#### 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. The SCGSR program is administered by the Oak Ridge
Institute for Science and Education for the DOE under contract number DE-SC0014664.
J.E. also thanks the Fermilab Theory Group for their hospitality.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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