Gravitationally Focused Dark Matter Around Compact Stars
GGravitationally Focused Dark Matter Around Compact Stars
Benjamin C. Bromley
Department of Physics & Astronomy, University of Utah,115 S 1400 E, Rm 201, Salt Lake City, UT 84112 [email protected]
ABSTRACT
If dark matter self-annihilates then it may produce an observable signal whenits density is high. The details depend on the intrinsic properties of dark mat-ter and how it clusters in space. For example, the density profile of some darkmatter candidates may rise steeply enough toward the Galactic Center that self-annihilation may produce detectable γ -ray emission. Here, we discuss the possi-bility that an annihilation signal may arise near a compact object (e.g., neutronstar or black hole) even when the density of dark matter in the neighborhoodof the object is uniform. Gravitational focusing produces a local enhancementof density, with a profile that falls off approximately as the inverse square-rootof distance from the compact star. While geometric dilution may overwhelmthe annihilation signal from this local enhancement, magnetic fields tied to thecompact object can increase the signal’s contrast relative to the background.
1. Introduction
Dark matter accounts for the majority of the mass in the Universe, yet its identityremains elusive. Candidates include weakly interacting massive particles (WIMPs) like theneutralino ( χ ), the supersymmetric partner of the neutrino (Pagels & Primack 1982), al-though their properties are only loosely constrained by theory and experiment. In somecases, plausible values of the mass and cross section suggest that self-annihilation signaturesmay be detectable in regions where the density of dark matter is high (Berezinsky et al. 1992;Bergstr¨om & Gondolo 1996; Bertone et al. 2004). For example, Bergstr¨om et al. (1998) cal-culate the gamma ray flux from neutralino self-annihilation in the Galactic center (see alsoZaharijas & Hooper 2006), while Tyler (2002) and Bergstr¨om & Hooper (2006) provide es-timates of the annihilation signal from the nearby Draco dwarf galaxy. Intermediate-massblack holes may yield a WIMP annihilation signal (Bertone et al. 2005), as may remnant dark a r X i v : . [ a s t r o - ph . H E ] D ec ρ ∼ r − γ in many astrophysical contexts. The power-law index γ is between 1 and 2 in the central regions of galaxies according to cosmological simulations(Navarro et al. 1996; Moore et al. 1999; Power et al. 2003), corresponding to a density “cusp.”If the simulations realistically describe the distribution of dark matter, then self-annihilationof WIMPS may indeed be observable in the centers of galaxies.The presence of massive black holes in the centers of galaxies may further enhance thesteep rise of a dark matter density profile. Gondolo & Silk (1999) model the adiabatic growthof a central black hole to show that a density “spike,” with a profile steeper than r − , formsaround the black hole. Dynamical processes, such as scattering by stars and capture theblack hole, may erode a spike over time, although it may remain largely intact (Bertone &Merritt 2005). More problematic is the inspiral of smaller massive black holes captured fromaccreted galaxies that may disrupt a density spike (Merritt et al. 2002). Binary black holecoalescence can destroy density structures in the vicinity of the central black hole. However,even if merger event disrupts a density spike, relaxation processes may regenerate a darkmatter “crest,” with a density profile that falls off as r − . (Merritt et al. 2007). The timescale for the growth of a dark matter crest can be long, ∼
10 Gyr in the case of the MilkyWay, so that its structure could reflect the merger history of the Galaxy.From an observational perspective, strong density cusps or spikes are not obviouslycommon. For instance, Kravtsov et al. (1998) find that dwarf and low surface brightnessgalaxies have shallow core profiles, with γ < .
5, consistent with results from van denBosch & Swaters (2001) in a study of dwarf galaxies. Milosavljevi´c et al. (2002) identify ananticorrelation between profile steepness and mass of the central black hole, lending supportfor a scenario in which the black hole grows through a sequence of merger events which tendto reduce or destroy any density cusp.Even if no density cusp or spike exists in a galactic nucleus, the presence of a massiveblack hole, such as Sgr A ∗ in the center of the Milky Way (Melia & Falcke 2001), can facilitateannihilation radiation. The reason is that gravitational focusing inevitably leads to a darkmatter density enhancement near the black hole. This phenomenon will occur for any starin a field of dark matter (Danby & Camm 1957; Griest 1988; Sikivie & Wick 2002; Alenazi& Gondolo 2006). The purpose of this paper is to review the gravitational focusing effectand discuss its observational consequences in terms of dark matter annihilation radiation. 3 –
2. Gravitational focusing
A point particle with mass M in a uniform bath of hot, dissipationless dark matterwill influence dark matter orbits, but in a manner which conserves phase-space densityalong each trajectory. Danby & Camm (1957) derived analytical expressions for the densityenhancement near the point mass in a bath of unbound particles with a Maxwellian velocitydistribution with velocity dispersion σ v . For a point mass which is at rest with respect tothe bulk dark matter, the density enchancement is ρρ ∞ = (cid:114) π q + e q / (cid:104) − erf( q/ √ (cid:105) , (1)where q = (cid:112) GM/σ v r is the ratio of escape velocity at distance r from the point mass. Thus,close to the point mass the density enhancement grows with decreasing radius according to r − . . Note that details of Dandy & Camm’s results have been contested by Griest (1988),and more recently by Sikivie & Wick (2002). The final word comes from Alenazi & Gondolo(2006), who resolve the discrepancies.To support the previous analytical work, and to provide a framework for studyinggravitational focusing about an arbitrary mass distribution, I developed a computer code totrack the flow of dark matter about a compact star. The code calculates Monte Carlo orbitsassuming that the dark matter has a Maxwellian velocity distribution at large distance fromthe star, and that the star may have some finite speed relative to the average rest frameof the dark matter. The resulting simulations show that the density enhancement fromgravitational focusing can exceed two orders of magnitude near the surface of a neutron star,and even higher within a factor of a few times the Schwarzschild radius of a black hole. Here,to account for the possibility that a dark matter particle may be captured by the black hole,the code does not track orbits inside of five times the Schwarzschild radius (see Gondolo& Silk 1999). The measure of particle trajectories that reach inside this distance is small,and numerical simulations demonstrate that these orbits do not greatly affect the densityenhancement at larger radii. Thus analytical calculations for point particles are broadlyapplicable to compact objects.Figure 1 illustrates the density enhancement at the Galactic center for a background darkmatter halo distribution similar to one calculated by Merritt et al. (2002). The backgroundmodel has a profile that falls off as r − . , as predicted by numerical simulations (Navarro etal. 1996). A “core” has been imposed to crudely mimic the effects of dark matter clearing byblack hole mergers of Merritt et al. (2002). To calculate the effect of gravitational focusing bySgr A ∗ , a black hole mass of 3 × M (cid:12) is assumed, and the velocity distribution is assignedto be isotropic with a dispersion of σ v = 155 km/s. The black hole mass, the assumption 4 –of an isotropic velocity distribution, and the value of σ v used here are all approximatelyconsistent with observed stellar kinematics in the Galactic center (Ghez et al. 1998; Genzelet al. 2000).The observed dark matter self-annihilation flux depend on the local emissivity, j = Y (cid:104) σ ann v (cid:105) ρ πm (2)where ρ dm /m dm is the dark matter number density, and (cid:104) σ ann v (cid:105) gives the self-annihilationrate per unit density. For the neutralino, typical values from the literature (e.g. Bergstr¨om& Gondolo 1996) are m χ ≡ m dm = 100 GeV, and (cid:104) σ ann v (cid:105) = 10 − cm /s, independentof pairwise closing speed v . The quantity Y specifies the yield of decay by-products; forexample, the bolometric yield corresponds to Y = m χ c . The emissivity per unit frequencyof photons produced by electrons in a magnetic field depends on electron-positron productionchannels, as well as synchrotron radiative efficiencies Tyler (e.g., 2002).In the case of certain neutralino decay products, namely neutrinos, the flux is a straight-forward line-of-sight integral over the emissivity, since self-absorption and diffusion do notoccur (Gondolo & Silk 1999). The line of sight integral along some sky direction ˆ n is conve-niently expressed in dimensionless form as (Bergstr¨om et al. 1998; Merritt et al. 2002) J (ˆ n ) = 18 . (cid:18) . / cm (cid:19) (cid:90) ˆ n d(cid:96) ρ . (3)Figure 2 gives J , averaged inside a circular aperture centered on Sgr A ∗ , as a function ofaperture radius. The density profile is the same as in Figure 1. The enhancement fromgravitational focusing is significant inside small apertures. Even so, Bertone et al. (2004)point out that the neutrino flux from the Galactic Center will be undetectable if the currentgamma-ray constraints are any indication of the annihilation rate. Erkoca et al. (2010) aremore hopeful from a theoretical perspective, while the observations are providing limits tothe neutrino flux (e.g., from IceCube Abbasi et al. 2011), but no Galactic Center signal atthis point.
3. Implications
The strength of a gravitationally focused density profile around a compact object isunfortunately insufficient to generally produce a strong flux enhancement above that fromthe background density. In a uniform bath of dark matter the flux only grows logarithmicallywith decreasing radius, and is much weaker than in the case of the density spike envisioned by 5 –Gondolo & Silk (1999). Still, gravitational focusing may have astrophysical relevance if theflux is enhanced by any distinctive properties of the environment around the compact object.This situation may occur in the case of synchrotron radiation from a region around a blackhole or neutron star. For supermassive black holes, magnetic field strengths can be ordersof magnitude higher than in the interstellar medium. Just as the gravitational focusing ispinned to the black hole, so is the magnetic field, as it is likely tied to gas accreting on tothe black hole.An estimate of the synchrotron flux from Sgr A ∗ follows from the density profile ofFigure 1 and an equipartition model for the magnetic field strength (Melia 1992), in which B = ( r/ pc) − . mG. The value for electron-positron yield may be obtained as in Tyler(2002); in the low energy limit, the electron energy distribution per neutralino annihilationis dN/dE ≈ E − . . An integration of the radiative transfer equation should include thepossible effects of synchrotron self-absorption. Synchrotron limits have been used by Bertoneet al. (2001) to place constraints on dark matter properties. However, these limits depend ona strong flux generated in a density cusp and spike, along with a relatively strong magneticfield, all of which must be centered on Sgr A ∗ . According to the results of Merritt et al.(2002), even a relatively minor merger between the Milky Way and a small galaxy can greatlyreduce any annihilation signal. For example, if the Milky Way were to accrete a galaxy witha black hole whose mass is one tenth of Sgr A ∗ , then a strong density enhancement nearSgr A ∗ would be destroyed.On the other hand, the more modest density peak produced by gravitational focusingwill persist, even after a merger event, presuming that dark matter particle orbits in thevicinity of Sgr A ∗ are randomized. Gravitational focusing will also occur around other,smaller compact objects in Galaxy. Consider, for example, a neutron star moving withconsiderable velocity through a thermal bath of dark matter. Figure 3 shows the densityenhancement along the star’s direction of motion, and Figure 4 illustrates the intensity ofannihilation radiation from material in a plane containing the path of the star. The signalis strongest behind the star, and weaker in front, but the angle-averaged signal strength isnot greatly different from the case where the star is at rest in the bulk dark matter frame.A neutron star is an improbable source of annihilation radiation in the form of syn-chrotron photons; the volume where there is significant gravitational focusing is too small tobe observable in the case of the neutralino. The physics is nonetheless remarkable. Neutralinoself-annihilation through a charged pion channel in a 10 G magnetic field will produce apion synchrotron signal, since the synchrotron cooling rate is fast ( < − s) compared withthe pion lifetime ( > − s). Furthermore, quantum effects (Baring 1989) will be importantin the case of a magnetar, with field strength in excess of 10 G. Then, the emitted photons 6 –will have energies comparable to the typical pion energy.
4. Conclusion
Gravitational focusing will generally enhance the annihilation signal around any massivecompact object. However, because of geometrical dilution, this signal may not be distin-guishable from the background annihilation radiation. Fortunately, local conditions arounda compact object may cause the annihilation signal to appear in wavebands which are differ-ent from the background. For example, magnetic fields tied to compact objects can generatea distinctive synchrotron radiation. Of the sources considered here—neutron stars, stellar-mass black holes, and massive black holes—only the latter seems capable of producing adetectable signal by gravitational focusing alone. Neutron stars, specifically magnetars, maybe the weakest sources, but they still promise interesting physics involving WIMP decayproducts.Theoretical models of WIMP annihilation in the Galactic Center, bolstered by recentobservations with the Fermi Gamma-ray Space Telescope (e.g., Abdo et al. 2010), havethe dark matter concentrated about Sgr A ∗ as a result of uncertain galactic evolution andblack hole growth dynamics. The main point of this work is that gravitational focusing willgenerate some signal, no matter what the formation history. While distinctive gamma raysignal could be lost in a uniform bath of WIMPs, synchrotron radiation may retain somecontrast with the background. The trade-off with synchrotron signal is that the accretionflows that presumably generate the localized magnetic field may contaminate the signal.Our best hope may be to find isolated intermediate mass black holes (Bertone et al. 2005).Alternatively we might seek an underluminous active galactic nucleus. Fortunately, our ownGalaxy has one. REFERENCES
Abbasi, R., Abdou, Y., Abu-Zayyad, T., et al. 2011, Phys. Rev. D, 84, 022004Abdo, A. A., et al. 2010, ApJS, 188, 405Alenazi, M. S., & Gondolo, P. 2006, Phys. Rev. D, 74, 083518Baring, M. G. 1989, A&A, 225, 260Berezinsky, V., Dokuchaev, V., & Eroshenko, Y. 2003, Phys. Rev. D, 68, 103003 7 –dm annihilation from small clumps as in cdm simulationsBerezinsky, V. S., Gurevich, A. V., & Zybin, K. P. 1992, NASA STI/Recon Technical ReportN, 93, 28473Bergstr¨om, L., & Gondolo, P. 1996, Astroparticle Physics, 5, 263Bergstr¨om, L., & Hooper, D. 2006, Phys. Rev. D, 73, 063510Bergstr¨om, L., Ullio, P., & Buckley, J. H. 1998, Astroparticle Physics, 9, 137Bertone, G., Hooper, D., & Silk, J. 2004, Phys. Rep., 405, 279Bertone, G., & Merritt, D. 2005, Phys. Rev. D, 72, 103502Bertone, G., Nezri, E., Orloff, J., & Silk, J. 2004, Phys. Rev. D, 70, 063503Bertone, G., Sigl, G., & Silk, J. 2001, MNRAS, 326, 799Bertone, G., Zentner, A. R., & Silk, J. 2005, Phys. Rev. D, 72, 103517Danby, J. M. A., & Camm, G. L. 1957, MNRAS, 117, 50Erkoca, A. E., Reno, M. H., & Sarcevic, I. 2010, Phys. Rev. D, 82, 113006Genzel, R., Pichon, C., Eckart, A., Gerhard, O. E., & Ott, T. 2000, MNRAS, 317, 348Ghez, A. M., Klein, B. L., Morris, M., & Becklin, E. E. 1998, ApJ, 509, 678Griest, K. 1988, Phys. Rev. D, 37, 2703Gondolo, P., & Silk, J. 1999, Physical Review Letters, 83, 1719Kravtsov, A. V., Klypin, A. A., Bullock, J. S., & Primack, J. R. 1998, ApJ, 502, 48Melia, F. 1992, ApJ, 387, L25Melia, F., & Falcke, H. 2001, ARA&A, 39, 309Merritt, D., Harfst, S., & Bertone, G. 2007, Phys. Rev. D, 75, 043517Merritt, D., Milosavljevi´c, M., Verde, L., & Jimenez, R. 2002, Physical Review Letters, 88,191301Milosavljevi´c, M., Merritt, D., Rest, A., & van den Bosch, F. C. 2002, MNRAS, 331, L51Moore, B., Quinn, T., Governato, F., Stadel, J., & Lake, G. 1999, MNRAS, 310, 1147 8 –Navarro, J. F., Frenk, C. S., & White, S. D. M. 1996, ApJ, 462, 563Power, C., Navarro, J. F., Jenkins, A., Frenk, C. S., White, S. D. M., Springel, V., Stadel,J., & Quinn, T. 2003, MNRAS, 338, 14Pagels, H., & Primack, J. R. 1982, Physical Review Letters, 48, 223Sandick, P., Diemand, J., Freese, K., & Spolyar, D. 2011, arXiv:1108.3820Sikivie, P., & Wick, S. 2002, Phys. Rev. D, 66, 023504Tyler, C. 2002, Phys. Rev. D, 66, 023509van den Bosch, F. C., & Swaters, R. A. 2001, MNRAS, 325, 1017Zaharijas, G., & Hooper, D. 2006, Phys. Rev. D, 73, 103501
This preprint was prepared with the AAS L A TEX macros v5.2. J (eq. 3), averaged inside acircular beam of radius θ , centered on Sgr A ∗ . As in Fig. 1, the dashed line represents acalculation without the effect of gravitational focusing, while the solid curve includes thefocusing effect. 11 –Fig. 3.— The density enhancement of thermal dark matter ( σ v = 155 km/s) in the wake ofa neutron star. The solid line shows the case where the neutron star is at rest with respect tothe frame of the dark matter bath; the dashed and dotted lines correspond to neutron starvelocities of 200 km/s and 1600 km/s, respectively. The upper curves show focusing of thedownstream flow, lower curves correspond to upstream density enhancement. In the limit ofinfinite star velocity, a downstream density caustic will emerge (Sikivie & Wick 2002). 12 – I/I ! ! -1.5 -1 -0.5
0 0.5 1 ! x (10 km) ! !!! y (10 km) ! -1.5 -1 -0.5 ! ! Fig. 4.— The intensity of self-annihilation radiation a plane containing the path of a high-velocity neutron star. In this instance, the neutron star (located at the origin) is moving at1000 km/s in a thermal bath of dark matter ( σ v = 200 km/s). The color scale (see inset)corresponds to the log of the intensity I (relative to the background signal I ) from darkmatter in a thin sheet containing this plane, as approxmated from a direct simulation of ∼9