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Fate of scalar dark matter solitons around supermassive galactic black holes

T0 review · 4 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A self-interacting scalar dark-matter soliton falling into a Schwarzschild black hole reaches a unique steady state with a critical flux of order $r_s^2 m^4/\lambda_4$, and in that state it survives many Hubble times.

desk verdict A careful analytic construction of a unique accretion flux for self-interacting scalar solitons; the branch-selection argument is the main soft spot, but it deserves a serious referee. read the letter →

arxiv 1909.02614 v1 pith:RKYC3QL2 submitted 2019-09-05 astro-ph.CO hep-ph

Philippe Brax , Jose A. R. Cembranos , Patrick Valageas This is my paper · ORCID
classification astro-ph.COhep-ph
keywords scalardarkmattersolitonsupermassiveblackholesquarticself-interactionJacobiellipticfunctionconstantfluxselectiontransonicaccretionlifetime
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper asks what happens to a scalar-field dark-matter soliton when a supermassive black hole sits at its centre and steadily pulls the field inward. The paper's central claim is that, for a repulsive quartic self-interaction in the large-mass limit, the infall cannot be arbitrary: the requirement of a constant flux and the boundary conditions at the soliton and at the horizon select one unique critical flux, analogous to the transonic accretion solution in hydrodynamics. This flux is small, scaling as $F_c \simeq 0.66\, r_s^2 m^4/\lambda_4$, so the soliton loses mass only slowly. The result matters because it determines whether such solitons can survive to the present day inside galaxies; the paper finds typical lifetimes $t_c \gg t_H$, so scalar solitons around supermassive black holes are not destroyed by accretion. It also predicts a specific density profile, $\langle\rho_\varphi\rangle \propto r^{-1}$ in the black-hole-dominated region, which is testable with stellar-dynamics constraints near galactic centres.

What carries the argument

The central object is the leading-order large-mass ansatz $\varphi=\varphi_0(r)\,\mathrm{cn}[\,\omega(r)t-K(r)\beta(r),k(r)]$, in which $\mathrm{cn}$ is the Jacobi elliptic function and $K(k)$ the complete elliptic integral of the first kind; it generalises the free-field cosine $\cos(mt-s)$ to the anharmonic oscillator generated by the quartic potential. Substituting this ansatz into the nonlinear Klein-Gordon equation turns the radial problem into two algebraic equations: one fixes the amplitude $\varphi_0$ in terms of the modulus $k$, and the other fixes the radial velocity $\beta'$. The argument is carried by the flux function $F(k,x)$ that comes from the averaged energy-momentum conservation; its peak value as a function of radius has a unique minimum $F_c$, and that minimum is the only flux at which the solution can switch from the outer soliton branch to the inner horizon branch, in direct analogy with the transonic accretion solution of spherical hydrodynamics.

What would settle it

A direct check would be to solve the full nonlinear Klein-Gordon equation for the quartic potential on a Schwarzschild background with a soliton-like outer boundary condition and see whether the steady infall approaches the predicted critical flux $F_c\simeq 0.66\, r_s^2 m^4/\lambda_4$ and a $1/r$ density profile down to $r_s/4$; a complementary check is to carry the large-mass expansion to next order and see whether the minimum of the peak flux shifts.

Watch

Extended reading notes

Core claim

Treating the scalar field as a coherent wave $\varphi = \varphi_0(r)\,\mathrm{cn}[\,\omega(r)t-K(r)\beta(r),k(r)]$ and keeping only the leading-order radial gradients in the large-mass limit, the authors reduce the relativistic Klein-Gordon equation to two algebraic conditions for the amplitude and the modulus $k(r)$. Imposing a time-independent averaged flux $F$ through each spherical shell gives a flux function $F(k,x)$ whose maximum over $k$ has a global minimum $F_* \simeq 0.66$ at $x_* \simeq 2.43$, where $x=r/r_s$. Matching to the static soliton at large radii (upper branch $k_2$) and to the free-fall horizon condition at small radii (lower branch $k_1$) forces the flux to take exactly this critical value $F_c$; no other constant flux admits a continuous branch-switching profile. The resulting density profile falls as $1/r$ in the region dominated by the black hole, and the associated soliton depletion time exceeds the Hubble time for the parameters of interest.

Load-bearing premise

The entire calculation rests on assuming that the scalar field can be written as a slowly varying amplitude and modulus times a rapidly oscillating elliptic wave all the way down to the black-hole horizon, even though the oscillation phase diverges logarithmically there; if the neglected smaller radial-gradient terms grow near the horizon, the critical flux would change.

Editorial extensions

If this is right

  • Galactic scalar solitons with $\rho_a\sim 1\,\mathrm{eV}^4$ are not swallowed by their central supermassive black holes: the critical accretion flux gives $t_c\sim 10^3\, t_H (\rho_s/\bar\rho_c)(\rho_a/1\,\mathrm{eV}^4)^{-5/2}(M/10^8M_\odot)^{-2}\gg t_H$.
  • In the black-hole-dominated region the time-averaged energy density and the radial velocity both decay as $1/r$, instead of the $r^{-3/2}$ free-infall profile, so the dark-matter mass inside radius $r$ grows as $M_\varphi(<r)\propto r^2$ and is tiny near the hole.
  • Self-interactions remain important down to the horizon: the field at the Schwarzschild radius is a nonlinear elliptic wave with all odd harmonics, not a simple cosine.
  • Current stellar-dynamics bounds on dark matter around the Milky Way and M87 central black holes are satisfied by orders of magnitude for the galactic-scale soliton parameters considered.
  • Stellar-mass black holes wandering through a soliton cannot deplete it either, adding at most a negligible mass-loss channel.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not explore attractive quartic self-interactions; if the same branch-switching selection applies there, the negative pressure might allow a different critical flux or no steady transonic-type solution, possibly giving much shorter soliton lifetimes.
  • The predicted $1/r$ density profile could be probed indirectly through precise orbital precession of the closest stars or through light-bending effects near Sgr A*, since the dark-matter distribution changes the effective potential at the few-percent level.
  • The analytic method should extend to higher-order self-interactions such as $\phi^6$: the elliptic ansatz would be replaced by a more general periodic solution, but the same minimum-of-the-peak-flux argument should select the accretion rate whenever a single branch switch exists.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. This paper analyzes steady-state accretion of a massive scalar field with a repulsive quartic self-interaction onto a Schwarzschild black hole, in the large scalar-mass limit. The authors introduce a Jacobi-elliptic ansatz, Eq. (51), reduce the nonlinear Klein-Gordon equation to algebraic conditions, and derive the flux function F(k,x), Eq. (84). They argue that the boundary conditions—matching to a static soliton at large radii and to an ingoing free-fall solution near the horizon—force the modulus k(x) to switch branches at the minimum of the maximum-flux curve, thereby selecting a unique critical flux Fc = F* Fs with F* ≈ 0.66, Eq. (91). They derive the density profile <ρ_φ> ∝ r^{-1} in the black-hole-dominated region, Eq. (110), and estimate the soliton lifetime tc ≫ t_H, Eqs. (118)–(119). The free-field case is treated separately and gives an unconstrained flux. The paper closes by noting that dedicated numerical simulations would be useful to confirm the nonlinear dynamics.

Significance. Should the central claim hold, the paper provides a concrete, essentially parameter-free prediction for scalar dark-matter soliton infall onto supermassive black holes: a unique flux scaling as F_c ∼ -r_s^2 m^4/λ4, a universal density slope in the BH-dominated region, and survival of the soliton for many Hubble times. The derivation is explicit and transparent, with the flux obtained from the equations of motion rather than fitted to observables. The comparison with the free-field case and with the hydrodynamic transonic solutions gives useful physical context. The main caveats are that the uniqueness argument is heuristic and the near-horizon validity of the large-mass ansatz is not fully controlled; the quantitative value F_* ≈ 0.66 should therefore be regarded as a well-motivated but unverified prediction until a numerical benchmark is provided.

major comments (4)
  1. [IV D 1, Eq. (84), Figs. 1–3] The uniqueness of the critical flux rests entirely on the assumptions that F(k,x) has a single maximum in k for every radius and that a continuous steady profile cannot switch branches except at the peak. These facts are illustrated numerically but not proven analytically. If F(k,x) had additional extrema for some x, or if the steady solution were allowed to jump between branches, the selection argument would fail. Since the value F_* ≈ 0.66 is the central quantitative output, this gap should be closed by an analytic argument or, preferably, by a targeted numerical solution of the full Klein-Gordon equation (49). The paper itself calls for such simulations in Sec. VI, which reinforces the need for this check.
  2. [IV D 2 and Eq. (98)] The near-horizon branch choice is not fully controlled. The text states that for x → 1/4 the modulus is on the lower branch and 'close to zero', yet Eq. (98) gives k_c(1/4) = k_s ≈ 0.54. The critical solution at the horizon is therefore not in the small-k regime used to justify why the flux is small. This inconsistency should be resolved, as it directly bears on the identification of k1 with the near-horizon branch and hence on the uniqueness argument.
  3. [II B, IV E] The large-mass ansatz (51) is assumed valid down to the Schwarzschild radius, but the phase β diverges logarithmically, Eq. (104), and β' diverges as (r − r_s/4)^{-1}. The paper checks density gradients, Eq. (28), but does not demonstrate that all subleading radial-gradient terms in the Klein-Gordon equation are negligible in this regime. If those terms become important near the horizon, the derived critical flux F_c could change. A controlled asymptotic expansion in powers of (m r_s)^{-1}, or a numerical integration, is needed to justify the leading-order solution all the way to the horizon.
  4. [IV D 2, Eq. (95)] The large-radius matching to the static soliton is explicitly approximate: the text notes that 'there remains a nonzero velocity β′' and a nonzero flux. This is acknowledged, but the uniqueness of the outer branch k2 and the quantitative value of the transition radius rsg depend on the smallness of this residual velocity. No estimate of the matching error is given. The authors should quantify how small the residual velocity is compared with the free-fall velocity at the matching radius, or explain why the precise value does not affect the selected flux.
minor comments (5)
  1. [Title and abstract] The title and abstract contain typographical artifacts ('gal actic'); the manuscript should be proofread before resubmission.
  2. [II B, Eq. (16)] The notation ⟨ρφ⟩ used in Eqs. (46) and (107) is not defined before its first appearance; please define the average over the fast oscillation period when it is introduced.
  3. [Fig. 3 caption] The caption is confusing: it describes both the F = Fc/3 curves and the F = Fc curves, but the phrase 'the inner dotted curves that meet at x⋆' could be read as referring to the dashed curves. Please make the distinction between dashed and dotted curves explicit.
  4. [VI] The discussion calls the study 'fully nonrelativistic' while also referring to 'relativistic infall at small radii'; this wording is internally inconsistent and should be clarified.
  5. [IV D 3, Eq. (100)] The identification v_r = πβ′/(2m) is introduced without explaining the factor π/2; a short justification after Eq. (59) would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

The critical flux is derived from the model equations and boundary conditions, not fitted or imported from prior work; no circular step is exhibited.

full rationale

Walked the derivation chain from Eq. (49) through Eqs. (84), (86), (91), (107), (110), and (118)-(119). Equation (84) is derived, not posited: the flux F is the conserved steady-state flux (83), rewritten using the leading-order Klein-Gordon conditions (58)-(59). The numerical constant Fstar about 0.66 is obtained as the minimum over radius of the maximum of the analytically defined function F(k,x)/Fs (Figs. 1-2, Eq. (91)); it is a computed model prediction, not a fit to any observable and not an input parameter. The only self-citation, [85], supplies the outer soliton boundary condition (Sec. IV B 1) and illustrative parameter values; it does not contain the critical-flux result, and the paper explicitly acknowledges that the matching to the static soliton is approximate and leaves a nonzero velocity (Sec. IV D 2). The branch-switching uniqueness argument is an internal mathematical claim conditional on the single-maximum shape of F(k,x), illustrated but not rigorously proven, and Sec. VI calls for dedicated simulations; that is an unverified assumption or correctness risk, not a circular reduction. No fitted parameter is renamed as a prediction, and no equation reduces by construction to an input of the paper.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters are fitted to data. The numerical constants (F*≈0.66, x*≈2.43, ks≈0.54) are computed from elliptic integrals. The calculation relies on the Schwarzschild test-field approximation, the large-mass limit, the elliptic-function ansatz, the prior soliton solution as an outer boundary condition, and the steady-state assumption.

assumptions (6)
  • domain assumption The metric near the black hole is the fixed Schwarzschild metric in isotropic coordinates, and the scalar field is a test field with negligible self-gravity for r < rsg.
    Used in Eqs. (7)-(10); the scalar cloud only contributes through the Newtonian potential for r > rsg. This neglects backreaction of the scalar energy-momentum on the spacetime near the black hole.
  • domain assumption The scalar mass is large enough that quantum pressure is negligible, m ≫ max(10^-21, 6.7e-19 (M/1e8 Msun)^-1) eV.
    Eqs. (4), (39), (40); this defines the regime of the paper and is required for the hydrodynamical or Madelung description and for the ansatz (51).
  • ad hoc to paper The scalar field takes the leading-order form φ = φ0(r) cn[ω(r)t - K(r)β(r), k(r)] with slowly varying φ0 and k, and β of order m; all subleading terms in the large-m expansion are neglected.
    Eq. (51) and Section IV A; the flux formula (84) and the critical solution are derived from this ansatz. Near the horizon the phase diverges logarithmically in isotropic coordinates, so control of subleading terms is argued but not fully proven.
  • domain assumption The large-radius boundary condition is the self-interaction-supported soliton ρ(r)=ρs sin(r/ra)/(r/ra) with ρa=4m^4/(3λ4), taken from the authors' earlier work [85].
    Section IV B 1; sets the outer branch k≈k+ and the matching radius rsg. A different soliton profile would change the outer boundary condition and the lifetime estimates.
  • domain assumption A steady state exists with a time-independent angle-averaged flux F and a common oscillation frequency ω0 at all radii.
    Sections III B and IV C; Eq. (83) defines the constant flux used to select the critical solution. The paper argues the soliton is quasi-static because the accretion timescale is much longer than the Hubble time.
  • standard math Standard identities and definitions of Jacobi elliptic functions and complete elliptic integrals are used.
    References [91-93]; these are standard mathematical tools, not new physical assumptions.

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Pith. "Pith review of Fate of scalar dark matter solitons around supermassive galactic black holes." pith.science (2026). https://pith.science/paper/RKYC3QL2

@misc{pith2026190902614,
  author       = {Pith},
  title        = {Pith review of: Fate of scalar dark matter solitons around supermassive galactic black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RKYC3QL2}},
  note         = {Machine review of arXiv:1909.02614}
}
read the original abstract

In scalar-field dark matter scenarios, a scalar-field soliton could form at the center of galactic halos, around the supermassive black holes that sit at the center of galaxies. Focusing on the large scalar-mass limit, where the soliton is formed by the balance between self-gravity and a repulsive self-interaction, we study the infall of the scalar field onto the central Schwarzschild black hole. We derive the scalar-field profile, from the Schwarzschild radius to the large radii dominated by the scalar cloud. We show that the steady state solution selects the maximum allowed flux, with a critical profile that is similar to the transonic solution obtained for the hydrodynamic case. This finite flux, which scales as the inverse of the self-interaction coupling, is small enough to allow the dark matter soliton to survive for many Hubble times.

Figures

Figures reproduced from arXiv: 1909.02614 by the authors.

Figure 1
Figure 1. FIG. 1: Normalized flux [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Peak value [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Moduli [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Scalar-field energy density computed in the Ed [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]

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Works this paper leans on

130 extracted references · 16 canonical work pages

  1. [85]

    Mukaida, M

    K. Mukaida, M. Takimoto, and M. Yamada, JHEP 03, 122 (2017), 1612.07750

    arXiv 2017

  2. [1]

    Therefore, we match the solution (

    Large-radius boundary condition At large radii, r ≫rsg, the gravitational field is small and set by the self-gravity of the scalar cloud. Therefore, we match the solution (

  3. [2]

    pressure

    Small-radius boundary condition Close to the Schwarzschild radius, we can expect the self-interaction “pressure” to be negligible and to recover 8 the free-scalar infall ( 36) (but we shall see below that the self-interaction plays a role for the scalar profile down to the Schwarzschild radius, as it dictates the constant flux of the steady state). Indeed, ...

  4. [3]

    low velocity) branch at large radii with the supersonic (i.e

    This is similar to the hydrody- namical case [ 94, 95], which selects the only value of the flux that provides a transonic solution that connects the subsonic (i.e. low velocity) branch at large radii with the supersonic (i.e. high velocity) branch at low radii

  5. [4]

    Thus, in this article we focus on scalar field masses in the range 10 −19 ≪m ≲ 1 eV

    associated with the growth of cosmological 5 structures. Thus, in this article we focus on scalar field masses in the range 10 −19 ≪m ≲ 1 eV. Our results also apply to the case of astrophysical BH, with M ∼ 1M⊙, if m ≫ 10−11 eV. C. Behavior at the Schwarzschild radius In the regime dominated by the BH gravity, r ≪ rsg, we can use the explicit expressions o...

  6. [5]

    L. Hui, J. P. Ostriker, S. Tremaine, and E. Witten, Phys. Rev. D95, 043541 (2017), 1610.08297

    arXiv 2017

  7. [6]

    Sarkar, R

    A. Sarkar, R. Mondal, S. Das, S. Sethi, S. Bharadwaj, and D. J. E. Marsh, JCAP 1604, 012 (2016), 1512.03325

    arXiv 2016

  8. [7]

    quantum pressure

    applies to all radii r > rs/4. Within ra- dius rsg the metric functions f (r) and h(r) are given by Eq.(8), whereas beyond rsg they are given by Eq.( 11) with the weak-gravity mapping ( 9). 3 In the static spherical metric ( 7) the scalar-field Klein- Gordon equation writes ∂2φ ∂t2 − √ f h3 ⃗∇ · ( √ fh⃗∇φ) +f∂V ∂φ = 0. (12) This also directly follows from ...

Show all 130 references
  1. [8]

    Abel et al., Phys

    C. Abel et al., Phys. Rev. X7, 041034 (2017), 1708.06367

    2017 arXiv

  2. [9]

    so as to give a regular expression in terms of ( ˜t, ˜r), once we use an appropriate time coordinate. In a similar fashion, the energy-momentum tensor as- sociated with the Schwarzschild or isotropic metrics di- verges at the Schwarzschild radius, but the one associated with t...

  3. [10]

    Hirano, J

    S. Hirano, J. M. Sullivan, and V. Bromm, Mon. Not. Roy. Astron. Soc. 473, L6 (2018), 1706.00435

    2018 arXiv

  4. [11]

    J. P. Conlon, F. Day, N. Jennings, S. Krippendorf, and F. Muia, Mon. Not. Roy. Astron. Soc. 473, 4932 (2018), 1707.00176

    2018 arXiv

  5. [12]

    Brito, S

    R. Brito, S. Ghosh, E. Barausse, E. Berti, V. Cardoso, I. Dvorkin, A. Klein, and P. Pani, Phys. Rev. Lett. 119, 131101 (2017), 1706.05097

    2017 arXiv

  6. [13]

    Brito, S

    R. Brito, S. Ghosh, E. Barausse, E. Berti, V. Cardoso, I. Dvorkin, A. Klein, and P. Pani, Phys. Rev. D96, 064050 (2017), 1706.06311

    2017 arXiv

  7. [14]

    Sarkar, S

    A. Sarkar, S. K. Sethi, and S. Das, JCAP 1707, 012 (2017), 1701.07273

    2017 arXiv

  8. [15]

    velocity

    remains finite. D. Density profile From Eqs.(16) and ( 23), the energy density associated with the Eddington coordinates is given, at leading order in the large-m limit, by ˜ρφ = ρ { sin2(mt −s) [ 2 −f + 1 f (1 −f − √ 1 −f )2 ] + cos2(mt −s) } . (45) In terms of the flux F , we o...

  9. [16]

    W. Hu, R. Barkana, and A. Gruzinov, Phys. Rev. Lett. 85, 1158 (2000), astro-ph/0003365

    2000 arXiv

  10. [17]

    M. S. Turner, Phys. Rev. D28, 1243 (1983)

    1983

  11. [18]

    (44) Thus, the scalar field is well defined at the Schwarzschild radius, provided we use regular coordinates, and as ex- pected we recover a fully ingoing solution

    and using the Eddington time ( 14) with the Schwarzschild radial coordinate, as in the metric ( 15), we obtain at leading order for ˜r →rs, φ = √ −2F m2r2s cos[m(˜t + ˜r +rs(1 − ln 4))] +... (44) Thus, the scalar field is well defined at the Schwarzschild radius, provided we use...

  12. [19]

    We show in Fig

    Function F (k, x) For each radiusx, Eq.(84) gives the flux F as a function of k. We show in Fig. 1 the normalized flux F/Fs as a function of the modulus k for several values of the radial coordinate x. The modulus k is constrained to range between 0 and the value k+(x)< 1/ √ 2 w...

  13. [20]

    J. A. R. Cembranos, A. L. Maroto, and S. J. N´ u˜ nez Jare˜ no, Phys. Rev. D87, 043523 (2013), 1212.3201

    2013 arXiv

  14. [21]

    velocity

    Boundary conditions To obtain the boundary condition at large radius for the modulus k(x), we consider the behavior of F (k,x ) at small k. Indeed, as seen in Fig. 1, at large radii the upper boundary k+(x) becomes much smaller than unity. Then, Eq.( 84) gives at leading order...

  15. [22]

    ´Alvarez-Luna and J

    C. ´Alvarez-Luna and J. A. R. Cembranos (2018), 1812.08501

    2018 arXiv

  16. [23]

    3, verifies F =Fc, k c(x) = k1(x) for x<x ⋆, kc(x) = k2(x) for x>x ⋆

    Critical solution Thus, the unique function kc(x), shown by the solid line in Fig. 3, verifies F =Fc, k c(x) = k1(x) for x<x ⋆, kc(x) = k2(x) for x>x ⋆. (97) At the Schwarzschild radius we obtain r =rs/4 : kc(1/4) ≡ks ≃ 0.54, (98) whilekc(x) decreases at large radius. From Eq.(...

  17. [24]

    (32) In the weak gravity regime, r ≫rs, we recover the usual Newtonian limit of the fluid equations, r ≫rs : ˙ ρ +⃗∇ · (ρ⃗ v) = 0, (33) ˙⃗ v+ (⃗ v·⃗∇)⃗ v= −⃗∇Φ

    we obtain ˙ρ + √ f h3 ⃗∇ · ( √ fhρ⃗ v) = 0, (31) ˙⃗ v+⃗∇ ( f h ⃗ v2 2 ) = − ⃗∇f 2 . (32) In the weak gravity regime, r ≫rs, we recover the usual Newtonian limit of the fluid equations, r ≫rs : ˙ ρ +⃗∇ · (ρ⃗ v) = 0, (33) ˙⃗ v+ (⃗ v·⃗∇)⃗ v= −⃗∇Φ. (34) This pressureless Euler equa...

  18. [25]

    Thus, from Eq.( 100) we obtain close to the Schwarzschild ra- dius r →rs/4 : πβ ′ 2m ∼ − 16(1 +α)mrs π(4r −rs) , (103) β ∼ − 4(1 +α)mrs π ln ( 4r −rs 4rs )

    we obtain finite quantities. Thus, from Eq.( 100) we obtain close to the Schwarzschild ra- dius r →rs/4 : πβ ′ 2m ∼ − 16(1 +α)mrs π(4r −rs) , (103) β ∼ − 4(1 +α)mrs π ln ( 4r −rs 4rs ) . (104) Substituting into Eq.( 51) and using the Eddington coor- dinates as in the metric ( 1...

  19. [26]

    J. A. R. Cembranos, A. L. Maroto, and S. J. N´ u˜ nez Jare˜ no, JHEP03, 013 (2016), 1509.08819

    2016 arXiv

  20. [27]

    (30) Taking the gradient of the second equation and substi- tuting the velocity field defined in Eq.(

    with respect to s, ˙ρ + √ f h3 ⃗∇ · ( √ fhρ ⃗∇s m ) = 0, (29) and with respect to ρ, ˙s m + f h (⃗∇s)2 2m2 = 1 −f 2 . (30) Taking the gradient of the second equation and substi- tuting the velocity field defined in Eq.(

  21. [28]

    (39) Using rs = 2GM , this reads as m ≫ 6.7 × 10−19 ( M 108M⊙ ) −1 eV

    of small density gradients is valid, as long as the Schwarzschild radius is large enough, rs ≫m−1. (39) Using rs = 2GM , this reads as m ≫ 6.7 × 10−19 ( M 108M⊙ ) −1 eV. (40) This lower bound is somewhat larger than the lower bound (

  22. [29]

    pressure

    and ( 31) take again the usual form ( 33), ˙ρ + ⃗∇ · (ρ⃗∇s) m = 0, ˙ρ +⃗∇ · (ρ⃗ v) = 0, (60) whereas the Euler equations ( 30) and ( 32) become ˙s m + (⃗∇s)2 2m2 = −(Φ + Φ I), ˙⃗ v+ (⃗ v·⃗∇)⃗ v= −⃗∇(Φ + Φ I), (61) where Φ I is given by [ 85] Φ I(ρ) = ρ ρa , ρ a ≡ 4m4 3λ4 . (62...

  23. [30]

    J. P. Ostriker and P. J. Steinhardt, Science 300, 1909 (2003), astro-ph/0306402

    2003 arXiv

  24. [31]

    The Euler equation (

    gives √ fhr 2ρvr =F, (35) where F < 0 is the constant inward flux per unit solid angle, which does not depend on the radius in a steady state. The Euler equation (

  25. [32]

    quantum pressure

    gives vr = − √ h(1 −f ) f , ∂s ∂r =mvr, (36) where we choose the boundary condition vr → 0 at r → ∞ to obtain the integration constant. Then, there is no additional uniform time dependence for s and we can choose s(r) = ∫ r drmvr, (37) and the complex scalar field ψ is given by...

  26. [33]

    R. D. Peccei and H. R. Quinn, Phys. Rev. Lett. 38, 1440 (1977)

    1977

  27. [34]

    Wilczek, Phys

    F. Wilczek, Phys. Rev. Lett. 40, 279 (1978)

    1978

  28. [35]

    Weinberg, Phys

    S. Weinberg, Phys. Rev. Lett. 40, 223 (1978)

    1978

  29. [36]

    D. J. E. Marsh, Phys. Rept. 643, 1 (2016), 1510.07633

    2016 arXiv

  30. [37]

    J. A. R. Cembranos, A. L. Maroto, S. J. N´ u˜ nez Jare˜ no, and H. Villarrubia-Rojo, JHEP 08, 073 (2018), 1805.08112

    2018 arXiv

  31. [38]

    Arbey, J

    A. Arbey, J. Lesgourgues, and P. Salati, Phys. Rev. D64, 123528 (2001), astro-ph/0105564

    2001 arXiv

  32. [39]

    Again, this means that one cannot use a perturbative approach in the scalar field, even in the large scalar mass limit

    is fulfilled, the radial derivatives of the phase s and of the wave functions ψ and φ are not small and even diverge at the Schwarzschild radius. Again, this means that one cannot use a perturbative approach in the scalar field, even in the large scalar mass limit. One must keep...

  33. [40]

    Kobayashi, R

    T. Kobayashi, R. Murgia, A. De Simone, V. Irˇ siˇ c, and M. Viel, Phys. Rev. D96, 123514 (2017), 1708.00015

    2017 arXiv

  34. [41]

    Banik, A

    N. Banik, A. J. Christopherson, P. Sikivie, and E. M. Todarello, Phys. Rev. D95, 043542 (2017), 1701.04573

    2017 arXiv

  35. [42]

    Schive, M.-H

    H.-Y. Schive, M.-H. Liao, T.-P. Woo, S.-K. Wong, T. Chi- ueh, T. Broadhurst, and W. Y. P. Hwang, Phys. Rev. Lett. 113, 261302 (2014), 1407.7762

    2014 arXiv

  36. [43]

    D. J. E. Marsh and A.-R. Pop, Mon. Not. Roy. Astron. Soc. 451, 2479 (2015), 1502.03456

    2015 arXiv

  37. [44]

    Calabrese and D

    E. Calabrese and D. N. Spergel, Mon. Not. Roy. Astron. Soc. 460, 4397 (2016), 1603.07321

    2016 arXiv

  38. [45]

    Chen, H.-Y

    S.-R. Chen, H.-Y. Schive, and T. Chiueh, Mon. Not. Roy. Astron. Soc. 468, 1338 (2017), 1606.09030

    2017 arXiv

  39. [46]

    stabilizes the scalar-field soliton obtained at large radii, as recalled in section IV B 1, and slows down the infall onto the central BH at smaller radii

    (dashed line) and the self- interaction case ( 107) (solid line), for the same value Fc of the flux. stabilizes the scalar-field soliton obtained at large radii, as recalled in section IV B 1, and slows down the infall onto the central BH at smaller radii. On the other hand, nea...

  40. [47]

    J. A. D. Diacoumis and Y. Y. Y. Wong, JCAP 1709, 011 (2017), 1707.07050

    2017 arXiv

  41. [48]

    A. X. Gonz´ alez-Morales, D. J. E. Marsh, J. Pe˜ narrubia, and L. A. Ure˜ na-L´ opez, Mon. Not. Roy. Astron. Soc.472, 1346 (2017), 1609.05856

    2017 arXiv

  42. [49]

    Thus, at this order, we can see that the radial derivatives do not change the structure of Eq.( 56)

    gives φ0 [ ω2 − f h (Kβ′)2 ] ∂2cn ∂u2 +fm 2φ0cn +fλ 4φ3 0cn3 = 0, (56) where we only kept the term of order m2 among the radial derivative contributions. Thus, at this order, we can see that the radial derivatives do not change the structure of Eq.( 56). This is why it again a...

  43. [50]

    M. C. Johnson and M. Kamionkowski, Phys. Rev. D78, 063010 (2008), 0805.1748

    2008 arXiv

  44. [51]

    to the soliton profile obtained for the self-gravitational nonrelativistic scalar cloud [ 85]. 7 a. Scalar-field soliton In this regime, we can decom- pose the scalar field φ as in Eq.(18) and use the Madelung transformation ( 23) for the complex field ψ. Taking into account the q...

  45. [52]

    J. A. R. Cembranos, C. Hallabrin, A. L. Maroto, and S. J. N. Jare˜ no, Phys. Rev. D86, 021301 (2012), 1203.6221

    2012 arXiv

  46. [53]

    velocity

    and the series expansions [ 93] K(k) = π 2 ( 1 + k2 4 +... ) , (70) q(k) = k2 16 ( 1 + k2 8 +... ) , (71) we obtain at leading order k ≪ 1 : φ =φ0 cos(ω0t −πβ/2) +... (72) The comparison with Eq.( 68) gives rsg ≪r ≪Rs : φ0(r) = √ 2ρs(0) m , β ≃ 0, (73) and ω0 = (1 + α)m. (74) ...

  47. [54]

    J. A. R. Cembranos, A. L. Maroto, and S. J. N´ u˜ nez Jare˜ no, JCAP1403, 042 (2014), 1311.1402

    2014 arXiv

  48. [55]

    Hwang and H

    J.-c. Hwang and H. Noh, Phys. Lett. B680, 1 (2009), 0902.4738

    2009 arXiv

  49. [56]

    Park, J.-c

    C.-G. Park, J.-c. Hwang, and H. Noh, Phys. Rev. D86, 083535 (2012), 1207.3124

    2012 arXiv

  50. [57]

    Hlozek, D

    R. Hlozek, D. Grin, D. J. E. Marsh, and P. G. Ferreira, Phys. Rev. D91, 103512 (2015), 1410.2896

    2015 arXiv

  51. [58]

    Chavanis (2018), 1810.08948

    P.-H. Chavanis (2018), 1810.08948

    2018 arXiv

  52. [59]

    J. A. R. Cembranos, A. L. Maroto, and S. J. N´ u˜ nez Jare˜ no, JHEP02, 064 (2017), 1611.03793

    2017 arXiv

  53. [60]

    Schive, T

    H.-Y. Schive, T. Chiueh, and T. Broadhurst, Nature Phys. 10, 496 (2014), 1406.6586

    2014 arXiv

  54. [61]

    In terms of the scalar fields ψ and φ this gives ψ = √ ρ me−iαmt, hence s = −αmt, (67) and φ = √2ρ m cos[(1 +α)mt]

    gives ⃗∇(Φ + Φ I) = 0, and we have r ≤Rs : Φ + Φ I =α, (65) where α is a constant, given by the value of the Newto- nian potential at the boundary of the soliton, α = Φ(Rs), (66) as Φ I(Rs) = 0. In terms of the scalar fields ψ and φ this gives ψ = √ ρ me−iαmt, hence s = −αmt, (...

  55. [62]

    Broadhurst, H

    T. Broadhurst, H. N. Luu, and S. H. H. Tye (2018), 1811.03771

    2018 arXiv

  56. [63]

    J. A. R. Cembranos, J. L. Feng, A. Rajaraman, and F. Takayama, Phys. Rev. Lett. 95, 181301 (2005), hep- ph/0507150

    2005

  57. [64]

    D. H. Weinberg, J. S. Bullock, F. Governato, R. Kuzio de Naray, and A. H. G. Peter, Proc. Nat. Acad. Sci. 112, 12249 (2014), [Proc. Nat. Acad. Sci.112,2249(2015)], 15 1306.0913

    2014 arXiv

  58. [65]

    Thus we also have r =rsg : Φ φ =α − Φ I ∼ −ρs ρa

    down to rsg, where the mass distribution starts to deviate from the flat soliton solution. Thus we also have r =rsg : Φ φ =α − Φ I ∼ −ρs ρa . (114) Then, we can check that we indeed have Φ φ ∼ Φ BH at the transition radius rsg given by Eq.( 112). From Eq.( 102) we find that at l...

  59. [66]

    Pontzen and F

    A. Pontzen and F. Governato, Nature 506, 171 (2014), 1402.1764

    2014 arXiv

  60. [67]

    Boylan-Kolchin, J

    M. Boylan-Kolchin, J. S. Bullock, and M. Kaplinghat, Mon. Not. Roy. Astron. Soc. 415, L40 (2011), 1103.0007

    2011 arXiv

  61. [68]

    Moore, S

    B. Moore, S. Ghigna, F. Governato, G. Lake, T. R. Quinn, J. Stadel, and P. Tozzi, Astrophys. J. 524, L19 (1999), astro-ph/9907411

    1999 arXiv

  62. [69]

    W. J. G. de Blok, Adv. Astron. 2010, 789293 (2010), 0910.3538

    2010 arXiv

  63. [70]

    Davoudiasl and P

    H. Davoudiasl and P. B. Denton, Phys. Rev. Lett. 123, 021102 (2019), 1904.09242

    2019 arXiv

  64. [71]

    Lesgourgues, A

    J. Lesgourgues, A. Arbey, and P. Salati, New Astron. Rev. 46, 791 (2002)

    2002

  65. [72]

    Chavanis, Phys

    P.-H. Chavanis, Phys. Rev. D84, 043531 (2011), 1103.2050

    2011 arXiv

  66. [73]

    P. H. Chavanis and L. Delfini, Phys. Rev. D84, 043532 (2011), 1103.2054

    2011 arXiv

  67. [74]

    Goodman, New Astron

    J. Goodman, New Astron. 5, 103 (2000), astro- ph/0003018

    2000

  68. [75]

    B. Li, T. Rindler-Daller, and P. R. Shapiro, Phys. Rev. D89, 083536 (2014), 1310.6061

    2014 arXiv

  69. [76]

    Su´ arez and P.-H

    A. Su´ arez and P.-H. Chavanis, Phys. Rev. D95, 063515 (2017), 1608.08624

    2017 arXiv

  70. [77]

    Su´ arez and P.-H

    A. Su´ arez and P.-H. Chavanis, Phys. Rev. D92, 023510 (2015), 1503.07437

    2015 arXiv

  71. [78]

    Schwabe, J

    B. Schwabe, J. C. Niemeyer, and J. F. Engels, Phys. Rev. D94, 043513 (2016), 1606.05151

    2016 arXiv

  72. [79]

    Veltmaat and J

    J. Veltmaat and J. C. Niemeyer, Phys. Rev. D94, 123523 (2016), 1608.00802

    2016 arXiv

  73. [80]

    Desjacques, A

    V. Desjacques, A. Kehagias, and A. Riotto, Phys. Rev. D97, 023529 (2018), 1709.07946

    2018 arXiv

  74. [81]

    V. H. Robles and T. Matos, Mon. Not. Roy. Astron. Soc. 422, 282 (2012), 1201.3032

    2012 arXiv

  75. [82]

    Bernal, L

    T. Bernal, L. M. Fern´ andez-Hern´ andez, T. Matos, and M. A. Rodr ´ ıguez-Meza, Mon. Not. Roy. Astron. Soc.475, 1447 (2018), 1701.00912

    2018 arXiv

  76. [83]

    P. Mocz, M. Vogelsberger, V. H. Robles, J. Zavala, M. Boylan-Kolchin, A. Fialkov, and L. Hernquist, Mon. Not. Roy. Astron. Soc. 471, 4559 (2017), 1705.05845

    2017 arXiv

  77. [84]

    goes to unity. Then, the small value of the flux F as compared with the local peak value Fmax(x) is reached by having a small value of k, thanks to the prefactor k2, rather than by having a large value of k close to the upper boundary k+ where the square root vanishes. This mea...

  78. [86]

    Vicens, J

    J. Vicens, J. Salvado, and J. Miralda-Escud´ e (2018), 1802.10513

    2018 arXiv

  79. [87]

    N. Bar, D. Blas, K. Blum, and S. Sibiryakov, Phys. Rev. D98, 083027 (2018), 1805.00122

    2018 arXiv

  80. [88]

    J. Eby, K. Mukaida, M. Takimoto, L. C. R. Wijeward- hana, and M. Yamada, Phys. Rev. D99, 123503 (2019), 1807.09795

    2019 arXiv

  81. [89]

    Bar-Or, J.-B

    B. Bar-Or, J.-B. Fouvry, and S. Tremaine, Astrophys. J. 871, 28 (2019), 1809.07673

    2019 arXiv

  82. [90]

    D. J. E. Marsh and J. C. Niemeyer (2018), 1810.08543

    2018 arXiv

  83. [91]

    Emami, T

    R. Emami, T. Broadhurst, G. Smoot, T. Chiueh, and H. N. Luu (2018), 1806.04518

    2018 arXiv

  84. [92]

    D. G. Levkov, A. G. Panin, and I. I. Tkachev, Phys. Rev. Lett. 121, 151301 (2018), 1804.05857

    2018 arXiv

  85. [93]

    Broadhurst, I

    T. Broadhurst, I. de Martino, H. N. Luu, G. F. Smoot, and S. H. H. Tye (2019), 1902.10488

    2019 arXiv

  86. [94]

    Hayashi and I

    K. Hayashi and I. Obata (2019), 1902.03054

    2019 arXiv

  87. [95]

    N. Bar, K. Blum, J. Eby, and R. Sato, Phys. Rev. D99, 103020 (2019), 1903.03402

    2019 arXiv

  88. [96]

    Kormendy and D

    J. Kormendy and D. Richstone, Ann. Rev. Astron. As- trophys. 33, 581 (1995)

    1995

  89. [97]

    Ferrarese and H

    L. Ferrarese and H. Ford, Space Sci. Rev. 116, 523 (2005), astro-ph/0411247

    2005 arXiv

  90. [98]

    Narayan, New J

    R. Narayan, New J. Phys. 7, 199 (2005), gr-qc/0506078

    2005 arXiv

  91. [99]

    M. C. Ferreira, C. F. B. Macedo, and V. Cardoso, Phys. Rev. D96, 083017 (2017), 1710.00830

    2017 arXiv

  92. [100]

    gives r ≫rsg : vr = −F⋆m4 λ4ρx2. (102) E. Behavior at the Schwarzschild radius As for the case of the free scalar field studied in section III, the radial velocity vr (100) and the den- sity ρφ (101), defined by the energy-momentum ten- sor associated with the isotropic metric, ...

  93. [101]

    Boˇ skovi´ c, F

    M. Boˇ skovi´ c, F. Duque, M. C. Ferreira, F. S. Miguel, and V. Cardoso, Phys. Rev. D98, 024037 (2018), 1806.07331

    2018 arXiv

  94. [102]

    Cardoso, ´O

    V. Cardoso, ´O. J. C. Dias, G. S. Hartnett, M. Middle- ton, P. Pani, and J. E. Santos, JCAP 1803, 043 (2018), 1801.01420

    2018 arXiv

  95. [103]

    Desjacques and A

    V. Desjacques and A. Nusser, Submitted to: Mon. Not. Roy. Astron. Soc. (2019), 1905.03450

    2019 arXiv

  96. [104]

    N. Bar, K. Blum, T. Lacroix, and P. Panci (2019), 1905.11745

    2019 arXiv

  97. [105]

    harmonic) expression ( 44) of the free case

    differs from the cosine (i.e. harmonic) expression ( 44) of the free case. We now ob- tain a nonlinear radial wave, with higher-order harmonics as given by the expansion ( 53). F. Density profile From Eqs.( 16) and ( 51), using Eqs.( 54)-(55), the en- ergy density associated wit...

  98. [106]

    pressure

    are of the same order. This means that the terms associated with the self-interaction potential are of the same order as those associated with the quadratic part. Thus, close to the BH the self-interaction potential can no longer be treated as a perturbation, which was the cas...

  99. [107]

    L. Hui, D. Kabat, X. Li, L. Santoni, and S. S. C. Wong, JCAP 1906, 038 (2019), 1904.12803

    2019 arXiv

  100. [108]

    The unboundedness of the free case is recovered by the fact that ⟨˜ρφ⟩ → ∞ when λ4 → 0

    is set by the char- acteristic density ρa defined in Eq.( 62), which measures the strength of the self-interactions. The unboundedness of the free case is recovered by the fact that ⟨˜ρφ⟩ → ∞ when λ4 → 0. We can see that all terms in Eq.(

  101. [110]

    (111) This BH-dominated regime stops at the radius rsg where the scalar field density has decreased down to the soliton core density ρs

    we obtain the more explicit scalings rs <r<r sg : ˜ ρφ ∼ρa rs r , v r ∼ −rs r . (111) This BH-dominated regime stops at the radius rsg where the scalar field density has decreased down to the soliton core density ρs. This gives rsg =rs ρa ρs . (112) From Eqs.(10) and ( 62) we fi...

  102. [111]

    For this analysis to be valid, we must check that the transition radius rsg is smaller than the soliton radius Rs

    and ( 115). For this analysis to be valid, we must check that the transition radius rsg is smaller than the soliton radius Rs. Using Eqs.( 112) and ( 64), with Rs ∼ ra, we find thatrsg <R s corresponds to M <Ms, where Ms ∼ρsr3 a is the soliton mass. The ratio M/Mh of the superm...

    2020

  103. [112]

    Su´ arez and P.-H

    A. Su´ arez and P.-H. Chavanis, Phys. Rev. D98, 083529 (2018), 1710.10486

    2018 arXiv

  104. [113]

    F. X. L. Cede˜ no, A. X. Gonz´ alez-Morales, and L. A. Ure˜ na-L´ opez, Phys. Rev. D96, 061301 (2017), 1703.10180

    2017 arXiv

  105. [114]

    Fan, Phys

    J. Fan, Phys. Dark Univ. 14, 84 (2016), 1603.06580

    2016 arXiv

  106. [115]

    Rindler-Daller and P

    T. Rindler-Daller and P. R. Shapiro, Mon. Not. Roy. As- tron. Soc. 422, 135 (2012), 1106.1256

    2012 arXiv

  107. [116]

    P. S. B. Dev, M. Lindner, and S. Ohmer, Phys. Lett. B773, 219 (2017), 1609.03939

    2017 arXiv

  108. [117]

    B. Li, P. R. Shapiro, and T. Rindler-Daller, Phys. Rev. D96, 063505 (2017), 1611.07961

    2017 arXiv

  109. [118]

    P. Brax, J. A. R. Cembranos, and P. Valageas, Phys. Rev. D100, 023526 (2019), 1906.00730

    2019 arXiv

  110. [119]

    Q. Yu, F. Zhang, and Y. Lu, Astrophys. J. 827, 114 (2016), 1606.07725

    2016 arXiv

  111. [120]

    Akiyama et al

    K. Akiyama et al. (Event Horizon Telescope), Astrophys. J. 875, L6 (2019), 1906.11243

    2019 arXiv

  112. [121]

    Poisson, A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics (Cambridge University Press, 2009)

    E. Poisson, A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics (Cambridge University Press, 2009)

    2009

  113. [122]

    Blau, Lecture notes on general relativity

    M. Blau, Lecture notes on general relativity

  114. [123]

    Madelung, Zeitschrift fr Physik 40, 322 (1927), ISSN 1434-601X, URL http://dx.doi.org/10.1007/BF01400372

    E. Madelung, Zeitschrift fr Physik 40, 322 (1927), ISSN 1434-601X, URL http://dx.doi.org/10.1007/BF01400372

    1927 doi

  115. [124]

    Kovacic and M

    I. Kovacic and M. Brennan, The Duffing Equation: Non- linear Oscillators and their Behaviour (Wiley, 2011)

    2011

  116. [125]

    I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, se- ries, and products (New York Academic Press, 1965), 4th ed., URL http://openlibrary.org/books/OL5955048M. 16

    1965

  117. [126]

    Byrd and M

    P. Byrd and M. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists (Springer, Berlin, Heidel- berg, 1971)

    1971

  118. [127]

    Bondi, Mon

    H. Bondi, Mon. Not. Roy. Astron. Soc. 112, 195 (1952)

    1952

  119. [128]

    F. C. Michel, Astrophysics and Space Science 15, 153 (1972)

    1972

  120. [129]

    Ferrarese, Astrophys

    L. Ferrarese, Astrophys. J. 578, 90 (2002), astro- ph/0203469

    2002

  121. [130]

    E. Y. Davies and P. Mocz (2019), 1908.04790

    2019 arXiv

  122. [131]

    J. F. Navarro, C. S. Frenk, and S. D. M. White, Astro- phys. J. 462, 563 (1996), astro-ph/9508025

    1996 arXiv

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