Neutron stars NSs can capture dark matter DM particles because of their deep gravitational potential and high density. The accumulated DM can affect the properties of NSs. In this work we use a general relativistic twofluid formalism to solve the structure of DMadmixed NSs DANSs and the surrounding spacetime. Specifically we pay attention to the situation where those DANSs possess DM halos. Due to the gravitational effect of the DM halo the pulse profile of an xray pulsar is changed. Our study finds a universal relation between the peak flux deviation of the pulse profile and inlineformulammlmath displayinlinemmlmsubmmlmiMmmlmimmlmrowmmlmihalommlmimmlmrowmmlmsubmmlmommlmommlmsubmmlmiRmmlmimmlmrowmmlmiBMmmlmimmlmrowmmlmsubmmlmathinlineformula which is the ratio of the DM halo mass inlineformulammlmath displayinlinemmlmsubmmlmiMmmlmimmlmrowmmlmihalommlmimmlmrowmmlmsubmmlmathinlineformula to the baryonic matter BM core radius inlineformulammlmath displayinlinemmlmsubmmlmiRmmlmimmlmrowmmlmiBMmmlmimmlmrowmmlmsubmmlmathinlineformula. Our results show that when inlineformulammlmath displayinlinemmlmsubmmlmiMmmlmimmlmrowmmlmihalommlmimmlmrowmmlmsubmmlmommlmommlmsubmmlmiRmmlmimmlmrowmmlmiBMmmlmimmlmrowmmlmsubmmlmommlmommlmn0.292mmlmnmmlmathinlineformula and the DM particle mass inlineformulammlmath displayinlinemmlmsubmmlmimmmlmimmlmifmmlmimmlmsubmmlmommlmommlmn0.3mmlmnmmlmtext mmlmtextmmlmtext mmlmtextmmlmiGeVmmlmimmlmathinlineformula the maximum deviation of the profile can be larger than 100 which has implications in xray pulsar observation.
['Astrophysics and astroparticle physics', 'Astrophysics - High Energy Astrophysical Phenomena']
Effects from dark matter halos on xray pulsar pulse profiles
{'E ff ects from Dark Matter Halos on X-ray Pulsar Pulse Profiles': 'Yukun Liu, 1 Hong-Bo Li, 2 Yong Gao, 3 Lijing Shao, 2, 4, ∗ and Zexin Hu 1, 2  \n1 School of Physics, Peking University, Beijing 100871, China  \n2  \nKavli Institute for Astronomy and Astrophysics, Peking University, Beijing 100871, China 3 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Muhlenberg 1, D-14476 Potsdam-Golm, Germany 4 National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China  \n(Dated: September 19, 2024)  \nNeutron stars (NSs) can capture dark matter (DM) particles because of their deep gravitational potential and high density. The accumulated DM can a ff ect the properties of NSs. In this work we use a general relativistic two-fluid formalism to solve the structure of DM-admixed NSs (DANSs) and the surrounding spacetime. Specifically, we pay attention to the situation where those DANSs possess DM halos. Due to the gravitational e ff ect of the DM halo, the pulse profile of an X-ray pulsar is changed. Our study finds a universal relation between the peak flux deviation of the pulse profile and M halo / R BM, which is the ratio of the DM halo mass, M halo, to the baryonic matter (BM) core radius, R BM. Our results show that, when M halo / R BM = 0 . 292 and the DM particle mass mf = 0 . 3 GeV, the maximum deviation of the profile can be larger than 100%, which has implication in X-ray pulsar observation.', 'I. INTRODUCTION': "Observational evidence has increasingly supported the existence of dark matter (DM). For example, DM facilitates the formation of early galaxies [1] and helps galaxies rotate at constant speeds in their outer reaches [2]. Measurements through gravitational lensing have revealed some possible DM substructures [3]. According to our current understanding, DM occupies about a quarter of the energy density of the present Universe, while the baryonic matter (BM) only accounts for about 5% [4]. The nature of DM is a vastly uncharted territory, and exploring it can help us better understand the constitution of the Universe.  \nHowever, direct methods of searching for DM particles have not yielded satisfactory results. Thus it is of great interest to constrain the properties of DM through indirect and complementary methods. Neutron stars (NSs) provide extreme celestial laboratories for fundamental physics [5]. They may capture DM particles due to their high density and deep gravitational potential [6-13]. Even given a cross section constraint between DM and BM ≲ 10 -45 cm 2 (say, from underground DM search experiments [14-17]), the mean free path of DM particles is still possible to be of the same order as the radius of a NS [7].  \nDi ff erent models of DM have di ff erent scenarios for DMadmixed NSs (DANSs) [6, 7, 18, 19]. A two-fluid model is often used for the equation of state (EOS) of DANSs [20]. In this model, one fluid is the BM while the other stands for DM, which can be asymmetric fermionic DM or self-interacting bosonic DM, and so on. After solving for the equilibrium equations, a DANS can either possess a DM core or a DM halo [8-11, 13], depending on which fluid has a larger radius. DMcansignificantly a ff ect the properties of NSs. Leung et al. [8] demonstrated that the existence of a fermionic DM core with the DM particle mass mf ∼ 1 GeV would lower the maximum mass of NSs. Hippert et al. [13] showed that DANSs  \nmay have the same mass and radius but di ff erent DM fractions. Kain [11] calculated the radical oscillation frequencies of DANSs and found that the frequencies of DANSs could be larger than the maximum possible frequencies of normal NSs. Axion DM particle stream away from NSs may transfer into photons and generate a large broadband contribution to the pulse [21]. Annihilating DM may a ff ect the thermal conductivity and nuetrino emission of NSs [22]. Numerous investigation suggests an appealing probability that NSs can be a laboratory for studying DM, concerning various observables of NSs that can be influenced by DM. To give a few examples, DM may play a role in binary systems [23-25]; radio telescope may find imprints of DM from NSs' magnetosphere [26-28]; NS surface temperature can also be impacted by DM [29, 30]. More studies on this topic can be found in Refs. [31-37].  \nIn this work we mainly focus on the e ff ects of DM halos on X-ray pulsar pulse profiles. A pulsar is a highly magnetized rotating NS that emits electromagnetic radiations. This radiation can only be observed when the emitting beam points to the Earth. Hot regions on a pulsar created by inward moving particles colliding with the NS surface can produce X-ray pulses, which have been observed through telescopes like the Neutron star Interior Composition ExploreR (NICER) [3841]. The flux of the X-ray pulse changes with time, which shapes the X-ray pulse profile. It carries information related to the mass and radius of the NS [42-44]. If there is a DM halo around a NS, the pulse profile will be changed due to the gravitational e ff ects of the massive halo. If such e ff ects are observed, the properties of DM can be better inferred.  \nMiao et al. [45] and Shawqi and Morsink [46] showed that the X-ray profile from a pulsar can be slightly changed if a di ff use DM halo exists. In our study, we compare the pulse profile of a DANS with a NS that has the same total mass, M , and the BM core radius, R BM, which-for some pulsars-are inferred observables from orbital dynamics and X-ray thermal emission, respectively [47]. Suppose that the total mass and the BM radius of a NS are obtained, we can compare its pulse profiles under the assumptions of a DANS or a normal NS. We find a universal relation between the peak flux deviation of the  \nDANS pulse profiles and the ratio M halo / R BM, where M halo is the mass of the DM halo. Compared with earlier studies, we consider more comprehensive models, incorporating a wider range of parameters to provide a thorough analysis of the impact of DM halos on pulse profiles. We change the position of the hot spots and the relative direction of observers, showing the fractional deviation of the peak flux as a function of the colatitude of the emitting region. In addition, we focus on extended emitting regions besides the point-like approximation, providing a more realistic representation of the actual situation. In this work, for the first time we obtain the universal relation that still holds for bosonic DM model.  \nThe paper is organized as follows. Section II provides an introduction to the formalism used to determine the structure of DANSs and their stability. Section III briefly presents some numerical results of the structure of DANSs. Section IV outlines general methods for the calculation of pulse profiles, and shows the influence of fermionic DM halos in detail. We also extend our study to bosonic DM in this section. Section V gives a summary of the paper.  \nThroughout the paper, we adopt geometric units with c = G = 1, where c and G denote the speed of light and the gravitational constant respectively. We adopt ( -, + , + , + ) for the metric signature.", 'A. The Two-fluid Model and Equation of State': 'We briefly review the derivation of the two-fluid Tolman-Oppenheimer-Volko ff (TOV) equations [48-50]. The Einstein field equation reads  \nG µν = 8 π T µν , (1)  \nwhere G µν and T µν are the Einstein tensor and energymomentum tensor of matters, respectively. Considering the results of underground DM search experiments [14-17], strict  \nconstraints were set for the DM-BM coupling strength [51]. For this reason, to the leading order we neglect all interactions between DM and BM except gravity. So the energymomentum tensor can be divided into the BM component, T µν BM, and DM component, T µν DM,  \nT µν = T µν BM + T µν DM . (2)  \nEach of them is conserved,  \nT µν BM; ν = T µν DM; ν = 0 . (3)  \nThe two energy-momentum tensors are written as  \nT µν BM = ( ϵ BM + p BM) u µ BM u ν BM + p BM g µν , (4)  \nT µν DM = ( ϵ DM + p DM) u µ DM u ν DM + p DM g µν . (5)  \nTaking the spherical symmetry, we can write the spacetime line element as d s 2 = -e 2 µ d t 2 + e 2 ν d r 2 + r 2 (d θ 2 + sin 2 θ d ϕ 2 ), where µ and ν are functions of r . From Eq. (1) and (3), the two-fluid TOV equations can be derived as  \nd mi d r = 4 π r 2 ϵ i , (6)  \nd pi d r = -4 π r 3 p + m r 2 (1 -2 m / r ) ( ϵ i + pi ) , (7)  \nwhere the subscripted i denotes the fluid (DM or BM), p = p DM + p BM, and m = m DM + m BM. The particle number of each component Ni is  \nd Ni d r = 4 π r 2 ni √ 1 -2 m / r , (8)  \nwhere ni is the particle number density.  \nWeuse the asymmetric fermionic DM model as a prototype, which is often approximately regarded as a free Fermi gas at zero temperature. From thermodynamic statistics, the energy density, pressure, and number density are [52]  \nϵ = 1 2 π 2 Z kF 0 k 2 q k 2 + m 2 f d k = 1 8 π 2             kF q m 2 f + k 2 F GLYPH<0> 2 k 2 F + m 2 f GLYPH<1> -m 4 f ln             kF + q m 2 f + k 2 F mf                         ,  \n(9) p = 1 6 π 2 Z kF 0 k 4 q k 2 + m 2 f d k = 1 24 π 2             kF q m 2 f + k 2 F GLYPH<0> 2 k 2 F -3 m 2 f GLYPH<1> + 3 m 4 f ln             kF + q m 2 f + k 2 F mf                         , (10) n = k 3 F 3 π 2 , (11)  \nwhere kF is the Fermi momentum. Considering these formulas and eliminating kF , we obtain p ( ϵ ). The EOS of bosonic DM can be found in Eq. (17) at the end of Section IV. For the EOS of BM, we use AP4 model for illustration [53]. With  \nthis setting, we can obtain the structure of DANSs and the surrounding spacetime via integration of the two-fluid TOV equations.', 'B. The Critical Curve': 'An important question is whether a DANS is stable under perturbations. In the case of pure NSs, each solution of the TOV equation is uniquely identified by a single quantity, the central pressure. The entire star becomes unstable if its central pressure exceeds the critical point that corresponds to the maximum mass [52]. However, when it comes to two-fluid systems, there are t w o kinds of fluids and t w o independent central pressures. One is the central pressure of DM, p c DM , and the other is the central pressure of BM, p c BM . Thus there is a critical curve instead of a critical point in the p c BM -p c DM plane.  \nThe way to get the critical curve is not simply searching for the maximum mass while fixing one central pressure and varying the other. The radical oscillation of a DANS under small perturbations is a problem of the Sturm-Liouville type, with eigenvalues ω 2 0 < ω 2 1 < ω 2 2 < . . . , where ω 0 is the eigenfrequency of the fundamental mode. The critical points with ω = 0 mark the onset of instability [52, 54]. Because N DM and N BM conserve in perturbations, Henriques et al. [55] proposed an equivalent yet more straightforward criterion,  \nd M d p = d N DM d p = d N BM d p = 0 , (12)  \nwhere M is the total mass, and p is a vector that is simultaneously tangent to both Ni contour lines in the central pressure parameter space. Jetzer [56] has proved that d M can be written as a linear combination of two d Ni at these points. So d M / d p = 0 holds naturally if d Ni / d p = 0.  \nUsing this criterion, we adopt the methods used by Kain [11] to compute the critical curve. This involves computing contour lines of N DM in the central pressure parameter space using Eq. (8) and Eq. (11). Once the contour lines are determined, we follow one line to search for the point where M reaches its maximum. The critical curve then consists of all these points.', 'III. STRUCTURE OF DARK-MATTER-ADMIXED NEUTRON STARS': 'By choosing a pair of central pressures as the initial value, we can solve the two-fluid TOV equations. The pressures will decrease from the center to the surfaces at Ri , which are defined by pi = 0. If R DM is less than R BM, the DANS has a DM core. Conversely, it has a DM halo. Figure 1 shows the p c BM -p c DM parameter space of DANSs for mf = 0 . 5 GeV. The critical curve is drawn with a thick black line. The parameter space below the critical curve gives stable static solutions. The orange area indicates static solutions with a DM halo, while the blue area indicates those with a DM core.  \nIn the one-fluid system, a 1-dimensional central pressure parameter space results in a mass-radius curve. Accordingly, a 2-dimensional p c BM -p c DM parameter space results in a 2-dimensional M -R BM diagram in the two-fluid system. Here M and R BM are the most essential and straightforward observables. In Fig. 2, we show the relation between M and R BM  \nFIG. 1. The p c BM -p c DM parameter space for solutions in the two-fluid model with mf = 0 . 5 GeV.  \n<!-- image -->  \nFIG. 2. The M -R BM relations of DANSs for mf = 0 . 5 GeV, with di ff erent colors indicating di ff erent p c DM . NICER constraints are included for PSRs J0740 + 6620 [40, 57, 58] (blue regions), J0437 -4715 [59, 60] (orange regions), and HESS J1731 -347 [61] (grey regions). Contours show 68.3% and 95.4% credible intervals.  \n<!-- image -->  \nfor mf = 0 . 5 GeV. The thick black dashed line is the massradius curve for pure NSs, and lines with di ff erent colors show DANSs. Each line maintains a constant p c DM while changing p c BM . Meanwhile, the solid dots are dividing points between solutions with a DM halo and a DM core. Points closer to the dashed line are solutions with a DM core, while the other corresponds to solutions with a DM halo. When p c BM gets large enough, a DANS has a DM core, and the lines in the figure get closer to the thick black dashed line. That means a DM core a ff ects the structure of DANSs slightly, just making M and  \nR BM a little smaller compared to the NSs with the same p c BM . When p c BM is small, a DANS has a DM halo and a significantly di ff erent structure. For a pure NS, a smaller p c BM will result in a larger radius and a smaller mass. However, the existence of DMwill increase the total mass, and make the BM core shrink further due to the gravity of DM. That is also why R BM tends to decrease while M tends to increase as p c DM increases. In addition, di ff erent central pressures can result in a same set of M and R BM, as the lines intersecting with each other in Fig. 2.  \nWhen mf = 0 . 5 GeV, DANSs with either a DM core or a DM halo can be generated while their masses remain near 1 M ⊙ . Two further examples with di ff erent mf are given in Fig. 3. When mf is smaller, the DM halo scenario (solutions with a DM halo) occurs near the dashed line and then move upwards. That is because that di ff use DM is easier to form a DMhalo and allow more DM to exist. When mf is larger, the DMhalo scenario will be compressed at the bottom.', 'A. Light Bending E ff ects of Gravitational Fields': "In this subsection, we briefly outline the methods to calculate the X-ray pulsar pulse profile. For detailed derivation, readers can refer to Silva and Yunes [62] and Xu et al. [63]. We assume there is a pair of hot spots on the BM surface of a DANS with a DM halo. The radiation from each of these point-like emitting regions is presumed to be isotropic [64]. Light emitted from them will be subject to additional light bending e ff ects due to the gravity of the DM halo. In the presence of DM, the metric of spacetime outside the BM core is no longer the Schwarzschild metric, though it is still spherical symmetric [65].  \nFollowing Refs. [43, 62, 63], we introduce some geometry quantities (see Fig. 8 in Ref. [63]) and the calculation of the flux for arbitrary spherically symmetric metric. In the model, a pair of hot spots is located on the BM surface with colatitudes of γ and π -γ measured from the spin axis; n is a unit vector pointing to one hot spot from the center of the star; k 0 is a unit vector in the direction of light emission; k is a unit vector in the direction of light after bending; β stands for the velocity of the emitting region in the local static frame. There are four other relevant angles: the angle ι between the line of sight and the spin axis, the angle ψ between the line of sight and n , the angle α between k 0 and n , and the angle ζ between k 0 and β [63].  \nThe observed di ff erential flux in terms of the quantities at the emitting region is [66]  \nd F = δ 5 g ( R BM) I ' ( ν ' ) d ν ' cos α sin α sin ψ d α d ψ d S ' D 2 , (13)  \nwhere δ is the Doppler factor, I ' ( ν ' ) is the radiation intensity as a function of frequency in the comoving frame, d S ' is the proper di ff erential area at the emitting region, D is the distance between the pulsar and the observer, and we have defined -g as the t -t component of the metric. The  \nDoppler factor, describing how the boost speed β caused by the pulsar rotation a ff ects the flux, can be written as δ = p 1 -β 2 / GLYPH<0> 1 -β cos ζ GLYPH<1> [67].  \nThrough the analysis in the local static frame [66], we can calculate β and cos ζ by β = R BM Ω sin γ/ p g ( R BM) and cos ζ = -sin α sin ι sin( Ω t 0) / sin ψ , where Ω is the angular velocity of pulsar rotation, and t 0 is the coordinate time when the ray is emitted and has been set to zero when the spot is the closest to the observer. The relation between ψ and α can be derived by doing an integral along the path of light propagation [68], ψ = R ∞ R BM p f g h r -2 σ -2 -r -4 g i d r , where f is the r -r component of the metric and σ ≡ R BM sin α/ p g ( R BM) is the impact parameter.  \nThe flux is normalized by a factor R I ' ( ν ' ) d ν ' d S ' / D 2 ,  \nF ≡ D 2 R dF d S ' R I ' ( ν ' )d ν ' = δ 5 g ( R BM) cos α sin α sin ψ d α d ψ . (14)  \nUsing geometric analysis, we obtain cos ψ = cos ι cos γ + sin ι sin γ sin( Ω t 0). Thus every emission time t 0 corresponds to an angle ψ , an impact parameter σ , and an emission angle α . Then we calculate the observer time, t obs, which is di ff erent from t 0 because the time of light propagation di ff ers. Taking this time delay into consideration, we finally obtained [68],  \nt obs = t 0 + Z ∞ R BM s f g         1 -σ 2 g r 2 ! -1 2 -1         d r . (15)  \nEventually, using Eq. (14) and Eq. (15), we obtain the relation between the normalized flux and the observer time, namely the pulse profile.", 'B. X-Ray Pulse Profile Flux': "Previously, Miao et al. [45] studied the pulse profiles of DANSs, fixing the BM core radius, R BM, and the core mass, M ( R BM). To characterize the e ff ects of DM, they defined a quantity, M halo / R DM, where M halo = M -M ( R BM), is the mass of the DM halo. In fact, it is the half of the increase of the t -t component of the metric, g ( R DM), when adding a halo. Here g plays an important role in calculating the flux intensity. In our study, as mentioned in Sec. I, we assume that M and R BM are potential observables, respectively from orbital dynamics and X-ray emission modeling. In this scenario, we find that the flux deviation-with respect to a normal NS-is better dependent on M halo / R BM than M halo / R DM. So we choose M halo / R BM as a characteristic quantity in describing the e ff ects of DM halos on pulse profiles. In Table I we choose mf = 0 . 5 GeV for illustration, and show two pairs of DANSs (DANS1 and DANS2) and normal NSs with the same M and R BM. DANS1 and DANS2 have similar M and R BM, but DANS1 has a lighter DMhalo than DANS2.  \nIn the upper panel of Fig. 4, we show the pulse profiles from point-like hot spots with γ = 45 · and ι = 45 · , from a pulsar with a rotational period T = 5 ms. The normalized flux deviation, ∆ F / F ≡ ( F DANS -F NS) / F NS, as a function of t obs, is  \n<!-- image -->  \nFIG. 3. The M -R BM relations of DANSs. The left panel is for mf = 0 . 3 GeV, showing the DM halo region near the dashed line and above. The right panel is for mf = 0 . 7 GeV, showing the DM halo region compressed at the bottom.  \n<!-- image -->  \nTABLE I. Relevant parameters of two DANSs and corresponding normal NSs, whose M and R BM are the same. The DM particle mass mf is fixed to 0 . 5 GeV.  \nshown in the lower panel of the figure, where F DANS and F NS are respectively fluxes of DANSs and normal NSs. As shown in this figure, the fraction M halo / R BM significantly a ff ects the DANS pulse profiles. The maximum deviation occurs near the peak of the pulse profile. Compared to a normal NS, DANS pulse flux tends to exhibit increased brightness during high-luminosity periods, and decreased brightness during low-luminosity periods. That can be explained by Eq. (14). F and g ( R BM) are positively correlated. For DANSs, g ( R BM) is larger because of the existence of DM halos. As a result, F DANS will fluctuate in a wider range. It always holds true for large M halo / R BM since the di ff erence of g ( R BM) between DNASs and NSs is big enough. However, it is not shown in this figure that when M halo / R BM drops to about 10 -4 , the amplitude of F DANS may become less than F NS.  \nMore realistically, we extend our study to a scenario where the emitting regions are extended. We assume their diameters to be d = 3 km. We consider a grid of emission spots on the surface [43, 69]. We use the same values for γ and ι , namely γ = 45 · and ι = 45 · . To show the di ff erences between pointlike emitting regions and extended emitting regions, we plot a comparison of pulse profiles from DANS1 in Fig. 5. An obvious feature of the curve representing the extended emitting region is the absence of sharp turns. The sharp turn in the point-like case comes from the emitting region with a colatitude γ = 135 · (the opposite patch). This region remains invisible when it turns away from the line of sight. It only contributes to the overall flux during a part of the rotational  \nperiod. Considering an extended emitting region as an ensemble of point-like sources, each of them has a distinct sharp turn. The addition of them gives a smooth pulse profile.  \nTo see the e ff ect from DM halos, taking DANS1 and NS1 as example, the pulse profiles from extended emitting regions are shown in Fig. 6. The upper panel is their pulse profiles, and the lower panel is ∆ F / F . Our previously derived conclusions based on point-like emitting regions remain qualitatively applicable here. Compared with the point-like case, the pulse profile from the extended emitting region exhibits a smaller range between its minimum and maximum fluxes. The di ff erence is about 20% in Fig. 5. To provide an explanation, we investigate the impact of the colatitude of the emitting region, γ , and the observer's relative position, represented by ι . To see the deviation more clearly, we take the example of DANS2 and NS2. The modification of the peak flux with di ff erent γ and ι are shown in Fig. 7. These two angles exhibit similar impacts on the pulse profile, as they always appear in pair in the formulas except for the Doppler factor, which is relatively small. When γ = ι , the most significant deviation of the pulse profile occurs. However, in the context of extended emitting regions, γ = ι does not hold at each point. As a result, the final pulse profile exhibits a gentler overall behavior.  \nTo provide a comprehensive overview of how DM halos a ff ect pulse profiles, we compute the peak flux deviations for di ff erent central pressures for mf = 0 . 5 GeV, assuming the presence of a DM halo. We only consider DANSs with R BM > 3 M , which is a reasonable assumption considering the  \nFIG. 4. ( Upper ) Pulse profiles from two point-like hot spots of the DANSs and NSs in Table I. The emitting region in the northern hemisphere is located at a colatitude γ = 45 · . The inclination angle of the line of sight is ι = 45 · . ( Lower ) Deviation of DANS pulse profiles with respect to the corresponding normal NS pulse profiles.  \n<!-- image -->  \nFIG. 5. Comparison of pulse profiles of DANS1 with point-like and extended emitting regions.  \n<!-- image -->  \nconstraints on the EOS [68]. Our results are depicted in Fig. 8, revealing the substantial influence of DM halos across most parameter space for DM halos. When the central pressures lie near the dividing line between the DM halo and the DM core (see Fig. 1), their pulse profiles exhibit minimal deviation. Overall, as expected, the peak flux deviation tends to increase with an increasing p c DM when p c BM is fixed.  \nFIG. 6. Pulse profiles of DANS1 and NS1 (see Table I) with a pair of the emitting region, γ , and the inclination angle of the line of sight, ι .  \n<!-- image -->  \nFIG. 7. Influence of the colatitude γ and the inclination ι to the peak flux deviation of the emitting region between DANS2 and NS2.  \n<!-- image -->", 'C. Universal Relations': "The impacts of central pressures on peak flux deviations have been shown in Fig. 8. However, we want to derive a more direct relation. As mentioned above, the deviation might be highly dependent on the quantity M halo / R BM. To verify this hypothesis, we calculate a series of flux deviations systematically with a varying mf . For each mf , we randomly select M and R BM from the M -R BM diagram where DANSs have halos. Existing observations have already established constraints on the mass and radius of NSs [57, 70-72]. To remain within these bounds, we only consider cases where M ∈ [1 , 3] M ⊙ and R BM ∈ [8 , 20] km. The colatitude of the northern point-  \n<!-- image -->  \n|  \n|  \nFIG. 8. Peak flux deviation in the p c BM -p c DM parameter space for mf = 0 . 5 GeV. Here all DANSs have a DM halo. The area with the smallest deviation is near the dividing line between the DM halo and the DM core (see Fig. 1).FIG. 9. Peak flux deviation as a function of M halo / R BM. Each point is a randomly chosen DANS. Di ff erent colors represent di ff erent mf . The solid line is a fitted curve given by Eq. (16).  \n<!-- image -->  \nlike emitting region γ and the inclination angle ι are randomly distributed in [30 · , 60 · ].  \nIn Fig. 9, we show | ∆ F peak | / F peak against M halo / R BM using logarithmic scales. Over a range of mf ∈ [0 . 1 , 0 . 7] GeV, M halo / R BM ∈ [1 . 1 × 10 -8 , 2 . 9 × 10 -1 ], all of the randomly selected data points can be loosely fitted by a cubic function. Defining fF ( x ) = lg( | ∆ F peak | / F peak), x = lg( M halo / R BM), the  \nFIG. 10. Peak flux deviation versus the DM particle mass. There are two branches of solutions in the figure: one spans all mf in the figure, while the other stops at mf = 0 . 3 GeV.  \n<!-- image -->  \nuniversal relation is represented by the following formula,  \nfF ( x ) = 0 . 012 x 3 + 0 . 290 x 2 + 2 . 351 x + 1 . 667 . (16)  \nIn particular, the maximum deviation of the peak flux exceeds 100% when M halo / R BM = 0 . 292 and mf = 0 . 3 GeV, which can potentially leads to a significant impact on observations. Take PSR J0740 + 6620 as an example [40, 57, 58]. A good constraint on the mass and radius was obtained using about three years' NICER data, with the total net exposure time about 10 6 sec. The uncertainty of the telescope was estimated to be about 10%, that means the flux variations might be observable. Analyzing X-ray pulses of DANSs represents a theoretically viable approach to searching for DM.  \nWe investigate how DM halos a ff ect pulsar pulse profiles of DANSs with typical NS masses and radii. For a normal NS with a mass of 1 . 4 M ⊙ , the AP4 EOS yields a radius of 11 . 4 km. We randomly choose DANSs with M ∈ [1 . 37 , 1 . 43] M ⊙ and R BM ∈ [11 . 2 , 11 . 6] km. The peak flux deviation versus the DM particle mass mf is shown in Fig. 10. When mf > 0 . 3 GeV, the peak flux deviation remains nearly constant at about 70%. To enhance clarity, we include a smaller inset figure that magnifies this region. When mf < 0 . 3 GeV, the data points are largely scattered .  \nIt is clear that there are two branches of solutions in Fig. 10. One of the branches spans all mf in the figure, while the other stops at mf = 0 . 3 GeV. To provide an explanation, we define a 'solution region' where M ∈ [1 . 37 , 1 . 43] M ⊙ and R BM ∈ [11 . 2 , 11 . 6] km. Following Figs. 2-3, for heavy DM particles, say, mf = 0 . 7 GeV, the parameter space for DM halo would not overlap with the solution region. When mf is smaller, the data points for DM halos move upwards, overlapping with the solution region. When mf continues to decrease, the dividing points get closer to the black dashed lines (see Figs. 2-3), and the second branch of solution occurs. These solutions are near the dividing points, which means a much more di ff use DM halo, resulting in a smaller flux deviation.  \nFIG. 11. Peak flux deviation as a function of M halo / R BM for bosonic DM. Di ff erent colors are for di ff erent A in Eq. (17). The solid line gives the universal relation in Eq. (18).  \n<!-- image -->  \nThe closer the solution is to the dividing points, the smaller the flux deviation is. When mf is small, the e ff ect of a di ff use DM halo is insignificant. For example, when M halo / R BM ∼ 10 -2 or 10 -6 , according to Fig. 9, the quantity | ∆ F peak | / F peak may di ff er within GLYPH<2> 10 -5 , 10 -2 GLYPH<3> . However, for a larger mf , there is no DANS with a di ff use DM halo in the solution region. The quantity | ∆ F peak | / F peak remains in GLYPH<2> 10 -0 . 3 , 10 -0 . 1 GLYPH<3> . Consequently, the peak flux deviation induced by a heavy DM mass can exceed the one induced by a light DM mass by orders of magnitude, due to significantly di ff erent DANS central pressures. These two varied scenarios manifest as either di ff usion of data points for a small mf or clustering of data points for a large mf when plotted on a logarithmic scale.  \nBosonic DM is another candidate in explaining the DM phenomenology. In a simple model, the EOS of bosonic DM can be written as [73, 74]  \np = A ρ 2 , (17)  \nwhere A = 2 π l / m 3 f with l the scattering length. The universal relation between lg( | ∆ F peak | / F peak) and lg( M halo / R BM) for bosonic DM halos is shown in Fig. 11. In particular, as shown by the solid line in the figure, the universal relation can be loosely represented by the following formula,  \nfB ( x ) = 0 . 010 x 3 + 0 . 270 x 2 + 2 . 364 x + 2 . 190 , (18)  \nwhere fB ( x ) = lg( | ∆ F peak | / F peak), and x = lg( M halo / R BM).  \nWederive two universal relations concerning fermionic DM and bosonic DM, respectively given in Eq. (16) and Eq. (18); also see Fig. 9 and Fig. 11. Striking similarities can be seen, which underscores the generality of our results.", 'V. CONCLUSIONS': 'In our study, we analyzed the structure of DANSs in a twofluid model. Di ff erent M -R BM diagrams for DANSs were constructed considering di ff erent DM particle mass, mf . Via integration of the TOV equations, a DANS possesses either a DM core or a DM halo. Our primary focus lies in investigating the impact of DM halos on X-ray pulsar pulse profiles. The pulse profiles of DANSs were compared with those of normal NSs. In contrast to earlier studies, we fixed the total mass M , instead of the core mass M ( R BM), considering that M is a more fundamental and direct observable. Similarly, we also fixed the BM core radius, R BM.  \nOur calculations revealed that the presence of dense DM halos will increase the amplitude of X-ray flux fluctuation. Besides examining point-like hot spots, we explored the effects of extended emitting regions. We found that extended emitting regions tend to reduce the gap between the minimum and maximum fluxes, by about 20% (see Fig. 5). The impacts of the position of emitting regions, represented by the colatitude angle γ , and the inclination angle to the line of sight, ι , were also studied. The influences of γ and ι on the pulse profiles are similar. When γ = ι , the influence reaches its maximum.  \nFurthermore, we observed a universal correlation between the peak flux deviation of DANSs with respect to normal NSs, and the dimensionless quantity, M halo / R BM. Remarkably, this relation remains consistent when considering both fermionic and bosonic DM models. In this paper, we did not consider self-interacting DM, which is a very interesting topic in DM studies (see e.g. Ref. [75]). However, focusing on the universal relation, Miao et al. [45] have studied fermionic selfinteracting DM via a repulsive Yukawa type, and the relation of theirs holds when the interaction strength is randomly chosen. Indepth studies concerning di ff erent interactions and rich phenomena are left to future investigation. We also studied DANSs with typical masses and radii ( M ≃ 1 . 4 M ⊙ and R BM ≃ 11 . 4 km). It is interesting that the peak flux deviation can be divided into two di ff erent branches, in which the larger deviation stays at about 70%. In all calculated examples, the maximum deviation of the pulse profiles can exceed 100% when M halo / R BM = 0 . 292 and mf = 0 . 3 GeV, leading to a potentially significant observable di ff erence. Our study suggests that X-ray observation of pulsars may be a feasible way to search for DM.', 'ACKNOWLEDGMENTS': "This work was supported by the Beijing Natural Science Foundation (QY23009, 1242018), the National SKA Program of China (2020SKA0120300), the National Natural Science Foundation of China (11991053), the Max Planck Partner Group Program funded by the Max Planck Society, and the High-Performance Computing Platform of Peking University.  \n- [1] P. Mocz et al. , Phys. Rev. Lett. 123 , 141301 (2019), arXiv:1910.01653 [astro-ph.GA].\n- [2] Y. Sofue and V. Rubin, Ann. Rev. Astron. Astrophys. 39 , 137 (2001), arXiv:astro-ph / 0010594.\n- [3] R. Fadely and C. R. Keeton, Mon. Not. Roy. Astron. Soc. 419 , 936 (2012), arXiv:1109.0548 [astro-ph.CO].\n- [4] N. Aghanim et al. 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