Analytical models of supermassive black holes in galaxies surrounded by dark matter halos

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I. INTRODUCTION
The overwhelming evidence for dark matter (DM) is clear and compelling on a wide range of scales [1][2][3][4][5].The motion of galaxies within clusters [6,7], galaxy rotation curves [8,9], bullet cluster systems [10], baryonic acoustic oscillation [11] and cosmic microwave background [12] all point to its existence.According to the latest Planck data [13], about 5/6 of the mass in the Universe is made of DM.However, despite of numerous efforts in the search of DM in the past decades, our understanding is still based only on their gravitational interactions.Therefore, uncovering the nature and origin of DM is arguably one of the greatest challenges of modern physics and cosmology [14][15][16].
According to the current galaxy formation scenario, every galaxy forms within a DM halo, and at the center of almost every galaxy a supermassive black hole (SMBH) is present [17].As a matter of fact, two such SMBHs have been already observed by the Event Horizon Telescope (EHT), one is the Sagittarius A * black hole (BH) at the centre of our own Milky Way galaxy [18], and the other called M 87 * is located at the centre of the more distant Messier 87 galaxy [19].With the arrival of the gravitational wave (GW) astronomy [20], the studies of SMBHs in galaxies with DM halos have gained further momenta, see, for example, [21][22][23][24][25] and references therein.This is because such studies bear important consequences for compact objects orbiting around a SMBH.The mass ratio q ≡ m/M of such binaries is typically of O(10 −3 ∼ 10 −4 ) for intemediate mass-ratio inspirals (IMRIs) [26], and O(10 −5 ∼ 10 −6 ) for extreme massratio inspirals (EMRIs) [27].GWs emitted by IMRIs can be detected by the upcoming ground-based detectors, such as the Einstein Telescope [28] and the Cosmic Explorer [29], and the ones from EMRIs are ideal targets for space-based detectors, such as LISA [30], TianQin [31], Taiji [32] and DESIGO [33].
When a compact object moves inside the DM halo, gravitational drags will be induced on the inspiraling object, changing its orbit and gravitational waveforms emitted by it.For IMRIs, such effects can be detected by LISA [34][35][36].Similar effects for EMRIs are also expected and can be used to study the nature of DM [37][38][39].For binaries with a very small q, the compact objects can be well described as test particles, moving under the influence of SMBHs and DM halos.Therefore, in this Letter we shall focus ourselves on the gravitational field produced by a SMBH in a galaxy with the presence of a DM halo.
Using Newtonian gravity, Gondolo and Silk first studied the effects of a massive BH sitting at the center of a galaxy on the distribution of the DM halo, and found that a spike is always produced at a distance of 4r s , where r s denotes the Schwarzschild radius of the SMBH [40].The orbits of the DM particles with radii less than 4r s become unstable, so that the density of the DM halo vanishes for r 4r s .Sadeghian, Ferrer and Will later took the relativistic effects into full considerations and found a spike is indeed developed [41].However, they found that the orbits of the DM particles now start to be unstable at 2r s , instead of 4r s , purely due to the relativistic effects, which is further confirmed by Speeney et al [42].Due to the complexity of the problem, such studies have been carried out mainly numerically, except one particular case [43], in which Cardoso et al constructed an analytic model with a density profile that asymptotically approaches the Hernquist density contribution [44].This solution has immediately attracted lots of attention, and its various properties and applications have already been explored, including its Love numbers, quasi-normal modes (QNMs), tails and grey-body factors [43,[45][46][47], shadows [48], and the detectability of GWs emitted by EMRIs by LISA, TianQin and Taiji [49], to name only a few of them.Despite of all its interesting properties and aspects, the model suffers from a couple of defects, for example, the density of the DM halo vanishes only inside the BH horizon, instead in the region r ≤ 2r s , as shown explicitly in [41,42].In addition, near the BH horizon, the DM halo does not satisfy the dominant energy condition [50].Extensions of this model to different circumstances have been also studied, see, for example, [51][52][53] and references therein.
In this Letter, we present three analytic solutions, each of which represents a SMBH located at the center of a galaxy surrounded by a DM halo, in which the energy density of the halo satisfies all the energy conditions, weak, strong and dominant [50], and identically vanishes 1for r ≤ 2r s .The spacetime is smoothly joint to the vacuum Schwarzschild BH solution across r = 2r s .The difference among the three models lies in the slopes of the density profiles in the far region from the galaxy center.

II. GENERAL SETUP
Let us consider a spherical Einstein cloud made of collisionless DM particles all moving tangentially with zero total angular momentum.Then, the cloud can be considered as a fluid with a negligible radial pressure [43].The corresponding energy-momentum tensor of the DM halo is given by T µ ν = (−ρ(r), 0, P (r), P (r)), here ρ and P denote respectively the energy density and tangential pressures of the halo.The spacetime is describe by where f (r) and m(r) are solutions of the Einstein field equations.Then, the Einstein field equations have three independent equations, given respectively by where a prime denotes the derivative wrt r.Note that in the current case we have four unknowns but only three independent equations.To determine them uniquely, the equation of state can be imposed.In this Letter, instead we consider the choice [42,54] where a represents the characteristic scale of the DM halo, the parameters (α, β, γ) control the slopes of the density profile in the near (r ≪ a), far (r ≫ a) and transition (r ≃ a) regions, and ρ 0 is a characteristic density of the halo, while M the BH mass.Observationally, it is found that 0.005kpc a 50kpc.Recall that r s ≡ 2M = 9.608×10 −11 (M/10 6 M ⊙ ) kpc.So, in general we have a ≫ r s .The above model includes several important cases [54], such as (α, β, γ) = {(1, 4, 1), (1, 3, 1)}, which correspond respectively to the Hernquist [44] and Navarro-Frenk-White (NFW) [55] profiles without BHs (M = 0).As mentioned above, when a central BH is present, the trajectories of the DM particles become unstable for r < 2r s and will eventually fall inside the BH, whereby a spike is developed [40][41][42].So, we must have ρ(r) = 0, f (r) = f o (1 − r s /r) and m(r) = M for r < 2r s , where f o is a constant, which will be determined by the matching conditions across r = 2r s .Note that in the rest of the Letter we shall use both M and r s to denote the BH mass and radius.From (1) we can see that the Einstein field equations contain only the first-order derivatives of f and m.Therefore, as long as they are continuous, these equations will be satisfied.For the choice (2), it is clear that ρ and P are already continuous, while f can be always made continuous by properly choosing f o .Therefore, in the following we shall impose the boundary condition m(2r s ) = M .With such choices of f o and m(r), the Einstein field equations ( 1) can be solved in the whole spacetime r ∈ (0, ∞).In addition, we shall also assume that the spacetime is asymptotically flat, so that f (∞) = 1 and m(∞) = M + M h , where M h is the total mass of the DM halo.Observations show that M ≪ M h ≪ a [1,42,55].

III. ANALYTICAL MODELS
With the above setup, we are ready to integrate (1) for the choice of the density profile (2), for which we find a large class of analytical solutions.In the following, we shall present only three of them, as the expressions of others are rather lengthy and shall present them in detail in the following-up paper [56].
Model I: The first solution corresponds to the Hernquist choice of (α, β, γ) and is given by where , and M h = 2πρ 0 a 4 /(a + 4M ), and where p ≡ B − A 2 /3, q ≡ −2A 3 /27 + AB/3 − C, ∆ ≡ q 2 + 4p 3 /27.It can be shown that ∆ > 0 for 10M < M h < a/10 [56], so that the two roots x 2,3 of D 1 (x) = 0 are complex conjugate, while x 1 is real.It can be also shown that the matching conditions at r = 2r s and the asymptotically-flat conditions are all satisfied.
Model II: In this model, we choose (α, β, γ) = (1, 5, 1).Then, we find but now with M h ≡ 2πa 5 ρ 0 /[3(a + 4M ) 2 ].Inserting the above expression into (1), we find  [56] in terms of a, M and M h .Then, we find that where n − 2y 3R y n .It can be shown that the above solution satisfies the asymptotically-flat and matching conditions mentioned above, by properly choosing the integration constant f o of f (r) in the region r < 2r s .
To show that all three energy conditions hold also in the current case, we first note that G ′′ (r) = 12M h (a + 4M ) 2 (3r − a − 16M )/(a + r) 5 , which vanishes at r min = (a + 16M )/3.This corresponds to a minimum of G ′ (r), where Hence, in this region G(r) is monotonically increasing and G(r) ≥ r s .Then, the DM halo also satisfies the weak and strong energy conditions.To check the dominant energy condition, we first note that the maximum of [m(r)/r] is 1/4 for 10M < M h < a/10.Then, we find that ρ − P = (5 − r/G)ρ/4 > (5 − 1/(1 − 0.5))ρ > 0. Hence, the dominant energy condition also holds in the present case.
Model III: In this model, we choose (α, β, γ) = (1, 5  2 , 1).Then we find where M ≡ 8πGa 3 ρ 0 , and , where v ≡ M/a, u ≡ 2M/a.Thus, D 3 (z) = 0 has three roots, denoted by z 1,2,3 .Depending on the values of u and v, the nature of these roots are different.In particular, if u < 0.1709, we find that all of them are real.If u > 0.1716, two of them are complex conjugate, and only one is real, say, z 1 .If 0.1709 < u < 0.1716, we find that whether the three roots are all real or not depends on the values of v.In the following, let us consider the two cases separately.
Before doing so, it is interesting to note that m(r) ∝ √ r → ∞ as r → ∞.Thus, now the mass of the DM halo becomes infinitely large.In fact, this is always the case for β ≤ 3, including the NFW model [55], for which β = 3 and m(r) ∝ ln r.In reality, a halo is always restricted to a finite region, so one needs to restrict the expression of ( 7) to be valid only up to a maximal radius, say, r ∞ .Then, the total mass of the halo will be finite.In our current case, the mass does not increase fast enough to close the whole spacetime, so we still have an asymptotically-flat spacetime, as now we still have f (∞) = 1 and m(r)/r → 0 as r → ∞, as to be shown below.
With the above in mind, let us first consider the case in which all three roots are real.Then, we find that where 2 )], with z 1,2,3 being the three real roots of D 3 (z) = 0 [56].The above expressions shows that indeed we have f (∞) = 1.On the other hand, from (7) we find that m(2r s ) = M .Therefore, by properly choosing the integration constant f o in the region r < 2r s , the solution can be smoothly matched to that of the Schwarzschild vacuum solution valid in the region r ≤ 2r s .We have also found that the corresponding DM halo satisfies all the three energy conditions.However, the proof is rather tedious and will be presented in detail in [56].So, here we shall omit the proof.The same will be done for the case in which only one of the three roots is really.
When only one of the three roots is real, which will be chosen as z 1 , we find where . Thus, we have f (∞) = 1 and m(r)/r → 0 as r → ∞.As a result, the spacetime is asymptotically flat, despite the fact m(r) ∼ M r/a for r ≫ a.

IV. CONCLUSIONS
In this Letter, we have presented three analytic models, each of which describe a SMBH sitting at the center of a galaxy and being surrounded by a DM halo.The halo satisfies all the three energy conditions, weak, strong and dominant [50], and vanishes within two Schwarzschild radii, r ≤ 2r s , of the SMBH.The latter is consistent with previous investigations of relativistic DM halo models in the presence of a SMBH [41,42], in which it was shown that the trajectories of DM particles become unstable when r < 2r s and these particles shall all fall inside the BH.As a result, we have ρ = 0 for r < 2r s .To our best knowledge, these are the only analytic models found so far that have such desirable properties.
We expect that the construction of these analytic models will significantly facilitate the studies of QNMs of SMBHs and GWs emitted by IMRIs and EMRIs, which are the ideal sources of the next-generation GW detectors, both ground-and space-based detectors [28][29][30][31][32][33].As the small compact companion of the SMBH passes through the DM halo, it feels gravitational dragging forces, whereby its orbits and gravitational waveforms emitted by it will be changed.Such effects can be detected by LISA and other next generation detectors [34][35][36][37][38][39].Hence.a completely different window to study the nature of DM is provided.
Certainly, the applications of these models are not restricted to the studies mentioned above but also equally applicable to other fields, such as shadows of BHs, which can be the sources of EHT [18,19], and the dynamics of galaxy systems, including our own Milky Way galaxy [1-5, 44, 55].