Confronting nuclear equation of state in the presence of dark matter using GW170817 observation in relativistic mean field theory approach
Arpan Das Tuhin Malik Alekha C. Nayak



















































    1.5cm-1cm^1Physical Research Laboratory, Ahmedabad 380009, India ^2Department of Physics, BITS-Pilani, K.K. Birla Goa Campus, Goa 403726, India.
        #1#1#11/2/#1_\ #1#⃗1
            We confront admixture of dark matter inside neutron star using gravitational wave constraints coming from binary neutron star merger.
            We consider a relativistic mean field model including σ-ω-ρ meson
            interaction with NL3 parameterization. We study fermionic dark matter interacting with nucleonic matter via
            Higgs portal mechanism. We show that admixture of dark matter inside the neutron star soften the equation state and lower the value of tidal deformability.
             Gravitational wave GW170817 observation puts an upper bound on tidal deformability of
            a binary neutron star with low spin prior at 90% confidence level, which disfavors stiff equation of state such as Walecka model with NL3 parameterization.
            However, we show that
             Walecka model with NL3 parameterization with a fermionic dark matter 
             component satisfy the tidal deformability bound coming from the GW170817 observation. 
            ....Confronting nuclear equation of state in the presence of dark matter using GW170817 observation in relativistic mean field theory approach 
                Alekha C. Nayak^1,[ alekha@prl.res.in]
                December 30, 2023
            ===========================================================================================================================================
            
            
            
            § INTRODUCTION
            
            
            
             Compact objects like neutron stars (NS) are nature's laboratory which can shed light directly or indirectly on the different branches of physics such as
             low energy 
             nuclear physics, QCD under extreme conditions, the general theory of relativity etc. A neutron star is one of the remnants of a dying star
             undergoing gravitational collapse. Gravitational collapse of stars with a mass range between 1.4-3.0M_⊙ evolve into a neutron star. 
             Neutron degeneracy pressure inside the neutron star makes it hydrostatically stable against the gravitational collapse. If the mass of a
             dying star is very large (beyond 10M_⊙) then the stellar remnant will overcome the neutron degeneracy pressure and gravitational
             collapse will produce a black hole. Matter density inside the neutron star can be  as large as few 
             times nuclear saturation density (n_B=0.16 fm^-3). Interior of a neutron star provides a situation to study the behavior of matter
             at extreme conditions. In this context, the possible equation of state (EoS) of infinite nuclear matter has been explored extensively
             (for a brief review see <cit.>). The main challenge in the description of matter at high densities inside neutron stars
             is to develop a model
             which  not only describes matter at high densities, but also the properties of matter observed at saturation densities <cit.>. Valid nuclear EoS has to satisfy presently well accepted empirical and experimental  constraints <cit.>,
             e.g. ground state properties of 
             spherical and deformed nuclei, saturation density, binding energy, symmetry energy, compression modulus etc. as well 
             as constraints coming from infinite nuclear matter e.g., neutron star mass radius relation, tidal deformability etc. Rotating neutron stars 
             or pulsars give important information about superfluid nature of nucleon inside the neutron star <cit.>. 
             Superfluidity of nucleons is important to explain timing irregularities (glitch) of pulsars. From the high energy nuclear physics point of view, neutron star provides an ideal condition where high density QCD matter (quark matter phase, color superconducting phase etc.) can exist
             <cit.>. 
             Historically 
             neutron star mass radius relationship coming from solving
             Tolman-Oppenheimer-Volkoff (TOV) equation have been studied extensively to put constraints on the nuclear EoS  <cit.>. 
             However recent observation of gravitational waves from neutron star mergers  opens up another dimension in the study of the nuclear EoS<cit.>. 
            
             
             On August 17, 2017, the Advanced LIGO and Virgo observatories detected the gravitational waves (GW) from a merging binary
             NS <cit.>. GW170817 data open up a new way to understand the neutron star structure and the underlying EoS of dense matter.
             Details of the internal structure of the 
             neutron stars in the binary mergers become important as the orbital separation become comparable with the size of the bodies. For neutron star 
             binary merger the tidal field of the companion induces a  quadrupole moment to the other neutron star. The relation of the induced quadrupole
             moment to the external tidal field is proportional to the tidal deformability. Tidal deformability is sensitive to the mass, radius and tidal
             love number, which intern depends on the nuclear EoS. Using observed tidal deformability parameters of the neutron star merger one can put
             strong constraints on the neutron star EoS, for details and related studies see <cit.>
             and references therein.
             The GW170817 observation puts an upper bound on tidal deformability of the combined binary NSs at 90% confidence <cit.>. Consequently,
             this can be used to rule out certain equation of states of  neutron stars. 
            
              Observations of the kinematics of self gravitating objects such as galaxies and clusters of galaxies give strong hint of the existence of
             dark matter (DM). Cosmological observations tell us that this invisible matter cannot consist of baryons, it must be a new kind of 
             matter which interacts with the rest of the standard model particles very weakly. The exact nature of the dark matter, its coupling between
             standard model particles and the mass is still not known. However extensive studies on the particle physics dark matter models have put 
             strong constraints on the coupling constant and mass of the dark matter particles (for a recent review on dark matter physics see <cit.>).
             Among different proposals of dark matter, weakly interacting massive particle (WIMP) scenario has gained favor because
             it gives the correct prediction of the measured relic abundance of the dark matter today very naturally. 
             
               
             Presence of dark matter inside  neutron star and its consequences have been discussed in the literature
             <cit.>. This discussions include dark matter capture 
             by neutron star and heating of old neutron star in the galactic halo to a temperature detectable by upcoming infrared telescopes <cit.>,
             trapped WIMPs inside the neutron star <cit.>, charged massive dark matter particle and its effect on neutron star <cit.>, 
             heating of a neutron star due to dark matter annihilation <cit.>, or the collapse of a 
             neutron star due to accretion of non annihilating dark matter <cit.> etc. In Ref.<cit.> authors have considered possible
             effects of a self interacting dark matter core on the maximum mass of a neutron star, mass-radius relation and on the NS 
             tidal deformability parameter. They have computed radial density and pressure profiles of the baryonic and dark matter components for different 
             nuclear EoS and different dark matter fractions. In Ref.<cit.>, the authors have considered  
             Walecka relativistic mean field model including σ-ω interaction for the nucleonic part <cit.> and  fermionic dark matter inside the neutron star. Using mean field approximation they
             have calculated effective nucleon mass, variation of 
             σ field, EoS  and the corresponding mass radius relation in this model. However, it is important to mention that the simple
             relativistic mean field model (RMF) model taken
             in this work is unsuccessful in producing nuclear saturation properties. This simple model is ruled out
             due to the fact that it gives high nuclear incompressibility (≈ 500MeV) and very low nucleon mass <cit.>.  
              Keeping this limitation of the simple σ-ω in this work we have considered the a generalized Lagrangian
             for the nucleonic sector including σ-ω-ρ meson interaction with NL3 parameterization <cit.>. The EoS of this model
             is disfavoured by GW170817 tidal deformability upper bound limit. However, we show that if we consider  fermionic dark matter  interaction
             via Higgs portal mechanism, then it can evade the GWs tidal deformability upper bound constraint. 
              We have also considered non gravitational interaction of the dark matter and the nucleon field  in a relativistic mean field approach.
             
             
             This paper is organized as follows: in Sec.(<ref>) we discuss generalized Walecka model with NL3 parameterization.
             In Sec.(<ref>), we discuss fermionic dark matter model and its interaction with nuclear matter. The EoS of total
             Lagrangian density of dark matter and nuclear matter is presented in Sec.(<ref>). Constraint from GWs tidal deformability
             on  EoS is discussed in Sec.(<ref>). Finally, we conclude in Sec.(<ref>).
             
             
            
            
            § WALECKA MODEL WITH NL3 PARAMETERIZATION
            
            In this section, we briefly summarize the relativistic mean field model (RMF) <cit.>, which is also known as Quantum Hadron Dynamics
             (QHD) <cit.>. In this framework, nucleons are quasiparticles with an effective medium dependent mass and baryon chemical potential.
             They move in the mean field of mesons. Simplest QHD model is known as σ-ω model. In this model nucleon-nucleon interaction is mediated by 
             the exchange of σ and ω mesons. Properties of  symmetric nuclear matter has been studied in this model. In general
             σ mesons give rise to a strong attractive central force and a spin-orbit nuclear force, on the other hand, ω-mesons are 
             responsible for the repulsive central force. However, this simple model does not reproduce nuclear saturation properties, e.g. 
             compressibility <cit.>.
             More advanced versions of QHD includes ρ meson exchange interaction between 
             nucleons <cit.>. Since protons and neutrons only differ in terms of their isospin projections,  ρ mesons are included
             to distinguish between 
             these baryons and to give a better account of the symmetry energy. These vector mesons are charged, hence the reaction between  ρ meson 
             and  proton will differ from the reaction between  ρ meson and  neutron. In general one can also include photon field, however, neutron 
             star is assumed to be charge neutral, hence the contribution from the photon field can be neglected. 
             
             
             Lagrangian including nucleon field, σ, ω and ρ mesons and their interactions can be written as <cit.>,
            
                ℒ =    ψ̅[γ^μ(i∂_μ-g_vV_μ-g_ρτ⃗.b⃗_⃗μ⃗)-(M_n+g_sϕ)]ψ
                 +1/2∂_μϕ∂^μϕ-1/2m_s^2ϕ^2-1/3g_2ϕ^3-1/4g_3ϕ^4
                    -1/4V^μνV_μν+1/2m_V^2V^μV_μ-1/4b⃗^⃗μ⃗ν⃗.b⃗_⃗μ⃗ν⃗+
                 1/2m_ρ^2b⃗^⃗μ⃗b⃗_⃗μ⃗
            
            In general one can also include terms quartic in ω meson and ω-ρ interactions. However in the present work we have used NL3 
            parameterization of the above Lagrangian. In this parameterization coupling of terms quartic in ω meson and ω-ρ interactions
            are taken to be zero. In the above equation ψ is  nucleon doublet, ϕ, V_μ and b⃗_⃗μ⃗ denotes σ, ω, ρ 
            meson field respectively. m_s, m_V and m_ρ are the 
              masses of the mesons and M_n denotes the nucleon mass. g_s, g_v and g_ρ are the scalar, vector and isovector coupling constants
              respectively. Field strength tensor of the vector and isovector mesons are given by, 
              
                V_μν=∂_μV_ν-∂_νV_μ,
            
              and,
              
                b⃗_μν=∂_μb⃗_ν-∂_νb⃗_μ
            ρ meson field can be written explicitly as, b⃗^μ=(b_1^μ,b_2^μ,b_3^μ). b_3^μ represents neutral ρ^0 meson and 
              ρ^± are the orthogonal linear superposition of b_1^μ and b_2^μ.
              
                b^μ_±=1/√(2)(b_1^μ± b_2^μ).
            τ⃗=(τ_1,τ_2,τ_3) are the Pauli matrices, these are also the isospin operators. Proton and neutron are the
               different projections of nucleon in isospin space. Operation of τ_3 on neutron and proton is as follows,
               
                τ_3|p⟩=+1|p⟩    &  τ_3|n⟩=-1|n⟩
            
               Numerical values of the all the parameter of the 
             the Lagrangian are given in the following table<cit.>, 
            
            
            § INTERACTION LAGRANGIAN BETWEEN NUCLEAR MATTER AND DARK MATTER
            
            Due to the galaxy rotation, the compact object like neutron star pass through the dark matter halo and capture dark matter particle from it. Because of the high baryon density inside neutron star, DM particle loose energy due to its  interaction with neutrons. The strong gravitational force of the NS trap the DM after it loses some energy <cit.>. 
            There are also other mechanism such as conversion  neutrons to scalar dark matter, scalar DM production via bremsstrahlung increases the dark matter density inside the neutron star<cit.>. Since dark matter composed of 95 % total matter density, one could possible imagine that many compact objects composed of DM. The amount of dark matter inside NS also depend on the evolution history of NS, the environment where it lives. In Ref.<cit.>, the authors have shown that the binary neutron star systems might enhance DM accumulation probability inside NS.
            
             We consider fermionic dark matter  (χ)  inside  the neutron star. 
             Here we consider the lightest neutralino which acts as a fermionic dark matter candidate <cit.>. Dark matter is not directly
             coupled with the nucleons rather they are coupled to the Higgs field h. Coupling 
             between the dark matter and the Higgs field is y. For neutralino mass (M_χ=200 GeV), the value of y
             varies from 0.001 - 0.1. We fix y= 0.07 for the rest of our analysis <cit.>. The Higgs field is also coupled
             to the nucleons through 
             effective Yukawa coupling fM_n/v, where f the proton-Higgs form factor and its value has been estimated to be approximately 0.35
             <cit.>.
             We have not considered h^3 and h^4 term in the Higgs potential, because in the 
             mean field approximation the value of the h is small and the only dominant term is the h^2. So, the dark sector 
             Lagrangian and its interaction with the nucleons and Higgs field is given by <cit.>
                ℒ_DM=  χ̅[iγ^μ∂_μ-M_χ+yh]χ+1/2∂_μh∂^μh-1/2M_h^2h^2
                 +fM_n/vψ̅hψ.
            
             
             Direct detection experiment such as LUX <cit.> and XENON <cit.> excluded dark matter-nucleon cross section above ∼  8 × 10 ^-47 cm^2  for dark matter mass range 30-50 GeV at 90 % C.L. . The invisible Higgs decay width tightly constraint the dark matter below M_h/2, hence the dark matter mass m_χ=200 GeV evades these constraints. It is important to mention that dark matter may not be a single component, it may well be multicomponent system. Dark matter can be consist of low mass as well as high mass particles. As an example in Ref.<cit.> authors have considered heavy dark matter particles inside 
            the neutron star.
             
             Here we have considered the average number density of nuclear matter is 10^3 times larger than the  average dark matter density (n_DM), which gives the ratio between mass of the dark matter inside neutron star and mass of the neutron star to be ∼1/6<cit.>.
            We know that nuclear saturation density n_B∼ 0.16fm^-3, so dark matter number density is n_DM∼ 10^-3n_B∼ 0.16× 10^-3fm^-3.
             Number density of dark matter n_DM=(k_F^DM)^3/3π^2, which gives k_F^DM∼ 0.033 GeV. In our calculations we have varied k_F^DM from 0.02 GeV to 0.06 GeV. For 
             these values of k_F^DM corresponding dark matter density will be different.
             
            
            
            
            
            § NEUTRON STAR EQUATION OF STATE AND BETA EQUILIBRIUM
            
            The Euler Lagrange Equation of motion for nucleon doublet (ψ), scalar(ϕ), vector(V^μ), isovector(b⃗^μ), DM particle (χ)
            and Higgs boson (h) can be derived from Lagrangian densities Eq.(<ref>) and Eq.(<ref>) as,
            
                [                                                      [γ^μ(i∂_μ-g_vV_μ-g_ρτ⃗.b⃗_μ)-(M_n+g_sϕ-fM_n/vh)]ψ=0,;                                                                    ∂_μ∂^μϕ+m_s^2ϕ+g_2ϕ^2+g_3ϕ^3+g_sψ̅ψ=0,;                                                                               ∂_μV^μν+m_V^2V^ν=g_vψ̅γ^νψ,;                                                                           ∂_μb⃗^μν+m_ρ^2b⃗^ν=g_ρψ̅γ^ντ⃗ψ,;                                                                                     (iγ_μ∂^μ-M_χ+yh )χ=0,;                                                                           ∂_μ∂^μh+M_h^2h =yχ̅χ+fM_n/vψ̅ψ, ]
            
            respectively.  The DM particle 
            mass and Higgs particle mass are denoted as M_χ and M_h, respectively. 
            Applying standard relativistic mean field approximation we get,
            
                [                                              ϕ_0 =1/m_s^2(-g_s⟨ψ̅ψ⟩-g_2ϕ_0^2-g_3ϕ_0^3),;                                               V_0 =g_v/m_V^2⟨ψ^†ψ⟩= g_v/m_V^2(ρ_p+ρ_n),;                                                          h_0 =y⟨χ̅χ⟩+fM_n/v⟨ψ̅ψ⟩/M_h^2,;                                             b_0 =g_ρ/M_ρ^2⟨ψ^†τ_3ψ⟩=g_ρ/M_ρ^2(ρ_p-ρ_n),;                                              (iγ^μ∂_μ-g_vγ^0V_0-g_ρτ_3γ^0b_0-M_n^⋆)ψ=0,;                                                                     (iγ^μ∂_μ-M^⋆_χ)χ=0, ]
            
            where M_n^⋆ and M_χ^⋆ are nucleon and dark matter effective mass respectively. ρ_p and ρ_n are the densities of proton and neutron respectively. 
            The effective mass of nucleon and dark matter can be given as,  
            
                [                              M_n^⋆= M_n+g_sϕ_0-fM_n/vh_0,;                                           M_χ^⋆=M_χ-yh_0. ]
            
            The baryon density (ρ), scalar density (ρ_s) and dark matter scalar density (ρ_s^ DM)
            are
            
                [                                                                                                   ρ= ⟨ψ^†ψ⟩ = γ/(2 π)^3∫_0^k_Fd^3k,;                                                                           ρ_s = ⟨ψ̅ψ⟩ = γ/(2 π)^3∫_0^k_FM_n^⋆/√(M_n^⋆^2+ k^2) d^3k,;                                                                   ρ_s^ DM= ⟨χ̅χ⟩ =  γ/(2 π)^3∫_0^k_F^ DMM_χ^⋆/√(M_χ^⋆^2+ k^2) d^3k, ]
            
            where k_F and k_F^ DM are the Fermi momentum for the nucleonic matter and dark matter respectively. 
            γ is the spin degeneracy factor of nucleon and  γ=2 for neutron and proton individually. 
            
            The masses for both nucleon and dark matter depends on baryon density for fixed values of
            dark matter Fermi momentum k_F^ DM and coupling constants. These masses and coupling values are discussed in Table.(<ref>) and Sec.(<ref>). To get the density 
            dependent profile for M_n^⋆ and M_χ^⋆, one needs to solve numerically Eq. (<ref>) together with
            the field equations Eq. (<ref>) in self consistent manner. 
            In this work, we have taken the average dark matter number density approximately 1000 time smaller 
            than the average neutron number density. This implies the dark matter mass fraction with respect to 
            the neutron star mass is ≃ 1/6.
            The expectation values of the energy-momentum tensor or the stress tensor provide the energy density and 
            pressure of the system in static case i.e., the EoS, which is given by,
            ϵ = ⟨ T^00⟩     and    
            P= 1/3⟨ T^ii⟩ .
            
            The expression for the total energy density (ϵ) and the pressure (P) can be
            obtained by combining the Lagrangian density Eq.(<ref>) and Eq.(<ref>) :
            ϵ = g_v V_0(ρ_p+ρ_n)+g_ρb_0(ρ_p-ρ_n)+1/π^2∫_0^k_pdk k^2√(k^2+(M_n^⋆)^2)
            
               +1/π^2∫_0^k_ndk k^2√(k^2+(M_n^⋆)^2)+1/π^2∫_0^k_F^DMdk k^2√(k^2+(M_χ^⋆)^2)
            
              +1/2m_s^2ϕ_0^2+1/3g_2ϕ_0^3+1/4g_3ϕ_0^4-1/2m_V^2V_0^2-1/2m_ρ^2b_0^2+1/2M_hh_0^2. 
              
            P = 1/3π^2∫_0^k_pk^4 dk/√(k^2+(M_n^⋆)^2) 
             +1/3π^2∫_0^k_nk^4 dk/√(k^2+(M_n^⋆)^2) + 1/3π^2∫_0^k_F^DMk^4 dk/√(k^2+(M_χ^⋆)^2)
            
             -1/2m_s^2ϕ_0^2-1/3g_2ϕ_0^3-1/4g_3ϕ_0^4+1/2m_V^2V_0^2+1/2m_ρ^2b_0^2-1/2M_hh_0^2,
             ρ_n and ρ_p are the neutron and proton number density with k_F^n and k_F^p are the corresponding 
            Fermi momentum of neutron and proton, respectively. The number densities and corresponding Fermi momenta are equal 
            for the symmetric nuclear matter. The matter inside neutron star mainly composed of neutrons. However, the neutron will 
            eventually β- decays as,
             
            n → p + e^- + ν̅, 
            
            n + ν→ p + e^- .
             
            To maintain the neutron star matter charge neutral, muons (μ) will appear when the chemical potential 
            of the electrons reaches the muon rest mass (m_μ = 106 MeV). For a given baryon density (ρ=ρ_n+ρ_p), 
            the charge neutrality is given as,
            ρ_p=ρ_e+ρ_μ 
            and the β- equilibrium condition is given as,
            μ_n=μ_p+μ_e                and          μ_e=μ_μ 
            Where, the chemical potentials μ_p, μ_n, μ_e and μ_μ are given as,
            
                μ_p    =   ∂ϵ/∂ρ_p=g_vV_0+g_ρb_0+√(k_p^2+(M_n^⋆)^2)
                μ_n    =   ∂ϵ/∂ρ_n=g_vV_0-g_ρb_0+√(k_n^2+(M_n^⋆)^2)
                μ_e    =   √(k_e^2+m_e^2)
                μ_μ   =   √(k_μ^2+m_μ^2)
            
            The particle fractions of neutrons and protons will depend on both charge neutrality and 
            the β- equilibrium condition as given above. The self consistent numerical solution
            of Eq.(<ref>) and Eq.(<ref>) will set the fraction of neutron, proton, electron and muon number density for a given baryon
            density. The total energy density and pressure of leptons are given as,
            ϵ_l    =   ∑_l=e,μ1/π^2∫_0^k_l  k^2 √(k^2+m_l^2) dk 
            
            P_l    =   ∑_l=e,μ1/3 π^2∫_0^k_lk^4 dk/√(k^2+m_l^2) 
            The total energy density and pressure for β- equilibrated neutron star matter are
            ϵ_ NM= ϵ_l + ϵ , 
            
            P_ NM=P_l + P .
             
              
               In Fig.(<ref>), we plot pressure (P_ NM) as function of the total energy density (ϵ_ NM) for different dark matter
               Fermi momentum k_F^DM=0.0-0.06 GeV.  Fermi momentum k_F^DM=0.0 GeV corresponds equation of 
               state without dark matter. Increasing the value of  k_F^DM from 0.02 GeV to 0.06 GeV, the EoS becomes softer,
               i.e. with increasing density of dark matter pressure reduces, which is consistent with earlier work <cit.>.
               This behaviour is evident from the expression of energy density and pressure from Eq.(<ref>) and Eq.(<ref>),
               i.e. with increasing  k_F^DM
               the dark matter contribution in energy density increases much faster than the pressure.
                
               
                Neutron star mass radius relationship can be obtained by solving TOV equation
                for a given nuclear matter EoS <cit.>. The EoS for the core is obtained from the Walecka Model with NL3 parameterization in the presence
                and absence of dark matter components. Crust EoS is modeled using the BPS EoS <cit.> for the range of density
                ρ∼ 4.8 × 10^-9 to 2.6× 10^-4 fm^-3.
                We use the polytropic form P_NM(ϵ_NM)=a_1+a_2ϵ_NM^γ'<cit.> to join the core and crust of the NS, 
                where a_1 and a_2 are
                obtained by matching the edge of the core at one end with the inner edge of the
                outer crust at other end, and γ' is taken  4/3<cit.>.    
                In Fig.(<ref>) we plot the mass  radius of NS using the EoS as shown in Fig.(<ref>).
                It is clear from the Fig.(<ref>) that the equation state is softer in the presence of larger dark matter density.
                A softer equation of state gives a lower value of the maximum mass of neutron star. From Fig.(<ref>), the maximum mass without
                dark matter (k_F^DM=0) ∼2.8 M_⊙ and the value of the maximum mass of NS decreases with increasing dark matter density.
                
               
               
               In the context of dark matter inside the neutron star 
            it is very natural to consider the formation of dark matter core like structure inside the neutron star with nonuniform distribution of the dark matter. This picture has been 
            explored in various literatures<cit.>. We would like to point out that we are
            focusing on the mainly qualitative aspect of the presence of dark matter
            inside the neutron star. Effect of non uniformly distributed dark matter
            inside the neutron star on its properties has been discussed in the Ref.<cit.>.
            In the Ref.<cit.> authors have invoked two-fluid picture of the neutron star
            containing nuclear matter fluid as well as dark matter fluid inside the neutron star.
             In that work, authors also have considered RMF picture for both nuclear fluid
            as well as dark matter fluid, the
            mass radius relation in the presence of dark matter for some specific EOS
            is qualitatively similar to our result, e.g. in the
            presence of dark matter, mass of neutron star can decrease. They have also shown 
            that for mass of the dark matter (M_DM) can be large e.g. 0.33  M_⊙. However, if one assumes RMF kind of situation for the dark matter sector then the coupling of the dark matter particle with a scalar, vector particles etc. will be free parameters of the theory. In the nuclear matter sector, the coupling
            constants have been fixed keeping in mind the finite nuclei properties, but for the dark matter, these constraints are not available. Hence these free
            coupling constants in the dark matter sector will make the model less predictive.   
               
               
               
               Tidal deformability also depends upon the compactness and equation of state. In the next section, 
               we study the effect of dark matter on nuclear matter EoS  using the tidal deformability  constraint from  GW170817 observation.
               
             
             
            
              
            
            
            § TIDAL DEFORMABILITY CONSTRAINT
             
             One of the  important measurable structural properties of a binary merger is the 
            tidal deformability. In a coalescing binary NS system, during the last stage of inspiral,  
            each NS develops a mass quadrupole due to the tidal gravitational 
            field induced by the other NS forming the binary. The tidal deformability describes 
            the degree of deformation of a NS due to the tidal field of the 
            companion NS and is sensitive to the nature of the EoS.
            The tidal deformability is defined as, 
            
                λ = 2/3k_2R^5,
            
            where R is the radius of the NS. The value of k_2 is typically in 
            the range ≃ 0.05-0.15<cit.> for NSs and depends on the stellar 
            structure. This quantity can be calculated using the following expression <cit.>       k_2 = 8C^5/5(1-2C)^2
            [2+2C(y_R-1)-y_R]
                  ×{2C(6-3 y_R+3 C(5y_R-8))
                  +4C^3[13-11y_R+C(3 y_R-2)+2
            C^2(1+y_R)] 
                    
            +3(1-2C)^2[2-y_R+2C(y_R-1)]log(1-2C)}^-1,
            
            where C(≡ M/R) is the compactness parameter of the star of
            mass M.  The quantity y_R(≡ y(R)) can be obtained by solving
            the following differential equation
            
            r d y(r)/dr + y(r)^2 + y(r) F(r) + r^2 Q(r) = 0
            
            with
            
            F(r) = r-4 π r^3 ( ϵ(r) - p(r)) /r-2
            M(r),  Q(r)    =   4 π r (5 ϵ(r) +9 p(r) +
            ϵ(r) + p(r)/∂ p(r)/∂ϵ(r) - 6/4 π r^2)/r-2M(r)
               -     4[M(r) + 4 π r^3
            p(r)/r^2(1-2M(r)/r)]^2, 
            along with TOV equation with proper boundary conditions <cit.>. One can then define the dimensionless tidal deformability: 
            Λ = 2/3k_2 C^-5.
            
              
             Individual dimensionless tidal deformability of two stars, Λ_1 and Λ_2
            cannot be extracted independently from the observed gravitational
            waveform. Instead, an effective dimensionless tidal deformability of the binary
            Λ̃ can be extracted, which is a mass-weighted average of the individual dimensionless tidal deformability
             Λ_1 and  Λ_2. The effective tidal
            deformability (Λ̃) of binary system in terms of Λ_1 and Λ_2 is defined as <cit.>Λ̃=16/13(m_1+12 m_2) m_1^4 Λ_1+(m_2+12m_1)m_2^4 Λ_2/(m_1+m_2)^5
            where m_1 and m_2 are the masses of the neutron stars in the binary.
            Similarly, masses of the two
            companion neutron stars cannot be  measured directly, rather
            the chirp mass, ℳ_c=(m_1m_2)^3/5(m_1+m_2)^-1/5, which is measured directly. By assuming low-spin
            prior which is consistent with the binary neutron star systems
            that have been observed in, GW170817 put an upper
            bound on the NSs combined dimensionless tidal deformability and
             chirp mass with 90% confidence <cit.>. This analysis predicts that the
            combined dimensionless tidal deformability of the NS merger is
            Λ̃≤ 800. It is important to note that a reanalysis of
            GW170817 observation has been done assuming the
            same EoS for both stars and this puts an upper limit on the dimensionless
            tidal deformability, Λ̃≤ 1000<cit.>. However
            lower bound on dimensionless tidal deformability can be put using the
            investigation of the UV/optical/infrared
            counterpart of GW170817 with kilonova models <cit.>. The lower bound of
            dimensionless tidal  deformability is Λ̃≥ 400.
            
            One of the important goals of this work is to study the structural properties of neutron stars 
            in the presence of the dark matter component. For the sake of arguments it is important to understand
            the behavior of dimensionless tidal deformability and the tidal Love number of the 
            companion neutron stars. Solving Eq.(<ref>) and TOV equation with appropriate boundary conditions, we get y_R.
            Using the value of y_R and compactness one can get  k_2 using the expression Eq.(<ref>).  Left plot in Fig.(<ref>)
            shows the dimensionless tidal deformability Λ and the right plot shows tidal Love number
            k_2 as a function of the NS mass for our EoS with different dark matter density. The value of k_2 is of the range 0.09-0.13 
            for a typical neutron star of mass 1.5M_⊙, which is expected <cit.>. For a given neutron star mass
            (say around 1.5M_⊙) EoS without the dark matter 
            predicts larger radius and with increasing dark matter density radius decreases. Since the dimensionless tidal deformability is inversely 
            proportional to the compactness (C=M/R), its value is larger in the absence of dark matter. 
            
            To study the tidal deformability constraint from the GW170817 observation on EoS of NS, we plot the combined tidal deformability of the 
            binary system in
            Λ_1, Λ_2 plane in Fig.<ref>. Λ_1 and Λ_2 are the individual dimensionless tidal deformability
            of the high mass m_1 and low mass
            m_2 neutron stars in a binary, respectively. The curves are corresponding to the EoS with different dark matter density and
            obtained by varying m_1 and m_2 independently. m_1 has been taken in the range 1.365<m_1/M_⊙<1.60 and the range of
            the m_2 is determined by 
            keeping the chirp mass ℳ_c fixed at 1.188M_⊙. The dashed and the
            dot lines represent, respectively, the 90% and 50% confidence limits of the combined dimensionless tidal deformability
            obtained from the GW170817 for the low spin prior. One can see from this plot that EoS given by the NL3 parameterization without 
            dark matter component can be excluded at 90% confidence level using the upper bound on tidal deformability of a binary system. However
            if we consider dark matter component in neutron stars, then 
            NL3 parameterized EoS comes within the 90% confidence level. Hence a small component of dark matter inside a neutron star can revive 
            well known EoS, which otherwise might be excluded by the GW170817 observation.
            
            
            
            § CONCLUSIONS
            
                  We have confronted the neutron star equation of state in the presence of dark matter 
                  component using the gravitational wave constraint from the binary star merger. We have shown that for a uniformly
                  distributed dark matter inside neutron star, the EoS becomes softer which eventually produces lower NS mass with 
                  increasing dark matter density. We have taken Walecka model with NL3 parameterization in the nuclear matter sector. Walecka model
                  with NL3 parameterization without dark matter admixture gives rise to a maximum mass of NS ∼ 2.8 M_⊙. By increasing dark matter density 
                  (Fermi momentum) inside neutron star reduces the value of maximum mass. Value of the maximum mass of neutron star changes from
                  2.8 M_⊙ to 2.1 M_⊙ by increasing dark matter Fermi momentum from 0.0 GeV to 0.06 GeV.
                  One of the striking results of our analysis is that the  stiffer equation of states such as relativistic mean field model (Walecka model)
                  with NL3 parameterization is ruled out at 90% C.L. 
                  using the GW170817 observation. However, in the presence of dark matter this constraint can be evaded and NL3 parameterization
                  can be brought within  90% C.L.. 
                  
                  
            
            § ACKNOWLEDGEMENTS
            
                  T.M. acknowledges the hospitality provided by Physical Research Laboratory (PRL), Ahmedabad, India, during his visit, where the work has been initiated. 
                  We would like to thank Prof. Hiranmaya Mishra for constant encouragement and support.
            
            
            
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