#### 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. (A.2), vanishes as
! 0, the self-interaction energy
approaches a constant in the limit
Because this result is nite, we are led to conclude that the kinetic energy, which diverges
! 0. The gravitational
interaction, which diverges as
energy is always bounded from below. Though we have here only proved this for the
Gaussian ansatz, this conclusion is signi cantly more general, and will be investigated in
detail in a future work.
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.
Lett. 38 (1977) 1440 [INSPIRE].
Instantons, Phys. Rev. D 16 (1977) 1791 [INSPIRE].
Lett. 40 (1978) 279 [INSPIRE].
Harmless Axion, Phys. Lett. B 104 (1981) 199 [INSPIRE].
Sov. J. Nucl. Phys. 31 (1980) 260 [Yad. Fiz. 31 (1980) 497] [INSPIRE].
103 [INSPIRE].
72 (1999) 105 [INSPIRE].
(1983) 127 [INSPIRE].
(1983) 133 [INSPIRE].
123509 [arXiv:1603.04249] [INSPIRE].
[INSPIRE].
Rev. D 27 (1983) 995 [INSPIRE].
of coherent oscillations of a primordial axion
eld in the universe, Phys. Atom. Nucl. 59
(1996) 1005 [Yad. Fiz. 59 (1996) 1050] [INSPIRE].
distribution in standard axionic CDM and its cosmological impact, Nucl. Phys. Proc. Suppl.
[24] J. Eby, P. Suranyi, C. Vaz and L.C.R. Wijewardhana, Axion Stars in the Infrared Limit,
[25] A.H. Guth, M.P. Hertzberg and C. Prescod-Weinstein, Do Dark Matter Axions Form a
Condensate with Long-Range Correlation?, Phys. Rev. D 92 (2015) 103513
[arXiv:1412.5930] [INSPIRE].
(1980) 253 [INSPIRE].
[38] C.J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge
Rev. A 56 (1997) 1424.
[INSPIRE].
Bose-Einstein condensates: Variational solutions of the Gross-Pitaevskii equations, Phys.
Axion Stars, arXiv:1609.05182 [INSPIRE].
arXiv:1605.04361 [INSPIRE].
formation from axion stars, arXiv:1609.04724 [INSPIRE].
484 (2000) 177 [astro-ph/0003388] [INSPIRE].
[1] R.D. Peccei and H.R. Quinn , CP Conservation in the Presence of Instantons , Phys. Rev.
[2] R.D. Peccei and H.R. Quinn , Constraints Imposed by CP Conservation in the Presence of [3] S. Weinberg , A New Light Boson?, Phys. Rev. Lett . 40 ( 1978 ) 223 [INSPIRE].
[4] F. Wilczek , Problem of Strong p and t Invariance in the Presence of Instantons , Phys. Rev.
[5] M. Dine , W. Fischler and M. Srednicki , A Simple Solution to the Strong CP Problem with a [6] A.R. Zhitnitsky , On Possible Suppression of the Axion Hadron Interactions (in Russian), [7] J.E. Kim , Weak Interaction Singlet and Strong CP Invariance, Phys. Rev. Lett . 43 ( 1979 ) [15] L.F. Abbott and P. Sikivie , A Cosmological Bound on the Invisible Axion, Phys. Lett. B 120 [16] S. Davidson and T. Schwetz , Rotating Drops of Axion Dark Matter, Phys. Rev. D 93 ( 2016 ) [17] M. Dine and W. Fischler , The Not So Harmless Axion, Phys. Lett . B 120 ( 1983 ) 137 [18] R. Holman , G. Lazarides and Q. Sha , Axions and the Dark Matter of the Universe , Phys.
[8] M.A. Shifman , A.I. Vainshtein and V.I. Zakharov , Can Con nement Ensure Natural CP Invariance of Strong Interactions?, Nucl . Phys . B 166 ( 1980 ) 493 [INSPIRE].
[9] M.Y. Khlopov , B.A. Malomed and Y.B. Zeldovich , Gravitational Instability of Scalar Field and Primordial Black Holes, Mon. Not. Roy. Astron. Soc . 215 ( 1985 ) 575 [INSPIRE].
[10] Z.G. Berezhiani , A.S. Sakharov and M.Y. Khlopov , Primordial background of cosmological axions , Sov. J. Nucl. Phys . 55 ( 1992 ) 1063 [Yad . Fiz. 55 ( 1992 ) 1918] [INSPIRE].
[11] A.S. Sakharov and M.Y. Khlopov , The Nonhomogeneity problem for the primordial axion eld , Phys. Atom. Nucl . 57 ( 1994 ) 485 [Yad . Fiz. 57 ( 1994 ) 514] [INSPIRE].
[12] A.S. Sakharov , D.D. Sokolo and M.Y. Khlopov , Large scale modulation of the distribution [13] M.Y. Khlopov , A.S. Sakharov and D.D. Sokolo , The nonlinear modulation of the density [14] J. Preskill , M.B. Wise and F. Wilczek , Cosmology of the Invisible Axion, Phys. Lett. B 120 [19] P. Sikivie , Axion Cosmology , Lect. Notes Phys. 741 ( 2008 ) 19 [astro-ph/0610440] [INSPIRE].
[20] I.I. Tkachev , Coherent scalar eld oscillations forming compact astrophysical objects , Sov. Astron. Lett . 12 ( 1986 ) 305 [Pisma Astron . Zh. 12 ( 1986 ) 726] [INSPIRE].
[21] E.W. Kolb and I.I. Tkachev , Axion miniclusters and Bose stars , Phys. Rev. Lett . 71 ( 1993 ) [22] C.J. Hogan and M.J. Rees , Axion Miniclusters , Phys. Lett . B 205 ( 1988 ) 228 [INSPIRE].
[23] J. Barranco and A. Bernal , Self-gravitating system made of axions, Phys. Rev. D 83 ( 2011 ) [26] P.-H. Chavanis , Mass-radius relation of Newtonian self-gravitating Bose-Einstein condensates with short-range interactions: I. Analytical results , Phys. Rev. D 84 (2011) 043531 [27] E. Braaten , A. Mohapatra and H. Zhang , Dense Axion Stars, Phys. Rev. Lett . 117 ( 2016 ) [28] E. Braaten , A. Mohapatra and H. Zhang , Nonrelativistic E ective Field Theory for Axions , Phys. Rev . D 94 ( 2016 ) 076004 [arXiv:1604.00669] [INSPIRE].
[29] P.-H. Chavanis , Collapse of a self-gravitating Bose-Einstein condensate with attractive self-interaction , Phys. Rev. D 94 ( 2016 ) 083007 [arXiv:1604.05904] [INSPIRE].
[30] J. Fan , Ultralight Repulsive Dark Matter and BEC, Phys. Dark Univ . 14 ( 2016 ) 84 [arXiv:1603.06580] [INSPIRE].
[31] J. Eby , C. Kouvaris , N.G. Nielsen and L.C.R. Wijewardhana , Boson Stars from Self-Interacting Dark Matter , JHEP 02 ( 2016 ) 028 [arXiv:1511.04474] [INSPIRE].
[32] J. Eby , P. Suranyi and L.C.R. Wijewardhana , The Lifetime of Axion Stars, Mod. Phys. Lett. A 31 ( 2016 ) 1650090 [arXiv:1512.01709] [INSPIRE]. Rev . D 42 ( 1990 ) 3918 [INSPIRE].
[33] Y. Nambu and M. Sasaki , Quantum Treatment of Cosmological Axion Perturbations , Phys.
[34] E.J.M. Madarassy and V.T. Toth , Evolution and dynamical properties of Bose-Einstein condensate dark matter stars , Phys. Rev. D 91 ( 2015 ) 044041 [arXiv:1412.7152] [INSPIRE].
[35] G. Grilli di Cortona , E. Hardy , J. Pardo Vega and G. Villadoro , The QCD axion, precisely, JHEP 01 ( 2016 ) 034 [arXiv:1511.02867] [INSPIRE].
[36] P. Di Vecchia and G. Veneziano , Chiral Dynamics in the Large-N Limit, Nucl . Phys . B 171 [37] T. Harko , Gravitational collapse of Bose-Einstein condensate dark matter halos , Phys. Rev.
[39] V.M. Perez-Garcia , H. Michinel , J.I. Cirac , M. Lewenstein and P. Zoller , Dynamics of [40] A.L. Fetter , Rotating trapped Bose-Einstein condensates , Rev. Mod. Phys . 81 ( 2009 ) 647 [41] P.-H. Chavanis and L. Del ni, Mass-radius relation of Newtonian self-gravitating Bose-Einstein condensates with short-range interactions: II. Numerical results , Phys. Rev. D 84 ( 2011 ) 043532 [arXiv:1103. 2054 ] [INSPIRE].
[42] E. Braaten , A. Mohapatra and H. Zhang , Emission of Photons and Relativistic Axions from [43] L.N. Chang , D. Minic , C. Sun and T. Takeuchi , Observable E ects of Quantum Gravity, [44] E.A. Donley , N.R. Claussen , S.L. Cornish , J.L. Roberts , E.A. Cornell and C.E. Wieman , Dynamics of collapsing and exploding Bose-Einstein condensates , Nature 412 ( 2001 ) 295 [45] D.G. Levkov , A.G. Panin and I.I. Tkachev , Relativistic axions from collapsing Bose stars , [46] T. Helfer , D.J.E. Marsh , K. Clough , M. Fairbairn , E.A. Lim and R. Becerril , Black hole [47] A. Riotto and I. Tkachev , What if dark matter is bosonic and sel nteracting? , Phys. Lett. B