Boson stars from selfinteracting dark matter
HJE
Boson stars from selfinteracting dark matter
Joshua Eby 1 2 3
Chris Kouvaris 1 2
Niklas Gr nlund Nielsen 1 2
L.C.R. Wijewardhana 1 2 3
0 Origins University of Southern Denmark
1 Campusvej 55 , DK5230, Odense M , Denmark
2 Cincinnati , OH 45221 , U.S.A
3 University of Cincinnati, Dept. of Physics
We study the possibility that selfinteracting bosonic dark matter forms starlike objects. We study both the case of attractive and repulsive selfinteractions, and we focus particularly in the parameter phase space where selfinteractions can solve well standing problems of the collisionless dark matter paradigm.
Cosmology of Theories beyond the SM; Classical Theories of Gravity

We
nd the mass radius
relations for these dark matter bosonic stars, their density pro le as well as the maximum
mass they can support.
1 Introduction 2 3
SIDM parameter space
2.1
DM scattering with boson stars
Bosonic dark matter
3.1
3.2
3.3
Noninteracting case
Repulsive interactions
Attractive interactions
4
Conclusions
drives the expansion of the universe in quintessence models [10].
These bosonic particles often make good Dark Matter (DM) candidates as well. One
reason for this is that unlike the Higgs, many of these new scalars would be stable or
longlived enough that they could coalesce into DM halos which constitute the seeds of galaxy
formation. Unlike the usual collisionless cold DM picture, however, we are interested in the
scenario where large collections of these bosons form bound states of macroscopic size due
to their selfgravitation (and selfinteraction generically). For this picture to be consistent,
the scalars are taken to be su ciently cold so that they may coalesce into a BoseEinstein
Condensate (BEC) state, and can thus be described by a single condensate wavefunction.
These wavefunctions can indeed encompass an astrophysically large volume of space and
have thus been termed \boson stars" [11].
It was shown many years ago that objects of this type are allowed by the equations of
motion, rst by Kaup [12] and subsequently by Ru ni and Bonazzola [13] in noninteracting
systems. They found a maximum mass for boson stars of the form Mmax
0:633MP2=m,
where MP = 1:22
1019 GeV is the Planck mass and m is the mass of the individual
bosons. (This is very di erent from the analogous limit for fermionic stars, termed the
Chandrasekhar limit, which scales as MP3=m2). Later, it was shown by Colpi et al. [11]
that self interactions in these systems can cause signi cant phenomenological changes. In
particular, they examined systems with repulsive selfinteractions, and show that the upper
{ 1 {
limit on the mass is Mmax
MP3=m2, where
is a dimensionless 4 coupling.1 This
extra factor of MP=m as compared to the noninteracting case makes it more plausible
that boson stars can have masses even larger than a solar mass. A di erent method of
constraining the boson star parameter space, which ts the coupling strength using data
from galaxy and galaxy cluster sizes, has been considered in [14, 15].
The situation for attractive selfinteractions is slightly more complex. The simplest
case involves a selfinteraction of the form
4, where
< 0 for attractive interactions. If
this were the highestorder term in the potential, then it would not be bounded below, and
so one typically stabilizes it by the addition of a positive
6 term. We will assume that
the contribution of such higherorder terms is negligible phenomenologically (we address
the validity of that assumption in section 3.3). Furthermore, in this scenario the typical
sizes of gravitationally bound BEC states is signi cantly smaller than the repulsive or
noninteracting cases. This is because the only force supporting the condensate against collapse
comes from the uncertainty principle. Gravity and attractive selfinteractions tend to shrink
the condensate. We will see in section 3 that the maximum mass for an attractive
condensate scales as Mmax
MP=p
j j. This result was originally derived using an approximate
analytical method [16], and was later con rmed by a precise numerical calculation [17].
DM selfinteractions have already been proposed and studied in di erent contexts [18{
37]. One of the main reasons why DM selfinteractions can play an important role is
due to the increasing tension between numerical simulations of collisionless cold DM and
astrophysical observations, the resolution of which (for the moment) is unknown. The rst
discrepancy, known as the \cuspcore problem", is related to the fact that dwarf galaxies
are observed to have
at density pro les in their central regions [38, 39], while Nbody
simulations predict cuspy pro les for collisionless DM [40]. Second, the number of satellite
galaxies in the Milkly Way is far fewer than the number predicted in simulations [41{46].
Last is the socalled \too big to fail" problem: simulations predict dwarf galaxies in a mass
range that we have not observed, but which are too large to have not yet produced stars [47].
The solution of these problems is currently unknown, but a particularly wellmotivated
idea involves selfinteracting DM (SIDM). Simulations including such interactions suggest
that they have the e ect of smoothing out cuspy density pro les, and could solve the other
problems of collisionless DM as well [25, 48, 49]. These simulations prefer a selfinteraction
cross section of 0:1 cm2/g .
=m . 10 cm2/g. There are, however, upper bounds on
=m
from a number of sources, including the preservation of ellipticity of spiral galaxies [50, 51].
The allowed parameter space from these constraints nonetheless intersects the range of cross
sections which can resolve the smallscale issues of collisionless DM, in the range 0:1 cm2/g
. =m . 1 cm2/g.
Selfgravitation and additionally extra selfinteractions among DM particles can lead
in some cases to the collapse of part of the DM population into formation of dark stars.
The idea of DM forming starlike compact objects is not new. Dark stars that consist of
annihilating DM might have existed in the early universe [52{54]. Dark stars have been
1Note that the Colpi et al. result does not reduce to the Kaup bound as
! 0 because the former is
also studied in the context of hybrid compact stars made of baryonic and DM [55{58] as
well as in the context of mirror DM [59{62]. Additionally some of the authors of the current
paper studied the possibility of dark star formation from asymmetric fermionic DM that
exhibits Yukawa type selfinteractions that can alleviate the problems of the collisionless
cold DM paradigm [63]. Unlike the dark stars of annihilating DM, asymmetric dark stars
can be stable and observable today. [63] displays the parameter space where it is possible
to observe such dark stars, providing mass radius relations, corresponding Chandrasekhar
mass limits and density pro les. Selfinteractions in dark stars have also been considered
in [64] for fermionic particles, as well as in [65] for bosonic ones.
In this paper we examine the dark stars composed of asymmetric selfinteracting
bosonic DM. The study is fundamentally di erent from that of [63] because unlike the
case of fermionic DM where the stability of the star is achieved by equilibrium between
the Fermi pressure and gravitation, bosonic DM does not have a Fermi surface. They
form a BEC in the ground state and it is the uncertainty principle that keeps the star
from collapsing. We are going to demonstrate how DM selfinteractions a ect the mass
radius relation, the density pro le and the maximum mass of these DM bosonic stars in
the context of the selfinteractions that reconcile cold DM with the observational ndings.
Note that we set ~ = c = 1 in what follows.
2
SIDM parameter space
As we mentioned above, galactic scale N body simulations of cold, noninteracting DM
indicate that the central regions of galaxies should have a \cuspy" density pro le, contrary
to the cored pro les one observes. This, along with the \missing satellites" and \too big
to fail" problems, has led some to question the noninteracting DM paradigm. While some
believe that the inclusion of baryonic physics could alleviate these issues [66{69], it remains
an open question. On the other hand, the inclusion of selfinteractions in the DM sector
could resolve these issues without creating tension with other astrophysical constraints.
These two conditions can be simultaneously satis ed if the cross section per unit mass for
DM satis es
Assuming a velocity independent cross section, [25] found that =m = 1 cm2/g tends to
over atten dwarf galaxy cores and that it is marginally consistent with ellipticity
constraints of the Milky Way. On the other hand a value of 0:1 cm2/g satis es all constraints
and attens dwarf galaxy cores su ciently. Let us consider a potential of the form
Note that
> 0 ( < 0) signi es a repulsive (attractive) interaction. The resulting DMDM
scattering crosssection is
0:1
cm2
g
.
m
at tree level. Plugging this into eq. (2.1), we get the constraint
that these DM particles coalese into boson stars at some point in early cosmology.
If a large fraction of DM is contained inside boson stars, the derived parameter space
may be signi cantly altered [71], since boson starDM interactions and boson star
selfinteractions may become signi cant. We will however assume that boson stars are rather
number density and selfinteraction cross section of free DM is taken to be m, n and .
The mean free path a DM particle travels before hitting another DM particle or a boson
star will be
DM = (n ) 1 and
BS
(nBS R2) 1, respectively. Scattering with boson
stars has to be much rarer than with other free DM in our approximation. Therefore we
require DM
BS. For the DM density we use the typical value of the solar system, i.e.
DM = M nBS + mn
0:3GeV/cm3. These requirements lead to the following condition
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
naBtSt
j jm DM
64 2MP2
2
10 5j j MeV
m
AU 3
;
where AU is an astronomical unit. The minimum mean distance between attractive boson
stars can therefore within this approximation be (naBtSt) 1=3
scenario with repulsive interactions the maximum mass scales as
p
40(j jm=MeV) 1=3AU. In the
MP3=m2. Therefore the
second term in the denominator of eq. (2.6) dominates. The number density must satisfy
nrBeSp
p
m2 DM
MP3
9
valid can at most be (nrBeSp) 1=3
nominator of eq. (2.6) tends to dominate. We obtain in the attractive scenario
MP=p
j j.
An important property of light scalar particles that has been examined extensively in
the literature [72, 73] is that large collections (particle number N
1) can transition to
a BEC phase at relatively high temperature, as compared to terrestrial experiments with
cold atoms. The critical temperature for condensate occurs when the de Broglie wavelength
is equal to the average interparticle distance, dB = [ (3=2)=n]1=3, where n is the average
number density of the particles and (x) is the Riemann Zeta function. This implies a
critical temperature for transition to the BEC phase of the form
kTc =
2
m
n
(3=2)
2=3
In this paper, we will assume that all relevant scalar eld particles are condensed, i.e.
that the system is in its ground state, a perfect BEC. The e ect of thermal excitations is
examined in [74] and they are expected to be negligible as long as T < Tc is satis ed.
3.1
Noninteracting case
It is instructive to begin with the case of boson stars bound only by gravity, rst analyzed
in [12]. In this seminal work, Kaup considers the free
eld theory of a complex scalar
in a spacetime background curved by selfgravity. The equations of motion2 were solved
numerically. The maximum mass of these solutions was found to be Mmax
the oftquoted Kaup limit for noninteracting boson stars. This value was later con rmed
by Ru ni and Bonazzola [13], who used a slightly di erent method by taking expectation
0:633MP2 =m,
values of the equations of motion in an N particle quantum state.
Interacting eld theories are more complex. In particular, for cross sections satisfying
eq. (2.1), the phenomenology of repulsive and attractive interactions are very di erent, and
accordingly, the methods required to analyze them are di erent as well. We outline the
relevant methods in the sections below.
3.2
Repulsive interactions
If the selfinteraction is repulsive, we can make use of the result of Colpi et al. [11]. Like
Kaup, their method begins with the relativistic equations of motion for a boson star,
the coupled Einstein and KleinGordon equations, but including a selfinteraction term
represented by :
A0
A2x
B0
B2x
00 +
+
+
2
x
1
x2
1
x2
+
B0
2B
1
1
1
A
1
A
A0
2A
2
2
B
B
=
=
0 + A
B
2 +
2
1
2
2
4 +
4 +
( 0)2
( 0)2
A
A
3
= 0;
2The noninteracting equations of motion are equivalent to eqs. (3.2) and (3.3) in the limit
(3.2)
! 0.
{ 5 {
where the rescaled variables are x = mr,
the particle energy), and
=
selfgravity of the condensate,
B(r) must be solved for; these represent the deviations from the at metric due to the
= p
( the scalar eld),
= !=m (!
MP2=(4 m2). In addition to the scalar eld itself, A(r) and
2M(x)=x] 1. In the limit that the interactions are strong (precisely,
In practice, one can trade the metric function A(r) for the mass M(x) by the relation
the system can be simpli ed signi cantly, as one can perform a further rescaling of the
1. In this limit the equations simplify to
equations:
=
section 2 suggest a value of
1=2, x = x
1=2, and M
= M
1=2. The relevant parameters of
= O(1040) or higher, so it is completely safe to neglect terms
(3.3)
1),
(3.4)
(3.5)
(3.6)
where the pressure p and density
are given by
B0
Bx
=
p =
1
1
1
16
16
2M
x
M
0 = 4 x
=
r
2
B
2
2M
x3
B
1
2
0:22. Restoring the appropriate dimensions, one
nds
M < M mreapx = 0:22
4
MP3 :
m2
This bound on the mass of repulsive boson stars was con rmed very precisely using a
hydrodynamic approach as well [75].
The branch to the left of the peak in
gure 1 represents unstable equilibria, where the
ground state energy is higher than the equilibrium on the right branch with the same
number of particles (and thus the same quantum numbers).
If we take the allowed range of
to be given by eq. (2.4), then we nd the following
range for M mreapx:
1 MeV
m
5=4
3:42
104M
. M mreapx .
6:09
104M ;
(3.7)
1 MeV
m
5=4
where M
= 1:99
1030 kg is the solar mass. The range of masses allowed by these
inequalities are represented in gure 3. Because of the strength of the repulsive interactions,
{ 6 {
corresponds to the pro le of the maximum mass equilibrium, while the blue and green are taken
on the stable branch of equilibria. The dimensionless variables in the plot are de ned in terms of
the dimensionful ones as
de ned in eq. (3.5) and x = mr
1=2.
these solutions can have masses several orders of magnitude above M . If there is a
signi cant number of such objects in the Milky Way, it could have important observational
signatures. However, a detailed analysis of the formation of these objects is required, in
order to give some indication of whether DM boson stars in galaxies have masses close to
the maximum value or lower.
3.3
Attractive interactions
If DM selfinteractions are attractive, then the method of [11] does not apply. However,
assuming relativistic corrections are negligible, one can instead solve the nonrelativistic
equations of motion numerically and analyze the solutions. To be precise, the dynamics
{ 7 {
as a function of DM particle mass m. The green band is the region consistent with solving the
small scale problems of collisionless cold DM. The blue region represents generic allowed interaction
strengths (smaller than 0:1 cm2/g) extending down to the Kaup limit which is shown in black.
The red shaded region corresponds to
& 4 . Note that the horizontal axis is measured in solar
masses M .
of a dilute, nonrotating BEC are governed by the GrossPitaevskii equation for a single
condensate wavefunction (r; t) =
(r)e iEt [76]
E (r) =
+ V (r) +
~ 2
r
2m
4 a
m j (r)j2
(r)
where V is the trapping potential, which in our case is the gravitational potential of the
BEC and satis es the Poisson equation
r~2V (r) = 4 Gm (r):
The swave scattering length a is related to a dimensionless 4 coupling
by a = =(32 m).
Here, (r) = m
n(r) = m j (r)j2 is the mass density of the condensate, which is
normalized such that R d3r (r) = M , the total mass. The three terms on the righthand
side of eq. (3.8) correspond to the kinetic, gravitational, and selfinteraction potentials,
respectively. As our notation signi es, we will assume that the density function is spherically
symmetric, i.e. (~r) = (r), which should be correct for a ground state solution.
Because the GrossPitaevskii + Poisson system (hereafter GP, de ned by eqs. (3.8)
+ (3.9)) cannot be solved analytically in general, we use a shooting method to integrate
the system numerically over a large range of parameters. As boundary conditions, we
{ 8 {
correspond to the density pro les in gure 5. The dimensionless variables in the plot are de ned in
terms of the dimensionful ones as M~ =
R99
m2
MP
R99.
(3.10)
(3.11)
HJEP02(16)8
choose the values of (0) and V (0) so that both functions are regular as r ! 0, and so
that asymptotically
(r) ! 0 and rV (r) ! 0 exponentially as r ! 1. Some examples of
integrated density functions are given in
gure 5. Our numerical procedure requires the
following rescaling of the dimensionful quantities:
=
r
m
1 ~
4 G ja~j
a = mGja~j
V
E =
r =
m ~
V
a~
j j
p a~
m
j j r~;
where the dimensionless quantities on the r.h.s. are denoted with a tilde. The equations
take the form
2 r
1 ~ 2 + V~
~2
j j
~ = 0
r~ 2V~ = j ~2j;
where r~ denotes a gradient with respect to r~, and we have explicitly taken a < 0. These
are the equations we solve. Similar rescaled equations were used in [77], but for repulsive
interactions, and unlike [77], we also scale away the scattering length a. This makes our
solutions valid for any generic a < 0.
In
gure 4 we show the massradius relation for the bosonic stars, which agrees well
with the results obtained in [17]. As in the repulsive case, there is a maximum mass for these
condensates, but this mass is signi cantly smaller for attractive interactions. For
parameters satisfying eq. (2.4), our analysis shows that condensates of this type would be light and
very dilute, having masses < 1 kg and radii R
O(km). (Our assumption that the General
Relativistic e ects could be neglected in this case is therefore well supported a posteriori.)
{ 9 {
0.100
˜ρ0.010
MP
corresponds to the pro le of the maximum mass equilibrium, while the blue and green are taken
on the stable branch of equilibria. The dimensionless variables in the plot are de ned in terms of
the dimensionful ones as ~ =
m4 and r~ =
One can arrive at a good, order of magnitude analytic estimate on the size and mass
of condensates by a variational method which minimizes the total energy. To this end, we
follow the approach of [16] by using the GP energy functional,
As input, we choose an ansatz for the wavefunction
(r), and subsequently compute the
energy of the condensate by integrating eq. (3.12) up to some maximum size R.
Minimizing the energy with respect to R should give a good estimate for the size of stable
structures. Note that the gravitational potential V (r) must be chosen selfconsistently to
satisfy eq. (3.9) for a given choice of (r).
In order to illustrate the salient features of the method, we will choose a simple ansatz
for the wavefunction:
which is normalized as above. Performing the energy integral gives the result
(r) = <
8q 43NR3 eir=R
:0
if r
R,
if r > R,
E = N
A
R2
BN
R
+
3AN a
R3
;
Rc =
A
BN
s
1
1 +
9a
A=B
!
where A
points
1=(2m) and B
6Gm2=5. Minimizing E(R) with respect to R gives two critical
(3.12)
(3.13)
(3.14)
(3.15)
In this calculation, a natural length scale X
A=B emerges. For any a 6= 0 (repulsive or
attractive), the minimum of the energy lies at the solution with the \+" sign, i.e.
R0 =
X
N
r
1 +
1 +
9a
X
!
In the case of attractive interactions, there is a critical number of particles Nmax
pX=(9jaj), above which the real energy minimum disappears and no stable condensate
exists. Using Mmax = mNmax, this analysis sets a value for the maximum mass for stable
condensates with attractive interactions:
j j
The corresponding limit on the radius is a lower bound, attractive boson stars being stable
only for
R > Rmatitn =
r 15
16 j j m2
MP
:
scaling relations M matatx
MP=pj j and Rmatitn
p
MP=m2 are completely generic.
Note that while the coe cient depends on the details of the wavefunction ansatz, the
(3.16)
(3.17)
(3.18)
Using eq. (2.4), we nd
1 MeV
m
3=4
1 MeV
m
3=4
7:37
10 9 kg . M matatx .
1:31
10 8 kg
(3.19)
The range of masses allowed by these inequalities is given by the green band in gure 6. We
plot the maximum masses over many orders of magnitude, between 1 eV and 1 GeV, but the
maximum mass of boson stars with such strong attractive selfinteractions is still < 1 kg.
Note that the numerical results agree well with the predictions of the variational
method to within an order of magnitude, even for the nave constant density ansatz in
eq. (3.13). These estimates can be improved further by a more robust ansatz for the
wavefunction.
As an example of a physical model, eld theories describing axions exhibit an attractive
selfcoupling through the expansion of the axion potential V (A) = m2f 2 1
cos(A=f ) ,
where A is the axion eld, m is the axion mass, and f is the axion decay constant.
Gravitationally bound states, particularly in the context of QCD axions, have become the topic
of much recent interest [78{80]. These states typically have maximum masses of roughly
10 11M , far below the bounds set in this section, because the couplings are typically
many orders of magnitude smaller.
As we pointed out in the introduction, in the case of attractive interactions the
potential is unbounded from below since
< 0. Therefore there must exist higher dimensional
operators suppressed by some cuto . The rst irrelevant operator with a Z2 symmetry is
6
= c2 where c is the cuto scale. We will now set a lower limit for c by requiring that
the
6 term is negligible with respect to the 4 term for typical boson star eld values.
Assuming that the kinetic energy of the eld is negligible, the energy density is roughly
103 1010 108 106 104 102 100
102
101
100
101
102
103
104
105
106
(3.20)
as a function of DM particle mass m. The green band is the region consistent with solving the
small scale problems of collisionless cold DM. The blue region represents generic allowed interaction
strengths (smaller than 0:1 cm2/g) extending up to the Kaup limit which is shown in black. The
red shaded region corresponds to
& 4 . Note that the horizontal axis is measured in grams.
equal to the potential. The maximum mass and minimum radius in eqs. (3.17) and (3.18)
can also be used to estimate the energy density as
can estimate the eld value ~ inside the boson star with attractive interactions to be
Mmax=Rm3in
m6=j j2MP2. Now we
~
j j
m
p2j j
0
4m2
j jMP2
1
2
j j
~6= c2 we obtain the inequality c
m=j j.
4
Conclusions
In this paper we studied the possibility that selfinteracting bosonic DM forms stars. We
assumed that selfinteractions are mediated by a
4 interaction and we investigated what
type of stars can be formed in the case of both attractive and repulsive selfinteractions,
giving particular emphasis to the parameter phase space of masses and couplings where the
DM bosons alleviate the problems of collisionless DM. We have considered DM particles
that populate the BEC ground state. We estimated the maximum mass where these dark
stars are stable, the massradius relation and the density pro le for generic values of DM
mass and selfinteracting coupling .
We leave several things for future work. The rst and most important is the mechanism
of formation for these bosonic dark stars. Su ciently strong selfinteractions can lead to the
gravothermal collapse of part or the whole amount of DM to dark stars [81]. In this case,
DM selfinteractions can facilitate the formation of bosonic stars because DM particles get
con ned to deeper selfgravitating wells simply by expelling high energetic DM particles out
of the core. As the core loses energy, the virial theorem dictates that the core shrinks and
heats up the same time. This leads to further energy loss and thus to the gravothermal
collapse. Such a scenario could also explain why the black hole at the center of the Galaxy is so
heavy, since DM bosonic stars could provide the initial seed required for the further growth
of the supermassive black hole [82]. It is interesting to note that boson stars can coexist in
equilibrium with black holes, as shown in [83, 84]. One should also notice that if the whole
density of DM collapses to dark stars, one does not have to be within the narrow band of
parameter space depicted in gures 3 and 6. Another possibility is the creation of high DM
density regions due to adiabatic contraction, caused by baryons [85, 86]. Moreover, bosonic
DM particles can get trapped inside regular stars via DMnucleon collisions. The DM
population is inherited by subsequent white dwarfs that, in case of supernovae 1a explosions,
can leave the bosonic matter intact, either alone or with some baryonic matter [87].
Asymmetric bosonic dark stars where no substantial number of annihilations take place
will not be very visible in the sky, although present. Gravitational lensing could be one way
to deduce the presence of such stars in the universe. Additionally, if the DM boson interacts
with the Standard Model particles via some portal (e.g. kinetic mixing between a photon
and a dark photon), thermal Bremmstrahlung could potentially produce an observable
amount of luminosity. This is particularly interesting since such a photon spectrum would
probe directly the density pro le of the boson star. Bosonic stars could also disguise
themselves as \odd" neutron stars. For example, it is hard to explain submillisecond
pulsars with typical neutron stars. XTE J1739285 could possibly be such a case, since it
allegedly rotates with a frequency of 1122Hz [88]. Compact enough bosonic stars would
have no problem to explain such high rotational frequencies. Another possibility is the
observation of compact stars with masses higher than the maximum mass a neutron star
can support. Such might be the case of the socalled \black widow" PSR B1957+20, with
a mass of 2.4 solar masses [89]. Therefore, abnormal neutron stars can well be the smoking
gun for the existence of asymmetric dark stars either with fermionic constituents like [63],
or with the bosonic ones studied here.
Acknowledgments
The research of C.K. and N.G.N. is supported by the Danish National Research Foundation,
Grant No. DNRF90. J.E. is supported by a Mary J. Hanna Fellowship. L.C.R.W.'s
research is partially supported by a faculty development award at UC. C.K. and L.C.R.W.
acknowledge support by the Aspen Center for Physics through the NSF Grant PHY1066293.
J.E. and L.C.R.W. also acknowledge M. Ma, C. PrescodWeinstein, and P. Suranyi for
valuable discussions about BoseEinstein Condensation.
Open Access.
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any medium, provided the original author(s) and source are credited.
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