Collapse of axion stars

Journal of High Energy Physics, Dec 2016

Abstract Axion stars, gravitationally bound states of low-energy axion particles, have a maximum mass allowed by gravitational stability. Weakly bound states obtaining this maximum mass have sufficiently large radii such that they are dilute, and as a result, they are well described by a leading-order expansion of the axion potential. Heavier states are susceptible to gravitational collapse. Inclusion of higher-order interactions, present in the full potential, can give qualitatively different results in the analysis of collapsing heavy states, as compared to the leading-order expansion. In this work, we find that collapsing axion stars are stabilized by repulsive interactions present in the full potential, providing evidence that such objects do not form black holes. In the last moments of collapse, the binding energy of the axion star grows rapidly, and we provide evidence that a large amount of its energy is lost through rapid emission of relativistic axions.

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Collapse of axion stars

Received: September Collapse of axion stars Joshua Eby 0 1 2 Madelyn Leembruggen 0 2 Peter Suranyi 0 2 L.C.R. Wijewardhana 0 2 Open Access 0 c The Authors. 0 0 2600 Clifton Ave , Cincinnati, OH, 45221 U.S.A 1 Fermi National Accelerator Laboratory 2 Department of Physics, University of Cincinnati Axion stars, gravitationally bound states of low-energy axion particles, have a maximum mass allowed by gravitational stability. Weakly bound states obtaining this maximum mass have su ciently large radii such that they are dilute, and as a result, they are well described by a leading-order expansion of the axion potential. Heavier states are susceptible to gravitational collapse. Inclusion of higher-order interactions, present in the full potential, can give qualitatively di erent results in the analysis of collapsing heavy states, as compared to the leading-order expansion. In this work, we axion stars are stabilized by repulsive interactions present in the full potential, providing evidence that such objects do not form black holes. In the last moments of collapse, the binding energy of the axion star grows rapidly, and we provide evidence that a large amount of its energy is lost through rapid emission of relativistic axions. Classical Theories of Gravity; Cosmology of Theories beyond the SM 1 Introduction 2 3 4 The non-relativistic expansion for axion stars Variational method Gaussian ansatz Cosine ansatz Decay of collapsing solutions A Total energy as Introduction The axion, a pseudoscalar particle originally associated with a solution of the strong CP problem in QCD [1{8], has been analyzed in a variety of astrophysical contexts, particularly in cosmological evolution [9{13] and as a candidate for dark matter [14{19]. Axions can condense into gravitationally bound objects, either in the early universe through largescale overdensities in a coherent axion eld (called \miniclusters"), or through gravitational cooling and collapse (called \axion stars") [20{22]. The masses of weakly bound axion stars have been computed previously [23, 24], and they are bounded above by gravitational stability [24{26]. Axion stars which exceed this maximum mass Mc have a fate which remains an open question. Some authors [27, 28] suggest that such con gurations collapse to a compact, very dense state. Recently, the author of [29] examined the collapse of a boson star with an attractive self-interaction which has M > Mc, using a dynamical equation derived from a Gaussian ansatz for its wavefunction. The author found that, as its potential is unbounded from below, a star of this kind collapses all the way to its Schwarzschild radius and forms a black hole. Indeed, the leading axion self-interaction is attractive; axionic or other bosonic objects with repulsive interactions have been considered by [30, 31]. However, the axion potential contains additional terms which become increasingly important as the axion density becomes large. It is thus plausible to ask whether these higher-order terms, some of which give rise to repulsive self-interactions, can stabilize the collapsing axion star prior to its formation of a black hole state. In this note, we will consider the consequences of including the full self-interacting axion potential in the collapse of heavy, weakly bound axion stars. In section 2, we review the nonrelativistic limit of axion eld theory in the description of axion stars; then in section 3, we outline the variational method used to nd energetically stable bound states, and the computation of the total collapse time for large mass solutions. We estimate the binding energy in section 4, in the initial and nal states, but also dynamically in time during collapse. As the binding energy increases, it is known [32] that the rate of decay for axion stars, through an annihilation process which ejects relativistic axions, rises quickly. We thus investigate whether collapsing axion stars emit a large fraction of their energy and decay due to quantum mechanical e ects. Finally, we outline our conclusions in section 5. The non-relativistic expansion for axion stars Axions are real scalar elds, but in the nonrelativistic limit can be described by a complex wavefunction , using the expansion [25, 33] = p which preserves the Hermiticity of the axion eld . At low temperatures, the wavefunc describes collectively a condensed state of N axions, termed an axion star, and is in the non-relativistic limit, yields the Lagrangian density L = = i for , where V ( ) = m2 f 2 1 is the low-energy axion potential, with m and f the mass and decay constant of the axion, respectively. The gravitational potential Vgrav(j j2) = representing the self-gravity of the condensate, can be added by hand [25, 34]. Then the H = 2 Vgrav(j j2) is conserved. Here W ( ) describes the quantum self-interactions of the axion eld, W ( ) = m2 f 2 1 Note that the mass term in the rst equality of eq. (2.5) is included to account for a cancellation in the non-relativistic limit between the potential and kinetic terms in L. In energy Etot=N m in the axion star is small. In that case, we can expand eq. (2.5) using eq. (2.1) and drop the rapidly oscillating terms containing extra factors of e imt. The resulting equation of motion for is the nonlinear Schrodinger equation. We derive the total energy from eq. (2.4) in the following way. The nth term in eq. (2.5) contains the factor 2n = where 2nCn are binomial coe cients. Dropping the rapidly oscillating pieces, we obtain W ( ) = = m2 f 2 1 = m2 f 2 1 where J0(x) is a Bessel function of the rst kind.1 Including also the kinetic and gravitational terms, the total energy functional has the form E( ) = 2 Vgrav j A minimum of the energy correponds to a stable bound state, an axion star. Typically, one expands the Bessel function in eq. (2.7) to obtain the leading self-interaction term, which is proportional to ( )2. This leading self-interaction is attractive, and as we will explain below, this implies that the potential appears unbounded from below as the axion star size decreases, R ! 0. There can nonetheless exist local energy minima, corresponding to metastable states which are dilute and weakly bound. However, there exists a critical particle number Nc above which no energy minimum exists, local or global. As a result, it is often assumed that an axion star with M > m Nc, being gravitationally unstable, will collapse all the way to a black hole state. A full description of this process can be found in [29], who used a Gaussian ansatz for the wavefunction and calculated the time for collapse to a black hole, which was on the order of an hour. The full axion self-interaction potential, given by eq. (2.6), contains additional terms beyond the attractive ( )2, which depend on increasing powers of the eld . Indeed, these higher-order terms, beginning with a repulsive ( )3 term, become increasingly relevant as the system increases in density, and we wish to investigate whether these terms have the e ect of stabilizing the potential against complete collapse. To this end, we will 1The J0 dependence of the axion self-interaction potential was pointed out in [24] and later in [27, 28]. examine the energy functional, including higher-order interactions, and determine whether the endpoint of collapse can lie at a radius greater than the Schwarzschild radius of the axion star. Such a result would be evidence that axion stars stabilize before they collapse to black holes. Variational method We will use a variational ansatz for the wavefunction to calculate the energy in eq. (2.7) as a function of the condensate size, in order to estimate the positions of any energy minima. Using the result of [24], we know how the macroscopic parameters of a weakly bound axion star, the radius R and the axion number N , scale with the dimensionful parameters of the theory; we thus de ne the dimensionless quantities and n by R = N = are m = 10 5 eV and f = 6 1011 GeV, implying = O(10 14) [35]. We will use a single variational parameter, the rescaled radius , at xed rescaled axion number n. Then the general form of a variational ansatz will be where at xed the function F ( ) is independent of , n, and . Then substituting the ansatz into the normalization condition of gives the normalization constant as (r) = w F w = 3=2 ; Ck = 4 where we introduced the notation Using (2.7) we obtain for the energy functional 2 C2 2 2 C22 v = 4 6 Z [(k + 2)!]2 2 C2 3 which, at n = nc, has a value of Gaussian ansatz v0 = vj =0 = nc = min n=nc 8 B4 (r) = leading-order of , which is The minimization of (3.5) with respect to locates the radii of metastable minima and maxima of the binding energy. The condition for the existence of metastable states constrains the reduced particle number n to a nite constant. Restricting ourselves to For n < nc, there exists a metastable minimum of the energy at a reduced radius min = 3 B4 C4 n2 8 C22 D22 and where we de ned the functions D2 = 4 B4 = 32 2 Note that just like Ck, also B4 and D2 are independent of the physical parameters n, , and . The leading-order approximation of (3.5) is obtained when we take the small limit, at which Following [26, 29], we use a Gaussian ansatz to approximate the axion star wavefunction: which corresponds to eq. (3.2) with w = p N =( 3=4 3=2) and F ( ) = e 2=2. Note that when we talk about the \size" of such a condensate (whose wavefunction extends to r ! 1), we refer to the conventional R99, inside which :99 of the mass is contained. For the Gaussian ansatz, this occurs not at , but at a value closer to 3 . Note also that we de ne using eq. (3.1) with R = , not R = R99. The energy functional, given by eq. (3.5), depends on the coe cients D2 = 3 3=2 B4 = Ck = 2 reduced particle number n. The global minimum always lies at a radius GM > eq, the position at which the kinetic energy computed using the Gaussian function in eq. (3.14). Written out explicitly, we have 3=2 3 k=0 [(k + 2)!]2 (k + 2)3=2 2 3=2 3 2 3=2 3 where in the second equality we have expanded J0 and integrated term by term. Though no closed form exists for the integral in eq. (3.16), we show in the appendix that it is nite as region. Consequently, the total energy is bounded from below in this formalism, and always has a global minimum.2 In gure 1, we show the position GM of this global minimum of the energy functional in eq. (3.16) as a function of n. The global minimum always lies at a very small radius: =(2 n ). so at n = nc, S = 10 13 GM, and this possible endpoint of collapse is not a The normalized energy per particle coming from the self-interaction is shown to be a 1=2 in the small limit, so we can estimate the value of 1 at which the 2It is possible that this conclusion would be modi ed by post-Newtonian corrections to the gravitational S = 2 n kinetic and self-interaction energies are of the same order; we nd comparable magnitudes at a radius of eq zontal black line in 10 7, corresonding to roughly Req eq is shown as a horigure 1. This radius Req is of the same order as the axion reduced Compton wavelength, c = ~=m c 2 cm. It should be noted that on length scales of O( c), neglecting higher powers of e imt in the expansion of eq. (2.1) would fail, as special relativistic corrections to the kinetic energy could be large. Nonetheless, weakly bound stars have radii much larger than this, and as we describe below, even collapsing stars are well described by the non-relativistic approximation until the last moments of collapse. We have estimated the leading correction to the kinetic energy, which is compared to the leading-order term. We will thus postpone any further consideration of these relativistic corrections to the energy, which will be addressed in a future publication. p4, and in the range In this work we analyze the low-energy axion potential in eq. (2.3), sometimes called the instanton potential. But it is well-known (see e.g. [35, 36]) that an improved approximation is the chiral potential V ( ) = m2 f 2 1 and f are the mass and decay constant of the QCD pion. This expression takes into account the non-perturbative e ects of up and down quark masses mu and md. We nd that substituting eq. (2.3) with this chiral potential does not qualitatively change the conclusions of this work: the global minimum of the energy in gure 1 shifts down by at most a few percent, still signi cantly larger than the corresponding black hole state. We put o any further discussion of the chiral potential to a future publication. We also consider the e ect of including a nite but increasing number of terms in the series of eq. (3.17). Because it has no closed form resummation, what is typically done is to k=0 [(k + 2)!]2 (k + 2)3=2 2 3=2 3 : (3.18) The minima of eK ( ) should, at su ciently large K, approximate well the stable bound states of the full energy function. The existence of a global minimum of the full energy functional in eq. (3.17) has important consequences. In particular, we have pointed out above that this minimum lies at a radius many orders of magnitude larger than the Schwarzschild radius of the axion star, providing evidence that such objects do not collapse to black holes. Further, we note that the terms contained in the series of eq. (3.18) alternate between attractive and repulsive interactions, even and odd k respectively. But as a result, a truncated energy N=.85Nc N=.9Nc N=.95Nc N=Nc for di erent for K > 0 makes a negligible di erence in this range of . eK ( ) in eq. (3.18) for any even K has no global minimum, and thus such a truncation removes the possiblity of approximating the stable radius of the full energy functional. We thus submit that when considering dense con gurations of axions or collapse of axion stars, it is important to truncate the series on a repulsive term to preserve the global minimum. The leading-order interaction term is contained in e0( ), and has been considered in great detail previously [24{26, 29]. It has been pointed out that there exists a maximum corresponds to a radius of R99 500 km for QCD axions [24], and is approximated to the correct order of magnitude by the Gaussian ansatz, which gives a radius R99 3 and at a radius we analyze the consequences of additional interaction terms in the axion potential. The energy functional in the vicinity of this minimum is shown in gure 2. It is also worth noting that the inclusion of additional terms in the self-interaction potential introduces negligible di erences in this range of good approximation in this region. But as noted above, any eK ( ) for even K (e.g. e0( )) is unbounded from below and will not be applicable in approximating the global energy We turn now to e1( ), including the leading repulsive interaction which originates from minimum of the full potential, which is at )3 term in the potential: e1( ) = In this case, the energy is bounded from below and has a minimum at a very small radius K=1 K=3 K=5 K=7 N=.9Nc is preserved at each order, but shifts to smaller radii as K increases. The repulsive kinetic term 1= 2 dominates the total energy at = eq D (in contrast to the result using only e0). At these small values of , the energy is well approximated by the self-interaction terms only (gravity and kinetic energy are negligible); thus we can use the analytic expression 37=2 sponding to R99 7 meters. Comparing with the global minimum of the full energy in gure 1, we nd a di erence of only about a factor of 3{4 near this value of n reasonable order of magnitude agreement. This justi es our truncation of the energy at the leading repulsive term, i.e. e1( ), in this analysis. The di erence between D and GM 10 6, corredoes become large if n increases far above nc. nd that the existence of a dense global energy minimum is preserved at any odd K in the approximation of eq. (3.18), and at increasing order, shifts to smaller radii (see gure 3). Nonetheless, the kinetic energy term dominates the full potential below 10 7, and the global minimum of the full energy is at GM > eq, for any n. The collapse of dark matter halos consisting of condensed scalar particles was examined by [37], using a time-dependent formalism that originated in [38], and utilized by [26, 39]. The application of this method to an axion star, at leading-order in the self-interaction potential, was recently performed by [29]. This collapse process is described by the dynamical Etot = : 0 = :1 , 0 = :5 , 0 = :8 , 0 = . At N < Nc, condensates can still collapse if the starting radius 0 < R(t) is the size of the condensate, which varies with time during collapse. For a condensate t = where in the second equality we have rescaled the dimensionful quantities. In the analysis of [29], E(R) was approximated by the leading-order expression E0(R), order of an hour. We wish to investigate the e ect of additional self-interactions in the axion potential on the collapse process. Including the rst non-leading interaction piece, i.e. using e1( ), we have found that a global energy minimum exists at D; thus, we integrate eq. (3.20) not from = 0 but rather from If the axion star begins its collapse at 0 = , then of course at n = nc the collapse time is formally in nite, because the potential is at at . We consider values of n which are slightly larger than nc and see how the collapse time changes. We also investigate the change in collapse time as the starting size 0 deviates from . This latter case could be of interest, say, if axion star collapse can be catalyzed by collisions with other astrophysical sources. In that case, even condensates with N < Nc can collapse, provided some catalyzing interaction which reduces its initial radius to R0 < R . These considerations are represented together in gure 4. n=2nc n=3nc n=4nc We can also track the radius of the axion star as a function of time, throughout the collapse process; see gure 5. For a large portion of the total collapse time, the radius changes little, as the star rolls slowly down a shallow potential, but later collapses fast to the dense minimum of radius D. Cosine ansatz The Gaussian ansatz is believed to be a reasonable approximation to the axion star wavefunction. However, in order to verify that our results are not an artifact of the wavefunction one chooses, we present a second ansatz for the variational analysis: (r) = 2 R (r < R): A comparison of the two ansatze we use is shown in gure 6 for the same total size.3 The energy functional, rescaled and truncated as above, depends on the coe cients D2 = C2 = B4 = 8 C4 = C6 = 3Note that while the cos2 wavefunction goes to 0 at some r and thus has a de nite edge, the Gaussian function (as we pointed out previously) does not. ansatz (black, solid), normalized to the same total size. This implies that, for the cosine ansatz, e1( ) = 1440(2 2 2304(2 2 691200(2 2 5369) n dense minimum is at approximately D from the Gaussian case, D 10 6n1=3. Decay of collapsing solutions As before, we minimize the approximated energy e1( ) with respect to , and nd both a dilute and a dense minimum. The dilute minimum disappears above a critical particle number, corresponding to nc 12:6, where the radius is :44 (around 200 km). The 10 6n1=3, within a factor of 2 of the result In a previous work [32], some of us found that axion stars can decay through repeated occurrences of the a process which ejects relativistic axions from the star. Such a process is not forbidden by any symmetry because axions, being Hermitian elds, do not have a conserved number, and because bound axions, along with the axion star itself, are not in momentum eigenstates. To describe this interaction, the spectrum of bound states describing the axion star was extended by a collection of scattering states, labeled by momentum p. The leading contribution to this process was an interaction of the form AN 3 + ap, where AN denotes an axion star with N axions and ap denotes a relativistic axion with momentum p. Without the addition of these scattering states, the matrix element for this and many other interactions are identically zero. Our analysis assumed a small binding energy in the axion star. A contrarian point of view was expressed We found in [32] that the lifetime of an axion star through emission of relativistic axions depends on a reduced binding energy parameter leading-order expansion in 1 is equivalent to the infrared limit of the theory, where only the marginal 4 term appears in the interaction potential [24]. For weakly bound stars, the leading process AN ! AN 3 + ap has a rate which, as a function of , is dominated (Etot=N m)2. The by an exponential factor, 1024 r3 m energies, corresponding to generally, we found that if a star has . :05{:06, then it is stable on timescales as long as the age of the universe, because the lifetime is a monotonically decreasing function of in the relevant range. The constant in eq. (4.2) has the value yM = 25:46. The dense energy minimum D has a large binding energy, corresponding (in the Gaussian case) to of [32] applies only in the weak binding limit. Further, eq. (4.2) takes into account only the 4 interaction, but this is a valid approximation throughout most of the decay process. Nonetheless, if valid, such a short lifetime would imply that these dense states, as the endpoint of collapse, would decay very quickly. However, our calculational method is not applicable to strongly bound systems, so we cannot make a de nite statement about We hope to investigate the decay of strongly bound states in greater detail in the future. Recent investigations of collapse using a classical collapse analysis have concluded that collapsing axion stars lose a signi cant fraction of their mass through emission of relativistic axions [45, 46]. Number-changing interactions of a similar type have also been suggested as a mechanism for limiting the core densities of dark matter halos [47]. In the weak binding region, where eq. (4.2) holds, we know that is a one-to-one function of , and thus also of the collapse time t as de ned in eq. (3.20). We nd that the binding energy obtains 10 4 (compared with function of time, the binding energy only changes appreciably in the last fraction of a second of the collapse, but rises quickly to a strongly bound nal state (see In these last moments, the decay rate in eq. (4.1) becomes astronomically large; emitted axion/sec at :0223, and rises to 1050 emitted axions/sec at therefore are led to the conclusion that axion stars, as they collapse, emit many highly n=2nc n=3nc n=4nc of a collapsing axion star using the approximate energy energetic free axions.4 Such an explosion, referred to as a Bosenova, has been observed experimentally by condensed matter physicists using cold atoms [44]. While this work was under review, a di erent group performing a numerical simulation also suggested that a large fraction of axion star energy is expelled during the collapse process through relativistic axion emission [45]. We emphasize again that the analysis of the decay process in [32] applies only at weak binding, when 1. This condition holds for the dilute state as well as throughout a large portion of the collapse process, but it is possible that some new dynamics take hold at truly strong binding :56. We are led to the conclusion that relativistic axion emission becomes important during collapse, but it is possible that a stable, strongly bound remnant remains. Conclusions The contribution of the axion self-interaction potential to the total energy in the variational method can be computed to arbitrary order using an expansion in powers of the axion eld. This expansion is equivalent to an expansion in the small parameter Because of the smallness of this parameter ( 10 14 for QCD axions), the potential is typically truncated at leading-order, including only the attractive ( truncation works extraordinarily well at large radii, and the dilute radius R found by multiple authors previously [24, 25] is preserved. In the regime of larger (e.g. axions :1 MP), some of these conclusions could be changed. While this work was being reviewed, an analysis performed in the classical limit [46] suggested that axion theories = f 2=MP2 )2 term. This 4If dark matter consists of axion stars, then this decay process could deplete the total amount of dark matter in galaxy clusters. This e ect is considered in a di erent context in [43]. indeed allow collapse to black holes in some regions of parameter space. We are working out the mass spectrum of axion stars in such theories, which will be the topic of future work. Going beyond the leading-order approximation, without truncation we have found that a global minimum of the full energy exists, which is not present in the leading-order expansion; we calculated its position, and it corresponds to a radius RGM many orders of magnitude larger than the corresponding Schwarzschild radius. We approximate this global minimum using a next-to-leading-order expansion, using the truncated energy of eq. (3.19), which has a global minimum at a radius RD using the Gaussian ansatz, the dilute radius R good order of magnitude estimate of RGM. R . For m = 10 5 eV QCD axions and 200 km, while RD 7 meters. RD is a Previous analyses of collapsing boson stars with an attractive self-interaction have concluded (correctly) that, with nothing to stabilize the potential as R ! 0, the endpoint of collapse is a black hole state. For the axion potential, we have found higher-order self-interactions, some of which are repulsive, stabilize axion stars as they collapse and there exist energetically stable con gurations at very small radii. These con gurations correspond to dense axion star states which are nonetheless not black holes, and resemble closely the type of the dense states found by the authors of [27] using a di erent method. Dense con gurations of this kind can exceed the maximum mass normally allowed for weakly bound axion stars, which is roughly Mc 1019 kg for m = 10 5 eV axions [24]. We have examined the collapse dynamically in time, and nd that masses M just radius changes slowly at rst, then drops rapidly as the slope of the potential becomes increasingly steep. Stars which begin collapse at a radius R0 < R were also considered, a case which is interesting if, for example, axion star collapse is catalyzed by collisions of two lighter axion stars. This could occur even if these stars do not become gravitationally bound to each other. This topic will be pursued in a future work. If stable, then heavy axion star states could be detectable via gravitational lensing experiments. Such states have large binding energies, and thus non-relativistic and nonperturbative corrections may become important in that regime. During collapse, however, when binding energies increase but are still su ciently small, previous calculations [32] suggest that the rate of emission of relativistic axions from an axion star will rise very rapidly. The rate of decay through the leading number-changing interaction AN ! AN 3 + ap rises to & 1050 emitted axions/sec in the nal moments of collapse, leading to rapid emission of axions in what is often called a Bosenova [44]. It is not clear in our analysis precisely what fraction of the energy of the star would be expelled through this process, or whether a stable dense state could remain. It would be interesting to investigate the energy spectrum of these collapses in detail, to determine if there are detectable consequences of such an explosion. Acknowledgments We thank P. Argyres, R. Gass, A. Kagan, D. Kulkarni, J. Leeney, M. Ma, and C. Vaz for conversations. M.L. thanks the WISE program and Professor U. Ghia for support and encouragement, and the University of Cincinnati and the Department of Physics for a summer research fellowship. The work of JE was partially supported by a Mary J. Hanna Fellowship through the Department of Physics at University of Cincinnati, and also by the U.S. Department of Energy, O ce of Science, O ce of Workforce Development for Teachers and Scientists, O ce of Science Graduate Student Research (SCGSR) program. The SCGSR program is administered by the Oak Ridge Institute for Science and Education for the DOE under contract number DE-SC0014664. Total energy as In this section we outline the proof that the contribution of the self-interaction potential to the total energy is nite in the ! 0 limit, and consequently that the kinetic energy ansatz is of the form VSI = 3=2 3 2 3=2 3 interested in the case of ! 0, corresponding to z ! 1. We proceed with the estimation of the integral I in the limit ! 0 in the following way. Break up the integral into two parts: (1) I1, integrated over the interval 0 < u < , < z, and (2) I2, integrated over the remaining < u < z. Consider rst I1: which goes quickly to 0 as u ! 0. At larger values u dominated by the term 1=z, on the other hand, I1 is and consequently, For I2, we consider large u, since 1 < u. Then the bracket in eq. (A.1) dominated Z ln(z= ) pte 2tdt 2 Erf r 2 ln z In the limit z ! 1, the rst term in the brackets ! 1, while the second term vanishes. Finally, since I1, given by eq. 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Joshua Eby, Madelyn Leembruggen, Peter Suranyi, L. C. R. Wijewardhana. Collapse of axion stars, Journal of High Energy Physics, 2016, 66, DOI: 10.1007/JHEP12(2016)066