Collisions of dark matter axion stars with astrophysical sources

Journal of High Energy Physics, Apr 2017

If QCD axions form a large fraction of the total mass of dark matter, then axion stars could be very abundant in galaxies. As a result, collisions with each other, and with other astrophysical bodies, can occur. We calculate the rate and analyze the consequences of three classes of collisions, those occurring between a dilute axion star and: another dilute axion star, an ordinary star, or a neutron star. In all cases we attempt to quantify the most important astrophysical uncertainties; we also pay particular attention to scenarios in which collisions lead to collapse of otherwise stable axion stars, and possible subsequent decay through number changing interactions. Collisions between two axion stars can occur with a high total rate, but the low relative velocity required for collapse to occur leads to a very low total rate of collapses. On the other hand, collisions between an axion star and an ordinary star have a large rate, Γ⊙ ∼ 3000 collisions/year/galaxy, and for sufficiently heavy axion stars, it is plausible that most or all such collisions lead to collapse. We identify in this case a parameter space which has a stable region and a region in which collision triggers collapse, which depend on the axion number (N ) in the axion star, and a ratio of mass to radius cubed characterizing the ordinary star (M s /R s 3 ). Finally, we revisit the calculation of collision rates between axion stars and neutron stars, improving on previous estimates by taking cylindrical symmetry of the neutron star distribution into account. Collapse and subsequent decay through collision processes, if occurring with a significant rate, can affect dark matter phenomenology and the axion star mass distribution.

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Collisions of dark matter axion stars with astrophysical sources

Received: February Collisions of dark matter axion stars with astrophysical sources Joshua Eby 0 1 2 Madelyn Leembruggen 0 2 Joseph Leeney 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 If QCD axions form a large fraction of the total mass of dark matter, then axion stars could be very abundant in galaxies. As a result, collisions with each other, and with other astrophysical bodies, can occur. We calculate the rate and analyze the consequences of three classes of collisions, those occurring between a dilute axion star and: another dilute axion star, an ordinary star, or a neutron star. In all cases we attempt to quantify the most important astrophysical uncertainties; we also pay particular attention to scenarios in which collisions lead to collapse of otherwise stable axion stars, and possible subsequent decay through number changing interactions. Collisions between two axion stars can occur with a high total rate, but the low relative velocity required for collapse to occur leads to a very low total rate of collapses. On the other hand, collisions between an axion star and an ordinary star have a large rate, 3000 collisions/year/galaxy, and for su ciently heavy axion stars, it is plausible that most or all such collisions lead to collapse. We identify in this case a parameter space which has a stable region and a region in which collision triggers collapse, which depend on the axion number (N ) in the axion star, and a ratio of mass to radius cubed characterizing the ordinary star (Ms=Rs3). Finally, we revisit the calculation of collision rates between axion stars and neutron stars, improving on previous estimates by taking cylindrical symmetry of the neutron star distribution into account. Collapse and subsequent decay through collision processes, if occurring with a signi cant rate, can a ect dark matter phenomenology and the axion star mass distribution. sources; Cosmology of Theories beyond the SM; Classical Theories of Gravity 1 Introduction Variational method for axion stars Collisions between two ASts Modi ed energy functional Collision of an ASt with an ordinary star Gravitational potential inside star Collisions of an ASt with a neutron star Axion stars [1{3] are macroscopic bound states of axion particles [4{11], and their existence can have astrophysical or cosmological implications [12{17]. In particular, axions could form all or part of dark matter in the universe, potentially in the form of axion stars [18{23]. Axions could also be connected to leptonic mass hierarchy and mixings as well [24, 25]. The masses of dilute, or weakly bound, axion stars [26{28] are bounded above by considerations of gravitational stability [29, 30]. The endpoint of collapse of a weakly bound axion star whose mass exceeds the critical value Mc has recently received a lot of attention. By analyzing the energy functional, the author of [31] found that boson stars which possess attractive self-interactions collapse to black holes when M exceeds Mc. Using a similar method, some of us [32] concluded that the full axion potential, which contains both attractive and repulsive interactions, is bounded from below, and the full energy functional is minimized at a dense radius RD. We concluded that the endpoint of collapse for an axion star was a dense state, but with a radius still larger than the corresponding Schwarzschild radius RS. The possibility of such dense states were also proposed earlier by [26, 33], and if they exist in nature, they can have interesting phenomenological consequences [34]. Dense states for axion stars have large binding energies, and following our analysis of axion star decay in [35], we also suggested in [32] that during collapse a large number of relativistic axions are emitted in what is often called a Bosenova [36]. The recent work of [37] and [38], using very di erent methods, seem to similarly indicate that relativistic axions are emitted from collapsing axion stars. The non-relativistic e ective eld theory of axion stars, as outlined in [39], can also have sensitivity to unique decay signatures, and such rates increase with the density of the axion star as well [40]. The dominant mechanism for the emission of relativistic axions is the subject of current debate, and we will not attempt to resolve it here. However, a recent paper on oscillon decay [41] supports the mechanism suggested in [35] of decay through emission of a single relativistic axion. A consensus seems to have emerged that as binding energy of an axion star increases, its decay rate through number changing interactions increases rapidly. This condition is satis ed by collapsing axion stars. Collisions of axion stars with astrophysical sources could occur with a relatively high rate, especially if axion stars compose a large fraction of dark matter. Because collisions can change the energy functional for a dilute axion star, they can lead to unique collapse scenarios which, in turn, can suggest high rates of relativistic axion emission. With this in mind, it is interesting to analyze collisions of dilute axion stars with two potentially copious astrophysical sources: ordinary stars and other axion stars. Axions couple at loop-level to photons, which allows decay of free or condensed axions through a process a ! 2 , but this rate is believed to be small enough to be ignored on cosmological timescales [42]. However, in collisions with neutron stars, strong magnetic elds can stimulate these interactions, leading to bursts of photons that are potentially observable [43{46]. The idea that such collisions could lead to the observed Fast Radio Bursts [47{50], which appeared originally several years ago, has been investigated as recently as this year [51]. Because of the unique detection signatures arising from these collisions, we also revisit the calculation of the collision rate of axion stars with neutron stars. In section 2, we review the variational method for determining the macroscopic parameters describing axion stars, which is used to analyze the axion star energy functional. We estimate collision rates of axion stars with other axion stars (section 3), and with ordinary stars (section 4), and in both cases, map the parameter space for collapse. In section 5, we calculate the collision rate of axion stars with neutron stars. We conclude in section 6. Variational method for axion stars Axion self-interactions can be described by the low-energy potential [32, 33] W ( ) = m2 f 2 1 where m and f are the mass and decay constant of the axion, and is the low-energy wavefunction describing an N -particle condensate of axions. For QCD axions, typical values are m = 10 5 eV and f = 6 1011 GeV, which implies the ratio f 2=MP 2 = 1, which will be used in what follows.1 The total self-energy of the axion star (hereafter \ASt") is EAS( ) = 2 Vgravj j2 + W ( ) ; where R is the radius of the star. The gravitational potential is taken to be Poissonian, and thus satis es r2 Vgrav = 4 The self-interaction potential in eq. (2.1) is typically expanded in powers of the axion wavefunction . Following [32], we expand W ( ) and truncate at the next-to-leading order, including both the attractive ( )2 and a repulsive ( )3 interaction. Because we will later consider possible ASt collapse, we believe that both of these interaction terms are of crucial importance. We will use a variational ansatz of the form and particle number N of the ASt can be rescaled as [32] giving a total self-energy of (r) = < 2 R ; r R r > R n = EAS( ) = m N A = C = B = D = 691200(2 2 for the cosine ansatz of eq. (2.2).2 The terms in eq. (2.4) correspond to kinetic, self-gravitational, attractive interaction, and repulsive interaction energies, respectively. suppressed by high powers of critical value of the particle number. In this range, the energy is well-approximated by only the rst three terms in eq. (2.4), and has a local minimum at a rescaled radius 2Note that the structure of eq. (2.4) is ansatz-independent, but the values of the coe cients A; B; C; D the local minimum of ASt 1 (2), ;1 ( ;2), and the isoenergetic point on the left of the maximum of ASt 1 (2), 0 ;1 ( 0 ;2). The black, dashed curve is the full energy of these two ASts occupying the same volume, which has no minimum in this range of , because the e ective particle number This local minimum is illustrated in gure 1 for two di erent values of n. One can easily read particle number n > nc is also illustrated in gure 1. Such states are referred to in the literature as dilute ASts, and correspond to small binding energies. This is parameterized by a parameter = p1 = O( p ), which is O(10 7) for the QCD axion parameters we use in our analysis [27]. At any rescaled particle number n, the global minimum of the energy in eq. (2.4) lies at a very small radius D 1 [32]. At these small radii, the last two terms of eq. (2.4) dominate and the energy minimum lies approximately at a rescaled radius of For QCD axions with the cosine ansatz, R 200 km and RD 7 meters when n 1019 kg. An ASt with n > nc (as illustrated by the dashed line in possesses only a dense energy minimum at D 1. Such a state is referred to as a dense ASt, and has large binding energy = O(1). There are various mechanisms for stimulating collapse from R to RD. For example, if ASts form against a dilute background of free axions, they could accrete such axions, thereby acquiring masses M > Mc and triggering collapse; however, it is not clear how e ciently such accretion would occur. We show below that collapse could also be catalyzed by interactions between ASts and other astrophysical sources, including stellar matter or other ASts. ASts can also be converted e ciently to photons through interactions with neutron stars and their strong magnetic elds [43{46, 51]. We are thus motivated to investigate the rate of collisions of ASts with these sources, how a collision could alter the energy landscape, and how the population of ASts might change as a result. ASts collapsing from the dilute radius R to the dense one RD move quickly from small to large binding energies [32]. When the binding energy becomes large, the rate of numberchanging interactions in the ASt grows quickly, and so the ASt emits many relativistic axions as it collapses [35, 37, 38]. In the end, we will speculate about the observable e ects of ASt decay, which could be stimulated by collisions and subsequent collapse. Collisions between two ASts Modi ed energy functional We consider rst collisions between two weakly-bound ASts in a single dark matter halo. Such collisions could lead to mergers of these ASts, which can trigger collapse, because the e ective energy functional is modi ed from eq. (2.4) to E2AS( ) = m (N1 + N2) B (n1 + n2) C (n1 + n2) where N1 and N2 are the number of axions in each of the two stars. If the sum n1 +n2 > nc, then both stars begin to collapse, as their combined energy no longer has a local minimum However, there is no guarantee that colliding ASts will merge; because they move with some relative velocity vrel, they may occupy the same volume for only a nite time. Another way to say this is that, in light of the weak self-interactions of axions, it is possible that such objects would pass right through one another. Indeed, no mechanism is known for dissipating energy during the collision | with the exception of gravitational waves, but the corresponding rate of energy dissipation is negligibly small.3 An important, related note is that we also ignore di erences in phase for the colliding ASts, which could increase the merger rate for condensates close to being in-phase, or lead to inelastic \bounces" when the condensates are out of phase [52]. For su ciently large velocities, these e ects are likely to be negligible, but could be relevant for ASts with low relative velocities. We hope to return to this point in the near future. Thus, for the purposes of this work, we will assume that colliding ASts do not dissipate energy and become bound, and so the energy functional changes for only a nite time; we 3This is the case because the masses of QCD ASts are too small to have a signi cant gravitational wave output, but the situation could be di erent in some more generic axion theory which allows very heavy bound states. We plan to return to this point in a future work. Nonetheless, collapse will begin when the stars occupy the same volume, as the energy functional changes. If the ASts overlap for a su ciently long time, then when the stars separate, they will already be gravitationally unstable and will continue to collapse. Collision rates In this section, we begin to examine the approximate rates for two types of astrophysical bodies colliding with each other in a single galaxy, where at least one of these objects is an ASt. We will perform this calculation in two ways: we begin by making the assumption that ASts are distributed with some constant density in galactic halos; later, we will take into account the nontrivial number density. For two populations of astrophysical objects, the rst being ASts and the other labeled by i, the general expression for the collision rate is where n(~r) denotes the number density of some population of astrophysical objects, i is the cross section for a collision, and v is the relative velocity between the objects. The Generically the expectation value i = 1 Z d3r nAS(~r) ni(~r)h vii fMB(v) = f0 exp ^i = This expression is signi cantly simpler than eq. (3.2), and often gives a good order of magnitude estimate, but as we will see in later sections, it sometimes vastly underestimates In this work we will use data for the Milky Way to approximate the collision rates for \typical" galaxies. For this reason we set Rgal 105 lightyears, which is the radius of the is an average over velocities in the halo, using a Maxwell-Boltzmann distribution of velocin the halo. In the special case where both distributions are trivial, that is, that the objects are distributed with constant density throughout the halo, eq. (3.2) simpli es greatly. In that for collisions of a species i with ASts in a dark matter halo is Milky Way. For the cross section, we will use i = (Ri + RAS) 2 G (Mi + MAS) v2 (Ri + RAS) with Ri the radius of the body which collides with the ASt, and Mi is its mass. This is the geometric cross section modi ed by an enhancement factor to account for classical capture through gravitational e ects. Note that we will assume in what follows that the initial con guration of the colliding ASt is the dilute state (not the dense state). This is because, rst, the collision rates for as de ned by eqs. (2.5) and (2.6), compared to dilute ASts; and second, because otherwise stable dilute ASts can collapse as a result of collisions, making them additionally interesting. The number of ASts in the galaxy NAS can be estimated, albeit roughly, by assuming that the total mass MDM with a xed mass of MAS = Mc of the dark matter in the Milky Way consists of ASts 1019 kg. (M = 2 1030 kg is a solar mass.) In truth, there are two prominent e ects that would modify this estimate. First, dark matter may consist only partially of ASts, the rest potentially being in a dilute background of axions or some other dark particles; we represent this e ect by a multiplicative factor 0 . FDM . 1, so that the total dark matter mass in ASts is FDMMDM. Secondly, ASts likely have a spread in their mass distribution, and so some ASts will have masses smaller than the maximum Mc; we introduce a second factor 0 . FAS . 1, so that the average mass of a single ASt is FASMAS. These competing e ects can, of course, compensate each other in the calculation of NAS. The total number of ASts in the Milky Way is NAS = FDMMDM = 2 FAS NFW(r) = is negligible in the ASt cross section in this case. Then a constant-density estimate of the rate in eq. (3.4) is year galaxy We can improve on this rough estimate by taking into account the number density distributions of ASts. If ASts are a component of dark matter, then it is reasonable to assume that they are distributed according to the Navarro-Frenk-White (NFW) pro le [53] where 0 and R0 are scale mass density and radius for the halo. An analysis of this kind was recently performed by [54], who considered primordial black holes as dark matter. Rvir = 1 Z Rvir vvir [300 km/s] values of FDM=FAS. Taking the Milky Way as a sample galaxy, we use the values of 0, R0, and virial mass Mvir from [55]. These values are reported in table 1. We will also take into account velocity dispersion by assuming the Maxwell-Boltzmann distribution of eq. (3.3), though the precise value of vvir is somewhat uncertain. We introduce an O(1) factor vvir = so xing the value of Mvir from [55], we have a virial radius of The velocity-averaged cross section is implying a total rate for collisions h viAS = FDM 2 Z Rvir r2 NFW(r)2 dr year galaxy collision rates. The table on the top-left describes the dark matter halo using an NFW pro le using data from [55]; the top-right table describes the distribution of stars using [56]; and the table along the bottom gives parameters describing the distribution of neutron stars using [57]. The -dependence in the last bracket comes from the uncertainty in Rvir in the upper bound of the integral. If = 1, we have ASj =1 = 107 FDM year galaxy Thus in this case, the constant density approximation in eq. (3.4) gives a reasonable estimate, that of eq. (3.5). The full decay rate AS over a range of , and for di erent values true collision rate could easily be larger or smaller by a few orders of magnitude from the approximate value 107 collisions/year/galaxy. Collapse time The total energy of two ASts as they pass through each other can be approximated by eq. (3.1), which is equivalent to eq. (2.4) with N ! N1 + N2. This is equivalent to an increase in the e ective particle number in the total energy, while the stars occupy the same volume. If two dilute ASts have N1 + N2 > Nc, their combined energy functional in eq. (3.1) will no longer possess a dilute energy minimum at , triggering the collapse process for both stars. We illustrate the change in the energy during the collision for a set of sample parameters in gure 1. During the time the two ASts occupy the same volume, if one of them collapses su ciently far, it will remain gravitationally unstable even after it passes through the other. A su cient condition for this would be that its radius decreases from to 0 , identi ed as the point isoenergetic with on the left branch of the energy curve (illustrated in gure 1). The collapse time from to some other point end is [32] t = Rend = R0 ). N1=0.3 Nc N1=0.4 Nc N1=0.5 Nc N1=0.6 Nc N1=0.8 Nc N1=0.995 Nc vrel [km/sec] lines are the required times t for an ASt with N1 particles to collapse from to 0 , given that it collides with a second ASt which has N2 partices at a relative velocity vrel. The blue curve represents the approximate time tin that one ASt spends traversing the other. If tin < t (the blue shaded region), then the ASt collapses. q = for the cosine ansatz. Thus we will analyze the time t to collapse from to 0 (i.e. using Full collapse from the dilute radius to the dense radius D was shown in [32] to last a We nd that collapse from to 0 , catalyzed by collision of two nearly-critical ASts, takes nearly as long as a collapse all the way to D. This is easy to understand if one notes that the potential is shallow at the beginning of the collapse, near , and becomes steeper towards the end (see gure 1). Because collapses occur slowly, we conclude that colliding ASts only collapse if they move with a su ciently low relative velocity, so that the two ASts occupy the same volume for a long enough time interval. We investigate next how small vrel must be for collapse to be induced. time t required for a star to collapse from to 0 , de ned in eq. (3.9), at di erent values of n. We observe that vrel . 1 km is required for nearly any two ASts to collapse as a result of the collision process. For the distribution of velocities in eq. (3.3), the probability that two ASts have a relative velocity vrel . V is P (vrel . V ) = For V = 1 km/sec, this gives P (vrel < 1 km/sec) 10 8. This implies that while the collision rate in eq. (3.8) is high, even on the assumption that all ASts are very close to critical :9 Nc), the total rate of collapses induced by collisions of two ASts is very small, FAS year galaxy The rate of induced collapses is even lower if the colliding ASts are lighter. We conclude that collapses induced by collisions between two ASts very likely have a negligible e ect on the overal mass distribution of ASts, or more generally, of dark matter. We also note in passing that if ASts collide but do not collapse fully from (e.g. a scenario outside of the shaded blue region of gure 3), they will oscillate around their respective dilute minima after emerging from the collision. With no mechanism for damping, such oscillations would continue inde nitely. Collision of an ASt with an ordinary star Gravitational potential inside star ASts can also collide with ordinary stars. As an ASt passes through an ordinary star, there will be an additional contribution to its gravitational energy due to the stellar potential Vs(r). To estimate this contribution, consider rst an ASt at rest at a point concentric with that of a star. The energy functional describing the ASt, given by eq. (2.4), acquires an additional contribution from the gravitational e ects of the stellar mass, EGS (R) = m Note that we have assumed that the external gravitational source has a radius Rs larger than the radius R of the ASt; this is appropriate for the types of QCD axions we discuss here, but could change in some more generic axion theory which produces very large ASts. Assuming a constant density for the star and requiring continuity of the full gravitational potential across each boundary gives the unique result Vs(r) = <> 2 Rs3 8 G Ms 2 3 G Ms 2 Rs where Ms is the mass of the star. ASt (R) and the star (Rs) are not to scale. In reality, ASts and ordinary stars will collide with some relative velocity vrel and the gravitational potential will be changing with time. However, the change in the total energy resulting from this e ect is a function of the distance from the ASt to the center of the ordinary star (not of the variational parameter R), and so will not a ect the existence or position of an energy minimum. To see that this is so, suppose the ASt sits at a point ~a away from the center of the star, and let ~x represent a position measured with respect to EGS (R) = 2 Rs3 2 Rs3 2 Rs3 (~a + ~x)2j (~a + ~x)j2d3x 3 G m Ms Z 3 G m Ms N (a2 + x2 + 2~a ~x cos )j (~a + ~x)j2d3x 3 G m Ms N s = 2 Rs3 = m N = m N 3 G m Ms N s = where we de ned the dimensionless quantities in analogy with the radius and mass of an ASt in eq. (2.3). The constant c1 is always < 0, and very large: for the sun, Rs = R = 7 = 1735 and = 1:6 1012, and then c1 Evaluating the integral in eq. (4.3), we have 105 km and Ms = M = 2 1030 kg, which 109 regardless of the value of . 2 Rs3 2 Rs3 3 G m Ms N; (x2 + 2~a ~x cos )j (~a + ~x)j2x2 dx d(cos )d where in the third line we introduced the angle between the vectors ~a and ~x. For a spherically symmetric ASt wavefunction, (x) does not depend on , and so the second integral vanishes. The last two terms of eq. (4.3) are constants with respect to R. For notational simplicity, we denote them by EGS (R) = 3 G Ms N m 2 Rs 50 2 + 315 R2 = m N F = Combining eqs. (2.4) and (4.6), we can write the rescaled total energy Es( ) = m N The constant c1 a ects only the magnitude of Es( ), and not the existence of a minimum, i.e. it will have no e ect on our analysis of collapse in the sections below. Collision rates When an ASt collides with an ordinary star, i in eq. (3.4) refers to stellar matter. We will take the number of stars in the Milky Way to be N = 1011. Taking the sun as a \typical" star, we use the radius R = 7 105 km in the cross section; we assume for now that stars also move at a virial velocity v 300 km/s. The corresponding enhancement factor in the cross section is Plugging everything in, the constant density estimate of the collision rate using eq. (3.4) is FAS year galaxy As before, we can improve on our estimation of the rate by allowing ASts to be distributed according to the NFW pro le in eq. (3.6). For the distribution of stars, we use the phenomenological t found in [56] assuming cylindrical symmetry; the authors used a double exponential to describe the disk and a power law for the baryonic halo component. Here we neglect the halo component, which constitutes a few percent correction and is thus negligible at the level of precision of this work. The distribution is n (`; z) = nd(L ; 0) exp + f nd(L ; 0) exp the galactic center in the galactic plane, and z the height above the galactic plane. These expressions depend on the t parameters L1;2, H1;2, and f , as well as the position of the sun (L ; Z ). We use the best t values reported in [56] and ignore uncertainties; these values are reproduced in table 1. The sample in this reference contained only about 108 stars, whereas the total number in the Milky Way is understood to be closer to 1011. To remedy this, we rescale the nal parameter in the t, nd(L ; 0), by requiring the total number of stars to be n d3r = N = 1011; The collision rate of ASts with ordinary stars using eq. (3.2) is then4 = 2 FASMAS For simplicity, we will not take into account the velocity dispersion of stars and use only the virial velocity vvir. Then is it easy to compute vvir = 2:4 z symmetry, allowing us to integrate only on z 2 f0; Rvirg and multiply by 2. Performing the integrals over ` and z gives the result5 = 3000 FAS year galaxy Comparing this result to the approximation in eq. (4.8), we see that in this case, taking the ASt and stellar distributions into account increases the rate by nearly an order of magnitude. This likely re ects the fact that the distibution of stars in the Milky Way is very far from a constant density; stars are packed very densely into the stellar disk, and the density is peaked strongly at low z. Collisions occur with a much higher frequency near the galactic center, where the density of both ASts and ordinary stars is very large; this fact is not captured by the constant-density approximation. Finally, note that the estimate in eq. (4.11) is likely an underestimate of the total collision rate of ASts with ordinary stars. This is because in the estimate of the cross section, we took the sun as a \typical" star, which is likely a reasonable approximation for the O(1011) stars contained in our sample. However, many stars have radii Rs much larger than R of the sun, and the cross sections for those stars is enhanced by a factor of roughly (Rs=R ) 1. We have also ignored smaller stars, like red dwarfs, but their contribution to the collision rate is likely to be a small correction. If an ASt collides with an ordinary star, the critical particle number changes as a result of the additional gravitational interaction in eq. (4.7). Recall that outside the star, there exists a dilute state only for n < nc 12:6. This e ective critical particle number decreases in the star. For the sun, we nd that the critical particle number decreases to nc; which implies that ASts with nc; n < nc will collapse when they enter the sun. For n < nc; , the dilute state binding energy increases as the ASt moves into the star (though it is still appropriate to consider it a weakly bound state). More generally, the energy landscape in eq. (4.7) is a function of ASt particle number N , and a density parameter D Ms=Rs3 characterizing the star. In gure 5, there exists a region in which ASts colliding with ordinary stars remain stable, though with larger binding energy (shown in yellow), and one in which they collapse (red). In the red region, collapse proceeds via the mechanism outlined in [32]: an initial slow roll, followed by a quick nal collapse towards a strongly bound dense state. As the binding energy increases rapidly in the last moments, a large number of relativistic axions are emitted. If the star has a su ciently large radius, or if the relative velocity is su ciently low, then collisions occurring in the red region of gure 5 allow the ASt enough time to collapse fully before it passes fully through the star. To test whether this is plausible, we have used eq. (3.9) to calculate the time tc needed for the ASt to fully collapse. For axion stars with We compare this to the time a colliding ASt spends inside the star, tin. Averaging over the impact parameter of the collision, the average distance an ASt travels through a star is 5In this expression, we have neglected the uncertainty of Rvir, represented by additional factors of , in the integration bounds. This approximation would not be appropriate if and a collapse region with no dilute minimum (red). tin = 4 R =3 = 52 min; so tc < tin. Thus, on the assumption that the sun is a \typical" star, and that a \typical" collapse. This implies a collapse rate of6 FAS year galaxy As we pointed out in section 4.2, eq. (4.11) is likely an underestimate of the true collision rate. For the same reason, we feel our conclusion that ASts colliding with ordinary stars will collapse is robust, because a larger stellar radius implies a longer transit time, and thus a greater time allowed for collapse. Finally, ASts which do not collapse fully on a single pass through a star also can dissipate kinetic energy through interactions with stellar matter, becoming bound to the star, resulting in repeated collisions and an increased probability for collapse. Axion star collisions with neutron stars are particularly interesting, because the strong magnetic elds in the latter can induce currents in the former, leading to stimulated axion conversion to radio-frequency photon bursts that could be observed. This mechanism has been proposed as the source of observed Fast Radio Bursts (FRBs) [47{50], and recent estimates suggest that the rate of collisions of ASts with neutron stars is compatible with the observed frequency of FRBs [45, 46, 51]. In this section we repeat the analysis of previous authors using eq. (3.4), but also improve on these estimates by taking into account the NFW distribution of ASts, and a nontrivial distribution of neutron stars as well. If ASts and neutron stars were distributed with uniform density in the galaxy, the rate of collisions is that of eq. (3.4). The gravitational enhancement of the cross section is 2 G (MNS + MAS) v2 (RNS + RAS) re ecting the fact that the neutron star is very compact and massive. The resulting cross NS = 2:7 ^NS = 1:5 FAS year galaxy This result was already known by previous authors [45, 46, 51] to be of the right order of magnitude to account for the reported frequency of observed FRBs. As with other astrophysical collisions, we can improve on this estimate by taking nontrivial distributions into account. For ASts, we use (again) the NFW pro le in eq. (3.6), while for neutron stars we use the phenomenological t of [57]. The number density nNS(`; z) of neutron stars can be written in terms of two probability distributions nNS(`; z) = N p`(`) pz(`; z)=2 `, where p`(`) = A0;` + A e `= and pz(`; z) = A0;z (z :1 kpc) + A1 e z=h1(`) + A2 e z=h2(`); (x) is a Heaviside function. The scale heights h1;2(`) are de ned by h1(`) = k1 ` + b1 h2(`) = The coordinates ` and z are de ned as in section 4.2, and the remaining constants are best- t parameters from the analysis of [57], reproduced in table 1. We normalize the distribution using N by the requirement ` nNS(`; z) d` d dz = NNS = 109: We again use vvir = the total collision rate to be v, and ignore velocity dispersion so that h v vvir. We nd NS = 2 N = 4 2 N FDM dz FAS year galaxy p`(`) pz(`; z)h viNS d` NFW(p`2 + z2) p`(`) pz(`; z) It should be noted that an estimate of this kind, using the neutron star distribution found in [57], was rst performed in [44] assuming spherical symmetry, whereas we have included the z-dependence in nNS. In this work we have remained agnostic about how axion stars form, preferring to parameterize our ignorance by the parameters FAS and FDM. It is sometimes argued that all axion stars are formed in the early universe through the collapse of axion miniclusters. However, it is also possible that axion stars continually form as the dilute background of axions thermalizes and clumps together; such accretion of axions into condensates might be especially e cient if they form in the cores of ordinary stars. In this analysis, we have strived to correctly capture the relevant physics, as well as the astrophysical uncertainties regarding axion star formation. Because of these important uncertainties, we do not comment here on the interpretation of Fast Radio Bursts as originating in ASt collisions with neutron stars. We have calculated the collision rates of axion stars (ASts) with each other, with ordinary stars, and with neutron stars, in dark matter halos similar to that of the Milky Way. In each case, we have taken into account the number density distributions of each class of astrophysical bodies, and by analyzing the energy functional, determined whether the ASt could survive the collision, or if it instead collapses. The rates of these collisions are large, some as frequent as O(107) collisions/year/galaxy, but each can be larger or smaller by a few orders of magnitude, due to the uncertainty in the calculations, which is absorbed into the free parameters FDM, FAS, and , de ned in section 3.2. The collision rate of ASts with ordinary stars is of O(103) collisions/year/galaxy. Because an ordinary star is so large and so massive compared to a typical ASt, the gravitational e ect it exerts is important during a collision. In particular, stable ASts approaching ordinary stars develop deeper energy minima as they pass through. ASts which are sufciently close to their critical mass Mc collapse under the added in uence of the star's gravitational eld, as seen in gure 5. Keeping in mind the relevant uncertainties in axion star formation and astrophysical data, it is plausible that a very large number of ASt collapses are induced through collisions with ordinary stars. On the other hand, collisions between two ASts, even with masses very close to critical, are unlikely to result in collapse; this is because a very low relative velocity is required to allow enough time for the ASts to collapse while occupying the same volume. We nd that this condition is satis ed only when vrel . 1 km/s, even for the heaviest stable ASts (see gure 3). The required velocity decreases as the mass of the ASts in question decreases further from criticality. We conclude that in spite of a large total collision rate, the small number of collapses induced in this way will likely have a negligible e ect on the dark matter or ASt mass distribution. As ASts collapse, they can emit relativistic axions in the form of radiation [32, 37, 38]. This is because the binding energy of a collapsing ASt increases rapidly [35]. It is possible that all, or a large fraction of, the mass of the ASt would be converted in this way. If this is the case then the total axion star contribution to dark matter in galactic halos could be decreasing with time, as relativistic axions escape from galaxies and galaxy clusters. If this decrease is substantial, then it could constrain the hypothesis that axion stars constitute a large fraction of dark matter. This could also lead to interesting possibilities for astrophysical observations, or other cosmological consequences; we will return to these topics in a future work. We nd for both of the above types of collisions that, over the majority of parameter space, only those axion stars with masses very close to the critical value Mc can collapse through collisions. Because neither the primordial nor current mass distribution of axion stars in dark matter halos is known to a high degree of certainty, it is unclear precisely how many such ASts should be expected. That said, we nd a high rate for collapses induced by ASt collisions with ordinary stars; this makes the collapse possibility interesting. Even if only a small fraction of dark matter is in the form of near-critical ASts (i.e. if FDM is possible that their collapse and subsequent conversion to relativistic axions could lead to observable consequences. This question could be fully answered by a dedicated study of the formation mechanism and timescale of ASts, a task we plan to undertake in the near future. Using the same framework, we have also revisited previous calculations of the collision rate betwen ASts and neutron stars. Such events are of particular interest because they can result in novel detection signatures. We improve on previous estimates by including the full dependence on the equatorial radius ` and polar distance z of the neutron star distribution. The oft-quoted result of NS 10 3 collisions/year/galaxy can be recovered when using a nontrivial distribution, but only given speci c assumptions about the fraction of dark matter in axion stars. We have attempted here to parameterize the uncertainties in this calculation, in the hopes that these estimates can be improved in the future. Finally, any condensate of bosons will have a maximum mass above which it collapses. Therefore, this framework can be applied to other condensates of bosons with attractive or repulsive interactions, whether they are star-like or galaxy-sized. Acknowledgments We thank P. Argyres, J. Berger, P. Fox, R. Gass, R. Harnik, 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. J.L. thanks the MUSE program and the Department of Physics at University of Cincinnati for support and nancial assistance. The work of J.E. 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. 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Joshua Eby, Madelyn Leembruggen, Joseph Leeney, Peter Suranyi, L. C. R. Wijewardhana. Collisions of dark matter axion stars with astrophysical sources, Journal of High Energy Physics, 2017, 99, DOI: 10.1007/JHEP04(2017)099