The Destiny of Dark Matter

The majority of baryons, which account for 15% of the matter in the Universe, will end their lives as carbon and oxygen inside cold black dwarfs. Dark matter (DM) makes up the remaining 85% of the matter in the Universe; however, the fate of DM is unknown. Here we show that the destiny of purely gravitationally interacting DM particles follows one of two possible routes. The first possible route, the “radiation-destiny” scenario, is that massive DM particles lose sufficient energy through gravitational radiation, causing them to spiral into a supermassive black hole that ultimately disappears through Hawking radiation. The second possible route, the “drifting-alone” destiny, applies to lighter DM particles, where only the central DM halo region spirals into the central BH, which is then Hawking radiated away. The rest of the DM halo is ripped apart by the accelerated expansion of the Universe.


INTRODUCTION
In approximately 5 billion years, our Sun will evolve into a red giant, expanding its radius by several hundred times and engulfing the innermost planets of the solar system, likely including the Earth, which will become a scorched and lifeless desert.During the same period, the Milky Way, our galaxy, will collide with the Andromeda galaxy, forming a resulting galaxy named Milkomeda.This new galaxy will continue as a large elliptical galaxy, with its central black holes merging into a supermassive BH (Schiavi et al. 2020).Due to the accelerated expansion of the Universe, only a few more galaxies will collide with Milkomeda, and after several tens of billions of years, all solarmass stars will fade away, and after trillions of years, all low-mass stars will have exhausted their fuel (Adams & Laughlin 1997).
The Universe comprises not only stars and gas but also large amounts of dark matter (DM).This has been observed in galaxies and clusters from the early 1930th (Lundmark 1930;Oort 1932;Zwicky 1937), thoroughly established with galactic rotations curves in the 1970th (Rubin et al. 1980), and confirmed on the scales of the full universe through the cosmic microwave background observations (Planck Collaboration et al. 2016).DM particles have a negligible collisional cross-section (Markevitch et al. 2004), which implies that they orbit the galaxy solely under the influence of gravity.The ultimate fate of DM particles depends on their annihilation rate or whether they decay.In some of the most popular particle scenarios, DM can annihilate when two DM particles come close enough to each other (Bertone et al. 2005) .These models establish a strong correlation between the DM annihilation cross-section and the DM abundance.A typical annihilation cross-section for such particles is of the order of σ ∼ 10 −32 m 1TeV 2 cm 2 , where the mass of 1TeV is often considered for thermally produced DM particles (Bertone et al. 2005) This means that a significant fraction of DM in a typical galaxy with a mass of 10 12 M ⊙ will annihilate within 10 17 years.
During DM annihilation, the resulting products may include photons and other high-speed particles that escape the galaxy.As more DM particles annihilate, the central part of the galaxy vanishes.This causes a reduction in the gravitational attraction, allowing DM particle velocities to exceed the escape velocity when ∼ 10% of the DM particles have annihilated (Binney & Tremaine 2008) (see Appendix A).As a result, if DM particles are thermally produced, and hence typically have annihilation cross sections of the order the weak interaction scale, then the DM in the galaxy disperse into the emptiness of the expanding space.The ultimate fate of these dispersed DM particles depends on their specific particle properties, including whether they decay into lighter particles or not.Other popular particle candidates for the DM considers decaying DM particle.For instance for the sterile neutrino (Dodelson & Widrow 1994) the decay time into active neutrinos or photons is approximately τ decay ∼ 10 19 10keV m 3 years.As the particles decay away the resulting less tightly bound galaxy will be ripped apart by the accelerated expansion of space, and the dispersed DM particles will decay.We will not consider neither thermally created DM particles or sterile neutrinos further.
In contrast, the destiny of the more general class of DM particles that only interact through gravity is generally unknown, and we will here show how it depends sensitively on their particle mass.
There is a long history of production of particles with no non-gravitational interactions (Parker 1969;Grib & Mamaev 1969;Parker 1971;Mamaev et al. 1976;Grib et al. 1976), and this production may even lead to abundances relevant for it being the DM (Dolgov & Kirilova 1990;Traschen & Brandenberger 1990;Dolgov & Hansen 1999;Kuzmin & Tkachev 1999;Bassett & Liberati 1998;Chung et al. 1998Chung et al. , 2001)).The DM can be created for instance by allowing it to have a coupling to a scalar inflaton field ϕ, such as L ∼ ϕ 2 χ 2 , where the DM is a scalar χ.The inflaton field oscillates towards the end of inflation, and the DM is produced due to the nonadiabatic expansion of spacetime during the transition to the matter or radiation dominated phase.An important restriction is that the DM must have properties to prevent subsequent thermalization, which is often achieved by considering very massive DM particles with no other coupling to standard model particles.Alternatively, the coupling of DM can be directly to gravity (and nothing else), through a conformal coupling like L ∼ ξRχ 2 , where R is the Ricci scalar curvature.The fundamental cause of particle production is that the expanding Universe breaks time-translation symmetry, which leads to non-conservation of energy in the quantum particles (Ford 2021).For recent lists of references, see for instance (Ford 2021;Lebedev 2021;Kolb et al. 2023).This wide range of production mechanisms have one thing in common, namely that the DM particles today will appear essentially sterile and non-interacting, except through gravity.Thus, for the rest of this paper, we are only considering the general class of DM particles that today only interact through gravity.
Massive objects circling each other have been predicted to emit gravitational waves (GW) (Einstein 1918).This effect was measured indirectly by the frequency change of pulsars (Weisberg et al. 1981), and more recently 100 years of search culminated with the direct observation of GW (Abbott et al. 2016).
The purely gravitationally interacting DM particles emit gravitational waves as they orbit the galaxy, and the power emitted by a single particle can be expressed as P m (r) = 32G 5c 5 Ω 6 m 2 r 4 , where G is the gravitational constant, c is the speed of light, Ω is the angular velocity of the particle, and r is the radius of its orbit (see appendix B).Notably, the power emitted depends on the square of the particle's mass.Thus, if we consider a wide range of DM candidates with masses ranging from that of the axion particle (m ∼ 10 −38 g) to a 100 solar-mass DM candidate (m ∼ 10 35 g), there will be a difference of a factor of 10 146 in the emitted power.The angular velocity of the DM particle depends strongly on the potential existence of a supermassive central black hole, since the typical circular velocity can be expressed as v circ (r) = GM (r)/r.Therefore, a large central object will allow the DM to emit more gravitational radiation and transit to a smaller orbit at a faster rate.A central BH is not a permanent fixture, as it is subject to Hawking radiation due to quantum effects (Hawking 1975).The timescale for a black hole to radiate away due to this effect is on the order of τ Hawking ∼ 10 −19 M BH g 3 years.Thus, a BH with a mass of 10 6 M ⊙ will completely evaporate in approximately 10 85 years.
Figure 1.The two main evolutionary tracks of dark matter haloes: for very massive DM candidates the large amount of gravitational radiation emitted leads to a quick inspiral onto the central BH.Subsequently this BH evaporates, and all DM thereby disappear in radiation.Alternatively, for light DM candidates, the central BH evaporates before a significant fraction of the DM has collapsed onto the central BH, and subsequently the remaining part of the DM halo will be dispersed into the expanding Universe.

THE TWO POSSIBLE DESTINIES
For a given DM particle mass, m, we can calculate the timescale for a fraction of the galaxy to inspiral due to energy loss from gravitational radiation.For instance, we can ask how long it would take for the innermost 10 −9 of the galaxy's mass to move on sufficiently small orbits that the DM particles will get absorbed by the central BH.For a 10 12 M ⊙ galaxy with a density profile in reasonable agreement with observations and numerical simulations (Hernquist 1990) (see Appendix B2), and with an initial central BH of 10 6 M ⊙ , this timescale is approximately 10 80 years for a DM particle with mass m = 100TeV.This is a shorter time than the Hawking radiation timescale of 10 85 yrs discussed above.For a DM candidate with mass m = 10 −5 eV, the corresponding timescale is on the order of 10 118 years.Thus, the destiny of DM particles in a 10 12 M ⊙ galaxy with a central BH of 10 6 M ⊙ is fundamentally different for different DM particle masses.The most massive DM candidates will lose enough energy through gravitational radiation to be entirely absorbed by the central BH in a sufficiently short time, and their fate is to end as Hawking radiation on timescales of the order of 10 103 years.This is shown as "radiation-destiny" in Figure 1.On the other hand, lighter DM candidates will lose energy through gravitational radiation slowly enough that the central BH will evaporate.Subsequently, the remaining galaxy will be dispersed into the vast empty space.In this case, shown as "drifting-alone destiny" in Figure 1, a significant fraction of the individual DM particles will survive.

THREE PARAMETERS M GAL , M BH AND M DM
As DM particles are treated as point-like, they only interact through 2-body gravitational interactions that are long-range.The corresponding relaxation process can affect their energy distribution, which may result in the ejection of DM particles if their velocities exceed the local escape velocity (Spitzer 1940).However, there is a counteracting effect of dynamical friction (Chandrasekhar 1943a), which reduces the velocity of the fastest particles.Since relaxation is a stochastic process and dynamical friction provides a systematic deceleration, the resulting energy distribution of DM particles will not contain particles that will evaporate from the cosmological structure (see appendix C for more details).
Galaxies can be distinguished based on the available gas and stellar matter, the total mass of the dark matter halo, and the mass of individual dark matter particles.Additionally, the initial mass of the central object can vary from a single stellar mass to the mass of the Milky Way's black hole, which is approximately 10 6 M ⊙ , or to supermassive black holes with masses exceeding several 10 9 M ⊙ .We have today observed a free-floating BH of mass ∼ 7M ⊙ (Sahu et al. 2022) and EHT took the first image of the black hole at the center of galaxy Messier 87 (Event Horizon Telescope Collaboration et al. 2019).The Milky Way has a BH of mass 4 • 10 6 M ⊙ (Ghez et al. 1998;Schodel et al. 2002), Andromeda has a BH of mass ∼ 1.4 • 10 8 M ⊙ (Al-Baidhany et al. 2020), and it is believed that most massive galaxies host a supermassive BH near its center (Kormendy & Ho 2013).
The gas and stellar matter will either be ejected from the galaxy or absorbed by the central black hole on much shorter timescales than those of the dark matter, and to avoid the details of this complication here, we simply allow these options to be covered by the seed BH mass to range from stellar mass to the entire mass of the cosmological structure (see Appendix C). 1 Consequently, we can reduce the number of important parameters to three: the mass of the seed black hole (which 1 In principle, one should also consider collisions between stars and the DM particles, in the case of sufficiently massive DM particles.For DM particles not thermally created one can still assume to ignore the non-gravitational scattering and hence just include gravitational interactions.In this case the physical extend of the DM particle becomes relevant, since a point-like DM particle with solar-mass could exhibit strong energy exchange with the white (or cold, black) dwarfs.It is worth remembering that DM substructures created from much smaller individual DM particles are very large, typically of the order 10 −2 pc for Earth mass DM haloes, and about 10 pc for solar-mass DM haloes (Diemand & Moore 2011;Wang et al. 2020).A few of these substructures typically survive the formation of the galaxies, and will hence also affect the stellar motion.For simplicity, we will ignore the effects of interactions between the black dwarfs and the DM particles in this paper.may include all the mass of present-day stars and gas), the total mass of the cosmological structure, and the mass of the individual dark matter particle.
In Figure 2, we present the fate calculation for a wide range of possible parameters.The general conclusion is that dark matter survives, i.e., it does not get absorbed by the central black hole, for small dark matter particle masses.At the second order, smaller seed black hole masses allow for more dark matter to survive.

CONCLUSION
The properties of DM particles are mostly unknown, and they may potentially decay or undergo annihilation.This paper examines the scenario where DM particles solely interact gravitationally over significantly longer timescales compared to the current age of the Universe.
We have demonstrated that the fate of DM is highly dependent on the mass of the DM particles.Extremely massive DM particles will promptly emit gravitational waves, leading to their gradual spiral towards the central black hole of the galaxies.Subsequently, the black hole will emit Hawking radiation, causing the DM particles to ultimately disappear as radiation.
In the case of lighter DM particles, the emission of gravitational radiation occurs at a significantly slower rate.As a result, only a minor portion of the DM particles will be absorbed by the central black hole.Once the central black hole ceases to exist, the potential of the galaxy is slightly reduced, and the remaining DM particles within these cosmological structures will gradually evaporate.Consequently, these DM particles will follow the "drifting-alone" destiny.

A. EVAPORATION OF DM PARTICLES WHEN CENTRAL REGION HAS ANNIHILATED AWAY
As discussed in the introduction, some DM particle candidates have annihilation cross section of the order the weak interaction scale.Such particle most often also annihilate when two particles get close to each other.
When DM particles have a non-zero annihilation cross section, then the central part of the halo will first disappear since the annihilation rate is proportional to ρ 2 .In this case the potential of the structure is reduced, and hence the high-energy tail of the DM distribution function may evaporate from the halo.The corresponding calculation is as follows.
Consider a halo in equilibrium, with a density profile given by ρ(r).One can integrate the Jeans equation to show that the radial velocity velocity dispersion is given by From numerical simulations it is known that the velocity distribution function does not have an exponential tail, but instead has a rapid decline which goes to zero around v = 2σ tot (Hansen et al. 2006).Thus, if the escape velocity where Φ(r) is the potential of the structure, becomes smaller than approximately 2 times the total velocity dispersion, then the high-energy particles will escape.Assuming that the velocity anisotropy is zero, one has σ 2 tot = 3σ 2 r , and we find that if 10% of the central mass (in a Hernquist structure) is removed, then the potential is reduced by slightly more than 10% at all radii.This will reduce the potential even further, leading to a run-away process where all the DM particles will evaporate.This is shown in figure 3.
Another popular particle candidate for DM is a decaying DM particle.For instance, in the case of sterile neutrinos (Dodelson & Widrow 1994), the decay time into active neutrinos or photons is Figure 3. Velocities as a function of radius in a typical DM halo, with scaled distances and velocities.The total velocity dispersion is given by σ (blue line) the circular velocity is v circ (orange line).The uppermost curve is the escape velocity (green line) for the full structure, and the reduced escape velocity (second from top, red line) is calculated when removing the central total mass.When v esc /σ ≈ 2 at a given radius, then the highest energy particles will escape, and through a run-away process the entire structure will disperse.
approximately τ decay ∼ 10 19 10keV m 3 years.As these particles decay, the resulting galaxy becomes less tightly bound, some of the particles now have velocities exceeding the escape velocity of the galaxy, and eventually the remaining part of the galaxy is ripped apart by the accelerated expansion of space.The dispersed DM particles will also decay.

B. LINEARIZED GRAVITY
To describe the GW emission we work under the assumptions of linearized gravity.This amounts to considering small GW amplitudes, large distances from the source and short wavelength GWs.
Linearized gravity implies assuming that gravitational waves (GWs) are a small perturbation to the Minkowski metric η αβ ≡ diag(−1, 1, 1, 1).The GWs will therefore be described by a metric perturbation h αβ such that the metric solving the Einstein equation can be written as with h αβ ≪ 1 for every α, β.
In linearized gravity, the Einstein equation assumes the form (Hartle 2003), in geometrized units (c = G = 1, mass measured in length).T αβ is the stress-energy tensor and the trace-reversed amplitude hαβ is defined by with h being the trace of the metric perturbation (i.e.h ≡ h γ γ ), and η αβ being the Minkowski metric.The d'Alembert operator □ is defined as following the convention (− + + +) for the metric signature.Imposing gauge conditions allow to close the system and uniquely solve the equation.We choose the Lorenz gauge, that can be conveniently expressed in terms of hαβ as It can be shown (Hartle 2003) that the spatial components of the trace-reversed GW amplitude can be written as where the second mass moment I ij , here evaluated at the retarded time t − r/c = t − r, is defined as The energy flux (energy per unit time per unit area) f GW of a linearized, plane GW is proportional to the square of the amplitude of the GW, let us call it a, times the square of its frequency ω (Hartle 2003): Since we are looking at the GW far away from the source, and the amplitude in Eq.B7 describes a spherical wave, the plane wave approximation is legitimate.The frequency dependence in Eq.B9, together with Eq.B7, suggests a dependence of f GW which is quadratic in the third time derivative of I ij .We can write: The right function ξ can be found by noticing that there is no radiation from a spherically symmetric mass distribution.The quadrupole moment tensor can be expressed in terms of the second mass moment as satisfies such requirements.The total power radiated can be found by integrating f GW over a surface encompassing the mass distribution, say a sphere, in the limit r −→ ∞, i.e.
Including units we can finally express the total power radiated by gravitational waves in the quadrupole approximation as with ⟨ • ⟩ denoting a time average over a period (Hartle 2003).

B.1. Test mass in central gravitational field
A DM mass m is orbiting a central mass M , with M ≫ m, in an elliptical orbit with such a low eccentricity that we can assume the orbit to be circular.Let R be the initial radius of the orbit, Ω the orbital frequency.By placing the origin of our Cartesian coordinate system to coincide with the position of the central object and choosing the orbit to lie the xy plane we can describe the trajectory of the test mass as: The mass density of the system can be written as and the components of the second mass moment can then be written I xy = mR 2 cos(Ωt) sin(Ωt) = 1 2 mR 2 sin(2Ωt) (B19) ) The remaining components are determined by the fact that the second mass moment is by definition a symmetric tensor, i.e., I ij = I ji .The third time derivative of each non-zero component is easily calculated to be Recalling the definition of the quadrupole moment B11 and noticing I The power radiated B13 is thus This is in agreement with the results of (Weinberg 1972).

B.2. The mass profile of galaxies
We assume the DM mass distribution to be spherically symmetric, with a density ρ(r) described by the Hernquist profile (Hernquist 1990) where M is the total mass and a a linear scale of the object.This profile approximates well the mass distribution of galactic bulges and elliptical galaxies, but also the DM distribution in haloes.We opt for this profile instead of the NFW profile since the Hernquist mass is finite without the need for a truncation at large radii.The cumulative mass profile, and the corresponding potential pertaining to the density profile B27 are, respectively (Hernquist 1990), The velocity dispersion σ 2 v is obtained by solving the 1D Jeans equation for a non-rotating, spherical system.Its radial component, σ 2 vr , is given by (Hernquist 1990) The corresponding angular velocity is given by being the transverse velocity dispersion.In fact, for a spherical system, vr .The single particle GW power loss is thus given by In figure 4 we show that for a DM particle of mass 1T eV and an initial seed BH of mass 10 6 M ⊙ the Hawking radiation timescale is longer than the inspiral time for the entire galaxy, and hence this structure will inspiral and gets absorbed by the central BH.In contrast a central initial seed BH of only 10 3 M ⊙ will evaporate away before a fraction 10 −6 of the galaxy has inspiraled onto the BH.This implies that all subsequent inspiraling DM will eventually evaporate through Hawking radiation, until the remaining DM halo is sufficiently dilute that it will disperse through the accelerated expansion of the Universe, and hence a significant fraction of the DM particles will remain as DM particles in the expanding Universe.C. EVAPORATION V.S. DYNAMICAL FRICTION It has almost become "common knowledge" that gravitational 2-body interactions lead to effectively relaxed systems, which implies a slow but steady evaporation of particles from the system (Spitzer 1940).The argument is that the 2-body interactions leads to an exponential distribution of energies, and since any exponential will have a high-energy tail beyond the systems escape-velocity, then this implies that particles will evaporate.This conclusion is, however, incorrect, as we will show now, since it ignores another important gravitational effect: Dynamical Friction (DF) (Chandrasekhar 1943a).
The relaxation time arises from long-range encounters causing a cumulative diffusion of a stars velocity.It is frequently estimated by following the trajectory of a subject star with initial velocity v, as it passes a field star with impact parameter b.The acceleration from the field star gives the subject star a perpendicular velocity of the order δv = 2Gm/(bv) (Binney & Tremaine 2008).If we consider a large spherical structure with radius R and N particles each with mass m, then we can calculate the number of long-range encounters during one crossing.Each encounter produces a small perturbation to the subject stars velocity, and since these are independent of each other we can add the δv 2 linearly.Hereby one can integrate over all impact parameters to find where the Coulomb logarithm comes from the maximum and minimum impact parameters b max ∼ R and b min ∼ R/N , giving logΛ ∼ logN .It is important to keep in mind that the standard trick of numerical N-body simulations of the inclusion of a softening merely leads to a slightly bigger value for b min , which only enters the expression through the logΛ.A typical velocity is given by and we hence have which implies that after N 8 logN crossings the totalt energy exchange is of the same level as the initial energy (the stars orbit has been completely randomized), and this gives the result This effect is possibly most famous for Globular clusters, where N ∼ 10 5 and crossing times of Myrs makes this 2-body relaxation important given the age of the globular clusters.If these repeated encounters set up a Maxwellian distribution of velocities, then the high-energy tail will contain particles moving beyond the esacape velocity, and these particles will hence evaporate.Given the small number of particles in the high-energy tail, one often expects that the entire cosmological structure may evaporate at time-scales around 100 times the relaxation time (Spitzer 1940).
There is, however, another gravitational effect, which also must be included, namely the Dynamical Friction (DF).This effect is often interpreted through the gravitational focusing behind the particles path, which slows the particle down, and hence transfers energy from the rapidly moving particles to the slow ones.By integrating over impact parameters the acceleration is often written by Chandrasekhars expression (Chandrasekhar 1943a) where the subject star has mass M and the field stars have mass m.From this formula it is clear that only the slower moving field particle contribute to slowing the subject particle down.For a rapidly moving subject particle the integral over the field particles is just the number density and hence the magnitude of the acceleration can be written as where we used M = m when considering only DM particles.To make the comparison with the relaxation time as explicit as possible we will again consider a sphere of radius R with N particles of mass m, where a typical velocity is still given by v 2 = GmN/R.If we are considering a fast moving particle, then we can ask the number of crossings (of crossing time τ cross = R/N ) the particle needs, in order to reduce its velocity by the order v dv which is solved by Comparing with eq.C36 we thus see that the timescale for reducing the velocity of fastmoving particles is the same (within a factor of 3/4) as that of evaporation.The process of relaxation/evaporation is a stochastic process, whereas DF has a systematic decelerating effect.Any given particle which happens to have a velocity slightly larger than the field particles will therefore have its velocity reduced by DF faster than the statistical process of relaxation can push it beyond the escape velocity.
The inclusion of DF in the calculation of stellar evaporation was first studied in (Chandrasekhar 1943b) by considering the stochastic process of relaxation as a diffusion process.The conclusions of (Chandrasekhar 1943b) was also that the effect of DF is crucial to include in order to calculate evaporation, even though the paper (Chandrasekhar 1943b) works under the assumption of Gaussian distributions of velocities, which is today known to be incorrect long before the onset of effects of both relaxation of DF (Hansen et al. 2006).It is expected that the very rapid process of violent relaxation (Binney & Tremaine 2008) is responsible for the appearance of the non-exponential shape of the velocity distribution function with no high-energy particles.As shown in Figure 4, the stochastic appearance of high-energy particles will immediately be damped by DF, and hence no DM particles will evaporate.This calculation only considers N = 10 3 particles, and the effect of DF is only calculated accurately for the high-energy tail of the energy distribution (the bulk of the particles have their energies adjusted accordingly to assure energy conservation in each time-step), and thus a more careful calculation is needed in order to address complicated dynamical systems like Globular clusters.The above argument (and simple calculation) is here partly used as an argument why we may allow the "initial seed" BH to cover everything from a single star, to the mass of the entire collection of stars.
It is a pleasure thanking the referee for very constructive suggestions which improved the paper.SHH thanks Jens Hjorth and Radek Wojtak for interesting discussions. of N = 10 3 particles with random energies between 0 and 1.By allowing the particles to exchange energy through elastic collision one obtains the orange line analytically.Numerically by allowing 10 4 times N/2 collisions with a maximal energy exchange of 1% of the particle energies, one gets the green (up-sloping hatched) histogram, which follows the analytical prediction.By including the effect of dynamical friction for the 5% of the particles with highest energy (corresponding to the 2σ tail), one get a reduction of the high-energy tail, as represented by the solid red histogram.Here we included the factor of 3/4 as derived above.Interestingly, it is visible how the high-energy particles are now slightly piled up at lower energies.

Figure 2 .
Figure2.The figure illustrates the fate of dark matter for a wide range of galaxy parameters.The blue-colored surface and the region under it represent the parameter space in which the central black hole evaporates rapidly enough that at least half of the dark matter particles in the cosmological structure end up dispersed in the Universe.Conversely, the non-colored region shows parameters for which more than half of the structure ends up being engulfed by the central black hole, which subsequently evaporates through Hawking radiation.The calculation spans over 70 orders of magnitude in the dark matter particle mass, m DM , and galaxy masses ranging from dwarf galaxies of 10 6 M ⊙ to galaxy clusters of 10 15 M ⊙ .It also allows for the possibility of the central seed black hole having a range of masses, from a single star to all the stars of the structure.The figure is cut short at log (m DM /g) ∼ 0 to better visualize the surface, as heavier dark matter particles lie out and above the blue-colored surface for any galaxy and seed black hole mass shown.

Figure 4 .
Figure 4. Timescales for inspiral and Hawking radiation as a function of the fraction of the galaxy.The four dashed lines show the dependence on the mass of the DM particle, where the upper-most curve are for the lightest DM particle (m DM = 10 −14 GeV) and the lowest curve is the most massive (m DM = 10 12 GeV).The four solid lines show the Hawking radiation timescale dependence of the initial seed BH mass.If the initial seed BH is small (lowest curve, 1M ⊙ ) then the Hawking radiation timescale is short, whereas a supermassive BH initial seed of 10 11 M ⊙ leads to very long radiation timescales (uppermost solid curve).

Figure 5 .
Figure5.The normalized energy distribution.The yellow dashed histogram shows an energy distribution of N = 10 3 particles with random energies between 0 and 1.By allowing the particles to exchange energy through elastic collision one obtains the orange line analytically.Numerically by allowing 10 4 times N/2 collisions with a maximal energy exchange of 1% of the particle energies, one gets the green (up-sloping hatched) histogram, which follows the analytical prediction.By including the effect of dynamical friction for the 5% of the particles with highest energy (corresponding to the 2σ tail), one get a reduction of the high-energy tail, as represented by the solid red histogram.Here we included the factor of 3/4 as derived above.Interestingly, it is visible how the high-energy particles are now slightly piled up at lower energies.