Axion Core–Halo Mass and the Black Hole–Halo Mass Relation: Constraints on a Few Parsec Scales

Mon. Not. R. Astron. Soc. 000, 1–8 (0000) Printed 7 April 2020 (MN LATEX style ﬁle v2.2)

Axion core–halo mass and the black hole–halo mass relation: constraints on a few parsec scales

Vincent Desjacques? and Adi Nusser

Physics department and Asher Space Science Institute, Technion, Haifa 3200003, Israel

ABSTRACT If the dark matter is made of ultra-light axions, stable solitonic cores form at the centers of virialized halos. In some range for the mass m of the axion particle, these cores are suﬃciently compact and can mimic supermassive black holes (SMBH) residing at galactic nuclei. We use the solitonic core–halo mass relation, validated in numerical simulations, to constrain a new range of allowed axion mass from measurements of the SMBH mass in (pseudo)bulge and bulgeless galaxies. These limits are based on observations of galactic nuclei on scales smaller than 10 pc. Our analysis suggests −18 that m . 10 eV is ruled out by the data. We brieﬂy discuss whether an attractive self-interaction among axions could alleviate this constraint. Key words: cosmology: theory, large-scale structure of Universe, dark matter

1 INTRODUCTION thereby, FDM scenarios can be further constrained using different astrophysical observables (see Hui et al. 2017, for In the fuzzy dark matter (FDM) scenario (see, e.g., Balde- a detailed overview), such as galactic rotation curves (Bar schi et al. 1983; Khlopov et al. 1985; Sin 1994; Hu et al. 2000; et al. 2018; Robles et al. 2018), the survival of star clusters Svrcek & Witten 2006; Amendola & Barbieri 2006; Chavanis in dwarf galaxies Marsh & Niemeyer (2018) or, even, the 2011; Marsh & Silk 2014; Hlozek et al. 2015; Hui et al. 2017, absence of black-hole superradiance in M87 (Davoudiasl & and references therein), a halo is made of a solitonic core Denton 2019). engulfed by a haze of fluctuating density granules resulting Here, we assess the extent to which the presence or ab- from the interference of (classical) waves. When the FDM sence of supermassive black holes (SMBHs) constrain FDM is ultra-light axions (see, e.g., Marsh 2016, for a recent re- scenarios. The paper is organized as follows. After a brief view), the solitonic core is dubbed “axion star” or, simply, an digression on the origin of the axion core – halo mass rela- axion core. Numerical simulations of the Gross-Pitaievskii- tion (§2), we demonstrate that measurements of SMBH and Poisson (GPP) system have established that the mass of host halo mass in bulge and, in particular, bulgeless galaxies the axion core M increases with the FDM halo mass M c h yield constraints at least as competitive as rotation curves (Schive et al. 2014; Schive et al. 2014; Schwabe et al. 2016). (§3). We conclude in §4. Furthermore, simulations have robustly demonstrated the In all illustrations, we assume a concordance ΛCDM existence of a haze of fluctuating granules extending much cosmology with Hubble parameter h = 0.7 and present-day farther than the embedded solitonic core (Schive et al. 2014). matter density Ω = 0.3. This quasi-particle picture has been explored further in Hui m arXiv:1905.03450v2 [astro-ph.CO] 4 Apr 2020 et al. (2017); Bar-Or et al. (2018) in the context of dynamical friction. It can also be used to understand the properties of the axion cores. Measurements from the Lyman-α forest power spec- 2 AXION CORE - HALO MASS RELATION trum set a lower bound on the axion mass of m & 2 × 10−21 eV at 95% C.L. (Irˇsiˇcet al. 2017; Armengaud et al. For sake of completeness, we shall discuss briefly the ori- 2017). While our own galaxy could still harbour a solitonic gin of the axion core–halo mass relation in the context of core for a axion mass as low as m ∼ 10−22 eV (De Martino virial equilibrium, and illustrate how it can be extended to et al. 2018; Broadhurst et al. 2019), this is quite unlikely in a non-vanishing (attractive) self-interaction. More thorough light of the large scale structure constraints (see, however, discussions can be found in Chavanis (2011); Schive et al. Zhang et al. 2017). The existence of solitonic cores and, (2014); Marsh & Pop (2015); Hui et al. (2017). We use natural units c = ~ = 1 throughout and 2 write Newton’s gravitational constant as G = 1/mP, where ? 19 Email: [email protected] mP = 1.22 × 10 GeV is the Planck mass. Furthermore, we

c 0000 RAS 2 Vincent Desjacques and Adi Nusser parametrize the axion mass m and decay constant f as As a result, the escape velocity vesc at the surface of the

−22 solitonic core is given by m = 10 m22 eV (1) 17 r s f = 10 f17 GeV . (2) GMc 2 m vesc = = √ 2 Mc (8) Rc 3 π m As a rule of thumb, the present-day axion energy density is P 2 1/2 M given by Ω ∼ 0.1f17m22 (Marsh 2016). Note also that f17 c −1 ' 138 m22 9 km s . quantifies the strength of the axion self-interaction, which 10 M we assume attractive . Since we will consider f < mP al- This relation reproduces the empirical scaling Mc ∝ ways, we are in the “strong regime” of axion self-interactions 1/2 (Chavanis 2018b). (|Eh|/Mh) , where Eh is the energy of the halo. This can also be understood in terms of a wave-like uncertainty principle (Schive et al. 2014), or diffusive equilibrium (Bar et al. 2018). 2.1 Virial equilibrium considerations The axion core–halo mass relation follows immediately Equilibrum configurations of FDM halos with a density and from combining Eqs. (5), (6) and (8): velocity profile (ρ,u) can be obtained by means of minimiz- 2/3 1/3 ing the total energy (Chavanis 2011) Mc = N Mc,min Mh . (9)

E = W + K + Q + U, (3) Here, Mc,min is a minimum core mass, where K and Q are the kinetic and the “quantum pressure” 2 1/2 1 3/4 3/8 −3/4 1/4 mPH0 contributions, W is the gravitational binding energy of the Mc,min = 3 π a (Ωm∆vir) (10) 2 m3/2 self-gravitating FDM halo, and U is the “internal energy” 7 −3/4 −3/2 arising from the self-interaction. Equilibrium configurations ' 2.51 × 10 a m22 M , also satisfy the quantum analog of the classical virial theo- and N = 0.25 is a empirical normalisation factor which ac- rem (Chavanis 2011; Hui et al. 2017) implying, in the steady counts for the fact that the mass assigned to an axion core state limit, in numerical simulations is computed from the central re- 0 = W + 2K + 2Q + 2U. (4) gion with R . Rc only. The virial overdensity ∆vir(z) is defined relative to the average matter densityρ ¯m(z). We Since K ≥ 0, this yields the inequality U + Q ≤ |W |/2, ignore the mild redshift dependence of ∆vir(z) and assume which is saturated in the axion core where K = 0. By con- ∆vir(z) = 200 throughout. Furthermore, a is the scale fac- trast, in the gaseous atmosphere of quasi-particles (see Hui tor. Since all the data considered here is at redshift z 1, et al. 2017, Appendix A), the quantum pressure and the self- we will simply set a = 1 in all subsequent illustrations. interaction can be neglected, so that we recover the usual For a present-day MW-size galaxy with Mh = 4 × 12 virial theorem W + 2K = 0. 10 M , the axion core mass would be Mc ∼ 5.4 × For virialized FDM halos, the velocity dispersion of the 8 −1 10 m22 M . The minimum core mass Mc,min originates quasi-particles surrounding the core is from the fact that a solitonic core with mass Mc = Mc,min would have the average density of the Universe (Marsh & 2 GMh hv i ≈ . (5) Pop 2015). In principle, there is a maximum stable core Rh mass reached when Rc equals the Schwarzschild radius As we shall see now, the core properties are determined Rs = 2GMc. For realistic CDM cosmologies however, there through the requirement that the quasi-particles are barely is not enough time by z = 0 to form virialized structures bound to the core, that is which could host axion cores with Mc ∼ Mc(Rc = Rs). Finally, one should bear in mind that, although we will hv2i ≈ v2 , (6) esc apply Eq. (9) unrestrictedly, it may be a good description where vesc is the escape velocity from the axion core. of the axion core – halo mass relation solely over a limited range of halo and axion masses (see Hui et al. 2017, for a discussion). 2.2 Without self-interaction The axion core is characterized by an approximately Gaus- sian density profile, which reaches a constant central density 2.3 With attractive self-interaction ρc on scales less than the core radius Rc. In the absence of a self-interaction, U = 0 and the virial equilibrium condition In the presence of an attractive self-interaction, U 6= 0 and −1 the relation between Rc and Mc is more involved. One finds W + 2Q = 0 inside the solitonic core yields Rc ∝ Mc . This 2 2 (Chavanis 2011) scaling arises from W ∝ Mc /Rc and Q ∝ Mc/Rc . A more detailed analysis gives (Chavanis 2011) √  s  2 2 √ 3 π mP 1 m 2 2 3 π mP Rc = 1 ± 1 − 2 Mc  (11) Rc = (7) 4Mc m 12π mPf 2Mc m 9 −2 10 M The stable branch corresponds to the plus sign. In this case, ' 227 m22 pc . Mc the core radius monotonically decreases with increasing Mc

c 0000 RAS, MNRAS 000, 1–8 Axion core - halo mass and the black hole–halo mass relation: constraints on a few parsec scales 3

√ 2 to reach 3 π mP at the maximum core mass axion masses considered in Fig.1. This scenario likely applies 4Mc m to major mergers during which the gravitational potential √ mPf Mc,max = 2 3π (12) fluctuates significantly on a short timescale and, thereby, m destroys the coherence of the axion core. 11 f17 Under this assumption, the relevant timescale is the ' 1.19 × 10 M , m22 two-body relaxation timescale between the FDM quasi- above which there is no stable solution. particles. As discussed in Hui et al. (2017) (and Bar-Or et al. In Appendix A, we show that the quasi-particle ap- 2018, in further details), this reads proach discussed above also holds in the presence of a self- 4 1010 yr v 2 r interaction. Applying the same hydrostatic considerations t = m3 . (15) relax f 22 100 km s−1 5 Kpc yield a core–halo mass relation given by relax 4/3 2/3 An FDM halo will develop a compact solitonic core from M M s 4M 2 c,min h c,max the mass bound to the descendant halo within a radius Rc Mc = N 4/3 2/3 − 1 . (13) 2Mc,max Mc,minMh if the condition trelax(Rc) . tmg is satisfied. Here, tmg is the time elasped since the merger. Setting v = V in the above For the normalisation, we shall adopt again N = 0.25. The circ expression, and using the core – halo mass relation Eq.(9), axion core mass reaches its maximum M = M for a c c,max the newly merged halo will develop an axion core of mass halo mass 1/3 Mc ∝ M provided that 3 h 3/2 Mc,max M˜ = 2 (14) 1/2 −3/2 10 1/2 h 2 a m 10 yr Mc,min 4 22 Mc & 3.5 × 10 M 1/2 . (16) tmg 18 3/2 3 frelax = 7.57 × 10 a f17 M Although trelax increases with the axion mass, the minimum independently of the axion mass m22. For Mh ≥ M˜ h, hy- −3/2 core mass scales like Mc ∝ m22 because of the core radius drostatic equilibrium cannot be satisfied. Note that Eq.(13) −2 Rc ∝ m22 shrinks rapidly as the axion mass is increased. recovers Eq.(9) in the limit f17 → ∞, that is, in the absence −1 Assuming frelax ∼ 1 and tmg = H for illustration, where of self-interactions. 0 H0 is the Hubble constant today, this condition is satisfied An attractive self-interaction lowers the minimum core for the whole range of circular velocities and masses shown mass obtained upon setting Mc = Mh. However, for values in Fig.1. of f17 & 0.01 compatible with all axions being the dark Requiring that the whole descendant FDM halo be in matter, this is at most a factor of 2 smaller than Mc,min: the virial equilibrium (which amounts to setting v = Vvir and axion self-interaction scale as ∝ ρ2 and, thus, is very weak r = Rvir in Eq.(15)) would ensure that the axion core mass at low densities. of the merged halo precisely falls on the relation Eq.(9). However, we found that such a condition cannot be satisfied 2.4 Mergers and the persistence of axion cores unless the core mass is close to Mc,min (so that the FDM atmosphere is tenuous). Therefore, one should expect some The equilibrium considerations above do not take into ac- scatter in Mc at fixed halo mass. count the evolution of Mc and Mh through mergers and Note that the sum of the progenitor core mass is always smooth accretion, which is an essential aspect of hierarchi- larger than the core mass expected if the final descendant cal structure formation. A related issue is the persistence halo reaches hydrostatic equilibrium. To see this, let Mh1, of the axion core – halo mass relation Eq. (9) through the Mh2 be the mass of the progenitor halos, with corresponding assembly history of the host FDM halos (see, e.g. Schwabe core mass Mc1 and Mc2; and Mh = Mh1 + Mh2 be the mass et al. 2016; Du et al. 2017) of the merged halo. Let us also define Mc = Mc1 + Mc2. Although numerical simulations indicate that cores are Assuming that the core - atmosphere of the progenitors is ubiquitous inside FDM halos (Schive et al. 2014; Veltmaat in hydrostatic equilibrium, so that Eq.(9) initially holds, we et al. 2018), the fate of solitonic cores during the merger of have two FDM halos is unclear. Therefore, a lack of evidence for a " 2/3 1/3 1/3 2/3 #1/3 central core does not necessarily translate into a constraint 2/3 1/3 Mh1 Mh2 + Mh1 Mh2 Mc = Mc,minMh 1 + 3 , on the axion mass, unless the characteristic timescale for the Mh formation of a new core following a merger event is shorter (17) than the age of the galaxy. 2/3 1/3 which shows that Mc1 + Mc2 > M M . The difference The cores of the progenitor FDM halos may i) remain c,min h is maximum for a major merger with Mh1 ≈ Mh2, in which intact, or ii) momentarily disappear during the merging pro- 1/3 case Mc ≈ 1.6 Mc,minM . cess. To determine whether a core forms in the central region h of the descendant FDM halo, one should thus consider either i) the dynamical friction timescale on which they sink to the 3 CONSTRAINTS ON AXION MASS FROM center of the merged halo, or ii) the relaxation timescale of M - V MEASUREMENTS FDM quasi-particle, which defines the region within which • CIRC virial equilibrium can be established. Furthermore, all this We discuss now the constraints on the axion mass m that could depend on the axion mass since the solitonic cores be- arise from measurements of the mass, M•, of SMBH residing come more compact as m22 increases and, therefore, are less at galactic nuclei, and from the galactic (asymptotic) circu- likely to be disrupted. For simplicity however, we will as- lar velocity Vcirc at larger radii. The asymptotic circular ve- sume that scenario ii) is the relevant picture for the range of locity is used as a proxy for the host halo mass Mh. The full

c 0000 RAS, MNRAS 000, 1–8 4 Vincent Desjacques and Adi Nusser rotation curve is irrelevant for the constraints derived here. as the thick black line. The powerlaw scaling reflects the For sufficiently small Rc, the axion core could masquerade relationship advocated by McConnell et al. (2011) (see also as a galactic SMBH. Hui et al. (2017) briefly discussed this Ferrarese & Merritt 2000). possibility for large galaxies. Here, we will show that small Furthermore, we display measurements from bulgeless galaxies actually give the strongest limits on m. galaxies as the green squares. Except for NGC 4395, for which a reverberation-mapping measurement gives M• = 5 (3.6±1.1)×10 M (Peterson et al. 2005), all these measure- 3.1 Strategy ments provides an upper limit on the mass of the SMBH. For NGC 300, 3423, 7424 and 7793, the limits on M are from Observational constraints on galactic SMBH masses are • Neumayer & Walcher (2012). V for NGC 7424 is from mainly obtained from studying the stellar kinematics within circ Sorgho et al. (2019) whereas, for NGC 7793, V is from small distances (R < 10 pc) of a few times the radius of in- cric e de Blok et al. (2008). Finally, M33 (a nearby spiral galaxy fluence of the SMBH. When an estimate of the host halo embedded in a dark matter halo) has an asymptotic circular mass M is available, the axion core radius R (M , m) and h c c velocity of V ≈ 125 km s−1 (Mayall & Aller 1942) and the mass M (M , m) can be obtained from the relations (7) and circ c h tightest upper limit on the SMBH mass: M < 1.5×103 M (9), respectively. More precisely, taking into account the de- • (Gebhardt et al. 2001). We have thus labelled the corre- pendence M ∝ m−1, cf. Eq.(10), we find c,min sponding data point on Fig. 1. The data of the bulgeless 1/3 galaxies is all summarized in Table 1. Mh 1 Mc ∝ ,Rc ∝ . (18) m mMh On the one hand, too low values for m imply large core 3.3 Constraints masses M , yet constraints cannot be obtained because the c It is instructive to first compare the data to the axion core core is too diffuse. On the other hand, too high m cannot be – halo mass relations discussed in §2. For this purpose, we ruled out either since they yield M M . Therefore, this c • overlay in Fig.1 the M - V relations inferred from Eq.(9) technique can constrain a limited, albeit interesting range of c circ (dashed lines, no self-interaction) and Eq.(13) (dotted lines, m where the core is sufficiently compact and massive. with self-interactions). The axion mass increases in steps of To compare the data to theoretical expectations based an order of magnitude, from m = 1 until m = 105 (from on the axion core – halo mass relation, we need to associate 22 22 top to bottom). The effect of a self-interaction is shown only the observed circular velocity V to the halo mass M . We circ h for a mass m = 102. The two dotted curves assume a adopt the following relation 22 decay constant f17 = 0.005 and 0.01 (from left to right). 1/3 The triangle marks the value of Vcirc at which Mc = Mc,max, −1 Mh Vcirc ≈ 144 km s 12 , (19) where Mc,max is given by Eq. (12). We have not shown the 10 M scatter expected around the Mc - Vcirc relation owing to the which assumes an overdensity threshold ∆vir = 200 (in imperfect relaxation after merger events etc. outlined in §2. unit of the critical density ρcr). This allows us to convert Numerical simulations indicate that this scatter is of order 9 11 Mc(Mh, m22) into a relation Mc - Vcirc once an axion mass 0.3 dex for halo masses 10 . Mh . 10 M (Schive et al. is assumed. 2014). Axion cores could mimic a point source like a SMBH provided their radius is smaller than the radius Re of the 3.2 Data central nuclear star cluster, the velocity dispersion of which The analysis requires a sample of measured black hole constrains the SMBH mass. Typical values of Re are in the masses and circular velocities of the respective host halos. range Re ∼ 1 − 10 pc. For illustration, the thick orange line Kormendy & Ho (2013) provides an excellent review of the shows the locus Mc(m22) for which the core radius is Rc = relevant techniques for measuring SMBH masses, as well as 10 pc, so that the shaded upper half of Fig.1 corresponds a discussion of the correlations between the inferred masses to axion cores with a radius Rc > 10 pc. Such axion cores and properties of their host galaxies. The tightest correlation cannot be approximated as a central point source similar to a SMBH. This excludes the possibility that the classical is between M• and the velocity dispersion σ of the central −1 stellar component. Fortunately, Kormendy & Ho (2013) also bulges with Vcirc & 250 km s actually harbor axion cores. Measurements of the Lyman-α forest rule out the range list the circular velocities Vcirc of many of the host galaxies −21 m . 2 × 10 eV (i.e. m22 . 20) at 95% C.L. (Irˇsiˇc given in their paper. For spiral galaxies, Vcirc is derived√ from the rotation curves while, for ellipticals, it is simply 2σ. At et al. 2017; Armengaud et al. 2017). Such low values of m a given galaxy mass, the least massive SMBH are found in yield an axion core – halo mass relation (at least partly) spirals with pseudobulges or no bulge at all. Thus, we ex- in the orange shaded region of Fig.1 and, therefore, typi- pect that the strongest constraints will be obtained using cally correspond to large core radii Rc which cannot mimic these galaxies, rather than ellipticals or galaxies with clas- a central point source. Furthermore, if the hypothetical ax- sical bulges. ion has a mass m22 & 20, then essentially all the classical In Fig.1, we display measurements from classical and bulges must correspond to SMBHs. Fig.1 also suggests that, pseudo-bulges as (filled) red and (empty) blue circles, re- if this axion would be self-interacting with a decay constant spectively, along with the empirical relation 0.005 . f17 . 0.01, the low mass compact objects harbored mainly by pseudobulges could actually be axion cores, while V 5.1 the more massive ones would have exceeded the threshold M ≈ 0.32 × 108 M circ (20) • 200 km s−1 Eq.(12) and collapsed to form black holes. In such a scenario,

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Figure 1. Measurements of central SMBH mass M• vs circular velocity Vcirc for diﬀerent types of galaxies. Filled red and empty blue circles designate the dynamical measurement of M• in classical and pseudobulges, respectively, while the green squares represent (mostly) upper limits for bulgeless galaxies (from the compilation of Kormendy & Ho 2013). The galaxy with the tightest black hole mass upper limit is M33 as indicated on the ﬁgure. The thick black line is the empirical M• - Vcirc relation, whereas the dashed and dotted curves indicate the axion core mass Mc vs. Vcirc expected for ultra-light axions w/o self-interactions (see text for details). The shaded orange area shows the region in which the core radius Rc is larger than 10 pc. two populations of central compact objects - axion cores and mass of SMBHs - could coexist over a certain range of Vcirc. How- 1/2 3/2 a M•,max 1 pc ever, our discussion thus far does not take into account the m22 9.7 √ (22) . 103 M R non-detection of central compact objects in nearly all the N e 2 low-V bulgeless galaxies. This yields the strongest con- 100 km s−1 circ × , straints on m as we shall see now. Vcirc To exclude a range of axion mass from bulgeless galax- in which we set a = 1 and√ N = 0.25 as advocated above. ies, we require that the core mass within the radius of the The multiplicative term N arises from the fact that, in nuclear cluster be less than the maximum black hole mass Eq.(21), Mc comes with one normalisation factor N (since inferred from the nuclear star cluster. More precisely, let Re Mc represents the mass enclosed within Rc solely), while Rc be the radius of the central stellar cluster. There are two does not. possibilities depending on whether Rc is larger or smaller If Rc < Re, then the core mass must satisfy than Re. If Rc > Re, we demand √ 2 3 π mP 1 < Mc < M•,max . (23) 2 m Re

3 This translates into a lower limit on the allowed axion mass Re Mc < M•,max , (21) of Rc 3 1/2 1/2 4 10 M 1 pc m22 & 1.5 × 10 (24) M•,max Re where M•,max is the upper limit on the SMBH mass as given independently of the host halo mass. in Table 1. Assuming the core is in hydrostatic equilibrium Values of Re are obtained from Gebhardt et al. (2001) so that relation Eq.(9) holds, we can express both Mc and for M33, and from Neumayer & Walcher (2012) for the re- Rc as a function of Mh or, equivalently, Vcirc. As a result, maining galaxies. They are all summarized in Table 1, along Eq. (21) translates into an upper limit on the allowed axion with the constraints on m. Taking into account the ﬁnite

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5 extent of the nuclear cluster, the actual limits on the axion together with the detection of a ∼ 10 M SMBH in NGC mass are diﬀerent from those directly inferred from Fig. 1 4395. 5 (e.g. we would read oﬀ m22 & 10 from M33). Notwith- standing, the range of low axion masses allowed by this data, m 10−21 eV, is incompatible with the Lyman-α for- . 5 ACKNOWLEDGMENTS est constraints. Therefore, if dark matter is an ultra-light axion, then its mass must exceed the lower limits given V.D. acknowledges support by the Israel Science Foundation in the last column of Table 1 in order to satisfy the con- (grant no. 1395/16). A.N. acknowledges support by the I- straints from bulgeless galaxies. The absence of a compact CORE Program of the Planning and Budgeting Committee, object at the center of M33 gives the strongest constraint: the Israel Science Foundation (grants No. 1829/12 and No. −18 4 m > 1.2 × 10 eV (or, equivalently, m22 > 1.2 × 10 ). 936/18) and the Asher Space Research Institute.

4 CONCLUSIONS APPENDIX A: ON THE QUASI-PARTICLE DESCRIPTION OF FDM HALOS We have assessed the extent to which measurements of the Motivated by numerical simulations, Hui et al. (2017) sug- SMBH mass M•, and the halo mass Mh for bulge and bulgeless galaxies can prove the mass m of an hypothetical gested that the atmosphere of FDM halos can be approxi- ultra-light axion dark matter. This data can constrain an mated as a gas of quasi-particles of characteristic size λdB, −1 interesting range of axion mass 10−20 − 10−18 eV for which where λdB = (mv) is the de Broglie wavelength of the −4 the axion cores are neither too diffuse nor too massive. axion particle. For a typical velocity v ∼ 10 , λdB ∼ −1 −1 While we have used the compilation of Kormendy & m22 h Kpc is on galactic scales. Ho (2013) for (pseudo)bulge galaxies, small bulgeless galax- Numerical simulations show that the binding energy of ies actually give the strongest constraints on m. In partic- these quasi-particles is negligible compared to their kinetic ular, the non-detection of a central compact object in M33 energy (Veltmaat et al. 2018). In other words, their self- – an isolated spiral galaxy from the local group without gravity can be neglected so that their dispersion relation is 2 any indication of recent mergers or interactions with other that of a free particle, ω(k) = k /2m. As a result, a quasi- −18 particle of initial width λ gradually spreads over with a galaxies (Verley et al. 2010) – gives m & 1.2 × 10 eV. dB −19 −18 p 4 2 The range of mass 10 − 10 eV is not easily accessi- rms width given by (λdB + (t/m) )/λdB. Quasi-particles 2 ble to measurements from rotation curves, which typically thus disperse after a time τ ∼ λdB/m, that is, probe scales r 1 pc (e.g. Slepian & Goodman 2012; Bar v 2 τ ∼ 2.1 × 107 m−1 yr , (A1) et al. 2018). Our constraints also improve on those inferred 22 10−4 by Marsh & Niemeyer (2018) from the presence of old star in agreement with the findings of Veltmaat et al. (2018). clusters in Eridanus II. We stress that our limits rely on the The number N of quasi-particles populating a FDM halo caveat that the axion core – halo mass relation Eq.(9) is is, therefore, not conserved. Nevertheless, we expect the av- valid in a range of axion masses for which it has, in fact, not erage number hNi to be conserved for FDM halos in virial been tested numerically. Therefore, our constraints would equilibrium. weaken, would the hypothetical axion core mass fall below Turning on the self-interaction should not affect this the relation Eq. (9). picture noticeably. To see this, we assume that the quasi- Instability of the axion core owing to an attractive self- particles are described by Gaussian wave packets of size λ interaction (e.g., Chavanis 2011; Visinelli et al. 2018) with dB and mass m = (2π)3/2ρλ3 as in Hui et al. (2017). Here, ρ f 0.01 (possibly amplified by external perturbers, cf. eff dB 17 . is the density in the FDM atmosphere surrounding the ax- Eby et al. 2018), could help relaxing the constraints on m ion core. The various energy contributions straightforwardly if, during the collapse to a black hole, a significant fraction follow from the Gaussian ansatz used by Chavanis (2011). of the axion core mass can be expelled. Fig.1 shows that, We find: for a galaxy with a mass comparable to M33, an axion de- 14 1 m σ m cay constant f . 5 × 10 GeV is required for the harbored eff 3 eff K = 2 2 ,Q = 2 2 (A2) axion core to be unstable. It is unclear whether such low 2m λdB m λdB values of f could still produce the right relic abundance, ζ m2 ν m2 U = 3 eff ,W = 3 eff , although temperature-dependent effects during the symme- 2 2 3 2 m f λdB mP λdB try breaking can help achieving Ω ∼ 0.1 (Diez-Tejedor & Marsh 2017). In any case, it is pretty clear that such a self- where interacting model would have to be somewhat fine-tuned in 3 1 1 σ = , ζ = − , ν = − √ (A3) order to satisfy both M33 and cosmological constraints. 3 8 3 128π3/2 3 2 π What is the fate of unstable axion cores ? Self-similar This gives solutions to the ”wave collapse” indicate that interactions −1 4 near the center create an outgoing stream of particles which K −2 2 ρ v ' 5.7 × 10 m22 −3 −4 , (A4) can carry away a large fraction of the axion core before the W M pc 10 formation of a black hole remnant (see, e.g., Levkov et al. 2017). It would be interesting to investigate whether this and −1 2 effect can produce a range of remnant SMBH masses broad K 2 2 ρ v ' 2.4m22f17 −3 −4 . (A5) enough to explain the non-detection of a SMBH in M33, U M pc 10

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Table 1. Constraint on axion mass from bulgeless galaxies. M• is the mass of the central SMBH (in M ), Re is the radius of the −1 −22 nuclear star cluster (in pc), Vcirc is the asymptotic circular velocity (in km s ), and the axion mass m22 is in unit of 10 eV. The constraints on the axion mass assume that the core radius Rc is either larger (left column) or smaller (right column) than Re. See text for details.

M• Re Vcirc constraint on m22 M33 < 1.5 × 103 1.0 125 < 15 > 1.2 × 104 NGC 300 < 105 2.9 90 < 48 > 880 NGC 3423 < 7 × 105 4.18 127 < 36 > 280 NGC 4395 (3.6 ± 1.1) × 105 – 90 – – NGC 7424 < 4 × 105 7.4 145 < 9.0 > 270 NGC 7793 < 8 × 105 7.7 86 < 34 > 190

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