The presence of Dark matter DM within a neutron star NS can substantially influence the macroscopic properties. It is commonly assumed that the pressure inside an NS is isotropic but in reality pressure is locally anisotropic. This study explores the properties of anisotropic NS with a subfraction of DM isotropic trapped inside. Implementing a twofluid formalism with three Equations of State EOS AP3 a realistic nucleonnucleon interaction model BSk22 modeling atomic nuclei and neutronmatter and MPA1 considering relativistic effects in nuclear interactions. The properties of NS such as mass M radius R and dimensionless tidal deformability Lambda for various DManisotropic configurations have been rigorously tested against observational constraints. These constraints include data from the binary NS merger GW170817 NICER xray measurements and pulsar massradius observations. We observe that with increasing DM subfraction higher anisotropies could also satisfy the observational constraints. Furthermore increasing the coupling g between DM and its mediator leads to the formation of a corehalo structure with a DM halo surrounding the baryonic matter BM. Specifically for coupling values of g  104 103.7 and 103.5 we observe that the maximum radius Rmax decreases with increasing anisotropy which contrasts with the behavior at g  105 and in scenarios with no DM. Our analysis indicates that binary pulsar systems could potentially constrain the extent of admixed anisotropic NS or more optimistically provide evidence for the existence of DMadmixed anisotropic NS.
['Astrophysics - Cosmology and Nongalactic Astrophysics', 'Astrophysics - High Energy Astrophysical Phenomena', 'General Relativity and Quantum Cosmology']
Implications of Fermionic Dark Matter Interactions on Anisotropic Neutron Stars
{'Implications of Fermionic Dark Matter Interactions on Anisotropic Neutron Stars': 'Premachand Mahapatra , 1, ∗ Chiranjeeb Singha , 2, † Ayush Hazarika , 3, ‡ and Prasanta Kumar Das 1, §  \n1 Department of Physics, Birla Institute of Technology and Science - Pilani,  \nK. K. Birla Goa Campus, NH-17B, Zuarinagar, Sancoale, Goa- 403726, India  \n2 Inter-University Centre for Astronomy and Astrophysics, PostBag 4, Ganeshkhind, Pune-411007, Maharashtra, India  \n3  \nDepartment of Physics, Tezpur University, Napaam, Tezpur, 784028, Assam, India  \n3  \nDepartment of Physics, Tezpur University, Napaam, Tezpur, 784028, Assam, India  \n(Dated: September 13, 2024)  \nThe presence of Dark matter (DM) within a neutron star (NS) can substantially influence the macroscopic properties. It is commonly assumed that the pressure inside an NS is isotropic, but in reality, pressure is locally anisotropic. This study explores the properties of anisotropic NS with a subfraction of DM (isotropic) trapped inside. Implementing a two-fluid formalism with three Equations of State (EOS): AP3 (a realistic nucleon-nucleon interaction model), BSk22 (modeling atomic nuclei and neutron-matter), and MPA1 (considering relativistic effects in nuclear interactions). The properties of NS, such as mass ( M ), radius ( R ), and dimensionless tidal deformability ( Λ ), for various DM-anisotropic configurations, have been rigorously tested against observational constraints. These constraints include data from the binary NS merger GW170817, NICER x-ray measurements, and pulsar mass-radius observations. We observe that with increasing DM subfraction, higher anisotropies could also satisfy the observational constraints. Furthermore, increasing the coupling ( g ) between DMand its mediator leads to the formation of a core-halo structure, with a DM halo surrounding the baryonic matter (BM). Specifically, for coupling values of g = 10 -4 , 10 -3.7 , and 10 -3.5 , we observe that the maximum radius ( Rmax ) decreases with increasing anisotropy, which contrasts with the behavior at g = 10 -5 and in scenarios with no DM. Our analysis indicates that binary pulsar systems could potentially constrain the extent of admixed anisotropic NS or, more optimistically, provide evidence for the existence of DM-admixed anisotropic NS.', 'I. INTRODUCTION': "Neutron stars (NSs) are among the densest and most compact celestial objects known to science. Understanding the internal composition of such compact objects is based on our knowledge of high-density matter [1, 2]. Precision in determining neutron star (NS) properties at densities exceeding those found in nuclei faces two main challenges: limited understanding of nuclear interactions [3-5] and gravitational interactions [6-8]. The repulsive nuclear equation of state (EOS), which describes the relationship between matter pressure and energy density, and the opposing attractive strong-field gravitational physics are interconnected through the Tolman-Oppenheimer-Volkoff (TOV) equations [1, 2, 9]. Uncertainties in these equations can affect predictions related to the structure and characteristics of NSs.  \nIn recent years, our understanding of NSs has significantly evolved due to multimessenger observations [10], particularly from binary neutron star mergers [11]. These observations have enabled a more detailed examination of various parameters influencing NS global properties, such as radius, tidal deformability, maximum mass, and other characteristics closely tied to the equation of state (EOS) governing NS behavior. There has been an abundance of observational data on NSs from both gravitational wave detectors, such as Advanced LIGO [12] and Advanced Virgo [13], and electromagnetic radiation, exemplified by the NICER mission [14]. These astrophysical observations have led to numerous efforts to constrain the EOS of NSs. These endeavors include using NICER's measurements of mass and radius [15, 16], assessing tidal deformability through gravitational waves, and applying joint constraints [15, 17-  \n- 19]. The detection of the binary neutron star merger GW170817 [20], in particular, has spurred further investigations in this field [21-23].  \nThe groundbreaking GW170817 event, marked by the first detection of gravitational waves from a neutron star merger, has been central to numerous investigations. Beyond its historical significance, it has spurred inquiries into the internal aspects of NSs. Studies have explored the influence of the NS crust on the equation of state (EOS) [24, 25], the impact of introducing an isovector-scalar meson into the quark-meson coupling description of nuclear matter [26, 27], and various Skyrme-like parametrizations [28-30]. Nonparametric inference has indicated a preference for a soft EOS [31, 32].  \nThe event has also prompted critical evaluations of the EOS for dense matter, incorporating nuclear physics within the chiral effective field theory framework [33, 34]. Despite progress, certain regions of the EOS, particularly at high densities, remain not fully understood. Identifying an electromagnetic counterpart to the event enabled the first joint gravitational wave-electromagnetic constraints on the NS EOS. Using the binary's tidal deformability parameter, simulations of electromagnetic observations within numerical relativity and kilonova models were conducted, ruling out extreme EOS models.  \nStatistical Bayesian methods were employed in the context of GW170817, with microscopic models of cold NSs using chiral effective models. Recent gravitational wave events with X-ray sources were also combined and studied with relativistic mean-field models [35-37]. Additionally, NASA's Neutron Star Interior Composition Explorer (NICER) made significant electromagnetic measurements, constraining the mass-radius relationship of the pulsar PSR J0030+0451 [15-17]. In this case, nonparametric inference suggested a preference for stiffer EOS.  \nMeanwhile, Earth-based experiments like the Lead Radius Experiment (PREX-2) have provided valuable insights into nuclear matter around saturation density, with implications for the neutron star crust. Extrapolating the data has constrained stellar radii to specific ranges, indicating a softening in the intermediate region and a stiffening at high densities, potentially leading to a phase transition in the stellar core. The expanding pool of observational data has been crucial in establishing further constraints on the EOS for dense matter, creating fertile ground for statistical and machine-learning models [38].  \nThe composition of neutron star (NS) cores has been a topic of debate and mystery in recent years. While astrophysical observations provide some insight, determining the exact nature of these dense interiors from observational data alone remains elusive. Hypotheses in the literature suggest the presence of free quarks, hyperons, delta isobars, or kaon condensation within NSs [39-41]. The presence of such exotic particles tends to soften the equation of state (EOS), affecting observables like the mass and tidal deformability of the star [42, 43]. Besides standard exotic particles, theoretical studies also propose the presence of dark matter (DM) particles within NSs. The ultra-dense profile and particle-antiparticle asymmetry in DM could lead to gravitational trapping of DM without self-annihilation in NS cores [44, 45]. Various DM candidates, such as neutralinos, axions, and technibaryons, have been theoretically explored [46-48]. Though the nature and interactions of DM with standard model particles are weak and uncertain, theoretical efforts have examined DM's effects on astral bodies [49, 50]. Gravitational wave events like GW170817 and GW190814 have provided new avenues to study DM's impact on the universe [51, 52].  \nThe universe is known to contain only about 6 % visible matter, with the remaining 94 % comprising DM and dark energy. Zwicky's mass estimation of the universe revealed this discrepancy, leading to the identification of DM (approximately 26 %) and dark energy (around 68 %) [53, 54]. Numerous theoretical, experimental, and observational efforts have been dedicated to unravelling the mysteries of DM and dark energy. Several DM candidates have been hypothesized, including weakly interacting massive particles (WIMPs), feebly interacting massive particles (FIMPs), neutralinos, and axions [55, 56]. WIMPs, produced in the early universe and annihilating in pairs, are considered thermal relics. Various experiments, such as DAMA [57], Xenon [58], and CDMS [59], aim to detect the interactions between WIMPs and nucleons, while indirect detection experiments like Fermi large area telescopes [60] and imaging air Cherenkov telescopes [61] also contribute to DM research.  \nIt is known that NSs, the densest known objects in the universe, have garnered attention because of their potential to provide insight into DM. With increasingly precise observational data, NSs are becoming valuable for studying DM properties. Theoretical models of DM-admixed NSs (DNSs) integrate nuclear matter (NM) equations of state (EOS) with hypothetical DM EOS using a general-relativistic two-fluid approach [62, 63]. A major challenge in  \nunderstanding DM in NSs is the complex mass-radius relationship influenced by DM content alongside the uncertain high-density NM EOS. This complexity makes it difficult to draw definitive conclusions about either DM or NM. For instance, observing a very massive 'NS' could indicate either a DM presence or a very stiff NM EOS. Therefore, new theoretical methods are needed to clearly identify and quantify DM in NSs.  \nCurrently, it is generally believed that most observed NSs are standard hadronic stars with minimal DM admixture [64]. Estimates suggest that NSs acquire negligible amounts of DM through accretion over their lifetimes, typically less than 10 -10 solar masses, which is unobservable. Significant DM fractions in DNSs, proposed in this study, would require exotic capture or formation mechanisms, which remain speculative.  \nRecent research has intensively investigated the effects of DM on NM and NS properties [65]. This includes DM's fundamental impact on NS mass and radius, the possibility of DM acquisition by NSs, and phenomena like heating, internal black hole formation, and collapse. Quantitative studies have examined DM's effects on NM properties and the validity of universal relations between NS compactness, moment of inertia, tidal deformability, and quadrupole moment in the presence of DM [64-68]. Some studies have considered the possibility that LIGO/Virgo events GW170817 and GW190425 might involve DNSs [69]. Other research has focused on DM's effects on NS tidal deformability and related observables, which are measurable by current and future gravitational wave detectors. Additionally, recent studies have explored the impact of DM on pulsar X-ray profiles and NS cooling processes [70].  \nUnderstanding DM's influence on NSs and other cosmic phenomena requires a multidisciplinary approach, incorporating various aspects of physics and leveraging both theoretical and observational advancements. The potential presence of exotic and DM particles within NSs highlights the need for continued exploration to unravel these cosmic mysteries. In summary, understanding DM's influence on NSs requires integrating various fields of physics and considering multiple DM candidates [71].  \nAnisotropic NSs represent a nuanced departure from the conventional TOV stars [72]. While their isotropic counterparts assume uniform pressure throughout their interiors, anisotropic stars permit pressure anisotropy, introducing an additional layer of complexity to their behaviour. This departure from isotropy is not merely theoretical; it has profound consequences for the stability, equilibrium, and observability of these cosmic bodies. Very high magnetic field [73], pion condensation [74], phase transitions [75], relativistic nuclear interaction [76], superfluid core [77], etc., are the main causes of the anisotropy inside a star. At very high pressures and densities, nuclear matter shows some irregularity in the tangential and radial components of pressure, which creates an anisotropic fluid [78]. One of the primary motivations for our exploration of anisotropic stars is their potential to serve as natural laboratories for the study of exotic states of matter. Within their ultra-dense cores, where nuclear densities prevail, exotic phases of matter, such as quark matter or hyperonic matter, may exist. The existence and properties of these phases hold profound implications for our comprehension of the fundamental forces governing the universe. Anisotropic stars [79], with their altered pressure profiles, may offer unique insights into the presence and properties of these elusive forms of matter. The complexity of anisotropic NS interiors and the potential presence of exotic and DM particles highlight the need for continued theoretical, observational, and experimental efforts to unravel these cosmic mysteries.  \nThe paper is organized as follows. In Section II, we briefly introduce the construction of the Nuclear Equation of State (EOS) and fermionic DM EOS to be studied here. Section III emphasizes the equation of stellar structure defining the hydrostatic equilibrium condition and anisotropic TOV equation in two fluid approaches along with the tidal love number equation. In Section IV, we provide a detailed examination of our results with a detailed discussion considering the GW170817 data, NICER x-ray data, and all pulsars data as our observational constraints for physical observables of NSs like mass, radius, and tidal deformability in subsection IV A, then in subsection IV B taking a fermionic DM at the core of the NS where the baryonic matter (BM) is assumed to be anisotropic, and the DM form is taken to be isotropic& it's validation with respect to the given observation constraints, followed by the neutron star properties with variation of the coupling strength g impacting core-halo transition in subsection IV C and in subsection IV D we provide the correlation analysis of the key physical parameters involved in our admixed-NS study, using both Pearson and Kendall correlation coefficients for the considered EOS taken. Finally, in Section V, we present our concluding remarks.", '1. Nuclear EOS': 'Numerous models have been suggested to describe the properties of nuclear matter in highly dense environments, particularly at densities exceeding the nuclear saturation density n 0 ∼ 0.16 fm -3 . These models can be based either on a Hamiltonian framework (potential models) or a Lagrangian framework (field-theoretical models).  \nBelow are the key characteristics and assumptions of the EOS analyzed in this study:  \n- · In this work, we utilize the Brussels-Montreal functional BSk22 [80], and for comparison, details of BSk19-21 can be found in Ref. [81]. This nuclear EOS is parametrized primarily by fitting the measured masses of atomic nuclei with Z , N ≥ 8, based on the 2012 Atomic Mass Evaluation [82]. BSk22 is characterized by its relative stiffness, allowing it to support heavy NSs in alignment with observational data.\n- · The relatively stiff MPA1 equation of state (EOS) is derived using the relativistic Dirac-Brueckner-Hartree-Fock (DBHF) formalism. This EOS incorporates the contributions to the energy arising from the exchange interactions between the π and ρ mesons [83].\n- · We have also considered the well-known AP3 EOS, which utilizes the Argonne 18 potential along with the inclusion of the delta-meson ( δ ) interaction, but without incorporating the UIX three-nucleon potential, leading to a slightly stiffer EOS compared to AP4 [84].  \nIt should be noted that this EOS ( P B ( ρ B ) ; where ρ B = Baryonic Energy density & P B = Pressure of BM) serves as a benchmark to illustrate the effects of new physics and comes with its own constraints; future research might favor alternative EOSs [85].  \nFig. 1 shows the pressure (MeV/fm 3 ) as a function of the energy density (MeV/fm 3 ) for the three chosen equations of state (EOS): AP3, BSk22, and MPA1, in the context of an isotropic neutron star. These EOS models provide insights into the behavior of matter under extreme conditions, such as those found in NSs. Furthermore, we have calculated the squared speed of sound, normalized by the squared speed of light ( c 2 ), to ensure that it remains below the causality limit ( c 2 s / c 2 = 1) for all EOS, as shown in Fig. 2.', '2. Fermions with Yukawa Interactions': 'To explore the effects of novel particles, we consider strongly self-interacting asymmetric fermionic dark matter with an equation of state described by [86, 87];  \nρχ = ρ kin ( x ) + ρ Y ( x ) = m 4 χ 8 π 2 ( x √ 1 + x 2 ( 1 + 2 x 2 ) -ln ( x + √ 1 + x 2 ) ) + g 2 x 6 m 6 χ 2 ( 3 π 2 ) 2 m 2 ϕ P χ = P kin ( x ) + P Y ( x ) = m 4 χ 8 π 2 ( x √ 1 + x 2 ( 2 3 x 2 -1 ) + ln ( x + √ 1 + x 2 ) ) + g 2 x 6 m 6 χ 2 ( 3 π 2 ) 2 m 2 ϕ (1)  \nHere ρ kin ( x ) and ρ Y ( x ) are the kinetic energy density and Yukawa potential (arising due to dark matter selfinteraction) energy density and P kin ( x ) and P Y ( x ) are the kinetic and Yukawa contribution to the pressure density, respectively. Here x = P χ m χ (where P χ denotes the Fermi momentum and m χ is the mass of the dark matter fermion). Here, we use natural units ¯ h = c = 1. The potential arising due to the self-interaction(repulsive) of two dark matter fermions 1 and 2 is considered to be of Yukawa type(repulsive in nature) -  \nV 12 = g 2 4 π e -m ϕ r 12 r 12  \nFIG. 1: The plot compares the pressure (MeV/fm 3 ) as a function of energy density (MeV/fm 3 ) for three EOS models: AP3 (solid blue line), MPA1 (dashed red line), and BSk22 (dash-dotted black line), highlighting the variation in the stiffness of the isotropic BM across different models.  \n<!-- image -->  \nFIG. 2: The profile shows the speed of sound squared normalized by the squared speed of light as a function of radius (Km) for three EOS models (as seen in Fig. 1). This illustrates the differences in the speed of sound within an isotropic NS, indicating how the internal structure varies depending on the chosen EOS.  \n<!-- image -->  \nwhere r 12 is the distance of separation of two dark matter fermions, g is the Yukawa coupling constant (of the dark matter and the scalar mediator interaction term g χχϕ ), and m ϕ is the mass of the scalar mediator. It is important to note that this EOS is consistent for repulsive self-interactions, which we consider; however, it may be inconsistent for relativistic fermions in the case of attractive self-interactions mediated by scalars [88, 89].', 'A. Anisotropic TOV': "We start with the action governing our system given by:  \nS = S G + S M , (2)  \nwhere S G represents the gravitational component, defined by the standard Einstein-Hilbert action, and S M encapsulates the matter components, modeled as a perfect fluid with properties determined by the energy density ρ , radial pressure Pr , tangential pressure P ⊥ , and an equation of state described by F ( ρ , Pr ) = 0 [90].  \nBy varying this action with respect to the metric, we derive the Einstein field equations (without a cosmological constant):  \nG µν = R µν -1 2 g µν R = 8 π T µν , (3)  \nwhere G , Newton's constant, is set to unity. The stress-energy tensor for the system is:  \nT µ ν = Diag ( -ρ , Pr , P ⊥ , P ⊥ ) . (4)  \nIn the comoving frame, the matter content is an anisotropic fluid with energy density ρ , radial pressure Pr , and tangential pressure P ⊥ .  \nThe general form of the metric is given by:  \nds 2 = -e ν dt 2 + e λ dr 2 + r 2 d Ω 2 , (5)  \nwhere ν ( r ) and λ ( r ) are functions of the radial coordinate r , and d Ω 2 ≡ d θ 2 + sin 2 θ d ϕ 2 denotes the solid angle element [91]. Solving Einstein's equations, we find:  \ne λ ( r ) = 1 1 -2 m r , (6)  \nand  \nν ' ( r ) = 2 m + 8 π r 3 Pr r 2 -2 mr . (7)  \nThe hydrostatic equilibrium equation, known as the generalized Tolman-Oppenheimer-Volkoff (TOV) equation, is given by [92, 93]:  \ndPr dr = -( ρ + Pr )( m + 4 π r 3 Pr ) r ( r -2 m ) + 2 Π r , (8)  \nwhere Π ≡ P ⊥ -Pr = pressure anisotropy. For isotropic matter, Π = 0, which implies P ⊥ = Pr . Thus, the anisotropic term vanishes, simplifying the TOV equation to describe hydrostatic equilibrium in isotropic stars. The expression for P ⊥ is given by,  \nP ⊥ = Pr + α 3 ( ρ + 3 Pr )( ρ + Pr ) r 2 1 -2 m / r (9)  \nwhere the factor α measures the degree of anisotropy in the fluid, similar to the Bowers-Liang (BL) model of anisotropy [94].  \nInterpreting the terms in equation (8), we see that the hydrostatic force F h = -dPr / dr , gravitational force Fg = -[( ρ + Pr )( m + 4 π r 3 Pr )] / [ r ( r -2 m )] , and anisotropic force Fa = 2 Π / r sum to zero:  \nFg + F h + Fa = 0. (10)  \nThe mass function m ( r ) is related to the density by the differential equation:  \ndm dr = 4 π r 2 ρ . (11)  \nIn our analysis, we assume that the interaction between BM and DM is negligible, allowing us to treat them as separate components: BM ( B ) and non-baryonic DM ( χ ). This leads to the separation of the fluid's energy density into ρ = ρ B + ρχ and similarly for pressures Pr = P rB + Pr χ .  \nConsequently, the equations (8) and (11) are split into two sets for baryonic ( B ) and DM ( χ ) components:  \ndP ri dr = -( ρ i + P ri )( m + 4 π r 3 Pr ) r ( r -2 m ) + 2 Π i r , (12)  \nwhere Π i = P ri -P ⊥ i , and Pr = P rB + Pr χ  \ndm i dr = 4 π r 2 ρ i . (13)  \nHere, the subscript i = B or χ indicates whether the quantity refers to BM or DM, respectively. For darkmatter matter, we assume isotropy, so P ⊥ χ = Pr χ and hence Π χ = 0.  \nThe total gravitational mass m ( r ) is the sum of the masses of both components:  \nm ( r ) = m B ( r ) + m χ ( r ) . (14)  \nWe solve equations (12) and (13) using appropriate central and boundary conditions. The central conditions specify the initial densities of B and χ . At the boundaries, hydrostatic equilibrium requires that the pressures vanish:  \nP i ( R i ) = 0, (15)  \n̸  \nwhere R B and R χ are the radii containing all the mass of baryonic and DM components, respectively. Typically, R B = R χ .  \nHere are two possible scenarios for a neutron star (NS) mixed with dark matter (DM):", '1. DMCondensed in the NS Core :': 'In this case, the dark matter is concentrated within the core of the neutron star, resulting in the radius of the DM component ( R χ ) being smaller than the radius of the baryonic matter (BM) component ( R B ), meaning R B > R χ .', '2. DMForming an Extended Halo :': 'Alternatively, the dark matter can form a halo around the neutron star, where the radius of the DM component ( R χ ) exceeds that of the baryonic matter component ( R B ), so R χ > R B .  \nThese configurations are valid when the interaction between baryonic matter and dark matter is purely gravitational, allowing for both components to mix within the core and possibly leading to one component extending into the outer regions of the combined object.  \nSince only the BM is directly observable, we identify the observed radius R of the neutron star with R B . This allows our results to be compared with observational data, where R is the visible radius of the neutron star, and the additional mass is due to the DM component.  \nThe total gravitational mass and the DM subfraction within the neutron star are given by:  \nM T = M B ( R B ) + M χ ( R χ ) , (16)  \nand  \nf χ = M χ ( R χ ) M T . (17)  \nBy varying the central densities of baryonic and DM, we can explore different combinations of total gravitational mass M T and neutron star radius R for a given DM subfraction f χ .', 'B. Tidal Deformability': "Apart from mass and radius, the tidal deformability of NSs plays a pivotal role in defining their structural attributes. During the final phases of inspiral in a binary neutron star system, the gravitational field exerted by one neutron star onto its companion induces quadrupole deformations. These deformations, resulting from tidal forces, quantify the extent of distortion experienced by the neutron star, known as 'tidal deformability'. This parameter provides insights into how much the neutron star is affected by the gravitational forces exerted by its companion.  \nThe tidal deformability parameter λ can be expressed as [95-97],  \nλ = -Q ij E ij , (18)  \nwhere Q ij represents the components of the induced quadrupole moment tensor and E ij denotes the components of the tidal field tensor. In terms of the Love number k 2 , the mass normalized dimensionless tidal deformability parameter is given by  \nΛ ≡ λ M 5 = 2 3 k 2 ( R M ) 5 ≡ 2 3 k 2 C -5 , (19)  \nwhere R and M are the radius and mass of the star, and C ≡ M / R is its compactness.  \nThe tidal Love number k 2 depends on the underlying EOS of the star, and it can be expressed in terms of the dimensionless compactness parameter C as [98, 99],  \nk 2 = 8 C 5 5 ( 1 -2 C ) 2 [ 2 + 2 C ( y R -1 ) -y R ] × { 2 C [ 6 -3 y R + 3 C ( 5 y R -8 ) ] + 4 C 3 [ 13 -11 y R + C ( 3 y R -2 ) + 2 C 2 ( 1 + y R ) ] + 3 ( 1 -2 C ) 2 [ 2 -y R + 2 C ( y R -1 ) ] ln ( 1 -2 C ) } -1 . (20)  \nThe function y R ≡ y ( r ) | r = R is related to the metric perturbation and satisfies the following differential equation:  \nr dy ( r ) dr + y ( r ) 2 + y ( r ) F ( r ) + r 2 Q ( r ) = 0, (21)  \nwith  \nF ( 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 ) + ∑ i ε i ( r )+ P i ( r ) ∂ P i ( r ) / ∂ε i ( r ) -6 4 π r 2 ) r -2 m ( r ) -4 [ m ( r ) + 4 π r 3 P ( r ) r 2 ( 1 -2 m ( r ) / r ) ] 2 .  \nFor a two-fluid system composed of DM and BM, the parameters ϵ , p and M are defined as  \np = ∑ i p i , ϵ = ∑ i ϵ i , M = ∑ i M i , i=BM, DM. (22)  \nTo calculate the Love number and the tidal deformability parameter Λ for a spherically symmetric star, one can solve Eq. (21) and the Tolman-Oppenheimer-Volkoff (TOV) equations (Eqs. (12)-(13)) simultaneously. This process involves utilizing boundary conditions such as P ( 0 ) = Pc and m ( 0 ) = 0, along with y ( 0 ) = 2 obtained from the perturbative expansion of the deformed metric up to the second order.", 'IV. RESULTS AND DISCUSSION': 'The introduction of additional degrees of freedom to a celestial body can alter its physical characteristics. Without exploring the specific production mechanisms, we will examine the properties of DNSs under varying degrees of anisotropy using three different EOSs (AP3, BSk22, MPA1). These DNSs incorporate a DM particle with a mass of 1 GeV, interacting through a light mediator particle with a mass of 1 keV, governed by a coupling constant of g = 10 -5 . The detection of gravitational waves from GW170817 has constrained the DM fraction to 10% for soft EOS and 1% for stiff nuclear EOS [100, 101]. In line with this, and as highlighted in [65, 102], a 5% DM subfraction can significantly influence the interpretation of NS properties. Consequently, we examine a 5% DM subfraction alongside two additional subfractions 1%, 0.1%, and compared them with the no DM scenario. The anisotropic parameter is varied from -2 to 2 in increments of 0.25. In our analysis, we evaluate M max, R max, R 1.4 , and Λ 1.4 , and compare the resulting parametric values with observational constraints as shown in IV A.', 'A. Observational Constraints': 'Wecompare the prediction of the model with recent observational data from various sources to validate our findings. Table I summarizes these constraints, which include measurements from the GW170817 event [103], NICER X-ray observations [17, 18, 104, 105], and mass-radius data from specific pulsars [106-108]. Our analysis adheres to these observational constraints to ensure that the proposed neutron star configurations, particularly those involving DM, remain consistent with empirical data.  \nTABLE I: Observational constraints', 'B. Neutron Stars properties with various DM subfraction for a fixed value of g': "In this subsection, we examine how varying subfractions of DM within the NS, specifically at its core, influence its physical properties. In the two-fluid model, we consider the visible radius R B as the star's outer radius, while the DMradius R χ lies within the BM fluid. We analyze the effects of DM fractions of 5%, 1%, 0.1%, and 0% across three different EOS models (AP3, BSk22, and MPA1), with the anisotropic parameter α ranging from -2 to + 2 (where α = 0 represents the isotropic BM case).  \nAs shown in Fig. 3 and Table II, the mass-radius stability profiles of DM-admixed anisotropic NSs, characterized by parameters g = 10 -5 , m χ = 1 GeV, and m ϕ = 1 KeV, are governed by the AP3 EOS. It is observed that increasing the value of α leads to an increase in the maximum masses and their corresponding radii, and vice-versa. The corresponding subplots highlights different fractions of DM, 5% (Fig. 3 (a)), 1% (Fig. 3 (b)), 0.1% (Fig. 3 (c)) and no DM(Fig. 3 (d)). In the no DM case, only negative values of α = ( -2, -1 ) satisfy the observational constraints, such as the Mmax values from all pulsar data and the R 1.4 region provided by the GW170817 and NICER x-ray data. As the DMsubfraction increases from 0.1% to 1% and 5%, significant changes occur, with the constraints being satisfied for α = ( -2, -1 ) , ( -2, -1, 0 ) , and ( -2, -1, 0 ) , respectively. Increasing the α value results in stiffer NSs while increasing the DM subfraction softens them.  \nTidal deformability is an important physical parameter to define NS properties. We get the value of Λ 1.4 for different α values get reduced corresponding to increased values of DM subfraction from no DM, 0.1%, 1%, and 5% suggesting that DM induces a reduction in tidal deformation between two coalescence bodies as the radius is squeezed as shown in Fig. 4. All these results are within the given Λ 1.36 = 720 value, defined by aLIGO detector for GW170817. Similarly, we perform this analysis for BSk22 and MPA1 EOS and get significant changes for BSk22 EOS, which has some typical behaviour while increasing (or decreasing) the DM fractions as mentioned below. From Fig. 5 and table III, the mass-radius profile for no DM case is found to be only fit for α = -1 satisfying all the given observational GW170817 and NICER x-ray R 1.4 region and all Pulsars maximum mass Mmax data which wasn't satisfied for isotropic case( α = 0) being a stiff EOS. While increasing the DM subfraction from 0.1%, 1%, and 5% a substantial change happens to the α values (like ( -1, 0 ) , ( -1, 0 ) and ( -1, 0, + 1 ) respectively) which satisfy all the constraints. Similarly, the dimensionless tidal deformability value Λ 1.4 which satisfied the Λ 1.36 ≃ 720 of GW170817 data irrespective of the α values with different DM fractions for AP3 EOS, doesn't satisfy the same for BSk22 EOS with no DM case. But, introducing a DM core with 5 % DM subfraction to the single fluid anisotropic baryonic model  \n(a) 5% DM  \n<!-- image -->  \nFIG. 3: Mass-radius stability profile for DM-admixed anisotropic NS with g = 10 -5 , m χ = 1 GeV, and m ϕ = 1 KeV using the AP3 EOS. The color contour highlights the range of α -values from -2 to 2. The subplots illustrate different DMfractions: (a) 5% DM, (b) 1% DM, (c) 0.1% DM, and (d) No DM. Observational constraints are shown with deep sky blue (PSR J0952-0607), red (PSR J0740+6620), orange (PSR J0348+0432), black (PSR J1614-2230), steel-blue solid patch (GW170817 90%), steel-blue dashed patch (GW170817 50%), red and black dashed line for PSR J0030+0451.  \n<!-- image -->  \n(c) 0.1% DM  \n(b) 1% DM  \n<!-- image -->  \n/uni00000040  \n/uni00000030  \n/uni0000003e  \n/uni00000003  \n/uni00000030  \n(d) No DM  \n<!-- image -->  \ndepicts that all the α values ( -2 to + 2 ) satisfy the aLIGO band as shown in Fig. 6.  \nConsidering MPA1 EOS for no DM case we get a typical behavior of mass-radius profile, being stiff EOS for isotropic case, i.e. α = 0 the maximum mass value doesn't satisfy the given pulsars Mmax data, where decrement of α values to -1 and -2 which gives Mmax = 2.29, 2.14 M ⊙ within the pulsars mass data and the radius at 1.4 M ⊙ i.e R 1.4 are 11.68 km (and 11.43 km) satisfying the GW170817 and NICER x-ray data PSRJ0030+0451 90 % CI region and varying the DMsubfraction from 0.1 %, 1 %, and 5 %, the physical curves satisfied for α = (-2, -1), (-2,-1), and (-2,-1, 0) satisfying all the mass-radius constraints as shown in Fig. 7. For this case, Λ 1.4 without DM and with DM always satisfy the aLIGO dimensionless tidal deformability Λ 1.36 band, only a reduction of Λ 1.4 value for greater DM subfraction case seen represented in Fig. 8 and table IV.  \nAnexcellent observation from this analysis is that we get the presence of DM inside the core of the neutron star soften in the EOS model, which gives a maximum mass and maximum radius with values not satisfying any observational  \nFig. 9 depicts the relation between dimensionless tidal deformability and the radius of a 1.4 M ⊙ i.e Λ 1.4 and R 1.4 for DM-admixed anisotropic NSs is investigated for all the 3 taken EOS (AP3, BSk22, MPA1) with g = 10 -5 , m χ = 1GeV, and m ϕ = 1KeV. The color contour represents the range of α -values from -2 to 2. The subplots illustrate different DM subfractions: (a) 5% DM, (b) 1% DM, (c) 0.1% DM, and (d) No DM. The GW170817 constraint (deep sky blue) reflects observational data from the binary neutron star merger. An important observation we get, for BSk22 EOSthe color contours for zero DM case with positive values of α are going beyond the observational bounds, but increasing DMsubfraction up to 5% gives a unique and robust relation satisfying between Λ 1.4 and R 1.4 in the GW170817 band. The other 2 EOS don't respond much to increment of DM subfraction; irrespective of those values, they consistently satisfy the LIGO tidal deformability band.  \n<!-- image -->  \n(a) 5% DM  \n<!-- image -->  \nFIG. 4: Tidal deformability-mass stability profile for DM-admixed anisotropic NS with g = 10 -5 , m χ = 1 GeV, and m ϕ = 1 KeV using the AP3 EOS. The color contour highlights the range of α -values from -2 to 2. The subplots illustrate different DM fractions: (a) 5% DM, (b) 1% DM, (c) 0.1% DM, and (d) No DM. The GW170817 constraint is shown in a black shape.  \n<!-- image -->  \nconstraints, suggesting a more compactness and lesser mass-radius value. As the tidal deformability depends on the O ( 5 ) power of the radius value significantly, its value at 1.4 M ⊙ , i.e., Λ 1.4 is reduced even if we increase the DM subfraction inside the core of the star.  \n<!-- image -->  \n(b) 1% DM  \nTABLE II: Maximum mass M max (in M ⊙ ), maximum radius R max (in km), the radius at 1.4 M ⊙ , i.e., R 1.4 (in km), and dimensionless tidal deformability at 1.4 M ⊙ , i.e., Λ 1.4 for AP3 EOS as a function of anisotropic parameter α for different DM fractions with g = 10 -5 , m χ = 1GeV, and m ϕ = 1KeV.  \nFrom table V, we represent the range of anisotropic parameter ( α ) satisfying the observational constraints are identified for different DM subfractions under the conditions of g = 10 -5 , m χ = 1GeV, and m ϕ = 1KeV. The α -values ranged from [ -2, 2 ] , with increments of 0.25. The symbol ( ✓ ) indicates that all α -values within this range satisfy the specified constraint. On the other hand, the symbol ( ✗ ) denotes that none of the α -values meet the constraints. Notably, NICER observations of PSR J0030+0451 rule out AP3, and MPA1 EOS, even when anisotropic configurations are taken into consideration. By increasing the DM subfraction up to 5 %, dimensionless tidal deformability Λ 1.4 of BSk22 EOS satisfies the range given by GW170817 Λ 1.36 value that wasn't satisfied for no DM case, so suggesting a DMadmixed NS. If the value of anisotropic parameters α is negative, it means that the corresponding model satisfies the given pulsar maximum mass constraint with 5 % DM inside the core of the star.", 'C. Coupling strength g variation effects': "Now, we will see how the variation of the coupling strength g affects the NS properties. For the first case where we fixed the value of g, we took the visible radius to be R B as DM was presented inside the core of the neutron star. Taking the log scale of g for our calculation purpose, we take g values varying from -5, -4, -3.7, and -3.5, respectively.  \nFrom Fig. 10(a) and 11(a), it is clear that for AP3 EOS when we increase the coupling strength to 10 -4 , for this value core-halo transition occurs, giving a maximum radius of 20.23 km, R 1.4 = 32.04 km and dimensionless tidal deformability i.e Λ 1.4 ≃ 13000 for the isotropic case (i.e α = 0). When we try to increase the value of the anisotropy parameter from -2, 0 to + 2, one important observation we get the value of Rmax decrease from 22.79, 20.23 to 18.06 km respectively. Again, further increasing the value of g i.e 10 -3.7 (and 10 -3.5 ) the characteristics properties of NS such as mass - radius profile & dimensionless tidal deformability - mass curves gives the maximum radius Rmax = 55.17 km (and 99.41 km) & Λ 1.4 ≃ 3 × 10 6 (and ≃ 2.9 × 10 8 ) respectively for the isotropic case. For this, 2 values of g also Rmax decrease while changing α values from -2, 0 to + 2 respectively as mentioned in table VI.  \n(a) 5% DM  \n<!-- image -->  \n<!-- image -->  \nFIG. 5: Same as Fig. 3, but for BSk22 EOS.  \n<!-- image -->  \nSimilarly, the mass-radius plots for BSk22 EOS and MPA1 EOS with g variation are shown in Fig. 10(b) and 10(c). Putting the anisotropic parameter α = 0, the maximum mass Mmax and maximum radius Rmax of BSk22 model are found to be ( 2.26 M ⊙ , 22.89 km), ( 2.33 M ⊙ , 58.15 km), and ( 2.36 M ⊙ , 102.33 km) for g values = 10 -4 , 10 -3.7 , and 10 -3.5 respectively.MPA1 EOS gives the same ( 2.46 M ⊙ , 20.69 km), ( 2.55 M ⊙ , 55.14 km), and ( 2.58 M ⊙ , 99.36 km) for g values = 10 -4 , 10 -3.7 , and 10 -3.5 . While increasing the values of α from -2, 0 to + 2, one important finding we get is the value of Rmax decrease for increasing value of α , while taking this g values to be varied. This unique characteristic of g variation suggests a dark matter halo formation with corresponding core - halo transition g value as the statement is reinforced by the information given in table VI.  \nThe values of dimensionless tidal deformability Λ 1.4 give a drastic increase up to the order of O ( 10 5 ), O ( 10 6 ), and O ( 10 7 ) for g values 10 -4 , 10 -3.7 , and 10 -3.5 irrespective of the anisotropic parameter α for all the 3 given EOS models as shown in Fig. 11.  \n(b) 1% DM  \n<!-- image -->  \nFig. 12 presents the correlation analysis of the key physical variables involved in our admixed-NS study, using both Pearson and Kendall correlation coefficients for the considered EOS (AP3, BSk22, MPA1). The left-hand side of Fig. 12 shows Pearson's correlation coefficient that measures the linear relationship between two variables, with values ranging from -1 to 1. Values close to 1 indicate a strong positive linear relationship, values close to -1 indicate a strong negative linear relationship, and values around 0 indicate no linear relationship. The key observations from the Pearson correlation matrix are as given.  \n<!-- image -->  \n(a) 5% DM  \n<!-- image -->  \nFIG. 6: Same as Fig. 4, but for BSk22 EOS.  \n<!-- image -->", 'D. Correlation between different Physical observables': '- · α has a very high positive correlation with Mmax indicating a strong linear relationship. In contrast, it has a near-zero correlation with Rmax , R 1.4 , Λ 1.4 , DM / BM and g , indicating no significant linear relationship.\n- · g shows an almost perfect correlation with Rmax , R 1.4 and Λ 1.4 , suggesting a near-perfect linear dependency among these variables. It is weakly correlated with Mmax and shows a moderate correlation with DM / BM .\n- · DM / BM exhibits moderate correlations with Rmax , R 1.4 and Λ 1.4 and near-zero with Mmax .\n- · Rmax , R 1.4 and Λ 1.4 indicates significant linear relationships among them. Conversely, they show a near-weakly  \n<!-- image -->  \n(b) 1% DM  \nTABLE III: Same as Table. II, but for BSk22 EOS', 'dependence on Mmax .': "The right-hand side of Fig. 12 shows the Kendall correlation coefficient that measures the ordinal association between two variables, with values ranging from -1 to 1. It is a measure of the correspondence between the rankings of the data. The key observations from the Kendall correlation matrix are as follows.  \n- · α has no ordinal relationships with DM / BM and g , displays moderate correlations with Rmax , R 1.4 and Λ 1.4 , and shows a strong ordinal association with Mmax .\n- · g demonstrates moderate correlations with DM / BM , strong with Rmax , R 1.4 , and Λ 1.4 , while showing negligible ordinal association with Mmax .\n- · DM / BM shows weak ordinal relationships with Rmax , R 1.4 , and Λ 1.4 , and a negligible correlation with Mmax .\n- · Rmax , R 1.4 and Λ 1.4 exhibit a high degree of ordinal association among them, but moderate with Mmax .  \nBoth matrices provide insights into the relationships between the various macroscopic variables, with the Pearson correlation highlighting linear dependencies and the Kendall correlation emphasizing ordinal associations. Both matrices highlight a strong positive association between α and Mmax , indicating α 's significant impact on the maximum mass of the admixed-NSs. Rmax , R 1.4 , and Λ 1.4 show strong linear and ordinal correlations, reflecting their tight linkage in NS structural properties. DM / BM exhibits moderate linear correlations with Rmax , R 1.4 and Λ 1.4 , but weaker ordinal relationships, indicating its lesser influence. g is strongly correlated with radius and deformability parameters, underlining its crucial role in the NS equation of state. Notably, Mmax has weak linear but moderate ordinal associations with Rmax , R 1.4 and Λ 1.4 , suggesting that while linear predictability is low, rank-based relationships provide meaningful insights. These insights are crucial for understanding the underlying physics of admixed-NSs, particularly in how different EOS parameters influence key structural properties like maximum mass, radius, and tidal deformability. The strong correlations among certain parameters across both matrices suggest robust dependencies that are critical for the theoretical modeling and observational analysis of NSs.  \n(a) 5% DM  \n<!-- image -->  \nFIG. 7: Same as Fig. 3, but for MPA1 EOS.  \n<!-- image -->  \n<!-- image -->  \n(b) 1% DM  \n<!-- image -->  \n(d) No DM", 'V. CONCLUSIONS': "This study illustrates the importance of considering both DM and anisotropy in NS modeling. The M-R curves highlight that even small amounts of DM or changes in α can significantly impact stability, which has profound implications for interpreting astrophysical observations and the underlying physics of NSs. The major findings worth mentioning here are as follows.  \n- · The mass-radius stability profile reveals that increasing (decreasing) the α value results in stiffer (softer) NSs while increasing (decreasing) the DM subfraction softens (stiffens) the NSs, with a fixed coupling strength.\n- · It is noteworthy that dimensionless tidal deformability decreases (increases) with incrementing (decrementing) DMsubfraction. Therefore, the presence of DM subfraction promotes more compact NSs that resist deformation.\n- · It is seen that increasing the coupling parameter ( g ) between DM and its mediator initiates the formation of a core-halo structure, where a DM halo surrounds the BM. For coupling values g = 10 -4 , 10 -3.7 , and 10 -3.5 , we observe a decrease in the maximum radius ( Rmax ) with increasing anisotropy. This behavior suggests  \nFIG. 8: Same as Fig. 4, but for MPA1 EOS.  \n<!-- image -->  \nthat stronger DM interactions lead to more compact neutron stars with significant core-halo differentiation, differing from the patterns observed at g = 10 -5 or in scenarios without DM, where such compactness is not as pronounced.  \n- · It is noticeable that with coupling values g = 10 -4 , 10 -3.7 , and 10 -3.5 , the Dimensionless Tidal deformability is not observationally feasible . Hence, such coupling values could be discarded.\n- · Out of these 3 models taken, the color contours only for BSk22 EOS in zero DM case with positive values of α are going beyond the observational bounds, but increasing DM subfraction up to 5% gives a unique and robust relation satisfying between Λ 1.4 and R 1.4 in the GW170817 band. The other 2 EOS don't respond much to increment of DM subfraction.\n- · The key observation from the Pearson correlation matrix is that α has a very high positive correlation with Mmax , indicating a strong linear relationship. In contrast, it has a near-zero correlation with Rmax , R 1.4 , Λ 1.4 , DM / BM and g , indicating no significant linear relationship, whereas the Kendall correlation matrix suggests that α has no ordinal relationships with DM / BM and g , displays moderate correlations with Rmax , R 1.4 and Λ 1.4 , and shows a strong ordinal association with Mmax .  \nTABLE IV: Same as Table. II, but for MPA1 EOS  \n- a Satisfying PSR (1.97-2.35).  \nIn our study, we assumed baryonic matter (BM) to be anisotropic and dark matter (DM) to be isotropic. However, alternative configurations-where BM is isotropic, DM is anisotropic, or both are anisotropic-are also possible and warrant exploration in future research. The stability of neutron stars (NS) is influenced by the densities of BM and DM, as discussed in [109]. Recent observations of high-mass pulsars, like PSR J0952+0607, challenge traditional isotropic models, which struggle to explain such extreme masses. Anisotropic models, on the other hand, provide a more realistic explanation, supporting stable configurations even at higher masses. Anisotropy also impacts the star's deformation and response to tidal forces, leading to more accurate predictions of gravitational waveforms.  \nIn summary, our proposed study is expected to make an important and fundamental contribution to the understanding of DM-admixed NSs. DM doesn't interact electromagnetically, and gravitational wave (GW) experiments do not differentiate between various forms of matter; the resulting tidal effects are determined only by the compactness of the object. Thus, more data from LIGO and future GW experiments such as LISA [110], Einstein Telescope [111], and Cosmic Explorer [112] will present opportunities to identify NSs that have subfractions of DM or particles of a similar type. Incorporating anisotropic pressure into DM-admixed NS models provides a more comprehensive understanding of their properties, structure, and behavior under extreme conditions. It not only aligns better with observational data but also allows for more sophisticated modeling of exotic matter and gravitational wave predictions, pushing the limits of our understanding of compact objects in the universe.", 'ACKNOWLEDGMENTS': "P.M. would like to thank BITS Pilani K K Birla Goa campus for the fellowship support, Prashant Thakur for discussing & sharing ideas on pulsars and GW170817 data, and comments on our EOS model and IUCAA, Pune, for providing a good computational facility for this project. C.S. and P.M. thank Apratim Ganguly for the rigorous discussion on the EOS models and relevant plots for this work. A.H. expresses gratitude to Michael Collier (Durham University, UK) for his assistance with the discussion on two-fluid code and to the International Centre for Theoretical Sciences (ICTS) for the opportunity to participate in the program 'Summer School on Gravitational-Wave Astronomy' (code: ICTS/GWA2024/07). This work makes use of the following Python packages: pandas [113],  \nFIG. 9: The relation between Tidal deformability and the radius of a 1.4 M ⊙ DM-admixed anisotropic NS is investigated for the labeled EOS with g = 10 -5 , m χ = 1GeV, and m ϕ = 1KeV. The color contour represents the range of α -values from -2 to 2. The subplots illustrate different DM fractions: (a) 5% DM, (b) 1% DM, (c) 0.1% DM, and (d) No DM. The GW170817 constraint (deep sky blue) reflects observational data from the binary neutron star merger.  \n<!-- image -->  \n(a) 5% DM  \n<!-- image -->  \n(c) 0.1% DM  \n(b) 1% DM  \n<!-- image -->  \n<!-- image -->  \n(d) No DM  \nscipy [114], seaborn [115], matplotlib [116], numpy [117].  \n<!-- image -->  \n/uni00000015/uni00000011/uni00000013  \n<!-- image -->  \n/uni00000015/uni00000011/uni00000013  \n(b) BSk22  \nFIG. 10: Mass-radius stability profile for 5% DM-admixed anisotropic NS with varying g -values, as labeled, and constant m χ = 1GeV, and m ϕ = 1KeV. The color contour represents the range of α -values from -2 to 2. The subplots correspond to (a) AP3 EOS, (b) BSk22 EOS, and (c) MPA1 EOS. Observational constraints are shown with deep sky blue (PSR J0952-0607), red (PSR J0740+6620), orange (PSR J0348+0432), black (PSR J1614-2230), steel-blue solid patch (GW170817 90%), steel-blue dashed patch (GW170817 50%), red and black dashed line for PSR J0030+0451.  \n<!-- image -->  \nTABLE V: The anisotropic parameter ( α ) satisfying the observational constraints are identified for different DM subfractions under the conditions of g = 10 -5 , m χ = 1GeV, and m ϕ = 1KeV. The α -values ranged from [ -2, 2 ] , with increments of 0.25. The symbol ( ✓ ) indicates that all α -values within this range satisfy the specified constraint. On the other hand, the symbol ( ✗ ) denotes that none of the α -values meet the constraints.  \nFIG. 11: Tidal deformability-mass stability profile, with the same setup as in Fig. 10  \n<!-- image -->  \nTABLE VI: Maximum mass M max (in M ⊙ ), maximum radius R max (in km), the radius at 1.4 M ⊙ , i.e., R 1.4 (in km), and dimensionless tidal deformability at 1.4 M ⊙ , i.e., Λ 1.4 for AP3, BSk22, and MPA1 EOS as a function of anisotropic parameter α for different DM fractions with varying g = 10 -4 , 10 -3.7 , 10 -3.5 , m χ = 1GeV, and m ϕ = 1KeV.  \n<!-- image -->  \n/uni00000033/uni00000048/uni00000044/uni00000055/uni00000056/uni00000052/uni00000051/uni00000003/uni00000026/uni00000052/uni00000055/uni00000055/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni00000030/uni00000044/uni00000057/uni00000055/uni0000004c/uni0000005b  \n/uni0000002e/uni00000048/uni00000051/uni00000047/uni00000044/uni0000004f/uni0000004f/uni00000003/uni00000026/uni00000052/uni00000055/uni00000055/uni00000048/uni0000004f/uni00000044/uni00000057/uni0000004c/uni00000052/uni00000051/uni00000003/uni00000030/uni00000044/uni00000057/uni00000055/uni0000004c/uni0000005b  \n<!-- image -->  \n(c) MPA1  \n- [1] James M Lattimer and Maddappa Prakash. 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