Axion stars in the infrared limit
Received: December
Axion stars in the infrared limit
Joshua Eby 0 1
Peter Suranyi 0 1
Cenalo Vaz 0 1
L.C.R. Wijewardhana 0 1
Open Access 0 1
c The Authors. 0 1
0 2600 Clifton Ave , Cincinnati, OH 45221 , U.S.A
1 Dept. of Physics, University of Cincinnati
Following Ruffini and Bonazzola, we use a quantized boson field to describe condensates of axions forming compact objects. Without substantial modifications, the method can only be applied to axions with decay constant, fa, satisfying = (fa / MP )2 1, where MP is the Planck mass. Similarly, the applicability of the RuffiniBonazzola method to axion stars also requires that the relative binding energy of axions satisfies of the star continues to rise, the mass of the star, M , attains a maximum at max ' 0.58. All stars are unstable for > max. We discuss the relationship of our results to current observational constraints on dark matter and the phenomenology of Fast Radio Bursts.
Cosmology of Theories beyond the SM; Classical Theories of Gravity

1, where Ea and ma are the energy and mass of the axion. The
equations, depending only on the parameter, =
marginal terms contribute to the equations of motion. The number of axions in the star is
1 Introduction 3 4 2.1
Axion field dynamics in the condensed state
Expectation values
Equations of motion
Expansion in the binding energy of axions
Analytic approximations to the physical parameters of axion stars
A Calculation of the quantum potential in tree approximation
Scalar fields, which give rise to spinzero quanta satisfying Bose statistics, are natural in
quantum field theories of phenomenological importance. The recently discovered Higgs
field, which generates masses for all the other particles, is the prime example. The axion,
a yet to be discovered pseudoscalar, postulated to solve the strongCP problem endemic
to QCD, is another well motivated spinzero particle. It is also generic in string theory
for 4dimensional axionlike degrees of freedom to arise as KaluzaKlein zero modes from
the compactification of antisymmetric tensor fields defined in 10 spacetime dimensions.
Quintessence, the almost massless scalar field invoked to drive the latetime acceleration of
the universe, is another hypothetical scalar degree of freedom. It would be quite interesting
to determine if such spinzero degrees of freedom could form stable compact structures due
to their selfgravitation.
Starting with the seminal works of Kaup [1] and Ruffini and Bonazzola [2], the study
of gravitationally bound bosonic degrees of freedom has received a great deal of attention.
It has been seen that complex scalar field configurations in the presence of gravity can
form stable compact objects, termed boson stars [310]. Kaup in his original work solved
the KleinGordon free field equation in an asymptotically flat spherically symmetric
spacetime background and found localized spherically symmetric field configurations which are
energy eigenstates. Adopting a slightly different approach, Ruffini and Bonazzola used
the expectation value of the energymomentum tensor operator of a second quantized
free hermitian scalar field system, evaluated in an N particle state where all the particles
occupy the lowest energy state, to be the source of gravitational interactions. Here the
basis states for second quantization were the wave functions of the KleinGordon equation.
This approach yielded an equation of motion for the wave function of the N particle state,
which represented a condensate of N bosons in a single state. They found stable localized
wave functions for the condensate, indicating the interesting possibility that lumps of
selfgravitating scalar field configurations could exist in nature.
In a recent set of publications, [1114] analyzed the selfgravitating field
theoretical system consisting of a real scalar field with an interaction potential of the form
a potential represents the interactions of the axion field at low energies, as derived using
the dilute instanton gas approximation, where fa represents the axion decay constant, and
ma the axion mass [1517]. In this paper we analyze the same system but present an
analytic expansion, which simplifies the equations of motion. We also consider a different
range of input parameters. This leads to interesting physical consequences. Without
substantial modification, our expansion method can only be applied to axions with fa
theoretic method of Ruffini and Bonazzola [2] can only be applied in this regime when
fa . We find analytic
than the central density of the star, has the advantage of being able to compare total
energies of solutions with equal numbers of axions as shown in figure 2. As expected, the
critical temperature of the condensed axion star is very large, Tc > 1010 GeV.
ma2 Ea2 / ma, is small.
Axion field dynamics in the condensed state
The dynamics of a selfgravitating, free hermitian scalar field was first analyzed in [2]. More
recently, the authors of [11, 12] used a similar procedure to describe the condensation of
interacting hermitian Bose fields. We revisit this analysis by evaluating the expectation
value of the axion potential in the N particle condensed state and simplifying the equations
of motion using the expansion method described in the previous section. As stated earlier
the range of parameters we explore differ from that of [1114].
Expectation values
with potential energy
We will see that the leadingorder deviation of the spacetime metric from flat space is
We start with the standard definition of the axion potential (2.1), but replace it by the
expectation value of a quantum potential in which the quantum field, expanded in modes
having definite radial, polar and azimuthal quantum numbers, is
where an,l,m is the creation operator for an axion with the appropriate quantum numbers.
To a very good approximation every axion in an axion star is in the ground state, so the star
is an almost perfect condensate. As we will see later, for QCD axions the critical
temperature of the condensate is 1011 GeV. Consequently, in what follows, we will use the expression
and, in appendix A, we calculate the expectation value of the axion potential exactly in
the tree approximation. Note that, to justify using the tree approximations one needs
to consider the fact that negative energy states and with them loop corrections can be
ma. We obtain in the large N limit
an effective potential that is different from the classical potential. We thus obtain the
equations of motion by taking the expectation value of the Einstein equation and the scalar
field equation. For this, we will also need the expectation values
Equations of motion
We write the spherically symmetric metric as
N R(r) ei E t.
ds2 = B(r) dt2 A(r) dr2 r2 d.
1. When = 0 then the solution of
order contributions can be safely neglected.
The expectation value of the Einstein equation and of the nonlinear equation of motion
A 1
A fa2 + m2 [1 J0(X)] ,
2 B 2 A
function X(z) = 2
N R(r) / fa, and defining
z
a0 = z
b0 =
X02 + 1 J0(X) ,
X02 1 + J0(X) ,
X00 =
+ (a0 b0) X0
2(1 + a b) X + 2 (1 + a) J1(X).
Expansion in the binding energy of axions
of the Compton wave length of the axion. We will show now that for axion stars with
equations depending on
1 2 '
p2(m E)/m
and through the combination =
1, (2.8) can be reduced to a system of
radial wave function, as
Substituting (2.9) into (2.8) the axion equation of motion takes the form
V (X) =
16
192
from each term are the engineering dimensions of the corresponding operator. All terms
of (2.10) are of dimension three, except for the irrelevant, nonrenormalizable terms, like
X5 and even higher powers of X, and of the relevant last term on the right hand side
a0(x) =
b0(x) =
Y (x)2
Y 00(x) = Y (x) x
Y 0(x) 8
relevant and marginal terms only is tantamount to taking the infrared limit of the theory.
The leading order equations for the dimensionless field Y (x), depending on the
dimensionless coordinate x are
number of axions in the system. N is the natural physical input parameter.
of (2.11) as fa / MP
1. These constraints can be combined together to give the range of validity
the system (2.11) provides a correct solution for a wide range of sizes of axion stars.
We used the shooting method to integrate (2.11) and calculate the function, Y (x).
which can be chosen as the value of Y (x) and b(x) at the center of the star. The boundary
conditions are that Y (x), a(x), and b(x) tend to zero at x
. a(x) and b(x) have
implement in the numerical calculations. However, notice that (2.11) implies
a(x) + b(x) = 2 x
1 Z
has similar behavior. Such a boundary condition can be imposed easily in a numerical
example, in figure 1 we plot our solution for the wave function, Y (x), as a function of x
shape. The initial values required for attaining the necessary asymptotic behavior are
Analytic approximations to the physical parameters of axion stars
The most important parameters describing axion stars are the total mass, M , the radius
inside which 99% of the matter contained in the star is concentrated, R99, and the number
of axions in the star, N . The mass of the star is given in leading order of the infrared limit
M = 3
V ().
V (y) =
V ()
R99 =
x99 =
R99 =
V ()
Furthermore, using the standard definition for boson stars, we define x99 as
Then restoring the appropriate scale, the radius of the axion star becomes
Combining (3.1) and (3.4) we obtain the relationship between its radius and its mass.
2 103
where RS is the Schwarzschild radius. This relationship, which implies gravitational
sta
1 / 500. As shown by (3.6), our formalism could even
be applied to axions with fa 1015
1016 GeV, because for axions with decay constant
1 a linear fit gives an excellent representation of R99,
are represented by the solid (dashed) line. The dotted line connects sample states with identical
numbers of axions. The mass difference of these stars is a small fraction of the mass.
106
105.
M =
V () ' 15
We plotted the dependence of the mass, scaled by the factor fa ma / MP , as a function
axion wave function have already been found in previous work [3, 612], but the maximum
in figure 2 has a physical significance due to the following considerations.2 Note that the
number of axions in the condensate is approximately equal to
2Seidel and Suen [5] have discussed the question of instability in a complex scalar field model of boson
N =
maximal number of axions in an axion star is
relationship between the masses is illustrated by the dotted line in figure 2. As a result,
dashed line, is unstable. In fact, using (3.12) it is easy to estimate the mass difference
between the two states of the axion star containing N axions. We obtain
macroscopic energy.
present paper we do not attempt to calculate the decay rate. That will be the subject of
In table 1 we provide the mass, mass difference between stars containing the same
101
where our approximations are valid.) For the input values of fa and ma we use QCD
of the choice of fa.
A final comment concerns the critical temperature of the axion condensate. To see that
the axion star is a pure condensate, with very little contribution from excited states one
3 M / (4 R939 ma), and the expression for the radius in (3.6) we obtain the following esti
mate for the critical temperature (neglecting interactions)
Tc =
As it has been pointed out in the previous section, to describe compact objects formed from
axions one needs to consider that, except possibly in the early universe, the temperature
of the object is many orders magnitudes smaller than the critical temperature of boson
condensation. In other words, in a good approximation one may assume that all the axions
are in the ground state. Therefore, along with Ruffini and Bonazzola [2] we substitute the
wave function of the axions by a quantized field. As the resulting complicated nonlinear
field theory cannot be solved analytically, one is restricted to use perturbation theory. A
similar approach was used by Barranco and Bernal [11, 12] in their application to axion
The most important observations of this paper are that the application of the method
of [2] to axion stars must be restricted in two different ways: (i) the decay constant of
the axions, fa must be much smaller than the Planck mass, or using the notations of this
1; (ii) only weakly bound axions should be considered, i.e. the
axion binding energy should satisfy ma E
ma. The reason for requiring small binding
energy is a requirement of using the RuffiniBonazzola method [2] for an interacting field
theory: if the binding energy is of O(ma) then the perturbation expansion of the axion field
theory breaks down. Pairs of positive and negative energy states contribute to the ground
state and to the expectation values of physical quantities, as well. To extend calculations
theory, which is beyond the scope of this paper.
1 requires using nonperturbative field
1 has allowed us to expand the equations of motion in the scale
transcendental axion potential do not contribute.
We have calculated the masses, radii and densities of axion stars as a function of the
1. We found that the radius of the star increases approximately linearly
independent of fa, while the mass is an increasing function of fa. The maximal number of
In the future we will also investigate rotating axion stars and analyze the maximal
mass and the question of stability as a function of angular momentum. We will investigate
collisions of axion stars in view of the existence of the maximal mass. Furthermore if an
stable state. We will compute rate of transition, and possible signatures of their decay.
The masses of compact axion star solutions found in our work are consistent with the
mass bounds derived by Tkachev for condensate formation through gravity [20]. Axion
stars, along with free axions may form all or part of dark matter. Various production
mechanisms of axions in the early universe are discussed in [2123]. Bounds on the axion
decay constant resulting from astrophysical and cosmological observations are discussed
in [21] and [2426]. We will investigate how our conclusions change if axion stars are in
equilibrium within a cloud of free axions after reaching the maximal mass. We will also
investigate the consequence if axions come in a multitude of flavors, as expected in theories
derived from compact extra dimensions.
The collisions of axion condensates with neutron stars have been studied in the past [13,
14, 28]. Recently axion stars have been proposed as progenitors for Fast Radio Bursts
(FRBs) [2932] via collisions with neutron star atmospheres [3335]. Whatever produces
FRBs should be able to generate large amounts of energy, on the order of approximately
1042 ergs/s, in a fairly tight frequency range around 1.4 GHz over time scales on the order
of milliseconds. Assuming that all of the mass of an axion star is converted into radiation
during a putative collision with a neutron star, one finds that an axion star mass on the
order of 1012 M
would be required. This mass is indeed compatible with our results
listed in table 1. However, as the radius of our boson star is at least an order of magnitude
larger that that of a neutron star we feel that the possibility of converting all the axions to
radiation in such a collision is highly unlikely. Therefore we plan to perform an accurate
estimation of this conversion rate in the near future.
The radius of the axion stars may be estimated from the duration of the bursts and
was found in [2932] to be on the order of 100 km, which is also compatible with our
results in table 1. Moreover, [2932] also showed that the frequency of the radiation can be
in greater detail and will report on the results of our investigation elsewhere.
3Note that Rmax is the radius of the heaviest axion star but not the axion star of largest radius. The
radii of unstable axion stars are always larger than Rmax.
As mentioned, one result of our calculations is that for a fixed number of axions
consequence is the possibility of tunneling from the state with a higher mass to the one
with a lower mass. We will investigate whether collisions with neutron stars could induce
such a tunneling process resulting in a fainter companion burst of approximately 1029 ergs,
detected, such a companion burst could serve to distinguish between axion star progenitors
of FRBs and other proposals [3638]. However, considering the faintness of the companion
burst, it is unlikely to be observed from events that occur outside our galaxy. The observed
this phenomenon even more difficult to observe.
Finally, we would like to emphasize again that there is no fundamental reason to
limit calculations to theories with decay constants satisfying fa
MP and stars with
sufficiently small binding energy. We just state that the method of taking the expectation
value of the equations of motion in free particle states, which are defined in flat space is
not permissible if the axion decay constant is comparable with the Planck mass and/or the
relative binding energy is not much smaller than 1. Using the current formalism to extend
to false results. Such an extension would require using exact solutions of an interacting
field theory, possibly in curved space, a calculation, which is beyond the scope of our
current analysis. Restricting ourselves to leading order contributions has the added benefit
of simplifying results to the extent of being able to obtain a clearer physical interpretation
of the properties of axion stars.
We thank P. Argyres, P.Esposito, A. Kagan, C. Kouvaris, and J. Zupan for discussions.
Calculation of the quantum potential in tree approximation
X Rn,l(r) [Ylm(, ) an,l,m + Ylm ?(, ) an,l,m],
where an,l,m is the annihilation operator of a state with radial quantum number n and
with the rest of the commutators vanishing.
particle condensate can be calculated exactly, without resorting to the Taylor expansion.
(k!)2 (N k)!
S =
at fixed R / fa, the sum is dominated by k
multiplier on the right hand side of (A.5) tends to N k giving our final result
N . Then the last
The BakerCampbellHausdorff lemma implies that, if [X +[ +
cos(X + + X ) = e 21 [X + ,X ] ei X + ei X + c.c.
If we introduce the rescaled field, X(z) = 2
N R / fa the exponent S acquires a factor N
in the denominator, while the N dependence cancels in all other terms of the equations
of motion. Thus, the in the limit of N
we can set S 0 leaving the tree level
different for the two potentials. For small X (i.e. for large distances), where quadratic
contributions dominate, Vc(X) / Vq(X) ' 2.
approximation one should evaluate
When one calculates the expectation value of the scalar equation in the semiclassical
Taking the expectation value of the operator of (A.3) between N particle condensates
using (A.1) one obtains4
hN  cos[(+ + ) / fa]N i = e 2 f1a2 [+ ,]hN ei + / fa ei / fa N i + c.c
Omitting loop corrections and using free particle states, the right hand side of (A.4) can
be readily calculated after expanding the two exponentials into power series of the creation
and annihilation operators. Owing to the fact that in the condensate only ground state
particles can be annihilated in the expectation values all contributions containing operators
of the excited states vanish on the right hand side of (A.4). It follows that
with V 0(X) = J1(X).
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