Dark compact objects: an extensive overview
Maksym Deliyergiyev, Antonino Del Popolo, Laura Tolos, Morgan Le Delliou, Xiguo Lee, Fiorella Burgio
DDark compact objects: an extensive overview
Maksym Deliyergiyev,
1, 2
Antonino Del Popolo,
3, 4, 5
LauraTolos,
6, 7, 8, 9
Morgan Le Delliou,
10, 11
Xiguo Lee, and Fiorella Burgio Institute of Modern Physics, Chinese Academy of Sciences, Department of High Energy Nuclear Physics,Post Office Box31, Lanzhou 730000, Peoples Republic of China Institute of Physics, Jan Kochanowski University, PL-25406 Kielce, Poland ∗ Dipartimento di Fisica e Astronomia, University Of Catania, Viale Andrea Doria 6, 95125, Catania, Italy INFN Sezione di Catania, Via S. Sofia 64, I-95123 Catania, Italy † Institute of Modern Physics, Chinese Academy of Sciences,Post Office Box31, Lanzhou 730000, Peoples Republic of China Institut f¨ur Theoretische Physik, Goethe Universit¨at Frankfurt, Max-von-Laue-Straße 1, 60438 Frankfurt, Germany Frankfurt Institute for Advanced Studies, Goethe Universit¨at Frankfurt,Ruth-Moufang-Str.1, 60438 Frankfurt am Main, Germany Institute of Space Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans, 08193, Barcelona, Spain Institut dEstudis Espacials de Catalunya (IEEC), 08034 Barcelona, Spain ‡ Institute of Theoretical Physics, Physics Department, Lanzhou University,No.222, South Tianshui Road, Lanzhou, Gansu 730000, P R China Instituto de Astrof´ısica e Ciˆencias do Espac¸o, Universidade de Lisboa,Faculdade de Ciˆencias, Ed. C8, Campo Grande, 1769-016 Lisboa, Portugal § (Dated: March 5, 2019)We study the structure of compact objects that contain non-self annihilating, self-interacting dark matteradmixed with ordinary matter made of neutron star and white dwarf materials. We extend the previous workPhys. Rev. D M (cid:12) constraintfor neutron star masses and the nominal 1 M (cid:12) for white dwarfs, while for larger dark matter particle masses orin the weakly interacting case the compact objects show masses in agreement or smaller than these constraints,thus hinting at the exclusion of strongly self-interacting dark matter of masses 1-10 GeV in the interior of thesecompact objects. Moreover, we observe that the smaller the dark matter particle mass, the larger the quantity ofdark matter captured is, putting constraints on the dark matter mass trapped in the compact objects so as to fullfill (cid:39) M (cid:12) observations. Finally, the inhomogeneity of distribution of dark matter in the Galaxy implies a massdependence of compact objects from the environment which can be used to put constraints on the characteristicsof the Galaxy halo DM profile and on particle mass. In view of the these results, we discuss the formation ofthe dark compact objects in an homogeneous and non-homogeneous dark matter environment. PACS numbers: 97.60.Jd, 97.10.Nf, 97.10.Pg, 95.35.+d, 26.60.-cKeywords: Neutron stars; white-dwarfs; dark matter; dark matter particles; stability; compact objects
I. INTRODUCTION
Astrophysical and cosmological observations strongly in-dicate that the mass content of our universe is dominated bynon-baryonic mass/energy [1, 2]. According to [3], the Uni-verse is composed of . of baryonic matter, . of invis-ible form of matter whose existence is inferred from its grav-itational effects, dubbed dark matter (DM). A further com-ponent, dubbed dark energy (DE), which is hypothesized topermeate all space, and whose existence is related to the ac-celerated expansion of the Universe [4, 5], constitutes . of the total mass. ∗ [email protected] † [email protected] ‡ [email protected] § Corresponding author: ([email protected],)[email protected]
In the Λ CDM model, a parameterization of the big-bangcosmology with six parameters, the DE is associated withthe cosmological constant Λ , and the material componentsof the Universe are the ones indicated previously. Thequoted Λ CDM paradigm, describes correctly many of theobservations[2, 6–9], but has some drawbacks. On large scaleCMB shows some anomalies, such as that the Planck 2015data are in tension with the CFHTLenS weak lensing [10],and σ [11]. There is also another tension with the value ofthe Hubble parameter measured by SNIa . Another big issueof the paradigm is the nature of DM. A ”zoo” of candidateshave been proposed, with masses in the range of − GeV(Fuzzy DM) to GeV (Wimpzillas). In that large ”zoo”, The Λ CDM paradigm has some other drawbacks, as the cosmological con-stant problem [12, 13], the unknown nature of DE [14–16] and the so called”small scale problems” [17]. a r X i v : . [ g r- q c ] M a r WIMPs, Axions, and sterile neutrinos, have received a pecu-liar attention [1, 2].Several different attempts to detect DM have been made.Direct detection attempts to detect DM particles through theirelastic scattering with nuclei (normal matter recoiling fromDM collisions). The recoil is not measured directly butthrough crystal or liquid scintillation (DAMA/LIBRA [18],KIMS[19], CRESST-II [20], ZEPLIN [21]), phonons genera-tion (CRESST-I [22]), ionization (CDMS [23], superCDMS[24], XENON100 [25], XENON1T [26], LUX [27]), ax-ion cavities (ADMX [28]), and several others. The indirectsearches aimt a detecting the products of WIMP annihila-tions (e.g., gamma rays, neutrinos, positrons, electrons, andantiprotons). The FERMI-LAT, DAMPE [29], CALET [30],HAWC [31], HESS [32], VERITAS [33], MAGIC [34] or theplanned CTA [35] are working in the photons channel, whileIceCube [36] in neutrinos and FERMI-LAT [37] in antimat-ter. Concerning particle accelerators, DM (WIMPs) produc-tion together with jets (other particles) should give rise to alarge amount of energy. In 2015, in LHC it was observed a750 GeV resonance in the di-photon final state, later disproved[38]. However, all these experiments based on direct or indi-rect detection [1, 2, 39], or detection in particle accelerators[40] have produced no evidence of the existence of DM par-ticles, apart some claims (e.g. DAMA/LIBRA) not confirmedby other experiments.In this context, different avenues for testing the possibleeffects of DM are welcome. For this purpose, compact ob-jects (COs), such as white dwarfs (WDs) and neutron stars(NSs), present the advantage of extreme densities, increasingthe probability of the interaction of DM with nucleons and theDM capture [41, 42].The DM (WIMPs) accumulation in COs modifies theirstructure. If the accumulated DM is larger than a critical value[44], DM can become self-gravitating forming a mini black-hole in the collapse. This could be used to infer constraintson the mass and cross section of DM (WIMPs) [45]. If DMis constituted by WIMPs trapped in the COs, its annihilationproduces a heating of the star, and an increase of the surfacetemperature and luminosity. In the limit case of COs locatedclose to the galactic center, the temperature can reach K,and luminosities − L (cid:12) [46]. The quoted changes are dif-ficult to observe, especially for objects located close to thegalactic center [47].An appealing alternative to WIMPs is the asymmetric darkmatter (ADM) model, based on the idea that the present DMabundance has a similar origin of visible matter [48]. A pecu-liar case of ADM is mirror matter. WIMPs are supersymmet-ric particles, based on the assumption of a symmetry betweenbosons and fermions. If one assumes that instead of the quotedsymmetry, one has that nature is parity symmetric, we have adifferent form of DM, mirror matter [49, 50]. The main mo-tivation for the existence of mirror matter is that of restoring The DM content of a CO depends on a) its formation process and b) theaccumulation through capture in the CO’s lifetime [43]. parity symmetry in nature laws . An interesting feature of thiskind of DM is that it is consistent with DAMA experiment[18, 51]. Since ADM is non-annihilating it can accumulateand thermalize in a small radius, producing changes in massand radius of the stars, with the possibility of forming extraor-dinary compact NSs. Comparing the mass-radius ( M − R )relation predicted by stars models with ordinary matter andwith ordinary matter admixed with DM in NSs, it is possibleto extract information on DM and the equation of state (EoS)of the NSs [43]. These extra compact NSs, having a DM core,could explain the discrepancy between the structure of (e.g.)4U − and the M − R relationship coming from nuclearmatter models [43].Moreover, the NSs behaviour is dictated by their EoS, con-strained at normal nuclear saturation density [52], but not atdensities larger than normal. At those densities the proper-ties of matter are unknown. This implies that fundamentalquantities like the mass cannot be known with certainty. Inorder to explain NSs with high masses, like the binary mil-lisecond pulsar PSR J1614-2230 of M = 1 . ± . M (cid:12) [53] and the PSR J0348+0432 of M = 2 . ± . M (cid:12) [54],one needs a stiff EoS. Then, whereas most of the phenomeno-logical models for EoS are able to reproduce the M (cid:12) obser-vations, these observations are in tension with microscopicalmodels that implement some possible exotic components ofthe EoS (e.g., quarks, mesons, hyperons) that soften the EoS.In recent years, it has been realized that the presence ofADM in NSs plays a similar role to that of exotic states [47].In Ref. [47] the authors considered the NSs composed of “nor-mal” matter admixed with mirror matter coupled only throughgravity, whereas in Ref. [55] a general relativistic two fluidapproach to study admixture of nuclear matter and degenerateDM was used. In both papers it was found that the presenceof DM gives rise to a different M − R relationship, charac-terized by smaller mass and radii when the increasing ratio ofthe DM and normal matter increased. In Ref. [56] quark mat-ter admixed DM was studied with a similar formalism, find-ing, among other results, a total mass of 1.95 M (cid:12) , close to the2 M (cid:12) observations. Li et al. [57] showed that a NS with ADMhas a M − R relationship similar to that obtained by [47], andthat small values of the mass of the DM particles produce anincrease in the final stellar mass reaching values even largerthan M (cid:12) .In the previous work of Ref. [58], the Tolman-Oppenheimer-Volkoff (TOV) equations were solved for NSand WD material admixed with ADM and particle mass equalto 100 GeV. It was found that in the case of weak self-interacting, non-annihilating DM, the TOV’s solutions cangive COs with Earth-like masses and radii a few km to a fewhundreds km, while in the case of strong self-interactions,Jupiter-like COs with radii of a few hundreds km were ob-tained. The maximum DM content that NSs with M (cid:12) andWDs of the nominal mass of M (cid:12) can sustain was also ana-lyzed. Weak interaction are not parity symmetric.
In the present paper we aim to extend this previous work byconsidering weak and strongly interacting DM with particlemasses in the range of 1-500 GeV. We find that the total massof the COs increases with decreasing DM particle mass, thushinting at excluding strongly self-interacting DM of masses1-10 GeV in the interior of these COs. Moreover, we obtainthat the smaller the mass of the DM particle, the more DM iscaptured in the COs, setting constraints on DM capture givenby M (cid:12) observations. This finding has to be tested by analyz-ing the quantity of DM captured during the formation of theseCOs inside a DM environment, thus again helping to constrainthe DM mass particle and its self-interaction.The paper is organized as follows. In Sec. II we describethe theoretical model, whereas in Sec. III we describe the re-sults. Sec. IV contains the description of the capture mecha-nism which can accrete the mass predicted by the TOV equa-tions solution, giving some constraints on the DM captured.Finally Sec. V is devoted to the discussion and conclusions. II. THEORETICAL MODEL
In the following we solve the TOV equations for an ad-mixture of ADM and ordinary matter (OM) coupled only bygravity. The aim is to understand what kind of COs can beformed, since the spherical configurations obtained by solvingthe TOV equations may yield to certain stable configurationswith unusual masses and radii. Therefore, we start from thedimensionless TOV equations [59], following [58] notation, dp (cid:48) OM dr = − ( p (cid:48) OM + ρ (cid:48) OM ) dνdr ,dm OM dr = 4 πr ρ (cid:48) OM ,dp (cid:48) DM dr = − ( p (cid:48) DM + ρ (cid:48) DM ) dνdr ,dm DM dr = 4 πr ρ (cid:48) DM ,dνdr = ( m OM + m DM ) + 4 πr ( p (cid:48) OM + p (cid:48) DM ) r ( r − m OM + m DM )) , (1)the quantity p (cid:48) = P/m f , and ρ (cid:48) = ρ/m f , are respectivelythe dimensionless pressure and energy density, being m f thefermion mass (i.e., DM particle mass, and neutron mass). In-dicated with M p the Planck mass [59], each one of the twospecies can give rise to an astrophysical object with radius, R = ( M p /m f ) r and mass M = ( M p /m f ) m .For OM we use the same EoSs as described in [58], whereasthe interacting Fermi gas EoS for DM is taken from Ref. [59].The DM particles are non-self annihilating [60–64], and self-interacting [65]. Differently from [58], where the only DMmass considered was 100 GeV, in this paper we take into ac-count DM particle masses equal to 1, 5, 10, 50, 100, 200, 500GeV. We study the case of weakly interacting, and strong in-teracting DM. The interaction strength is expressed in termsof the ratio of the fermion mass m f , and scale of interaction m I , y = m f /m I . The two values of y are y = 10 − forweakly interacting DM ( m I ∼
100 MeV ), and y = 10 for strongly interacting DM ( m I ∼
300 GeV ). The ratio betweenthe DM pressure and that of the ordinary matter, p DM /p OM ,is assumed to be in the range of − to , a range largerthan the one considered in Ref. [58].For the determination of the COs and their characteristics,one has to perform a stability analysis to determine the pos-sible stable configurations. Our results are given in the nextSection, whereas we discuss the stability criteria in AppendixA. III. RESULTS
The stable configurations for COs are determined by per-forming the stability analysis described in Appendix A andshown in Figs. 1-4. Whereas Fig. 1 shows the stability analy-sis for OM for different ratios of the dimensionless DM pres-sure ( p DM ) versus the dimensionless OM pressure ( p OM ),i.e. p DM /p OM , each of the Figs. 2-4 shows the analysis forDM for different values of the DM particle mass but for onlytwo p DM /p OM ratios in each figure. In those figures we dis-play the mass (upper panels) and radius (lower panels) of OM(DM) as a function of the central pressure of OM, P OM , forweakly interacting matter ( y = 10 − ) in the left panels, whilethe strongly interacting case ( y = 10 ) is shown in the rightpanels.After performing the stability analysis described in Ap-pendix A for OM only, the stable regions in OM are delim-ited by vertical lines in both mass and radius plots in Fig.1.As indicated previously, we show the solutions of mass andradius for different p DM /p OM . The non-straight feature ofthe vertical lines in this figure results from joining the differ-ent P OM values, for the various p DM /p OM ratios, at whicha stable solution turns unstable and vice–versa. We find thatthe regions in pressure below the first vertical dashed line andabove the second vertical dashed one give rise to stable mass-radius configurations, for both weakly and strongly interactingDM cases.For P OM (cid:28) − MeV fm − , the mass-radius stable con-figurations on Fig.1, with M (cid:28) M (cid:12) and M ∝ R , are ofplanet-like type [66]. Subsequently, the mass rises with cen-tral pressure while the radius decreases, leading to WD con-figurations, one of the two stable M − R branches in compactobjects. In this WD configuration and for the case y = 10 − ,the OM central pressure lies below − MeV fm − , whilefor the case y = 10 it remains below − MeV fm − , asclearly displayed in Fig.1. Next, the M ( P OM ) curves presentan interval with a series of extrema, with the radius curveschanging their slope and giving rise to unstable configura-tions. Following the stability criteria, the last sharp downturnresults in having the last unstable mode turning stable, so thatthe NS configuration is reached. In this latter region, the OMcentral pressure is approximately above − MeV fm − for y = 10 − , and − MeV fm − for y = 10 . For typicalordinary matter EoS, further increase in central pressure in-duces the M − R curve to spiral counterclockwise, and moreand more modes become unstable. With DM, instead of spi-raling at high central pressure, another stable “twin” or “third S t a b l e r e g i o n S t a b l e r e g i o n S t a b l e r e g i o n S t a b l e r e g i o n ORDINARY MATTER y=10 -1 P OM [MeVfm -3 ] S t a b l e r e g i o n S t a b l e r e g i o n S t a b l e r e g i o n S t a b l e r e g i o n p DM /p OM =10 p DM /p OM =10 p DM /p OM =10 p DM /p OM =10 p DM /p OM =10p DM /p OM =10 -1 p DM /p OM =10 -2 p DM /p OM =10 -3 p DM /p OM =10 -4 p DM /p OM =10 -5 p DM /p OM =1 ORDINARY MATTER y=10 P OM [MeVfm -3 ] FIG. 1. Mass and radius of OM content of the astrophysical objects as a function of the central pressure of OM for the DM interaction strengthparameters y = 10 − (left panels) and y = 10 (right panels). The solutions go through both stable and unstable regions. The separations aredelimited by the roughly vertical dashed curves. The coloured lines indicate solutions for M OM and R OM over OM central pressures for a setof dimensionless ratio, p DM /p OM , running from − to . Those lines turn grey in the unstable regions. Note that for the case y = 10 − ,lines of p DM /p OM = 10 − to p DM /p OM = 10 − for the M OM and R OM distributions lie on top of each other. family” branch may arise [67–70].Once we have restricted the values of P OM for all differ-ent p DM /p OM ratios that lead to stable WD and NS con-figurations with only OM, we analyze the common stabilityregions for both OM and DM. In this manner, we can applycuts for the set of the examined DM particle masses (from 1to 500 GeV). These results are displayed in Figs. 2 to 4. InFig. 2 we present the results for the mass and radius as func-tion of P OM for p DM /p OM = 10 − and for weakly (leftpanel) and strongly (right panel) interacting DM for DM par-ticle masses ranging from 1 to 500 GeV, whereas in Fig. 3we present the cases for p DM /p OM = 10 − and , and p DM /p OM = 10 − and p DM /p OM = 10 are displayed in Fig. 4.Each curve in the panels stands for a DM particle mass go-ing from 1 GeV to 500 GeV from top to bottom. Black andred lines correspond to the two sets of p DM /p OM , as indi-cated in these panels. Namely, black curves and black verticallines correspond to p DM /p OM = 10 − (Fig. 2), − (Fig. 3),and − (Fig. 4); whereas red curves and red vertical linescorrespond to p DM /p OM = 10 (Fig. 2), (Fig. 3), and (Fig. 4). As mentioned previously, the vertical lines aredetermined by imposing having stable OM and DM simulta-neously. The regions below the first vertical line and abovethe second one, as indicated by the arrows, fulfill the sta-bility criteria for both species. By analyzing the stability P OM [MeVfm -3 ] S t a b l e r e g i o n S t a b l er e g i o n S t a b l e r e g i o n S t a b l er e g i o n
DARK MATTER p DM /p OM =10 p DM /p OM =10 -5 m D M = G e V m D M = G e V m D M = G e V m D M = G e V m D M = G e V m D M = G e V DARK MATTER p DM /p OM =10 p DM /p OM =10 -5 P OM [MeVfm -3 ] m D M = G e V m D M = G e V m D M = G e V m D M = G e V m D M = G e V m D M = G e V m D M = G e V S t a b l e r e g i o n S t a b l er e g i o n S t a b l er e g i o n S t a b l e r e g i o n FIG. 2. Mass and radius of DM content of the astrophysical objects as a function of the OM central pressure for the DM interaction strengthparameter y = 10 − (left panels), and y = 10 (right panels). The vertical black and red dashed lines delimit the stable regions for two setsgiven by different p DM /p OM , denoted by black and red curves. Each curve in each set stands for a DM particle mass going from 1 GeV to500 GeV (1, 5, 10, 50, 100, 200, 500) from top to bottom. In this figure black curves correspond to p DM /p OM = 10 − , whereas red curvescorrespond to p DM /p OM = 10 . region below the lower vertical line, we observe that from p DM /p OM = 10 − up to p DM /p OM = 10 , the stable re-gion extends up to P OM ≈ − MeV fm − , showing a slightdependence on the ratio p DM /p OM , and on the DM inter-action strength y . By increasing p DM /p OM , the size of thestable region shows a strong dependence on p DM /p OM and y , and starts well below p OM ≈ − MeV fm − . A simi-lar behaviour is displayed also for the stable region which liesabove the second vertical line.For completeness, so as to show the range of OM centraldensities of our astrophysical objects, we display in Fig. 5the total mass, M T , and the visible radius of OM, R OM , of the compact objects as a function of the OM central density, ρ c /ρ , for y = 10 − (left panels) and y = 10 (right panels).We take m DM = 1 GeV. As in Fig. 1, the coloured lines in-dicate solutions for a set of dimensionless ratio, p DM /p OM ,running from − to . These lines turn grey in the unsta-ble regions.Once the stability analysis is performed, the features of theCOs with DM can be studied from the total mass as a functionof the visible radius of OM, that is, M T vs R OM , for the NS(high central pressure branch) or WD (low central pressurebranch), in a similar manner as done in Ref. [58]. The M T − R OM relation is presented in Figs. 6-9 for the range of central DARK MATTER p DM /p OM =10 p DM /p OM =10 -3 S t a b l e r e g i o n S t a b l e r e g i o n S t a b l er e g i o n S t a b l er e g i o n m D M = G e V m D M = G e V m D M = G e V m D M = G e V m D M = G e V m D M = G e V P OM [MeVfm -3 ] p DM /p OM =10 p DM /p OM =10 -3 m D M = G e V m D M = G e V m D M = G e V m D M = G e V m D M = G e V P OM [MeVfm -3 ] S t a b l e r e g i o n S t a b l e r e g i o n S t a b l er e g i o n S t a b l e r e g i o n DARK MATTER
FIG. 3. Same as in Fig.2, for the dimensionless ratios, p DM /p OM = 10 − (black curves) and (red curves). pressure ratios ( p DM /p OM = 10 − − ), particle mass (1-500 GeV), and DM interaction strength y = 10 − , and y =10 . The cases of weakly interacting DM are shown in Fig. 6for the WD branch and Fig. 7 for the NS one, while stronglyinteracting DM yields Fig. 8 for WD and Fig. 9 for NS. Ineach figure the panels are ordered left to right and then top tobottom with respect to the sequence of fixed value of pressureratios, while each panel contains the curves for all the selectedDM masses, marked by different colours and symbols.For ratios of p DM /p OM < − and weakly interactingDM, the two resulting stable M − R configurations are equiv-alent to the WD and NS branches with only OM (top panelsof Figs. 6 and 7). While WDs exhibit typical masses of 1 M (cid:12) and radii of few thousand km, NSs are characterized bymasses of 1-2 M (cid:12) and radius of 10 km. With increasing DMcentral pressure, the NS branch becomes unstable, and the re- maining WD branch presents OM densities below the neutrondrip line, and unconventional masses and radii (middle andbottom panels). Indeed, we confirm the results of Ref. [58] onCOs with Earth-like masses and radii from a few km to a fewhundred km in the case of p DM /p OM = 10 (bottom middlepanel).As for the strongly interacting case and ratios of p DM /p OM below − , we observe that the smaller the DM particlemass, the larger the total mass of the CO is, as seen in Ref. [47]for mirror DM. In fact, our results are compatible with the val-ues obtained from Eq. (47) of Ref. [59] for the total mass ofan object with DM, that strongly depends on the interactionparameter, and Ref. [71]. We again reproduce the COs withJupiter masses with − − − M (cid:12) masses and few hundredkm radii, as found in Ref. [58].Finally, we show the maximum total masses of the COs in DARK MATTER S t a b l e r e g i o n S t a b l er e g i o n S t a b l er e g i o n m D M = G e V m D M = G e V m D M = G e V m D M = G e V p DM /p OM =10 p DM /p OM =10 -1 P OM [MeVfm -3 ] S t a b l e r e g i o n P OM [MeVfm -3 ] p DM /p OM =10 p DM /p OM =10 -1 m D M = G e V m D M = G e V m D M = G e V m D M = G e V m D M = G e V S t a b l e r e g i o n S t a b l e r e g i o n S t a b l er e g i o n S t a b l e r e g i o n DARK MATTER
FIG. 4. Same as in Fig.2, for the dimensionless ratios, p DM /p OM = 10 − (black curves) and (red curves). the NS and the WD branches as function of the DM mass con-tent, M max T ( M DM ) , for weakly interacting DM (Fig. 10) andstrongly interacting DM (Fig. 11). The upper panels in thesefigures show the NS branch, whereas the low panels displaythe WD one. The different curves in each panel are given fora fixed value of the DM mass particle. The maximum massvalues of the CO are obtained from the maximum masses ofall possible stable NS and WD configurations given by a fixed p DM /p OM ratio, but varying the DM particle masses (1-500GeV) and for weakly and strongly-interacting DM matter, ascan be extracted from Figs. 6-9.As seen in Ref. [58], we observe that, for the weakly in-teracting DM case, the reduction of the total mass in the WDbranch from the nominal value of 1 M (cid:12) takes place for lowerDM mass content than for the case of 2 M (cid:12) in NSs. There-fore, WDs sustain less DM than the most massive NSs. This is also the case for strongly interacting DM for DM particleswith masses above 50 GeV. On the other hand, for DM parti-cle masses of 10 GeV and less, we obtain an increase in thetotal mass of CO when DM content exceeds ∼ − M (cid:12) forthe NS branch and ∼ − M (cid:12) for WDs. This might excludestrongly self-interacting DM of masses 1-10 GeV in the inte-rior of COs if formation mechanisms of COs allow for theseDM mass contents. The possible formation of these COs andtheir DM content is discussed in the following section. IV. MATTER IN THE COS: ACCUMULATION ANDENVIRONMENT.
In order COs having terrestrial or Jupiter-like masses, theyshould acquire from the environment a precise quantity of -4 -3 -2 -1 -11 -9 -7 -5 -3 -1 -2 -1 R O M [ k m ] M T [ ] r c / r WEAKLY INTERACTING DM y=10 -1 p DM /p OM =10 p DM /p OM =10 p DM /p OM =10 p DM /p OM =10 p DM /p OM =10p DM /p OM =10 -1 p DM /p OM =10 -2 p DM /p OM =10 -3 p DM /p OM =10 -4 p DM /p OM =10 -5 p DM /p OM =1 -2 -1 -11 -9 -7 -5 -3 -1 -2 -1 R O M [ k m ] M T [ ] r с / r STRONGLY INTERACTING DM y=10 p DM /p OM =10 p DM /p OM =10 p DM /p OM =10 p DM /p OM =10 p DM /p OM =10p DM /p OM =10 -1 p DM /p OM =10 -2 p DM /p OM =10 -3 p DM /p OM =10 -4 p DM /p OM =10 -5 p DM /p OM =1 FIG. 5. Total mass, M T , and radius of OM content, R OM , of the astrophysical objects as a function of the central density of OM for theDM interaction strength parameters y = 10 − (left panels) and y = 10 (right panels). We fix m DM = 1 GeV. The coloured lines indicatesolutions for M T and R OM over OM central densities for a set of dimensionless ratio, p DM /p OM , running from − to . These linesturn grey in the unstable regions. Note that for the case y = 10 − , lines from p DM /p OM = 10 − to p DM /p OM = 10 − for the M T and R OM distributions lie on top of each other. DM. As an example, for an object like Jupiter having a mass (cid:39) − M (cid:12) , the content of DM must be in the range of − − − M (cid:12) , as found in Ref. [58] from the solution ofthe TOV. One question that can naturally arise is if, in nature,there are processes that allow the accretion of that quantity ofDM by a CO.To have an idea of the quoted quantity, we have to considerthe DM acquired during the different phases of the CO for-mation. In the case of a Earth-like, or Jupiter-like CO, thereare two phases of accretion: a) accretion during the collapsephase; b) accretion of DM after collapse due to the capturein the CO by interaction with the CO’s nuclei. In the case ofNSs or WDs, one must take account of a) the DM acquiredduring the collapse phase, b) the change of DM inside the starduring the star evolution till the supernova explosion phase, c)the DM acquired by the NS.In order to have a good estimate of the DM in NSs and WDs, one should perform simulations similar to those of [72],but for the inner part of the NSs and WDs, not only the dis-tribution of DM external (mini-halo) to the stars. Simplifiedcalculation consider just the DM capture during the NS, WDphase [42, 44], or estimate the accretion by the CO progenitor,and the CO phase [73]. One can obtain an estimate of the DMtrapped in the star by using Eq. (4) in [74] for the capture rate F = 1 . × s − (cid:18) ρ dm . / cm (cid:19) (cid:18) / s v (cid:19) (cid:18) TeV m (cid:19)(cid:18) MM (cid:12) (cid:19) (cid:18) RR (cid:12) (cid:19) (cid:16) − e − E v (cid:17) f, (2)where ρ dm is the local dark matter density, v the averageWIMP velocity, f the probability that in the star one hasat least one WIMP-proton scattering and E the maximumenergy of the WIMP per WIMP mass leading to capture WHITE DWARF BRANCH y=10 -1 m DM = 1 GeV m DM = 5 GeV m DM = 10 GeV m DM = 50 GeVm DM = 100 GeV m DM = 200 GeV m DM = 500 GeV p DM /p OM =10 p DM /p OM =10 p DM /p OM =10 p DM /p OM =10 p DM /p OM =10p DM /p OM =1 p DM /p OM =10 -1 p DM /p OM =10 -3 p DM /p OM =10 -5 M T M T M T FIG. 6. Mass-radius relations of the equilibrium configuration of DM-admixed WD branch for weakly interacting DM, y = 10 − . Resultsare shown for DM particle mass m DM ranging from 1 to 500 GeV (1, 5, 10, 50, 100, 200, 500) from top to bottom. Each colour correspondsto the DM particle mass, indicated in the legend. Each panel corresponds to a different p DM /p OM , indicated in legend. Note that for somevalues of p DM /p OM , some mass curves are not visible because they overlap with others. (see [74]). In the case of a typical NS (WD), Eq.(2) gives (cid:39) − (10 − )M (cid:12) , for the DM trapped in a NS (WD),and values even smaller in the case of the planet-like COs.The capture rate has been estimated by several other authors[44, 75–77] and the results are more or less in agreement with that of [74]. A comparison of the DM contained in the planet-like COs, and the DM that can be trapped into them by accre-tion shows that this mechanism cannot explain, at first glance,their existence (see however the following), in the case DMis uniformly distributed in the halo. The previous discussion0 M T M T M T NEUTRON STAR BRANCH y=10 -1 m DM = 1 GeV m DM = 5 GeV m DM = 10 GeV m DM = 50 GeVm DM = 100 GeV m DM = 200 GeV m DM = 500 GeV p DM /p OM =10 p DM /p OM =10 p DM /p OM =10 p DM /p OM =10 p DM /p OM =10p DM /p OM =1 p DM /p OM =10 -1 p DM /p OM =10 -3 p DM /p OM =10 -5 FIG. 7. Same as in Fig. 6 for the NS branch. shows a discrepancy between what the TOV equation allows,and the quantity of DM that the accretion mechanism can trapin the star.As many studies [78–83] pointed out, the DM distributioninto an halo is not homogeneous, and super-dense dark mat-ter clumps (SDMC), i.e. bounded DM objects virialized atthe radiation dominated era, and ultra compact mini-haloes(UCMH) formed from secondary accretion on SDMCs [81], are present in the halo.According to those studies, a SDMC of (cid:39) (cid:12) can havea density (cid:39) × larger than the local DM density, andlarger for smaller masses. A NS located in such a SDMCwould trap through the accretion process a DM mass of theorder of − M (cid:12) , − M (cid:12) for a SMDC of 0.1 M (cid:12) [81, 83].The maximum density in the center of clumps can beestimated by means of the annihilation criterion [81, 83] and1 WHITE DWARF BRANCH y=10 m DM = 1 GeV m DM = 5 GeV m DM = 10 GeV m DM = 50 GeVm DM = 100 GeV m DM = 200 GeV m DM = 500 GeV p DM /p OM =10 p DM /p OM =10 p DM /p OM =10 p DM /p OM =10 p DM /p OM =10p DM /p OM =1 p DM /p OM =10 -1 p DM /p OM =10 -3 p DM /p OM =10 -5 M T M T M T FIG. 8. Same as in Fig. 6 for the strongly interacting DM, y = 10 . is (cid:39) larger that the local DM density, implying that aNS would acquire a DM mass equal to (cid:39) . × − M (cid:12) .In the previous scenario, we considered COs planet-like,either NSs or WDs, which trapped DM from the environmentby means of the accretion mechanism. However, another pos-sibility, actually the more natural, is that mini-halo forms anda planet-like CO is born by the following collapse of baryons on the potential well of the mini-halo. The last correspond tothe phase a. (accretion during the collapse phase) previouslymentioned. After formation, the planet could continue to ac-quire DM by accretion from the environment (phase b). Wewill study this aspect in a future work.Finally, even if DM was distributed in a homogeneous fash-ion in the Galaxy, its density increases going toward the centerof the Galaxy, similarly to what happens in a mini-halo. The2 NEUTRON STAR BRANCH y=10 m DM = 1 GeV m DM = 5 GeV m DM = 10 GeV m DM = 50 GeVm DM = 100 GeV m DM = 200 GeV m DM = 500 GeV p DM /p OM =10 p DM /p OM =10 p DM /p OM =10 p DM /p OM =10 p DM /p OM =10p DM /p OM =1 p DM /p OM =10 -1 p DM /p OM =10 -3 p DM /p OM =10 -5 M T M T M T FIG. 9. Same as in Fig. 8 for the NS branch.
DM density profile of our Galaxy is not known. In particularit is not known if the profile is cored, as in several dwarf spiralgalaxies, or cuspy. N-body simulations predict cuspy profilesparameterized by an Einasto profile: ρ = ρ − e − α (cid:104)(cid:104) rr − (cid:105) α − (cid:105) , (3)where α is related to the curvature degree of the profile, r − is the distance at which d ln ρd ln r = − , and ρ − is the density at r − . In the case of the more cuspy profile (see [46]), the DMdensity at − pc is × GeV/cm , a factor (cid:39) largerthan in the Sun neighborhood. This means that while a NSlocated at the Sun neighborhood will accrete (cid:39) − M (cid:12) , at − (0 . pc will accrete − (10 − )M (cid:12) . This implies thateven in a homogeneous halo, planet-like CO objects can format about . pc to the center of the Galaxy.3 ] [M DM M - - - - - - - - - - - -
10 1 ] [ M T M ] [M DM M - - - - - - - - - - - - - ] [ M T M WHITE-DWARF y=10 -1 NEUTRON STAR y=10 -1 m DM = 1 GeVm DM = 5 GeVm DM = 10 GeVm DM = 50 GeVm DM = 100 GeVm DM = 200 GeVm DM = 500 GeV FIG. 10. The total maximum mass of the CO as a function of the DM mass for weakly interacting DM, y = 10 − . The coloured curves, builtfrom the extrema at fixed p DM /p OM curves (see text) at a given DM particle mass, show the evolution of the star mass with increasing DMmass content. Each colour corresponds to one DM particle mass, as indicated in legend. Top panel
The neutron-star branches.
Bottom panel
The white-dwarf branches.
The previous discussion has some consequences on the NSsand WDs structure, and generates a relationship between theCOs masses and the distance from galactic center, which willbe discussed in a forthcoming paper.
V. CONCLUSIONS AND DISCUSSIONS
In this paper, we have studied how DM, non-self annihilat-ing, self-interacting dark matter, admixed with ordinary mat-ter in COs changes their inner structure, and discussed the for-4 ] [M DM M - - - - - - - - - -
10 1 10 ] [ M T M ] [M DM M - - - - - - - - - - -
10 1 10 ] [ M T M WHITE-DWARF y=10 NEUTRON STAR y=10 m DM = 1 GeVm DM = 5 GeVm DM = 10 GeVm DM = 50 GeVm DM = 100 GeVm DM = 200 GeVm DM = 500 GeV FIG. 11. Same as Fig. 10 for the strongly interacting DM, y = 10 . Note that lines shift horizontally to the left with increase of DM particlemass, while exhibiting a growth with increasing DM content for m DM = 1 − GeV. mation of planet-like COs. We consider DM particle massesfrom 1 to 500 GeV, while taking into account both weakly andstrongly interacting DM.The total mass of the COs depends on the DM particle massas well as the quantity of DM in its interior. In the strong inter-acting case, some combinations of DM particle mass and DMmass content lead to COs much heavier than the known limits given by NSs and WDs. This puts constraints in the parameterspace y , M D , p DM /p OM , and R OM . Also, constraints on themass trapped in the pulsar come from the observations of pul-sars with masses (cid:39) M (cid:12) . At the same time, it is possible toexplain the existence of NSs with masses (cid:39) M (cid:12) through theincrease of the total COs mass with decrease of particle mass.Following the discussion in Section IV, on the inhomogene-5ity of DM distribution in the halo, and minihaloes of the MilkyWay, in a next paper we will discuss how the inhomogeneityaffects the COs structure. As we indicate in Section IV, weknow that DM is not uniformly distributed inside the galac-tic haloes for two reasons: a) the presence of clumps (mini-haloes) randomly located inside the galactic halo, b) the in-crease of DM density going towards the center of the galactichalo. The difference from outskirts to the halo, or mini-halo,center is usually of , . This implies the accumulationmechanism can trap much more mass close to the halo center,giving rise to different COs structures. ACKNOWLEDGMENTS
M.D. work was supported by the Chinese Academy ofSciences Presidents International Fellowship Initiative UnderGrant No. 2016PM043 and by the Polish National ScienceCentre (NCN) grant 2016/23/B/ST2/00692. MLeD acknowl-edges the financial support by Lanzhou University startingfund. L.T. acknowledges support from the FPA2016-81114-P Grant from Ministerio de Ciencia, Innovacion y Universi-dades, Heisenberg Programme of the Deutsche Forschungs-gemeinschaft under the Project Nr. 383452331 and PHAROSCOST Action CA16214.
Appendix A: Appendix: Stability Determination
The stability of TOV solutions is commonly determined us-ing one of two methods. The first is the Bardeen, Thorne,and Meltzer (BTM) method [84], based on counting the mass-radius ( M − R ) relation extrema, which curves are generatedby varying the central pressure P c . Each curve point is a sta-tionary configuration, but only those with stable radial modesare stably stationary. The simple rules proposed by Ref. [84]are: at each extremum, with increasing central pressure, onemode changes from i) stable to unstable where the M ( R ) curve rotates counter-clockwise; ii) unstable to stable wherethe M ( R ) curve rotates clockwise. The other method consistsin solving the relevant Sturm-Liouville equation to explicitlyobtain the radial oscillation eigenmodes, following [85]. Thisis used by [58, 86] and by us in what follows.The time-dependent displacement describes the radial os-cillations: δr n ( r, t ) = e ν ( r ) r u n ( r ) e iω n t , (A1)where ν ( r ) comes from the double radial metric definition, n marks the mode spectrum index, and u n ( r ) is a solution with eigenvalue w n to the Sturm-Liouville problem ddr (cid:18) Γ P r ddr ( r ξ ) (cid:19) − r dPdr ξ + ω ρ c ξ = 0 , (A2)where Γ = d log P ( r ) /drd log ρ ( r ) /dr gives the adiabatic index governingthe pressure-density equilibrium relation. More precisely, ityields its pressure-weighted average, i.e. the fractional changein pressure per fractional change in comoving volume, at con-stant entropy and composition. The quantity ξ gives the radialcomponent of the perturbations (cid:126)ξ ( (cid:126)x, t ) , ρ c is the central massdensity, and P is the electrostatic pressure.The boundary conditions for Eq. (A2) are ξ = 0 at r = 0 , and ξ finite at r = R , where R is the surface of the star.The eigenvalues of Eq. (A2) are given by ω = (cid:82) R { Γ P r (cid:2) ddr ( r ξ ) + 4 rξ dPdr (cid:3) } dr (cid:82) R ρ c ξ r dr , (A3)where we have integrated by parts, using the boundary condi-tions ξ = 0 at r = 0 and ∆ P = 0 at r = R . The physicalinterpretation of some terms in Eq.(A3) follows. Since we al-ways have Γ ≥ , the first term is stabilizing. It arises fromthe electric field “compression” or the electrostatic pressure.It can be equivalently interpreted as due to the acoustic modes.The second term reflects gravity destabilizing effect.The Sturm-Liouville eigenvalue problem, Eq. (A2), pro-duces a discrete set of solutions eigenfunctions u n ( r ) , foreigenvalues ω from Eq. (A3), the squared frequencies ofthe oscillation modes. Those eigenvalues form a real lower-bounded infinite sequence ω n < ω n +1 , for n = 0 , , ... . Anymode n is stable and oscillatory if ω n > , so the frequencyis real. However, for ω n < , the purely imaginary frequencyleads to an unstable mode which exponentially grows or de-cays.The overall stability of the star is sufficiently determinedby the fundamental radial mode, ω . Indeed, if ω > , then all ω n > and the star is stable. For ω < , there is (at least) oneunstable mode and the star is unstable. The sign of ω thussufficiently ascertain the overall stability. It derives from theanalysis of the star mass versus mass density or radius [85].The typical behavior of the lowest eigenvalues, ω and ω canbe found in Ref. [86].Sturm-Liouville properties of the perturbation equationyield general arguments showing that the mode stabilitychanging extremum occurs for even modes ( n = 0 , , ... ) if dR/dρ c < and for odd modes ( n = 1 , , ... ) if dR/dρ c > .Using this method and starting from low-mass densities whereall modes are positive, the stability analysis is obtained forhigher-mass densities studying the change of the sign of thedifferent modes while keeping their hierarchy. [1] G. Bertone, D. Hooper, and J. Silk, Physics Reports , 279(2005), arXiv:hep-ph/0404175.[2] A. Del Popolo, International Journal of Modern Physics D , 1430005 (2014), arXiv:1305.0456 [astro-ph.CO].[3] Planck Collaboration, P. A. R. Ade, N. Aghanim, M. Arnaud,M. Ashdown, J. Aumont, C. Baccigalupi, A. J. Banday, R. B. Barreiro, J. G. Bartlett, and et al., A&A , A13 (2016),arXiv:1502.01589.[4] A. G. Riess, A. V. Filippenko, P. Challis, and et al., AJ ,1009 (1998), arXiv:astro-ph/9805201.[5] S. Perlmutter, G. Aldering, G. Goldhaber, and et al., ApJ ,565 (1999), arXiv:astro-ph/9812133.[6] E. Komatsu, K. M. Smith, J. Dunkley, and et al., ApJS , 18(2011), arXiv:1001.4538 [astro-ph.CO].[7] Planck Collaboration, P. A. R. Ade, N. Aghanim, C. Armitage-Caplan, M. Arnaud, M. Ashdown, F. Atrio-Barandela, J. Au-mont, C. Baccigalupi, A. J. Banday, and et al., A&A , A16(2014), arXiv:1303.5076.[8] A. Del Popolo, in
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