On (Not)-Constraining Heavy Asymmetric Bosonic Dark Matter
aa r X i v : . [ a s t r o - ph . H E ] D ec On (Not)-Constraining Heavy Asymmetric Bosonic Dark Matter
Chris
Kouvaris ∗ CP -Origins, University of Southern Denmark andDanish Institute for Advanced Study, Campusvej 55, Odense 5230, Denmark Peter
Tinyakov † Service de Physique Th´eorique, Universit´e Libre de Bruxelles, 1050 Brussels, Belgium
Recently, constraints on bosonic asymmetric dark matter have been imposed based on the ex-istence of old neutron stars excluding the dark matter masses in the range from ∼ Preprint: CP -Origins-2012-034 & DIAS-2012-35. PACS numbers: 95.35.+d 95.30.Cq
The last few years stellar observations have been usedin order to constrain specific dark matter candidates orpredict interesting phenomena related to dark matter [1–13]. Specifically, compact stars such as white dwarfsand neutron stars have been found to impose severe con-straints on some dark matter models [14–27]. In princi-ple, there are two types of effects that can take place incompact stars and can give rise to constraints on darkmatter. The first type is related to the thermal evolutionof compact stars [15, 16, 19, 20]. In this case, annihilationof trapped weakly interacting massive particles (WIMPs)inside a compact star can produce significant amount ofheat that can change the thermal evolution of the star atlater times. As a result, stars old enough to be quite coldmight maintain higher temperature due to the releasedheat.The second type of constraints is related to asymmet-ric dark matter [14, 21, 23–27]. In this case WIMPs carrya conserved quantum number and there is an asymme-try between the populations of WIMPs and anti-WIMPs,so that the annihilation is impossible in the present-dayuniverse where only the WIMPs remain. Such kind ofWIMPs can accumulate in a compact star quite effi-ciently as long as the WIMP-nucleon cross section ex-ceeds a certain critical value which in the case of a neu-tron star is quite small ( σ ∼ − cm ) [15], severalorders of magnitude smaller than current limits from di-rect searches. Under certain circumstances, the amountof WIMPs accumulated during the lifetime of the staris sufficient to result in a gravitational collapse of thethe WIMPs into a black hole. In the case of fermionic ∗ Electronic address: [email protected] † Electronic address: [email protected] dark matter the amount of WIMPs needed for collapse is ∼ M /m , M pl being the Planck mass and m the massof the WIMP. Unless one assumes very heavy WIMPsand/or extremely high background dark matter densityat the location of a neutron star, this amount of WIMPsis difficult to accumulate during the lifetime of the star(although the situation may change in the presence ofattractive self-interactions [25]).On the contrary, if WIMPs are of bosonic nature, thereis no Fermi pressure and the amount of WIMPs neededfor gravitational collapse is significantly lower [28], M > M min = 2 M πm = 9 . × GeV (cid:16) m (cid:17) − . (1)This amount of WIMPs can be easily accumulated evenby nearby known old neutron stars with the standardassumption about dark matter density near the Earth.As it was pointed out in [23], once ∼ WIMPs havebeen accreted and thermalized within the neutron star,a Bose-Einstein condensate (BEC) forms and all newlyaccreted WIMPs fall into the ground state. This stateis very compact, so the WIMPs in the condensate startto self-gravitate way before the condition of Eq. (1) issatisfied. Thus, once the amount of WIMPs in the con-densate meets the criterion for the gravitational collapse(Eq. (1)), they form a black hole. Note that only a frac-tion of all WIMPs (namely, those in the BEC) goes into ablack hole. If the resulting black hole starts to grow andcan destroy the host star in a reasonable time, constraintson dark matter arise as follows from the well known ex-istence of old (more than a few billion years old) neutronstars.The fate of the black hole inside the neutron star is de-termined by the two competing processes: the accretionof surrounding nuclear matter that increases the mass ofthe black hole and scales as M ( M BH being the blackhole mass), and Hawking radiation that scales as M − .Therefore, there exists a critical value M crit such thatheavier black holes grow until the whole star is consumed,while lighter black holes evaporate completely. It followsfrom Eq. (1) that the mass of the black hole formed bythis mechanism is inversely proportional to the WIMPmass, so that any black hole formed from WIMPs heav-ier than a few GeV is lighter than M crit and evaporatescompletely, in which case no constraints arise.In Refs. [24, 26, 27] it was suggested that for WIMPsheavier than a few TeV the above mechanism is modifiedand the constraints are recovered. The argument makesuse of the observation that for sufficiently heavy WIMPs,the self-gravitation of the WIMP sphere sets in beforethe formation of BEC. At this point the total amount ofaccumulated WIMPs exceeds by many orders the amountof Eq. (1) required for the black hole formation, as wellas the critical value M crit . So if one assumes that thecollapse happens to the whole WIMP sphere at once theresulting black hole is heavy enough to grow and destroythe star. Hence the constraints reappear.In this paper we demonstrate that this is not whathappens. Although it is true that for heavy WIMPs theself-gravitation sets in without BEC formation, the col-lapse of the self-gravitating WIMP sphere is hamperedby the released gravitational energy and happens on atimescale set by the cooling of the WIMPs. As the WIMPsphere gradually shrinks, its density increases and BECforms on a timescale much shorter than the cooling time.From this stage on, the original scenario is reproduced:the BEC region grows, becomes self-gravitating and col-lapses into a black hole as soon as Eq. (1) is satisfied.The resulting black hole is too small to grow. Instead,it evaporates on a time scale much shorter than it takesfor the rest of the WIMP sphere to collapse. The netresult is the formation of tiny black holes one after theother which evaporate shortly after their birth withoutproducing a heavy stable black hole. Thus no constraintson heavy WIMPs arise.Let us consider this argument more quantitatively.The WIMPs become self-gravitating when their densitybecomes larger than that of the nuclear matter in thecore of the neutron star. This happens when the totalmass of the WIMPs exceeds M sg = 4 π r ρ c =2 . × GeV (cid:18) m (cid:19) / (cid:18) T K (cid:19) / , (2)where r th ≃ . (cid:18) T c K (cid:19) / (cid:16) m (cid:17) − / , (3)is the thermal radius of the WIMP sphere inside the neu-tron star at temperature T and ρ c = 5 × GeV/cm is the star core density. The density of WIMPs requiredto form BEC is [23] ρ crit ∼ . × GeV cm − (cid:16) m (cid:17) / (cid:18) T K (cid:19) / , (4)which implies that the total WIMP mass necessary forthe formation of BEC is M BEC = 2 . × GeV (cid:16) m (cid:17) (cid:18) T K (cid:19) . (5)From the dependence of Eqs. (2) and (5) on the WIMPmass m one can see that for m &
10 TeV the self-gravitation sets in prior to the formation of BEC. Notethat the total mass of the WIMPs in this case is largeenough to overcome the constraint imposed by the un-certainty principle of Eq. (1).Once the self self-gravitation sets in, the WIMP spherestarts to collapse. However, it cannot collapse directlyinto a black hole because the WIMPs have to lose theirenergy and momentum. Instead, the collapse of WIMPsis likely to resemble the formation of dark matter halos,with the essential difference that constant interactionswith nucleons provide an extra energy loss mechanismfor the WIMPs. One may expect, therefore, that theWIMPs develop a cuspy profile similar to the dark matterhalos, which shrinks as the WIMPs lose their energy ininteractions with nucleons.Even without the cusp, the shrinking WIMP spherewould form BEC long before it reaches the size compa-rable to its Schwarzschild radius. To make the argu-ment as robust as possible, consider the worst (unreal-istic) case where the WIMP sphere contracts maintain-ing a uniform density. In order to form a black hole ofmass M the WIMPs have to reach the density ρ BH ∼ πG M ) − ∼ GeV/cm ( M/ GeV) − , whilethe density required for BEC formation is much lower,c.f. Eq. (4). Note that the smaller is the black hole, thehigher density is required, so the uniform contraction isindeed the worst case. Thus, BEC will unavoidably beformed before the WIMPs collapse into a black hole.Consider now the formation of BEC in some more de-tail. Once the density of Eq. (4) is reached, the formationof BEC happens on time scales of order [29] t BEC ∼ ~ k B T ∼ − s , (6)i.e. practically instantaneously. Further shrinking ofthe WIMP sphere results in increasing the mass of thecondensate rather than the density of non-condensedWIMPs.The size of the condensate can be estimated by not-ing that once the WIMPs become self-gravitating theydominate the gravitational potential. Taking their den-sity approximately constant and equal to that requiredfor the condensate formation gives r BEC = (cid:18) π Gρ crit m (cid:19) − / = 1 . × − cm (cid:16) m (cid:17) − / (cid:18) T K (cid:19) − / . (7)This is much smaller than the size of the WIMP sphere.As the mass of the BEC grows, it eventually becomesself-gravitating itself. This happens when the densityof the condensate becomes equal to the non-condensedWIMP density, i.e., when its mass exceeds M BEC , sg = 4 π ρ crit r = 9 . × GeV (cid:16) m (cid:17) − / . (8)Although self-gravitating, the condensate cannot yet col-lapse because it does not satisfy the constraint of Eq. (1).The latter is satisfied when the total mass of the con-densed state reaches M min . Beyond that point stableconfigurations of the self-gravitating condensate do notexist and it collapses into a black hole [28]. The resultingmass of the black hole is given by Eq. (1). Note that thismass is much smaller than the total mass of the WIMP-sphere which is of order M sg , Eq. (2). Note also that theblack hole mass becomes smaller as the mass of the darkmatter particle, m , increases.The black hole of mass M min is too small to survivethe Hawking radiation. It evaporated on the time scale τ = 5 × s (cid:18) m (cid:19) . (9)To conclude the argument, let us show that such burn-ing of dark matter by the formation and evaporation ofsmall black holes is more efficient than both the accretionof new dark matter onto the neutron star and the creationof new black holes by the above mechanism. To this endconsider the relevant time scales. The accumulation ofWIMPs by the neutron star proceeds at a constant rate F that depends on the local dark matter density ρ dm andequals [19] F = 1 . × GeV / s (cid:18) ρ dm GeV / cm (cid:19) . (10)The time needed to accumulate the amount of dark mat-ter equal to M min is t acc = 7 . × s (cid:16) m (cid:17) − (cid:18) ρ dm GeV / cm (cid:19) − . (11)This is larger by an order of magnitude than the evapo-ration time even for m = 10 TeV and even considering a10 GeV / cm local dark matter density. For nearby neu-tron stars where the dark matter constraints are morereliable, this will give 5 orders of magnitude higher for10 TeV WIMP mass. Consider now the time needed to form a new black holein the course of collapse of the self-gravitating WIMP-sphere. According to the picture described above, thecontracting WIMP-sphere reaches at some point the den-sity ρ crit at which the formation of the BEC becomes pos-sible. Further contraction happens at a constant WIMPdensity ρ = ρ crit as the excess of the WIMPs goes into theBEC state. To form a black hole, the amount of WIMPsequal M min has to be moved from the WIMP-sphere tothe BEC. The corresponding release of potential energyis equal to the black hole mass times the difference ofthe gravitational potential between the surface and thecenter of the sphere, δE = 12 GM M min r , (12)where M is the total WIMP mass and r is the radius ofthe WIMP-sphere, r = (cid:18) M sg πρ crit (cid:19) / = 2 . (cid:18) m (cid:19) / . (13)The released energy has to be dispersed via collisions ofWIMPs with nucleons, which sets the time scale of theprocess. Taking into account the Pauli blocking factor,the WIMP-nucleon collision time is τ col = 1 nσv p F m N v = 2 p F m ρ c σǫ , (14)where p F is the Fermi momentum of the nucleons, m N ≃ v is the WIMP velocity and ǫ is the WIMP kinetic energy estimated as ǫ ∼ GM m r (15)from the virial theorem. Taking into account that thetypical energy loss per collision is δǫ = 2 m N ǫ/m andassembling all the factors, the time required to lose theexcessive energy and form a black hole is t cool = τ col mδEM δǫ = 43 π p F m N r M ρ c σM . (16)One can see that this time is shorter for larger mass M .Substituting M = M sg into Eq. (16) one gets t cool ≃ . × s × (cid:16) m (cid:17) / (cid:18) T K (cid:19) − (cid:16) σ − cm (cid:17) − . (17)This time is shorter by a factor of a few than the blackhole evaporation time of Eq. (9). Note, however, thestrong dependence of both quantities on the WIMP mass m . Already for masses m &
13 TeV the black hole evap-oration time becomes shorter.Three comments are in order. The first comment isrelated to the black hole evaporation. The fate of a blackhole of mass M BH is dictated by the equation dM BH dt = CM − fM , (18)where the first term in the left hand side corresponds tothe Bondi accretion, and the second term corresponds tothe Hawking radiation. Here C = 4 πλρ c G /c s , λ be-ing a constant of order one and c s the speed of sound.As we mentioned earlier, using Eq. (1) one can find thatfor WIMP masses above roughly 10 GeV, Hawking ra-diation wins over accretion and the evaporation time isgiven by Eq. (9). However, the newly formed black holecan also accrete from the dark matter population that, inthe case examined here, can have densities significantlyhigher than the nuclear matter density ρ c , as it can beseen from Eq. (4). The accretion of non-interacting colli-sionless particles in the non-relativistic limit onto a blackhole is given by [30] F = 16 πG M ρ dm v ∞ , (19)where ρ dm is ρ crit of Eq. (4), and v ∞ is the averageWIMP velocity far away from the black hole. Using thevirial theorem, we take v ∞ = p GM/r , M being the to-tal mass of the WIMP sphere and r the radius of theWIMP sphere. It is understood that for the first blackhole M = M sg , and r = r . We have checked that forWIMP sphere masses ranging from Eq. (1) to Eq. (2)the Hawking radiation dominates overwhelmingly overthe WIMP accretion despite the large WIMP density,and therefore the black hole evaporation time is givenaccurately by Eq. (9).The second comment is that the cooling time derivedabove refers to the formation of the first black holeof mass given by Eq. (1). Subsequent black holes re-quire progressively longer times. This is easily seen fromEq. (16) because t cool scales as 1 /M / (recall that r scales as M / from Eq. (13)). Thus, the more blackholes have formed and evaporated, the smaller is the re-maining WIMP mass M and the longer the time neededto form the next one.The third comment has to do with the neutron startemperature. As one can see in Eq. (17), there is a strongdependence in T . So it is essential that 10 K is a goodestimate for the temperature of the neutron star at thetime the star has accumulated M sg mass of WIMPs. Notethat M sg depends on the temperature, cf. Eq. (2). Theneutron star does not always has the same temperature,but becomes colder with time due to different coolingmechanisms. The cooling curves (i.e. the temperature ofthe star as a function of the age) for typical neutron starscan be found, e.g. in [31] and [15]. Knowing the cool-ing curves, M sg ( T ) and the WIMP accretion rate fromEq. (10), one can easily estimate the time it takes for thestar to accumulate a total WIMP mass of M sg . We found that in all cases of interest (i.e. for m ranging from 10 to1000 TeV and ρ dm from 0.3 to 10 GeV / cm ), the accu-mulation time is at least ∼ . × years. By that timethe temperature of the star should be below 10 K, whichjustifies the use of this temperature in our estimates.Finally, let us examine one more factor that can poten-tially affect our estimates. As the first black hole evapo-rates, it emits particles via Hawking radiation that reheatthe core of the star. In principle such a deposit of heatcan increase the temperature of nucleons and WIMPsand thus could potentially change our estimates. Weshow here that the increase in the temperature is smalland the whole effect is negligible.The energy produced by the black hole evaporation,once transferred to nucleons, propagates according to thediffusion equation, ∂u∂t − D ∇ u = Q, (20)where u is the energy density due to the black hole evap-oration, D is the diffusion coefficient and Q is the energyinjection rate.Since the black hole is tiny and the mean freepath of the Hawking-emitted particles is short ( ∼ − cm assuming a typical nucleon cross section σ N ∼ π − cm ), one may take Q to be a delta-function inspace, Q = Q ( t ) δ ( ~x ). We will also assume that the en-ergy is injected at a constant rate. As we will see below,the time scale associated with the heat conductivity ofnuclear matter is extremely short, so a constant rate isa good approximation for most stages of the black holeevaporation.The diffusion coefficient D = κ/c V can be expressedin terms of the specific heat capacity c V = 1 . × − GeV ( T / K) of the nuclear matter [32] and thethermal conductivity k . The latter can also be found inRef. [32], k ∼ GeV .Assuming that the energy release by the black holestarts at t = 0, the solution to the above diffusion equa-tion is u ( r, t ) = Q πDr erfc (cid:18) r √ Dt (cid:19) . (21)The time-dependent factor saturates to one as soon as r/ √ Dt ≪ t st = r D ≃ . × − s (cid:18) m (cid:19) / (cid:18) T K (cid:19) . (22)This time scale is much shorter than t cool for all casesof interest. The same time scale determines the onset ofthe equilibrium once the heat source is switched off (theblack hole has evaporated).When the stationary solution is reached, the tempera-ture of the nuclear matter is given by T = Q πkr + T ∞ , (23)where T ∞ = 10 K. The first term describes the effect ofthe evaporating black hole.At distances comparable to the size of the WIMP-sphere r ∼ r , the change in the temperature during theblack hole evaporation is ∼
10 K ( m/
10 TeV) , where wehave estimated Q by dividing the black hole mass by itsevaporation time. However, in cases where t cool is longerthan the evaporation time, it is more appropriate to es-timate the value of Q by dividing the mass of the blackhole by the t cool of Eq. (17). In this case, we find anincrease in the temperature at most δT . (cid:16) m
10 TeV (cid:17) − / . (24)Thus, the effect of the black hole evaporation on the tem-perature in the center of the neutron star is completelynegligible. I. CONCLUSIONS
We demonstrated that it is not possible to obtainconstraints on asymmetric bosonic dark matter withmasses in the TeV range or higher from the collapse ofWIMPs into black holes inside neutron stars and de- struction of the latter. Although the self-gravitationof the accumulated WIMPs starts before the BEC for-mation, we showed that the WIMP sphere unavoidablypasses through the stage of the BEC formation as it con-tracts. As a result, black holes with masses much smallerthan the total WIMP mass form and evaporate in timesshorter than both the accretion time scale and the WIMPsphere cooling time scale. This means that the WIMPsphere does not collapse to a single large black hole whichwould grow and consume the host star, but instead col-lapses piece by piece in such a way that every black holethat forms evaporates before another gets formed. Theoverall effect is the (unobservable) heating of the neutronstar, but not its destruction. Thus no constraints on theTeV WIMP mass range can be imposed.We would like to stress that the scenario we have stud-ied is the most conservative one. We assumed a uniformconstant density for the WIMP sphere. In reality, oneshould expect a more cuspy profile towards the center ofthe WIMP sphere. As a result, ρ BEC will be achievedearlier, and the cooling time for formation of subsequentblack holes can increase substantially because the earlierthe BEC forms, the smaller will be the energy loss ofWIMPs colliding to nucleons due to smaller differencesin the temperatures of the two species. [1] J. Casanellas and I. Lopes, Astrophys. J. , 135 (2009)[arXiv:0909.1971 [astro-ph.CO]].[2] M. A. Perez-Garcia, J. Silk and J. R. Stone, Phys.Rev. Lett. , 141101 (2010) [arXiv:1007.1421 [astro-ph.CO]].[3] J. Casanellas and I. Lopes, Mon. Not. Roy. Astron. Soc. , 535 (2011) [arXiv:1008.0646 [astro-ph.CO]].[4] I. Lopes, J. Casanellas and D. Eugenio, Phys. Rev. D ,063521 (2011) [arXiv:1102.2907 [astro-ph.CO]].[5] J. Casanellas and I. Lopes, Astrophys. J. , L51 (2011)[arXiv:1104.5465 [astro-ph.SR]].[6] J. Casanellas, P. Pani, I. Lopes and V. Cardoso, Astro-phys. J. , 15 (2012) [arXiv:1109.0249 [astro-ph.SR]].[7] M. A. Perez-Garcia and J. Silk, Phys. Lett. B , 6(2012) [arXiv:1111.2275 [astro-ph.CO]].[8] M. A. Perez-Garcia, J. Silk and J. R. Stone, AIP Conf.Proc. , 525 (2012) [arXiv:1108.5206 [astro-ph.CO]].[9] L. Brayeur and P. Tinyakov, Phys. Rev. Lett. ,061301 (2012) [arXiv:1111.3205 [astro-ph.CO]].[10] F. Iocco, M. Taoso, F. Leclercq and G. Meynet, Phys.Rev. Lett. , 061301 (2012) [arXiv:1201.5387 [astro-ph.SR]].[11] C. J. Horowitz, arXiv:1205.3541 [astro-ph.HE].[12] F. Capela, M. Pshirkov and P. Tinyakov,arXiv:1209.6021 [astro-ph.CO].[13] I. Lopes and J. Silk, Astrophys. J. , 130 (2012)[arXiv:1209.3631 [astro-ph.SR]].[14] I. Goldman and S. Nussinov, Phys. Rev. D , 3221(1989).[15] C. Kouvaris, Phys. Rev. D , 023006 (2008)[arXiv:0708.2362 [astro-ph]]. [16] G. Bertone and M. Fairbairn, Phys. Rev. D , 043515(2008) [arXiv:0709.1485 [astro-ph]].[17] F. Sandin and P. Ciarcelluti, Astropart. Phys. , 278(2009) [arXiv:0809.2942 [astro-ph]].[18] M. McCullough and M. Fairbairn, Phys. Rev. D ,083520 (2010) [arXiv:1001.2737 [hep-ph]].[19] C. Kouvaris and P. Tinyakov, Phys. Rev. D , 063531(2010) [arXiv:1004.0586 [astro-ph.GA]].[20] A. de Lavallaz and M. Fairbairn, Phys. Rev. D , 123521(2010) [arXiv:1004.0629 [astro-ph.GA]].[21] C. Kouvaris and P. Tinyakov, Phys. Rev. D , 083512(2011) [arXiv:1012.2039 [astro-ph.HE]].[22] P. Ciarcelluti and F. Sandin, Phys. Lett. B , 19 (2011)[arXiv:1005.0857 [astro-ph.HE]].[23] C. Kouvaris and P. Tinyakov, Phys. Rev. Lett. ,091301 (2011) [arXiv:1104.0382 [astro-ph.CO]].[24] S. D. McDermott, H. -B. Yu and K. M. Zurek, Phys. Rev.D , 023519 (2012) [arXiv:1103.5472 [hep-ph]].[25] C. Kouvaris, Phys. Rev. Lett. , 191301 (2012)[arXiv:1111.4364 [astro-ph.CO]].[26] T. Guver, A. E. Erkoca, M. H. Reno and I. Sarcevic,arXiv:1201.2400 [hep-ph].[27] Y. -z. Fan, R. -z. Yang and J. Chang, arXiv:1204.2564[astro-ph.HE].[28] E. W. Mielke and F. E. Schunck, Nucl. Phys. B , 185(2000) [gr-qc/0001061].[29] H. T. C. Stoof, Phys. Rev. A , 8398 (1992).[30] Ya.B. Zeldovich and I.D. Novikov in Relativistic As-trophysics,vol. 1, The University of Chicago Press,Chicago,1972[31] D. G. Yakovlev, O. Y. Gnedin, M. E. Gusakov, A. D. Kaminker, K. P. Levenfish and A. Y. Potekhin,Nucl. Phys. A , 590 (2005) [astro-ph/0409751].[32] J. M. Lattimer, K. A. van Riper, M. Prakash and M. Prakash, Astrophys. J.425