Syddansk Universitet Boson Stars from Self-Interacting Dark Matter

Syddansk Universitet

Boson stars from self-interacting dark matter Eby, Joshua; Kouvaris, Christoforos ; Nielsen, Niklas Grønlund ; Wijewardhana, Rohana

Published in: The Journal of High Energy Physics (Online)

DOI: 10.1007/JHEP02(2016)028

Publication date: 2016

Document version Final published version

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Citation for pulished version (APA): Eby, J., Kouvaris, C., Nielsen, N. G., & Wijewardhana, R. (2016). Boson stars from self-interacting dark matter. The Journal of High Energy Physics (Online), 1-18. [28]. DOI: 10.1007/JHEP02(2016)028

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Download date: 19. Apr. 2017 JHEP02(2016)028 a Springer February 3, 2016 January 13, 2016 December 8, 2015 : : : 10.1007/JHEP02(2016)028 Received Published Accepted doi: , and L.C.R. Wijewardhana b Published for SISSA by [email protected] [email protected] Niklas Grønlund Nielsen , , b . 3 1511.04474 The Authors. Chris Kouvaris, c Cosmology of Theories beyond the SM, Classical Theories of Gravity

We study the possibility that self-interacting bosonic dark matter forms star- a , [email protected] -Origins University of Southern Denmark, 3 University of Cincinnati, Dept.Cincinnati, of OH Physics, 45221, U.S.A. CP Campusvej 55, DK-5230, Odense M,E-mail: Denmark [email protected] b a Open Access Article funded by SCOAP ArXiv ePrint: relations for these dark mattermass bosonic they stars, can their support. density profile as well as theKeywords: maximum Abstract: like objects.we We focus study particularly both instanding the the problems case parameter of phase of space the attractive where collisionless and self-interactions dark repulsive can matter self-interactions, solve paradigm. well and We find the mass radius Joshua Eby, Boson stars from self-interacting dark matter JHEP02(2016)028 ] , 11 /m 2 P M 633 . 0 ≈ ], but many other 4 ] in non-interacting max – 1 13 ] or the scalar which M 9 – 5 ]. 10 is the mass of the individual m ). Later, it was shown by Colpi et al. [ 2 – 1 – /m 3 P ]. 4 M 11 7 5 3 5 5 GeV is the Planck mass and ] and subsequently by Ruffini and Bonazzola [ 19 12 10 × 1 12 22 . = 1 P M It was shown many years ago that objects of this type are allowed by the equations of These bosonic particles often make good Dark Matter (DM) candidates as well. One 2.1 DM scattering with boson stars 3.1 Non-interacting case 3.2 Repulsive interactions 3.3 Attractive interactions where bosons. (This isChandrasekhar very limit, different which from scalesthat the as self analogous interactions limit in for theseparticular, they systems fermionic examined can systems stars, cause with termed repulsive significant self-interactions, the phenomenological and show changes. that In the upper Condensate (BEC) state, and canThese thus wavefunctions be can described indeedhave by encompass thus a an been single astrophysically termed condensate large “boson wavefunction. volume stars” of [ space and motion, first by Kaup [ systems. They found a maximum mass for boson stars of the form lived enough that they couldformation. coalesce Unlike the into usual DM collisionless halosscenario cold where which DM large constitute picture, collections however, the we of seedsto are these their interested of bosons in self-gravitation galaxy the (and form self-interaction boundthe generically). states scalars For of are this macroscopic taken picture size to to due be be consistent, sufficiently cold so that they may coalesce into a Bose-Einstein scalar degrees of freedomscales. have These been include proposeddrives (among to the many expansion extend others) of particle the the physics universe axion to in of quintessence high QCD modelsreason energy [ [ for this is that unlike the Higgs, many of these new scalars would be stable or long- 1 Introduction Bosonic degrees ofphysics, freedom both arise in the genericallyportance, Standard being and Model the naturally and only beyond. fundamental in scalar The theories in the Higgs of Standard boson Model fundamental is [ of paramount im- 4 Conclusions 3 Bosonic dark matter Contents 1 Introduction 2 SIDM parameter space JHEP02(2016)028 – ]. ]. ]. /g 2 18 46 47 ]. 51 This σ/m – , 17 1 41 50 1 cm . coupling. ], while N-body 4 φ 39 , 0 because the former is 38 ]. → ]. Dark stars have been λ 15 , 54 term. We will assume that – 14 6 52 0 for attractive interactions. If φ ]. Second, the number of satellite λ < 40 is a dimensionless λ , where ]. These simulations prefer a self-interaction ). Furthermore, in this scenario the typical 4 , where /g. There are, however, upper bounds on 49 λφ , 2 – 2 – 2 3.3 48 that the maximum mass for an attractive conden- /m , 3 P 3 25 10 cm λM . √ . This result was originally derived using an approximate | 02 λ . | σ/m 0 p . / ≈ P as compared to the noninteracting case makes it more plausible /g M 2 max ], and was later confirmed by a precise numerical calculation [ ∼ M /m 16 P . 1 cm . 3 max M /g. 2 M 1 cm . Self-gravitation and additionally extra self-interactions among DM particles can lead The solution of these problems is currently unknown, but a particularly well-motivated DM self-interactions have already been proposed and studied in different contexts [ The situation for attractive self-interactions is slightly more complex. The simplest Note that the Colpi et al. result does not reduce to the Kaup bound as 1 σ/m ]. One of the main reasons why DM self-interactions can play an important role is The idea of DMannihilating forming DM star-like might compact have objects existed is in not the new. early Darkderived universe by stars [ rescaling that the consist equationsas of we of motion see and in dropping section higher-order terms in the strong coupling limit, from a number of sources, includingThe allowed the parameter preservation space of from ellipticity these of constraintssections nonetheless spiral which intersects galaxies can the [ resolve range of the cross . small-scale issues of collisionless DM, in the range 0 in some cases to the collapse of part of the DM population into formation of dark stars. idea involves self-interacting DM (SIDM).that Simulations they including have the such effect interactionsproblems of suggest of smoothing collisionless out DM cuspy as densitycross well profiles, section [ and of could 0 solve the other are observed to havesimulations flat predict cuspy density profiles profiles forgalaxies collisionless in in DM their the [ central MilklyLast Way regions is is [ the far so-called fewer “too thanrange big that the to we number have fail” not predicted problem: observed, in but simulations which predict simulations are dwarf [ too galaxies large to in have a not mass yet produced stars [ 37 due to the increasingastrophysical tension observations, between the numerical resolution of simulationsdiscrepancy, which of known (for collisionless the as cold moment) the is DM “cusp-core unknown. and problem”, The is first related to the fact that dwarf galaxies interacting cases. This is becausecomes the from only the uncertainty force principle. supporting Gravity thethe and condensate attractive condensate. against self-interactions tend collapse We to will shrink seesate in scales section as analytical method [ case involves a self-interaction ofthis the were the form highest-order termso in the one potential, typically then stabilizes itthe would it not contribution by be of the bounded such below, additionthe higher-order and of validity terms of a is that positive negligiblesizes assumption of phenomenologically in gravitationally (we bound section address BEC states is significantly smaller than the repulsive or non- extra factor of that boson stars canconstraining have the masses boson even starfrom larger parameter galaxy space, than and which a galaxy fits cluster solar the sizes, mass. has coupling been strength A considered using different in data method [ of limit on the mass is JHEP02(2016)028 ] as (2.3) (2.1) (2.2) 58 – 55 /g tends to ], it remains 2 69 – 66 = 1 cm ] because unlike the 63 σ/m /g satisfies all constraints 2 1 cm . . . 4 2 φ g ] found that cm λ 4! 1 2 25 + . 2 2 ] for bosonic ones. πm λ φ σ m 2 65 64 2 – 3 – -body simulations of cold, non-interacting DM m . ]. Additionally some of the authors of the current = N 2 ] displays the parameter space where it is possible 62 σ g ) = – 63 cm φ ( 59 1 . V 0 = 1 in what follows. c = ]. Unlike the dark stars of annihilating DM, asymmetric dark stars ~ 0) signifies a repulsive (attractive) interaction. The resulting DM-DM 63 λ < 0 ( λ > ] for fermionic particles, as well as in [ Note that we set In this paper we examine the dark stars composed of asymmetric self-interacting 64 Note that scattering cross-section is Assuming a velocity independentover-flatten cross dwarf section, galaxy [ coresstraints and of the that Milky it Way.and On is flattens the dwarf marginally other galaxy consistent hand cores a with sufficiently. value ellipticity Let of us 0 con- consider a potential of the form could resolve these issuesThese two without conditions creating can tension beDM simultaneously with satisfies satisfied other if astrophysical the cross constraints. section per unit mass for indicate that the central regionsto of galaxies the should cored have profiles ato “cuspy” one fail” density observes. problems, profile, has contrary This, ledbelieve along some that with to the question the inclusion the of “missing non-interacting baryonican satellites” DM physics open could and paradigm. alleviate question. “too While these some big issues On [ the other hand, the inclusion of self-interactions in the DM sector 2 SIDM parameter space As we mentioned above, galactic scale the Fermi pressure andform gravitation, a bosonic BEC in DMfrom the does collapsing. ground not We state haveradius are and a relation, going it the Fermi to is density surface. demonstratethe profile the context how and of They uncertainty DM the the principle self-interactions self-interactions maximum that affect that mass reconcile keeps the of cold the mass these DM star with DM the bosonic observational stars findings. in to observe such dark stars,mass providing limits mass and radius density relations,in profiles. corresponding [ Chandrasekhar Self-interactions in dark stars have also beenbosonic considered DM. Thecase study of is fermionic fundamentally DM different where from the that stability of of [ the star is achieved by equilibrium between well as in the context ofpaper mirror DM studied [ the possibilityexhibits of Yukawa dark type star self-interactions formation thatcold from can DM asymmetric paradigm alleviate [ fermionic thecan DM problems be that of stable the and collisionless observable today. [ also studied in the context of hybrid compact stars made of baryonic and DM [ JHEP02(2016)028 . | . 2 λ | σ /m p (2.5) (2.6) (2.7) (2.8) (2.4) / P and P M | M n λ | , AU. In the ∼ 3 . The mass, m p / 1 , which would BS − π ∼ . Therefore the n 2 4 R . /m 3 P λ MeV) . 3 m/ λM − , | 3 . λ √ , the first term in the de- | pc − 2 P / 2 3 40( M AU pc. 3 ≈ / m 2 3 m MeV . / − m . MeV 1 | − and number density M 1 MeV λ 2 , respectively. Scattering with boson ) | / 1 5 Mσ 1 + − att BS M 3 − MeV) − ) + n 2 P 2 m λ DM | DM 10 < 9 2 λ M m/ 100 MeV. In this mass range, it is plausible | ρ − ( σρ 3 πR × 2 | 3 − . / π 10 λ 2 ), and the boson star radius to be comparable to ) dominates. The number density must satisfy | BS 2 , mass – 4 – mπR 10 λ n 64 × m ≈ R ( 2 2.3 2.6 ), we get the constraint 9 < 2 P 10 ∼ 2 ≈ DM 2.1 / M × BS 3 . These requirements lead to the following condition 2 BS 3 P BS n 3 5 π mρ n λ | DM ], since boson star-DM interactions and boson star self- ρ λ ≈ 64 | λM 2 3 71 ]. For perturbativity, we should restrict m / √ m and 1 70 1 MeV − 1 ) − att BS 3GeV/cm ) . ) becomes n rep BS 0 rep BS n nσ n ) tends to dominate. We obtain in the attractive scenario 2.5 ≈ . For the DM density we use the typical value of the solar system, i.e. 2.6 = ( BS mn λ + DM )), eq. ( λ BS 3.18 DM λ Mn = If a large fraction of DM is contained inside boson stars, the derived parameter space DM second term in the denominator of eq. ( The minimum mean distance between repulsivevalid boson can stars which at leaves most our approximation be ( where AU is an astronomicalstars unit. can therefore The within minimum this approximation mean bescenario distance ( with between repulsive attractive interactions the boson maximum mass scales as The maximum mass of a boson starSince with this non-negligible attractive scaling interactions is isnominator only of proportional eq. ( to a single power of Taking self-interactions to be that of eq.the ( minimum radius (which scales the(see same eq. for ( both signs of interaction) star will be stars has to berequire much rarer than withρ other free DM in our approximation. Therefore we 2.1 DM scatteringTo with quantify boson how scarce stars bosonboson stars stars have to have be anumber within characteristic density this radius approximation, and we self-interaction assumeThe cross that mean section free of path free a DM DM is particle taken travels before to hitting be another DM particle or a boson may be significantly alteredinteractions may [ become significant.scarce We and will the however DM assume self-interactions that are boson dominated stars by are DM-DM rather scattering. This matches the results of [ imply that our resultsthat are these valid DM only particles for coalese into boson stars at some point in early cosmology. at tree level. Plugging this into eq. ( JHEP02(2016)028 , /m (3.2) (3.1) 2 P 0. ]. Like M → 11 633 were solved . 0 2 is the average ≈ is satisfied. n c , max 1) can transition to 2 2 ) ) = 0 0 0 ) in the limit Λ M A A σ σ T < T ( ( 3.3 3 , where 3 σ N / + + 1 Λ ] 4 4 σ σ − ) and ( /n 2 2 Λ Λ σ 2) 3.2 / + − . 1 2 2 3 (3 / σ σ ζ − 2 2 = [ B Ω 2) + 1 + 1 / dB n 2 2 λ (3 B B ζ Ω Ω A – 5 – + -particle quantum state. π 0 ) is the Riemann Zeta function. This implies a = = m 2 σ N x ( ζ = 0 1 1 A A A c A 2 − − kT − 1 1 0 B B ], who used a slightly different method by taking expectation 2 2 2 1 1 x x 13 + + + 2 x x x 0 0 2 2 A ] is that large collections (particle number B A B + 00 73 , σ ] and they are expected to be negligible as long as 72 74 ), the phenomenology of repulsive and attractive interactions are very different, and ]. In this seminal work, Kaup considers the free field theory of a complex scalar 2.1 Interacting field theories are more complex. In particular, for cross sections satisfying 12 The non-interacting equations of motion are equivalent to eqs. ( 2 the coupled Einstein andrepresented by Klein-Gordon Λ: equations, but including a self-interaction term relevant methods in the sections below. 3.2 Repulsive interactions If the self-interaction isKaup, repulsive, we their can method make use begins of with the result the of relativistic Colpi equations et al. of [ motion for a boson star, by Ruffini and Bonazzola [ values of the equations of motion in an eq. ( accordingly, the methods required to analyze them are different as well. We outline the It is instructive to beginin with [ the case ofin boson a stars bound spacetime only backgroundnumerically. by curved gravity, The first by maximum analyzed self-gravity. mass of Thethe these oft-quoted equations solutions Kaup of was limit motion found for to non-interacting be boson stars. This value was later confirmed In this paper,that we the will system assume isexamined in that in its all [ ground relevant state, scalar a perfect field BEC. particles The3.1 are effect condensed, of thermal i.e. Non-interacting excitations case is number density of thecritical particles temperature for and transition to the BEC phase of the form An important propertythe of literature light [ scalara particles BEC that phase at hascold relatively been atoms. high The temperature, examined critical as temperature extensivelyis for compared in condensate equal to occurs to terrestrial when the the experiments de average with Broglie interparticle wavelength distance, 3 Bosonic dark matter JHEP02(2016)028 ω ( 1), (3.5) (3.6) (3.7) (3.3) (3.4) ) and r ( ω/m A ,

M 4 ) by the relation x 10 ( × M 09 . 6 4 . / , ), then we find the following 2 5 ∗ . The relevant parameters of Ω 2 . d 2.4 1 / πp 2 1 3 2 P 1 r − − m m M ∗ − = 8 Λ 2 + Φ (Φ the scalar field), Ω = ρ B 2 π Ω 2 ∗ 1 MeV 2 ∗ λ ) for the mass 4 M B 3 ∗ Ω r . πG dr ( x πx M ) 4 r 2 = r 2 A r . represents unstable equilibria, where the ( √ ∗ 22 1 = = 4 . − A 1 + 1 = ∗ 0 ∗ rep M max − 22. Restoring the appropriate dimensions, one + 2 . σ ∗ = 0 σ – 6 – ). In addition to the scalar field itself, 2 0 2 M M B 2 ∗ , B 3Ω Ω x M dt ≈ . ) or higher, so it is completely safe to neglect terms ) rep 2 max to be given by eq. ( πm , and mr r are given by

40 . Because of the strength of the repulsive interactions, ( 2 π π λ − / (4 ∗ max 1 1 3 = ]. 1 B M / ∗ ρ 1 16 16 4 (10 2 − P − x 75 Λ O M 10 = = ∗ = x M < M 0 ∗ ∗ λM × 2 B p ρ = Bx kg is the solar mass. The range of masses allowed by these ds 42 ∗ . . In the limit that the interactions are strong (precisely, Λ 3 x 30 1 4 , / − and density 2 5 ] 10 / ∗ 1 . In this limit the equations simplify to /x p × show the mass-radius relation and selected density proﬁles, respectively. 1 Λ ) − 2 σ x 99 ( m . : = M 1 MeV and ∗ 2 rep max = 1 1 σ − M

suggest a value of Λ = M 2 If we take the allowed range of Figures ) = [1 ) must be solved for; these represent the deviations from the flat metric due to the r x ( ( where inequalities are represented in figure The branch to theground left state of energy thenumber is peak of higher particles in than (and figure thus the the equilibrium same quantum on numbers). range the for right branch with the same finds This bound onhydrodynamic the approach mass as well of [ repulsive boson stars was confirmed very precisely using a In this limit, the equationsmaximum (dimensionless) do mass not depend on Λ, and one finds numerically that there is a where the pressure proportional to Λ In practice, one canA trade the metricthe function system can beequations: simplified significantly, assection one can perform a further rescaling of the the particle energy), and ΛB = self-gravity of the condensate, where the rescaled variables are JHEP02(2016)028 . If there is a

M . 2 / 1 − Λ ] does not apply. However, mR 2.2 . 11 2 = 2.0 / 1 ∗ − 2.0 X Λ 1.5 mr 1.8 and = P ∗ * x 1.6 x * /M X 2 1.0 / 1 − 1.4 – 7 – ) and Λ 2 . The dimensionless variables in the plot are deﬁned in 0.5 3.5 1.2 2 mM = 1.0 ∗ 0.0 -4 -4 M 0.100 0.050 0.010 0.005 0.001

5.× 10 1.× 10 0.05 0.00 0.20 0.15 0.10

* * ρ M defined in eq. ( ∗ ρ . Three examples of density profiles in the case of repulsive interactions. The red profile . The mass-radius relation for a boson star with strong repulsive coupling. The 3 circles 3.3 Attractive interactions If DM self-interactions areassuming attractive, relativistic then corrections the are methodequations of negligible, of [ motion one numerically can and instead analyze solve the the solutions. nonrelativistic To be precise, the dynamics these solutions cansignificant have number of masses such several objectssignatures. in orders the Milky of However, Way, a it magnitude could detailedorder have above to analysis important give observational of some the indicationthe formation of maximum of value whether or these DM lower. objects boson stars is in required, galaxies in have masses close to Figure 2 corresponds to the profileon of the the stable maximum branchthe mass of dimensionful equilibrium, equilibria. ones while The as the dimensionless blue variables and in the green plot are are taken defined in terms of Figure 1 correspond to the density profilesterms in of figure the dimensionful ones as JHEP02(2016)028 ) ), ). 3.8 2.4 0 -1 -2 -3 -4 -5 -6 3 2 1 (3.9) πm 10 10 10 10 10 10 10 10 10 10 (32 λ/ 11 11 = 10 10 a 9 9 by 10 10 λ ] 7 7 ☉ ) (3.8) 10 10 r [M ( ψ 5 5 max coupling 10 10 2 M 4 | ) φ r self-interactions satisfying eq. ( 3 3 ( ψ . 10 10 | ) r ( 1 1 πa m 4 10 10 repulsive ] ) + πGmρ 76 r [ ( -8 -8 V – 8 – . Note that the horizontal axis is measured in solar is the mass density of the condensate, which is iEt π ) = 4 10 10 + − 4 r 2 | e ( 2 ) ) m & -10 -10 , the total mass. The three terms on the right-hand r V r ~ ∇ 2 ( ( 2 λ 10 10 . The green band is the region consistent with solving the M ψ ψ ~ ∇ − m · | -12 -12 ) = is related to a dimensionless ) = m /g) extending down to the Kaup limit which is shown in black. 10 10 r 2 a ( ) = r, t ( r -14 -14 rρ ( ), which should be correct for a ground state solution. ) = φ 3 r r 10 10 1 cm d ( . ( ρ Eψ n R /g 2 -16 -16 · /g 2 10 10 ) = m ~r ( Kaup Limit 0.1 cm 1 cm λ=4π -18 -18 ρ ) correspond to the kinetic, gravitational, and self-interaction potentials, re- ) = 10 10 r ( 3.8 ρ . -20 -20 is the trapping potential, which in our case is the gravitational potential of the

. The maximum mass of a boson star with 10 10 M V 0 3 2 1 )) cannot be solved analytically in general, we use a shooting method to integrate -5 -6 -1 -2 -3 -4

10 10 10 10 Because the Gross-Pit¨aevskii+ Poisson system (hereafter GP, deﬁned by eqs. ( Here,

10 10 10 10 10 10 3.9

] [ MeV m spectively. As our notation signifies,symmetric, we i.e. will assume that the density function is spherically + ( the system numerically over a large range of parameters. As boundary conditions, we The s-wave scattering length normalized such that side of eq. ( where BEC and satisfies the Poisson equation masses of a dilute, nonrotatingcondensate BEC wavefunction are governed by the Gross-Pitäevskiiequation for a single Figure 3 as a function ofsmall scale DM problems particle of collisionless mass strengths cold DM. (smaller The than blue region 0 The represents generic red allowed shaded interaction region corresponds to JHEP02(2016)028 (3.10) (3.11) 0. These 0, and so → a < r . This makes our ], but for repulsive a . . Some examples of ˜ r, 77 99 | ˜ ˜ a R V | | m 2 P → ∞ ˜ a m | p m M r π = = 1000 λ r 32 , E | 500 2 r − ˜ ψ | = = 0 = 99 ˜ ˜ R ˜ ψ 100 V (km). (Our assumption that the General . Our numerical procedure requires the 2 | ˜ 5 ∇ 50 2 and 99 ˜ ψ , and we have explicitly taken ˜ R ∼ O r P 0 exponentially as M R − | M – 9 – → ˜ π V . The dimensionless variables in the plot are defined in ) 10 λ r 32 5 + ( 5 2 r ˜ 0. rV ∇ ˜ = ψ V 1 2 (0) so that both functions are regular as | ˜ 1 ˜ a − M V | a < ]. As in the repulsive case, there is a maximum mass for these 1 | 1 kg and radii 0 and

0.10 0.05 0.01 0.50

˜ a 17

M ], we also scale away the scattering length | m πG ˜ < 4 → 77 ) mG r r (0) and ( ), our analysis shows that condensates of this type would be light and = = ψ ψ a ψ 2.4 we show the mass-radius relation for the bosonic stars, which agrees well 4 denotes a gradient with respect to ˜ . The mass-radius relation for a boson star with attractive interactions. The three circles ˜ ∇ In figure with the results obtained in [ condensates, but this mass isters satisfying significantly eq. smaller ( for attractivevery interactions. dilute, having For masses parame- Relativistic effects could be neglected in this case is therefore well supported a posteriori.) where are the equations weinteractions, solve. and Similar unlike rescaled [ equationssolutions were valid used for in any [ generic take the form where the dimensionless quantities on the r.h.s. are denoted with a tilde. The equations choose the values of that asymptotically integrated density functions arefollowing given rescaling in of figure the dimensionful quantities: Figure 4 correspond to the density profiles in figure terms of the dimensionful ones as JHEP02(2016)028 (3.12) (3.13) (3.14) (3.15) . Mini- R gives two critical R . # 4 | ψ . | , , , ! R 10 2 πa m ), and subsequently compute the 2 ≤ N r 3 5 ( r r > R ) with respect to + R ) must be chosen self-consistently to a ψ ANa R 2 if r 9 | 3 ( ( A/B . ) up to some maximum size ψ E r | V + 2 P V r ˜ 1 + 3.12 m ir/R M R 1 + e BN s π 3 2 λ | N 32 – 10 – − πR 3 ψ m 0.5 4 ). 1 2 2 r ~ ∇ r | A should give a good estimate for the size of stable

( R q 0 if = " ψ r    r R A 5. Minimizing 3 N BN / d 2 = ) = and ˜ = 0.1 1 Z r -4 c ( ρ E 10 Gm

4 ψ

0.100 0.010 0.001

R 6

ρ λ ] = ˜ m ψ ] by using the GP energy functional, ≡ [ = E 16 B ρ ) and m ) for a given choice of (2 / 3.9 1 ≡ . Three examples of density profiles in the case of attractive interactions. The red profile A In order to illustrate the salient features of the method, we will choose a simple ansatz One can arrive at a good, order of magnitude analytic estimate on the size and mass where points which is normalized as above. Performing the energy integral gives the result structures. Note that thesatisfy gravitational eq. potential ( for the wavefunction: As input, we chooseenergy an of ansatz the for the condensatemizing wavefunction by the integrating energy eq. with ( respect to of condensates by a variationalfollow method the which approach minimizes of the [ total energy. To this end, we Figure 5 corresponds to the profileon of the the stable maximum branch mass of equilibrium, equilibria.the while dimensionful The the ones dimensionless blue as variables and in ˜ the green plot are are taken defined in terms of JHEP02(2016)028 , ≡ ) . We 1 kg. (3.16) (3.18) (3.17) max 6 A/f < N cos( symmetry is − 2 1 Z kg (3.19) 6= 0 (repulsive or by requiring that 2 8 a f c − 2 µ 10 m × ) = . 31 . | A 1 ( λ P | 4 are completely generic. V / M . 3 p 2 ! . 2 2 P /m 27 is the axion decay constant. Gravi- 320 N P m M emerges. For any f a m | r X 9 λ | λM 1 MeV term for typical boson star field values. = √ | 4 15 16 A/B 1 + a φ | ∼ X . r ≡ 9 r = s X att min – 11 – att max 1 + 0. Therefore there must exist higher dimensional R m M att min

= . X N λ < and | att = kg max λ | R > R 0 9 − R ]. These states typically have maximum masses of roughly p , this analysis sets a value for the maximum mass for stable is the axion mass, and / 10 80 P – m max × M < M M 78 37 ∼ . mN 7 4 = / att max 3 ), we find M is the cutoff scale. We will now set a lower limit for max 2.4 c M µ m 1 MeV ), above which the real energy minimum disappears and no stable condensate , far below the bounds set in this section, because the couplings are typically | is the axion field, ). These estimates can be improved further by a more robust ansatz for the a

term is negligible with respect to the | where A M 6 (9 2 c 3.13 As we pointed out in the introduction, in the case of attractive interactions the poten- As an example of a physical model, field theories describing axions exhibit an attractive Note that the numerical results agree well with the predictions of the variational Using eq. ( In the case of attractive interactions, there is a critical number of particles φ 11 /µ − X/ 6 tial is unbounded fromoperators below suppressed since by someφ cutoff. The first irrelevantthe operator with a Assuming that the kinetic energy of the field is negligible, the energy density is roughly where tationally bound states, particularly inof the much context recent of interest QCD10 [ axions, have become themany topic orders of magnitude smaller. method to within aneq. order ( of magnitude,wavefunction. even for the constant na¨ıve density ansatz in self-coupling through the expansion of the axion potential The range of masses allowed byplot these the inequalities maximum is masses given over by manymaximum orders the of green mass magnitude, band of between in boson 1 figure eV stars and 1 with GeV, such but strong the attractive self-interactions is still Note that whilescaling the relations coefficient depends on the details of the wavefunction ansatz, the The corresponding limit on theonly radius for is a lower bound, attractive boson stars being stable exists. Using condensates with attractive interactions: attractive), the minimum of the energy lies at the solution with the “+” sign, i.e. p In this calculation, a natural length scale JHEP02(2016)028 ) ), 2.4 0 -1 -2 -3 -4 -5 -6 3 2 1 3.18 (3.20) 10 10 10 10 10 10 10 10 10 10 26 26 . Now we 10 10 2 P /g ) and ( 2 M /g 24 24 2 2 | 10 10 λ 3.17 | λ=4π Kaup Limit 0.1 cm 1 cm / 22 22 6 10 10 m . ≈ | 20 20 λ [g] | 10 10 m 3 min p . max | 18 18 M λ /R ≈ | 10 10 1 2 self-interactions satisfying eq. ( max m/   16 16 M 1 2 10 10 interaction and we investigated what ≈ c 4 2 P µ ρ 2 14 14 λφ M m attractive | 10 10 4 λ | − 2 2 – 12 – 1 10 10 . Note that the horizontal axis is measured in grams. 0 0 1 + . π 10 10 4 . The green band is the region consistent with solving the λ   | m & λ -2 -2 | λ inside the boson star with attractive interactions to be 2 m /g) extending up to the Kaup limit which is shown in black. The 10 10 ˜ 2 φ p -4 -4 we obtain the inequality 2 c 10 10 1 cm | ≈ . ˜ φ | /µ -6 -6 6 ˜ φ 10 10 -8 -8 4 ˜ 10 10 φ | λ | -10 -10 . The maximum mass of a boson star with 10 10 0 3 2 1 -5 -6 -1 -2 -3 -4

10 10 10 10

10 10 10 10 10 10

] [ MeV m giving particular emphasis to the parameterDM phase bosons space of alleviate masses the andthat problems couplings populate where of the the collisionless BECstars DM. ground state. are We have stable, We considered estimated themass the DM mass-radius and maximum relation particles mass self-interacting and coupling where the these density dark profile for generic values of DM 4 Conclusions In this paper weassumed studied that the self-interactions possibility are that mediatedtype self-interacting by of a bosonic stars DM can forms be stars. formed We in the case of both attractive and repulsive self-interactions, Requiring equal to the potential.can The also maximum be mass used and to minimumcan estimate radius estimate the in the energy eqs. field density ( value as Figure 6 as a function ofsmall scale DM problems particle of collisionless mass strengths cold DM. (smaller The than blue 0 regionred represents shaded generic allowed region interaction corresponds to JHEP02(2016)028 ], 63 ]. 87 ]. In this case, 81 ]. Moreover, bosonic 86 , 85 ]. Compact enough bosonic stars would ]. One should also notice that if the whole 88 84 , . Another possibility is the creation of high DM 83 – 13 – 6 and 3 ]. It is interesting to note that boson stars can coexist in 82 ]. Therefore, abnormal neutron stars can well be the smoking 89 Asymmetric bosonic dark stars where no substantial number of annihilations take place We leave several things for future work. The first and most important is the mechanism Grant No. DNRF90.search is J.E. partially is supported by supported aknowledge by faculty support development a award by at the Mary UC. Aspen C.K. J. CenterJ.E. and for Hanna and L.C.R.W. Physics L.C.R.W. ac- Fellowship. through also the acknowledge L.C.R.W.’s M. NSFable Ma, Grant re- C. discussions PHY-1066293. Prescod-Weinstein, about and P. Bose-Einstein Suranyi Condensation. for valu- 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, have no problem toobservation explain of compact such stars highcan with rotational support. masses frequencies. higher Such might than Anothera be the mass the possibility maximum of case mass is 2.4 of solar a the gun the masses neutron for so-called [ star the “black widow” existence PSR of B1957+20, asymmetric with dark stars either with fermionic constituents like [ amount of luminosity. This isprobe particularly interesting directly since the suchthemselves a density photon as profile spectrum “odd” would ofpulsars neutron the with stars. boson typical neutron star. For stars.allegedly example, rotates XTE Bosonic with J1739-285 it could a stars is possibly frequency could hard be of also to such 1122Hz a disguise explain [ case, sub-millisecond since it will not be very visible into the deduce sky, the although presence present. of such Gravitationalwith stars lensing in the could the Standard be universe. one Model Additionally, way and if particles the via a DM boson some dark interacts portal photon), (e.g. kinetic thermal mixing Bremmstrahlung between could a photon potentially produce an observable parameter space depicted in figures density regions due to adiabatic contraction,DM caused particles by can baryons get [ trappedulation inside is regular stars inherited via by DM-nucleoncan subsequent collisions. white leave The the dwarfs DM bosonic that, pop- matter in intact, case either of alone supernovae or 1a with explosions, some baryonic matter [ heats up the same time.lapse. This Such leads a to scenario further could energyheavy, also since loss explain DM and why the bosonic thus black to stars hole the couldof at gravothermal provide the the col- the center supermassive initial of black the seed hole Galaxy requiredequilibrium [ is for with so the black further holes, growth asdensity shown of in DM [ collapses to dark stars, one does not have to be within the narrow band of of formation for these bosonic darkgravothermal stars. collapse Sufficiently of strong self-interactions part can orDM lead to the self-interactions the can whole facilitate amount the ofconfined formation to DM of deeper to bosonic self-gravitating dark wells stars simply stars becauseof by [ expelling DM the high particles core. energetic get DM As particles the out core loses energy, the virial theorem dictates that the core shrinks and JHEP02(2016)028 , , ]. D Sov. Gen. (1964) SPIRE , IN Phys. Rev. ]. 13 ][ , Phys. Rev. Phys. Rev. , , Phys. Rev. Lett. 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