Ghostly as Solitons of Bose-Einstein Dark Matter

Tom Broadhurst,1, 2, 3, ∗ Ivan De Martino,2, † Hoang Nhan Luu,4, ‡ George F. Smoot,4, 5, 6, 7, § and S.-H. Henry Tye4, 8, ¶ 1Department of Theoretical Physics, University of the Basque Country UPV/EHU, E-48080 Bilbao, Spain 2Donostia International Physics Center (DIPC), 20018 Donostia-San Sebastian (Gipuzkoa) Spain. 3Ikerbasque, Basque Foundation for Science, E-48011 Bilbao, Spain 4Institute for Advanced Study and Department of Physics, Hong Kong University of Science and Technology, Hong Kong 5WF Chao Foundation Professor, IAS, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, 999077 Hong Kong 6Paris Centre for Cosmological Physics, APC, AstroParticule et Cosmologie, Universite´ Paris Diderot, CNRS/IN2P3, CEA/lrfu 7Universite´ Sorbonne Paris Cite,´ 10, rue Alice Domon et Leonie Duquet, 75205 Paris CEDEX 13, France 8Department of Physics, Cornell University, Ithaca, NY 14853, USA (Dated:) The large dark cores of common dwarf galaxies are unexplained by the standard heavy particle interpretation of dark matter. This puzzle is exacerbated by the discovery of a very large but barely visible, dark matter dominated II orbiting the , uncovered by tracking motions with the  satellite. Although Antlia II has a low mass, its visible radius is more than double any known , with an unprecedentedly low density core. We show that Antlia II favors dark matter as a Bose-Einstein condensate, for which the ground state is a stable soliton with a core radius given by the de Broglie wavelength. The lower the galaxy mass, the larger the de Broglie wavelength, so the least massive galaxies should have the widest soliton −22 cores of lowest density. An ultra-light boson of mψ ∼ 1.1 × 10 eV, accounts well for the large size and slowly moving within Antlia II, and agrees with boson mass estimates derived from the denser cores of more massive dwarf galaxies. For this very light boson, Antlia II is close to the lower limiting Jeans scale for galaxy formation permitted by the Uncertainty Principle, so other examples are expected but none significantly larger in size. This simple explanation for the puzzling dark cores of dwarf galaxies implies dark matter as an ultra-light boson, such as an axion generic in String Theory.

PACS numbers:

Dark matter (DM) is understood to be non- al. (2018) [7] used LIGO detection to obtain a tight up- relativistic even in the early , otherwise ini- per limit on the abundances of primordial black holes tial density perturbations destined to become galax- demonstrating that they can only contribute to a level of ies would be smoothed away by free streaming of a few percent to the dark matter abundance, which has the dark matter. This cold dark matter (CDM) has been independently confirmed in [8, 9]. Finally, and long been synonymous with heavy particle interpre- more in general, bounds placed on the abundance of tations beyond standard particle physics, but no such MACHO content in the from microlens- particles have been detected in stringent laboratory ing surveys such as MACHO, OGLE, and EROS, rule experiments[5]. Black holes also qualify as CDM, out any their significant contribution [10]. and although LIGO/ has claimed an abundance of arXiv:1902.10488v2 [astro-ph.CO] 25 Mar 2020 30M black holes, their space density is limited to less Enthusiasm for CDM has long been tempered by the than 5% of all dark matter by micro-lensing measure- shallow mass profiles of common dwarf galaxies that ments through massive lensing clusters [6]. Wang et appear to be cored rather than “cuspy” as predicted by N-body simulations. It has been hard to make sense of this apparent contradiction in the context of CDM with- out invoking hypothetical forces or implausibly trans- ∗Electronic address: [email protected] formational gas outflows. The very low density of stars †Electronic address: [email protected] in Antlia II and their low metallicities argues empir- ‡ Electronic address: [email protected] ically against an explanation that repeated episodes of §Electronic address: ¶Electronic address: [email protected] somehow flatten a CDM cusp into a core [12, 13], an idea that is disfavoured by accurate high 2

resolution simulations [11]. pendent and compelling argument for ψDM implied The above contradiction does not arise for a very dif- by the presence of ancient globular clusters on large ferent form of non-relativistic dark matter, as a Bose- orbits around . This is unexpected for discrete Einstein condensate. Pioneering simulations in this dark matter, such as CDM or black holes, that would be context reveal dark cores of very light bosons corre- focused gravitationally by an orbiting globular cluster sponding to the ground state of this quantum, wave-like into a ”wake”, so the globular clusters should have mi- form of dark matter[2], termed ψDM. These ψDM sim- grated long ago to the center of Fornax [19, 20]. This ulations simply evolve a coupled Schrodinger-Poisson¨ “dynamical friction” is not significant for ψDM, which equation describing a non-relativistic self-gravitating cannot be confined to less than the de Broglie scale be- condensate [14, 15] requiring only one free parameter, cause of the Uncertainty Principle [21, 22], leaving the the boson mass, mψ, where the smaller mψ the larger Fornax globular cluster orbits little affected [21]. the de Broglie wavelength. Rich non-linear structure is revealed by the ψDM simulations on the de Broglie 109

scale, including a prominent, standing wave core at the 108 center of every galaxy that is a stable soliton, represent- ] 107 ing the ground state, surrounded by a halo of turbulent, - 3

self interfering excited states [2, 3] and confirmed by kpc 6

⊙ 10 M independent simulations[16]. [ DM r r 5 r r r c ,Sculptor c , c , Antlia II c ,Fornax 10 c , Here we compare Antlia II with the central predic- ρ

tion of ψDM, that the least massive galaxies should 4 Antlia II 10 Sextans have the widest soliton cores of lowest density. The Fornax Sculptor de Broglie wavelength is of course larger at lower mo- 103 Draco mentum, so the soliton radius depends inversely with 10-1 100 101 −1 r [kpc] soliton mass, Rsol ∝ Msol . Also the ψDM simu- lations have established Msol increases with the total 1/3 FIG. 1: Here we show how the low mean density reported galaxy mass, as M ∝ M [3, 16] so the central sol gal for Antlia II is readily reproduced by the wide solitonic core DM density of soliton cores scales with galaxy mass 9 density profile of a low mass galaxy of ∼ 10 M in the con- as: text of ψDM. For comparison we compare the reported core mass densities of other well studied dwarf spheroidal galax- 4/3 2 ies with the density profiles for ψDM, for the same boson  M   m  M −22 ρ = 1.9×106 gal ψ mass, mψ = 10 eV, demonstrating the consistency with sol 109M 10−22 eV kpc3 the family of predicted ψDM profiles, where soliton radius (1) and mass are inversely related. Hence, at fixed mψ the core density of a dwarf 9 −4 galaxy of 10 M , should be much smaller, 10 , than The newly discovered Antlia II galaxy has an excep- 12 the central DM density of a massive galaxy of 10 M , tionally large core radius of ' 3 kpc, which together like the Milky Way. This contrasts with a predicted with its small velocity dispersion ' 6 km/s, corre- 5 −3 increase in density of ' 30 for standard CDM, from sponds to a mean DM density of only ' 10 M kpc , the “concentration-mass relation” of N-body simula- that is an order of magnitude lower than any known tions, where lower mass galaxies are predicted to have dwarf galaxy [1] and ' 30 times lower than Fornax, as denser dark matter profiles. shown in Figure 1. Antlia II extends the trend of recent Despite the general tendency to accommodate CDM, discoveries towards larger, lower surface brightness all well studied dwarf spheroidal (dSph) galaxies are galaxies of low velocity dispersion, including consistently claimed to have large dark cores, traced II and other large dSph galaxies in orbit around An- by a diffuse distribution of old stars of low velocity dromeda [23–25]. At face value these ”ghostly” galax- dispersion. In particular, the best studied Fornax dSph ies are encouraging for the ψDM interpretation of DM, is determined by several different methods to have a particularly Antlia II[1], and so here examine whether core radius of ' 1.0kpc [17, 18] with a density pro- the well studied dwarf spheroidals follow the distinc- file that is accurately fitted by the soliton form [2] of tive density-radius relation of eqn 1, that has the oppo- ψDM, as shown in Figure 1, for which a Jeans analy- site sign to the behaviour expected for CDM. It will −22 sis yields a boson mass of, mψ = 0.8 ± 0.2 × 10 also be of interest to examine the inner regions of more eV [2]. Furthermore, Fornax provides another inde- massive galaxies, requiring high resolution to resolve 3 the smaller, soliton scale predicted of ' 100 pc. It somewhat steeper decline for low boson mass. A bo- is encouraging that careful treatment of complex ”3D” son mass is favoured of 0.4 − 1.2 × 10−22 eV, (Figure gas dynamics of intermediate mass galaxies reveals ev- 2 and Appendix C) below which the observed mean idence of steep inner profiles and much smaller cores dispersion of ' 6.5 km/s is under predicted and above than typically derived using standard disk models [62]. which the soliton radius falls short of 2.5 kpc radius of In Figure 1 we show a family of ψDM profiles as a the outer bin, so the central velocity dispersion exceeds function of galaxy mass, following eqn 1, with the bo- the measured value, shown in Figure 2. The opposite −22 son mass mψ set to the canonical 10 eV,appropriate behavior is expected for CDM (Figure 2), where the for ψDM [2, 15, 26]. These model profiles can be seen continuously rising central density predicts a rising ve- to match the reported mean densities of the well stud- locity dispersion that exceeds the data in Figure 2, for ied dwarf spheroidal, listed in Table 1 (Appendix C) the best fitting NFW profile derived in the analysis of showing that Antlia II is fully consistent with a soli- Torrealba et al. (2018) [1]. tonic core that is 2.5 times larger than Fornax mass and We now jointly constrain the boson mass and soliton hence 40% lower mass than the core mass of Fornax, of core radius with galaxy mass in Figure 3. We com- 7 5 × 10 M , and in good agreement with the mass esti- pare Antlia II with Fornax and Sextans dSph’s because mated by Torrealba et al. (2018) based on the observed no tidal effects have been detected in deep imaging velocity dispersion of 6.5 km/s at the half light radius and careful dynamical work[27–29], so we need not be of 2.8 kpc. We can also compare the measured radial overly concerned that their masses are underestimated. In any case, these galaxies are established to have large r [deg] orbits about the Milky Way extending to ≥ 100 kpc, in- 0.0 0.2 0.4 0.7 0.9 1.1 cluding Antila II which is presently at a radius of 130 109 kpc with a sizeable estimated pericenter of 50 kpc im- 12 7 ] 10 3 - plying tidal effects are marginal[1, 30]. 5

kpc 10 10 ⊙ In Figure 3 we define a contour corresponding to the

[M [ ρ 102 ] 7

s 5.4 ± 2.1 × 10 M

/ measured mass of interior to a lim- 100 8 10- 1 100 101

km r [kpc] iting radius of ' 2.5 kpc within which the velocity dis- [ ) r

( 6 persion profile of Antlia II is observed to be flat[1], as

LOS shown in Figure 2, so the soliton core extends to at least σ 4 mψ,22=0.6 mψ,22=1.4 this radius. Importantly, Figure 3 shows that despite mψ,22=0.8 mψ,22=1.6 the considerable differences in mass and core radius 2 mψ,22=1.0 mψ,22=1.8

mψ,22=1.2 NFW between these three galaxies a common boson mass 0 can evidently be inferred, of ' 1.1 × 10−22 eV within 0.0 0.5 1.0 1.5 2.0 2.5 r [kpc] the uncertainties and for all dwarf spheroidal galaxies listed in Table 1 that have been analysed in this context FIG. 2: The measured velocity dispersion (data points) [1], (see Appendix C). compared with the predictions of ψDM mass profiles for a The Uncertainty Principle not only sets the soliton range of light boson masses - see legend where mψ,22 ≡ scale above, but also provides another fundamental pre- −22 mψ/10 . The central decline in dispersion velocity for diction of a sharp minimum halo mass by imposing a the lighter bosons matches well the data, and reflects the low “quantum Jeans condition” [31, 32, 34] because the central mass density. This contrasts with the relatively more DM cannot be confined within the de Broglie wave- concentrated best fitting NFW profile (dashed curve) where length thereby preventing galaxy formation below a an enhancement is expected, unlike the data. limiting Jeans mass (for more details see [21] and ref- erence therein): velocity dispersion profile observed for Antlia II [1] 1 − 3 with ψDM predictions by solving the Jeans equation in ! 4 ! 2 3 Ω h2 (1 + z)3 m projection (see Eq. (B5)). We assume the commonly a 70 ψ 7 MJ = −22 10 M , adopted Plummer profile appropriate for the stellar dis- 2 0.27 10 eV tribution of dwarf spheroidal galaxies, normalized to (2) the measured half light radius of 2.8 kpc [1]. The pre- dicted profiles shown in Figure 2 have a slow centrally as shown in Figure 3, where h70 ≡ −1 −1 declining velocity dispersion due to the declining inte- (H0/70 km s Mpc ) is the dimensionless Hubble rior mass, M(< r) for constant density cores, with a constant and Ωψ is the cosmological fraction of 4

II has likely been stripped off tidally as indicated by surface brightness distortions and because of its small inferred pericenter of only 10 kpc [23] where stripping is generally expected to significantly reduce the veloc- ity dispersion and truncate the stellar radius, depending on the phase of the orbit [27, 36, 37] and so previously Crater II may have resembled more Antlia II with a larger radius and higher velocity dispersion. Another large dwarf spheroidal galaxy, at 800 kpc and is determined to be one of only a few “isolated” galaxies that has not suffered significant interaction with other galaxies [38]. This galaxy extends to at least 3 kpc in deep imaging, with no detectable tidal truncation radius [38]. The velocity dispersion of Ce- tus is close to 10 km/s and can traced beyond its its half light radius to' 1.4 kpc[38]. The extended stellar profile may indicate a soliton core similar to Fornax (in Figure 1) surrounded by a lower density DM halo as predicted for ψDM in the absence of tidal truncation, extending to several kpc, as shown in Figure 1. FIG. 3: The plane of core radius versus mψ as a function of galaxy mass (colour coded) predicted for ψDM that are The above large core ”classical” dSph galaxies may limited to the hatched regions derived from the observations be contrasted with the newly discovered class of much of Antlia II summarized in Table 1 (Appendix C). Specifi- smaller 20 − 50 pc ”ultra faint dwarfs” uncovered in cally, we have used the observed core radius to obtain those wide field surveys[39, 40] on relatively small orbits, regions. The Fornax, Sextans dwarf galaxies are included for < 50 kpc, within the Milky Way. These relatively comparison because their density profiles are understood not small objects are very DM dominated, given their typ- to have been modified significantly by tidal stripping. In this ical velocity dispersion of ' 3 km/s and low lumi- case, the hatched regions are derived using the constraints on nosities, of typically only ' 1000 L and so together the core radius and the boson mass listed in Table 1. The lim- with their small orbits they are generally considered to iting Jeans mass is also indicated as a dashed black curve that be heavily stripped “remnants” [41–43] of originally places an upper limit on the radius of Antlia II at fixed mψ. Despite the wide range of mass and radius these three galax- large dSph galaxies. In the context of ψDM, tidal strip- ies which span well over an order of magnitude in density are ping is estimated to be significantly more efficient than −22 compatible with the same boson mass, mψ ∼ 10 eV. for CDM, as the soliton core expands in response to the loss of outer stripped halo mass pushing more DM beyond the tidal radius as the soliton expands in ra- dius in response to the reduced mass, in a runaway the dark matter critical density. This limit depends process[44]. This ”remnant” origin for the UDF galax- only weakly on formation redshift as the power ies may be supported by the serendipitous discovery of spectrum cuts off below a Jeans-like wavelength, central star clusters within several dSph galaxies, with 1/4 λJ ∝ (1 + z) [3, 32, 33], defining a relatively sharp sizes and luminosities similar to the UDF galaxies[45– lower limiting galaxy mass for ψDM [3, 15, 34, 35]. 48] that may be clarified with deep velocity dispersion Hence, low mass galaxies should be abundant towards measurements. A wider ψDM context may provide this limit and firmly absent below it. Furthermore, a natural origin for such dense central stars clusters, the presence of this mass limit predicts a maximum and in general for the puzzling presence of nuclear star soliton core radius of ' 3 kpc, as shown in Figure clusters commonly found in all types of galaxy, where 3, because the lowest mass galaxies should have the a minority contribution to the universal DM from a widest solitons of lowest mass density, with Antlia II heavier boson of ' 10−20eV may sink within the wide lying closest to this existential limit. solitons of the lighter, dominant DM (derived above of Another ghostly galaxy is the ”feeble giant” Crater 10−22eV), resulting in a smaller dense DM structure II, orbiting the Milky Way at 50 kpc, with a sizeable that helps explain the puzzling origin and character- radius of 1.1 kpc and surprisingly low velocity disper- istic scale of nuclear star clusters [49]. This ”multi- sion of only ' 3 km/s[23]. Most of the DM of Crater ple ultra-light” bosonic solution for the Universal dark 5 matter has the attraction of being underpinned theoreti- radius r can be straightforwardly computed by integrat- cally by String Theory, where a wide discrete spectrum ing the density profile of eqn. A1 as of axion-like particles extending to very light masses is Z r generically predicted [50–53], depending on the details 2 MDM (r) = 4π x ρDM (x)dx . (B1) of dimensional compactification. 0

Finally, the velocity dispersion can be obtained by Acknowledgements integrating the Jeans equation

2 2 d(ρ∗(r)σr (r)) MDM (r) ρ∗(r)σr (r) I.D.M. acknowledge the contribution of INFN (In- = −ρ∗(r) − 2β , dr r2 r iziativa Specifica QGSKY). This article is based upon (B2) 2 2 work from COST Action CA1511 Cosmology and σφ+σθ where β ≡ 1 − 2 is the anisotropy parameter, Astrophysics Network for Theoretical Advances and 2σr Training Actions (CANTATA), supported by COST and we have considered stars with a Plummer density (European Cooperation in Science and Technology) profile

5  2 − 2 3M∗ r ρ∗(r) = 1 + 2 (B3) Appendix A: ψDM Halo model 4πrh rh

A general solution of eqn. B2 for β constant is given ψ Recently, numerical simulations of DM model by [56] were carried out by Schive et al. (2014) [2], and were subsequently used to show the presence of a cored dark Z ∞ 2 −2β 2β−2 matter region in the inner part of each virialized halo. ρ∗(r)σr (r) = Gr x ρ∗(x)MDM (x)dx , r This cored region of the dark matter halo satisfies the (B4) ground state of the Schroedinger-Poisson equations, and, it must be projected along the line of sight to be and can be described by [2]: compared with data:

−1 −23 −2 −4 1.9 a (mψ/10 eV) (rc/kpc) −3 Z ∞ 2  2 ρ (r) ∼ M pc . 2 2 R σr (r)ρ∗(r) c −2 2 8 σ (R) = 1 − β rdr . [1 + 9.1 × 10 (r/rc) ] los Σ(R) r2 (r2 − R2)1/2 (A1) R −23 −2 (B5) where c1 = 1.9, c2 = 10 = , c3 = 9.1 × 10 are constants; and rc is the core radius which scales with the halo mass of the galaxy and the mass boson obeying Appendix C: Data, data analysis, and results to the following the scaling relation which have been calibrated with cosmological simulations [3]: The Antlia 2 dwarf galaxy has been recently dis- 10−22  ζ(z)−1/6 M −1/3 covered [1]. It is located at the distance of 130 kpc 1/2 h ∼ 2.9kpc rc = 1.6 a 9 from the Milky Way and extended . This mb ζ(0) 10 M galaxy has the lowest measured surface brightness (A2) (32.3mag/arcsec2) that is ∼ 100 times more diffuse than the ultra diffuse galaxies. The satellite was recognised when combining pho- Appendix B: Jeans Analysis tometric, astrometric and variability information from the satellite Gaia in its second data release, while kine- In order to use the stellar kinematics of Antlia II to matic information were subsequently obtained with constrain the boson mass, mψ, we assume the galaxy spectroscopic follow-up using AAOmega spectrograph to be spherically symmetric and supported by veloc- on the Anglo-Australian Telescope which allowed a ity dispersion. Then, we decompose the galaxy in the measurement of the velocity dispersion profile. ψDM halo and a stellar population which serves as We have modeled these data within ψDM picture tracer of the gravitational potential well dominated by described in the previous section. Retaining the ob- the DM halo. Thus, the mass enclosed in a sphere of served values of the half light radius and the observed 6

TABLE I: In columns 2 and 3, are listed the half-light radius and the corresponding mass from [1, 54, 55]. In column 4 and 5, we report the boson mass and soliton radius analyzed in [2, 26]. In the columns 6 and 7 are the predicted values of the masses −22 within the soliton radius and the virial radius, Msol and M200, assuming a boson mass 10 eV.

Galaxy rh M(< rh) mψ rc Msol M200 Ref. 7 −22 7 9 (kpc) (10 M )(10 eV) (kpc) (10 M )(10 M ) Antlia II 2.90 ± 0.31 5.4 ± 2.1 [0.6 − 1.4][1.8 − 3.4] 2.7 0.19 This work & [1] +0.16 +0.15 Fornax 0.67 ± 0.34 5.3 ± 0.9 0.81−0.17 0.92−0.11 5.7 1.1 [2, 54, 55] +1.08 Sextans 0.68 ± 0.12 2.5 ± 0.9 2.13−0.64 [1.5 − 3.0] 4.6 0.63 [26, 54, 55] +0.41 +0.11 Sculptor 0.26 ± 0.39 1.3 ± 0.4 1.23−0.33 0.60−0.12 6.9 2.1 [26, 54, 55] Draco 0.20 ± 0.12 0.94 ± 0.25 1.12 ± 0.52 0.56 ± 0.13 8.6 4.3 [26, 54, 55]

stellar mass within rh = 2867 ± 312 pc [1], the the- oretical counterpart is fully determined once the bo- son mass and the total halo mass are specified. Thus, we have explored the parameter space with the Monte Carlo Markov Chain (MCMC) pipeline employing the Metropolis-Hastings sampling algorithm. We allowed for an adaptive step size in order to reach an accep- tance rate between 20% and 50%, and we ensured the convergence relying on the Gelman-Rubin crite- ria. We adopted flat priors covering the range [0.1, 3]× −22 8 10 eV for the boson mass, and [0.5, 13] × 10 M for the total halo mass, while β is set to zero [1], and we run four different chains with random starting points. Then, once the convergence criteria is satisfied, the dif- ferent chains are merged to compute the total likeli- FIG. 4: 2D marginalized contours of the model parameters hood, and the 1D marginalized likelihood distribution [mψ,Mhalo] obtained from our MCMC analysis of Antlia II. with the corresponding the expectation value and vari- The 68%, 95% and 99% confidence levels are shown with the ance. 1D marginalized likelihood distribution.

The best fit values obtained with our analysis are −22 +0.41 8 mψ(10 eV) = 0.81−0.21 and Mhalo(10 M ) = +3.48 6.61−2.68, respectively. The best fit value of the bo- son mass is consistent at 1σ with other analysis carried out in [2, 26]. The 68%, 95%, and 99% of confidence levels of the posterior distribution are shown in Fig. 4. We have also translated these constraints in a one on the core radius using Eq. A2. We carried out 1000 Monte Carlo simulations randomly choosing the val- ues of mψ and Mhalo from their posterior distribution, and obtaining rc = 2.7 ± 0.63 kpc.

Finally, in Fig. 5, we compare our constraint on the boson mass with other independent estimates (listed in Table 1) assuming a gaussian posterior distribution. From the comparison, we obtain that a boson mass of mψ = 1.07 ± 0.08 for all the published boson mass analyses including our new measurement for Antlia II. 7

son mass ∼ 8% accuracy showing that the combina- tion of galaxy data favor an ultralight boson with mass ∼ 10−22eV. A previous analysis that used the UV-luminosity function to predict the reionization history of the uni- verse found a 3σ tension with boson masses ∼ 10−22 eV [35]. Nevertheless, since they used the optical depth as measured by Planck in the 2013 data release, those constraints are invalidated by the significant change of such a fundamental parameter in subsequent Planck re- sults. Furthermore, from the existence of the star clus- ter in II, a boson mass of the order of ∼ 10−19 eV has been inferred [61]. Nevertheless, the existence of this may be explained by a multiple ax- FIG. 5: Joint likelihood constraint on the boson mass from ion scenario as recently shown in [49]. our Antlia II analysis (solid blue line), together with previ- ous published work on Fornax (dotted red), Sextans (dashed Finally, we note recent claims of significantly higher cyan), Sculptor (dot dashed magenta), Draco (triple dotted bounds on the boson mass obtained from the high fre- dashed green). The final joint probability distribution for the quency behaviour of the Ly-α power spectrum [59], boson mass is shown as the black solid curve. made by analogy with Warm Dark Matter, under much debate where differences in the matter power spec- trum between ψDM and WDM may play a critical role Appendix D: Further Discussions and conclusions [21, 60]. Dedicated simulations are really required for ψDM that include hydrodynamics and cooling to pro- We have used the dispersion velocity of the stars in vide a more definite answer on whether or not Lyman- the newest discovered ghostly galaxy Antlia II to con- α excludes bosons with masses ∼ 10−22 eV, as it is strain the ψDM model. Such a galaxy has pointed out conceivable that excess variance in the small scale as several inconsistencies of the CDM model which may distribution arises from the pervasive de- Broglie scale be solved by changing the DM paradigm to explain the structure uncovered in the ψDM simulations caused by existence of such a large stellar core in a barely visible the inherent interference in this context that fully mod- galaxy. Our best estimation of the boson mass, namely ulates the density of halos and filaments, ranging from −22 +0.41 mψ(10 eV) = 0.81−0.21, is still comfortably con- constructive to destructive interference. Furthermore, sistent with other constraints coming from other inde- gas outflows and shocks are observed to be widespread pendent analyses (see for example [26, 56, 57]). Fur- at high redshift that can also be expected to enhance the thermore, as we use other constraints on ψDM coming variance in power on small scales in the gas, but these from the analysis of the dispersion velocity profiles in are not included in the simulations relied on for Ly-α other dwarf galaxies, we are able to estimate the bo- constraints even for Warm Dark Matter.

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Monthly Notes of Royal Astronomical Society, 491, 4993-5014 (2020)