Sergey Sibiryakov "Galactic rotation curves vs. ultra-light dark matter: Implications of the soliton—host halo relation"

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

eVmDM ,
109
What is dark matter ?
1012
1028
106810 23

eVmDM ,
WIMP’s
}
109
1012
1028
106810 23

eVmDM ,
WIMP’s
}
SM
SM
DM
DM
109
1012
1028
106810 23

eVmDM ,
WIMP’s WIMPZILLA’s
}
SM
SM
DM
DM
109
1012
1028
106810 23

eVmDM ,
primordial
black holes
}
SM
SM
DM
DM
109
1012
1028
106810 23

eVmDM ,
primordial
black holes
}
SM
SM
DM
DM
}
inﬂation ?
109
1012
1028
106810 23

eVmDM ,
primordial
black holes
}
SM
SM
DM
DM
}
inﬂation ?
Ultra-Light
Dark Matter
109
1012
1028
106810 23

eVmDM ,
primordial
black holes
}
SM
SM
DM
DM
}
inﬂation ?
Ultra-Light
Dark Matter
109
1012
1028
106810 23
misalignment

The simplest DM model: free massive scalar
0
e.o.m. in expanding Universe:
L =
1
2
( µ )2
m2 2
¨ + 3H ˙ + m2
= 0

0
• = const
L =
1
2
( µ )2
m2 2
¨ + 3H ˙ + m2
= 0
H > m

p = 0
density:
pressure:
0
• = 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

p = 0
density:
pressure:
0
• = 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

• Theoretical pedigree: pseudo-Goldstone boson
e.g. global periodic variable with period 2 fU(1)

non-zero mass can be generated by non-perturbative

eﬀects
examples: QCD axion, string theory ALP’s, relaxion,

quasi-dilaton, ....
m exp( Sinst)
V ( ) = m2
f2
1 cos( /f)

non-zero mass can be generated by non-perturbative

eﬀects
examples: QCD axion, string theory ALP’s, relaxion,

quasi-dilaton, ....
after inﬂation
• Density fraction:
f
m exp( Sinst)
V ( ) = m2
f2
1 cos( /f)
0.05
f
1017GeV
2 m
10 22eV
1/2

• 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

• aﬀects structures on small scales, could address
the problems of particle CDM (?) Hu, Barkana, Gruzinov (2000)
Hui et al. (2016)
m 10 22
eV
m 10 21
eV
m 10 23
eV

Hui et al. (2016)
• leads to black-hole superradiance:
probed by black-hole spins, gravitational waves
Arvanitaki, Dubovsky (2011)
Arvanitaki et al. (2016)
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

Hui et al. (2016)
focus of this talk:
NB. Can be axion-like particle, but not QCD axion
• leads to black-hole superradiance:
probed by black-hole spins, gravitational waves
Arvanitaki, Dubovsky (2011)
Arvanitaki et al. (2016)
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

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

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

Probing ULDM with
galactic rotation curves

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
Schrödinger-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

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

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

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

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
eV,
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

eV,
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
predictions vs data
m = 10 22
eV
eV,
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]

eV,
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
predictions vs data
m = 10 22
eV
eV,
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
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
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]

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
0
5
10
15
#ofgalaxies
m=10
-21
eV
soliton resolved
0 0.5 1 1.5 2
0
1
2
3
4
5
#ofgalaxies
45
26
5
< 1 < 0.55 < 0.33
4

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
0
5
10
15
20
#ofgalaxies
soliton resolved
0 0.5 1 1.5 2
0
5
10#ofgalaxies
m=10-21
eV
soliton resolved
0 0.5 1 1.5 2
0
1
2
3
4
5
#ofgalaxies
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
0
5
10
15
20
#ofgalaxies
soliton
m=10-21
eV
allow unresolved
0 0.5 1 1.5 2
0
2
4
6
8
10
#ofgalaxies
5
93
42
20
5

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

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

Caveats:

consistently


in the inner part
• Self-interaction and direct interaction with baryons

- negligible for minimal models
Msol > 10 Mbaryons

Caveats:

consistently


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

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

Summary
ULDM is a simple (perhaps, the simplest) option for dark
matter with interesting phenomenology. Theoretically
motivated

Summary
motivated
If soliton — host halo relation holds for real halos,

is disfavoured by galactic rotation curves

(also Ly )
m 10 21
eV

Summary
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

Sergey Sibiryakov "Galactic rotation curves vs. ultra-light dark matter: Implications of the soliton—host halo relation"

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Sergey Sibiryakov "Galactic rotation curves vs. ultra-light dark matter: Implications of the soliton—host halo relation"