards positive with the increasing of DMmomentum. From this, we conclude that a tiny amount ofDM can accumulate inside the NS. The more percentage ofthe DM heat the NS, and it accelerates the urca process,which enhanced the cooling of the NS, and it makes the NSunstable. ACKNOWLEDGEMENTS
The computations is supported in part by the SAMKHYA:High Performance Computing Facility provided by Instituteof Physics, Bhubaneswar. SKB is supported by the NationalNatural Science Foundation of China Grant No. 11873040
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