physical review d 98, 083027 (2018)

Galactic rotation curves versus ultralight dark matter:Implications of the soliton-host halo relation

Nitsan Bar,1 Diego Blas,2,3 Kfir Blum,1,2 and Sergey Sibiryakov2,4,51Weizmann Institute of Science, 234 Herzl Street, Rehovot 7610001, Israel

2Theory department, CERN, CH-1211 Geneve 23, Switzerland3Theoretical Particle Physics and Cosmology Group, Department of Physics,

King’s College London, Strand, London WC2R 2LS, United Kingdom4LPPC, IPHYS, Ecole Politechnique Federale de Lausanne, CH-1015 Lausanne, Switzerland

5Institute for Nuclear Research of the Russian Academy of Sciences, 60th October Anniversary Prospect,7a, 117312 Moscow, Russia

(Received 15 May 2018; published 31 October 2018)

Bosonic ultralight dark matter (ULDM) would form cored density distributions at the center of galaxies.These cores, seen in numerical simulations, admit analytic description as the lowest energy bound state solution(“soliton”) of the Schroedinger-Poisson equations. Numerical simulations of ULDM galactic halos findempirical scaling relations between themass of the large-scale host halo and themass of the central soliton.Wediscuss how the simulation results of different groups can be understood in terms of the basic properties of thesoliton. Importantly, simulations imply that the energy per unitmass in the soliton and in thevirialized host haloshould be approximately equal. This relation lends itself to observational tests because it predicts that the peakcircular velocity, measured for the host halo in the outskirts of the galaxy, should approximately repeat itself inthe central region. Contrasting this prediction with the measured rotation curves of well-resolved nearbygalaxies, we show that ULDM in the mass range m ∼ ð10−22 ÷ 10−21Þ eV, which has been invoked as apossible solution to the small-scale puzzles ofΛCDM, is in tension with the data. We suggest that a dedicatedanalysis of the Milky Way inner gravitational potential could probe ULDM up to m≲ 10−19 eV.

DOI: 10.1103/PhysRevD.98.083027

I. INTRODUCTION

Two forms of matter could provide the cosmologicalequation of state, required for dark matter (DM): non-relativistic massive particles (e.g., WIMPs), or the classicalbackground of an ultralight bosonic field oscillating arounda minimum of its potential (e.g., axions or axionlikeparticles). On cosmological scales, WIMPs and ultralightdark matter (ULDM) behave similarly. However, on scalescomparable to its de Broglie wavelength, ULDM behavesmarkedly different from WIMPs [1–13].Axionlike particles with exponentially suppressed

masses arise in the context of string theory [12,14–16].They are produced by misalignment and for mass in therange m ∼ ð10−22 ÷ 10−18Þ eV their cosmic abundancecould naturally match the observed DM density.ULDMwith massm ∼ ð10−22 ÷ 10−21Þ eV is particularly

motivated due to several puzzles, facing the standard WIMPparadigm on galactic scales (see, e.g., [12,17] for recent

reviews). This range ofm is in some tension with the matterpower spectrum, inferred from Ly-α forest analyses [18–22],which yields a boundm≳ 10−21 eV. Nevertheless, since thestrongest Ly-α bound constraints come from the smallest andmost nonlinear scales, where systematic effects are chal-lenging, we believe it prudent to seek additional methods toconstrain the model for m≳ 10−22 eV.Numerical simulations [6,7,10,11,13] show that ULDM

forms cored density profiles in the inner region of galactichalos, roughly within the de Broglie wavelength,

xdB ≈ 190

�v

100 km=s

�−1�

m10−22 eV

�−1

pc: ð1Þ

The core, referred to as “soliton” in the literature [4,5,8,9],corresponds to a coherent quasi-stationary solution of theULDM equations of motion and its density profile can bederived analytically.1 This match between analytic andnumerical simulation results offers a unique opportunity to

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.

1Strictly speaking, the soliton solution is only known numeri-cally. With some abuse of the word, we refer to it here asanalytical to emphasize that the solution can be found byintegrating a simple differential equation in a procedure thattakes < 1 sec on a standard laptop.

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understand and extend the simulations, making the ULDMmodel potentially more predictive than the WIMP para-digm. For nearby dwarf galaxies (v ∼ 10 km=s) orMilkyWay (MW)-like galaxies (v ∼ 100 km=s), the solitoncore could be resolved with current observational tools,offering a test of the model.Attempts to detect, constrain, or fit ULDM soliton cores

using galactic rotation velocity and velocity dispersion datawere discussed in the literature [2,3,6–8,23–26]. Some ofthese analyses fitted a soliton profile to describe entiregalaxies [2,3,23]. This exercise leads to smallULDMparticlemass, m≲ 10−23 eV, in strong tension with cosmologicalbounds [18–22]. Furthermore, numerical simulations[6,7,10,11,13] have shown that galactic halos above a certainmass would exhibit a more complex structure, with thecentral soliton transiting into a large-scale host halo com-posed of an incoherent superposition of multiple scalar fieldwavepackets. Consequently, otherworks [8,24–26] analyzedgalactic profiles using a solitonþ host halo description.A key question in comparing the solitonþ host halo

model to kinematical data is how to model the transitionbetween the central soliton and the host halo. Previousphenomenological analyses [8,24–26] defined separate freeparameters for the host halo and for the soliton, for eachindividual galaxy in the sample. On the numerical simu-lation side, several studies focused on finding a soliton–host halo relation [6,10,11,13]. The point in finding asoliton–host halo relation is, of course, that it could tietogether the behavior of ULDM in the large-scale host haloto predict the central core with fewer free parameters.In this work, we consider the soliton–host halo relations,

reported by different numerical simulation groups[6,7,10,11,13]. Our first observation is that properties ofthe analytic soliton solution can provide important insighton the numerical results. We show that: (i) the soliton–hosthalo relation, reported in [13], essentially attributes the totalenergy (kineticþ gravitational) of the halo to the dominantsoliton. This energetic dominance of the soliton is unlikelyto hold for realistic galaxies above a certain size; (ii) thesoliton–host halo relation, reported in [7], essentiallyequates the energy per unit mass in the soliton to theenergy per unit mass in the virialized halo. This relation canapply to real galaxies.Assuming the soliton–host halo relation of [7], we show

that it leads to a prediction: the peak circular velocitycharacterizing the host halo on large scales (few kpc fortypical (109 ÷ 1010) M⊙ galaxies) should repeat itself in thecore on small scales (≲1 kpc), insensitive to the details of thehost halo density profile. This implies an observationalconstraint that can be tested without free parameters.Applying this test to high-resolution rotation curves oflate-type galaxies from [27,28], we find that ULDM inthe mass rangem ∼ ð10−22 ÷ 10−21Þ eV is disfavored by thedata. Baryonic physics is unlikely to cure the tension:the discrepancy between the predictions of ULDM (with

the soliton–host halo relation) and the data are too large. Inmany galaxies in the sample we analyze, baryonic feedbackwould need to overcome a dark matter-to-baryon mass ratioof ≳10∶1, to destroy the soliton. As a result, if the soliton–host halo relation of [6,7] holds for real systems, ULDM isdisfavored belowm ∼ 10−21 eV and is unlikely to play a rolein solving the small scale puzzles of ΛCDM.We also discuss observational imprints of the soliton in

big galaxies, such as the MW. In this case, the shape of thesoliton is modified due to the gravitational potential ofbaryonic matter and supermassive black hole (SMBH).Preliminary numerical simulations of ULDM halos withstars [29] indicate that in the presence of baryons, thesoliton mass stays the same or increases compared to thesoliton–host halo prediction of pure ULDM. If these resultsare confirmed, solitons formed by ULDM with massesm≲ 10−19 eV will produce order-one contribution into themass budget of the inner MW. A dedicated analysis of innerMW kinematics—including simultaneous modeling of thebaryonic mass and the soliton—could potentially testULDM up to m≲ 10−19 eV.The outline of this paper is as follows. In Sec. II, we review

some basic properties of the soliton. In Sec. III, we discussthe soliton–host halo relations found in simulations, andshow that these relations can be understood in terms offundamental properties of the soliton and the halo. Thesimulations of [6,7] are considered in Sec. III A; they can besummarized by the statement ðE=MÞjsoliton ¼ ðE=MÞjhalo,where E is the total energy (kineticþ gravitational, withinthe virial radius, for the halo) andM is themass (againwithinthe virial radius, for the halo). The simulations of [13] areconsidered in Sec. III B; their soliton–halo result can besummarized by the statementEjsoliton ¼ Ejhalo.We argue thatthis result is unlikely to represent realistic galaxies above acertain size.In Sec. IV, adopting the soliton–host halo relation of [6,7],

we work out its observational consequences. We show thatin ULDM galaxies satisfying this relation, the rotationvelocity in the inner core should be approximately as highas the peak rotation velocity in the outer part of the galaxy. InSec. IVA, we compare this analysis to numerical profilestaken directly from the published simulation results, findinggood agreement. In Sec. IV B, we compare this predictionto real galaxies from [27,28], finding tension for m∼ð10−22 ÷ 10−21Þ eV. We show that the intrinsic scatter inthe soliton–host halo relation does not resolve thediscrepancy.In Sec. V, we calculate how baryonic effects could

modify the soliton solution. In Sec. VA, we consider asmooth fixed distribution of baryonic mass, deferring thecase of a super-massive black hole to App. B. These resultsare applied to the MW in Sec. V B.In Sec. VI, we compare our results to previous literature

and discuss some caveats to our analysis. Sec. VII containsa summary of our results and open questions.

BAR, BLAS, BLUM, and SIBIRYAKOV PHYS. REV. D 98, 083027 (2018)

083027-2

II. SOLITON PROPERTIES:ANALYTIC CONSIDERATIONS

In this section, we review the relevant properties of thesoliton that help to understand results from numericalsimulations.We consider a real, massive, free2 scalar field ϕ,

satisfying the Klein-Gordon equation of motion and min-imally coupled to gravity. In the nonrelativistic regime, it isconvenient to decompose ϕ as

ϕðx; tÞ ¼ 1ffiffiffi2

pme−imtψðx; tÞ þ c:c:; ð2Þ

with complex field ψ that varies slowly in space and time,such that j∇ψ j ≪ mjψ j and j _ψ j ≪ mjψ j. The field ψsatisfies the Schroedinger-Poisson (SP) equations [32]

i∂tψ ¼ −1

2m∇2ψ þmΦψ ; ð3Þ

∇2Φ ¼ 4πGjψ j2: ð4Þ

We look for a quasi-stationary phase-coherent solution,described by the ansatz3

ψðx; tÞ ¼�mMplffiffiffiffiffiffi

4πp

�e−iγmtχðxÞ: ð5Þ

The ULDM mass density is

ρ ¼ ðmMplÞ24π

χ2

≈ 4.1 × 1014�

m10−22 eV

�2

χ2 M⊙=pc3: ð6Þ

The parameter γ is proportional to the ULDM energy perunit mass. Validity of the nonrelativistic regime requiresjγj ≪ 1. Since we are looking for gravitationally boundconfigurations, γ < 0.Assuming spherical symmetry and defining r ¼ mx, the

SP equations for χ and Φ are given by

∂2rðrχÞ ¼ 2rðΦ − γÞχ; ð7Þ

∂2rðrΦÞ ¼ rχ2: ð8Þ

Finding the ground state solution amounts to solvingEqs. (7)–(8) subject to χðr → 0Þ ¼ const, χðr → ∞Þ ¼ 0,with no nodes. Given the boundary value of χ at r → 0, thesolution is found for a unique value of γ.

It is convenient to first solve Eqs. (7)–(8) with theboundary condition χð0Þ ¼ 1. Let us call this auxiliarysolution χ1ðrÞ, with γ1. A numerical calculation gives [4,5,8]

γ1 ≈ −0.69; ð9Þand the solution is plotted in Fig. 1. The mass of the χ1soliton is

M1 ¼M2

pl

m

Z∞

0

drr2χ21ðrÞ

≈ 2.79 × 1012�

m10−22 eV

�−1

M⊙: ð10Þ

Its core radius, defined as the radius where the mass densitydrops by a factor of 2 from its value at the origin, is

xc1 ≈ 0.082

�m

10−22 eV

�−1pc: ð11Þ

Other solutions of Eqs. (7)–(8) can be obtained fromχ1ðrÞ, Φ1ðrÞ by a scale transformation. That is, thefunctions χλðrÞ, ΦλðrÞ, together with the eigenvalue γλ,given by

χλðrÞ ¼ λ2χ1ðλrÞ; ð12ÞΦλðrÞ ¼ λ2Φ1ðλrÞ; ð13Þ

γλ ¼ λ2γ1; ð14Þalso satisfy Eqs. (7)–(8) with correct boundary conditionsfor any λ > 0. The soliton mass and core radius for χλ are

Mλ ¼ λM1; ð15Þxcλ ¼ λ−1xc1: ð16Þ

A mnemonic for the numerical value of λ is given by

FIG. 1. Profile of the “standard” χ1 soliton with λ ¼ 1 (bluesolid). We also show the corresponding gravitational potential(orange dashed) and circular velocity of a test particle(dotted green).

2Analyses of interacting fields can be found in, e.g.,[4,5,30,31].

3Mpl ¼ 1=ffiffiffiffiG

p.

GALACTIC ROTATION CURVES VERSUS ULTRALIGHT … PHYS. REV. D 98, 083027 (2018)

083027-3

λ ¼ 3.6 × 10−4�

m10−22 eV

��Mλ

109 M⊙

�: ð17Þ

The product of the soliton mass and core radius isindependent of λ,

Mλxcλ ≈ 2.27 × 108�

m10−22 eV

�−2

kpcM⊙: ð18Þ

Formally, solutions exist for any positive value of λ andhence for any soliton mass. However, if we select λ≳ 1we reach jγλj > 1, outside of the regime of validity ofthe nonrelativistic approximation. Thus, self-consistentsolutions are limited to λ ≪ 1 and their eigenvaluejγλj ¼ λ2jγ1j ≪ 1, consistent with the nonrelativisticapproximation.The energy in an arbitrary nonrelativistic ULDM con-

figuration is

E ¼Z

d3x

�j∇ψ j22m2

þΦjψ j22

�¼ Ek þ Ep; ð19Þ

with kinetic (potential) energy Ek (Ep). For the ansatz (5),integrating by parts and using Eqs. (3)–(5), we have

E ¼ 1

3Mγ: ð20Þ

Note that spherical symmetry is not needed for Eq. (20)to hold.Considering the χλ solitons, we find Ep;λ ¼ −2Ek;λ ¼

2Eλ with

Eλ ≈ −0.476λ3M2

pl

m; ð21Þ

Mλ ≈ 2.06λM2

pl

m: ð22Þ

This leads to a relation for an isolated soliton [4,5],

Mλ

ðM2pl=mÞ ≈ 2.64

�� Eλ

ðM2pl=mÞ

��13

: ð23Þ

Another useful relation gives the energy per unit mass fromthe scaling parameter λ,

jEλjMλ

≈ 0.23λ2; ð24Þ

which can also be written as

Mλ ≈ 4.3

�jEλjMλ

�12 M2

pl

m: ð25Þ

The circular velocity curve for a test particle in thesoliton gravitational potential is given by

V2circ;λðrÞ ¼ r∂rΦλðrÞ: ð26Þ

The circular velocity rises as Vcirc;λ ∝ r at small r anddecreases as Vcirc;λ ∝ r−

12 at large r, see Fig. 1. The peak of

Vcirc is obtained at

xpeak;λ ≈ 0.16λ−1�

m10−22 eV

�−1

pc

≈ 460

�m

10−22 eV

�−2�

Mλ

109 M⊙

�−1

pc; ð27Þ

and the peak velocity is

maxVcirc;λ ≈ 2.3 × 105 λ km=s

≈ 83

�m

10−22 eV

��Mλ

109 M⊙

�km=s: ð28Þ

III. MAKING CONTACT WITHNUMERICAL SIMULATIONS

We now discuss results from the numerical simulationsof three different groups, Refs. [6,7], Ref. [13], andRefs. [10,11].The first point to note is that soliton configurations, in a

form close to the idealized form discussed in Sec. II,actually occur dynamically in the central region of the haloin the numerical simulations.4 In Fig. 2, we collectrepresentative density profiles from Ref. [6] (blue),Ref. [13] (orange), and Ref. [10] (green). We refer tothose papers for more details on the specific set-ups in eachsimulation. To make Fig. 2, in each case, we find the λparameter that takes the numerical result into the χ1 soliton,rescale the numerical result accordingly and present it incomparison with the analytic χ21ðrÞ profile.While different groups agree that solitons form in the

centers of halos, they do not appear to agree on thematching between the inner soliton profile and the hosthalo. Refs. [6,7] and Ref. [13] reported scaling relationsbetween the central soliton and the host halo. As we showbelow, the scaling relations found by both groups areconnected to properties of a single, isolated, self-gravitatingsoliton (part of these observations were made in [10,11]).

A. Soliton vs host halo: The simulations of Ref. [6,7]

At cosmological redshift z ¼ 0, the numerical simula-tions of [6,7] yield approximately NFW-like halos whichtransit, in the central region, into a core with core radius andmass density

xc ≈ 160

�Mh

1012 M⊙

�−13

�m

10−22 eV

�−1

pc; ð29Þ

4The first simulations of cosmological ULDM galaxies [33]did not have sufficient resolution to resolve the central core.


083027-4

ρðxÞ ≈190ð m

10−22 eVÞ−2ð xc100 pcÞ−4

ð1þ 0.091ð xxcÞ2Þ8M⊙ pc−3; ð30Þ

where Mh is the virial mass of the host halo. As noted in[6–8], Eqs. (29)–(30) are an excellent numerical fit for asoliton χλ. The mass of this soliton is

M ≈ 1.4 × 109�

m10−22 eV

�−1�

Mh

1012 M⊙

�13

M⊙; ð31Þ

so its λ parameter is

λ ≈ 4.9 × 10−4�

Mh

1012 M⊙

�13

: ð32Þ

Note that Eq. (31) is applicable only as long as the haloexceeds a minimal mass,

Mh;min ∼ 5.2 × 107�

m10−22 eV

�−3=2

M⊙: ð33Þ

Smaller mass halos would be dominated by the soliton.Reference [7] showed that Eq. (31) is consistent with the

relation,

Mc ≈ α

�jEhjMh

�12 M2

pl

m; ð34Þ

where Mc is the core mass (mass within x < xc); Mh, Ehare the virial mass and energy of the simulated halo; andα ¼ 1 provides a good fit to the data. Reference [7] gave aheuristic argument, pointing out that Eq. (34) relates thesoliton scale radius (chosen as the core radius xc in [7])with the velocity dispersion of particles in the host halo,

in qualitative agreement with a wavelike “uncertaintyprinciple.”However, there is another way to express Eq. (34). The

core mass of a χλ soliton is related to its total mass viaMcλ ≈ 0.236Mλ. Thus, using Eq. (25) we have an analytic

relationMcλ ≈ 1.02ðjEλjMλ

Þ12 M2pl

m . This allows us to rephrase theempirical Eq. (34) by a more intuitive (though equallyempirical) expression:

EM

��soliton

≈EM

��halo

: ð35Þ

Therefore, the soliton–host halo relation in the simulationsof Ref. [6,7] can be summarized by the statement that theenergy per unit mass of the soliton matches the energy perunit mass of the host halo.

B. Soliton vs host halo: The simulations of Ref. [13]

The simulations of Ref. [13] pointed to an empiricalscaling relation between the soliton mass M and the totalenergy of the ULDM distribution in the simulation box, Eh,

MðM2

pl=mÞ ≈ 2.6

�� Eh

ðM2pl=mÞ

��13

: ð36Þ

However, this is just Eq. (23), if we replace the halo energyEh by the energy of the soliton. Because the central densityprofile found in [13] was a χλ soliton, to a good approxi-mation, it must be the case that the total energy of the haloin the simulations of [13] was dominated by the centralsoliton contribution. This situation is unlikely to hold forrealistic cosmological host halos with Mh significantlyabove Mh;min.How could this have happened? The initial conditions in

the simulations of [13] were a collection of N solitons,which were then allowed to merge. It appears that theseinitial conditions were constructed such that one initial statesoliton—the soliton of initially largest mass—grew toabsorb the entire energy of the system. Differently fromRef. [7] that considered initial conditions of N identicalinitial solitons, the simulations of [13] initiated their Nsolitons with a random flat distribution in soliton radius.Such distribution would be skewed towards large solitonenergy because Eλ ∝ x−3cλ . Considering the initial conditionset-up as explained in [13], we find that the most massiveinitial state soliton typically needed to grow in mass by onlya factor of 1.5 ÷ 2, to absorb the entire energy of the halo.Note that energy dominance of the central soliton over

the host halo, implied by Eq. (36), is not the same, ofcourse, as equating the energy per unit mass of the solitonand the halo, implied by Eq. (35). Halos in [6,7] attainedmasses up to two orders of magnitude larger than thecentral soliton mass, meaning their halo energy was twoorders of magnitude larger than the energy of the soliton.

FIG. 2. Review of results from numerical simulations bydifferent groups. Markers show density profiles of simulatedhalos from Schive et al. [6] (blue circles), Mocz et al. [13](orange squares), and Schwabe et al. [10] (green triangles). Thecentral regions of the halos are described by the soliton(solid line).


083027-5

C. Comments

As far as we can currently determine, Eq. (35) mayindeed reflect a realistic soliton–host halo relation for largeenough cosmological halos. In the following sections, wetake a leap of faith and assume that the simulations of [6,7]produced the correct scaling relation. We stress thatEq. (35) is an empirical result and was only tested in[6,7] for host halo masses ranging from ∼108 M⊙ to∼1011 M⊙. Our key numerical analysis will concernsystems in this range of mass.We defer a theoretical study of the origin of Eq. (35) to

future work. Here we give only a few comments. We stressthat the discussion in the rest of this section does not affectany of our results.For a soliton, E=M ¼ γ=3. On the other hand, γm can be

associated with the chemical potential of ULDM particlesin the soliton (see e.g., [34] and references therein). Thismay appear to hint that Eq. (35) corresponds to thermo-dynamic equilibrium between the ULDM particles in thehost halo and in the soliton. However, there is someevidence to the contrary from simulations.Reference [29] simulated ULDM, adding collisionless

point particles (“stars”). The stars aggregated dynamically ina cuspy profile, resulting in amoremassive soliton comparedto the pure ULDM simulations [6,7] with a given host halomass. Testing the reversibility of the system, Ref. [29]adiabatically “turned off” the stars after the initial systemvirialized. When eliminating the stars, the solitonþ halosystem did not relax back to Eq. (35). Instead, the excessULDMmass that was contained in the soliton in the presenceof stars remained captured in the soliton, and did not return tothe host halo. The final state of the system was not describedby Eq. (35): the soliton ended up containing larger (negative)E=M than the halo, and larger mass compared with Eq. (31).

IV. SOLITON-HOST HALO RELATION ANDGALACTIC ROTATION CURVES

As we have seen, the soliton–host halo relation found inthe simulations of [6,7] can be summarized by Eq. (35),equating the energyper unitmass of thevirialized host halo tothat in the soliton component. For a virialized system, theenergy per unit mass maps to kinetic energy density: inparticular, the characteristic circular velocity (or, up to anOð1Þ geometrical factor, the velocity dispersion) of testparticles in the halo and in the soliton shouldmatch. The peakcircular velocity of the soliton, given by Eqs. (27)–(28),occurs deep in the inner part, x < 1 kpc, of the galaxy; whilethe peak circular velocity of an NFW-like halo occurs far outat x ∼ 2Rs, with Rs the NFW characteristic radius, of order10 kpc for a MW-like galaxy. Thus, if the scaling derivedfrom the simulations of [6,7] is correct, ULDM predicts thatthe peak rotation velocity in the outskirts of a halo shouldapproximately repeat itself in the deep inner region. We nowdiscuss this result quantitatively.

Consider a halo with an NFW density profile

ρNFWðxÞ ¼ρcδc

xRsð1þ x

RsÞ2 ; ð37Þ

where

ρcðzÞ ¼3H2ðzÞ8πG

; δc ¼200

3

c3

lnð1þ cÞ − c1þc

: ð38Þ

The profile has two parameters: the radius Rs and theconcentration parameter c ¼ R200=Rs, where R200 is theradius where the average density of the halo equals 200 timesthe cosmological critical density, roughly indicating thevirialradius of the halo. The gravitational potential of the halo is

ΦNFWðxÞ ¼ −4πGρcδcR3

s

xln

�1þ x

Rs

�: ð39Þ

Near the origin, x ≪ Rs, ΦNFW is approximately constant,ΦNFWðx ≪ RsÞ ≈Φh, and is related to the mass of the halo,M200 ¼ 200ρc

4π3c3R3

s , via

Φh ¼ −G�

4πδcðlnð1þ cÞ − c

1þcÞ2�1

3

ρ13c M

23

200: ð40Þ

We can estimate the energy per unit mass of the virializedhalo by

EM

��halo

≈ π

R R200

0 dxx2ρNFWðxÞΦNFWðxÞM200

: ð41Þ

This gives

EM

��halo

≈c4Φh; ð42Þ

where

c ¼ c − lnð1þ cÞð1þ cÞ lnð1þ cÞ − c

: ð43Þ

Typical values of the concentration parameter are in therange c ∼ 5 ÷ 30 [35]. In this range, c varies betweenc ∼ 0.55 ÷ 0.35, respectively. (For reference, fits of theMW outer rotation curve give c ∼ 10 ÷ 20 [36].)Plugging Eq. (42) into the soliton–host halo relation

Eq. (35), the scaling parameter λ is fixed as

−0.23λ2 ≈Eλ

Mλ≈c4Φh; ð44Þ

which implies5

5In the numerical estimates, we use H0 ¼ 70 km=s=Mpc forthe present-day Hubble constant.


083027-6

Mλ ≈ 2.1ffiffiffiffiffiffiffiffiffiffiffiffi−cΦh

p M2pl

m

≈ 2.4 × 109�

m10−22 eV

�−1

×

�HðzÞH0

�13

�M200

1012 M⊙

�13

fðcÞM⊙; ð45Þ

with

fðcÞ ¼ 0.54

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�c

1þ c

�c − lnð1þ cÞ

ðlnð1þ cÞ − c1þcÞ2

s:

Eq. (45) depends weakly on the NFW concentration param-eter, via the factor fðcÞ that varies in the range 0.9 ÷ 1.1 forc ¼ 5 ÷ 30. It agrees parametrically with the simulationresult, Eq. (31) [including the redshift dependence, whichwe have suppressed in Eq. (31)]. It also agrees quantitativelyto about 20%; to see this, we need to account for the slightlydifferent definition of the halo massMh, used in [7], and ourM200. We do this comparison in App. A.Consider the rotation velocity curve of an ULDM galaxy

satisfying Eq. (35). The NFW rotation curve is given by

V2circ;hðxÞ

V2circ;hðRsÞ

¼ 2ð1þ ξÞ lnð1þ ξÞ − 2ξ

ξð1þ ξÞðlnð4Þ − 1Þ ; ξ≡ xRs

: ð46Þ

This halo rotation curve peaks at x ≈ 2.16 Rs with a peakvalue

maxVcirc;h ≈ 1.37 × 105ð−ΦhÞ12 km=s: ð47Þ

On the other hand, in the inner galaxy x ≪ Rs, the circularvelocity due to the soliton peaks to a local maximum of

maxVcirc;λ ≈ 1.51 × 105�

c0.4

�12ð−ΦhÞ12 km=s; ð48Þ

where we used Eq. (44) to fix λ and Eq. (28) to relate itto maxVcirc;λ.As anticipated in the beginning of this section, Eq. (35)

predicts approximately equal peak circular velocities forthe inner soliton component and for the host halo,

maxVcirc;λ

maxVcirc;h≈ 1.1

�c0.4

�12

; ð49Þ

independent of the particle massm, independent of the halomassM200, and only weakly dependent on the details of thehalo via the factor ðc=0.4Þ12. Equation (49) is plotted inFig. 3 as function of the concentration parameter.While maxVcirc;λ and the approximate equality Eq. (49)

are m-independent, the soliton peak velocity occurs in anm-dependent location,

xpeak;λ ≈ 184

�10−22 eV

m

��maxVcirc;λ

200 km=s

�−1

pc: ð50Þ

Figure 4 shows the circular velocity curve for the NFWhaloþ soliton system, following from Eq. (35), withULDM particle mass m ¼ 10−22 eV. The solid black,dot-dashed orange, and dashed blue lines show the con-tributions to Vcirc due to the total system, the soliton only,and the halo only. Results are shown for three differentvalues of the NFW concentration parameter, c ¼ 10, 15,25, withM200 ¼ 1012 M⊙ and 5 × 1010 M⊙ on the top andbottom panels, respectively. For larger m > 10−22 eV, thesoliton bump in the rotation curve would shift to smaller xaccording to Eq. (50), but would maintain its height.In Fig. 4, to define the rotation velocity for the total

system, we set the ULDMmass density for the total systemto be ρðxÞ ¼ max fρλðxÞ; ρNFWðxÞg, calculate the resultingmass profile MðxÞ, and use spherical symmetry to findVcircðxÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiGMðxÞ=xp

. This prescription for matchingbetween the soliton and NFW parts is ad hoc and onlyroughly consistent with the simulations of [6,7]. The truetransition region between the NFW part and the soliton partprobably deviates from the pure NFW form. Reference [37]considered this transition region and concluded that thedensity profile in this region should follow approximatelyρ ∼ x−

53, steeper than the usual inner NFW form ρ ∼ x−1.

This would affect the detailed shape of the rotation curve inthe intermediate region between the two peaks, but not ourgeneral results.Equation (49) was derived for an NFW host halo, but it is

the manifestation of Eq. (35) that is not tied to a particularparametrization of the halo profile. Building on Eq. (35),we expect in general that for DM-dominated galaxies, thesoliton peak circular velocity should roughly equal thepeak circular velocity in the host halo. The NFW exampledemonstrates that details of the host halo profile affect thisresult at the 10% level or so.

FIG. 3. Ratio between halo and soliton peak circular velocitiesas a function of the halo concentration.


083027-7

In the rest of this paper, when we refer to Eq. (49), we setthe RHS to unity. Approximating the RHS of Eq. (49) byunity, and replacing maxVcirc;h instead of maxVcirc;λ inEq. (28), the peak circular velocity of a host halo allows usto predict the scale parameter λ and thus the soliton relevantfor that host halo.

A. Comparison to numerical simulations

In Fig. 5, we compare our results to two solitonþ haloconfigurations from the simulations of [6,29] (for [6], wetake the largest halo, and for [29] we take the initial state ofCase C). To calculate the soliton, we read maxVcirc;h fromthe large-scale peak (at x ∼ 20 kpc) of the numericallyextracted halo rotation curves (solid lines). FollowingEq. (49), we use maxVcirc;h instead of maxVcirc;λ inEq. (28), and read off the value of λ. The predicted solitonbump is shown in dashed lines. It gives the correct solitonpeak rotation velocity to ∼20% accuracy in both cases.

In Fig. 6, we show the velocity profiles for 11 simulatedhalos, calculated for 6 halos from [7] (solid lines) and 5halos from [6] (dashed lines). The rotation curves are scaledand normalized such that xpeak;λ ¼ 1 and maxVcirc;λ ¼ 1 ineach case. All of the halos satisfy 0.65 < maxVcirc;λ

maxVcirc;h< 1.4; for

later reference, the shaded band highlights the range0.5 < maxVcirc;λ

maxVcirc;h< 1.5.

B. Comparison to real galaxies

We now consider some observational consequences ofour analysis. We choose to do so by examining the rotationcurves of nearby disc galaxies with halo masses in the rangecovered by the simulations of [6,7], and above the minimalmass of an ULDM halo, see Eq. (33). We divide ouranalysis into two parts. First, in Sec. IV B 1, we concentrate

FIG. 5. Comparison of the prediction of Eq. (49) (dashed lines)to the numerical simulation results (solid lines) of Schive et al.(2014) [6] and Chan et al. (2017) [29].

FIG. 6. Circular velocity profiles for halos, calculated forsimulation results published in [7] (solid lines) and [6](dashed lines), scaled and normalized such that xpeak;λ ¼ 1and maxVcirc;λ ¼ 1. Shaded band highlights the range

0.5 < maxVcirc;λ

maxVcirc;h< 1.5.

FIG. 4. Rotation curves for the ULDM solitonþ halo system,obtained for a DM-only NFW halo using the soliton–halo relationEq. (35) withm ¼ 10−22 eV. Solid black, dot-dashed orange, anddashed blue show Vcirc due to the total solitonþ halo system, thesoliton only, and the halo only. Results are shown for NFWconcentration parameter c ¼ 10, 15, 25, with M200 ¼ 1012 M⊙and 5 × 1010 M⊙ on the upper and lower panels, respectively.


083027-8

on a small set of galaxies for which high-resolutionkinematical and photometric data are available. For thissmall set of galaxies, we present detailed rotation curvedata and compare them to the ULDM prediction. We findclear tension between the data and the ULDM prediction.Next, in Sec. IV B 2, we extend the analysis to include alarge sample of galaxies, confirming and reinforcing thetension between the ULDM prediction and the rotationcurve data.

1. A few specific examples

In our first analysis of the data, we study a set of fourrepresentative rotation curves from Ref. [27] (see Ref. [28]for a recent rendering of these and many other rotationcurves), for which high-resolution kinematical data areavailable. Our goal is to check if the ULDM prediction, thatwe demonstrated in the previous section using simulatedgalaxies, actually holds in real observed galaxies. The keyprediction we check is summarized by Eq. (49), and wasdemonstrated for simulated galaxies in Figs. 5 and 6: thisprediction states that given a measurement of the large-radius halo rotation curve, ULDM prescribes a soliton-induced peak in the inner part of the halo, with heightspecified by Eq. (49), at a location specified by Eq. (50).The four measured rotation curves are shown in

Figs. 7–10, as blue markers. For each of these rotationcurves, we use the measured circular velocity at the farthestradius, to serve as an input for maxVcirc;h in Eq. (49). Inturn, Eq. (49) gives as output the predicted soliton-inducedpeak rotation velocity, maxVcirc;λ. Given the soliton-induced peak rotation velocity, the soliton λ parameter isfixed by Eq. (28) and with it, the full soliton-inducedrotation curve. The result is plotted as dashed line inFigs. 7–10; the upper panels show the result for m ¼10−22 eV and the lower panels show it for m ¼ 10−21 eV.We find that the ULDM soliton–host halo relation

significantly overestimates the rotation velocity in the innerpart of all of the galaxies in Figs. 7–10. This puts thepredictions of ULDM in the mass range m ∼ ð10−22 ÷10−21Þ eV in tension with the data.We emphasize that in using Eq. (49) to predict the

soliton, we set the RHS of that equation to unity, and thuswe ignore any details of the shape of the host halo. We alsoneglect corrections due to baryons. On the one hand, as wehave learned from the NFW analysis, this prescription forderiving the soliton profile would suffer Oð10%Þ correc-tions from the detailed halo shape. The baryonic (stellar andgas) contribution to the gravitational potential affects thehalo velocity at a similar level: in Ref. [28], the baryoniccontribution to the halo peak rotation velocity of UGC1281, UGC 4325, and NGC 100 was estimated fromphotometric data to be well below the DM contribution,6

VðbarÞcirc;h=V

ðobsÞcirc;h < 0.5. This means that the observed velocity

VðobsÞcirc;h is equal to the DM-induced velocity VðDMÞ

circ;h to betterthan 15%. On the other hand, this simple procedure relievesus from the need to fit for the virial mass or other details ofthe host halo. All that is needed is the peak halo rotationvelocity, a directly observable quantity.7 For completeness,we will return to the issue of baryonic effects and deal withit more systematically in the second part of this section,considering SPARC data [28].Equation (49) (represented by the dashed line in

Figs. 7–10) corresponds to the central value of the sol-iton–host halo relation. Reference [6,7] showed a scatter ofabout a factor of two around Eq. (34) between simulatedhalos. This translates to a factor of two scatter in the soliton λparameter (we have illustrated this scatter for a sub-sample ofsimulated halos in Fig. 6). In Figs. 7–10, we represent this

FIG. 7. Measured rotation curve of UGC 1281 superimposedon the prediction from Eq. (49) following from the soliton–hosthalo relation. The ULDM mass is m ¼ 10−22 eV (upper panel)and m ¼ 10−21 eV (lower panel). The shaded band accounts forthe intrinsic scatter of the soliton–host halo relation.

6For this estimate, we use 3.6 μm mass-to-light ratio ϒ� ¼0.5 M⊙=L⊙.

7The rotation curves in Figs. 7–10 do not show a clearpeak within the range of the measurement; this means thatour soliton bump, derived from the maximal velocity seen in thedata, underestimates the true predicted soliton and is thusconservative.


083027-9

scatter by a shaded band, showing the results when the λparameter inferred from Eq. (49) is changed by a factor of 2.

2. A large sample of galaxies: SPARC

It is important to check if scatter between differentgalaxies could explain the discrepancy, with the fourgalaxies in Figs. 7–10 being accidental outliers. To addressthis question, we analyze the 175 rotation curves containedin the SPARC data base [28]. This sample includes, inparticular, the galaxies UGC 1281, UGC 4325, and NGC100, shown in Figs. 7–10.Our SPARC analysis is as follows. For each galaxy, we

make a crude estimate of the halo mass contained within theobserved rotation curve profile byMgal ∼ RV2=G, whereR isthe radial distance of the last data point in the rotation curve,and V the corresponding velocity. We keep only galaxieswith 5×1011M⊙>Mgal>5×108ðm=10−22 eVÞ−3=2M⊙. Wedo this in order to limit ourselves to galaxy masses that arecomfortably above the minimal halo mass (33), and notabove the range simulated in [6,7]. Our results are notsensitive to the details of this mass cut.Next, for each galaxy we determine the observed

maximal halo rotation velocity maxVcirc;h, and use it tocompute the soliton prediction from Eq. (49). To avoidconfusion between halo peak velocity and soliton peakvelocity, we search for the halo peak velocity restricting to

radial distance x > 3 ðm=10−22 eVÞ−1 kpc. Galaxies withno data above x ¼ 3 ðm=10−22 eVÞ−1 kpc are discarded.Our results are not sensitive to this criterion; defining thehalo cut anywhere at ≳1 ðm=10−22 eVÞ−1 kpc guaranteesthat such confusion is avoided.SPARC galaxies comewith photometric data, allowing us

to model the baryonic contribution to the gravitationalpotential [28]. We use this information to limit baryoniceffects on our analysis, and to explore the sensitivity of ourresults to baryonic corrections. For each galaxy, we estimatethe baryonic contribution to the observed rotation velocityusing the mass models of [28] with 3.6 μm mass-to-

light ratio ϒ� ¼ 0.5 M⊙=L⊙. Setting VðbarÞ;2circ;h þ VðDMÞ;2

circ;h ¼VðobsÞ;2circ;h , we calculate the ratio fbar2DM ¼ VðbarÞ

circ;h=VðDMÞcirc;h .

We present results when cutting on different values offbar2DM < 1; 0.5; 0.33.Our first pass on the data includes only galaxies for

which the predicted soliton is resolved, namely, xpeak;λ from

Eq. (50), with maxVcirc;λ ¼ maxVðDMÞcirc;h , lies within the

rotation curve data. For these galaxies, we compute fromdata the ratio

Vcirc;obsðxpeak;λÞmaxVðDMÞ

circ;h

: ð51Þ

FIG. 8. Same as Fig. 7 for UGC 4325. FIG. 9. Same as Fig. 7 for NGC 1560.


083027-10

Here, Vcirc;obsðxpeak;λÞ is the measured velocity at theexpected soliton peak position.The results of this first pass on the data are shown in

Fig. 11. Red, blue, and green histograms show the resultwhen imposing fbar2DM < 1; 0.5; 0.33, respectively. Form ¼ 10−22 eV, we find 45, 26, and 5 galaxies that passthe resolved soliton cut for fbar2DM < 1; 0.5; 0.33. Form ¼ 10−21 eV, only 4 galaxies pass the resolved solitoncut for fbar2DM < 1, and none for fbar2DM < 0.5; 0.33.Including only galaxies with a resolved soliton causes us

to lose many rotation curves with discriminatory power.For example, UGC 4325 drops out of the analysis form ¼ 10−21 eV, though it clearly constrains the model, asseen from the lower panel of Fig. 8. To overcome thiswithout complicating the analysis, we perform a secondpass on the data. Here, we allow galaxies with unresolvedsoliton, as long as the innermost data point is located notfarther than 3 × xpeak;λ. We need to correct for the fact thatthe soliton peak velocity is outside of the measurementresolution. To do this, we modify our observable as

Vcirc;obsðxpeak;λÞmaxVðDMÞ

circ;h

→Vcirc;obsðxmin;dataÞ

maxVðDMÞcirc;h

×ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffixmin;data

xpeak;λ

r; ð52Þ

where xmin;data is the radius of the first data point. Thiscorrection is conservative, because it takes the fall-off of the

soliton gravitational potential at x > xpeak;λ to be the sameas for a point mass. In reality, the potential decays slowerand the soliton-induced velocity decreases slower. Keepingthis caveat in mind, Fig. 12 presents our results inclu-ding unresolved solitons. For m ¼ 10−22 eV, withfbar2DM < 1; 0.5, we find 48 and 16 galaxies with unre-solved soliton, that can be added to the sample of Fig. 11.No galaxy is added for fbar2DM < 0.33. For m ¼ 10−21 eV,16 and 5 galaxies are added with fbar2DM < 1; 0.5, andnone for fbar2DM < 0.33.In Figs. 11–12, vertical dashed line indicates the soliton–

host halo prediction. The shaded region shows the range ofthe prediction, modifying the RHS of Eq. (49) between0.5–1.5, consistent with the scatter seen in the simulations.

FIG. 10. Same as Fig. 7 for NGC 100.

FIG. 11. Distribution of SPARC galaxies [28] with respect tothe ratio of observed circular velocity at the soliton peak to themaximal circular velocity of the halo. The vertical dashed lineshows the prediction for the mean implied by the soliton-hosthalo relation and the shaded region accounts for the intrinsicscatter in this relation. The ULDM mass is m ¼ 10−22 eV (upperpanel) and m ¼ 10−21 eV (lower panel). Red, blue, green histo-grams correspond to the cuts fbar2DM < 1; 0.5; 0.33, respectively.Only rotation curves with resolved solitons are included (see themain text for details).


083027-11

We conclude that the four galaxies in Figs. 7–10 are notoutliers: they are representative of a systematic discrepancy,that would be difficult to attribute to the scatter seen in thesimulations. If the soliton-host halo relation of [6,7] iscorrect, then ULDM in the mass range m ∼ 10−22 eV tom ∼ 10−21 eV is in tension with the data.We have limited our attention to the range

m ¼ ð10−22 ÷ 10−21Þ eV, for which we believe the resultsare clear. We leave a detailed study of the precise exclusionrange to future work. We note that for lower particle mass,m≲ 10−23 eV, the soliton contribution extends over muchof the velocity profile of many of the SPARC galaxies,leaving little room for a host halo. This limit, where thegalaxies are essentially composed of a single giant soliton,was considered in other works. We do not pursue it further,one reason being that this range of small m is in significanttension with Ly-α data [18,19].For higher particle mass,m≳ 10−21 eV, the soliton peak

is pushed deep into the inner 100 pc of the rotation curve.Although high-resolution data (e.g., NGC 1560, Fig. 9) aresensitive to and disfavors this situation, a more carefulanalysis would be needed to draw a definitive conclusion.Note that in this range of m, ULDM ceases to offer a

solution to the small-scale puzzles of ΛCDM (see, e.g.,review in [12]).

V. BARYONIC EFFECTS

A. A fixed distribution of baryons

We now consider the soliton solution, and some aspectsof the soliton-host halo relation, with a coexisting distri-bution of baryonic mass. In this section, we consider asmooth mass distribution, deferring the analysis of theeffect of a super-massive black hole (SMBH) to App. B.The analysis of baryonic effects is qualitatively impor-

tant, and quantitatively relevant to ULDM in, e.g., MW-likegalaxies. We will see that in order to make a significantimpact, baryons need to constitute an Oð1Þ fraction of thetotal mass in the soliton region. For many of the galaxiesthat we analyzed in Sec. IV B, the predicted solitoncontribution to the rotation velocity exceeds the observedvelocities by a factor of 3-5, implying that the soliton over-predicts the mass in the central region of the galaxy byabout an order of magnitude. This leaves little room forbaryons (and, we think, baryonic feedback) to significantlyaffect the dynamics of the inner ULDM halo.Proceeding to the analysis, we denote the gravita-

tional potential due to baryons by ΦbðrÞ. We assumethat the distribution of baryonic mass is sphericallysymmetric8 and dies off at infinity sufficiently fast, so thatΦbðr → ∞Þ ¼ −GMb=r, where Mb is the total baryonicmass. Adding ΦbðrÞ changes Eqs. (7)–(8) into [2]

∂2rðrχÞ ¼ 2rðΦþΦbðrÞ − γÞχ; ð53Þ

∂2rðrΦÞ ¼ rχ2: ð54Þ

It remains convenient to solve the problem using boundaryconditions with χð0Þ ¼ 1. Let us denote this solution(satisfying χð0Þ ¼ 1) by χ1ðr;ΦbÞ, accompanied by thesoliton potential Φ1ðr;ΦbÞ. The general solution χλðr;ΦbÞsatisfying Eqs. (53)–(54) with boundary conditionχλð0;ΦbÞ ¼ λ2 is given by

χλðr;ΦbÞ ¼ λ2χ1ðλr; λ−2Φbðλ−1rÞÞ; ð55Þ

Φλðr;ΦbÞ ¼ λ2Φ1ðλr; λ−2Φbðλ−1rÞÞ: ð56Þ

Defining the soliton mass and energy in the presence of thebaryons by MλðΦbÞ, EλðΦbÞ, we have

MλðΦbÞ ¼ λM1ðλ−2Φbðλ−1rÞÞ; ð57Þ

FIG. 12. Same as Fig. 11, including galaxies with unresolvedsolitons (see the main text).

8The assumption of spherical symmetry is not realistic; someof the galaxies considered in Sec. IV B may in fact exhibitmaximal discs. Nevertheless, we expect our simplified analysis togive us correct order of magnitude estimates and defer theanalysis of nonspherical baryonicþ solitonþ halo systems tofuture work.


083027-12

EλðΦbÞ ¼ λ3E1ðλ−2Φbðλ−1rÞÞ; ð58Þ

where the ULDM energy is

EðΦbÞ ¼Z

d3x

�j∇ψ j22m2

þ�Φ2þΦbðxÞ

�jψ j2

�: ð59Þ

Given the functionΦb, and plottingMλðΦbÞ and EλðΦbÞ vsλ, we can find the value of λ and hence the profile for asoliton solution of any desired mass or energy.Solutions of Eqs. (55)–(56) can be compared with the

results of numerical simulations. Reference [29] added adistribution of mass in the form of point particles (“stars”)to ULDM simulations with m ¼ 0.8 × 10−22 eV. The firsttoy model they studied included an isolated soliton (with-out an ULDM host halo) of mass M ¼ 3.3 × 108 M⊙, towhich a stellar distribution was added and evolved to avirialized state. From Fig. 1 of Ref. [29], we derive the

baryonic contribution to the gravitational potential, andsolve for the distorted soliton at the stated ULDM mass. Inthe top panel of Fig. 13, we show Mλ vs λ, derived fromEq. (57). Blue solid (green dashed) lines show Mλ with(without) the stellar potential. Black horizontal line denotesthe value of M that was fixed in the simulation: theintersection of the black and the blue lines gives theparameter λ describing the distorted soliton. In the bottompanel, we show the density profiles. Red lines are takenfrom [29], while green lines show the analytic solitonsolutions. Clearly, the numerical profile from [29] closelymatches the analytically-derived distorted soliton ofEq. (55).Reference [29] also simulated the soliton in an ULDM

halo, the output of a DM-only cosmological simulation.Then, a stellar mass distribution was introduced as beforeand the system allowed to evolve.When embedded in a halo, the mass of the soliton is not

constant anymore but can grow by absorbing mass fromthe halo. However, regardless of the large-scale halo, in thecore region the perturbed soliton profile is still fixed by theSP equations up to the ambiguity of λ. We checked that inall of the virialized solitonþ haloþ stellar mass simula-tions, presented in Ref. [29], the soliton profile matches thatof the analytic distorted soliton, once accounting for thestellar potential. In the top panel of Fig. 14, we illustratethis point for the t ¼ tH snapshot of Case C, described inFig. 5 of Ref. [29].With stars included, the total t ¼ tH mass distribution,

given by summing the red (DM) and green (stars) solidlines in the upper panel of Fig. 14, leads to the rotationcurve shown by the red solid line in the lower panel of thesame figure. This rotation curve is peaked at small radiusdue to the distorted soliton, that is more massive than thesoliton-halo prediction of the DM-only simulations. Theinner velocity exceeds the prediction of Eq. (49) by a factorof two. We also show, by green dashed line, the rotationvelocity at t ¼ tH due to ULDM alone. To compare, theblue dotted curve shows the rotation curve of the initialULDM system, extracted from the cosmological simula-tions. This is the same curve we showed in Fig. 5; as wediscussed, it satisfies Eq. (49) to 20%.These numerical results suggest that the presence of

baryonic matter tends to make the soliton somewhat morecompact and massive than predicted by the pure ULDMsoliton–host halo relation (31). This, in turn, increases thepeak in the rotation curve due to the soliton, suggesting thatour constraints on ULDM obtained in Sec. IV B are robustwith respect to inclusion of baryons. We now consideran example, where baryonic effects are expected to beimportant: the Milky Way.

B. The Milky Way: Nuclear bulge vs soliton

References [6,7] commented that for a MW-like galaxy,Eq. (31) predicts that ULDM with m ∼ 10−22 eV should

FIG. 13. Top: Soliton mass vs λ parameter, determined usingEq. (57) with the stellar gravitational potential as in the numericalsimulation shown in Fig. 1 of Ref. [29]. Blue solid (green dashed)lines show the Mλ vs λ with (without) the stellar potential. Blackhorizontal (dot-dashed) line denotes the mass M chosen in thesimulation. Bottom: Soliton profiles with and without stars,describing the results of numerical simulation shown in Fig. 1of Ref. [29] (ρbkg is the cosmological background matter density).


083027-13

produce a bump in the rotation curve at a radial distance∼200 pc, consistent at face value with a feature observed inthe MW. Our results in Sec. IV B show that this value of mis in tension with observations of small disc galaxies.Nevertheless, the MW provides an interesting example forstudying the impact of baryons and of a SMBH on anULDM soliton, and developing intuition as to what extentthis effect can be important. Our goal in this section is toprovide a preliminary study along these lines, using photo-metric baryonic mass estimates. As an interesting outcome,we find that precision kinematical studies of the MW innerbulge could in principle be sensitive to ULDM with10−21 eV≲m≲ 10−19 eV, where the analysis along thelines of Sec. IV B may become challenging.Figure 15 shows a spherically averaged enclosed mass

profile, derived for the MW via various dynamical tracers[38–51]. Flattening of the enclosed mass at small radii

reflects the contribution of the SMBH. From the largeradius (r≳ 10 kpc) part of the rotation curve, the param-eters of the large-scale DM host halo can be extracted bystandard analysis. We demonstrate this in Fig. 15 by ablack dashed line, showing an NFW profile fitted inRef. [36] to r≳ 10 kpc SDSS data. The large-scale analysisyields for the MW an host halo mass roughly in therange Mh ¼ ð0.8 ÷ 2Þ × 1012 M⊙.Given the host halo mass Mh, the soliton–host halo

relation, Eq. (31), predicts the soliton massM. This, in turn,allows us to extract the soliton scaling parameter λ (fromEq. (17), with Mλ ¼ M) and with it the entire predictedsoliton contribution to the enclosed mass. In Fig. 15,shaded bands show the resulting soliton-induced rotationcurves for m in the range ð10−22 ÷ 10−19Þ eV. The width ofthe shaded bands comes from varying the host halo massMh in the range ð0.8 ÷ 2Þ × 1012 M⊙. The approximateeffects of baryon mass contributions, which cannot beneglected in the inner MW and which distort the solitonprofile, are included and discussed below.We stress that the purpose of Fig. 15 is to illustrate the

possible signature of ULDM in the inner MW, and not forstatistical analyses of the MW mass distribution. Modelingthe inner MW is a complicated task. The measurement ofinner kinematics of the galaxy, below a few kpc, is subjectto large systematic uncertainties due, among other issues, tothe effects of the Galactic bar and spiral arm structures [51],which impact tangent-point velocity measurements likethose utilized in [50,52]. Our simplified derivation of thespherically averaged mass profile in Fig. 15 combinesmany tracers with different systematics, and accounts fornone of these subtleties.Reference [53] analyzed the MW central gravitational

potential using a large set of observational constraints. Inaddition to the classical bulge and disc, Ref. [53] founddynamical evidence for the presence of a mass componentof ∼2 × 109 M⊙ extending to ∼250 pc. This mass

FIG. 14. Top: Soliton profiles with and without stars, describ-ing the inner part of the halo obtained in the numerical simulation(Case C in Fig. 5 of Ref. [29]). Green and red solid lines show thestellar and ULDM densities, respectively, taken from the simu-lation. Blue dashed show a distorted soliton solution, computedincluding the stellar potential. Dotted black shows an un-distortedsoliton with the same mass. Bottom: Rotation curve for the totalULDMþ stellar system (red solid) and including only theULDM contribution (green dashed), at t ¼ tH . We also showthe initial ULDM only distribution at t ¼ 0 (blue dotted).

10-4 10-2 100 102 104

r [pc]

10 5

10 6

10 7

10 8

10 9

10 10

10 11

10 12

encl

osed

mas

s [M

]

m=10-19 eV

m=10-20 eV

m=10-21 eV

m=10-22 eVGhez 2003McGinn 1989Fritz 2016Lindqvist 1992Schodel 2014Sofue 2009Sofue 2012Sofue 2013Chatzopoulos 2015Deguchi 2004Oh 2009Trippe 2008Gilessen 2008

Nuclear Bulge (disc+star cluster)from photometry, Launhardt (2002)

NFW fit,Piffl (2015)

FIG. 15. Spherically averaged mass profile in the Milky Way,vs ULDM soliton contributions. See text for details.


083027-14

component is visible as a mass bump in Fig. 15 (see, e.g.,green data points extracted from [50]). Consistent withcomments in [6,7], the bump is in tantalizing agreementwith the soliton prediction of Eq. (31) for m ¼ 10−22 eV(blue shaded band).Unfortunately, there are about a billion stars in there, too:

the bump in the mass profile at r ∼ 200 pc has beenassociated in the literature with the nuclear bulge (NB).Reference [54] fitted the NB mass and light by a dense discof stars, with mass density ρ� ∼ 200 M⊙=pc3, scale height≲45 pc and scale radius ∼230 pc. In all, the NB is thoughtto contain ð1.4� 0.6Þ × 109 M⊙ in stars, roughly enoughto match the dynamically inferred mass. Subsequentkinematic detection supporting the stellar mass and disclikemorphology of this component was given in [55].Microlensing analyses [56] lend further support to theresults of [53–55] down to r ≳ 220 pc.The photometrically-derived NB mass model of [54] is

superimposed as purple line in Fig. 15. We stress that thephotometric derivation is subject to large uncertainties dueto the need to correct for very strong extinction and due tounknown stellar mass-to-light ratios. What we learn fromthis photometric mass model, therefore, is that stars couldplausibly account for all of the kinematically inferred massin this region.Assuming that the NB is due to stars, we now use a toy

model of this mass distribution to see its effect on anULDM soliton. We replace the disclike morphology of theNB in [54] by a spherical model with the same radiallyaveraged mass. The nominal model, containing the NB andadditional subleading components described in [54], con-tains ∼1.7 × 109 M⊙ in stars inside of r ¼ 300 pc. Addinga SMBH of MBH ¼ 4.3 × 106 M⊙ [38], we calculatesoliton solutions in this baryonic potential.Figure 16 shows the soliton mass as function of the λ

parameter, for m ¼ 10−22 eV. Green dashed line shows theunperturbed Mλ vs λ relation. Solid, dashed, and dotted

black lines show the relation for the nominal NB model andfor two other models, obtained by scaling the NB massdensity by an over-all factor of 0.5 and 2, respectively.For orientation, shaded blue band shows the solitonmass predicted by Eq. (31) for a host halo with massMh ¼ ð0.8 − 2Þ × 1012 M⊙.For Mλ ≳ 3MNB ∼ 5 × 109 M⊙, the NB makes a negli-

gible impact on the soliton. For larger ratio of the stellar toULDM mass, Mλ ≲MNB, the NB becomes important,contracting the soliton profile. For the MW, this is theparametric region predicted by Eq. (31), implying that thesolitons would receive significant distortion. In Fig. 15, weillustrated this effect by presenting, in shaded bands, thesoliton mass profiles computed accounting for the nominalNB model. We observe that the solitons for m in the rangeð10−22 ÷ 10−19Þ eV are expected to affect the potential at anorder unity level. Thus, a dedicated analysis using ourformalism to calculate steady state soliton profiles con-sistent with a given baryonic mass model could be sensitiveto ULDM solitons all the way up to m ∼ 10−19 eV. Forlarger m, the expected soliton becomes subdominant withrespect to the SMBH.Another limitation comes from absorption of the ULDM

from the soliton by SMBH. As discusses in App. B 2, theaccretion rate is negligible as long as m≲ 5 × 10−20 eV.However, for higher ULDM masses the time scale ofaccretion can become shorter than the age of the universeand our steady-state Newtonian analysis in this section maynot apply.

VI. DISCUSSION

A. Comparison to previous work

An earlier analysis of ULDM halos including the effectof baryons was done in [2] (see also [3]). This workattempted to fit the rotation curves of spiral galaxies from[57], assuming that the entire ULDM halo is contained in agiant soliton (see also [58] for a similar approach inmodeling low surface brightness galaxies). This exerciseled [2] to report m < 10−23 eV, in tension with Ly-α data.This exercise is different from the current work. Assuggested by numerical simulations, we expect heavierULDM particles with m≳ 10−22 eV to produce galaxieswith a large scale host halo following roughly the usualNFW profile, in which the soliton affects the rotation curveonly in the inner part of the halo.References [6,8] performed a Jeans analysis, fitting a

solitonþ halo configuration to a small sample ofdispersion-dominated MW-satellite dwarf spheroidal gal-axies (dSph). They found that the modeling of stellarkinematics in the dSph is consistent with a cored innerregion, as expected in ULDM, with good fits to the data form≲ 10−22 eV. The fitted core found in [6] was consistentwith expectations from the soliton–host halo relation. In[8], the transition between an assumed host halo NFW

FIG. 16. Soliton mass vs λ parameter, accounting for thespherically averaged gravitational potential due to stars [54].The ULDM particle mass is m ¼ 10−22 eV. See text for moredetails.


083027-15

profile and the soliton was modeled using correspondingfree parameters. To conclude, these analyses considered asmall sample of dispersion-dominated galaxies, as opposedto our larger sample of rotation-dominated galaxies. Theirbest-fit ULDM mass m≲ 10−22 eV is disfavored by ourfindings.Reference [26] fitted ULDM solitonþ halo profiles to

SPARC [28] galaxies. Without a soliton–host halo relation,Ref. [26] assigned separate free parameters to the NFWhalo and to the soliton, for each galaxy. As a result of thisfreedom, it is difficult to understand from that analysisif ULDM is ruled out or not in the mass rangem ¼ ð10−22 ÷ 10−21Þ eV. Nevertheless, Ref. [26] did statea best fit ofm ≈ 0.5 × 10−23 eV, and noted the tension withLy-α data, in agreement with our findings. Our work hereshows that the soliton–host halo relation reduces themodeling freedom, and leads to disagreement with rotationcurve data.Recently, Ref. [59] addressed the problem from a

different angle. It builds on the results of Ref. [60] thatfitted rotations curves of a sample of galaxies assumingthat their halos are described by the Burkert profile.Reference [59] points out a correlation between theparameters of the fitted profiles—the central dark matterdensity ρc and the size of the core Rc,

ρc ∝ R−βc ; ð60Þ

with β ∼ 1. This is different from β ¼ 4 predicted by ULDMif the cores are identifiedwith the solitons, seeEq. (30). Thus,Ref. [59] concludes that ULDM cannot explain the origin ofthe halo cores, and cannot solve the core–cusp problem ofΛCDM. Our results are consistent with these findings andstrengthen them by showing that the soliton–host halorelation implies a tension between predictions of ULDMand the data for m ∼ ð10−22 ÷ 10−21Þ eV, disfavoring thismass range altogether.

B. Caveats

1. Soliton formation time

Our analysis has been based on the assumption thatULDM solitons satisfying the soliton–host halo relationexist in the centers of all (or most) galactic halos. Formationof such solitons within the lifetime of the universe has beendemonstrated in numerical simulations [6,61] for ULDMmassesm∼10−22 eV and halos with mass ð109 ÷ 1011Þ M⊙.Here we discuss whether such solitons have enough time toform for higher values of m and larger halos.Recently, Ref. [62] considered the scenario, where

solitons form as a result of gravitational relaxation ofthe inner part of the halo. In this case, the relaxation timecan be estimated as

τrelax ¼b

ffiffiffi2

pm3v6

12π3G2ρ2Λ; ð61Þ

where v and ρ are the mean velocity and density in the innerpart of the halo, Λ ¼ logðmvRÞ is the Coulomb logarithmdepending on the size R of the relaxation region, and b is anumerical coefficient close to one. The formula has beenconfirmed in [62] by numerical simulations that found b torange between 0.7 ÷ 0.9 depending on the precise form ofthe initial conditions. Substituting into Eq. (61) the valuesfor a typical halo and taking Λ ∼ 4, we obtain,

τrelax ∼ 7 × 108�

m10−22 eV

�3

×

�v

100 km=s

�6�

ρ

0.1 M⊙=pc3

�−2

yr: ð62Þ

We observe that the relaxation time has strong dependenceon the velocity and the density in the central region of thehalo, that have large uncertainties. Fixing the fiducialvalues of v ¼ 100 km=s and ρ ¼ 0.1 M⊙=pc3 one wouldconclude that the relaxation takes longer than the age of theuniverse if m≳ 3 × 10−22 eV.However, it can be incorrect to use Eq. (61) for the

estimate of the soliton formation time inside ULDM halosin the cosmological context. Indeed, this equation has beenderived and tested in the kinetic regime corresponding tolarge velocity dispersion of ULDM particles, such that theirde Broglie wavelength is much shorter than the size of thesystem. Starting from initial conditions of a gas of wave-packets with large velocity dispersion in a box, Ref. [62]observed a fast formation of a virialized halo, followed by along period of kinetic evolution populating the low-lyingenergy levels, until the system was able to form the solitonthrough Bose–Einstein condensation. The time scale (61)refers to the long second stage of the process. On the otherhand, the cosmological initial conditions prior to virializa-tion are characterized by very low velocities, where thekinetic description does not apply. In this situation, a longrelaxation seems unnecessary and the formation of thesoliton can proceed much faster, on the scale of the halofree-fall time; see discussion in [62]. This is supported bythe simulations with cosmological initial conditions whichappear to imply formation of the soliton in every halo at themoment of virialization [6,61]. We believe that the issue ofthe soliton formation time requires further investigation,including possible effects of baryonic matter.

2. Nongravitational interactions

In our analysis, we have neglected any nongravitationalinteractions of the ULDM particles. Let us discuss underwhich conditions this approximation is justified. As anexample we consider quartic self-interaction of the scalarfield ϕ with coupling constant κ,


083027-16

δVðϕÞ ¼ κϕ4

4: ð63Þ

The self-interaction is negligible as long as the correspond-ing potential energy is much smaller than the gradientenergy of the field,

jκjϕ4

4≪

ð∇ϕÞ22

: ð64Þ

For a soliton with the scale parameter λ, this translates into abound,

jκj < 2m2

x2cλρcλ: ð65Þ

Using the soliton–host halo relation in the form ofEqs. (29), (30) we obtain the condition,

jκj < 4 × 10−93�

m10−22 eV

�2�

Mh

1012 M⊙

�−23

: ð66Þ

If this condition is satisfied, our analysis is applicable.Otherwise, the effects of the self-interaction on the solitonproperties must be taken into account.While the condition (66) appears stringent, it is naturally

fulfilled in a broad class of theories containing axionlikeparticles. These typically have a periodic potential of theform,

VðϕÞ ¼ m2f2ð1 − cosðϕ=fÞÞ: ð67Þ

For the masses of interest m ∼ ð10−22 ÷ 10−18Þ eV, theaxion periodicity should be in the range f ∼ ð1016 ÷1017Þ GeV to produce the right dark matter abundancevia the misalignment mechanism (see e.g., [12]).Expanding the potential (67) in powers of ϕ we obtainthe value of the coupling,

κ ¼ −m2

6f2

≈ −1.7 × 10−97�

m10−22 eV

�2�

f1017 GeV

�−2: ð68Þ

We see that (66) is indeed satisfied for all galactic halosconsidered in this paper, independently of the ULDMmass.Similar analysis can be performed in the case of non-

gravitational interactions between ULDM and baryonicmatter, though in this case it will be strongly model-dependent.

VII. SUMMARY

Bosonic ultralight dark matter (ULDM) is an interestingparadigm for dark matter. For particle mass m ∼ 10−22 eV,

ULDMwould form solitonic cores in the center of galaxies,and it has been suggested that this could solve severalpuzzles of ΛCDM on small scales.We analyzed the results of numerical simulations of

ULDM, which have found scaling relations between themass of the central soliton and the mass or energy of thehost halo. Simulations by different groups converge on asoliton profile in good agreement with the analyticalsolution of the Schroedinger-Poisson (SP) equation, admit-ting important analytic insight into the numerical results.We showed that the simulations of Ref. [13]

contain a central soliton that dominates the total energyof the entire solitonþ halo system. This situation isunlikely to describe realistic galactic halos9 with mass above∼5 × 107ðm=10−22 eVÞ−3=2M⊙.We have demonstrated that the soliton–host halo relation

found in the simulations of Refs. [6,7] can be summarizedby the statement, that ðE=MÞjsoliton ¼ ðE=MÞjhalo. Thesimulations of [7] show a small spread, less than a factorof two, around this relation. This E=M relation could applyto real galaxies.The E=M soliton–host halo relation implies that the peak

circular velocity of the large-scale halo should approxi-mately reproduce itself in the inner soliton region. This canbe tested against observations without free parameters.Contrasting this prediction with high-resolution disc galaxydata, we showed that ULDM in the mass range m ∼ð10−22 ÷ 10−21Þ eV is disfavored by observations. Giventhat smaller m is in tension with cosmological measure-ments, this disfavors ULDM as a solution to the small scalepuzzles of ΛCDM.We also analyzed the effect of a fixed background

distribution of baryons and of a super-massive black hole(SMBH) on the soliton solution. We showed that thisanalysis would be important towards using high-resolutionkinematical data in the Milky Way or Milky Way-likegalaxies. Mapping the Milky Way gravitational potentialdown to r ∼ 10−2 pc may allow us to probe ULDM withmass up to m ∼ 10−19 eV.It should be stressed that the E=M soliton–host halo

relation (35) is an empirical result, deduced from numericalsimulations without baryons and tested in a limited range ofULDM and halo masses. The importance of its phenom-enological implications motivates further investigation tobetter understand the physics underlying this relation andmap out its domain of validity.Throughout the paper we have assumed that ULDM

comprises the total amount of dark matter. We leave forfuture the study of scenarios where only a fraction of darkmatter is in the form of ULDM.

9This does not invalidate the beautiful simulations of [13]. Weonly think that their setup for initial conditions does not matchreal galaxies.


083027-17

ACKNOWLEDGMENTS

We thank James Hung-Hsu Chan, Josh Eby, Lam Hui,Dmitry Levkov, Moti Milgrom, Eran Ofek, AlexanderPanin, and Igor Tkachev for discussions, and Yossi Nir,Scott Tremaine, and Eli Waxman for comments on an earlyversion of the draft. K. B. thanks the participants of theVBSM Polynesia workshop, where part of this work wasconducted, for useful discussions. K. B. is incumbent of theDewey David Stone and Harry Levine career developmentchair. The work of K. B. and N. B. is supported by GrantNo. 1937/12 from the I-CORE program of the Planning andBudgeting Committee and the Israel Science Foundationand by Grant No. 1507/16 from the Israel ScienceFoundation. S. S. is grateful to Columbia University forhospitality during completion of this work.

APPENDIX A: SOLITON MASS–HOST HALOMASS RELATION, Mh vs M200

Reference [7] defined the halo virial mass Mh andradius Rv,

Mh ¼4π

3R3vζðzÞρm0; ðA1Þ

where ρm0 is the present matter density and ζðzÞ ¼ð18π2 − 82ð1 −ΩmðzÞÞ − 39ð1 −ΩmðzÞÞ2Þ=ΩmðzÞ. Thehalo energy per unit mass was estimated as Eh=Mh ¼−3GMh=ð10RvÞ ¼ −G 3

10ð4π3ζΩm0Þ13ρ

13cM

23

h. In Sec. IV, weprefer to work with M200 and R200 and have calculatedthe halo energy per unit mass, using Eq. (42), as

E200=M200 ¼ −G c4ð 4πδcðlnð1þcÞ− c

1þcÞ2Þ13ρ1

3cM

23

200.

To express the soliton mass–host halo mass relation,Eq. (45), in terms of Mh of Ref. [7], we need to do so atfixed E200=M200 ¼ Eh=Mh ≡ E=M. This gives:

�M200

Mh

�13

��E=M

¼� 3

10ð4π3ζΩm0Þ13

c4ð 4πδcðlnð1þcÞ− c

1þcÞ2Þ13�1

2

: ðA2Þ

For c in the range 5 ÷ 30, the RHS of Eq. (A2) variesbetween 0.58 and 0.47. Plugging this result into Eq. (45)shows that it agrees with Eq. (31) of [7] to 20%.

APPENDIX B: Adding a super-massive black hole

Most galaxies, if not all, host a SMBH, and we shouldcheck how it affects the ULDM soliton. This is animportant point to verify and we do this analysis here insome detail. The upshot of our discussion is that a SMBHwould not affect our results for small disc galaxies, whereasin the MW it becomes relevant at the upper end ofconsidered ULDM masses, m≳ 5 × 10−20 eV.We start with a Newtonian analysis, dealing with the

ULDM configuration far from the BH Schwarzschildradius. Then we will consider the limitations of this

analysis imposed by absorption of ULDM on SMBH thatleads to the decay of the solitonþ SMBH configuration.

1. Soliton shape in the presence of SMBH

Adding a SMBH, coincident with the soliton center ofmass, changes Eqs. (7)–(8) into

∂2rðrχÞ ¼ 2r

�Φ −

Ar− γ

�χ; ðB1Þ

∂2rðrΦÞ ¼ rχ2; ðB2Þ

where

A ¼ GMBHm

≈ 3 × 10−6�

MBH

4 × 106 M⊙

��m

10−22 eV

�: ðB3Þ

We chose the reference value forMBH to represent the caseof the MW.It remains convenient to solve the problem using

boundary conditions with χð0Þ ¼ 1. Let us denote thissolution (satisfying χð0Þ ¼ 1) by χ1ðr;AÞ, accompanied bythe potential Φ1ðr;AÞ. Having found χ1ðr;AÞ and Φ1ðr;AÞfor any value of A, the physical solution χλðr;AÞ satisfyingEqs. (B1)–(B2) with boundary condition χλð0;AÞ ¼ λ2 isgiven by

χλðr;AÞ ¼ λ2χ1ðλr;A=λÞ; ðB4ÞΦλðr;AÞ ¼ λ2Φ1ðλr;A=λÞ: ðB5Þ

DefiningMλðAÞ, with obvious notation, the soliton mass is

MλðAÞ ¼ λM1ðA=λÞ: ðB6ÞFigure 17 shows the density profile χ21ðr;AÞ for different

values of A. For A≲ 10−2 the solution converges to theunperturbed χ1. For A≳ 1 the solution approachesχ1ðr;A → ∞Þ → e−Ar which is nothing, but the wave-function of the ground state in the Coulomb potential.This means that for A=λ≳ 1, solitons with different valuesof λ have the same profile up to over-all normalization:χλðr;A≳ λÞ → λ2e−Ar. In this limit, the soliton core radius

is rcλðAÞ → lnð2Þ=ð2AÞ, and the mass is MλðAÞ → λ4

4A3

M2pl

m .More generally, for any A the mass-radius relation is

MλðAÞrcλðAÞ ¼ M1ðA=λÞrc1ðA=λÞ: ðB7ÞThe mass-radius product is shown by the solid blue curvein Fig. 18 (top panel). The large-A analytic solution,

M1ðAÞrc1ðAÞjA→∞ → lnð2Þ8A4

M2pl

m (recall r ¼ mx), is shown indotted black, normalized to the A ¼ 0 result.As discussed in Sec. III A, Eqs. (31) and (35) are

equivalent when the ULDM soliton is self-gravitating,but this equivalence is broken by the SMBH gravitationalpotential. It is unlikely that Eq. (31) remains realistic when


083027-18

the SMBH becomes important. Instead, Eq. (35) may stillbe valid. In computing the ULDM energy, we need toaccount for the SMBH contribution,

EðAÞ ¼Z

d3x

�j∇ψ j22m2

þ�Φ2−

Amx

�jψ j2

�; ðB8Þ

which means that

EλðAÞ ¼ λ3E1ðA=λÞ: ðB9ÞThe large A limit (attained for a χλðr;AÞ soliton at A≳ λ) is

EλðA → ∞Þ → − λ4

8A

M2pl

m . In this limit, Eλ=Mλ tends to a λ-independent constant, Eλ=MλðA → ∞Þ → −A2=2, becausethe soliton is not held together by self gravity, but by theSMBH potential. The E=M relation for the χ1ðAÞ soliton isshown in the bottom panel of Fig. 18.In Fig. 19, solid lines show E=M for the soliton with

different values of particle mass m, in the presence of aSMBH with MBH ¼ 4 × 106 M⊙. The dashed lines showthe undistorted result, obtained from Eq. (25). The large Alimit with constant Eλ=Mλ is attained when the soliton massbecomes smaller than the SMBH mass.To further clarify this point we relate A=λ to the

unperturbed soliton mass using Eq. (15),

A=λ ≈ 2.06

�MBH

MλðA ¼ 0Þ�

≈ 0.8

�MBH

4 × 106 M⊙

��MλðA ¼ 0Þ107 M⊙

�−1: ðB10Þ

Next, we see from Eqs. (B4), (B5) that A=λ gives aneffective A-parameter that must be substituted in the χ1-soliton to obtain the actual solution. Thus, whether theeffects of SMBH are important or not can be read from theratio of the SMBH mass to the mass of the undistortedsoliton.To analyze the rotation curve for the solitonþ SMBH

system, it is useful to view Vcirc for the χ1ðAÞ soliton,obtained for different values of A. This is shown in Fig. 20.

Because χλðAÞ solitons are a scale transformation ofχ1ðA=λÞ, plotting χ1ðAÞ with different values of A isequivalent to plotting the rescaled velocity curve of solitonsat different values of MBH=Mλ. For example, usingEq. (B10), the A ¼ 0.2 curve in Fig. 20 corresponds to asoliton withMλ ≈ 10MBH; while the A ¼ 10−2 correspondsto Mλ ≈ 200MBH.For Mλ ≳ 10MBH, the soliton’s circular velocity peak

and the peak location are affected very little by the SMBH.

FIG. 17. χ21ðAÞ profiles, with A ¼ 0, 0.1, 1, 2 shown in blue,orange, green and red, respectively. For the A ¼ 1 and A ¼ 2profiles, the large-A analytic solution is shown in dotted black.

FIG. 18. Mass-core radius relation (top panel) and E=M(bottom panel) for the χ1ðAÞ soliton, as function of A. Thelarge-A analytic solution is shown in dashed black.

FIG. 19. Solid lines: energy per unit mass vs mass for a soliton,including the effect of a SMBH with MBH ¼ 4 × 106 M⊙.Dashed lines: switching off the SMBH.


083027-19

However, at small x approaching the SMBH, the velocitycurve turns back up reflecting the gravitational potential ofthe SMBH itself. This implies that the rotation curve of agalaxy hosting a SMBH may not decrease appreciablybelow xpeak;λ, even whenMBH is much smaller thanMλ. Forexample, even forMBH ¼ 0.05Mλ (A ¼ 0.1 in Fig. 20), thecircular velocity decreases by only ∼25% at x < xpeak;λbefore it goes back up due to the BH. For the sameparameters, Fig. 18 shows that Eλ=Mλ is essentiallyunperturbed, meaning that the soliton-host halo relationof Eq. (35) and the rotation curve properties discussed inSec. IV remain unaffected. Such a galaxy would have anapproximately flat rotation curve all the way in to the regionof SMBH dominance.The impact of a SMBH on the analysis of Sec. IV can be

concluded as follows. For Mλ ≳ 4MBH (corresponding, byusing Eq. (B10), to effective A=λ ≈ 0.5), Fig. 18 shows thatEλ=Mλ is corrected by ≲25% compared to its unperturbedvalue. Equations (28) and (44) then imply that λ andmaxVcirc;λ are corrected by an insignificant ∼12%. For alarge SMBH with MBH ≳Mλ, on the other hand, theSMBH gravitational potential itself must produce highrotation velocity at xpeak;λ, comparable or higher than whatthe soliton itself would provide. As a result, besides fromthe fact that the rotation curve may not decrease towardslower x below the soliton peak, our analysis in andobservational limits from Sec. IV should be robust againstthe effect of a SMBH.10

2. Absorption of soliton by SMBH

We now discuss accretion of ULDM from the solitononto SMBH. To estimate the lifetime of the solitonþSMBH configuration we follow the approach of [12] (seealso [64,65] for the study of scalar bound states in external

Schwarzschild metric and [66] for a numerical analysis ofthe lifetime of solitonþ BH system). Unruh [67] hasderived the cross section for absorption of a scalar particlewith mass m and momentum k by a Schwarzschild BH,whose size is much smaller than the Compton wavelengthof the particle. In the nonrelativistic limit, k ≪ m, it hasthe form,

σ ¼ 32π2ðGMBHÞ3m3

k2ð1 − e−ζÞ ; ðB11Þ

where

ζ ¼ 2πGMBHm2=k: ðB12Þ

As the cross section is s-wave dominated, it is appropriatefor calculation of accretion from a spherically symmetricscalar field configuration.The growth of BH mass due to an infalling flux of

particles with density ρ is,

dMBH

dt¼ 32π2ðGMBHÞ3m2ρ

kð1 − e−ζÞ : ðB13Þ

In the case of the soliton, we can estimate the absorptionrate by substituting into this expression the central densityof the soliton and the characteristic momentum of particlesin the soliton that can be read from the soliton profile (30),

k ∼ 0.3x−1c : ðB14Þ

This approximation is justified as long as the mass of thesoliton is bigger than the BH mass, so that the effect of BHon the soliton shape can be neglected.The expression for the accretion rate simplifies in the

limiting cases when the parameter ζ is small or large. UsingEq. (29) (which is consistent with the soliton–host halorelation) we obtain,

ζ ¼ 0.16

�m

10−22 eV

��MBH

4 × 106 M⊙

��Mh

1012 M⊙

�−13

:

If ζ ≪ 1, the characteristic accretion time, during which theBH mass grows by a factor 2 is,

τζ≪1 ∼ 2.4 × 1017�

m10−22 eV

�−2

×

�MBH

4 × 106 M⊙

�−1�

Mh

1012 M⊙

�−43

yr: ðB15Þ

In the opposite regime, ζ ≫ 1, the characteristic time is

FIG. 20. Rotation curve for the χ1ðAÞ soliton.

10This conclusion seems in contradiction with the claim ofRef. [63], who proposed that ULDM solitons distorted bySMBHs could explain the MBH − σ correlation, observed be-tween SMBHs and the stellar bulge of the galaxies hosting them.


083027-20

τζ≫1 ∼ 1.5 × 1018�

m10−22 eV

�−3

×

�MBH

4 × 106 M⊙

�−2�

Mh

1012 M⊙

�−1

yr: ðB16Þ

In deriving these expressions, we made use of Eqs. (29)and (30). Both expressions give the accretion time longerthan the age of the universe for MW’s SMBH with MBH ≈4.3 × 106 M⊙ and m≲ 5 × 10−20 eV. Our Newtonian

analysis, addressing stationary ULDM soliton solutions,is adequate in this case. For galaxies with a more massiveSMBH, e.g., M31 with MBH ∼ 108 M⊙ [68], Eq. (B16)implies that absorption of the ULDM field into the SMBHbecomes important already for m≳ 6 × 10−21 eV, and ourstationary Newtonian analysis is not expected to capturethe physics correctly for such values of m. Finally, for thereference value of m ¼ 10−22 eV, our analysis should bevalid as long as MBH ≲ 1010 M⊙.

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