Chaotic enhancement of dark matter density in binary systems
^{1} Institut UTINAM, Observatoire des Sciences de l’Univers THETA, CNRS & Université de FrancheComté, 25030 Besançon, France
email: rollin@obsbesancon.fr, jose.lages@utinam.cnrs.fr
^{2} Laboratoire de Physique Théorique du CNRS, IRSAMC, Université de Toulouse, UPS, 31062 Toulouse, France
email: dima@irsamc.upstlse.fr
Received: 23 December 2014
Accepted: 9 February 2015
We study the capture of galactic dark matter particles (DMP) in twobody and fewbody systems with a symplectic map description. This approach allows modeling the scattering of 10^{16} DMPs after following the time evolution of the captured particle on about 10^{9} orbital periods of the binary system. We obtain the DMP density distribution inside such systems and determine the enhancement factor of their density in a center vicinity compared to its galactic value as a function of the mass ratio of the bodies and the ratio of the body velocity to the velocity of the galactic DMP wind. We find that the enhancement factor can be on the order of tens of thousands.
Key words: chaos / celestial mechanics / binaries: general / dark matter
© ESO, 2015
1. Introduction
In 1890, Henri Poincaré proved that the dynamics of the threebody gravitational problem is generally nonintegrable (Poincaré 1890). Even 125 yr later, many aspects of this problem remain unsolved. Thus the capture crosssection σ of a particle that scatters on the binary system of Sun and Jupiter has only recently been determined, and it has been shown that σ is much larger than the area of the Jupiter orbit (Khriplovich & Shepelyansky 2009; Lages & Shepelyansky 2013). The capture mechanism is described by a symplectic dynamical map that generates a chaotic dynamics of a particle. The scattering, capture, and dynamics of a particle in a binary system recently regained interest with the search for dark matter particles (DMP) in the solar system and the Universe (Bertone et al. 2005; Garrett & Dūda 2011; Merritt 2013). Thus it is important to analyze the capture and ejection mechanisms of a DMP by a binary system. Such a system can be viewed as a binary system with a massive star and a light body orbiting it. This can be the Sun and Jupiter, a star and a giant planet, or a super massive black hole (SMBH) and a light star or black hole (BH). In this work we analyze the scattering process of DMP galactic flow, with a constant space density, in a binary system. One of the main questions here is whether the density of captured DMPs in a binary system can be enhanced compared to the DMP density of the scattering flow.
The results obtained by Lages & Shepelyansky (2013) show that a volume density of captured DMPs at a distance of the Jupiter radius r<r_{p} = r_{J} is enhanced by a factor ζ ≈ 4000 compared to the density of Galactic DMPs which are captured after one one orbital period around the Sun and which have an energy corresponding to velocities km s^{1} ≪ u. Here, m_{p},M are the masses of the light and massive bodies, respectively, u ≈ 220 km s^{1} is the average velocity of a Galactic DMP wind for which, following Bertone et al. (2005), we assume a Maxwell velocity distribution: .
Our results presented below show that for an SMBH binary system with v_{cap}>u there is a large enhancement factor ζ_{g} ~ 10^{4} of the captured DMP volume density, taken at a distance of about a binary system size, compared to its galactic value for all scattering energies (and not only for the DMP volume density at low velocities v<v_{cap} ≪ u, as discussed by Lages & Shepelyansky 2013). We note that the Galactic DMP density is estimated at ρ_{g} ~ 4 × 10^{25} g cm^{3}, while the typical intergalactic DMP density is estimated to be ρ_{g0} ~ 2.5 × 10^{30} g cm^{3} (Garrett & Dūda 2011; Merritt 2013). At first glance, this high enhancement factor ζ_{g} ~ 10^{4} seems to be rather unexpected because it apparently contradicts Liouville’s theorem, according to which the phase space density is conserved during a Hamiltonian evolution. Because of this, it is often assumed (Gould & Alam 2001; Lundberg & Edsjö 2004) that the volume (or space) DMP density cannot be enhanced for DMPs captured by a binary system, and thus ζ_{g} ~ 1. Below we show that this restriction is not valid for the following reasons: first, we have an open system where DMPs can escape to infinity, being ejected from the binary system by a timedependent force induced by binary rotation. This means that the dynamics is not completely Hamiltonian. Second, DMPs are captured (or they linger, or are trapped) and are accumulated from continuum at negative coupled energies near the binary during a certain capture lifetime (although not forever). Thus, the longer the capture lifetime, the higher the accumulated density. Third, we obtain the enhancement for the volume density and not for the density in the phase space, for which the enhancement is indeed restricted by Liouville’s theorem. We discuss the details of this enhancement effect in the next sections.
The scattering and capture process of a DMP in a binary system can be an important element of galaxy formation. This process can also be useful to analyze cosmic dust and DMP interaction with a supermassive black hole binary. This is expected to play a prominent role in galaxy formation, see Graham et al. (2015). Thus we hope that analyzing this process will be useful for understanding the properties of velocity curves in galaxies, which was started by Zwicky (1933) and Rubin et al. (1980). We note that the velocity curves of captured DMPs in our binary system have certain similarities with those found in real galaxies.
2. Symplectic map description
Following the approach developed by Petrosky (1986), Chirikov & Vecheslavov (1989), Malyshkin & Tremaine (1999), and Lages & Shepelyansky (2013), we used a symplectic dark map description of the DMP dynamics in one orbital period of a DMP in a binary system (1)where x_{n} = t_{n}/T_{p}(mod1) is given by time t_{n} taken at the moment of DMP nth passage through perihelion, T_{p} is the planet period, and . Here E,m_{d},andv_{p} are the energy, mass of the DMP, and the velocity of the planet or star. The amplitude J of the kick Ffunction is proportional to the mass ratio J ~ m_{p}/M. The shape of F(x) depends on the DMP perihelion distance q, the inclination angle θ between the planetary plane (x,y) and DMP plane, and the perihelion orientation angle ϕ, as discussed by Lages & Shepelyansky (2013). In the following we use for convenience units with m_{d} = v_{p} = r_{p} = 1 (here m_{d} is the DMP mass, which does not affect the DMP dynamics in gravitational systems).
For q>r_{p} the amplitude J drops exponentially with q and F(x) = Jsin(2πx), as shown by Petrosky (1986). This functional form of F(x) is significantly simpler than the real one at q<r_{p}, while it still produces chaotic dynamics at 0 <w ≪ 1 and integrable motion with invariant curves above a chaos border w>w_{ch}. In this regime the map takes the form (2)The same map describes a microwave ionization of excited hydrogen atoms that is called the Kepler map (see Casati et al. 1987; Shepelyansky 2012). There, the Coulomb attraction plays the role of gravity, while a circular planet rotation is effectively created by the microwave polarization. The microwave ionization experiments performed by Galvez et al. (1988) were made for threedimensional atoms, but the ionization process is still well described by the Kepler map (see Casati et al. 1990; Shepelyansky 2012). These results provide additional arguments in favor of a simplified Kepler map description of DMP dynamics in binary systems. The dynamics of the Kepler map can be locally described by the Chirikov standard map (see Chirikov 1979). We note that the approach based on the Kepler map has recently been used to determine chaotic zones in gravitating binaries, see Shevchenko (2015).
The similarity of dynamics of dark (1) and Kepler (2) maps is also well visible from comparing their Poincaré sections, shown in Fig. 1, for the typical dark map parameters corresponding to the Halley comet (see Fig. 1a in Lages & Shepelyansky 2013) and the corresponding parameter J of the Kepler map.
To take into account that J decreases with q, we use the relation J = J_{0} = const. for q<q_{b} and J = J_{0}exp( − α(q − q_{b}))) for q ≥ q_{b} (below J is used instead of J_{0}). We use q_{b} = 1.5 and α = 2.5, corresponding to typical dark map parameters (see Fig. 1 in Lages & Shepelyansky 2013), but we checked that the obtained enhancement is not affected by a moderate variation of q_{b}orα. The simplicity of map (2) allows increasing the number N_{p} of injected DMPs by a factor one hundred compared to map (1). The correspondence between (1) and (2) is established by the relation J = 5m_{p}/M, which works approximately for the typical parameters of Halley comet case.
Of course, as discussed by Lages & Shepelyansky (2013), the dark map and moreover the Kepler map give an approximate description of DMP dynamics in binary systems. However, this approach is much more efficient than the exact solution of Newton equations used by (Peter 2009a,b), and Sivertsson & Edsjö (2012) and allows obtaining results with very many DMPs injected during the lifetime of the solar system (SS) t_{S} = 4.5 × 10^{9} yr. The validity of such a map description is justified by the results obtained by Petrosky (1986), Chirikov & Vecheslavov (1989), Malyshkin & Tremaine (1999), Lages & Shepelyansky (2013), Rollin et al. (2015), and Casati et al. (1990).
3. Capture crosssection
Fig. 1 Poincaré sections for the dark map (1) (top) and the Kepler map (2) (bottom) for parameters of the Halley comet case in Eq. (1) and J = 0.007 in Eq. (2) (see text). 

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The capture crosssection σ is computed as previously described by Lages & Shepelyansky (2013) with , where h is a fraction of DMPs captured after one map iteration from w< 0 to w> 0, given by an interval length inside the F(x) envelope at  w = const., . The equation for σ(w) is based on the expression for the scattering impact parameter . For the Kepler map the hfunction only depends on q, and the numerical computation is straightforward. The differential energy distribution of captured DMPs is dN/ dw = σ(w)n_{g}f(w)/2 with n_{g} = ρ_{g}/m_{d}.
The results for σ(ω) and dN/dw/N_{p}, obtained for maps (1) and (2), are shown in Fig. 2. Here is the number of DMPs crossing the planet orbit area per unit of time. The results of Fig. 2 show that both maps give similar results, which provides additional support for the Kepler map description. The theoretical dependence σ ∝ 1/w , predicted by Khriplovich & Shepelyansky (2009), is clearly confirmed. The only difference between maps (1) and (2) is that the kick amplitude J ≈ 5m_{p}/M for (2) is restricted, and thus after one kick we may have only  w ≤ J, while for (1) some orbits can be captured with  w >J = 5m_{p}/M as a result of close encounters. However, the probability of such events is low.
Fig. 2 a) Dependence of the capture crosssection σ on DMP energy w for SunJupiter (black curve, data from Ref. (8)) and for the Kepler map at J = 0.005 (red curve); the dashed line shows the dependence σ ∝ 1/w . b) Dependence of the rescaled captured number of DMPs on energy w for the models of the left panel. Here w_{cap} = 0.001. 

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4. Chaotic dynamics
The injection, capture, evolution, and escape of DMPs is computed as described by Lages & Shepelyansky (2013): we numerically modeled a constant flow of scattered DMPs with an energy distribution d per time unit (we used q ≤ q_{max} = 4r_{p}). For Jupiter we have u ≈ 17 ≫ 1 and dN_{s} ∝ dqdw. However, for an SMBH we can have u^{2}<J so that one kick captures almost all the DMPs from the galactic distribution f(w). In this case, we used the whole distribution f(w) (w = v^{2}). Map (2) is simpler than (1) since the kick function only depends on q, which allows performing simulations with more DMPs.
Fig. 3 a) Number N_{cap} of captured DMPs as a function of time t in years for the energy range w> 0 (black curve), w> 4 × 10^{5} corresponding to half the distance between Sun and the Alpha Centauri system (red curve), w> 1/20 corresponding to r< 100 AU (blue curve); N_{J} = 4 × 10^{11} DMPs are injected during SS lifetime t_{S}; data are obtained from the map (2) at J = 0.005, u = 17 corresponding to the SunJupiter case. b) Top part: density distribution ρ(w) ∝ dN/ dw in energy at time t_{S} (normalized as ), bottom part: Poincaré section of the map (2); inset: density distribution of the captured DMPs in w (black curve), the red line shows the slope –3/2. 

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The scattering and evolution processes were followed during the whole lifetime t_{S} of the SS. The total number of DMPs, injected during time t_{S} for  w ≤ J and all q is N_{J}. For the Kepler map the highest value is N_{J} = 4 × 10^{11}, which is 100 times higher than for the dark map.
Fig. 4 a) Stationary radial density ρ(r) ∝ dN/ dr from the Kepler map at J = 0.005 with u = 17 at time t_{S} (red curve) and u = 0.035 at time t_{u} ≈ 4 × 10^{8}T_{p} (black curve); data from the dark map at m_{p}/M = 10^{3} are shown by the blue curve at u = 17 and time t_{S} for the SunJupiter case, and by the green curve at u = 0.035 and t_{S} for the SMBH; the normalization is fixed as , r_{p} = 1. b) Volume density ρ_{v} = ρ/r^{2} from the data of panel a), the dashed line shows the slope −2. 

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The time dependence N_{cap}(t) for the Kepler map, shown in Fig. 3, is very similar to that found for the dark map by Lages & Shepelyansky (2013). For a finite SS region w> 1/20 the growth of N_{cap}(t) saturates after a time scale of t_{d} ≈ 10^{7} yr. This scale approximately corresponds to a diffusive escape time t_{d} ~ 12 yr /D ~ 10^{6} yr, where the diffusion rate is taken in a random phase approximation to be D ≈ J^{2}/ 2 (see, e.g., Casati et al. 1987). The diffusive spreading extends from w ~ 0 up to chaos border w_{ch} ≈ 0.3. This value agrees well with the theoretical value w_{ch} = (3πJ)^{2/5} = 0.29 obtained from the Chirikov criterion (Chirikov 1979; see discussion for DMP dynamics in Petrosky 1986; Casati et al. 1987; Khriplovich & Shepelyansky 2009). The validity of the Chirikov criterion in this system was also demonstrated in Shevchenko (2015). As for the dark map, we obtain a density distribution of ρ(w) ∝ 1 /w^{3/2}, corresponding to the ergodic estimate according to which ρ(w) is proportional to time period at a given w. The results of Figs. 1–3 confirm the close similarity of dynamics described by maps (1) and (2).
5. Radial variation of the dark matter density
To compute the DMP density, we considered captured orbits N_{AC} with w> 4 × 10^{5}. The radial density ρ(r) was computed by the method described by Lages & Shepelyansky (2013): N_{AC} were determined at instant time t_{S}; for them the dynamics in real space was recomputed during a time period Δt ~ 100 yr of planet. The value of ρ(r) was computed by averaging over k = 10^{3} points randomly distributed over Δt for all N_{AC} orbits.
We also checked that a semianalytical averaging, using an exact density distribution over Kepler ellipses for each of N_{AC} orbits, gives the same result: assuming ergodicity ρ_{w,q}(r)dr = w^{3/2}dt/ 2π and using Kepler’s equation, the radial density of the DMPs on a given orbit is ρ_{w,q}(r) = (rw^{2}/ 2π)((1 − qw)^{2} − (1 − rw)^{2})^{− 1/2}, then adding the radial density of each N_{AC} orbit, we retrieve the DMP radial density ρ(r) shown in Fig. 4. From the obtained space distribution we determine a fraction η_{ri} of N_{AC} DMP orbits located inside a range 0 ≤ r ≤ r_{i} by computing η_{ri} = ΔN_{i}/ (kN_{AC}), where ΔN_{i} is the number of points inside the above range (we used r_{i}/r_{p} = 0.2,1,and6).
In Fig. 4 we show the dependence of radial ρ(r) and volume ρ_{v} = ρ/r^{2} densities on distance r. For the Kepler map data, the density ρ(r) has a characteristic maximum at r_{max} that is determined by the chaos border position r_{max} ≈ 2 /w_{ch} (this dependence, as well as the relation w_{ch} = (3πJ)^{2/5}, is numerically confirmed for the studied range 10^{3}<J< 10^{2} for the Kepler map with a given fixed J). The density profile ρ(r) is not sensitive to the value of u and remains practically unchanged for u = 17, 0.035. For the dark map a variation of the kick function with q and angles leads to a variation of w_{ch} that leads to a slow growth of ρ at large r. A powerlaw fit of ρ_{v} ∝ 1 /r^{β} in a range 2 <r< 100 gives β ≈ 2.25 ± 0.003 for the Kepler map data and β = 1.52 ± 0.002 for the dark map. We attribute the difference in β values to a larger fraction of integrable islands for the dark map, as is visible in Fig. 1 for typical parameters. We note that an effective range of radial variation is bounded by the kick amplitude with r<r_{cap} ≈ 1 /J, and in the range r_{p}<r<r_{cap} the data are compatible with ρ ~ const. (dashed line in Fig. 4b).
We note that the value of u does not significantly affect the density variation with r, as is clearly seen in Fig. 4. The spacial density distribution of computed from the dark map at u = 0.035 shown in Fig. 5 is also very similar to those at u = 17 (see Fig. 5 by Lages & Shepelyansky 2013). This independence of u arises because ρ(r) is determined by the dynamics at w> 0, which is practically insensitive to the DMP energies at −J<w< 0 that are captured by one kick.
Fig. 5 Density of captured DMPs at present time t_{S}/T_{p} ≈ 4 × 10^{8} for the dark map at m_{p}/M = 10^{3} and u/v_{p} = 0.035. Top panels: DMP surface density ρ_{S} ∝ dN/ dzdr_{ρ} shown at the left in the cross plane (0,y,z) perpendicular to the planetary orbit (data are averaged over ), at the right in the planet plane (x,y,0); only the range  r ≤ 6 around the center is shown. Bottom panels: corresponding DMP volume density ρ_{v} ∝ dN/ dxdydz at the left in the plane (0,y,z), at the right in the planet plane (x,y,0); only the range  r ≤ 2 around the SMBH is shown. The color is proportional to the density with yellow/black for maximum/zero density. 

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6. Enhancement of dark matter density
To determine the enhancement of the DMP density captured by a binary system we followed the method developed by Lages & Shepelyansky (2013). We computed the total mass of DMP flow crossing the range q ≤ 4r_{p} during time t_{S}: , where we used the crosssection for injected orbits with q ≤ 4r_{p}, w = v^{2}, k is the gravitational constant. For SS at u/v_{p} ≈ 17 we have M_{tot} ≈ 0.5 × 10^{6}M.
Fig. 6 Dependence of the DMP density enhancement factor ζ = ρ_{v}(r_{i}) /ρ_{gJ} on J at u/v_{p} = 17 (Jupiter); here ρ_{gJ} is the galactic DMP volume density for an energy range of 0 < w <J and r_{i} = 0.2r_{p},r_{p},6 r_{p} (blue, black, red); points and squares show results for map (2) with the number of injected particles N_{J} = 4 × 10^{9} and 4 × 10^{11}, respectively; crosses show data for map (1) with N_{J} = 4 × 10^{9} and J = 5m_{p}/M. b) Dependence of the galactic enhancement factor ζ_{g} = ρ_{v}(r_{i}) /ρ_{g} on u/v_{p} at r_{ζ} = r_{p} and J = 0.005 in (2) (points) and m_{p}/M = 0.001 in (1) (crosses), here ρ_{g} is the global galactic density; lines show dependencies ζ_{g} ∝ 1 /u (red) and ζ_{g} ∝ 1 /u^{3} (blue). c) Dependence of ζ_{g} on J at u/v_{p} = 17; d) the same at u/v_{p} = 0.035, parameters of symbols are as in a), b). The green curve shows theory (3) in all panels. 

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From the numerically known fractions η_{ri} of the previous section and the fraction of captured orbits η_{AC} = N_{AC}/N_{tot} we find the mass M_{ri} = η_{ri}η_{AC}M_{tot} inside the volume of radius r<r_{i} (r_{i} = 0.2r_{p};r_{p};6r_{p}). Here N_{tot} is the total number of injected orbits during the time t_{S}, while the number of orbits injected in the range  w <J (only those can be captured) is . For J ≪ u^{2} we have κ = N_{tot}/N_{J} = 2u^{2}/ (3J) ≈ 3.8 × 10^{4} for u/v_{p} = 17 and κ = 1 for u/v_{p} = 0.035 at J = 0.005. Thus for u/v_{p} = 17 the number of orbits, injected at 0 < w <J, N_{J} = 4 × 10^{11}, corresponds to the total number of injected orbits N_{tot} ≈ 1.5 × 10^{16}. Finally, we obtain the global density enhancement factor ζ_{g}(r_{i}) = ρ_{v}(r_{i}) /ρ_{g} ≈ 16πη_{ri}η_{AC}(r_{p}/r_{i})^{3}τ_{S}v_{p}/u, where τ_{S} = t_{S}/T_{p} is the injection time expressed in the number of planet periods T_{p} = 2πr_{p}/v_{p}. For u^{2} ≫ J it is useful to determine the enhancement ζ = ρ_{v}(r_{i}) /ρ_{gJ} of the scattered galactic density in the range 0 < w <J, whose density is ρ_{gJ} ≈ 1.38ρ_{g}J^{3/2}(v_{p}/u)^{3}. Thus ζ = 0.72ζ_{g}(u/v_{p})^{3}/J^{3/2}.
The results of the DMP density enhancement factors ζ and ζ_{g} are shown in Fig. 6. At (u/v_{p})^{2} ≫ J we have ζ ≫ 1 and ζ_{g} ≪ 1. At u/v_{p} = 17 we find that ζ ∝ 1 /J (the fit gives exponent a = 1.04 ± 0.01) and (the fit exponent is a = 0.46 ± 0.1) in agreement with the above relation between ζ and ζ_{g}. In general, we have ζ_{g} ∝ 1 /u for and ζ_{g} ∝ 1 /u^{3} for . There is only weak variation of ζ_{g} with J for . The values of ζ and ζ_{g} have similar values for the dark and Kepler maps (a part of the fact that at r_{i} = 0.2r_{p} and r_{i} = r_{p} the dark map has approximately the same ζ since there ρ_{v}(r) ~ const. for r ≤ r_{p}).
All these results can be summarized by the following formula for the chaotic enhancement factor of DMP density in a binary system: (3)Here ζ_{g} is given for DMP density at r_{i} = r_{p} and A ≈ 15.5, B ≈ 0.7. This formula describes the numerical data of Fig. 6 well. For (u/v_{p})^{2} ≫ J we have ζ_{g} ≪ 1, but we still have an enhancement of ζ = 0.72ζ_{g}(u/v_{p})^{3}/J^{3/2} ≈ 0.72A/J ≫ 1. For (u/v_{p})^{2} ≪ J we have the global enhancement . The color representation of dependence (3) is shown in Fig. 7.
Fig. 7 Logarithm of DMP density enhancement factor log _{10}ζ_{g} from (3), shown by color and log valuelevels, as a function of u/v_{p} and J; two points are for J = 0.005, u/v_{p} = 17 (SS) and u/v_{p} = 0.035 (SMBH; such v_{p} is about 2% of the light velocity). 

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Equation (3) can be understood on the basis of simple estimates. The total captured mass M_{cap} ≈ M_{AC} is accumulated during the diffusive time t_{d} and hence , where τ_{d} = t_{d}/T_{p}, and we omit numerical coefficients. This mass is concentrated inside a radius r_{cap} ~ 1 /J so that at r ~ 1 /J the volume density is , where we use a relation . (Our modeling of the injection process in the Kepler map with a constant injection flow in time, counted as the number of map iterations, shows that the number of absorbed particles scales as N_{K} ~ τ_{d} ~ J^{− 6/5} at small J.) It is important to stress that ρ_{v}(r = 1 /J) ≪ ρ_{gJ} in contrast to the naive expectation that ρ_{v}(r = 1 /J) ~ ρ_{gJ}. Using our empirical density decay ρ_{v} ∝ 1 /r^{β} with β ≈ 2.25 for the Kepler map, we obtain ζ ∝ 1 /J^{0.95}, which is close to the dependence ζ ~ 1 /J and ζ_{g} ~ J^{1/2}/ (u/v_{p})^{3} from (3) at u^{2} ≫ J. For the dark map we have β ≈ 1.5 but w_{ch} ~ const. as a result of the sharp variation of F(x) with x, which again gives ζ ~ 1 /J. It is difficult to obtain the exact analytical derivation of the relation ζ ~ 1 /J due to contributions of different q values (which have different τ_{d}) and different kick shapes in (1) that affect τ_{d} and the structure of chaotic component. In the regime (u/v_{p})^{2} ≪ J the entire energy range of the scattering flow is absorbed by one kick, and M_{cap} is increased by a factor (u/v_{p})^{2}/J, leading to an increase of ζ_{g} by the same factor, which yields in agreement with (3).
We note that for galaxies the value of exponent β is debated (see Merritt 2013). For the adiabatic growth model, we have 2.25 ≤ β ≤ 2.5, which is close to the value obtained from our symplectic map simulations.
Fig. 8 Density distribution of DMPs ρ(q) over q obtained from the Kepler map at J = 0.005 and time t_{u} ≈ 4 × 10^{8}T_{p}: a) u/v_{p} = 17; b) u/v_{p} = 0.04; the density is normalized to unity (). 

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The nontrivial properties of the distribution of the captured DMPs in q are shown in Fig. 8 in a stationary regime at times t_{S}/T_{p} ≈ 4 × 10^{8} for the Kepler map. While for u/v_{p} ~ 17 ≫ 1 we have a smooth drop of DMP density ρ(q) at q> 1.5r_{p}, for u/v_{p} = 0.04 ≪ 1 we have an increase of ρ(q) by a factor 3 for q/r_{p} ≈ 2.5 compared to q/r_{p} ≈ 1. We attribute this variation to different capture conditions at , where only DMPs at low velocities are captured by one kick, and , where practically all DMPs are captured by one kick. As a result of the dependence of J on q, we also have various diffusive timescales t_{d} ∝ 1 /J^{2} that can affect the contribution of the DMPs at different q values in the volume density distribution on r.
Finally, we stress the importance of the obtained result of large enhancement factors ζ and ζ_{g}. This result is drastically different from the frequent claims that there is no enhancement of the DMP density in the center vicinity of a binary system compared to its galactic value because of the Liouville theorem, which implies that the density of DM in the phase space is conserved during the evolution (Gould & Alam 2001; Lundberg & Edsjö 2004). However, this statement does not take into account the actual dynamics of captured DMPs. Indeed, the galactic space density ρ_{g} is obtained from all energies of DMPs in the Maxwell distribution. The analysis of symplectic DMP dynamics shows that DMPs at large q ≫ 1 are not captured, while DMPs with q ~ 1 are captured, and by diffusion, they penetrate up to high values w ~ w_{ch}, thus accumulating DMPs with typical distance values r ~ 1 /w_{ch}. The symplectic map approach also determines an effective size of our binary system of r_{cap} ~ 1 /J corresponding to an energy range w ~ J. If we assume that the DMP density in this range is the same as its galactic value, then we should conclude that the enhancement factor should be ζ_{g} ~ (r_{cap}/r_{p})^{β} ~ 1 /w_{cap}^{β} ~ 1 /J^{β} ~ 1.5 × 10^{5} for typical values J = 0.005 and β = 2.25 (we consider here the case ). This estimate gives a value ζ_{g} that is even higher than that given by relation (3). In fact, relation (3) takes into account that only bounded values of q are captured, it also estimates the chaos region, where DMPs are accumulated during the chaotic diffusion process, populating a part of the phase space volume from w ~ 0 up to w ~ w_{ch} ~ 1. This gives a lower value of ζ_{g} than the above simplified estimate. We also note that at the typical kinetic energy of an ejected DMP is significantly higher than the typical DMP energy u^{2} in the galactic wind. For these reasons, there is no contradiction with the Liouville theorem, and a large enhancement of the captured DMP density is possible.
7. Fewbody model
Fig. 9 a) Radial density ρ(r) ∝ dN/ dr for the Kepler models of SS (red curve) and SMBH binary (black curve) at t_{S}/T_{p} ≈ 4 × 10^{8}; the normalization is fixed as , r_{p} = 1 for the fifth body. b) Volume density ρ_{v} = ρ/r^{2} from the data of a), the dashed line shows the slope −2 (see text for details). 

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Above we considered the DMP capture in a twobody gravitating system. We expect that a central SMBH binary dominating the galaxy potential can be viewed as a simplified galaxy model. Recent observations of Graham et al. (2015) indicate that such systems may exist. Within the Kepler map approach it is easy to analyze the whole SS (an SMBH binary) including all eight planets (eight stars) with given positions r_{i} and velocities v_{i} measured in units of orbit radius r_{p} and velocity v_{p} of Jupiter for SS at u/v_{p} = 17 (and of, e.g., the fifth star for an SMBH binary at u/v_{p} = 0.035). Thus in (2) we have now for the SS eight kick terms with J_{i} ~ (m_{i}/M)(v_{i}/v_{p})^{2}. For the SMBH binary model we consider eight stars modeled by map (2) with the values J_{1} = 2.5 × 10^{4}, J_{2} = 5 × 10^{4}, J_{3} = 7.5 × 10^{4}, J_{4} = 10^{3}, J_{5} = 2.5 × 10^{3}, J_{6} = 6.25 × 10^{4}, J_{7} = 5 × 10^{4}, and J_{8} = 1.25 × 10^{4} with the same ratio r_{i}/r_{p} as for the SS. In both cases we injected N_{J} = 2.8 × 10^{10} particles considering evolution during τ_{S} orbital periods of Jupiter (fifth star). The steadystate density distribution is shown in Fig. 9. For the SS, ρ(r) is very close to the case of only one Jupiter discussed above. This result is natural since its mass is dominant in the SS. For the SMBH binary model we also find a similar distribution (see Fig. 4) with a slightly slower decay of ρ_{v}(r) with r (β = 2.06 ± 0.002) due to the contribution of more stars. We obtain ζ = 3000 (SS) and ζ_{g} = 3 × 10^{4} (SMBH). These two examples show that the binary model captures the main physical effects of the DMP capture and evolution.
8. Discussion
Our results show that DMP capture and dynamics inside twobody and fewbody systems can be efficiently described by symplectic maps. The numerical simulations and analytical analysis show that in the center of these systems the DMP volume density can be enhanced by a factor ζ_{g} ~ 10^{4} compared to its galactic value. The values of ζ_{g} are highest for a high velocity v_{p} of a planet or star rotating around the system center. We note that our approach based on a symplectic map description of the restricted threebody problem is rather generic. Thus it can also be used to analyze comet dynamics, cosmic dust, and freefloating constituents of the Galaxy.
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All Figures
Fig. 1 Poincaré sections for the dark map (1) (top) and the Kepler map (2) (bottom) for parameters of the Halley comet case in Eq. (1) and J = 0.007 in Eq. (2) (see text). 

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In the text 
Fig. 2 a) Dependence of the capture crosssection σ on DMP energy w for SunJupiter (black curve, data from Ref. (8)) and for the Kepler map at J = 0.005 (red curve); the dashed line shows the dependence σ ∝ 1/w . b) Dependence of the rescaled captured number of DMPs on energy w for the models of the left panel. Here w_{cap} = 0.001. 

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In the text 
Fig. 3 a) Number N_{cap} of captured DMPs as a function of time t in years for the energy range w> 0 (black curve), w> 4 × 10^{5} corresponding to half the distance between Sun and the Alpha Centauri system (red curve), w> 1/20 corresponding to r< 100 AU (blue curve); N_{J} = 4 × 10^{11} DMPs are injected during SS lifetime t_{S}; data are obtained from the map (2) at J = 0.005, u = 17 corresponding to the SunJupiter case. b) Top part: density distribution ρ(w) ∝ dN/ dw in energy at time t_{S} (normalized as ), bottom part: Poincaré section of the map (2); inset: density distribution of the captured DMPs in w (black curve), the red line shows the slope –3/2. 

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In the text 
Fig. 4 a) Stationary radial density ρ(r) ∝ dN/ dr from the Kepler map at J = 0.005 with u = 17 at time t_{S} (red curve) and u = 0.035 at time t_{u} ≈ 4 × 10^{8}T_{p} (black curve); data from the dark map at m_{p}/M = 10^{3} are shown by the blue curve at u = 17 and time t_{S} for the SunJupiter case, and by the green curve at u = 0.035 and t_{S} for the SMBH; the normalization is fixed as , r_{p} = 1. b) Volume density ρ_{v} = ρ/r^{2} from the data of panel a), the dashed line shows the slope −2. 

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In the text 
Fig. 5 Density of captured DMPs at present time t_{S}/T_{p} ≈ 4 × 10^{8} for the dark map at m_{p}/M = 10^{3} and u/v_{p} = 0.035. Top panels: DMP surface density ρ_{S} ∝ dN/ dzdr_{ρ} shown at the left in the cross plane (0,y,z) perpendicular to the planetary orbit (data are averaged over ), at the right in the planet plane (x,y,0); only the range  r ≤ 6 around the center is shown. Bottom panels: corresponding DMP volume density ρ_{v} ∝ dN/ dxdydz at the left in the plane (0,y,z), at the right in the planet plane (x,y,0); only the range  r ≤ 2 around the SMBH is shown. The color is proportional to the density with yellow/black for maximum/zero density. 

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In the text 
Fig. 6 Dependence of the DMP density enhancement factor ζ = ρ_{v}(r_{i}) /ρ_{gJ} on J at u/v_{p} = 17 (Jupiter); here ρ_{gJ} is the galactic DMP volume density for an energy range of 0 < w <J and r_{i} = 0.2r_{p},r_{p},6 r_{p} (blue, black, red); points and squares show results for map (2) with the number of injected particles N_{J} = 4 × 10^{9} and 4 × 10^{11}, respectively; crosses show data for map (1) with N_{J} = 4 × 10^{9} and J = 5m_{p}/M. b) Dependence of the galactic enhancement factor ζ_{g} = ρ_{v}(r_{i}) /ρ_{g} on u/v_{p} at r_{ζ} = r_{p} and J = 0.005 in (2) (points) and m_{p}/M = 0.001 in (1) (crosses), here ρ_{g} is the global galactic density; lines show dependencies ζ_{g} ∝ 1 /u (red) and ζ_{g} ∝ 1 /u^{3} (blue). c) Dependence of ζ_{g} on J at u/v_{p} = 17; d) the same at u/v_{p} = 0.035, parameters of symbols are as in a), b). The green curve shows theory (3) in all panels. 

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In the text 
Fig. 7 Logarithm of DMP density enhancement factor log _{10}ζ_{g} from (3), shown by color and log valuelevels, as a function of u/v_{p} and J; two points are for J = 0.005, u/v_{p} = 17 (SS) and u/v_{p} = 0.035 (SMBH; such v_{p} is about 2% of the light velocity). 

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In the text 
Fig. 8 Density distribution of DMPs ρ(q) over q obtained from the Kepler map at J = 0.005 and time t_{u} ≈ 4 × 10^{8}T_{p}: a) u/v_{p} = 17; b) u/v_{p} = 0.04; the density is normalized to unity (). 

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In the text 
Fig. 9 a) Radial density ρ(r) ∝ dN/ dr for the Kepler models of SS (red curve) and SMBH binary (black curve) at t_{S}/T_{p} ≈ 4 × 10^{8}; the normalization is fixed as , r_{p} = 1 for the fifth body. b) Volume density ρ_{v} = ρ/r^{2} from the data of a), the dashed line shows the slope −2 (see text for details). 

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In the text 