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- 1. Sergey Sibiryakov Galactic rotation curves vs. ultra-light dark matter: Implications of the soliton — host halo relation BW2018, Niš, Serbia with Nitsan Bar, Diego Blas, Kﬁr Blum, arXiv:1805.00122
- 2. eVmDM , 109 What is dark matter ? 1012 1028 106810 23
- 3. eVmDM , WIMP’s } 109 What is dark matter ? 1012 1028 106810 23
- 4. eVmDM , WIMP’s } SM SM DM DM 109 What is dark matter ? 1012 1028 106810 23
- 5. eVmDM , WIMP’s WIMPZILLA’s } SM SM DM DM 109 What is dark matter ? 1012 1028 106810 23
- 6. eVmDM , WIMP’s WIMPZILLA’s primordial black holes } SM SM DM DM 109 What is dark matter ? 1012 1028 106810 23
- 7. eVmDM , WIMP’s WIMPZILLA’s primordial black holes } SM SM DM DM } inﬂation ? 109 What is dark matter ? 1012 1028 106810 23
- 8. eVmDM , WIMP’s WIMPZILLA’s primordial black holes } SM SM DM DM } inﬂation ? Ultra-Light Dark Matter 109 What is dark matter ? 1012 1028 106810 23
- 9. eVmDM , WIMP’s WIMPZILLA’s primordial black holes } SM SM DM DM } inﬂation ? Ultra-Light Dark Matter 109 What is dark matter ? 1012 1028 106810 23 misalignment
- 10. The simplest DM model: free massive scalar 0 e.o.m. in expanding Universe: L = 1 2 ( µ )2 m2 2 ¨ + 3H ˙ + m2 = 0
- 11. The simplest DM model: free massive scalar 0 e.o.m. in expanding Universe: • = const L = 1 2 ( µ )2 m2 2 ¨ + 3H ˙ + m2 = 0 H > m
- 12. The simplest DM model: free massive scalar p = 0 density: pressure: 0 e.o.m. in expanding Universe: • = const • L = 1 2 ( µ )2 m2 2 ¨ + 3H ˙ + m2 = 0 p = cos(2mt) = m2 2 0 2 = 0 cos(mt) H > m H < m
- 13. The simplest DM model: free massive scalar p = 0 density: pressure: 0 e.o.m. in expanding Universe: • = const • behaves as DM on times longer than L = 1 2 ( µ )2 m2 2 ¨ + 3H ˙ + m2 = 0 p = cos(2mt) = m2 2 0 2 = 0 cos(mt) m 1 H > m H < m
- 14. • Theoretical pedigree: pseudo-Goldstone boson e.g. global periodic variable with period 2 fU(1)
- 15. • Theoretical pedigree: pseudo-Goldstone boson non-zero mass can be generated by non-perturbative eﬀects examples: QCD axion, string theory ALP’s, relaxion, quasi-dilaton, .... e.g. global periodic variable with period 2 fU(1) m exp( Sinst) V ( ) = m2 f2 1 cos( /f)
- 16. • Theoretical pedigree: pseudo-Goldstone boson non-zero mass can be generated by non-perturbative eﬀects examples: QCD axion, string theory ALP’s, relaxion, quasi-dilaton, .... e.g. global periodic variable with period 2 fU(1) after inﬂation • Density fraction: f m exp( Sinst) V ( ) = m2 f2 1 cos( /f) 0.05 f 1017GeV 2 m 10 22eV 1/2
- 17. • from CMB and LSS : otherwise too much suppression of structure Ly forest: based on complicated modelling Kobayashi et al. (2017) m 10 21 eV m 10 23 eV Observational probes
- 18. • from CMB and LSS : otherwise too much suppression of structure • aﬀects structures on small scales, could address the problems of particle CDM (?) Hu, Barkana, Gruzinov (2000) Hui et al. (2016) Ly forest: based on complicated modelling Kobayashi et al. (2017) m 10 22 eV m 10 21 eV m 10 23 eV Observational probes
- 19. • from CMB and LSS : otherwise too much suppression of structure • aﬀects structures on small scales, could address the problems of particle CDM (?) Hu, Barkana, Gruzinov (2000) Hui et al. (2016) Ly forest: based on complicated modelling • leads to black-hole superradiance: probed by black-hole spins, gravitational waves Arvanitaki, Dubovsky (2011) Arvanitaki et al. (2016) Kobayashi et al. (2017) m 10 14 ÷ 10 10 eV m 10 22 eV m 10 21 eV m 10 23 eV can probably be extended down to m 10 18 eV Observational probes
- 20. • from CMB and LSS : otherwise too much suppression of structure • aﬀects structures on small scales, could address the problems of particle CDM (?) Hu, Barkana, Gruzinov (2000) Hui et al. (2016) focus of this talk: NB. Can be axion-like particle, but not QCD axion Ly forest: based on complicated modelling • leads to black-hole superradiance: probed by black-hole spins, gravitational waves Arvanitaki, Dubovsky (2011) Arvanitaki et al. (2016) Kobayashi et al. (2017) m 10 14 ÷ 10 10 eV m 10 22 eV m 10 21 eV m 10 23 eV can probably be extended down to m 10 18 eV m 10 22 ÷ 10 18 eV Observational probes
- 21. Challenges to particle CDM at sub-kpc scales ? 14 Oh et al. Fig. 5.— Upper-left panel: The (DM only) rotation curves (small dots) of the 21 LITTLE THINGS (including 3 THINGS galaxies) for which Spitzer 3.6µm image is available. These are all scaled with respect to the rotation velocity V0.3 at R0.3 where the logarithmic slope of the rotation curve is dlogV/dlogR = 0.3 as described in Hayashi & Navarro (2006). The ‘⇥’ symbol represents the median values of the rotation curves in each 0.1R/R0.3 bin. The error bars show the 1 scatter. Lower-left panel: The scaled rotation curves of the seven THINGS, and the two simulated dwarf galaxies (DG1 and DG2 in Governato et al. 2010) which are overplotted to the median values of the LITTLE THINGS rotation curves. The grey solid and black solid lines with small dots indicate the CDM NFW dark matter rotation curves with V200 which is > 90 km s 1 and < 90 km s 1, respectively. Right panels: The corresponding dark matter density proﬁles derived using the scaled rotation curves in the left panels. The grey (V200 > 90 km s 1) and black solid lines with small dots (V200 < 90 km s 1) represent the CDM NFW models with the inner density slope ↵⇠ 1.0. See Section 4 for more details. proﬁles. This is much like the THINGS dwarf galaxies, and the simulated dwarfs (DG1 and DG2) with baryonic feedback processes as shown in the lower-right panel of Fig. A.3), we perform a least squares ﬁt (dotted lines) to the inner data points (grey dots) within a ‘break radius’. As described in de Blok & Bosma (2002; see also Oh - cores vs. cusps - missing satellites - too big to fail from Oh et al., arXiv:1502.01281 perhaps are explained by baryonic physics
- 22. Dynamics of ULDM in the Newtonian limit slowly varying amplitude }Schroedinger -- Poisson system leads to suppression of ﬂuctuations at short scale --- “quantum pressure” = (x, t)e imt + h.c. i ˙ 2 2m + m (x, t) = 0 2 = 2Gm2 | |2
- 23. Probing ULDM with galactic rotation curves
- 24. ULDM in the halo Figure 2: A slice of density ﬁeld of ψDM simulation on various scales at zzz === 000...111. This scaled sequence (each of thickness 60 pc) shows how quantum interference patterns can be clearly seen everywhere from the large-scale ﬁlaments, tangential fringes near the virial boundaries, to the granular structure inside the haloes. Distinct solitonic cores with radius ∼ 0.3 − 1.6 kpc are found within each collapsed halo. The density shown here spans over nine orders of magnitude, from 10−1 to 108 (normalized to the cosmic mean density). The color map scales logarithmically, with cyan corresponding to density 10. graphic processing unit acceleration, improving per- formance by almost two orders of magnitude21 (see Supplementary Section 1 for details). Fig. 1 demonstrates that despite the completely different calculations employed, the pattern of ﬁl- aments and voids generated by a conventional N- body particle ΛCDM simulation is remarkably in- distinguishable from the wavelike ΛψDM for the same linear power spectrum (see Supplementary Fig. S2). Here Λ represents the cosmological constant. This agreement is desirable given the success of stan- dard ΛCDM in describing the statistics of large scale structure. To examine the wave nature that distin- guishes ψDM from CDM on small scales, we res- imulate with a very high maximum resolution of 60 pc for a 2 Mpc comoving box, so that the dens- est objects formed of 300 pc size are well re- solved with ∼ 103 grids. A slice through this box is shown in Fig. 2, revealing ﬁne interference fringes the boundaries of virialized objects, where the de Broglie wavelengths depend on the local velocity of matter. An unexpected feature of our ψDM simula- tions is the generation of prominent dense coherent standing waves of dark matter in the center of every gravitational bound object, forming a ﬂat core with a sharp boundary (Figs. 2 and 3). These dark matter cores grow as material is accreted and are surrounded by virialized haloes of material with ﬁne-scale, large- amplitude cellular interference, which continuously ﬂuctuates in density and velocity generating quan- tum and turbulent pressure support against gravity. The central density proﬁles of all our collapsed cores ﬁt well with the stable soliton solution of the Schr¨odinger-Poisson equation, as shown in Fig. 3 (see also Supplementary Section 2 and Fig. S3). On the other hand, except for the lightest halo which has just formed and is not yet virialized, the outer proﬁles of other haloes possess a steepening loga- Schive, Chiueh, Broadhurst, arXiv: 1406.6586 3 Schive, Chiueh, Boardhurst, arXiv:1407.7762 soliton
- 25. Properties of the soliton ok for a quasi-stationary phase-coherent solution, bed by the ansatz3 (x, t) = ✓ mMpl p 4⇡ ◆ e i mt (x). (5) LDM mass density is ⇢ = (mMpl) 2 4⇡ 2 (6) ⇡ 4.1 ⇥ 1014 ⇣ m 10 22 eV ⌘2 2 M /pc3 . arameter is proportional to the ULDM energy it mass Validity of the non-relativistic regime re- | | ⌧ 1, and since we are looking for gravitation- und conﬁgurations, < 0. uming spherical symmetry and deﬁning r = mx, equations for and are given by 2 -1.5 -1.0 -0.5 Φ1 FIG. 1. Proﬁle of the solid). We also show (orange dashed) and green). Other solutions 1(r), 1(r) by a sc tions (r), (r), by m 22 eV ⌘2 2 M /pc3 . rtional to the ULDM energy he non-relativistic regime re- we are looking for gravitation- < 0. metry and deﬁning r = mx, are given by 2r ( ) , (7) r 2 . (8) solution amounts to solving ! 0) = const, (r ! 1) = 0, nitial value of at r ! 0, the ue value of . olve Eqs. (7-8) with the initial us call this auxiliary solution al calculation gives [4, 5, 8] green). Other solutions of Eqs. (7-8) can b 1(r), 1(r) by a scale transformation. tions (r), (r), together with the eig by (r) = 2 1( r), (r) = 2 1( r), = 2 1, also satisfy Eqs. (7-8) with correct bou for any > 0. The soliton mass and c are M = M1, xc = 1 xc1. A mnemonic for the numerical value of looking for gravitation- . and deﬁning r = mx, e given by ) , (7) (8) ion amounts to solving = const, (r ! 1) = 0, value of at r ! 0, the lue of . qs. (7-8) with the initial l this auxiliary solution ulation gives [4, 5, 8] 9, (9) g. 1. The mass of the 1 tions (r), (r), together with the eige by (r) = 2 1( r), (r) = 2 1( r), = 2 1, also satisfy Eqs. (7-8) with correct boun for any > 0. The soliton mass and co are M = M1, xc = 1 xc1. A mnemonic for the numerical value of = 3.6 ⇥ 10 4 ⇣ m 10 22 eV ⌘ ✓ 10 The product of the soliton mass and cor ) , (7) (8) amounts to solving nst, (r ! 1) = 0, ue of at r ! 0, the of . (7-8) with the initial is auxiliary solution ion gives [4, 5, 8] (9) The mass of the 1 (10) (r) = 1( r), = 2 1, also satisfy Eqs. (7-8) with correct bo for any > 0. The soliton mass and are M = M1, xc = 1 xc1. A mnemonic for the numerical value o = 3.6 ⇥ 10 4 ⇣ m 10 22 eV ⌘ ✓ 1 The product of the soliton mass and c pendent of , 8 ⇣ m ⌘ 3 r 2 4 6 8 10 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 χ1 Φ1 Vcirc,1 . rc1 FIG. 1. Proﬁle of the “standard” 1 soliton with = 1 (blue solid). We also show the corresponding gravitational potential /pc3 . ULDM energy istic regime re- or gravitation- ﬁning r = mx, y (7) (8) unts to solving (r ! 1) = 0, at r ! 0, the Other solutions of Eqs. (7-8) can 1(r), 1(r) by a scale transformation tions (r), (r), together with the e by (r) = 2 1( r), (r) = 2 1( r), = 2 1, also satisfy Eqs. (7-8) with correct b for any > 0. The soliton mass and are c = 4 c1
- 26. Soliton - host halo relationwhere Mh is the virial mass of the host halo. As noted in [6–8], Eqs. (29-30) are an excellent numerical ﬁt for a soliton . The mass of this soliton is M ⇡ 1.4 ⇥ 109 ⇣ m 10 22 eV ⌘ 1 ✓ Mh 1012 M ◆1 3 M , (31) so its parameter is ⇡ 4.9 ⇥ 10 4 ✓ Mh 1012 M ◆1 3 . (32) Note that Eq. (31) is applicable only as long as the halo exceeds a minimal mass, Mh,min ⇠ 5.2 ⇥ 107 ✓ m 10 22eV ◆ 3/2 M . (33) Smaller mass halos would be dominated by the soliton. Ref. [7] showed that Eq. (31) is consistent with the relation, ✓ |Eh| ◆1 2 M2 pl However, this is just ergy Eh by the energ density proﬁle found approximation, it mu of the halo in the sim the central soliton c likely to hold for rea Mh signiﬁcantly abov How could this hav in the simulations of which were then allow initial conditions wer state soliton – the so to absorb the entire from Ref. [7] that con tical initial solitons, t N solitons with a ran dius. Such distribut soliton energy becau tial condition set-up the most massive init Schive, Chiueh, Boardhurst, arXiv:1407.7762 Mh,min ⇠ 5.2 ⇥ 107 ✓ m 10 22eV ◆ 3/2 M . (33) Smaller mass halos would be dominated by the soliton. Ref. [7] showed that Eq. (31) is consistent with the relation, Mc ⇡ ↵ ✓ |Eh| Mh ◆1 2 M2 pl m , (34) where Mc is the core mass (mass within x < xc); Mh, Eh are the virial mass and energy of the simulated halo; and ↵ = 1 provides a good ﬁt to the data. Ref. [7] gave a heuristic argument, pointing out that Eq. (34) iden- tiﬁes the soliton scale radius (chosen as the core radius xc in [7]) with the inverse velocity dispersion in the host in [7]) with the inverse velocity dispersion in the hos lo, in qualitative agreement with a wave-like “uncer nty principle”. However, there is another way to express Eq. (34). Th re mass of a soliton is related to its total mass vi c ⇡ 0.236M . Thus, using Eq. (25) we have an an ytic relation Mc ⇡ 1.02 ⇣ |E | M ⌘1 2 M2 pl m . This allows u rephrase the empirical Eq. (34) by a more intuitiv hough equally empirical) expression: E M soliton ⇡ E M halo . (35 herefore, the soliton–host halo relation in the simula ns of Ref. [6, 7] can be summarised by the statemen
- 27. Exercise for NFW halo ss this result quantitatively. er a halo with an NFW density proﬁle ⇢NF W (x) = ⇢c c x Rs ⇣ 1 + x Rs ⌘2 , (37) z) = 3H2 (z) 8⇡G , c = 200 3 c3 ln(1 + c) c 1+c . (38) le has two parameters: the radius Rs and the tion parameter c = R200/Rs, where R200 is the ere the average density of the halo equals 200 cosmological critical density, roughly indicating radius of the halo. The gravitational potential o is NF W (x) = 4⇡G⇢c cR3 s x ln ✓ 1 + x Rs ◆ . (39) with f(c) = 0.54 v u u t Eq. (45) depends w rameter, via the 0.9 ÷ 1.1 for c = 5 the simulation resu pendence, which w agrees quantitative to account for the mass Mh, used in [ ison in App. A. Consider the ro galaxy satisfying E given by peak rotation velocity in the outskirts of a halo should approximately repeat itself in the deep inner region. We now discuss this result quantitatively. Consider a halo with an NFW density proﬁle ⇢NF W (x) = ⇢c c x Rs ⇣ 1 + x Rs ⌘2 , (37) where ⇢c(z) = 3H2 (z) 8⇡G , c = 200 3 c3 ln(1 + c) c 1+c . (38) The proﬁle has two parameters: the radius Rs and the concentration parameter c = R200/Rs, where R200 is the adius where the average density of the halo equals 200 imes the cosmological critical density, roughly indicating he virial radius of the halo. The gravitational potential of the halo is NF W (x) = 4⇡G⇢c cR3 s x ln ✓ 1 + x Rs ◆ . (39) Near the origin, x ⌧ Rs, NF W is approximately con- tant, NF W (x ⌧ Rs) ⇡ h, and is related to the mass of the halo, M200 = 200⇢c 4⇡ 3 c3 R3 s, via ⇥ ✓ H(z) H0 ◆3 ✓ with f(c) = 0.54 v u u t ✓ c 1 + c Eq. (45) depends weakly on rameter, via the factor f( 0.9 ÷ 1.1 for c = 5 ÷ 30. I the simulation result, Eq. (3 pendence, which we have su agrees quantitatively to abo to account for the slightly d mass Mh, used in [7], and ou ison in App. A. Consider the rotation v galaxy satisfying Eq. (35). given by V 2 circ,h(x) V 2 circ,h(Rs) = 2(1 + ⇠) l ⇠(1 + ⇠ scale radius concentration c 5 ÷ 30 value maxVcirc,h ⇡ 1.37 ⇥ 105 ( h) 1 2 km/s. (47) e other hand, in the inner galaxy x ⌧ Rs, the circu- locity due to the soliton peaks to a local maximum maxVcirc, ⇡ 1.51 ⇥ 105 ✓ ˜c 0.4 ◆1 2 ( h) 1 2 km/s, (48) we used Eq. (44) to ﬁx and Eq. (28) to relate it xVcirc, . anticipated in the beginning of this section, Eq. (35) cts approximately equal peak circular velocities for ner soliton component and for the host halo, maxVcirc, maxVcirc,h ⇡ 1.1 ✓ ˜c 0.4 ◆1 2 , (49) endent of the particle mass m, independent of the mass M200, and only weakly dependent on the de- of the halo via the factor (˜c/0.4) 1 2 . Eq. (49) is plot- Fig. 3 as function of the concentration parameter. the soliton bump in the rotation c smaller x according to Eq. (50), bu height. c = 25 c = Vcirc [km/s] 10 100 1000 10450 100 150 200 250 300 c = 25 c = 15 Vcirc [km/s] 75 100 125 150 where we used Eq. (44) to ﬁx and Eq. (28) to relate it to maxVcirc, . As anticipated in the beginning of this section, Eq. (35) predicts approximately equal peak circular velocities for the inner soliton component and for the host halo, maxVcirc, maxVcirc,h ⇡ 1.1 ✓ ˜c 0.4 ◆1 2 , (49) independent of the particle mass m, independent of the halo mass M200, and only weakly dependent on the de- tails of the halo via the factor (˜c/0.4) 1 2 . Eq. (49) is plot- ted in Fig. 3 as function of the concentration parameter. 5 10 15 20 25 30 1.05 1.10 1.15 1.20 1.25 1.30 c maxVcirc, maxVcirc,h
- 28. bottom panels, respectively. For larger m > 10 22 eV, the soliton bump in the rotation curve would shift to smaller x according to Eq. (50), but would maintain its height. M200 = 1012 M c = 25 c = 15 c = 10 Vcirc [km/s] x [pc] 10 100 1000 104 105 10650 100 150 200 250 300 c = 25 c = 15 c = 10 Vcirc [km/s] x [pc] M200 = 5 ⇥ 1010 M 10 100 1000 104 105 25 50 75 100 125 150 FIG. 4. Rotation curves for the ULDM soliton+halo system, predictions vs data m = 10 22 eV bottom panels, respectively. For larger m > 10 22 eV, the soliton bump in the rotation curve would shift to smaller x according to Eq. (50), but would maintain its height. M200 = 1012 M c = 25 c = 15 c = 10 Vcirc [km/s] x [pc] 10 100 1000 104 105 10650 100 150 200 250 300 c = 25 c = 15 c = 10 Vcirc [km/s] x [pc] M200 = 5 ⇥ 1010 M 10 100 1000 104 105 25 50 75 100 125 150
- 29. bottom panels, respectively. For larger m > 10 22 eV, the soliton bump in the rotation curve would shift to smaller x according to Eq. (50), but would maintain its height. M200 = 1012 M c = 25 c = 15 c = 10 Vcirc [km/s] x [pc] 10 100 1000 104 105 10650 100 150 200 250 300 c = 25 c = 15 c = 10 Vcirc [km/s] x [pc] M200 = 5 ⇥ 1010 M 10 100 1000 104 105 25 50 75 100 125 150 FIG. 4. Rotation curves for the ULDM soliton+halo system, predictions vs data m = 10 22 eV bottom panels, respectively. For larger m > 10 22 eV, the soliton bump in the rotation curve would shift to smaller x according to Eq. (50), but would maintain its height. M200 = 1012 M c = 25 c = 15 c = 10 Vcirc [km/s] x [pc] 10 100 1000 104 105 10650 100 150 200 250 300 c = 25 c = 15 c = 10 Vcirc [km/s] x [pc] M200 = 5 ⇥ 1010 M 10 100 1000 104 105 25 50 75 100 125 150 tive to the details of this mass cut. Of the 175 galaxies in [28], 162 pass the Mgal cut for m = 10 22 eV, and all 175 pass it for m = 10 21 eV. Vcirc [km/s] 0 1 2 3 4 5 6 0 20 40 60 80 UGC 1281 m = 10 22 eV x [kpc] Vcirc [km/s] 0 1 2 3 4 5 6 0 20 40 60 80 UGC 1281 m = 10 21 eV x [kpc]
- 30. bottom panels, respectively. For larger m > 10 22 eV, the soliton bump in the rotation curve would shift to smaller x according to Eq. (50), but would maintain its height. M200 = 1012 M c = 25 c = 15 c = 10 Vcirc [km/s] x [pc] 10 100 1000 104 105 10650 100 150 200 250 300 c = 25 c = 15 c = 10 Vcirc [km/s] x [pc] M200 = 5 ⇥ 1010 M 10 100 1000 104 105 25 50 75 100 125 150 FIG. 4. Rotation curves for the ULDM soliton+halo system, predictions vs data m = 10 22 eV bottom panels, respectively. For larger m > 10 22 eV, the soliton bump in the rotation curve would shift to smaller x according to Eq. (50), but would maintain its height. M200 = 1012 M c = 25 c = 15 c = 10 Vcirc [km/s] x [pc] 10 100 1000 104 105 10650 100 150 200 250 300 c = 25 c = 15 c = 10 Vcirc [km/s] x [pc] M200 = 5 ⇥ 1010 M 10 100 1000 104 105 25 50 75 100 125 150 tive to the details of this mass cut. Of the 175 galaxies in [28], 162 pass the Mgal cut for m = 10 22 eV, and all 175 pass it for m = 10 21 eV. Vcirc [km/s] 0 1 2 3 4 5 6 0 20 40 60 80 UGC 1281 m = 10 22 eV x [kpc] Vcirc [km/s] 0 1 2 3 4 5 6 0 20 40 60 80 UGC 1281 m = 10 21 eV x [kpc] Our analysis is as follows. For each SPARC galaxy, we ake a crude estimate of the halo mass contained within e observed rotation curve proﬁle by Mgal ⇠ RV 2 /G, here R is the radial distance of the last (highest dis- nce) data point in the rotation curve, and V the cor- sponding velocity. We keep only galaxies with Mgal > 9 m/10 22 eV 3/2 M . We do this in order to limit urselves to galaxy masses that are comfortably above e minimal halo mass (33). Our results are not sensi- ve to the details of this mass cut. Of the 175 galaxies [28], 162 pass the Mgal cut for m = 10 22 eV, and all 5 pass it for m = 10 21 eV. Vcirc [km/s] 0 1 2 3 4 5 6 0 20 40 60 80 UGC 1281 m = 10 22 eV x [kpc] Vcirc [km/s] 60 80 UGC 1281 ing the halo peak with a soliton peak, which would bias our analysis. Deﬁning the halo cut anywhere at & 1 m/10 22 eV 1 kpc guarantees that such confusion is avoided. Our ﬁrst pass on the data includes only galaxies for which the predicted soliton is resolved, namely, xpeak, from Eq. (50), with maxVcirc, = maxVcirc,h, lies within the rotation curve data. For these galaxies, we compute from data the ratio Vcirc, obs(xpeak, ) maxVcirc,h . (51) Here, Vcirc, obs(xpeak, ) is the measured velocity at the expected soliton peak position. The results of this ﬁrst pass on the data are shown in Fig. 11. A total of 46 galaxies pass the resolved soliton Vcirc [km/s] m = 10 22 eV 0 1 2 3 4 5 0 20 40 60 80 100 120 140 UGC 4325 x [kpc] Vcirc [km/s]
- 31. Analyzing SPARC data 175 high resolution rotation curves cuts: fbar2DM = V (bar) circ,h V (DM) circ,h 5 108 m 10 22eV 3 2 M < Mhalo < 5 1011 M nresolved soliton, that can be added to the sample of g. 11. No galaxy is added for fbar2DM < 0.33. For m = 0 21 eV, 16 and 5 galaxies are added with fbar2DM < 0.5, and none for fbar2DM < 0.33. In Figs. 11-12, vertical dashed line indicates the liton–host halo prediction. The shaded region shows e range of the prediction, modifying the RHS of q. (49) between 0.5 1.5, consistent with the scatter en in the simulations. We conclude that the four galaxies in Figs. 7-10 are not utliers: they are representative of a systematic discrep- ncy, that would be di cult to attribute to the scatter en in the simulations. If the soliton-host halo rela- on of [6, 7] is correct, then ULDM in the mass range soliton resolved m=10-22 eV 0 0.5 1 1.5 2 (velocity at soliton peak)/(max halo velocity) 0 5 10 15 #ofgalaxies Vcirc, obs(xpeak, )/maxVcirc,h data. We have limited our attention to the range m (10 22 ÷ 10 21 ) eV, for which we believe the result clear. We leave a detailed study of the precise exclu range to future work. We note that for lower pa mass, m . 10 23 eV, the soliton contribution ext over much of the velocity proﬁle of many of the SP galaxies, leaving little room for a host halo. This l where the galaxies are essentially composed of a s giant soliton, was considered in other works. We do pursue it further, one reason being that this ran small m is in signiﬁcant tension with Ly-↵ data [18 For higher particle mass, m & 10 21 eV, the so peak is pushed deep into the inner 100 pc of the rot curve. Although high resolution data (e.g. NGC 1 Fig. 9) is sensitive to and disfavours this situation, a careful analysis would be needed to draw a deﬁnitive clusion. Note that in this range of m, ULDM ceas o↵er a solution to the small-scale puzzles of ⇤CDM e.g., review in [12]). soliton m=10 -22 eV allow unresolved10 15 20galaxies soliton resolved m=10-22 eV 0 0.5 1 1.5 2 (velocity at soliton peak)/(max halo velocity) 0 5 10 15 #ofgalaxies Vcirc, obs(xpeak, )/maxVcirc,h m=10 -21 eV soliton resolved 0 0.5 1 1.5 2 (velocity at soliton peak)/(max halo velocity) 0 1 2 3 4 5 #ofgalaxies Vcirc, obs(xpeak, )/maxVcirc,h 45 26 5 < 1 < 0.55 < 0.33 4
- 32. Analyzing SPARC data cntd. if soliton peaks inside the innermost point m = 10 22 eV 0 1 2 3 4 5 0 20 40 60 x [kpc] Vcirc [km/s] m = 10 21 eV 0 1 2 3 4 5 0 20 40 60 80 100 120 140 UGC 4325 x [kpc] FIG. 8. Same as Fig. 7 for UGC 4325. -to-light ratio ⌥⇤ = 0.5 M /L . Set- DM),2 rc,h = V (obs),2 circ,h , we calculate the ratio /V (DM) circ,h . We present results when cut- values of fbar2DM < 1, 0.5, 0.33. on the data includes only galaxies for ted soliton is resolved, namely, xpeak, th maxVcirc, = maxV (DM) circ,h , lies within e data. For these galaxies, we compute io Vcirc, obs(xpeak, ) maxV (DM) circ,h . (51) eak, ) is the measured velocity at the peak position. his ﬁrst pass on the data are shown in e, and green histograms show the result bar2DM < 1, 0.5, 0.33, respectively. For e ﬁnd 45, 26, and 5 galaxies that pass C 1560 model, as seen from the lower panel of Fig. 8. To over- come this without complicating the analysis, we perform a second pass on the data. Here, we allow galaxies with unresolved soliton, as long as the innermost data point is located not farther than 3 ⇥ xpeak, . We need to correct for the fact that the soliton peak velocity is outside of the measurement resolution. To do this, we modify our observable as Vcirc, obs(xpeak, ) maxV (DM) circ,h ! Vcirc, obs(xmin,data) maxV (DM) circ,h ⇥ r xmin,data xpeak, , (52) where xmin,data is the radius of the ﬁrst data point. This correction is conservative, because it takes the fall-o↵ of the soliton gravitational porential at at x > xpeak, to be the same as for a point mass. In reality, the potential decays slower and the soliton-induced velocity decreases slower. Keeping this caveat in mind, Fig. 12 presents our results including unresolved solitons. For m = 10 22 eV, Vcirc [km/s] 80 100 120 NGC 100 mass, m . 10 23 eV, the soliton contribution extends ver much of the velocity proﬁle of many of the SPARC alaxies, leaving little room for a host halo. This limit, where the galaxies are essentially composed of a single iant soliton, was considered in other works. We do not ursue it further, one reason being that this range of mall m is in signiﬁcant tension with Ly-↵ data [18, 19]. For higher particle mass, m & 10 21 eV, the soliton eak is pushed deep into the inner 100 pc of the rotation urve. Although high resolution data (e.g. NGC 1560, ig. 9) is sensitive to and disfavours this situation, a more areful analysis would be needed to draw a deﬁnitive con- lusion. Note that in this range of m, ULDM ceases to ↵er a solution to the small-scale puzzles of ⇤CDM (see, .g., review in [12]). soliton m=10-22 eV allow unresolved 0 0.5 1 1.5 2 (velocity at soliton peak)/(max halo velocity) 0 5 10 15 20 #ofgalaxies Vcirc, obs(xpeak, )/maxVcirc,h soliton resolved 0 0.5 1 1.5 2 (velocity at soliton peak)/(max halo velocity) 0 5 10#ofgalaxies Vcirc, obs(xpeak, )/maxVcirc,h m=10-21 eV soliton resolved 0 0.5 1 1.5 2 (velocity at soliton peak)/(max halo velocity) 0 1 2 3 4 5 #ofgalaxies Vcirc, obs(xpeak, )/maxVcirc,h G. 11. Distribution of SPARC galaxies [28] with respect the ratio of observed circular velocity at the soliton peak the maximal circular velocity of the halo. The vertical shed line shows the prediction for the mean implied by the iton-host halo relation and the shaded region accounts for e intrinsic scatter in this relation. The ULDM mass is m = 22 21 e.g., review in [12]). soliton m=10 -22 eV allow unresolved 0 0.5 1 1.5 2 (velocity at soliton peak)/(max halo velocity) 0 5 10 15 20 #ofgalaxies Vcirc, obs(xpeak, )/maxVcirc,h soliton m=10-21 eV allow unresolved 0 0.5 1 1.5 2 (velocity at soliton peak)/(max halo velocity) 0 2 4 6 8 10 #ofgalaxies Vcirc, obs(xpeak, )/maxVcirc,h 5 93 42 20 5
- 33. Conclusion: ULDM with is disfavoured by rotation curves of disk galaxies m (10 22 ÷ 10 21 )eV cannot play a role in solving small-scale problems of LambdaCDM
- 34. Caveats:
- 35. Caveats: • Baryonic eﬀects - stars tend to increase the soliton mass (J.H.H.Chan et al. (2017)); their potential can be taken into account self- consistently - baryonic feedback unlikely to destroy the soliton: in the inner partMsol > 10 Mbaryons
- 36. Caveats: • Baryonic eﬀects - stars tend to increase the soliton mass (J.H.H.Chan et al. (2017)); their potential can be taken into account self- consistently - baryonic feedback unlikely to destroy the soliton: in the inner part • Self-interaction and direct interaction with baryons - negligible for minimal models Msol > 10 Mbaryons
- 37. Caveats: • Baryonic eﬀects - stars tend to increase the soliton mass (J.H.H.Chan et al. (2017)); their potential can be taken into account self- consistently - baryonic feedback unlikely to destroy the soliton: in the inner part • Self-interaction and direct interaction with baryons - negligible for minimal models • Accretion on supermassive black hole - negligible for and , but quickly increases with the mass of the ﬁeld Msol > 10 Mbaryons m 10 21 eV MSMBH < 1010 M
- 38. Future: probing higher masses computed holding their mass ﬁxed by Eq. (31), but in- cluding the e↵ects of baryons as we discuss below. An NFW proﬁle, ﬁtted in Ref. [36] to r & 10 kpc SDSS data, is shown in dashed black. 10 -4 10 -2 10 0 10 2 10 4 r [pc] 10 5 10 6 10 7 10 8 10 9 10 10 10 11 10 12 enclosedmass[M] m=10-19 eV m=10-20 eV m=10-21 eV m=10-22 eV Ghez 2003 McGinn 1989 Fritz 2016 Lindqvist 1992 Schodel 2014 Sofue 2009 Sofue 2012 Sofue 2013 Chatzopoulos 2015 Deguchi 2004 Oh 2009 Trippe 2008 Gilessen 2008 Nuclear Bulge (disc+star cluster) from photometry, Launhardt (2002) NFW fit, Piffl (2015) to the re The p superim the phot ties due and due we learn that sta ically in Assum toy mod an ULD of the N radially the NB in [55], 300 pc. we calcu Fig. 1 paramet the unp • Inner dynamics of the Milky Way
- 39. Summary
- 40. Summary ULDM is a simple (perhaps, the simplest) option for dark matter with interesting phenomenology. Theoretically motivated
- 41. Summary ULDM is a simple (perhaps, the simplest) option for dark matter with interesting phenomenology. Theoretically motivated If soliton — host halo relation holds for real halos, is disfavoured by galactic rotation curves (also Ly ) m 10 21 eV
- 42. Summary ULDM is a simple (perhaps, the simplest) option for dark matter with interesting phenomenology. Theoretically motivated Further understanding of structure formation with ULDM (baryonic eﬀects, supermassive black hole) More probes: Inner Milky Way, 21 cm, pulsar timing (see talk by Diego Blas) If soliton — host halo relation holds for real halos, is disfavoured by galactic rotation curves (also Ly ) Outlook m 10 21 eV