The distribution and_annihilation_of_dark_matter_around_black_holes

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Draft version June 23, 2015Preprint typeset using LATEX style emulateapj v. 01/23/15

THE DISTRIBUTION AND ANNIHILATION OF DARK MATTER AROUND BLACK HOLES

Jeremy D. SchnittmanNASA Goddard Space Flight Center, Greenbelt, MD 20771 and

Joint Space-Science Institute, College Park, MD 20742

Draft version June 23, 2015

ABSTRACT

We use a Monte Carlo code to calculate the geodesic orbits of test particles around Kerr black holes,generating a distribution function of both bound and unbound populations of dark matter particles.From this distribution function, we calculate annihilation rates and observable gamma-ray spectra fora few simple dark matter models. The features of these spectra are sensitive to the black hole spin,observer inclination, and detailed properties of the dark matter annihilation cross section and densityprofile. Confirming earlier analytic work, we find that for rapidly spinning black holes, the collisionalPenrose process can reach efficiencies exceeding 600%, leading to a high-energy tail in the annihilationspectrum. The high particle density and large proper volume of the region immediately surroundingthe horizon ensures that the observed flux from these extreme events is non-negligible.Keywords: black hole physics – accretion disks – X-rays:binaries

1. INTRODUCTION

Prompted by the recent paper by Banados et al. (2009)[BSW], there has been a great deal of interest in thepotential of Kerr black holes to accelerate particles toultra-relativistic energies and thus to probe a regime ofphysics otherwise inaccessible. The vast majority of thiswork has been analytic and thus largely limited to themost simple photon and particle trajectories in the equa-torial plane. Here we present a more numerical approachthat focuses on calculating the fully relativistic distribu-tion function of massive test particles around a spinningblack hole. With this distribution function and a sim-ple model for the dark matter annihilation mechanism,we can then calculate the annihilation rate and observedspectrum as a function of black hole spin and observerinclination.It has been noted repeatedly in recent works that the

net energy gained through the Penrose process is quitemodest, as is the fraction of collision products that mightescape, and thus the astrophysical importance of theBSW effect is questionable (Jacobson & Sotiriou 2010;Banados et al. 2011; Harada et al. 2012; Bejger et al.2012; McWilliams 2013). We argue here that two pri-mary factors (to our knowledge largely neglected in pre-vious work) could greatly enhance the astrophysical rele-vance and observability of this annihilation. The firstis an energy-dependent cross section for dark matter(DM) annihilation. This could take many forms, thesimplest of which are p-wave annihilation (Bertone et al.2005; Chen & Zhou 2013; Ferrer & Hunter 2013), wherethe cross section scales like the relative velocity, or athreshold energy, above which the cross section increasesgreatly. This latter assumption is a natural choice fora model that includes multiple DM species, with themore massive particles intermediate products in the an-nihilation process towards gamma rays [see, e.g., Zurek(2014)]. Because gravity is the only known force capa-ble of accelerating dark matter particles to high energies,it is possible that new annihilation channels could occur

[email protected]

around black holes that are completely inaccessible ev-erywhere else in the universe.The other effect considered here is the relativistic en-

hancement of the density close to the black hole. Thisis due to the time dilation of observers near the hori-zon. In a steady-state system, one can think of droppingparticles into the black hole from infinity at a constantrate Γ∞ as measured by coordinate time t. To an ob-server near the black hole measuring proper time τ , anenhanced rate Γ∞(dt/dτ) is seen, with dt/dτ > 1. Forannihilation rates that scale like the density squared, thelocal annihilation rate will be enhanced by (dt/dτ)2. Ofcourse, the products will get redshifted on their way backout to an observer at infinity (McWilliams 2013), but weare still left with a net enhancement of dt/dτ .Even without this relativistic enhancement, numer-

ous models also predict an astrophysical enhancement ofthe dark matter density in the galactic nucleus. Adi-abatic growth of the central black hole will capturea large number of particles onto tightly bound orbits,growing a steep density spike as the black hole grows(Gondolo & Silk 1999; Sadeghian et al. 2013). Gravita-tional scattering off the dense nuclear star cluster willalso lead to a dark matter spike (Gnedin & Primack2004), similar to the classical two-body scattering re-sult of Bahcall & Wolf (1976). At the same time, self-annihilation (Gondolo & Silk 1999) and elastic scattering(Shapiro & Paschalidis 2014; Fields et al. 2014) will actto flatten out this spike into a shallow core more similarto the unbound population.Because our approach to this problem is predominantly

numerical, we can easily include treatment of a rangeof black hole spins, particle distributions, and cross sec-tions, and not limit ourselves to special cases with an-alytic solutions. Therefore, we can calculate how oftenthose extreme cases are likely to occur in a real astrophys-ical setting [for a notable exception to the analytic ap-proaches of earlier work, see the exhaustive Monte Carlocalculations of Williams (1995, 2004) that explored thelimits of the Penrose process in the context of Comp-ton scattering and pair production in accretion disks and

2 Schnittman

jets]. Of particular interest has been the following ques-tion: for two particles each of mass mχ falling from restat infinity and colliding near the black hole, what is themaximum achievable energy for an escaping photon? Wefind that this limit exceeds 12mχ for an extremal blackhole with a/M = 1, significantly higher than previouslypublished values of 2.6mχ (Bejger et al. 2012). We ex-plain the underlying reason for this discrepancy in a com-panion paper (Schnittman 2014).

2. POPULATING THE DISTRIBUTION FUNCTION

2.1. Initial conditions

The primary goal of this paper is to calculate the 8-dimensional phase-space distribution function df(x,p) ofDM particles around a Kerr black hole. Two of thesedimensions are immediately removed due to the assump-tion of a steady-state solution and stationarity of themetric, and the mass-shell constraint of the particle mo-mentum, leaving us with df(r, θ, φ, pr, pθ, pφ). This func-tion is further reduced to five dimensions because ax-isymmetry removes the dependence on φ.To calculate the distribution function, we first distin-

guish between two basic populations: the particles grav-itationally bound and unbound to the black hole. Theproperties of the bound populations are more sensitive tounderlying astrophysical assumptions, and will be dis-cussed below in Section 2.4. The unbound populationis more straightforward: we simply assume an isotropic,thermal distribution of velocities at a large distance fromthe black hole. Here, “large distance” is taken to be theinfluence radius rinfl of a supermassive black hole withmass M , and the DM velocity dispersion is set equal tothe stellar velocity dispersion σ0 of the bulge (thus the“unbound” population considered in this paper is stillgravitationally bound to the galaxy, just not the blackhole). From the “M-sigma” relation (Ferrarese & Merritt2000) we take

M ≈ 2× 107M⊙

(

σ0

100 km/s

)4

(1)

and

rinfl ≡ GM

σ20

≈ 8 pc

(

σ0

100 km/s

)2

. (2)

In units of gravitational radii rg = GM/c2, the influence

radius is typically quite large: rinfl ≈ 107M−1/27 rg, where

M7 ≡ (M/107M⊙).Given this outer boundary condition, we shoot test

particles towards the black hole with initial velocitiesdrawn from an isotropic thermal distribution with char-acteristic velocity σ0. As we are only interested inthe distribution function relatively close to the blackhole, we can ignore any particle with impact param-eter greater than ≈ 1000 rg. For those particles thatwe do follow, we calculate their geodesic trajectorieswith the Hamiltonian approach described in detail inSchnittman & Krolik (2013) and used in the radiationtransport code Pandurata. A schematic of this pro-cedure is shown in Figure 1. As the particle movesaround the black hole and passes through different fi-nite volume elements, the discretized distribution func-tion df(ri, θj ,p) is updated with appropriate weights.

The great advantage of this Hamiltonian approach isthat the integration variable is the coordinate time t inBoyer-Lindquist coordinates (Boyer & Lindquist 1967).Because this is the time measured by an observer at infin-ity, it determines the rate at which particles are injectedinto the system in the steady-state limit. Then the distri-bution function can be populated numerically by assign-ing a weight to each bin in phase space through whichthe test particle passes, with the weight proportional tothe amount of time t spent in that volume. The processis repeated for many Monte Carlo test particles until the5-dimensional distribution function is completely popu-lated.

2.2. Geodesics and Tetrads

Following Schnittman & Krolik (2013), we define localorthonormal observer frames, or tetrads, at each point inthe computational volume. Depending on the populationin question (i.e., bound vs. unbound), it is convenient touse either the zero-angular-momentum observer (ZAMO;Bardeen et al. (1972)) or the “free-falling from infinityobserver” (FFIO) tetrads. In all cases we use Boyer-Lindquist coordinates (Boyer & Lindquist 1967), wherethe metric can be written

gµν =

−α2 + ω22 0 0 −ω2

0 ρ2/∆ 0 00 0 ρ2 0

−ω2 0 0 2

. (3)

This allows for a relatively simple form for the inversemetric:

gµν =

−1/α2 0 0 −ω/α2

0 ∆/ρ2 0 00 0 1/ρ2 0

−ω/α2 0 0 1/2 − ω2/α2

, (4)

with the following definitions:

ρ2≡ r2 + a2 cos2 θ (5a)

∆≡ r2 − 2Mr + a2 (5b)

α2≡ ρ2∆

ρ2∆+ 2Mr(a2 + r2)(5c)

ω≡ 2Mra

ρ2∆+ 2Mr(a2 + r2)(5d)

2≡[

ρ2∆+ 2Mr(a2 + r2)

ρ2

]

sin2 θ . (5e)

Unless explicitly included, we adopt units with G = c =1, so distances and times are often scaled by the blackhole mass M .The ZAMO tetrad can be constructed by

e(t)=1

αe(t) +

ω

αe(φ) (6a)

e(r)=

√

∆

ρ2e(r) (6b)

e(θ)=

√

1

ρ2e(θ) (6c)

e(φ)=

√

1

ϕ2e(φ) , (6d)

Dark Matter around Black Holes 3

Figure 1. Schematic of our method for populating phase space with geodesic trajectories. The test particles are injected at large radius(r0 = 107rg) with thermal velocities with dispersion σ0 ≪ c. Those particles passing within 1000 rg of the black hole contribute to thetabulated distribution function in each volume element (ri, θj) through which they pass, with a weight proportional to the amount ofcoordinate time t spent in that zone.

i+1ri

θj+1θj

σ0 n0

r

where we designate tetrad basis vectors by µ indices,while coordinate bases have normal indices.To construct the FFIO tetrad, the time-like basis vec-

tor e(t) is given by the 4-velocity uµ = gµνuν correspond-ing to a geodesic with ut = −1, uθ = uφ = 0, and fromnormalization constraints,

ur = −[

(α−2 − 1)ρ2

∆

]1/2

. (7)

Then the spatial basis vectors e(i) are constructed via

a standard Gram-Schmidt method and aligned roughlyparallel to the Boyer-Lindquist coordinate bases.Any vector can be represented by its components in

different tetrads via the relation

u = e(µ)uµ = e(µ)u

µ , (8)

whereby the components are related by a linear transfor-mation Eµ

µ :

uµ=Eµµu

µ , (9a)

uµ=[E−1]µµuµ . (9b)

These uµ are the components that we use for the tabu-lated distribution function.1 Because of the normaliza-tion constraints, we need only store three components ofthe 4-momentum in each spatial volume element, makingthe total dimensionality of the distribution function five:two space and three momentum.In Pandurata, the geodesics are integrated with

a variable time step 5th order Cash-Karp algorithm(Schnittman & Krolik 2013). This technique very nat-urally matches small time steps to regions of high cur-vature and thus areas of high resolution in the spatial

1 While these contravariant indices technically refer to 4-velocities, and not 4-momenta, we use the terms interchangeablyin the locally flat tetrad basis, where most of our calculations takeplace.

grid. For each time step, a weight proportional to thecoordinate time spent on that step is added to the dis-tribution function for that particular volume of phasespace. Because the particle typically remains within asingle volume element for many time steps, we find thatinterpolation errors are small.

The spatial momentum components γβ i can be posi-tive or negative and span many orders of magnitude. Toadequately resolve the phase space and capture the rel-ativistic effects immediately outside the black hole hori-zon, we find that on order ∼ 103 bins are required ineach dimension. If the entire phase-space volume wereoccupied, this would correspond to an unfeasible quan-tity of data. Fortunately, this volume is not evenly filled,so such a hypothetical 5-dimensional array is in fact ex-ceedingly sparse. In practice, we are able to use a dy-namic memory allocation technique that only stores thenon-zero elements of the distribution function. Yet evenso, a well-resolved calculation can easily require multipleGB of data for a single distribution function, and to ad-equately sample this phase space requires on the orderof ∼ 109 test particles, with each geodesic sampled overthousands of time steps. Fortunately, this is a triviallyparallelizable problem, so it is relatively simple to achievesufficient resolution in a reasonable amount of time witha small computer cluster.

2.3. Unbound Particles

As mentioned above, for the unbound population, theouter boundary condition for the phase space density atrinfl is relatively well-understood. The velocity distribu-tion is thermal with characteristic speed σ0

2. The spatialdensity of dark matter is measured from galactic rotation

2 While there could be some small anisotropy in the dark mattervelocity distribution at rinfl, it is unlikely to be correlated withthe black hole spin. Thus the predominantly radial velocities ofincoming particles will be independent of polar angle, and thereforefor all intents and purposes appear isotropic from the black hole’spoint of view. Similarly, even if the DM velocity distribution at theinfluence radius is not strictly Maxwellian, this too will have little

4 Schnittman

curves at kpc distances from the nucleus, and then mustbe extrapolated in to pc distances with a combinationof observations and stellar profile modeling. For exam-ple, in the Milky Way the DM density near the Sun is0.3 GeV/cm3, and the radial profile can be reasonablywell-modeled with a simple ρ ∼ R−1 profile, giving adensity of ∼ 103 GeV/cm3 at rinfl. Inside of rinfl there isalmost certainly an additional bound component to theDM distribution (Gondolo & Silk 1999), so the unboundpopulation described here can best be understood as astrict lower bound on the phase space density.Outside of ∼ 100 rg the unbound population can be

treated as a collisionless gas of accreting particles, as inZeldovich & Novikov (1971). In the Newtonian limit, thedensity and velocity dispersion can be written

n(r) = n0

(

1 +2GM

σ20r

)1/2

(10)

and

σ2(r) = σ20

(

1 +2GM

σ20r

)

. (11)

Figure 2. Spatial density (a) and mean relative momentum〈γβ〉rel (b) of unbound particles as measured in the FFIO framein the equatorial plane of a Kerr black hole with a/M = 1. Thedashed lines are the Newtonian solutions of equations (10, 11),while the solid curves come from the fully relativistic Monte Carlocalculation.

1 10 100r/M

10-5

10-4

10-3

n (a

rb. u

nits

)

GR

Newtonian

(a)

impact on the results presented here, because the initial velocitiesare so small compared to the orbital velocities near the black hole,the trajectories are indistinguishable from particles injected frominfinity with zero velocity.

1 10 100r/M

0.1

1.0

10.0

<γβ

>re

l

(b)

In Figure 2 we show the spatial density of unboundparticles as measured by a FFIO around a Kerr blackhole with spin parameter a/M = 1, as well as the meanparticle momentum as measured in that frame. We findvery close agreement to the Newtonian results all theway down to r ∼ 10rg. The deviation of the momentumfrom the Newtonian solution is due largely to the specialrelativistic terms proportional to the Lorentz boost γ.The proper density is governed by two competing rel-

ativistic effects. One is time dilation and the other isspatial curvature. Close to the black hole, the parti-cle’s proper time τ slows down relative to the coordinatetime t measured by an observer at infinity, giving a largedt/dτ . This has the effect of increasing the number den-sity because, in a steady state, particles are injected intothe system at a constant rate—as measured by an ob-server at infinity. The injection rate measured by anobserver close to the black hole is higher by a factor ofdt/dτ , leading to her seeing a larger proper density.

Figure 3. Proper volume measured in the FFIO frame. Thedashed line is the Newtonian value dV/dr = 4πr2, and the solid

curve measures the FFIO’s proper volume dV /dr.

1 10 100r/M

101

102

103

104

105

106

dV/d

r

In fact, the proper density would be even higher ifit weren’t for another important relativistic effect: thestretching of space around a black hole. Specifically,the Boyer-Lindquist radial coordinate element dr cor-responds to a greater and greater proper distance as

Dark Matter around Black Holes 5

the observer approaches the horizon. This naturallygives a greater proper volume dV , shown as a solidcurve in Figure 3. Again, we show the Newtonian valuedV/dr = 4πr2 as a dashed curve. Because the parti-

cle interaction rates scale like n2 v dV , all these effectscombine to increase the importance of reactions near theblack hole.In Figure 4 we plot the momentum distributions of un-

bound dark matter particles, as observed by a FFIO inthe equatorial plane, at a relatively large distance fromthe black hole: r = 100M . Each 1-dimensional distribu-tion is calculated by integrating over the other two mo-mentum dimensions. We also plot the momentum mag-nitude γ|β| in panel (a). Because the particles all haverelatively small velocities at infinity β0 ≈ σ0/c ≪ 1, theirvelocities in the weakly relativistic region rg ≪ r ≪ r0are given by v ≈

√

2GM/r, corresponding to v ≈ 0.14cfor r = 100M .For the three spatial components of the momentum

distribution, we see a nearly isotropic velocity distribu-tion with a few subtle but interesting deviations. First,we note how there is a slight deficit of particles withpositive pr. This is due to capture by the black hole ofparticles coming in from infinity with nearly radial tra-jectories. By definition, these particles also have small

values of pθ and pφ, depleting the distribution function inthose dimensions around β = 0. While the distributionin the θ dimension is symmetric, note that the depletionin the φ distribution is offset to slightly negative values

of pφ. This is due to the well-known preferential captureby Kerr black holes of retrograde particles with angularmomenta aligned opposite to the black hole spin.In Figure 5, we plot the phase-space distribution for

the same boundary conditions as in Figure 4, but nowat r = 2M . The difference is quite dramatic, but all thefeatures are essentially due to the same physical mecha-nisms. This close to the horizon, there is a very strongdepletion of outgoing particles with pr > 0, as most par-ticles are captured by the black hole. The only particlesthat can avoid capture at this radius have prograde tra-jectories in the equatorial plane. Thus, the distribution

is now peaked around pθ = 0 instead of showing a deficit.

There is also a strong peak near pφ = 1 due to therelatively stable, long-lived prograde orbits that circlethe black hole multiple times before getting captured orescaping back out to infinity. In fact, the distributionof coordinate momentum is significantly more lopsidedto pφ > 0, but this is masked in Figure 5d because thisdistribution is measured by an observer with uφ > 0herself. The sharp fall-off of the azimuthal distribution

above pφ ≈ 1 is due to the angular momentum barrier of

the black hole. Particles with higher values of pφ simplynever reach this small radius.To the best of our knowledge, these distribution func-

tions have never been calculated before for a Kerr blackhole. However, the particle number density can be de-termined analytically for a non-spinning Schwarzschildblack hole, in the limit of σ0 ≪ c. This allows atleast one test of our numerical methods, although admit-tedly not a very strong one, as most of the interestingfeatures are related to the far more complicated orbits

around a spinning black hole. We follow the approachof Baushev (2009), who integrates the distribution func-tion with fixed energy, carefully setting the angular mo-mentum integration bounds based on which orbits arecaptured from a given radius. The results are shown inFigure 6, with our numerical calculation plotted as a redcurve and the analytic result in black, showing perfectagreement. Note that Baushev’s expression is given fora coordinate density rather than a proper density, whichalso explains the sharper peak at small r.

2.4. Bound Particles

As mentioned above, the unbound population can bethought of as a lower limit on the total DM density.There will also likely be a substantial population of par-ticles that are gravitationally bound to the black hole.As described in Gondolo & Silk (1999), the origin of thebound population is the adiabatic growth of the super-massive black hole on a timescale much longer than thetypical orbital time. This physical mechanism can be un-derstood as follows: as a marginally unbound DM parti-cle passes within rinfl, a small amount of baryonic matteris accreted into this region, deepening the potential welljust enough to capture the particle onto a marginallybound orbit. Once captured, the particle continues toorbit the black hole while conserving its orbital angu-lar momentum as the black hole continues to gain mass.This has the effect of shrinking the radius of the orbit.Over time, more particles are captured and subse-

quently migrate closer to the black hole, building upa steep density spike (Gondolo & Silk 1999). Inside ofthe inner-most stable circular orbit (ISCO), there is asharp falloff in the density spike due to plunging trajec-tories (Sadeghian et al. 2013). Here we do not attemptto solve for the slope of the density spike at large radiibut leave it as a free parameter, and fix the densityat the influence radius as for the unbound population:nbound(r) = n0(r/r0)

−α. Following Gondolo & Silk(1999), we also allow for the possibility of a density up-per bound nannih due to annihilation losses occurring oververy long timescales.To populate the phase-space distribution for the bound

population, we follow a similar method as describedabove for the unbound particles, but instead of launch-ing them from large radius with a limited range of im-pact parameters, now we launch them in situ with aisotropic thermal velocity distribution, as measured bya local ZAMO. These particles begin much closer to theblack hole, so the relativistic Maxwell-Juttner velocitydistribution is used (Juttner 1911), with the characteris-tic virial temperature Θ(r) = 1/2[1 − ǫZAMO(r)], whereǫZAMO(r)− 1 is the specific gravitational binding energyof the ZAMO.Because many of the particles launched close to the

black hole get captured, we first integrate their trajecto-ries for a few orbital periods to ensure they are in fact onstable orbits. Only then do they contribute to the tabu-lated distribution function. Additionally, a small fractionof the test particles from the tail end of the velocity dis-tribution will in fact be unbound, and these are similarlydiscarded.As with the unbound distribution, for each step along

its trajectory, the test particle contributes to the phasespace distribution a small weight proportional to the

6 Schnittman

Figure 4. Momentum distribution of unbound particles observed by a FFIO in the equatorial plane at radius r = 100M . All particles

have nearly unitary specific energy at infinity, so the average particle speed is very close to√

2GM/r =√0.02c (panel a). In panels (b-d)

we show the distribution of the individual momentum components, which are nearly isotropic this far from the black hole.

0.0 0.1 0.2 0.3 0.4γ|β|

0

50

100

150

f(p)

(a)

-0.2 -0.1 0.0 0.1 0.2γβr

0

5

10

15

f(p)

(b)

-0.2 -0.1 0.0 0.1 0.2γβθ

0

5

10

15

f(p)

(c)

-0.2 -0.1 0.0 0.1 0.2γβφ

0

5

10

15

f(p)

(d)

amount of time spent on that step. Yet now, instead ofusing the coordinate time dt, we use the proper time ofthe ZAMO frame from which the particles are launched,including an additional weight to ensure the appropriateradial form of the density distribution at larger radii.In Figure 7 we plot the radial density distribution and

mean relative momentum of the bound particles, as mea-sured in the ZAMO frame, in the equatorial plane arounda Kerr black hole with spin a/M = 1. The density pro-file is constructed so that ρ(r) ∼ r−2 at large radii. Weclearly see major differences relative to the unbound pop-ulation shown in Figure 2. Because of the lack of sta-ble orbits close to the black hole, the bound populationdeclines inside r ≈ 4M , which corresponds roughly tothe mean radius of the ISCO for randomly inclined or-bits around a maximally spinning black hole. This effectwas described in detail for non-spinning black holes inSadeghian et al. (2013). For equatorial circular orbits,only prograde trajectories are allowed inside of r = 9M .This leads to all particles moving in roughly the samedirection closer to the black hole, and explains why therelative momentum 〈γβ〉rel does not increase nearly asfast for the bound population as it does for the unboundpopulation, which allows plunging retrograde trajecto-ries, and thus more “head-on” collisions.In Figure 8 we show the 2D density profile in the

x − z plane for both bound and unbound populations,

for a/M = 0 and a/M = 1. The horizon in Boyer-Lindquist coordinates is plotted as a solid black line. Forcomparison purposes, the density scale is normalized tothe mean value at r = 10M . In reality, the density ofthe bound particles could be orders of magnitude greaterat these radii (Gondolo & Silk 1999). The most obviousdifference here is the depletion of bound orbits inside ofthe ISCO, which lies at r = 6M for non-spinning blackholes. For spinning black holes, the radius of the ISCOis a function of the particle’s inclination angle, rangingfrom r = 1M for prograde orbits in the equatorial plane,to r = 5.2M for polar orbits, and r = 9M for retrogradeequatorial orbits.Inside of the ISCO, there is also the “marginally

bound” radius, where particles with unity specific energycan exist on unstable circular orbits. This radius is alsoa function of inclination angle, and is plotted in Figure 8as dotted curve. Inside of this orbit, no bound particleswill be found (for improved visibility, we have left thisregion white, not black, as would be required by a strictadherence to the color scale). One interesting feature ofFigure 8 is that the density of the unbound populationaround spinning black holes doesn’t show any obviousθ-dependence. It appears that the enhanced density dueto long-lived prograde orbits is almost exactly counteredby the lack of retrograde orbits at the same latitude.In Figure 9 we show the phase space distribution

Dark Matter around Black Holes 7

Figure 5. Momentum distribution of unbound particles observed by a FFIO in the equatorial plane at radius r = 2M . Unlike Figure 4,here we see a decidedly non-thermal and highly anisotropic distribution.

0 2 4 6 8γ|β|

0.0000

0.0005

0.0010

0.0015

f(p)

(a)

-4 -2 0 2 4γβr

0.0000

0.0005

0.0010

0.0015

f(p)

(b)

-4 -2 0 2 4γβθ

0.0000

0.0005

0.0010

0.0015

f(p)

(c)

-4 -2 0 2 4γβφ

0.0000

0.0005

0.0010

0.0015

f(p)

(d)

Figure 6. Comparison of our numerical results (red) with the ana-lytic expression (black) for the particle density derived by Baushev(2009) for a Schwarzschild black hole. The density here is definedin the coordinate, not proper, frame, leading to a much steeperrise at small r. In Boyer-Lindquist coordinates, the horizon for anon-spinning black hole is at r = 2M .

1 10 100r/M

10-5

10-4

10-3

10-2

n (c

oord

fram

e)

analytic

numerical

for each of the momentum components, as measuredby a ZAMO in the equatorial plane at large radius(r = 100M). Compared to the equivalent plot for theunbound distribution (Fig. 4), we see a number of sig-nificant differences. First, the fact that these particles

are bound requires that E < 1, and the imposed virialenergy distribution results in mean velocities that aresmaller than those of the unbound population by a fac-tor of ∼

√2. Second, because we require stable, long-

lived orbits, there is a larger depletion around pθ = 0

and pφ = 0, as these trajectories are all captured by theblack hole and thus do not contribute at all to the dis-tribution function. Similarly, we see a larger asymmetrydue to the preferential capture of retrograde orbits with

pφ < 0.In Figure 10 we plot the same momentum distribution

functions, now at r = 2M . Here the contrast with theunbound population (Fig. 5) is even greater. The onlystable orbits at this radius are prograde, nearly circular,nearly equatorial orbits. This results in a relatively nar-row distribution clustered around uµ = [

√2, 0, 0, 1] in the

ZAMO frame. This narrower range in allowed velocitieswill have a profound impact on the shape of the annihi-lation spectrum, as we will see in the following section.

3. ANNIHILATION PRODUCTS

Once we have populated the distribution function,we can calculate the annihilation rate given a simpleparticle-physics model for the dark matter cross section.Again, it is simplest to work in the local tetrad frame.Including special relativistic corrections (Weaver 1976),

8 Schnittman

Figure 7. Spatial density (a) and mean relative momentum〈γβ〉rel (b) of bound particles in the equatorial plane of a Kerrblack hole with a/M = 1, as measured in the ZAMO frame. Thedashed lines are the Newtonian solutions when n(r) ∼ r−2 far fromthe black hole.

1 10 100r/M

10-8

10-7

10-6

10-5

10-4

10-3

n (a

rb. u

nits

)

GRNewtonian (a)

1 10 100r/M

0.1

1.0

10.0

<γβ

>re

l

(b)

the local reaction rate is given by the following:

R(x) =

∫

d3p1

∫

d3p2 f(x,p1) f(x,p2)γrelγ1γ2

σχ(γrel) vrel ,

(12)where γ1 and γ2 are the Lorentz factors of two parti-cles as measured in the tetrad frame, vrel is their relativevelocity, and σχ is the annihilation cross section (poten-tially a function of the relative velocity). R(x) has unitsof [events per unit proper volume per unit proper time],so we multiply by dτ/dt to get the rate observed by adistant observer.The distribution function f(x,p) is calculated numeri-

cally using the methods of Section 2. As discussed there,the numerical representation of f can have upwards of108 elements, so the direct integration of equation (12)is generally not computationally feasible. Instead, we usea Monte Carlo sampling algorithm to pick random mo-menta for each particle with an appropriate weight basedon the magnitude of f and the size of the discrete phasespace volume.The spatial integration, however, is carried out di-

rectly, looping over coordinates r and θ. This is shownschematically in Figure 11. For each volume element, alarge number (typically ∼ 106) of pairs of particles aresampled, and for each pair, a center-of-mass tetrad is

created. The total energy in the center-of-mass frame isgiven by

Ecom = mχ

√

2(1 + u1 · u2) , (13)

where mχ is the rest mass of the DM particle, and u =p/mχ is the particle 4-velocity.The 4-velocity of the center-of-mass frame is then given

byucom = (u1 + u2)/Ecom . (14)

The center-of-mass tetrad is constructed with e(t) =ucom. The spatial basis vectors are totally arbitrary, asthey are only needed to launch photons with an isotropicdistribution in the center-of-mass frame. Two photons,labeled k3 and k4 in Figure 11, are launched in oppositedirections, each with energyEcom/2 in the center-of-massframe. We then transform back to a coordinate basis forthe geodesic integration of the photon trajectories to adistant observer.As in Schnittman & Krolik (2013), for the photons

that reach infinity, Pandurata can generate an image andspectrum of the emission region. An example in shownin Figure 12 for the annihilation signal from the unboundpopulation around an extremal black hole, limiting theemission signal to the region r < 100M . While the fluxclearly increases towards the center of the image, becausethe density and velocity profiles are relatively shallow(see Fig. 2 above), the net flux is actually dominated byemission from large radii. These annihilation events arenot very relativistic, so produce a strong, narrow peakin the observed spectrum, centered at the DM rest massenergy.The annihilation events occurring closer to the hori-

zon sample a much more energetic population of parti-cles. Restricting ourselves to only those events where thecenter-of-mass energy is greater than 1.5× the combinedrest mass of the annihilating particles, we can zoom into the center of Figure 12. The result is shown in Figure13, now focusing on the inner region within r < 6M . Atthese small radii, the effects of black hole spin becomemuch more evident. One such effect is the characteristicshape of the Kerr shadow, defined by the impact pa-rameter of critical photon orbits (Chandrasekhar 1983).The observed flux is clearly asymmetric, as the progradephotons originating from the left side of the image havea much greater chance of escaping the ergosphere andreaching a distant observer.There is another interesting feature of Figure 13 that

we believe is novel to this work. Namely, the purple lobesemerging from the “mid-latitude” regions near the centerof the image. These are regions of greater photon flux,albeit very highly redshifted. Recall, this image is cre-ated by considering only annihilations with moderatelyhigh center-of-mass energy. Near the equatorial plane,extreme frame dragging ensures that the velocity disper-sion is highly anisotropic, with most of the DM particlesand their annihilation photons getting swept along onprograde, equatorial orbits. Above and below the plane,the DM distribution is more isotropic, leading to a moreisotropic distribution of outgoing photons. Yet if onegoes two far off the midplane, it becomes more difficultfor the photons to escape. At the mid-latitudes, there isjust enough frame dragging for photons to escape, yet notso much that they get deflected away from the observer.

Dark Matter around Black Holes 9

Figure 8. Spatial density of test particles in the x− z plane, for both bound and unbound populations, for a/M = 0 and a/M = 1. Foreach case, we show the unbound distribution on the left side and the bound distribution on the right side of the plot, and all distributionfunctions are normalized to the mean density at r = 10M . The horizon is plotted as a solid curve and the radius of the marginally boundorbits is shown as a dotted curve. The spin axis of the black hole is in the +z direction.

-15 -10 -5 0 5 10 15x/M

-15

-10

-5

0

5

10

15

z/M

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15unbounda/M=0

bound

ρ/ρ0

10

1.0

0.1

.01

-15 -10 -5 0 5 10 15x/M

-15

-10

-5

0

5

10

15

z/M

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15

-15 -10 -5 0 5 10 15-15

-10

-5

0

5

10

15unbounda/M=1

bound

ρ/ρ0

10

1.0

0.1

.01

The spectrum corresponding to this image is also plot-ted in Figure 13. Not surprisingly, the red and bluewings of the annihilation line shown in Figure 12 comefrom the most relativistic events. As pointed out byPiran & Shaham (1977), even reactions with very highcenter-of-mass energies will typically lead to photonswith low energies as measured at infinity, thus explain-ing the red tail of the annihilation spectrum. The high-energy tail above E = 2mχ is due exclusively to Penrose-process reactions where one of the annihilation photonshas negative energy and gets captured by the black hole(Penrose 1969; Piran et al. 1975).Earlier analytic work predicted that the maximum

energy attainable from the collisional Penrose processwas 2.6mχ for particles falling from rest at infinity(Harada et al. 2012; Bejger et al. 2012). Because our cal-culation is fully numerical, it was able to reveal previ-ously unknown trajectories leading to very high efficien-cies with E > 10mχ, as seen in Figure 13. Closer inspec-tion revealed that these high-energy photons are createdwhen an infalling retrograde particle collides with a out-going prograde particle that has just enough angular mo-mentum to reflect off the centrifugal barrier, providingthe necessary energy and momentum for the annihila-tion photon to escape the black hole (Schnittman 2014;Berti et al. 2014).Due to the strong forward-beaming effects within

the ergosphere, the escaping photon flux is highlyanisotropic, with the peak flux and highest-energy pho-tons emitted in the equatorial plane. Figure 14 shows thepredicted annihilation spectra for observers at differentinclination angles for the same DM profile as shown inFigure 13. Again, we restrict ourselves to the highest-energy reactions with Ecom > 3mχ.It is also instructive to plot the annihilation flux as a

function of the emission radius. In Figure 15 we showboth the observed flux (solid curves) and the flux thatgets captured by the black hole (dashed curves) as a func-tion of radius, integrated over all observing angles. Theemission is further subdivided by the center-of-mass en-ergy of the annihilating particles. Of course, the photonsemitted closer to the black hole have a greater chance of

getting captured. For the unbound population, the to-tal escape fraction ranges from fesc = 93% at r = 10Mdown to fesc(2M) = 14%, and fesc(1.1M) = 0.25%. Atsmall radius, these numbers are somewhat smaller thanthose calculated by Banados et al. (2011), who only con-sidered critical trajectories in the equatorial plane, wherethe escape probability is greatest. Yet at large radius,our distribution includes particles with typically greaterimpact parameters, and thus greater chance for escape.Another interesting feature of the curves in Figure 15

is the very sharp cutoff above a critical radius for eachenergy bin. This is a natural consequence of conservationof energy. Because all unbound particles come in fromrest at infinity with E = mχ, the available kinetic energyin the center-of-mass frame is simply the gravitationalpotential energy Mmχ/r at that radius. For example,to reach a center-of-mass energy of 10% above the restmass energy, the particles must fall within r ≈ 10M .Also note that inside r ≈ 4M , most of the photons arecaptured, while outside of this radius, most escape. Thisis in close agreement with what we found for plungingorbits inside of the ISCO of a Schwarzschild accretionflow in Schnittman et al. (2013).On the other hand, for the bound population of DM

particles (Fig. 15b), which by definition are not plunging,we find that the photon escape fraction is more than 90%at all radii, greatly increasing the relativistic effects ob-servable from infinity. This is consistent with the classiccalculation by Thorne (1974) which found that for thinaccretion disks limited to circular, planar orbits outsidethe ISCO, the fraction of emission ultimately captured bythe black hole was never more than a few percent, evenfor maximally spinning black holes where the majorityof the flux emerges from extremely close to the horizon.As we showed in Schnittman (2014), the peak en-

ergy attainable from particles falling in from infinity isa strong function of the black hole spin. Now, consid-ering the full phase-space distribution function of theparticles, we can see how the shape of the spectrum de-pends on spin. In Figure 16 we plot the flux seen byan equatorial observer, again limited to the high-energyannihilations with Ecom > 3mχ. For even marginally

10 Schnittman

Figure 9. Momentum distribution of bound particles observed by a ZAMO in the equatorial plane at radius r = 100M . Unlike theunbound distribution in Figure 4, the energy distribution is much broader here, yet with a smaller mean momentum (panel a). In panels(b-d) we show the distribution of the individual momentum components.

0.0 0.1 0.2 0.3 0.4γ|β|

0.000

0.005

0.010

0.015

0.020

0.025

f(p)

(a)

-0.2 -0.1 0.0 0.1 0.2γβr

0.000

0.002

0.004

0.006

0.008

0.010

f(p)

(b)

-0.2 -0.1 0.0 0.1 0.2γβθ

0.000

0.002

0.004

0.006

0.008

0.010

f(p)

(c)

-0.2 -0.1 0.0 0.1 0.2γβφ

0.000

0.002

0.004

0.006

0.008

0.010

f(p)

(d)

sub-extremal spins, the peak photon energy falls precipi-tously. As the spin decreases further, the number of colli-sions with Ecom > 3mχ also decreases, thereby reducingthe total flux observed. Lastly, the decreasing spin alsoincreases the critical impact parameter for capturing pro-grade photons, making it harder for the annihilation fluxto escape to infinity.Recall from Section 2.3 above that the density of the

unbound distribution scales like n ∼ r−1/2. From therate calculation in equation (12) we see that the annihi-lation rate [events/s/cm3] scales like R(r) ∼ r−3/2. In-cluding the volume factor dV = 4πr2dr we can writethe differential annihilation rate as dR/dr ∼ r1/2. Inother words, the unbound contribution to the annihila-tion signal diverges at large radius. In practice, the outerboundary can be set as the black hole’s influence radius,typically 106−7rg . This means that the observed signalwill essentially be a delta function in energy, with onlysmall perturbations from the relativistic contributions atsmall r, and thus measuring spin from annihilation lineswould be a very challenging prospect indeed.Two possible effects provide a way around this prob-

lem, each with its own additional uncertainties. One pos-sibility is that the annihilation cross section is a strongfunction of energy, increasing sharply above some thresh-old energy. This is admittedly rather speculative, andin conflict with leading DM models of self-annihilation

(Bertone et al. 2005). On the other hand, we do noteven know what the dark matter particle is, or if thereare many DM species making up a rich “dark sector,”with all the beauty and complexity of the standard modelparticles (Zurek 2014). One could easily imagine a DManalog of pion production via the collision of high-energyprotons, in which case the only reactions could occur im-mediately surrounding a black hole, the ultimate gravita-tional particle accelerator. In this case, by constructionthe annihilation rate is dominated the region immedi-ately surrounding the black hole.Another possibility is that the DM density is domi-

nated by a population of bound particles. As describedabove in section 2.4, this population arises throughthe adiabatic growth of the black hole through accre-tion, capturing marginally unbound particles while alsomaking the bound particles ever more tightly boundGondolo & Silk (1999); Sadeghian et al. (2013). Thisprocess will generally lead to a much steeper density pro-file, such as the n ∼ r−2 distribution we use here. In thiscase, the differential reaction rate scales like dR/dr ∼r−5/2 so the annihilation spectrum is now dominated bythe particles at smallest radii. In both cases—energy-dependent cross sections and a large bound population—the relativistic effects described in Section 2.3 (expandedproper volume and time dilation) push the most impor-tant interaction region to even smaller radii, and thus

Dark Matter around Black Holes 11

Figure 10. Momentum distribution of bound particles observed by a ZAMO in the equatorial plane at radius r = 2M . Compared toFigure 5, here we actually see a more symmetric, thermal distribution making up a thick torus of stable, roughly circular orbits near theequatorial plane.

0 2 4 6 8γ|β|

0

1×10−5

2×10−5

3×10−5

φ(π

)

-4 -2 0 2 4γβr

0

5.0×10−6

1.0×10−5

1.5×10−5

φ(π

)

−4 −2 0 2 4γβθ

0

5.0×10−6

1.0×10−5

1.5×10−5

φ(π

)

−4 −2 0 2 4γβφ

0

1×10−5

2×10−5

3×10−5

φ(π

)

Figure 11. For a given phase-space distribution f(x,p), the an-nihilation rate is calculated in each discrete volume element aroundthe black hole. Every annihilation event samples the distributionfunction to get the momenta for the two dark matter particles p1

and p2 and produces two photons k3 and k4 with isotropic dis-tribution in the center-of-mass frame. The product photons thenpropagate along geodesics until they reach a distant observer orget captured by the black hole.

f( )

p2

k4

p1 k3

x,p

the annihilation spectra are even more sensitive to theblack hole spin.In Figure 17 we show the annihilation spectra for both

the bound and unbound populations for a variety of

spins, now including emission out to r = 1000M . Therelative amplitudes are somewhat arbitrary, because wedon’t know what the relative densities of the two pop-ulations might be (see discussion below in Sec. 4), butit is almost certain that the bound population shoulddominate, possibly even by many orders of magnitude(Gondolo & Silk 1999). At the same time, the unboundsignal will be even narrower and have a greater ampli-tude peak than shown here, as it is dominated by low-velocity particles at large radius. So while their over-all amplitudes are uncertain, the detailed shapes of thespectra away from the central peak are relatively robust,depending only on the properties of geodesic orbits nearthe black hole.In this broad part of the spectrum, the bound and

unbound signals show very different behavior. For non-spinning black holes, no particle can remain on a boundorbit inside of r = 4M (see Fig. 8), so there are no an-nihilation photons coming from just outside the horizon,and these are the photons that produce the most stronglyredshifted tail of the spectrum. As the spin increases andthe ISCO moves to smaller and smaller radii, the line be-comes steadily broader. On the other hand, the unboundparticles are found all the way down to the horizon, wherethey can annihilate to highly redshifted photons regard-less of the black hole spin.Comparing Figures 5 and 10, we see that the unbound

particles probe a much greater volume of momentum

12 Schnittman

Figure 12. Simulated image and spectrum of the annihilationsignal from unbound dark matter out to a radius r = 100M arounda Kerr black hole. The observer is located in the equatorial plane.While the brightness peaks towards the black hole, the total fluxis dominated by annihilations at large radii. The central shadowis clearly seen, blocking emission coming from the far side of theblack hole. The photon energy E is scaled to the dark matter restmass mχ.

10-5

10-4

10-3

10-2

10-1

1

I/Imax

200M

0.1 1.0 10.0E/mχ

10-16

10-14

10-12

10-10

10-8

10-6

10-4

Flu

x (E

FE)

space at small radii. This in turn leads to a greaterchance of producing the extreme Penrose particles thatcharacterize the blue tail of the spectrum. Because allthe bound particles are essentially on the same prograde,equatorial orbits, it is much more difficult to achieveannihilations with large center-of-mass energies, so thehigh-energy cutoff in the spectrum is much closer to theclassical result for a single particle decaying into two pho-tons in the ergosphere (Wald 1974). In short, for boundparticles the red tail of the spectrum is a better probe ofblack hole spin, while for the unbound population, theblue tail is the more sensitive feature. But in both cases,higher spin leads to a broader annihilation line.

4. OBSERVABILITY

In addition to the dependence on the dark matterdensity profile, the amplitude of the annihilation spec-trum will also depend on the unknown dark matter massand annihilation cross section. At this point, it is onlypossible to use existing observations to set upper lim-its on these unknown parameters. One major obstaclethat has plagued nearly all observational efforts to detectdark matter annihilation is the existence of more conven-tional astrophysical objects such as active galactic nuclei

Figure 13. Simulated image and of the annihilation signal aroundan extremal Kerr black hole, now considering only annihilationswith Ecom > 3mχ. The observer is located in the equatorialplane with the spin axis pointing up. While the image appearsoff-centered, it is actually aligned with the coordinate origin. Thephoton energy E is scaled to the dark matter rest mass mχ.

10-5

10-4

10-3

10-2

10-1

1

I/Imax

12M

0.1 1.0 10.0E/mχ

10-16

10-14

10-12

10-10

10-8

10-6

10-4

Flu

x (E

FE)

Figure 14. Observed annihilation spectrum for the unbound DMpopulation, as a function of observer inclination angle, consideringonly annihilations with Ecom > 3mχ. The black hole spin is a/M =1.

0.1 1.0 10.0E/mχ

10-16

10-14

10-12

10-10

10-8

Flu

x (E

FE)

i=90o

60o

0o

(AGN), pulsars, and supernova remnants, all of which arepowerful sources of high energy gamma rays. One solu-tion to this problem is to focus on nearby dwarf galaxies,

Dark Matter around Black Holes 13

Figure 15. Flux reaching infinity (solid curves) and getting cap-tured by the black hole (dashed curves), as a function of the center-of-mass energy and radius of annihilation, for both bound and un-bound populations. The black hole spin is maximal. Note that thescale on the y-axis is arbitrary, and depends strongly on the anni-hilation cross sections and peak density. The radial flux profile, onthe other hand, is a robust result for these populations.

1 10 100r/M

10-12

10-10

10-8

10-6

10-4

10-2

Flu

x dF

/d(lo

g r)

unboundescapedcaptured

Ecom

= [2.0-2.2]mχ2

[2.2-3.0]mχ2

[3.0-4.0]mχ2

[>4.0]mχ2

1 10 100r/M

10-12

10-10

10-8

10-6

10-4

10-2

Flu

x dF

/d(lo

g r)

boundescapedcaptured

Ecom

= [2.0-2.2]mχ2

[2.2-3.0]mχ2

[3.0-4.0]mχ2

[>4.0]mχ2

Figure 16. Observed spectrum as a function of black hole spin, foran observer at inclination i = 90◦, considering only annihilationswith Ecom > 3mχ.

0.1 1.0 10.0E/mχ

10-16

10-14

10-12

10-10

10-8

Flu

x (E

FE)

a/M=1.00.9990.990.90.50.0

which are thought to have a high DM fraction and arenot typically contaminated by AGN activity or signifi-cant star formation (Ackermann et al. 2011) (note, how-

Figure 17. Comparison of annihilation spectra from bound andunbound populations, including all emission out to r = 1000M .The peak of the unbound signal will actually be even narrower, asit is dominated by annihilations at large radii with small relativevelocities.

0.1 1.0 10.0E/mχ

10-14

10-12

10-10

10-8

10-6

10-4

10-2

Flu

x (E

FE)

a/M=1.00.9990.990.90.50.0

r<1000M, unbound

0.1 1.0 10.0E/mχ

10-14

10-12

10-10

10-8

10-6

10-4

10-2

Flu

x (E

FE)

a/M=1.00.9990.990.90.50.0

r<1000M, bound

ever, the recent work by Gonzalez-Morales et al. (2014),which focuses on the contribution of black holes in dwarfgalaxies).Yet for our purposes, it turns out that the strongest

upper limits actually come from the most massive galax-ies with the most massive central black holes. Massiveelliptical galaxies have the added advantage of being rel-atively quiescent both in nuclear activity and star for-mation [e.g., Schawinski et al. (2007)]. As mentionedabove, the annihilation signal from the unbound pop-ulation will be dominated by flux at large radius. It isdifficult enough to spatially resolve even nearby blackholes’ influence radii with HST, much less gamma-raytelescopes, so any potential annihilation signal will tellus little about the black hole itself.Prospects for detection of an unambiguous black hole

signature improve if we consider annihilation modelsthat include an energy dependence to the dark mattercross section. For example, p-wave annihilation mecha-nisms will have cross sections proportional to the rela-tive velocity between the two annihilating particles [seeChen & Zhou (2013); Ferrer & Hunter (2013) and refer-ences therein]. Unfortunately, from equation (11) we seethat this would only lead to an additional factor of r−1/2

in the integrand of equation (12), which would still bedominated by the contributions from large r.

14 Schnittman

Figure 18. Comparison of annihilation spectra from unboundpopulations, for two simple models of the dark matter cross sec-tion. All spectra are normalized to their peak intensity. For thiscomparison, all emission within r = 104M is included.

0.1 1.0 10.0E/mχ

10-10

10-8

10-6

10-4

10-2

100

Flu

x (E

FE)

a/M=1.0a/M=0.0

σx(v) = constσx(v) ~ v

This effect is shown in Figure 18, which plots the pre-dicted spectra for two annihilation models: σχ(v) =const (black curves) and σχ(v) ∝ v (red curves). Theblack hole spins considered are a/M = 0 (dashed curves)and a/M = 1 (solid curves), and in all cases only theunbound population is included. Integrating out tor = 104M , we see only a slight difference in the shapeof the spectrum, with the σχ(v) ∝ v model leading to aslightly broader peak (all curves are normalized to givea peak amplitude of unity).Another possible annihilation model is based on a res-

onant reaction at some energy above the DM rest mass,as suggested in Baushev (2009). If the cross sectionincreases sharply around a given center-of-mass energy,this would have the effect of focusing in on a relativelynarrow volume of physical space around the black hole,as in Figure 15.Alternatively, the cross section could abruptly increase

above a certain threshold energy, if new particles in thedark sector become energetically allowed, analogous topion production via proton scattering. In either theresonant or threshold models for the annihilation crosssection, one might imagine a pair of heavier, intermedi-ate dark particles getting created and then annihilatingto two photons as in the direct annihilation model. If,for example, the mass of these intermediate particles is1.5mχ, then the observed spectrum would look like thoseplotted in Figures 13 and 16. With a significant increasein the cross section above such an energy threshold, theserelativistically-broadened spectra could in fact dominateover the narrow line component produced by the rest ofthe galaxy.A less exotic option would be the simple density en-

hancement due to the bound population. If this is suf-ficiently large, it would easily dominate over the rest ofthe galaxy and also produce a characteristically broad-ened line sensitive to both black hole spin magnitude andorientation relative to the observer. Somewhat ironically,one of the things that could ultimately limit the strengthof the annihilation signal from bound dark matter is an-nihilation itself. If the adiabatic black hole growth oc-curred at high redshift, then in the subsequent ∼ 1010

years, the bound population will get depleted via self-annihilation at an accelerated pace due to its high density(Gondolo & Silk 1999; Gonzalez-Morales et al. 2014).On the other hand, if the black hole grows through

mergers, or experiences even a single merger since thelast extended accretion episode, it is quite likely that thebound dark matter population could get completely dis-rupted. The details of such an event are beyond the scopeof this paper, but could be modeled by following test par-ticles bound to each black hole through the merger, viapost-Newtonian calculations (Schnittman 2010) or nu-merical relativity (van Meter et al. 2010).The observational challenge is readily apparent: the

black holes with the largest bound populations will tendto be in gas-rich galaxies with a lot of accretion andhigh-energy nuclear activity that could overwhelm theDM annihilation signal. The more massive black holes,residing in gas-poor quiescent galaxies, are also morelikely to have lost their cloud of bound dark matterthrough a history of mergers. Even in the event thata gas-rich spiral galaxy hosts a quiescent nucleus, theblack holes in those galaxies tend to have lower masses(Kormendy & Ho 2013).While the relation between black hole mass and dark

matter density is quite complicated for the bound popu-lation, it is relatively straightforward to calculate for theunbound population, which we can take as a lower boundon the DM density. Recall the influence radius rinfl is thedistance within which the gravitational potential is dom-inated by the black hole, as opposed to the nuclear starcluster or dark matter halo. From equation (2) we seethat the influence volume scales like r3infl ∼ M3/2, whilethe total mass enclosed is—by definition—of the order ofM . If the dark matter and baryonic matter have similarprofiles (by no means a certainty!), then more massiveblack holes should have lower surrounding DM density,with ninfl ∼ M−1/2.Because the unbound DM density falls off more rapidly

outside the central core, the annihilation flux Funbound

will be dominated by the contribution from around rinfl,so we can estimate

Funbound ≈ 1

D2n2inflσχvinflr

3infl ∼ M3/4 D−2 , (15)

with D the distance to the black hole and the mean ve-locity at the influence radius vinfl = σ0.If we consider a threshold energy annihilation model

where all the flux comes from inside a critical radiusrcrit ∼ few × rg, then the density scales like ncrit ∼ninfl(rinfl/rcrit)

1/2 ∼ M−3/4 while the relative velocityscales like vcrit ∼ σ0(rinfl/rcrit)

1/2 ∼ M0. The net fluxthen scales like

Funbound ≈ 1

D2n2critσχvcritr

3crit ∼ M3/2 D−2 . (16)

In both cases, it appears that the brightest sources willbe the closest, as opposed to the most massive.Now consider the case where the annihilation signal is

dominated by the bound contribution, the bound den-sity is in turn limited by a self-annihilation ceiling as inGondolo & Silk (1999), and there is a threshold energyabove which the cross section greatly increases. In thiscase, the flux is simply proportional to the total volume

Dark Matter around Black Holes 15

within the critical radius, so Fbound ∼ M3 D−2. Withthis scaling, the greatest flux will actually come frommore distant, more massive black holes. For example,NGC 1277, with a mass of 1.7 × 1010M⊙ and at a dis-tance of 20 Mpc (van den Bosch et al. 2012), could givean observed flux over a thousand times greater than ourown Sgr A∗!Recent works by Fields et al. (2014) and

Gonzalez-Morales et al. (2014) have argued thatcurrent Fermi limits of gamma-ray flux from Sgr A∗

and nearby dwarf galaxies with massive black holesalready place the strongest limits on annihilation fromDM density spikes. Based on the arguments above, webelieve that even stronger limits should come from moredistant, massive galaxies. The other important advancepresented in the present work is that, for either theenergy-dependent cross sections, or the steep densityspikes, the annihilation signal will be dominated bythe region closest to the black hole, and thus a fullynumerical, relativistic rate calculation is absolutelyessential.Lastly, we should mention that gamma-rays, while the

primary observable feature explored in this work, are notthe only promising annihilation product. High-energyneutrinos could also be produced in some annihilationchannels, particularly those with energy-dependent crosssections like p-wave annihilation (Bertone et al. 2005).While neutrinos obviously present many new detectionchallenges, the successful commissioning of new astro-nomical observatories like IceCube make this approachan exciting prospect (Aartsen et al. 2013). Furthermore,the non-DM backgrounds may contribute significantlyless confusion in the neutrino sky.

5. DISCUSSION

As apparent in the previous section, there are still fartoo many unknown model parameters to allow for quan-titative predictions of the annihilation flux from darkmatter around black holes. Sadeghian et al. (2013) putit best: “There are uncertainties in all aspects of thesemodels. However one thing is certain: if the centralblack hole Sgr A∗ is a rotating Kerr black hole and ifgeneral relativity is correct, its external geometry is pre-cisely known. It therefore makes sense to make use ofthis certainty as much as possible.” We have attemptedto follow their advice to the best of our ability.Thus, in order of decreasing confidence, the results in

this paper can be summarized by the following:

• For a given DM density ninfl and velocity dispersionσ0 at the black hole’s influence radius, the fullyrelativistic, 5-dimensional phase-space distributionhas been calculated exactly for any black hole spinparameter, covering the region from rinfl all the waydown to the horizon.

• Given this distribution function and a model fordark matter annihilation, the observed gamma-rayspectrum can be calculated by following photonsfrom their creation until they are either capturedby the black hole or reach the observer. Two im-portant relativistic effects serve to increase the an-nihilation rate as compared to a purely Newtoniantreatment: time dilation near the black hole effec-tively raises the density of the unbound population

in a steady-state distribution being fed from infin-ity; and transforming from coordinate to properdistances greatly increases the interaction volumein the region immediately around the black hole(see Fig. 3).

• Our numerical approach has unveiled previouslyoverlooked orbits that can produce annihilationphotons with extreme energies, far exceeding pre-vious estimates for the maximum efficiency of thecollisional Penrose process (Schnittman 2014). Thepeak energy attainable for escaping photons is astrong function of the black hole spin.

• The population of bound dark matter has alsobeen calculated numerically, although this dependson two additional physical assumptions: a localisothermal velocity distribution with a virial-liketemperature; and an overall radial power-law forthe density, as found in Gondolo & Silk (1999) andSadeghian et al. (2013). Including only the long-lived stable orbits, we found that the density peaksin the equatorial plane somewhat outside of theISCO, forming a thick, co-rotating torus aroundthe black hole spin axis. Because the bound pop-ulation is not plunging towards the horizon, theemerging flux has a much greater chance of escap-ing the black hole.

• The annihilation spectra from both the bound andunbound populations are sensitive to the spin pa-rameter, but in opposite ways: the unbound spec-trum varies mostly in the high-energy cutoff, withhigher spins allowing higher-energy annihilationproducts; the bound population moves closer andcloser to the horizon with increasing spin, givinga stronger red-shifted tail to the annihilation spec-trum. Both bound and unbound spectra becomemore sensitive to observer inclination with increas-ing spin, as the spherical symmetry of the systemis broken.

• For dark matter particle physics models with anenergy-dependent cross section (particularly onethat increases with center-of-mass energy), the an-nihilation spectrum will be a more sensitive probeof the black hole properties. For DM models incor-porating a rich population of dark sector species,black holes may be the most promising way to ac-celerate these particles and observe their interac-tions.

• The shape of the annihilation spectra is relativelyrobust, but the normalization is highly dependenton uncertain parameters such as the dark matterdensity profile and cross section. If the unbounddensity profile follows the baryonic matter, withthe shallow slopes seen in core galaxies, the ob-served flux should be a relatively weak function ofblack hole mass. If, on the other hand, the anni-hilation signal is produced by the most relativisticpopulation within rcrit ∼ few× rg, then the signalcould scale like M3 and thus be dominated by themost massive black holes in the local Universe.

16 Schnittman

While this paper has treated the bound and un-bound particles separately, future work will also con-sider the self-interaction between these two populations(Shapiro & Paschalidis 2014; Fields et al. 2014), whichmay lead to a single, self-consistent steady-state distri-bution with density slope between −1/2 and −2. Futurework will also focus on developing a robust framework inwhich we can use existing and future gamma-ray obser-vations to constrain various parameters of the particlephysics (e.g., mχ, σχ(E), and the annihilation mecha-nism, i.e., line vs continuum) and astrophysical models(ninfl, the bound distribution normalization and slope,and the black hole mass, spin, and inclination). Whileinitial work will focus on setting upper limits on reac-tion rates by looking at quiescent galaxies, our ultimateambition is nothing short of an unambiguous detectionof dark matter annihilation around supermassive blackholes.

ACKNOWLEDGMENTS

We thank Alessandra Buonanno, Francesc Ferrer,Ted Jacobson, Henric Krawczynski, Tzvi Piran, LalehSadeghian, and Joe Silk for helpful comments and dis-cussion. Special gratitude is due to HKB”H for provid-ing us a world full of elegant wonder and beauty. Thiswork was partially supported by NASA grants ATP12-0139 and ATP13-0077.

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