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JCAP11(2008)014 hysics $30.00 18 P and and 3 stroparticle stroparticle A [email protected] , , Douglas Spolyar 2 , 1 3 0802.1724 The annihilation of weakly interacting massive particles can provide , formation osmology and and osmology stacks.iop.org/JCAP/2008/i=11/a=014 [email protected] C Michigan Center for Theoretical Physics, Physics Department, Perimeter Institute for Theoretical Physics, 31 CarolinePhysics Street, Department, North University of California, Santa Cruz, CA 95060, USA Anthony Aguirre an important heat sourcestars, for potentially the first leadingstar’. (Pop to III, When a dark ‘Pop’ matter new standing (DM) from for phase capture DM via ‘population’) annihilation of scattering may off dominate stellardepending over baryons the on is evolution luminosity the included, due known DM to the fusion, density asto and a scattering capture cross may ‘dark section. thusthe The prolong influx ambient the of DM DM dark due the density star is Eddington phase of luminosity highmetallicity stellar for enough. evolution . the as Comparison star long Alternatively,they of may as if might DM constrain be sufficiently luminosity the used massive to with stellar Pop bound mass III dark of matter stars properties. zero- are found, Keywords: ArXiv ePrint: doi:10.1088/1475-7516/2008/11/014 Abstract. 1 2 3 [email protected] Received 16 May 2008 Accepted 25 October 2008 Published 18 November 2008 Online at University of Michigan, Ann Arbor, MI 48109,Waterloo, USA ON, N2L 2Y5, Canada E-mail: ournal of ournal

An IOP and SISSA journal 2008 IOP Publishing Ltd and SISSA 1475-7516/08/11014+ c 

power source and limit on Dark matter capture in the first stars: a J JCAP11(2008)014 2 2 5 at 10 12 16 16 17 M 6 ) –10 ]intoasingle M 6 5 ]–[ ]). 4 10 ]–[ ...... 10 8 GeV–TeV to give the correct ∼ ...... 10 ]. 27 ] examining the effect of dark matter stacks.iop.org/JCAP/2008/i=11/a=014 ]–[ 1 25 s of the work that we consider here. WIMP behind this relic density. Probably the best 11 (2008) 014 ( ...... 10 Dark matter capture in the first stars: a power source and limit on stellar mass ction cross section and mass ...... 5 ]. These halos consist of 85% dark matter and 15% baryons in the 3 ...... 7 , 2 ] at the center of the halo (for reviews see e.g. [ 7 = 10–50 [ z 2.3.1. Dark2.3.2. matter density before Dark capture matter is density included. including capture. . The , the supersymmetric partner of the W, Z, and Higgs bosons, 4 This same annihilation process is the basi In this paper we continue our previous work [ 2.2. Capture rate 2.3. Dark matter density 2.1. Annihilation rate Acknowledgments References If the lightest supersymmetric particle were the or gravitino, it would not have weak interaction self-annihilation is relevant wherever the WIMP density is sufficiently high. Such regimes 4 3. Luminosity due to4. WIMP annihilation 5. Conclusion The first stars in theand Universe mark provide the the end ofimportant enriched the cosmic as gas dark required precursors ages,The reionize for to the first Universe, later stars black stellar are holes generations. thought that to coalesce They form may inside and also dark power matter be bright halos early of quasars. mass 10 1. Introduction 2. WIMP abundance 1. Introduction Contents Journal of Cosmology and Astroparticle Physics annihilations and would not produce the phenomena described in this paper. has the required weak intera amount of dark matterof and SUSY would and play other an dark matter important candidates role see in [ the first stars. For reviews redshifts form of metal freebaryonic matter hydrogen and cools helium and gas. collapses via Theoretical molecular calculations hydrogen indicate cooling that [ the small [ example of a WIMP is theparticle neutralino, which in many models is the lightest supersymmetric (DM) on the particles(WIMPs), on the which first stars. arethey We the focus automatically on favorite provide weaklyof dark interacting approximately massive the matter particles the current candidatesantiparticles, right energy of in density amount many which of ofwith case physicists the weak dark interaction they Universe. because matter, cross can sections, i.e. Many annihilate leaving WIMP 24% with candidates themselves are in their the own early Universe JCAP11(2008)014 3 (1) ]and 13 ) = 100 GeV for χ m ], in the [ 12 , 11 lly well to other WIMP candidates lies within an order of of ]) and would have even more drastic stacks.iop.org/JCAP/2008/i=11/a=014  16 other sources of heating. The first stars σv  number density of the collapsing protostar and allow the dark star phase to continue: e to form stars, they come to dominate the e very first stellar objects might be ‘dark batic contraction. Once the DM due to this ugh ambient DM passing through the star, pture of more DM from the ambient medium. nd. On the other hand, there is another effect r the star through DM annihilation. We are as , 1 ). These densities are still not enough: paper I 3 − ) s 11 (2008) 014 ( ] two of us (together with P Gondolo) considered z 3 1 for a 100 GeV WIMP mass): at this point WIMP 3 cm − 26 − cm Dark matter capture in the first stars: a power source and limit on stellar mass ]. Non-thermal particles can have annihilation cross sections 10 13 18 × , 17 =3 to 50 GeV–2 TeV, while ann χ (except at the low end of the mass range where it could be several orders  ], and in the first stars. As our canonical values, we will use the standard m 1 ]; the effects that we find apply equa σv − 15  s 19 , 3 14 cm 26 − In this paper, we consider the effects of WIMP annihilation on the first stars. In a Dark stars, powered by DM annihilation, require three key ingredients, as shown in The interaction strengths and masses of the neutralino depend on a large number of 10 × 3 previous paper (hereafter, paper I) [ with comparable cross sections for self-annihilation and scattering off nucleons. the effects of darka matter crucial annihilation transition on takes the place formation when of the the first gas stars. We found that exceeds a critical value (10 Journal of Cosmology and Astroparticle Physics value for the annihilation cross section the Earth [ include those of the early Universe, in galactic halos today [ paper I: (1)star, high and dark (3) matter DM density, (2) heating annihilation dominates products over remain the trapped in the annihilation heating dominatescollapse over of all thestars’, cooling star. a new mechanisms phase andmass We of density—provides stellar suggested prevents the evolution power that the in sourceyet which fo th further uncertain the of DM—while the onlythe supplying lifetime star 1% of runs of out, thesepossible. the the theoretical star objects. However, could as contract Oncefor the and repopulating the star heat the DM reaches up DM contained nuclear to inside inside density, the the there point star: is where ca an fusion additional becomes mechanism causing DM annihilation inside the star to continue, possibly even to today. exist at the right placecriterion: and at they the exist righthigh in time redshifts to the (density have high the scales density best as chance centers (1 of of + achieving dark the first matter halos, and they form at This new sourceprocess of can DM continue can as extend long the as there lifetime is of eno the dark star. Indeed the capture of magnitude smaller) [ that are many orders of magnitude larger (e.g. [ effects. Given the presentcould state apply of the [ field there are many types of DM candidates which as this gives the right WIMP relic density today, as well as taking showed that the DMto density become must important: be driven as up the still baryons condens further in order for the DM heating that can repopulate thecapture of DM more inside DM the from stars the ambient medium. To get a significant amount of captured potential well and pulldensities the DM were in computed with ineffect them paper runs and I out, drive from the up adia dark the star density. phase The might resultant e DM model parameters. Inbounds the restrict minimal supergravity model, experimental and observational our canonical value ofWIMP the masses WIMP (1 particle GeV–10 mass TeV) but and will cross also sections. consider a broader range of JCAP11(2008)014 4 ) ]. ]), a dark star powered ] in the conclusion that 1 29 23 stacks.iop.org/JCAP/2008/i=11/a=014 nnihilation on early zero- (Pop ars, the scattering cross section must be t scattering. The annihilation provides a DM annihilation luminosity to the fusion k star merges with other objects, it is not rmation and earlier stages of these stars. ffect the IR background, the re-ionization lear densities where fusion can take place. r of the stars, where the DM can annihilate unavoidable. Thus, even if DM annihilation densities due to adiabatic contraction. We mains undisturbed, and it is not clear how ngton luminosity. Of course, the presence of ep further and discuss the possibility that the ity for capture to be important, and find that ile after the star is created, but it is not clear next round of DM detection experiments. The d in the earlier protostellar phase. 11 (2008) 014 ( ]. 22 , 21 Dark matter capture in the first stars: a power source and limit on stellar mass ), on theoretical grounds the scattering cross section can vary across 1 ] and more generally [ 20 ]. We have carried the analysis further. We agree with [ We begin by discussing the equilibrium WIMP abundance in the first stars, by In this paper, we consider the effects of DM a Just as we were preparing to submit our paper, a very similar work was submitted Previous work on DM annihilation powering stars has also been done in the context The two key uncertainties in this work are: (i) the scattering cross section must 23 computing the numberthe of annihilation WIMPs rate of captured WIMPs by in the the first first stars. stars, A and discussion equating of this adiabatic with contraction, DM power source maygrowing exceed beyond the a Eddington limited luminosity mass. and prevent This the would first a stars from DM annihilation may dominatefor over this fusion, conclusion. and In addition, we we illustrate go the one DM st densities required of the Universe, thethe first number stars; of this supernovae, will and be potentially addressed the in nature a of separate publication supernovae [ of Journal of Cosmology and Astroparticle Physics material into the star requires the enhanced will estimate the required ambient DM dens it is somewhat lower than what is require by [ of high DMburners densities [ near the supermassive black holes in galactic centers, e.g. WIMP clear how long thelong central the DM dark in starindefinitely, the remains so halo at a re dark this star central could point. still In exist principle today, the but this capture is could very continue unclear. III) stars, once theyof do WIMPs; have as fusion thethe inside WIMPs stars. their move The through cores. capturedvery the DM These efficiently. stars, sinks stars This to some live has the ofof cente inside the the a effect a WIMPs star, of reservoir are compared dramaticallyheat captured to increasing source by DM the for annihilation annihilation the rate withou stars, inside and we compare this within a few ordersshould of be magnitude experimentally of accessible thesecond current in criterion experimental the is limits. likelyfor Such to a how be cross long. true section Once for the a wh halo containing the dar many orders of magnitude.bounds. As discussed later, For it scattering is, to however, constrained matter by in experimental the first st be at (orenough near) for the capture experimentalclose to limit, to take and equation place. (ii) ( the Whereas the ambient DM annihilation density cross must section be is fixed high to be luminosity of the stars,DM as well (with as high to densitiesimportant the due Eddi during to (and adiabaticHere, seriously contraction it or affect) is capture) the simply wouldthe fo importance already our of become intent DM to heatingfails in show Pop to that III stop under stars a some Pop circumstances III DM star captureAgain, from makes we forming wish (asdark to was star considered note and in that yet [ the may be DM responsible supplies for less its than luminosity. one percent of the mass of the by DM annihilation may exist at the high nuc JCAP11(2008)014 5 (6) (5) (2) (3) (4) )and ). The τ 1 ) −  (s t C inside of the star. In this section r ]. Other indirect searches such as 30 stacks.iop.org/JCAP/2008/i=11/a=014 ]. DM could be detected through sion luminosity. Finally, we compare the DM nd find a maximum stellar mass as a function 15 , , 2 , 11 (2008) 014 ( 14 N A ann ) C ). This process was previously noticed as important for σv − 1 ( 2 in the star is then determined by a competition between − C ) (s r Dark matter capture in the first stars: a power source and limit on stellar mass ( N ≡ 2 A χ 2 / A . Although most of the WIMPs travel right through the star, 1 c − rn σ ) /N 3 2Γ ). A ] and AMANDA (which did not find a signal and placed bounds) A d 5 − C. 32 2 1  ), where CC C -independent annihilation coefficient. Solving this equation, we find =2Γ = = N = t/τ =( A A A ( ] and in the Earth by [ ˙ 2 τ Γ C Γ N 13 ) is the density of captured DM at a point is the annihilation rate (and the factor of 2 appears because two particles are r tanh ( A χ C n 1 2 The number of WIMPs = A we will show that the WIMPs quickly thermalize with the core of the star, so we can treat we may use equation ( where Journal of Cosmology and Astroparticle Physics 2.1. Annihilation rate The annihilation rate As WIMPs travelscattering through cross a section star, they can scatter off the nuclei in the star with the 2. WIMP abundance of DM density. luminosity with the Eddington luminosity a which may drive up theannihilation DM luminosity density and near compare the with baryons, fu follows. Then, we compute the DM the Sun by [ annihilated in each event), and As we will show, for the case of Pop III stars, equilibrium is quickly reached ( some of them lose enough energy to be captured. We call the capture rate where Γ is defined asΓ an is the equilibration timescale.and Equilibrium annihilation corresponds rates, to i.e. a balance between the capture GLAST and PAMELA couldrespectively from detect DM the annihilating gamma-rayfrom in DM and the annihilation Milky positron in Way halo annihilation thenot (the Sun products make gamma-rays or it and Earth to positrons would our be detectors). trapped inside the objects and would capture and annihilation via the differential equation produced from the DM annihilationSuper-Kamiokande products in [ the Sun or Earth with experiments such as and ICECUBE (which is starting to take data) [ WIMPs then sink towith the the annihilation center rate of Γ the star, where they can annihilate with one another JCAP11(2008)014 6 = (9) (7) (8) χ (12) (10) (11) ,and m 2 )inthe − . ) p ann cm M  )isthemass 39 r σv − (  5 . 1 χ M m =10 c 1GeV,thetimescale ]. ∝ σ A 24  C χ m and also the total annihilation 1 is the central temperature of the V . Assuming a flat distribution and . Roughly, thermalization requires le to be very short, roughly three c 2 χ T n -independent annihilation coefficient 2] stacks.iop.org/JCAP/2008/i=11/a=014 / N /m = ) n timescale inside the star. The amount p p M N M and then we will compute the annihilation + for Pop III stars of different masses. From χ A scatters. Thus, for with respect to the center, and m is the core mass density of the star. The name C , p 11 (2008) 014 ( r. [( r 2 c / d / ρ M 3 p ) to find the ) for the annihilation cross section and mass ( 2 2 )] 1 / φ/T M c r , , χ χ χ ρ d χ . H m χ j ) m m 2 . r − m M V ( φ/kT M 2 Dark matter capture in the first stars: a power source and limit on stellar mass e 2 2 0 1 2 ≤ χ 2 r V n V (2j r m H ∗ − n GM ann e R T/ ann c  r  1 0 esc E/E 2 pl 0 n  v ):  c σv Δ m σv 3 π   = σ ≤ χ = , the average energy loss is 2 )= ) ≈ =4 =[3 , we find the thermalization timesca r χ r 3 A ( ( j j th , we list the computed − m φ V τ V n C lost by a WIMP in a scattering event with a nucleus (proton mass 1 ], we have obtained the properties of zero-metallicity stars when they are . One can define effective volumes cm E is the Planck mass, and  is the velocity of escape of a DM particle from the surface of the star, and r 1; i.e. there must be 28 is the central number density of DM and p 24 pl c esc ∼ M v m n First let us examine the WIMP thermalizatio In table One can then solve equation ( Thus we can use an isothermal distribution for the DM: =10 is the average density of the star. For a hundred GeV WIMP, E/E H H star, Journal of Cosmology and Astroparticle Physics rate. Our work closely follows the approach previously given by [ the DM density distribution as isothermal, of energy Δ n is the gravitational potential at radius interior to for thermalization can be estimated as Δ ‘effective volume’ is suggestive since we have where Upon integration this gives Woosley [ halfway through hydrogenour burning canonical on values the in main equation ( sequence. In the table we have used where where n 100 GeV); the results can easily be scaled to other values since taking rate is given as Γ = months. star ranges over 0 defined in equation ( JCAP11(2008)014 ] 2 2 7 is A N 31 χ [ (13) (15) (14) )for A n 3 2 C ]. The − 1 2 ) 28 = A (g cm ρ ,  ) 54 55 56 56 57 2 − − − − − ) B 1 10 10 10 10 10 as needed. − − × × × × × 2 (s 1 − B 16 31 14 06 33 A s . . . . . exp( 5 5 4 2 9 C 3 − ) cm 1 3 ]. The first stars are made only − 28 29 30 31 31 26 − − − − − − 39 − from the center of the star, for an , 1 (cm 10 10 10 10 10 r 10  2 38 1 × × × × × stacks.iop.org/JCAP/2008/i=11/a=014 2 × ) 2 /V r 77 38 86 11 72 2 ¯ . . . . . v ( V v =3 ly the spin-dependent contribution, though ) (K) and central baryon density ¯ v ) c 3 ann T  σ − )( σv r  11 (2008) 014 ( = 100 GeV [ ( χ (g cm χ n is also shown. The DM annihilation rate is Γ ρ 225.848.6331.8819.72 1 1 6 3 ]. The latter, which are the most constraining, assume ) m enhancement for the spin-independent contribution is 2 r 1 ( 37 2 7 8 8 8 . The bound on the spin-independent (SI) scattering is ) is the velocity of escape of WIMPs from the star at a n /V c for A r 2 ]–[ 2 10 10 10 10 , σ ( is a ‘velocity dispersion’ of WIMPs in the DM halo, and / 2 V Dark matter capture in the first stars: a power source and limit on stellar mass , 1 v 2 − × × × × 35 2 μ μ  cm (K) 55 13 18 23 ], [ 2 6 π . . . . /M cm ) T 2 χ 42 = 100 GeV and r  33 ¯ v Central temperature ( 39 − is the number of WIMPs in the star. Please note that the entry marked , χ v kT − ) 3 10 m 39 2 3 N )= , r M ≡ ≤ ( ( ≡ 32 =10 2 , 2 SI C V v c ] 1050 9 1 d σ d where ‘Sun’ refers to thevalues present day Sun forM comparison. We have usedSun the fiducial 100 —250 1 — 1 1 Table 1. σ B various masses of metaleffective volume free stars halfway through hydrogen burning [ ,¯ 31 r 40 , is the number of nucleons in the nucleus), and spin dependent, which require the 41 is the number density of nucleons (here, we only consider hydrogen because we A n The capture rate per unit volume at a distance given radius of hydrogen or helium, so the Journal of Cosmology and Astroparticle Physics we illustrate bounds onKamiokande [ the SD componentthat from a direct significant detection fractionSUSY as of particles); well the as if annihilationbe from energy the several Super- goes orders intospin-dependent of component neutrinos cross magnitude (as is section higher. is small likely As then for our the fiducial SD value, cross in this section paper, could we use the (where nucleon to have aweakest spin. for Currently, the the spin-dependentscattering experimental (SD) off bounds protons contribution since on (to the elastic be stars scattering are precise are comprised we primarily the are of hydrogen). considering In only figure 2.2. Capture rate WIMP interactions with nuclei are of two kinds: spin independent, which scale as not substantial. In thisin paper, principle we for consider any on specific candidate WIMP one should self-consistently include both. observer at rest with respect tohere), the is WIMP distribution [ (as should be a good approximation much tighter, the WIMP number density, which is consistent with allof experimental any bounds, result but we on will the always show value the of dependence where are not considering spin independent scattering and helium does not have a spin), JCAP11(2008)014 . 8 vir (22) (16) (20) (19) (21) (17) (18) Z in the 2. For s ) / r +1) . μ ]. With M 3 =( 6 , 2 − –10 μ 5 is the fraction of the star =10 ) is very close to 1 (justified H and the relative velocity as f ]fortheDM, 14 = 10–50 [ B halo 46 vir M , Z ) by treating the Pop III star as a polytrope r χ ( χ v ρ stacks.iop.org/JCAP/2008/i=11/a=014 m , known as the central density, depends nservative in considering only the spin- 2 use we believe this contribution to be 0 ]and  ρ . As our fiducial value, we will take 42 ¯ esc v 1 v r − d 2  esc nt contribution of scattering off hydrogen and r v ) ) ¯ v r c ≡ 11 (2008) 014 ( ( , and on the redshift when the halo virializes σ , 2  ( ) ) r s  halo = (1–10) [ ( vir  is the proton mass and R H C M GM C V p f r/r vir 2 d d  p ]. m C 0 halo r = Dark matter capture in the first stars: a power source and limit on stellar mass m = (1–15) km s M ρ ρ 40 d ], we take the virial velocity of the DM halo: . 2 2 1 v  , ) )(1 + 43  −  2 s | / πr 1 R 4 halo ( W r/r πG  v | ( 4 M R is the ratio of WIMP to nucleon mass and 6 π 0 ,wemaytake = − =   5 as per [ = N 2 ) =  v = = 2 r =10kms /M ( halo ¯ v χ ¯ C  v C v W ρ ) becomes a more complicated function of m is the stellar mass, 14 = 3 (a good approximation), and confirmed that the conservative capture rate presented here differs is the radius of the star. To obtain a conservative and fairly accurate estimate = is the scale radius. The normalization  n  , assume the term in square brackets in equation ( s μ M r R r The capture rate for the entire star is then To estimate ¯ We use a Navarro, Frenk and White (NFW) profile [ We have subsequently obtained much more accurate values for and the typical DM halo containing a Pop III star has of the capture rate 5 where shown in equation (2.24) of [ below), and take a uniform dark matter density. In this case the integral simplifies to give helium in thesubdominant. stars; In we any case have the not current done work is so co beca in hydrogen. (Notein that principle, hydrogen has consider spin the while spin-independe helium generally does not.dependent We scattering.) could, where on the concentration parameter with index Journal of Cosmology and Astroparticle Physics (it is too low) by only a factor of a few. where where an observer movingequation with ( respect to the WIMPs, the quantity in square brackets in for all where These parameters range from range (15–100) pc, we find ¯ JCAP11(2008)014 , . 9 τ ), 3 1 1 − )in M − (24) (23) ,we  s v esc t 24 V ) 10 ;thenfor GeV cm 1 ≈ 1 9 − from table − C ] it is roughly   A ), again using 1 ;theresultcan C 28 − 2 =10 18 χ χ ) using the capture ρ cm 4 m χ using 39 ρ =10kms 100 GeV − τ GeV cm v 9  1 =10 ) is very close to 1, and this 10 is extremely short, compared − ]. In obtaining these numbers, c  14 τ σ 1 (yr) 28 63 − τ 152 190 160 126 χ  ρ 1 − ), and surface escape velocity ( GeV cm 34 34 35 36 36  . For general values we find that stacks.iop.org/JCAP/2008/i=11/a=014 = 100 GeV, and with a velocity ¯ ) χ 9 R 1 10 10 10 , and calculated 10 10 ¯ v χ 2 − 10 × × × × × /m (s using our fiducial values m r that replaces the term in square brackets 9 9 1 8 9 c  cm ...... σ 1 C 10 km s C through the WIMP halo (rather than being he lifetime of the zero-metallicity stars, χ  − ). 39  ρ 2  − 2 ) 1 = 100 GeV, and  ∝ ), radius ( M − cm 11 (2008) 014 (  V 1 is much larger today (due to the fact that today’s c χ ( 12 − C σ =10 39 M v m ¯ v star, with − c esc , σ V 3 esc , which is much larger than ¯ 10 1 v −  10 km s M − ) . 1  Dark matter capture in the first stars: a power source and limit on stellar mass −  100.) Thus we may ignore the term in square brackets. The 618 km s R ). 55 ) and noting that for the Pop III models of [ 2  ( . 5  , the entry marked Sun refers to the present day Sun and not a 1 ∼ 1 17   cm R GeV cm c B Stellar mass ( χ   σ 9 39 m ) M M , we find that approximately M − M 45 = 618 km s   . 10 100 GeV M 0 = 100 GeV, and  1 1 ( =10   − − χ esc M s s χ 10 1.16 2.49 8 50 4.76 3.24 6 , we give the capture rate evaluated using equation ( v ρ 1, the term in square brackets in equation ( 100250 7.04 11.8 3.77 4.60 1 6 × M Sun 1 1 4 Table 2. As in table zero-metallicity star. Thein capture the rate true for present day the Sun Sun the still true capture uses rate the is fiducial much smaller values; ( solar units, foralso metal calculated free the stars capture rate halfway through hydrogen burning. We have mostly due to the much lower DM densities in the solar neighborhood. m × 2 ∝ 34 34 also shows the equilibrium timescale given by equation (   10 10 2 R B × × 9 9 . . 4 For In table Table = 100 GeV we find ≈ =4 χ bracketed term changesstationary for as a we star haveis assumed), moving O(1). but the (For facto example, for a 1 andwemayuseequation( properties (including stellar radius) ofwe Pop have III used stars from [ Journal of Cosmology and Astroparticle Physics star, we have C C easily be scaled to other values since holds for all stellar and WIMP masses that we are considering. (For example, for a 1 and annihilation rates determinedto above. the We lifetime can of see a that star. Hence for most of t m find that the factorstars is than 0.66.) in We the note first that stars this because factor is ¯ much more important in today’s Further, using equation ( true that galactic halos are much larger, e.g. 10 JCAP11(2008)014 10 ]to (25) star) )DM which 45 3 3 ) − − M cm cm / 13 ]forbothDM 10 46 GeV ∼ 81 . n 0 ) 3 − ] (see their figure 2). They dark star, before capture is cm 50 n/ M as numerous as the baryons can ]). Our adiabatically contracted 5( 1 )andlatest( 12

3 First we need estimates of the DM − − χ stacks.iop.org/JCAP/2008/i=11/a=014 ρ cm 3 e capture rate. Simulations have unfortu- 10 The amount of dark matter in the dark star ion approach. The highest DM density found e numbers. Still, as we will show in the next baryons in the star for every WIMP particle. ∼ ].) As the gas collapses, we allowed the DM to 16 n 11 (2008) 014 ( ]). The final DM density profiles were computed 48 , (which we doubt), the DM density would be as high ] obtained DM density profiles in and found 51 pc and 0.1 pc. The slope of these two curves is the , 3 49 = constant). After contraction, we found a DM density 52 3 − 50 r − ) , ) r cm 10 ( χ Dark matter capture in the first stars: a power source and limit on stellar mass 22 × m M . Should the adiabatic contraction continue all the way to the 10 (2 with a Navarro–Frenk–White (NFW) profile [ 3 A due to capture. Of course the density of WIMPs is more centrally − ∼ Γ ], two of us (with P Gondolo) used adiabatic contraction [ 4 1 f n outside the core (see figure 1 in [ M 6 9 = . .Wenotethat[ 1 3 –10 − Gev cm − DM r baryons for every WIMP particle; i.e., the fraction of WIMP particles has 8 L M ∝ 5 12 χ ], which has a DM core. A Burkert profile has been shown to be a good fit for ρ 47 GeV cm ] was 10 18 50 Journal of Cosmology and Astroparticle Physics 2.3.2. Dark matter density including capture. 2.3.1. Dark matter density before capture is included. 2.3. Dark matter density To study the effectsDM of passing dark matter through on the the stars first to stars, determine we th need to know the density of the nately not (as yet)these resolved numbers this are issue. unknown. Below we will use a variety of DM densities, since section, it is remarkable that particles which are 10 concentrated, so theannihilation ratio rate of peaks, WIMPs is to higher baryons than near thes the center of the star, where the at the outer edge of the baryonic core of roughly and gas, whereprofile the [ gas contributionthe is dynamics 15%. ofrespond today’s (For to galaxies comparison, the [ we changingwere also baryonic taken used gravitational from a potential, simulationswith Burkert adiabatic where of contraction the ( [ gas density profiles as 10 that adiabatic contraction produces3, densities even when that radial arespherical symmetry. orbits too Below are high we included, byTo will give or only use a a in a sense variety the factor of of presence DM of the densities of 2 numbers, due bars, we or to compute or the that in uncertainties. in the a absence 10 of NFW profiles match the DMpresent profile their obtained numerically earliest in [ (gas core density obviously increases significantly due to capture. We find that (again for a 10 We may now compute the luminosity due to WIMP annihilation, 3. Luminosity due to WIMP annihilation provide the dominant heat source for the star. same as ours.DM densities If as one withby our extrapolates adiabatic [ them contract inward to smaller radii, one obtains the same taken into account, there are roughly 10 profiles, as far inward as 5 there are 10 grown by a factor of 10 small stellar cores at density in theIn region a where previousobtain paper the estimates [ star of the forms,region DM of prior profile. 10 to Prior including to the this contraction, effects we of assumed an capture. overdense scales as JCAP11(2008)014 ) 5 11 (27) (28) (26) . 1 − ] ) DM 1  L ]forthese )whereas 1 1 due to DM − = 28 L ¯ v DM L 10 km s , and is dominant for to determine the value 3  (see figure χ 2 ρ )forwhich that is required in order 2 3 − cm M c crit σ 39 χ, 7 10 10 10 10 ∝ − ρ 10 10 10 10 10  10 × × × × × (GeV cm L ]. We stress that the DM densities 5 3 5 3 5 χ . . . . .  ower mass stars. The DM luminosity 7 1 7 8 8 ρ 3 29 stacks.iop.org/JCAP/2008/i=11/a=014 ) may be rewritten using equation ( − provided by the models of [ ) 25 1  − he ordinary stellar luminosity, which will ue, the star’s luminosity is dominated by L ilibrium), and consequently the luminosity. for a variety of stellar masses together with χ 33 35 36 36 37 ρ 3 nt density that determines the capture rate, ry fusion). We note that the luminosity due 3 goes into electrons, positrons, and photons ]scalesas 10 10 10 10 10 / (erg s GeV cm 28 9 × × × × × due to fusion versus luminosity 11 (2008) 014 ( 2 2 8 6 8 DM . . . . . 10  5 3 2 7 2 L . We stress that the DM densities listed here are those in is the fraction of annihilation energy that goes into L   L given in the last column of table )wemaywrite f ) 55 . 1 1 39 40 − 24 crit 33 37 39  . ) 10 10 χ,  10 10 10 Dark matter capture in the first stars: a power source and limit on stellar mass χ ρ × × 3 of the annihilation energy is lost to neutrinos that stream M × × × M (erg s exceeds m / 2 0 9 45 31 . . . . .  (2 Luminosity L 1 DM C − L . 2 f ) 3 / = 2 M erg s ( ∼ 33 DM 1050 4 2 M Sun100 3 250 6 2 Table 3. the ambient medium (NFW plusafter adiabatic capture; contraction) rather it than is thethus densities these the ambient equilibrium densities luminosity. that determine the capture rate and L f at which annihilation in zero-metallicity stars (asvalues defined for in DM previous tables), properties. using In fiducial the final column we vary 10 , so DM heating is more important for l 55 × . , which is quickly reached, equation ( 1 2 τ . M  ∝ =5 The WIMP luminosity is given in table The WIMP luminosity depends linearly on the WIMP density passing through the t DM DM is dominant for the values of as Journal of Cosmology and Astroparticle Physics that are trapped in the star and have their energy thermalized. Hence we take [ the luminosity. Roughlyright 1 out of the star, whereas the other 2 where we takeannihilate the per energy annihilation. per Here annihilation to be twice the WIMP mass; two WIMPS L L for the WIMP annihilation energy to equal t dramatically alter the properties of the first stars [ annihilation energy (rather than by ordina listed here are those inthe the ambient densities medium (NFW after plus capture; adiabatic contraction) it rather than isFor this any ambie WIMP densities higher thanto this fusion val in the zero-metallicity stars [ and also therefore the annihilation rate (in equ For stars. Roughly, using equation ( the ordinary fusion-powered stellar luminosity stars. We have also computed the WIMP energy density JCAP11(2008)014 . 12 M ;the 1 3 ≤ ) , and the top diamond line 3 − where they can be compared 1 ; thus for dark matter densities in ; in this final case, the DM luminosity 3 GeV cm 3 (squares are the examples listed in the − − 2 stacks.iop.org/JCAP/2008/i=11/a=014 12 cm DM annihilation remains subdominant. 39 − GeV cm GeV cm but we can nonetheless place an approximate 14 11 =10 10 11 (2008) 014 ( c σ × 9 ). If we keep the cross section fixed, the middle dot–dashed 3 and  3 χ − ρ Dark matter capture in the first stars: a power source and limit on stellar mass Log–log plot of stellar luminosity versus stellar mass (in solar units). GeV cm 9 =10 χ solid line is a fit toluminosity these for points. different DM The density remaining assumptions.listed lines indicate We here stress the that DM are the annihilation DM thoserather densities in than the the ambientcapture densities medium rate after (NFW and capture; plusρ the it adiabatic luminosity. is contraction) The these dotted densities line that is determine our the fiducial example with Figure 1. The dashed line shows Eddingtonshow luminosity the for data Thompson scattering. points for Triangles the luminosity of zero-metallicity stars from table third column of table line corresponds to a DM density of 10 would dominate over the Eddington luminosity for a star with a mass of would have a DM density of 10 Journal of Cosmology and Astroparticle Physics all relevant stellar masses for upper bound on the stars’ mass if we assume that they must be sub-Eddington. In the We, as yet, knowon little the about structure the of effect zero-metallicity that stars, significant dark matter heating would have excess of this value,star it is will a dark . thewith The DM one two another. heating are that If plotted determines theto in first have stellar figure stars the properties, are properties and observed predictedconstrain (e.g., the by of WIMP the fusion-driven properties James stars, Webb to then Space make Telescope) one sure could that use these results to 4. Eddington luminosity JCAP11(2008)014 , , p χ 13 ρ κ 55 . 1 Edd  (29) (30) M ) >L ∝ DM DM L L . (dotted line), and for these objects, L 2 ) Edd cm L /M 41  − M ( 4 =10 is the mass of the star, and 10 c  σ × M 5 . stacks.iop.org/JCAP/2008/i=11/a=014 ]) to be then constitutes an approximate upper essentially compare luminosity )=3 44 ld address a different problem: one should structure as they accrete mass on their way wo curves will cross for some stellar mass. Edd e first stars’ stellar atmospheres are hot and (solid line), r), and the Eddington luminosity. Since the re we are comparing dark star luminosity to on, the luminosity due to WIMP annihilation /M 2  >L M cm ted by Thompson scattering, so we take ( whereas the DM luminosity scales as 11 (2008) 014 ( 1 DM  39 − − L M is the speed of light, ∝ erg s c =10 , L 38 c  Dark matter capture in the first stars: a power source and limit on stellar mass (dashed line). σ 2 10 p × κ cm Maximum stellar mass due to the Eddington limit as a function of 4 πcGM . 43 4 − =1 = =10 Edd Edd c , we have plotted three luminosities as a function of stellar mass for zero- L Figure 2. for a fixed cross section.cross The sections: different lines correspond to different spin-dependent σ L 1 is Newton’s constant, G In figure The Eddington luminosity is defined (e.g. [ Journal of Cosmology and Astroparticle Physics nearly metal free, the opacity is domina is the opacity of . Since th current paper, we simplyor take not the the ZAMS stars, resultingstars add dark of the this stars DM mass are luminosity, to self-consistent. and exist? ask In We whether compute other the words, Eddington is luminosity it possible for dark where and ascertain whether or not the DM luminosity is in excess of this value. If for a large enough dark matter density the t the Eddington luminosity. The lightest stellar mass for which then a star ofblow this off mass some cannot ofreally exist: the follow mass. the the pressure In and from reality compute the one their shou star would(the be value so of large whichaccretion) as would to to depend the on Eddington the luminosity; nature he of the accretion, e.g. spherical or disk to becoming ZAMSs. Such a calculation would (for a variety of WIMP densities in the sta metallicity stars: the luminosity due to fusi Eddington luminosity scales as JCAP11(2008)014 , 3 .) 3 14 − − max 55 and  . (31) that 1  3 M c − M σ ) ), we get ∝ GeV cm GeV cm 29 18 DM 13 GeV cm . for a variety of L 8 10 . 15 =10 1 illustrates and would lead to − × )and( χ 10 M 3  ρ 2 1 − 28 . 2 × cm =1 c =3 σ 39 χ χ − ρ ρ GeV cm 10 13  .Figure 8 10 . star is Eddington limited by DM 1 × − M  1 . stacks.iop.org/JCAP/2008/i=11/a=014 3 M ]. The three horizontal lines are bounds . − 2 37 =1 nnihilation. If one assumes that accretion cm χ ]–[ ρ χ 44 35 ρ ar of that mass is unable to accrete any further ; and DS III corresponds to − GeV cm 2 ], [ 10 9 ; DS II corresponds to 11 (2008) 014 ( 32 cm 2 × , ellar masses is straightforward since 10 at which a 1 3 41  31 cm χ [ − < ρ 2 39 c 10 M − σ 8 cm ≤ 10 Dark matter capture in the first stars: a power source and limit on stellar mass c 10 38 σ ≤ − × c Bounds on spin-dependent scattering cross section as a function of 1 σ is a plot of WIMP scattering cross section versus WIMP mass. . 3 =1 max  and would lead to Figure 3. WIMP mass fromWIMP experiments, annihilation as well onand as the indirect potential first detection boundsspin-dependent stars. experiments from as cross the Displayed labeled; sections effectat limits the of are around include tightest 10 from present various bounds Super-Kamiokande direct on (labeled Super-K), that would result from the discovery of Pop III stars of 1 M the bound DM densities inside thesewould stars otherwise (due prevent their tocorresponds formation). the to fact The ambient that lines density DM are labeled annihilation pressure as follows: DS I wouldleadto In principle, these arguments can be turned around to place a bound on the scattering For example, we find that for a dark matter density of Journal of Cosmology and Astroparticle Physics Experimental bounds are shown. The horizontal lines indicate the values of as asection. function of WIMP energycross density section. for Figure several values of the scattering cross mass limit for the first stars, because the st due to the radiation pressure from the WIMP a efficiently drives up theuniquely mass determined of by the any properties Pop of III the star, DM. then Using the equations mass ( of the first star will be annihilation. (The scaling to other st Once the mass of the first stars is determined, then one could rule out any combination correspond to different values of the first stars cannot be more massive than 1 JCAP11(2008)014 , 3 15 − .In 3 cm ) 22 10 ∼ n . In this case the Eddington 3 − ill merge with other halos and that stacks.iop.org/JCAP/2008/i=11/a=014 s in this way depends on detailed under- GeV cm rguments. Thes. background density Instead, will for a given DS, the density n which the dark star resides. Clearly this g limits on spin-independent scattering; see for our fiducial scattering cross section. If, r and one may hope someday to address this 18 M 11 (2008) 014 ( 1 =10 ].  χ 37 ρ , ), which would be the tightest known bound on the on WIMPs for almost any mass (the equilibrium DM χ 34 , 2 m Dark matter capture in the first stars: a power source and limit on stellar mass 39 , cm 38 44 Same as the previous figure but for WIMPs with spin-independent − 10 × halos containing the dark stars w 3 for a given WIMP mass that would preclude stars of such a mass from Figure 4. interactions [ < M χ 6 c ρ σ . 4 and Obtaining bounds on the WIMP parameter It is of course true that we do not know exactly the ambient DM around any c σ luminosity is independent of Journal of Cosmology and Astroparticle Physics of standing of the ambientis WIMP a density very difficult withi that problem the which will 10 not be solved in the near future. Indeed, it is likely this extreme case wefigure could also put interestin spin-dependent scattering cross section by several orders of magnitude; see figure limit would be reached for masses forming. As an extremeall example, the if one way were to to believe the that adiabatic limit contraction where occurs the protostellar core has gas density of course notdepends be on the theresides. detailed same evolution and for However,of merger all in history DM dark principle, ofof star the from in halo many the the within simulationsthe DS, which neighborhood one the observations one DS can of withproperties. could obtain what many the perform is dark distribution a learned stars. statistical from comparison the of If simulations the to one DS learn also properties about had WIMP in observations one DS, but one can construct statistical a the stars will notstructure remain formation forever are in becomingquestion. regions ever of bette high DM density. However, simulations of instead, the firstand stars such are anbound observed enormous of to WIMP form density with were masses found larger than to one be solar sensible, mass, one could place a the WIMP density would reach JCAP11(2008)014 16 ) dark matter M 6 10 − 5 stacks.iop.org/JCAP/2008/i=11/a=014 lly affect the structure of the first stars. ill lower the Eddington stellar mass limit nalytic models of adiabatically contracted nto the star runs out as a source of fuel, it expected degree of wandering, the detailed dels of zero-metallicity stars. We also thank matter. And conversely, inferred properties n that when these rates are in equilibrium, rough a region of some radius significantly remain in a high density environment for a al burning time of the dark stars will be left ). The maximal burning time is determined s the time for which a dark star will be able 4 the ambient DM density must be high enough ar how long the central DM in the halo remains 11 (2008) 014 ( ] discussed the importance of dark matter annihilation in Dark matter capture in the first stars: a power source and limit on stellar mass 1 as given by equation ( τ ], suggesting that DM heating may radica 1 The two key uncertainties in this work are: (i) the scattering cross section must be at A potentially important consideration i For high enough DM density, DM heating w Journal of Cosmology and Astroparticle Physics In summary, Pop III stars are expected to reside at the core of a 10 5. Conclusion halo. Previously twothe of first stars, us proposing [ the the DM existence initially of ‘dark collapsing stars’ with powered the by baryons DM i heating. Even once (or near) the experimental limit, and (ii) can be replenished bythrough capture the of dark more star).self-annihilation DM In from in this the such paper, ambient stars. wethe medium have accompanying We (as estimated have heating the the show DM would ratesand passes provide of prolong WIMP an the capture energy darkSuch and source star densities that phase, seem can if plausiblehalos rival the on [ nuclear ambient the fusion dark basis matter of density a is sufficiently high. for sufficient capture towhile take after place. the starlong While has high this formed, enough second we DM wish criterion densities to is last. comment likely further to hereto be on burn the true DM question at forin of the a a how calculated time rate, of given order that it will annihilate its captured store of DM larger than that ofthe the expense star. offiducial somewhat This DM lower makes density average much numbers that DM morebut we density. DM those assume mass are (It for not available shouldstar over those for merges expected some then burning, with in significantly be other the at objects, larger innermost noted it core region.) that is the not Once cle the halo containing the dark by the totalmass mass could of be DMHowever, small we in expect if that the thesome given star region the non-zero were complexities of velocity of fixed phase and the exactly collapse space thus at process, ‘wander’ that the the th intersects star central will the cusp have star. of the This DM halo. undisturbed, and it isin not principle, clear the how darklong long star time the could (even dark until continue now), star to estimation remains of near the the center. While, DM density profile, and therefore the maxim for future and more detailed study. We thank S Woosley for sharing with us his mo Acknowledgments of Pop III starsDM (or masses even and future interaction cross direct sections. observations) might be used to strongly constrain C Church, J Primack,support S from: Profumo the DOE and and MCTP C via Savage the for University of useful Michigan, the discussions. Perimeter Institute We acknowledge to provide an upperformation properties mass of cutoffdetermined Pop for by III Pop the stars, III; particle the properties because of gross DM dark properties heating of might these also objects affect may the well be JCAP11(2008)014 ] ] ] 17 ] Prog. SPIRES [ [astro-ph] ) [astro-ph] ] , 1971 ] ] astro-ph/0509269 hep-ph/9506380 ] 0706.0039 481 Mon. Not. R. Astron. ] ] ] 0708.1883 astro-ph/0408346 344 ] ][arXiv: SPIRES ][arXiv: [ ]arXiv: ] ] JCAP04(2003)001 hep-ph/0404175 ]arXiv: SPIRES astro-ph/0702654 SPIRES ][arXiv: [ [ astro-ph/0501589 J. Cosmol. Astropart. Phys. 7305 ] SPIRES SPIRES [ 87 [ SPIRES SPIRES SPIRES [ 23 [ hep-ph/0701197 ][arXiv: SPIRES ][arXiv: [ 173 SPIRES ] [ ][arXiv: 011302 79 ] 195 555 SPIRES astro-ph/0701858 [ 2079 2nd edn (Berlin: Springer) 96 42 011102 ] 161 ] (erratum) 41 ] 267 021303 [ ] ] ][arXiv: 33 SPIRES 73 SPIRES [ [ SPIRES 804 stacks.iop.org/JCAP/2008/i=11/a=014 [astro-ph] ] ] SPIRES [ D 123513 653 SPIRES [ 100 D [ [astro-ph] ][arXiv: B ] Mon. Not. R. Astron. Soc. D 70 279 SPIRES 279 SPIRES [ SPIRES 891 L29 [ [ ] SPIRES B SPIRES D SPIRES 679 389 [ ] [ [ (Princeton, NJ: Princeton University Press) SPIRES SPIRES [astro-ph] Phys. Rep. 405 [ [ 154 J. Cosmol. Astropart. Phys. 659 296 0705.0521 455 433 SPIRES 1719 [ 29 141 Stellar Interiors 681 1034 Class. Quantum Grav. ] Phys. Rev. ] Phys. Rev. Lett. 27 Phys. Rev. 0711.0991 125 SPIRES 83 [ 570 SPIRES Phys. Rev. Phys. Lett. Dark matter limit plot generator 11 (2008) 014 ( [ 453 508 283 296 301 B 287 Phys. Rev. 349 Nucl. Phys. (Proc. Suppl.) B B Nucl. Phys. Nature 0709.1485 ] 28 SPIRES Phys. Rep. astro-ph/0406204 403 SPIRES [ [ astro-ph/0507130 Astrophys. J. Suppl. Ser. [astro-ph] 083523 Astrophys. J. Astrophys. J. Primordial turbulence and thermal history of the Universe 214 Astrophys. J. 295 70 919 Ann. Rev. Astron. Astrophys. ] B Phys. Rep. Dark matter capture in the first stars: a power source and limit on stellar mass Galactic Dynamics D ][arXiv: [astro-ph] Nucl. Phys. Phys. Rep. Astrophys. J. 167 328 Nucl. Phys. Nucl. Phys. Astrophys. J. Phys. Rev. Lett. B ] ] 0704.0423 ] SPIRES [ SPIRES astro-ph/0504038 [ Astropart. Phys. , 1986 Phys. Lett. 0802.0941 Phys. Rev. 416 (CDMS Collaboration), 2006 et al (CDMS Collaboration), 2006 46 , 2007 Phys. Lett. Astrophys. J. , 1988 (XENON Collaboration), 2008 , 2007 arXiv: , 2004 et al et al 393 [arXiv: et al et al astro-ph/0509259 astro-ph/0511789 hep-ph/0301106 et al et al et al 363 http://dmtools.berkeley.edu/limitplots/ [arXiv: see: Soc. [arXiv: [arXiv: JCAP07(2004)008 Theor. Phys. Griest K and Seckel D, 1988 [1] Spolyar D,[2] Freese K Yoshida and N, Gondolo Abel[3] P, T, 2007 Reed Hernquist D arXiv: L S, and Bower Sugiyama R, N, Frenk C 2003 S, Gao L, Jenkins A, Theuns T and White S D M, 2005 [4] Peebles P J E and Dicke R H, 1968 [8] Ripamonti E[9] and Abel Barkana T, R 2005 and arXiv: Loeb A, 2001 [6] Hollenbach D and McKee C F, 1979 [7] Omukai K and Nishi R, 1998 [5] Matsuda T, Sato H and Takeda H, [41] Press W H and Spergel D N, 1985 [39] Angle J [40] Gould A, 1988 [37] Gaitskill R, Mandic V and Filippini J, 2008 [38] Akerib D S [26] Bertone G,[27] Hooper D Hooper and D Silk[28] and J, Profumo Heger 2005 S, A 2007 [29] and Woosley Work S in E, progress [30] 2008 in Hubert preparation D (IceCube Collaboration), 2007 [33] Alner G J (ZEPLIN-II Collaboration), 2007 [36] Akerib D S [34] Alner G[35] J Lee H S [25] Jungman G, Kamionkowski M and Griest K, 1996 [23] Iocco F,[24] 2008 arXiv: Griest K and Seckel D, 1987 [21] Scott P,[22] Edsjo J Bertone and G Fairbairn and M, Fairbairn 2007 M, arXiv: 2007 arXiv: [20] Moskalenko I V and Wai L L, 2007 [43] Binney J and Tremaine S, 1987 [42] Diemand J, Moore B and Stadel J, 2005 References (KF), NSF grant AST-0507117 andof GAANN the (DS, Physics AA). Department KF at acknowledges the the University hospitality of Utah. [32] Desai S [19] Fargion D, Khlopov M and Stephan C A, 2006 [18] Gondolo P, Edsjo J, Ullio P, Bergstrom L, Schelke M and Baltz E A, 2004 [17] Edsjo J, Schelke M, Ullio P and Gondolo P, 2003 Journal of Cosmology and Astroparticle Physics [44] Hansen C J, Kawaler S D and Trimble V, 2004 [45] Blumenthal G R [31] Savage C, Gondolo P and Freese K, 2004 [10] Bromm V and Larson R B, 2004 [15] Krauss L M, Srednicki M and Wilczek F, 1986 [11] Ellis J R [16] Moroi T and Randall L, 2000 [13] Srednicki M,[14] Olive K Freese A K, and 1986 Silk J, 1987 [12] Gondolo P and Silk J, 1999 JCAP11(2008)014 18 ) ] ] astro-ph/0507589 SPIRES [ ][arXiv: 563 Mon. Not. R. Astron. Soc. 462 SPIRES [ vol 171, p 175 stacks.iop.org/JCAP/2008/i=11/a=014 70 93 L9 634 295 537 449 Astrophys. J. ] 378 11 (2008) 014 ( Science SPIRES [ Astrophys. J. L25 ] Dark matter capture in the first stars: a power source and limit on stellar mass Astrophys. J. Lett. 447 Mon. Not. R. Astron. Soc. astro-ph/0403154 IAU Symposium/Symp-Int. Astron. Union Astrophys. J. , 2007 [arXiv: et al 903 351 Burkert A, 1995 Journal of Cosmology and Astroparticle Physics [47] Burkert A, 1996 [46] Navarro J F, Frenk C S and White S D M, 1996 [49] Gentile G, Salucci P, Klein U, Vergani D and Kalberla P, 2004 [52] Sellwood J A and McGaugh S S, 2005 [48] Salucci P and Burkert A, 2000 [50] Abel T, Bryan G L and Norman M L, 2002 [51] Gao L