11institutetext: Department of Physics and Astronomy, Iowa State University,
Ames IA 50011

# Electromagnetic radiation from axion condensates in a time dependent
magnetic field

Srimoyee Sen<EMAIL_ADDRESS>1 Lars Sivertsen<EMAIL_ADDRESS>
###### Abstract

Time dependent magnetic fields can be sourced by spinning neutron stars,
orbiting binaries and merging neutron stars. We consider electromagnetic
radiation from axion condensates in the background of an alternating magnetic
field. We find that a resonant peak in radiation can occur when the frequency
of the alternating magnetic field is comparable with the axion mass scale.
More interestingly, in situations where the frequency of the alternating
magnetic field itself changes with time, as can be the case in binary mergers
due to steady increase in orbital frequency, the resonant peak in radiation
may occur for a range of axion mass scales scanned by the time-varying
magnetic field frequency.

subsection

##  1 Introduction

Axions were originally postulated to solve the strong CP problem
PhysRevLett.38.1440 ; PhysRevLett.40.223  and are considered to be one of the
prime dark matter candidates Preskill:1982cy ; Abbott:1982af ; Kim:1986ax ;
Cheng:1987gp ; Raffelt:1990yz ; Duffy:2009ig . Owing to their Bose statistics,
they can form coherently oscillating Bose-Einstein condensates (BECs)
Visinelli:2017ooc ; Kolb:1993hw ; PhysRevLett.117.121801 ; Eby:2015hyx ;
Eby:2016cnq ; PhysRevLett.118.011301 ; Helfer:2016ljl ; Braaten:2016dlp ;
Bai:2016wpg ; Eby:2017xaw ; Schiappacasse:2017ham . In theories with non-
negligible axion-photon coupling these condensates can decay to photons via
stimulated or spontaneous emission Ikeda:2018nhb ; Sen:2018cjt ;
Boskovic:2018lkj ; Boskovic:2019qao ; Rosa:2017ury ; Hertzberg:2018zte . They
can also decay by emitting electromagnetic radiation in the presence of an
external magnetic field. The rate of depletion of the axion condensate in this
case depends on the strength of the magnetic field, the frequency of the
oscillation of the axion BEC and the axion photon coupling. There have been
some recent studies of electromagnetic radiation from axion condensates in a
static background magnetic field Amin:2021tnq ; Dietrich:2018jov . In this
paper we take steps to understand how a time-varying magnetic field affects
this radiation. More specifically we consider periodically varying magnetic
fields (alternating magnetic fields) and investigate how the frequency of this
periodic variation affects electromagnetic radiation.

It is clear that if the frequency of the alternating external magnetic field
is much smaller compared to the axion mass scale, then it is a good
approximation to treat the magnetic field as static. However when the two
scales are comparable we can expect the magnetic field frequency to leave its
imprints on the radiation from the condensate. Some astrophysical scenarios
where we find time varying magnetic field include spinning or orbiting stars,
in particular neutron stars due to their strong magnetic fields. The magnetic
field of a neutron star is typically dipolar and the axis of rotation of the
star is often misaligned with respect to the axis of the dipole. As a result,
a spinning neutron star can give rise to a rotating magnetic field which can
be described as an alternating magnetic field with a frequency set by the spin
rate of the star. Similarly, orbital motion of strongly magnetized neutron
stars can also give rise to time varying magnetic fields in its environment.
Note that periodic variation is not the most general form of time dependence
found in neutron star environments. In fact, neutron star mergers are
speculated to give rise to exponentially growing magnetic fields, growing from
$10^{12}$ Gauss to $10^{18}$ Gauss in a matter of a few milliseconds
Kiuchi:2015sga ; Skoutnev:2021chg . We do not analyze such rapidly growing
magnetic fields in this paper and leave it for future work.

In our analysis we assume the axion photon coupling to be weak which allows us
to compute electromagnetic radiation from axion condensates perturbatively. We
ignore the effect of axion back reaction on the condensate. This is justified
so long as the back reaction time scale is larger compared to the time period
of outgoing electromagnetic radiation. Along with an external alternating
magnetic field we also take into account the effect of a background plasma on
the radiation.

Note that electromagnetic radiation from an axion condensate in the background
of a spatially constant and static electromagnetic field was considered in ref
Amin:2021tnq . It was found that the radiated power is highly sensitive to the
size of the axion condensate. More precisely, for an axion mass of $m_{a}$,
the radiation from an axion condensate of size $R\gg\frac{1}{m_{a}}$ was found
to be exponentially suppressed. For $m_{a}R\sim 1$ the condensates radiated
efficiently. One interesting feature of this analysis was that the presence of
a background plasma was found to increase electromagnetic radiation from large
condensates with $R\gg\frac{1}{m_{a}}$. In fact, a resonant enhancement of the
radiated power was obtained when the plasma frequency was very close to the
axion mass scale. Our analysis in this paper demonstrates that a similar
enhancement in radiated power can occur when the background magnetic field has
a periodic variation in time irrespective of whether there exists a background
plasma. In order for this resonance to take place, the frequency of the
alternating field must be close to the axion mass scale. If we now introduce a
background plasma with a plasma frequency of the order of the frequency of the
alternating magnetic field, then the axion mass scale at which a resonant
enhancement takes place depends on both the plasma frequency and the
alternating magnetic field frequency. In this case, all three scales in the
problem, namely the axion mass, the plasma frequency and the frequency of the
alternating magnetic field are of the same order. In general , there is no
reason for plasma frequency of a medium and the frequency of the alternating
magnetic field to coincide. However, such a coincidence can be found if one
compares the plasma frequency of the interstellar medium with the spin
frequency of a pulsar. The interstellar medium has a plasma frequency of about
a few kHz which is of the order of the spin frequency of fast spinning
pulsars. Thus a condensate of ultralight axions of mass $10^{-12}$ eV
(frequency $\sim$kHz) submerged in the interstellar medium (ISM) can emit
significant electromagnetic radiation when it comes in contact with the
alternating magnetic field of a pulsar. This serves as one of the motivations
for taking into account the effects of both an alternating magnetic field and
a plasma frequency on electromagnetic radiation from axion condensates.

Note that, although resonant enhancement of radiation can take place in the
absence of an alternating magnetic field when axion mass is of the order of
the plasma mass of the medium, to realize this enhancement, the axion mass has
to be very close to the plasma mass for large axion condensates. E.g. for an
axion condensate of size $R\sim 100m_{a}^{-1}$, the plasma mass must be within
a few percent of the axion mass. This precise coincidence may not in general
be realized even when the order of magnitude of the axion mass is the same as
that of the plasma mass. However, when an alternating magnetic field is
introduced, the resonant axion mass not only depends on the plasma mass, but
also on the frequency of the magnetic field which itself can vary with time.
111For example, in a binary neutron star merger, the orbital frequency of the
merging objects can increase. Similarly, accretion can increase the rate of
spin of a star contributing to an increasing frequency of rotation for the
magnetic field. As a result, a changing frequency of the magnetic field
changes the resonance condition: the axion mass scale which can experience
resonant enhancement in radiation changes with time. Therefore, even if the
axion mass does not exactly coincide with the resonant mass at a certain
instant in time, it can do so at a later instant due to the time variation of
the resonant mass. Thus, condensates of a range of mass scales can decay
through radiation in the presence of an alternating background magnetic field
of time dependent frequency.

Motivated by the above considerations, we take the first step towards
understanding the effect of alternating magnetic fields on electromagnetic
radiation from axion condensate in this paper. Note that the spatial
dependence of magnetic fields in astrophysical environment can be complicated
in general. We do not take into account such complexities here. Instead we
model the external field as a spatially constant background magnetic field
which alternates in time. We however do take into account the finite extent of
the axion condensate. We leave the analysis of a spatially inhomogeneous
magnetic field for future work.

The organization of the paper is as follows: We begin with a review of
electromagnetic radiation from axion condensates in the presence of a
spatially constant time independent magnetic field. We then discuss the
resonant enhancement in radiation when the external magnetic field has a
periodic time variation. This analysis is followed by a discussion of how a
background plasma and an alternating magnetic field together affect the
resonance in radiation. In the next part of the paper we introduce an external
alternating magnetic field with time varying frequency and demonstrate how a
large axion condensate which was not radiating at a certain instant in time
begins to radiate efficiently at a later instant. We conclude with a section
on how resonant enhancement in radiation affects axion decay (back reaction)
time scale.

##  2 Electromagnetic radiation from axion consates in a background magnetic
field

The axion-photon Lagrangian of interest to us is given by

$\mathcal{L}=-\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+J_{m}^{\mu}A_{\mu}+\frac{C\beta}{4\pi
f_{a}}\phi\epsilon^{\mu\nu\lambda\rho}F_{\mu\nu}F_{\lambda\rho}+\frac{1}{2}(\partial_{\mu}\phi)(\partial^{\mu}\phi)-\frac{1}{2}m_{a}^{2}\phi^{2}-V(\phi)+\cdots.$
(1)

Here $\phi$ is the pseudo-scalar axion field, $m_{a}$ is the mass of the
axion, $V(\phi)$ includes self interaction of the axion field and $\beta$ is
the fine-structure constant. In the axion-photon interaction term, $C$ is a
model dependent number and $f_{a}$ is the axion decay constant. We have also
included in the Lagrangian a coupling between the electromagnetic field
$A_{\mu}$ and matter fields via a current density $J_{m}^{\mu}$. This operator
will describe the physics of any background plasma around the axion condensate
as well as any matter source generating external electromagnetic fields. In
the absence of any electromagnetic coupling ($\beta=0$), local gravitational
interactions combined with axion self interaction $V(\phi)$ can give rise to a
stable axion BEC: a coherently oscillating axion condensate with frequency
$\sim m_{a}$ Schiappacasse:2017ham . These axion BECs have fixed particle
numbers. When electromagnetic interactions are turned on, particle number can
change and it may be possible for the axion BEC to decay to photons. In the
absence of any external electromagnetic field there are two mechanisms via
which axions can decay to photons: spontaneous emission and stimulated
emission Sen:2018cjt . The spontaneous emission rate for axions go as
$\frac{m_{a}^{3}}{f_{a}^{2}}$ and for light axions this rate is minuscule. For
example, for a QCD axion ($\Lambda_{\text{QCD}}\sim 100$ MeV) of mass
$10^{-12}$ eV, which corresponds to a frequency scale of about $1$ kHz, the
spontaneous emission rate is about $10^{-88}s^{-1}$ and the corresponding
decay time scale is much larger than the age of the universe making this
process irrelevant for time scales of interest to us. The rate of stimulated
emission for a homogeneous condensate on the other is given by
$\frac{C\beta|\tilde{\phi}|}{\pi f_{a}}m_{a}$ where $|\tilde{\phi}|$ is the
amplitude for the oscillation of the axion field. This emission rate can be
much larger than the rate of spontaneous emission. For example, with
$\frac{C\beta|\tilde{\phi}|}{\pi f_{a}}\sim 1$, for a homogeneous QCD axion
condensate of axion mass $m_{a}\sim 10^{-12}$eV, the decay time scale is of
the order of $10^{-3}$s. Note however, for a BEC of a finite size, the rate of
decay can be very sensitive to the size of the condensate. It was shown in
Hertzberg:2018zte  that stimulated emission can only take place for
$\frac{C\beta|\tilde{\phi}|m_{a}}{\pi f_{a}}>\frac{1}{R}$ where $R$ was the
size of the condensate. We can therefore imagine coherently oscillating axion
condensates having a long life-time despite their coupling to electromagnetic
fields, as long as there are no external electromagnetic fields present and
the axion-photon coupling is not too strong.

With this we can now proceed to understand the response of axion BECs to
external electromagnetic fields. Since we are specifically interested in
external magnetic fields we will set external electric fields to zero. Our
analysis can however be extended to include external electric fields in a
rather straightforward manner.

###  2.1 Radiation in a constant magnetic field

To understand the response of axion BECs to electromagnetic fields we first
write down the axion-photon equation of motion

$\displaystyle\nabla\times\mathbf{B}(\mathbf{x},t)-\partial_{t}\mathbf{E}(\mathbf{x},t)-\mathbf{J}_{m}(\mathbf{x},t)$
$\displaystyle=-\frac{C\beta}{\pi
f_{a}}\Big{[}\big{(}\partial_{t}\phi(\mathbf{x},t)\big{)}\mathbf{B}(\mathbf{x},t)+\nabla\phi(\mathbf{x},t)\times\mathbf{E}(\mathbf{x},t)\Big{]},$
(2) $\displaystyle\nabla\times\mathbf{E}(\mathbf{x},t)$
$\displaystyle=-\partial_{t}\mathbf{B}(\mathbf{x},t),$ (3)
$\displaystyle\nabla\cdot\mathbf{E}(\mathbf{x},t)$
$\displaystyle=\rho_{m}(\mathbf{x},t)+\frac{C\beta}{\pi
f_{a}}\nabla\phi(\mathbf{x},t)\cdot\mathbf{B}(\mathbf{x},t),$ (4)
$\displaystyle\nabla\cdot\mathbf{B}(\mathbf{x},t)$ $\displaystyle=0,$ (5)

where $\mathbf{J}_{m}$ and $\rho_{m}$ are the current and charge density
corresponding to some matter fields interacting with the electromagnetic
field. We can divide up the matter four-current into two contributions,
$J_{m}^{\mu}=J_{s}^{\mu}+J_{p}^{\mu}$ where $J_{s}^{\mu}$ will describe any
matter current sourcing background electromagnetic fields and $J_{p}^{\mu}$
will describe the physics of any plasma medium if present. In the absence of
an electromagnetic interaction, coherently oscillating axion BECs of frequency
very close to the axion mass can form owing to their gravitational interaction
and the axion self interaction $V(\phi)$ Schiappacasse:2017ham . These
interactions also determine the spatial profile for these axion BECs. We don’t
discuss the details of the derivation of these spatial profiles and the
stability criterion of these BECs in this paper. We will instead use the
profile of the axion condensate found in Schiappacasse:2017ham  to understand
the response of axions to background magnetic fields. We focus on spherically
symmetric axion clumps in this paper which can be described with the following
ansatz

$\displaystyle\phi(\mathbf{x},t)=\tilde{\phi}\cos(\mu
t)\operatorname{sech}\Big{[}\frac{\lvert\mathbf{x}\rvert}{R}\Big{]},$ (6)

where $R$ is the radius of the axion condensate and $\mu$ is the oscillation
frequency of the axion condensate with $\mu\sim m_{a}$. With this ansatz we
can now proceed to analyze the physics of a coherently oscillating axion
condensate in an external magnetic field. We will first review the radiation
from axion condensates in the background of a spatially constant time
independent magnetic field which will closely follow the analysis in
Amin:2021tnq . After this discussion we will move on to discussing alternating
time dependent magnetic fields.

In the presence of an external field, Maxwell’s equation as written in Eq. 5
will effectively include an oscillating current source due to the coupling of
axion condensate with the background field. This alternating current source in
turn produces electromagnetic radiation which can take away energy from the
axion condensate. To elaborate on this, let us write the electromagnetic
fields as $\bf{E}=\bf{E}_{0}+\bf{E}_{r}$ and $\bf{B}=\bf{B}_{0}+\bf{B}_{r}$.
Here $\bf{E}_{0}$ and $\bf{B}_{0}$ correspond to external and $\bf{E}_{r}$ and
$\bf{B}_{r}$ to radiated electromagnetic fields. Writing the matter current as
$J^{\mu}_{m}=J_{p}^{\mu}+J_{s}^{\mu}$ we can re-express Maxwell’s equations as

$\displaystyle\nabla\times\mathbf{B}_{0}(\mathbf{x},t)-\partial_{t}\mathbf{E}_{0}(\mathbf{x},t)-\mathbf{J}_{s}(\mathbf{x},t)=0,$
$\displaystyle\nabla\times\mathbf{E}_{0}(\mathbf{x},t)=-\partial_{t}\mathbf{B}_{0}(\mathbf{x},t),$
$\displaystyle\nabla\cdot\mathbf{E}_{0}(\mathbf{x},t)=\rho_{s}(\mathbf{x},t),$
$\displaystyle\nabla\cdot\mathbf{B}_{0}(\mathbf{x},t)=0,$ (7)

and

$\displaystyle\nabla\times\mathbf{B}_{r}(\mathbf{x},t)-\partial_{t}\mathbf{E}_{r}(\mathbf{x},t)-\mathbf{J}_{p}(\mathbf{x},t)$
$\displaystyle=-\frac{C\beta}{\pi
f_{a}}\Big{[}\big{(}\partial_{t}\phi(\mathbf{x},t)\big{)}\mathbf{B}_{0}(\mathbf{x},t)+\nabla\phi(\mathbf{x},t)\times\mathbf{E}_{0}(\mathbf{x},t)\Big{]},$
$\displaystyle\nabla\times\mathbf{E}_{r}(\mathbf{x},t)$
$\displaystyle=-\partial_{t}\mathbf{B}_{r}(\mathbf{x},t),$
$\displaystyle\nabla\cdot\mathbf{E}_{r}(\mathbf{x},t)$
$\displaystyle=\rho_{p}(\mathbf{x},t)+\frac{C\beta}{\pi
f_{a}}\nabla\phi(\mathbf{x},t)\cdot\mathbf{B}_{0}(\mathbf{x},t),$
$\displaystyle\nabla\cdot\mathbf{B}_{r}(\mathbf{x},t)$ $\displaystyle=0.$ (8)

Eq. 7 describes the the external electric and magnetic field sourced by some
matter current $J^{\mu}_{s}$, the details of which are uninteresting for the
problem at hand. Eq. 8 describes the radiated electromagnetic fields and how
this radiation depends on the background electromagnetic fields $\bf{E}_{0}$
and $\bf{B}_{0}$, the axion condensate and the matter current of the medium
$J_{p}^{\mu}$. In writing Eq. 8 we have assumed that the radiated
electromagnetic fields are small compared to the background electric and
magnetic fields, i.e. $\bf{E}_{r},\bf{B}_{r}\ll\bf{E}_{0},\bf{B}_{0}$. This is
justified as long as the axion-photon coupling is small so that a weak
coupling analysis holds. As is clear from Eq. 8, the axion couplings to
external electric and magnetic fields act like an oscillating current source
for the radiated fields $\bf{E}_{r}$ and $\bf{B}_{r}$. If we set the matter
current of the medium $J_{p}^{\mu}$ and the external electric field
$\bf{E}_{0}$ to zero, we can solve Eq. 8 with a current and charge density

$\displaystyle\mathbf{J}(\mathbf{x},t)$ $\displaystyle=$
$\displaystyle-\frac{C\beta}{\pi
f_{a}}\Big{[}\big{(}\partial_{t}\phi(\mathbf{x},t)\big{)}\mathbf{B}_{0}(\mathbf{x},t)\Big{]}\sim\frac{C\beta}{\pi
f_{a}}m_{a}\tilde{\phi}\sin(m_{a}t)\operatorname{sech}\Big{[}\frac{\lvert\mathbf{x}\rvert}{R}\Big{]}B_{0}\hat{\mathbf{x}}_{z},$
$\displaystyle\rho(\mathbf{x},t)$ $\displaystyle=$
$\displaystyle\frac{C\beta}{\pi
f_{a}}\nabla\phi(\mathbf{x},t)\cdot\mathbf{B}_{0}(\mathbf{x},t)\sim-\frac{C\beta}{\pi
f_{a}}\tilde{\phi}\cos(m_{a}t)\frac{\operatorname{sech}\big{[}\frac{\lvert\mathbf{x}\rvert}{R}\big{]}\tanh\big{[}\frac{\lvert\mathbf{x}\rvert}{R}\big{]}}{R}B_{0}\,\,\hat{\mathbf{x}}\cdot\hat{\mathbf{x}}_{z},$

where we have substituted $\mathbf{B}_{0}(x,t)$ with a constant magnetic field
in the $z$ direction $\mathbf{B}(\mathbf{x},t)=B_{0}\hat{\mathbf{x}}_{z}$.

Using the standard retarded Green’s function

$\displaystyle
G(x,t;x^{\prime}t^{\prime})=-\frac{\delta(t-t^{\prime}-|\mathbf{x}-\mathbf{x}^{\prime}|)}{4\pi|\mathbf{x}-\mathbf{x}^{\prime}|},$
(10)

we can now obtain the radiated electric and magnetic fields in the radiation
zone (at a distance much larger than the size of the axion condensate)

$\displaystyle\mathbf{E}_{\text{r}}(\mathbf{x},t)$
$\displaystyle=-\partial_{t}\mathbf{A}(\mathbf{x},t)-\nabla
A^{0}(\mathbf{x},t)$ $\displaystyle\approx\frac{C\beta}{\pi
f_{a}}\frac{\tilde{\phi}B_{0}m_{a}\pi^{2}R^{2}}{4\lvert\mathbf{x}\rvert}\bigg{[}\frac{\tanh(\pi
k_{m_{a}}R/2)}{\cosh(\pi
k_{m_{a}}R/2)}\cos(m_{a}t-k_{m_{a}}\lvert\mathbf{x}\rvert)\bigg{]}\hat{\mathbf{x}}_{z}$
$\displaystyle+\bigg{(}\frac{C\beta}{\pi
f_{a}}\bigg{)}\frac{\tilde{\phi}B_{0}k_{m_{a}}\pi^{2}R^{2}}{4\lvert\mathbf{x}\rvert}\bigg{[}\frac{\tanh(\pi
k_{m_{a}}R/2)}{\cosh(\pi
k_{m_{a}}R/2)}\cos(m_{a}t-k_{m_{a}}\lvert\mathbf{x}\rvert)\bigg{]}\hat{\mathbf{x}},$
$\displaystyle\mathbf{B}_{\text{r}}(\mathbf{x},t)$
$\displaystyle=\nabla\times\mathbf{A}(\mathbf{x},t)$
$\displaystyle\approx\frac{C\beta}{\pi
f_{a}}\frac{\tilde{\phi}B_{0}m_{a}\pi^{2}R^{2}}{4\lvert\mathbf{x}\rvert}\bigg{[}\frac{\tanh(\pi
k_{m_{a}}R/2)}{\cosh(\pi k_{m_{a}}R/2)}\cos(m_{a}t-\lvert\mathbf{x}\rvert
k_{m_{a}})\bigg{]}(\hat{\mathbf{x}}\times\hat{\mathbf{x}}_{z}),$ (11)

where $k_{m_{a}}=m_{a}$. The corresponding time-averaged radiated power is
given by

$\displaystyle P=\left(\frac{C\beta}{\pi
f_{a}}\right)^{2}\frac{\tilde{\phi}^{2}B_{0}^{2}m_{a}^{2}R^{4}\pi^{5}}{12}\left(\frac{\tanh(\pi
k_{m_{a}}R/2)}{\cosh(\pi k_{m_{a}}R/2)}\right)^{2}.$ (12)

Note that even though $\mathbf{E}_{r}=\partial_{t}\mathbf{A}-\nabla A_{0}$,
$\nabla A_{0}$ does not contribute to the outgoing radiated power. Thus, if we
define $\bf{E}_{r}^{\prime}=\mathbf{E}_{r}+\nabla A_{0}$ and
$\mathbf{B}_{r}^{\prime}=\mathbf{B}_{r}$, the outgoing radiated power can be
re-expressed as $|\mathbf{x}|^{2}\int
d\omega\,\hat{\mathbf{x}}\cdot\left(\bf{E}_{r}^{\prime}\times\mathbf{B}_{r}^{\prime}\right)$
where $\omega$ denotes the solid angle. It is clear from this expression that
the power radiated from axion condensates peaks when the radius of the clump
is of the order of the inverse axion mass. For condensates that are much
larger than the inverse mass scale of the axion, radiation is exponentially
suppressed. It was shown in Amin:2021tnq  that upon introduction of a
background plasma, this exponential suppression can be eliminated when the
plasma frequency is very close to the axion mass scale. In that case, it may
be possible for axion condensates of sizes much larger than the inverse axion
mass scales to radiate efficiently. In the next section we will see that a
similar enhancement can be achieved with an alternating background magnetic
field even in the absence of a plasma.

###  2.2 Radiation in an alternating magnetic field

We will begin with an external background field which is alternating in time,
but is spatially constant. We plan to relax this restriction in future work.
Since the purpose of this analysis is to understand the response of axion
condensates to the time-dependence of the external field specifically, we
refrain from introducing more complex spatial dependence in the background
field profile. In order to proceed, in Eq. 8 we assume that the contribution
to the current density from the coupling between any external electric field
and the axion condensate is much smaller than the contribution to the current
density from the magnetic field coupling to axions. Thus we work in the limit

$\displaystyle\nabla\phi\times\bf{E}_{0}\ll(\partial_{t}\phi)\mathbf{B}_{0}.$
(13)

The magnetic field is chosen to be of the form

$\displaystyle\mathbf{B}_{0}=B_{0}\cos(\Omega t+\gamma)\hat{\mathbf{x}}_{z},$
(14)

where $\gamma$ is some arbitrary phase shift. We also include in the axion
ansatz an arbitrary phase shift of the form

$\displaystyle\phi=\tilde{\phi}\cos(m_{a}t+\alpha)\operatorname{sech}\Big{[}\frac{\lvert\mathbf{x}\rvert}{R}\Big{]},$
(15)

in order to explore how a phase difference between the axion field and the
external magnetic field affects outgoing radiation. The corresponding current
density in Maxwell’s equation can be written as

$\displaystyle\mathbf{J}(\mathbf{x},t)$ $\displaystyle=$
$\displaystyle-\frac{C\beta}{\pi
f_{a}}\Big{[}\big{(}\partial_{t}\phi(\mathbf{x},t)\big{)}\mathbf{B}_{0}(\mathbf{x},t)\Big{]}$
$\displaystyle\sim$ $\displaystyle\left(\frac{C\beta}{\pi
f_{a}}m_{a}\tilde{\phi}\right)\left(\frac{\sin((m_{a}+\Omega)t+\alpha+\gamma)+\sin((m_{a}-\Omega)t+\alpha-\gamma)}{2}\right)\operatorname{sech}\Big{[}\frac{\lvert\mathbf{x}\rvert}{R}\Big{]}B_{0}\hat{\mathbf{x}}_{z}$
$\displaystyle\rho(\mathbf{x},t)$ $\displaystyle=$
$\displaystyle\frac{C\beta}{\pi
f_{a}}\nabla\phi(\mathbf{x},t)\cdot\mathbf{B}_{0}(\mathbf{x},t)$
$\displaystyle\sim$ $\displaystyle-\left(\frac{C\beta}{\pi
f_{a}}\tilde{\phi}\right)\left(\frac{\cos((m_{a}+\Omega)t+\alpha+\gamma)+\cos((m_{a}-\Omega)t+\alpha-\gamma)}{2}\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad\times\left(\frac{\operatorname{sech}[\frac{\lvert\mathbf{x}\rvert}{R}]\tanh[\frac{\lvert\mathbf{x}\rvert}{R}]}{R}\right)B_{0}\,\hat{\mathbf{x}}\cdot\hat{\mathbf{x}}_{z}.$

Without a loss of generality we can take $\Omega>0$. Note that when we compute
the radiated power in outgoing waves we will have to be careful in taking into
account both $m_{a}>\Omega$ or $m_{a}<\Omega$. The corresponding retarded
Green’s function is given by

$\displaystyle G(xt;x^{\prime}t^{\prime})$ $\displaystyle=$
$\displaystyle-\int\frac{d\omega}{2\pi}\frac{1}{4\pi|\vec{x}-\vec{x}^{\prime}|}\left(e^{i|\omega||\vec{x}-\vec{x}^{\prime}|-i\omega(t-t^{\prime})}\theta(\omega)+e^{-i|\omega||\vec{x}-\vec{x}^{\prime}|-i\omega(t-t^{\prime})}\theta(-\omega)\right)$

which can be used to find the radiated electric and magnetic fields

$\displaystyle\mathbf{E}^{\prime}_{\text{r}}$ $\displaystyle\approx$
$\displaystyle\left(\frac{C\beta}{\pi
f_{a}}\frac{B_{0}\tilde{\phi}m_{a}}{8|x|}R^{2}\pi^{2}\right)\left[\frac{\tanh\left(\frac{\pi(m_{a}+\Omega)R}{2}\right)}{\cosh\left(\frac{\pi(m_{a}+\Omega)R}{2}\right)}\cos((m_{a}+\Omega)(t-|\bf{x}|)+\alpha+\gamma)\hat{\mathbf{x}}_{z}\right.$
$\displaystyle\left.+\text{sgn}(m_{a}-\Omega)\frac{\tanh\left(\frac{\pi|(m_{a}-\Omega)|R}{2}\right)}{\cosh\left(\frac{\pi|(m_{a}-\Omega)|R}{2}\right)}\cos(|(m_{a}-\Omega)|(t-|{\bf
x}|)+\text{sgn}(m_{a}-\Omega)(\alpha-\gamma))\hat{\mathbf{x}}_{z}\right],$

and

$\displaystyle\mathbf{B}_{\text{r}}$ $\displaystyle\approx$
$\displaystyle(\hat{\mathbf{x}}\times\hat{\mathbf{x}}_{z})\left(\frac{C\beta}{\pi
f_{a}}\frac{B_{0}m_{a}\tilde{\phi}}{8|x|}R^{2}\pi^{2}\right)\left[\frac{\tanh\left(\frac{\pi(m_{a}+\Omega)R}{2}\right)}{\cosh\left(\frac{\pi(m_{a}+\Omega)R}{2}\right)}\cos((m_{a}+\Omega)(t-|\bf{x}|)+\alpha+\gamma)\right.$
$\displaystyle\left.+\text{sgn}(m_{a}-\Omega)\frac{\tanh\left(\frac{\pi|(m_{a}-\Omega)|R}{2}\right)}{\cosh\left(\frac{\pi|(m_{a}-\Omega)|R}{2}\right)}\cos(|(m_{a}-\Omega)|(t-|{\bf
x}|)+\text{sgn}(m_{a}-\Omega)(\alpha-\gamma))\right].$

Here the $\text{sgn}(\sigma)$ stands for the sign of the variable $\sigma$.
The approximate signs are there to remind us that we have ignored
$\sim|\bf{x}|^{-n}$ terms where $n>1$ since they don’t contribute to
radiation. For $m_{a}<\Omega$, the instantaneous power radiated is given by

$\displaystyle P_{i}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{3}\left(\frac{C\beta}{\pi
f_{a}}\frac{B_{0}m_{a}\tilde{\phi}}{8}\pi^{2}R^{2}\right)^{2}\left[\frac{\tanh\left(\frac{\pi(m_{a}+\Omega)R}{2}\right)^{2}}{\cosh\left(\frac{\pi(m_{a}+\Omega)R}{2}\right)^{2}}\left(1+\cos(2(m_{a}+\Omega)(t-|\mathbf{x}|)+2\alpha+2\gamma)\right)\right.$
$\displaystyle\left.-2\frac{\tanh\left(\frac{\pi(m_{a}+\Omega)R}{2}\right)}{\cosh\left(\frac{\pi(m_{a}+\Omega)R}{2}\right)}\frac{\tanh\left(\frac{\pi|m_{a}-\Omega|R}{2}\right)}{\cosh\left(\frac{\pi|m_{a}-\Omega|R}{2}\right)}\left(\cos(2\Omega(t-|\mathbf{x}|)+2\gamma)+\cos(2m_{a}(t-|\mathbf{x}|)+2\alpha)\right)\right.$
$\displaystyle\left.+\frac{\tanh\left(\frac{\pi|m_{a}-\Omega|R}{2}\right)^{2}}{\cosh\left(\frac{\pi|m_{a}-\Omega|R}{2}\right)^{2}}\left(1+\cos(2|m_{a}-\Omega|(t-|\mathbf{x}|)-2(\alpha-\gamma))\right)\right],$

where the subscript $i$ stands for instantaneous. For $m_{a}>\Omega$ the
instantaneous radiated power is

$\displaystyle P_{i}$ $\displaystyle=$
$\displaystyle\frac{4\pi}{3}\left(\frac{C\beta}{\pi
f_{a}}\frac{B_{0}m_{a}\tilde{\phi}}{8}\pi^{2}R^{2}\right)^{2}\left[\frac{\tanh\left(\frac{\pi(m_{a}+\Omega)R}{2}\right)^{2}}{\cosh\left(\frac{\pi(m_{a}+\Omega)R}{2}\right)^{2}}\left(1+\cos(2(m_{a}+\Omega)(t-|\mathbf{x}|)+2\alpha+2\gamma)\right)\right.$
$\displaystyle\left.+2\frac{\tanh\left(\frac{\pi(m_{a}+\Omega)R}{2}\right)}{\cosh\left(\frac{\pi(m_{a}+\Omega)R}{2}\right)}\frac{\tanh\left(\frac{\pi|m_{a}-\Omega|R}{2}\right)}{\cosh\left(\frac{\pi|m_{a}-\Omega|R}{2}\right)}\left(\cos(2\Omega(t-|\mathbf{x}|)+2\gamma)+\cos(2m_{a}(t-|\mathbf{x}|)+2\alpha)\right)\right.$
$\displaystyle\left.+\frac{\tanh\left(\frac{\pi|m_{a}-\Omega|R}{2}\right)^{2}}{\cosh\left(\frac{\pi|m_{a}-\Omega|R}{2}\right)^{2}}\left(1+\cos(2|m_{a}-\Omega|(t-|\mathbf{x}|)+2(\alpha-\gamma))\right)\right].$

There are quite a few lessons we can learn from the expression for the
radiated power. To begin with, one can concentrate on the two limits of
$\Omega\gg m_{a}$ and $m_{a}\gg\Omega$. In the former case the expression for
power radiated obtained from Eq. LABEL:power1 should be recoverable by
ignoring the axion frequency in the sinusoidal variation of the current
density, i.e. by writing

$\displaystyle\mathbf{J}(\mathbf{x},t)$ $\displaystyle=$
$\displaystyle-\frac{C\beta}{\pi
f_{a}}\left(\partial_{t}\phi\right)\mathbf{B}_{0}$ (23) $\displaystyle=$
$\displaystyle\frac{C\beta}{\pi
f_{a}}\tilde{\phi}\sin(\alpha)\operatorname{sech}\Big{[}\frac{\lvert\mathbf{x}\rvert}{R}\Big{]}B_{0}\cos(\Omega
t+\gamma)\hat{\mathbf{x}}_{z}.$

This amount to substituting the axion field $\tilde{\phi}$ by
$\tilde{\phi}\sin(\alpha)$ in Eq. 12. Similarly, for the latter case we should
be able to ignore the magnetic field frequency in the current density and the
radiated power should be obtainable from replacing the magnetic field $B_{0}$
by $B_{0}\cos(\gamma)$ in Eq. 12. In order to see how these expression for
radiated power emerge from Eq. LABEL:power1, let’s consider the two limits
$m_{a}\ll\Omega$ and $m_{a}\gg\Omega$ sequentially. For $m_{a}\ll\Omega$, in
the expression for instantaneous power in Eq. LABEL:power1 we should set
$\frac{m_{a}}{\Omega}\rightarrow 0$ and then average the instantaneous power
over the time scale of $2\pi/\Omega$. This produces the expected result for
the limit of $m_{a}\ll\Omega$ obtained by substituting $\tilde{\phi}$ by
$\tilde{\phi}\sin(\alpha)$ in Eq. 12. In the case of $\Omega\ll m_{a}$, one
should replace $\Omega\rightarrow 0$ in the expression for the instantaneous
power in Eq. LABEL:power2, and then average the instantaneous power over a
period of time $2\pi/m_{a}$. This produces the expected result for the
radiated power in the limit of $\Omega\ll m_{a}$ obtained from Eq. 12 by
substituting $B_{0}$ by $B_{0}\cos(\gamma)$.

Finally, if we are interested in the limit where $m_{a}$ and $\Omega$ are not
very different, we can average the expression for the power radiated in Eq.
LABEL:power1 and LABEL:power2 over a period of time much larger than either
$\Omega$ or $m_{a}$ to obtain the time averaged radiated power
$P_{\text{av}}$. We find that the time average radiated power in both cases,
$m_{a}>\Omega$ and $\Omega>m_{a}$ are approximately the same and given by

$\displaystyle P_{\text{av}}$ $\displaystyle\approx$
$\displaystyle\frac{4\pi}{3}\left(\frac{C\beta}{\pi
f_{a}}\frac{B_{0}m_{a}\tilde{\phi}}{8}\pi^{2}R^{2}\right)^{2}\left[\frac{\tanh\left(\frac{\pi(m_{a}+\Omega)R}{2}\right)^{2}}{\cosh\left(\frac{\pi(m_{a}+\Omega)R}{2}\right)^{2}}+\frac{\tanh\left(\frac{\pi|m_{a}-\Omega|R}{2}\right)^{2}}{\cosh\left(\frac{\pi|m_{a}-\Omega|R}{2}\right)^{2}}\right].$

The approximate sign in LABEL:power3 takes into account the fact that the
radiated power has contributions from electromagnetic waves of two different
frequencies $|m_{a}-\Omega|$ and $m_{a}+\Omega$ and the time average computed
in Eq. LABEL:power3 is taken over a time interval longer than the longest time
period of the two frequency scales, i.e. $\frac{2\pi}{|m_{a}-\Omega|}$. As we
observe from Eq. LABEL:power3 and the discussion preceding it, the average
radiated power is independent of the initial phase shift of the magnetic field
and the axion condensate unless $\frac{\Omega}{m_{a}}\ll 1$ or
$\frac{m_{a}}{\Omega}\ll 1$. The dependence of radiation on the phase shift in
the limits $m_{a}\ll\Omega$ and $\Omega\ll m_{a}$ can however be reinterpreted
as a dependence on the amplitude of an approximately time independent axion
condensate or a magnetic field respectively.

We see from Eq. LABEL:power3 that radiated power depends sensitively on the
size of the condensate as before. However, as opposed to the case in Eq. 12
when $\Omega=0$, the peak in radiation occurs for two different values of the
condensate size: one at $R\sim\frac{1}{|m_{a}-\Omega|}$, the other at
$R\sim\frac{1}{m_{a}+\Omega}$. When $m_{a}\sim\Omega$, if
$\frac{|m_{a}-\Omega|}{m_{a}}\ll 1$, these two length scales can be very
different. In this case, one of the peaks in radiation lies at a radius of the
order of $\frac{1}{m_{a}+\Omega}\sim\frac{1}{m_{a}}\sim\frac{1}{\Omega}$. The
other peak is at a much larger radius of the order of
$\frac{1}{m_{a}-\Omega}\gg\frac{1}{m_{a}}\sim\frac{1}{\Omega}$. Thus, we find
that there is a resonant enhancement of radiation for large axion condensates
when the frequency of the oscillating magnetic field coincides with the axion
mass. This enhancement is similar to the resonant enhancement found in
Amin:2021tnq  in the presence of a background plasma. In the next section we
will introduce a background plasma medium and analyze its effects on radiation
combined with the effects of an alternating magnetic field.

###  2.3 Effects of a background plasma

The effects of a background plasma on radiation can be modeled effectively by
replacing the matter current of the medium $J_{p}$ by $\sigma E$ where
$\sigma$ is the conductivity of the plasma. If we are interested in an
electron ion plasma as in the case of the interstellar medium, $\sigma$ has a
frequency dependence which can be modeled using the Drude model as

$\displaystyle\sigma(\omega)=\frac{\sigma_{0}}{1-i\omega\tau}.$ (25)

Here $\tau$ is the collision time for electrons. In the collision less limit,
where the collision frequency is much smaller than the frequency of the
electromagnetic radiation, i.e. $\omega\tau\gg 1$, we can write the
conductivity as

$\displaystyle\sigma\approx i\frac{\sigma_{0}}{\omega\tau}.$ (26)

If we substitute this expression for conductivity in Maxwell’s equations, we
find that the photon acquires a plasma mass
$\omega_{P}=\sqrt{\frac{\sigma_{0}}{\tau}}$ with a dispersion relation of the
form $\omega^{2}=k^{2}+\omega_{P}^{2}$. The conductivity $\sigma_{0}$ can be
expressed as $\sigma_{0}=\frac{4\pi n_{e}e^{2}\tau}{m_{e}}$ where $m_{e}$ is
the electron mass, $e$ is the electron’s charge and $n_{e}$ is the density of
electrons. Thus, the plasma mass is given by $\omega_{P}=\sqrt{\frac{4\pi
n_{e}e^{2}}{m_{e}}}$. As shown in Amin:2021tnq , the electromagnetic radiation
emitted from an axion condensate in the presence of a constant external
magnetic field and a plasma in the collision-less limit is given by

$\displaystyle P_{\text{av}}=\left(\frac{C\beta}{\pi
f_{a}}\right)^{2}\frac{\tilde{\phi}^{2}B_{0}^{2}m_{a}k_{m_{a}}R^{4}\pi^{5}}{12}\left(\frac{\tanh(\pi
k_{m_{a}}R/2)}{\cosh(\pi k_{m_{a}}R/2)}\right)^{2}$ (27)

with $k_{m_{a}}=\sqrt{m_{a}^{2}-\omega_{P}^{2}}$. This expression also makes
it clear how a plasma can enhance electromagnetic radiation from a large axion
condensate. The radiated power is dependent on the size of the condensate such
that the radiation is peaked for condensates of size
$R\sim\frac{1}{k_{m_{a}}}$ and exponentially suppressed otherwise. When
$\omega_{P}$ is negligible or even of the order of $m_{a}$, the peak in
radiation generally occurs for axion condensates of size
$\frac{1}{k_{m_{a}}}\sim\frac{1}{m_{a}}$. When
$\frac{\sqrt{m_{a}^{2}-\omega_{P}^{2}}}{m_{a}}\ll 1$, however, condensates of
much larger size than $\frac{1}{m_{a}}$ can radiate efficiently. More
precisely, radiation peaks when the condensate radius $R$, the axion mass
$m_{a}$, and the plasma frequency $\omega_{\text{P}}$ are related by
$R=\frac{\frac{2}{\pi}\log(\sqrt{2}+1)}{\sqrt{m_{a}^{2}-\omega_{P}^{2}}}$.
Thus, the larger the condensate, the closer the axion mass and the plasma
frequency have to be in order for enhanced radiation to take place.

In general though, there is no reason to expect the axion mass and the plasma
frequency to precisely coincide even if they are of the same order of
magnitude. Thus we don’t expect the resonance in radiation to take place for
very large axion condensates. However, in the presence of an alternating
background magnetic field with time varying frequency, it may be possible to
achieve this resonant conversion even when plasma frequency and axion mass
don’t exactly coincide. This is because the axion mass suited for resonant
conversion should depend both on the plasma frequency and the frequency of the
alternating magnetic field. Thus, with an alternating magnetic field whose
frequency changes with time, the condition for resonant radiation can be met
over a certain interval of time as the frequency changes. So long as the
plasma frequency, the mass of the axion and the frequency of the alternating
magnetic field are of the same order of magnitude, a time varying frequency
for the magnetic field is likely to scan a range of axion masses for which
enhanced or resonant radiation can take place. It is therefore useful to
analyze the combined effects of a plasma background and an alternating
magnetic field on the radiation from axion condensate as we will do next. In
our analysis here, we will treat the frequency of the alternating magnetic
field $\Omega$ to be a constant in time. This analysis should correctly model
a resonant enhancement in radiation even for time varying $\Omega$ as long as
the time variation of $\Omega$ over a single time period of oscillation of the
radiated field remains small compared to the frequency of the radiated field.
In this case, again to account for both of the possibilities $\Omega>m_{a}$
and $m_{a}>\Omega$ we can write the retarded Green’s function as

$\displaystyle G(xt;x^{\prime}t^{\prime})$ $\displaystyle=$
$\displaystyle-\int\frac{d\omega}{2\pi}\frac{\theta\left(\omega^{2}-\omega_{p}^{2}\right)}{4\pi|\vec{x}-\vec{x}^{\prime}|}\left(e^{i\sqrt{\omega^{2}-\omega_{p}^{2}}|\vec{x}-\vec{x}^{\prime}|-i\omega(t-t^{\prime})}\theta(\omega)+e^{-i\sqrt{\omega^{2}-\omega_{p}^{2}}|\vec{x}-\vec{x}^{\prime}|-i\omega(t-t^{\prime})}\theta(-\omega)\right)$
(28) $\displaystyle-$
$\displaystyle\int\frac{d\omega}{2\pi}\frac{\theta\left(-\omega^{2}+\omega_{p}^{2}\right)}{4\pi|\vec{x}-\vec{x}^{\prime}|}e^{-\sqrt{|\omega^{2}-\omega_{p}^{2}|}|\vec{x}-\vec{x}^{\prime}|-i\omega(t-t^{\prime})}.$

Here we have also taken into account of the fact that radiation for
frequencies smaller than the plasma frequency is exponentially damped. We can
now write down the time averaged radiated power

$\displaystyle P_{\text{av}}$ $\displaystyle\approx$
$\displaystyle\frac{4\pi}{3}\left(\frac{C\beta}{\pi
f_{a}}\frac{B_{0}m_{a}\tilde{\phi}}{8}\pi^{2}R^{2}\right)^{2}\left[\frac{\tanh\left(\frac{\pi\kappa_{m_{a}+\Omega}R}{2}\right)^{2}}{\cosh\left(\frac{\pi\kappa_{m_{a}+\Omega}R}{2}\right)^{2}}+\frac{\tanh\left(\frac{\pi\kappa_{m_{a}-\Omega}R}{2}\right)^{2}}{\cosh\left(\frac{\pi\kappa_{m_{a}-\Omega}R}{2}\right)^{2}}\right],$

where $\kappa_{\omega}=\sqrt{\omega^{2}-\omega_{P}^{2}}$. Again, the time
average of the instantaneous power is taken over a time scale larger than the
longer time period, $\frac{2\pi}{|m_{a}-\Omega|}$. As expected, the condition
for resonant radiation depends on both the frequency of the magnetic field and
the plasma frequency. The radiation peaks for
$R\sim\frac{1}{\kappa_{m_{a}+\Omega}}$ and for
$R\sim\frac{1}{\kappa_{m_{a}-\Omega}}$. For $m_{a}\sim\Omega\sim\omega_{P}$,
the first term in Eq. LABEL:power4 peaks for radius
$R\sim\frac{1}{\kappa_{m_{a}+\Omega}}\sim\frac{1}{m_{a}}$. If we now have
$\kappa_{m_{a}-\Omega}\ll m_{a}\sim\Omega$, the second term in Eq.
LABEL:power4 peaks for condensate sizes much larger than $\frac{1}{m_{a}}$. If
the condition, $\kappa_{m_{a}-\Omega}\ll m_{a}\sim\Omega$ is not satisfied,
the second term in Eq. LABEL:power4 is maximized for condensates of size
$\frac{1}{m_{a}}$ as well.

In figures 1-3 we show the radiated power as a function of $\Omega/m_{a}$ for
the three cases $R=10m_{a}^{-1}$, $R=m_{a}^{-1}$ and $R=0.01m_{a}^{-1}$,
respectively, for several different values of the plasma frequency. The power
is averaged over several periods of oscillation for the axion condensate
$\frac{2\pi}{m_{a}}$. Our goal here is to understand how the radiated power
changes with the magnetic field frequency $\Omega$ for various sizes of axion
condensate. Let us first discuss the figures in Fig. 1 and 2. The values of
plasma frequency that we have considered are smaller than the axion mass
scale. For these values of the plasma frequency, axion condensates of size
$R=m_{a}^{-1}$ and $R=10m_{a}^{-1}$, never realize the resonance condition
$\kappa_{m_{a}+\Omega}=\frac{2(\log(\sqrt{2}+1))}{\pi R}$. In other words, the
first term in square brackets in Eq. LABEL:power4 does not reach its peak for
any value of $\Omega>0$. However, the second term is maximized for
$\kappa_{m_{a}-\Omega}=\sqrt{(\Omega-
m_{a})^{2}-\omega_{P}^{2}}=\frac{2(\log(\sqrt{2}+1))}{\pi R}$ and the
corresponding resonance is observed for two different values of $\Omega$ with
$\frac{\Omega}{m_{a}}=1\pm\sqrt{\left(\frac{2(\log(\sqrt{2}+1))}{\pi
Rm_{a}}\right)^{2}+\frac{\omega_{P}^{2}}{m_{a}^{2}}}$. Note that, each of
these two peaks contain an abrupt drop in radiation on one side. This drop
corresponds to the fact that for
$1-\frac{\omega_{P}}{m_{a}}<\frac{\Omega}{m_{a}}<1+\frac{\omega_{P}}{m_{a}}$,
there is no electromagnetic radiation as the frequency of radiation is below
the photon’s plasma mass. For $R\sim 10m_{a}^{-1}$, the peaks are found
symmetrically distributed around $\Omega=m_{a}$. This is because, for $R\sim
10m_{a}^{-1}$ the first term in square brackets in Eq. LABEL:power4 is
negligible and the peaks observed in Fig. 1 reflect the behavior of the second
term in the square brackets. For $R\sim m_{a}^{-1}$ on the other hand, the
peaks are not as pronounced and they are not symmetrically distributed around
$\Omega=m_{a}$. To understand this, note that although for $R=m_{a}^{-1}$, the
first term in square brackets in Eq. LABEL:power4 does not reach its peak for
any $\Omega>0$, it is not completely negligible. Its tail contributes to Fig.
2 accounting for the qualitative differences observed in the nature of the
peaks located at $\Omega>m_{a}$ and $\Omega<m_{a}$ between Fig. 1 and Fig. 2.
We also note that as $R$ decreases, the peaks become wider: the peaks for
$R\sim 10m_{a}^{-1}$ are narrower than the ones for $R\sim m_{a}^{-1}$. To
understand this, we have to remember that the peak width in Fig. 1 and 2 is
controlled by the second term in the square brackets in Eq. LABEL:power4. The
contribution to radiated power from this term is substantial only for
$|\Omega-m_{a}|>\omega_{P}$ and $\sqrt{(\Omega-
m_{a})^{2}-\omega_{P}^{2}}<\frac{1}{R}$. The lower bound is set by the absence
of propagating electromagnetic fields below the plasma frequency and the upper
bound is set by the exponential drop off of radiation for wavelengths much
larger than the radius of the condensate. The width of the peaks is thus
$\delta\Omega\sim
m_{a}\bigg{(}\sqrt{\left(\frac{1}{m_{a}R}\right)^{2}+\frac{\omega_{P}^{2}}{m_{a}^{2}}}-\frac{\omega_{P}}{m_{a}}\bigg{)}$.
For $Rm_{a}\gg\frac{m_{a}}{\omega_{P}}$, the width goes as
$\delta\Omega\sim\frac{1}{\omega_{P}R^{2}}$, whereas for
$Rm_{a}\sim\frac{m_{a}}{\omega_{P}}$, the peak width scales as
$\delta\Omega\sim\omega_{P}$. Thus, the peaks in Fig.1 are much narrower than
the peaks in Fig. 2.

In the case of $R=0.01m_{a}^{-1}$ shown in Fig. 3, the resonance condition is
realized for both terms in the square bracket in Eq. LABEL:power4 around
$\Omega\sim\frac{2(\log(\sqrt{2}+1))}{\pi R}\gg m_{a},\omega_{P}$. Here, the
effect of the plasma frequency on the peaks is negligible and thus, the plots
for different plasma frequencies overlap. The width of the peaks is set by the
scale $\delta\Omega\sim\frac{1}{R}$ making them much wider than those in Fig.
1 and 2.

Figure 1: Radiated power as a function of a constant magnetic field frequency
$\Omega$ for the case $R=10m_{a}^{-1},$ averaged over 500 periods of the axion
condensate. The power is measured in units of
$\frac{4\pi}{3}\big{(}\frac{C\beta B_{0}m_{a}\tilde{\phi}\pi^{2}R^{2}}{8\pi
f_{a}}\big{)}^{2}$. Figure 2: Radiated power as a function of a constant
magnetic field frequency $\Omega$ for the case $R=m_{a}^{-1},$ averaged over
500 periods of the axion condensate. The power is measured in units of
$\frac{4\pi}{3}\big{(}\frac{C\beta B_{0}m_{a}\tilde{\phi}\pi^{2}R^{2}}{8\pi
f_{a}}\big{)}^{2}$. Figure 3: Radiated power as a function of a constant
magnetic field frequency $\Omega$ for the case $R=0.01m_{a}^{-1},$ averaged
over 500 periods of the axion condensate. The power is measured in units of
$\frac{4\pi}{3}\big{(}\frac{C\beta B_{0}m_{a}\tilde{\phi}\pi^{2}R^{2}}{8\pi
f_{a}}\big{)}^{2}$. Figure 4: Instantaneous radiated power in the absence of a
medium as a function of the retarded time $(t-\lvert\mathbf{x}\rvert)$
measured in units of the inverse axion mass for a linearly increasing magnetic
field, $\Omega(t)=\Omega_{0}+at$ (pink line), together with its corresponding
envelope (black dashed line). We have here set $a=0.001m_{a}^{2}$,
$\Omega_{0}=0.3m_{a}$ and $R=10m_{a}^{-1}$. The power is measured in units of
$\frac{4\pi}{3}\big{(}\frac{C\beta B_{0}m_{a}\tilde{\phi}\pi^{2}R^{2}}{8\pi
f_{a}}\big{)}^{2}$.

####  2.3.1 Dissipation

Previously while working with the frequency dependence of the conductivity, we
took the collision-less limit $\omega\tau\gg 1$. This resulted in a simple
dispersion relation between the photon frequency and its wavelength
$k=\sqrt{\omega^{2}-\omega_{P}^{2}}$ where we find propagating electromagnetic
waves for $\omega>\omega_{P}$ and damped waves for $\omega<\omega_{P}$. As we
will see, this conclusion is perfectly valid as long as
$\sqrt{\omega^{2}-\omega_{P}^{2}}$ is not much smaller than $\omega_{P}$. When
$\omega^{2}-\omega_{P}^{2}\ll\omega_{P}^{2}$, we have to be more careful. In
order to understand why this is the case, note that in the presence of a
frequency dependent conductivity in the collision-less limit we can write the
dispersion relation for the photon as

$\displaystyle
k^{2}=\omega^{2}-\omega_{P}^{2}-i\frac{\omega_{P}^{2}}{\omega\tau}+\cdots.$
(30)

We had previously ignored the third term on the RHS of Eq. 30, in the limit of
$\frac{1}{\omega\tau}\ll 1$. However, for $\omega\sim\omega_{P}$ we may not be
able to ignore this term if
$\frac{\omega^{2}-\omega_{P}^{2}}{\omega_{P}^{2}}\sim\frac{1}{\omega\tau}$. If
$\frac{\omega^{2}-\omega_{P}^{2}}{\omega_{P}^{2}}\sim\frac{1}{\omega\tau}$,
the wave vector has a non-negligible imaginary part and as a result the
electromagnetic wave of frequency $\omega$ experiences significant damping.
Since we found that the resonance in radiation can take place for long
wavelengths with wave vectors $k_{m_{a}-\Omega}$ where
$k_{m_{a}-\Omega}^{2}=(m_{a}-\Omega)^{2}-\omega_{P}^{2}\ll\omega_{P}^{2}\sim
m_{a}^{2}$, we need to examine whether such wavelengths experience any
significant damping more carefully. In order for resonance condition to be
satisfied, and for radiation to peak for a frequency $\omega$ we know that
$\omega^{2}-\omega_{P}^{2}\sim\frac{1}{R^{2}}$ and that
$R\gg\frac{1}{m_{a}}\sim\frac{1}{\omega_{P}}$. However, for this wave to
propagate without significant damping we will also need the condition
$\omega^{2}-\omega_{P}^{2}\sim\frac{1}{R^{2}}\gg\frac{\omega_{P}^{2}}{\omega\tau}\sim\frac{\omega_{P}}{\tau}\implies\tau\gg\omega_{P}R^{2}$.
So, in order for condensates of size much larger than
$\frac{1}{m_{a}},\frac{1}{\omega_{P}}$ for
$\frac{1}{m_{a}}\sim\frac{1}{\omega_{P}}$ to radiate efficiently we need a
scale separation between $\tau$ and $\frac{1}{m_{a}}$ such that

$\displaystyle\tau\gg\omega_{P}R^{2}\sim\frac{\omega_{P}}{(m_{a}-\Omega)^{2}-\omega_{P}^{2}}\gg\frac{1}{m_{a}}.$
(31)

In section 3 we will examine to what extent such a condition is met in the
interstellar medium. This will of course depend on the mass of axions we are
interested in. The scale separation in question will be difficult to achieve
if we lower the axion mass sufficiently such that the collision frequency of
electrons becomes comparable with the axion mass scale. However, we will find
that for the most interesting axion mass scales, this condition is met rather
comfortably.

###  2.4 Time dependent frequency of alternating magnetic field

At this stage we will attempt to understand the effect of a time dependent
magnetic field frequency for the external magnetic field on radiation from
axion condensates. We will simplify calculations by setting $\omega_{P}=0$
which amounts to neglecting the effects of a plasma. Incorporating a time
dependence in the magnetic field frequency while also taking into account a
finite plasma frequency for the photon is a rather involved calculation and we
reserve this analysis for future work. As mentioned before the time dependence
of the frequency of the magnetic field can be very complicated. However, our
goal in this section is to demonstrate that in the presence of a time
dependent external magnetic field frequency, it is possible for a condensate
that is not radiating at an instant in time $t$ to radiate resonantly at
another instant $t+\Delta t$. For the purpose of this demonstration, we don’t
need to consider the most general form for the frequency as a function of
time. We will instead consider a linearly increasing magnetic field frequency
$f(t)$ such that $\frac{d}{dt^{n}}f=0$ for $n>1$ and $\frac{df}{dt}\neq 0$. We
will restrict our calculations to a regime where the change in the external
magnetic field frequency over a single time period of the outgoing radiation
is small compared to the frequency of the outgoing radiation.

As an example, we begin with an external magnetic field of the form

$\mathbf{B}(\mathbf{x},t)=B_{0}\cos(\Omega(t)t)\hat{\mathbf{x}}_{z},$ (32)

with

$\Omega(t)=\Omega_{0}+at,$ (33)

where $\Omega_{0}$ is the frequency at time $t=0$. The effective frequency of
this time varying magnetic field is given by $\frac{d}{dt}(\Omega(t)t)$ which
in this case is $\Omega_{0}+2at$. We will assume that
$|(m_{a}-\Omega_{0})|R\gg 1$ such that there is no significant radiation at
any time $t$ when $a=0$. We will also assume $m_{a}>\Omega_{0}>0$ for
convenience. For an axion condensate of the form
$\phi=\tilde{\phi}\cos(m_{a}t)\operatorname{sech}\left(\frac{r}{R}\right)$,
the current density is given by

$\displaystyle\mathbf{J}(\mathbf{x},t)$
$\displaystyle=-\bigg{(}\frac{C\beta}{\pi
f_{a}}\bigg{)}\big{[}(\partial_{t}\phi(\mathbf{x},t))\mathbf{B}(t)\big{]}$
$\displaystyle=\bigg{(}\frac{C\beta}{\pi
f_{a}}\bigg{)}\frac{\tilde{\phi}m_{a}B_{0}}{2}\operatorname{sech}(\lvert\mathbf{x}\rvert/R)\Big{\\{}\sin\big{[}(m_{a}-\Omega(t))t\big{]}+\sin\big{[}(m_{a}+\Omega(t))t\big{]}\Big{\\}}\hat{\mathbf{x}}_{z}.$
(34)

Using the retarded Green’s function the vector potential can then be expressed
as

$\displaystyle\mathbf{A}(\mathbf{x},t)$
$\displaystyle=-\int\text{d}^{3}x^{\prime}\text{d}t^{\prime}\frac{\delta(t-t^{\prime}-\lvert\mathbf{x}-\mathbf{x}^{\prime}\rvert)}{4\pi\lvert\mathbf{x}-\mathbf{x}^{\prime}\rvert}\bigg{(}\frac{C\beta}{\pi
f_{a}}\bigg{)}\frac{\tilde{\phi}m_{a}B_{0}}{2}\operatorname{sech}(\lvert\mathbf{x}^{\prime}\rvert/R)$
$\displaystyle\quad\quad\quad\times\Big{\\{}\sin\big{[}(m_{a}-\Omega(t^{\prime}))t^{\prime}\big{]}+\sin\big{[}(m_{a}+\Omega(t^{\prime}))t^{\prime}\big{]}\Big{\\}}\hat{\mathbf{x}}_{z}$
$\displaystyle=-\bigg{(}\frac{C\beta}{\pi
f_{a}}\bigg{)}\frac{\tilde{\phi}m_{a}B_{0}}{8\pi}\int
2\pi\,\text{d}\lvert\mathbf{x}^{\prime}\rvert\,\text{d}(\cos\theta^{\prime})\lvert\mathbf{x}^{\prime}\rvert^{2}\frac{\operatorname{sech}(\lvert\mathbf{x}^{\prime}\rvert/R)}{\lvert\mathbf{x}-\mathbf{x}^{\prime}\rvert}$
$\displaystyle\quad\quad\quad\times\Big{\\{}\sin\big{[}(m_{a}-\Omega_{0})(t-\lvert\mathbf{x}-\mathbf{x}^{\prime}\rvert)-a(t-\lvert\mathbf{x}-\mathbf{x}^{\prime}\rvert)^{2}\big{]}$
$\displaystyle\quad\quad\quad\quad+\sin\big{[}(m_{a}+\Omega_{0})(t-\lvert\mathbf{x}-\mathbf{x}^{\prime}\rvert)+a(t-\lvert\mathbf{x}-\mathbf{x}^{\prime}\rvert)^{2}\big{]}\Big{\\}}\hat{\mathbf{x}}_{z}.$
(35)

For $a=0$, the vector potential simplifies significantly in the radiation zone
$|\mathbf{x}|\gg|\mathbf{x}^{\prime}|$. However, with a time dependent
frequency $a\neq 0$, the expression for the vector potential is somewhat more
complicated and is given by

$\displaystyle\mathbf{A}(\mathbf{x},t)$
$\displaystyle\approx-\bigg{(}\frac{C\beta}{\pi
f_{a}}\bigg{)}\frac{\phi_{0}m_{a}B_{0}}{4\lvert\mathbf{x}\rvert}\int\text{d}\lvert\mathbf{x}^{\prime}\rvert\text{d}(\cos\theta^{\prime})\lvert\mathbf{x}^{\prime}\rvert^{2}\operatorname{sech}(\lvert\mathbf{x}^{\prime}\rvert/R)$
$\displaystyle\quad\times\Big{\\{}\sin\big{[}(m_{a}-\Omega_{0})(t-\lvert\mathbf{x}\rvert+\lvert\mathbf{x}^{\prime}\rvert\cos\theta^{\prime})-a(t-\lvert\mathbf{x}\rvert)^{2}-2a(t-\lvert\mathbf{x}\rvert)\lvert\mathbf{x}^{\prime}\rvert\cos\theta^{\prime}-a\lvert\mathbf{x}^{\prime}\rvert^{2}(\cos\theta^{\prime})^{2}\big{]}$
$\displaystyle\quad+\sin\big{[}(m_{a}+\Omega_{0})(t-\lvert\mathbf{x}\rvert+\lvert\mathbf{x}^{\prime}\rvert\cos\theta^{\prime})+a(t-\lvert\mathbf{x}\rvert)^{2}+2a(t-\lvert\mathbf{x}\rvert)\lvert\mathbf{x}^{\prime}\rvert\cos\theta^{\prime}+a\lvert\mathbf{x}^{\prime}\rvert^{2}(\cos\theta^{\prime})^{2}\big{]}\Big{\\}}\hat{\mathbf{x}}_{z}.$
(36)

Here $\theta^{\prime}$ is the angle between $\mathbf{x}$ and
$\mathbf{x}^{\prime}$. At this point to make further progress we will assume
$(t-|\mathbf{x}|)\gg R$ where $R$ is the size of the axion condensate. This
allows us to ignore
$\lvert\mathbf{x}^{\prime}\rvert^{2}(\cos\theta^{\prime})^{2}$ in Eq. 36 and
compute the integral in $\mathbf{x}^{\prime}$ to find

$\displaystyle\mathbf{A}(\mathbf{x},t)$
$\displaystyle=-\bigg{(}\frac{C\beta}{\pi
f_{a}}\bigg{)}\frac{\tilde{\phi}m_{a}B_{0}R^{2}\pi^{2}}{8\lvert\mathbf{x}\rvert}$
$\displaystyle\quad\quad\quad\times\bigg{\\{}\frac{\tanh\big{[}\frac{\pi
R}{2}\big{(}(m_{a}-\Omega_{0})-2a(t-\lvert\mathbf{x}\rvert)\big{)}\big{]}}{\cosh\big{[}\frac{\pi
R}{2}\big{(}(m_{a}-\Omega_{0})-2a(t-\lvert\mathbf{x}\rvert)\big{)}\big{]}}\frac{\sin\big{[}(m_{a}-\Omega_{0})(t-\lvert\mathbf{x}\rvert)-a(t-\lvert\mathbf{x}\rvert)^{2}\big{]}}{m_{a}-\Omega_{0}-2a(t-\lvert\mathbf{x}\rvert)}$
$\displaystyle\quad\quad\quad+\frac{\tanh\big{[}\frac{\pi
R}{2}\big{(}(m_{a}+\Omega_{0})+2a(t-\lvert\mathbf{x}\rvert)\big{)}\big{]}}{\cosh\big{[}\frac{\pi
R}{2}\big{(}(m_{a}+\Omega_{0})+2a(t-\lvert\mathbf{x}\rvert)\big{)}\big{]}}\frac{\sin\big{[}(m_{a}+\Omega_{0})(t-\lvert\mathbf{x}\rvert)+a(t-\lvert\mathbf{x}\rvert)^{2}\big{]}}{m_{a}+\Omega_{0}+2a(t-\lvert\mathbf{x}\rvert)}\bigg{\\}}\hat{\mathbf{x}}_{z}.$
(37)

We will justify the validity of considering $(t-|\mathbf{x}|)\gg R$ at a later
stage in the calculation. We can now write the electric field in terms of the
following variables

$\displaystyle
k^{(1)}_{\pm}=m_{a}\pm\big{[}\Omega_{0}+a(t-\lvert\mathbf{x}\rvert)\big{]},$
(38) $\displaystyle
k^{(2)}_{\pm}=m_{a}\pm\big{[}\Omega_{0}+2a(t-\lvert\mathbf{x}\rvert)\big{]},$
(39)

as

$\displaystyle\mathbf{E}_{\text{r}}^{\prime}(\mathbf{x},t)$
$\displaystyle=-\bigg{(}\frac{C\beta}{\pi
f_{a}}\bigg{)}\frac{\phi_{0}m_{a}B_{0}R^{2}\pi^{2}}{8\lvert\mathbf{x}\rvert}\frac{\operatorname{sech}\big{(}\frac{\pi
R}{2}k_{-}^{(2)}\big{)}}{(k_{-}^{(2)})^{2}}$
$\displaystyle\times\bigg{\\{}a\pi
Rk_{-}^{(2)}\operatorname{sech}^{2}\big{[}\frac{\pi
R}{2}k_{-}^{(2)}\big{]}\sin\big{[}(t-\lvert\mathbf{x}\rvert)k_{-}^{(1)}\big{]}-(k_{-}^{(2)})^{2}\tanh\big{[}\frac{\pi
R}{2}k_{-}^{(2)}\big{]}\cos\big{[}(t-\lvert\mathbf{x}\rvert)k_{-}^{(1)}\big{]}$
$\displaystyle\quad+2a\tanh\big{[}\frac{\pi
R}{2}k_{-}^{(2)}\big{]}\sin\big{[}(t-\lvert\mathbf{x}\rvert)k_{-}^{(1)}\big{]}-a\pi
Rk_{-}^{(2)}\tanh^{2}\big{[}\frac{\pi
R}{2}k_{-}^{(2)}\big{]}\sin\big{[}(t-\lvert\mathbf{x}\rvert)k_{-}^{(1)}\big{]}\bigg{\\}}\hat{\mathbf{x}}_{z}$
$\displaystyle\quad+(a\leftrightarrow-a,\Omega_{0}\leftrightarrow-\Omega_{0}).$
(40)

In the limit of $a\ll(k_{-}^{(2)})^{2}$ and $aR\ll k_{-}^{(2)}$ the electric
field can be simplified further

$\displaystyle\mathbf{E}_{\text{r}}^{\prime}(\mathbf{x},t)$
$\displaystyle\approx\bigg{(}\frac{C\beta}{\pi
f_{a}}\bigg{)}\frac{\phi_{0}m_{a}B_{0}R^{2}\pi^{2}}{8\lvert\mathbf{x}\rvert}.$
$\displaystyle\quad\quad\quad\times\bigg{\\{}\frac{\tanh(\pi
Rk_{-}^{(2)}/2)}{\cosh(\pi
Rk_{-}^{(2)}/2)}\cos\big{[}(t-\lvert\mathbf{x}\rvert)k_{-}^{(1)}\big{]}+\frac{\tanh(\pi
Rk_{+}^{(2)}/2)}{\cosh(\pi
Rk_{+}^{(2)}/2)}\cos\big{[}(t-\lvert\mathbf{x}\rvert)k_{+}^{(1)}\big{]}\bigg{\\}}\hat{\mathbf{x}}_{z}$
(41)

Similarly, the magnetic field is given by

$\mathbf{B}_{\text{r}}(\mathbf{x},t)=\lvert\mathbf{E}_{\text{r}}(\mathbf{x},t)\rvert(\hat{\mathbf{x}}\times\hat{\mathbf{x}}_{z}).$
(42)

We now arrive at the expression for the instantaneous radiated power

$\displaystyle P_{i}$ $\displaystyle=\frac{4\pi}{3}\bigg{(}\frac{C\beta}{\pi
f_{a}}\bigg{)}^{2}\bigg{(}\frac{\tilde{\phi}m_{a}B_{0}R^{2}\pi^{2}}{8}\bigg{)}^{2}$
$\displaystyle\quad\quad\quad\times\bigg{\\{}\bigg{(}\frac{\tanh(\pi
Rk_{-}^{(2)}/2)}{\cosh(\pi
Rk_{-}^{(2)}/2)}\bigg{)}^{2}\cos^{2}\big{[}(t-\lvert\mathbf{x}\rvert)k_{-}^{(1)}\big{]}+\bigg{(}\frac{\tanh(\pi
Rk_{+}^{(2)}/2)}{\cosh(\pi
Rk_{+}^{(2)}/2)}\bigg{)}^{2}\cos^{2}\big{[}(t-\lvert\mathbf{x}\rvert)k_{+}^{(1)}\big{]}$
$\displaystyle\quad\quad\quad\quad+2\frac{\tanh(\pi Rk_{-}^{(2)}/2)}{\cosh(\pi
Rk_{-}^{(2)}/2)}\frac{\tanh(\pi Rk_{+}^{(2)}/2)}{\cosh(\pi
Rk_{+}^{(2)}/2)}\cos\big{[}(t-\lvert\mathbf{x}\rvert)k_{-}^{(1)}\big{]}\cos\big{[}(t-\lvert\mathbf{x}\rvert)k_{+}^{(1)}\big{]}\bigg{\\}}.$
(43)

It is easy to see that the radiated power for an external magnetic field of
constant frequency is recovered in the limit of $a=0$. Note that, even though
the sinusoidal variation in the radiated power is expressed as a function of
$k_{\pm}^{(1)}$, the effective frequency and wavelengths of these waves are
set by
$\frac{d}{d(t-|\mathbf{x}|)}\left(k_{\pm}^{(1)}(t-|\mathbf{x}|)\right)\sim
k_{\pm}^{(2)}$. As is clear from Eq. 43, for condensates of size
$R\gg\frac{1}{m_{a}}\sim\frac{1}{k_{+}^{(2)}}$ a resonant enhancement in
radiation can take place in the limit of $k_{-}^{(2)}\sim
2\frac{\log(\sqrt{2}+1)}{\pi R}$ when the first term inside the curly brackets
in Eq. 43 dominates over the others which are exponentially suppressed. We can
define a time averaged radiated power by averaging the instantaneous power
over a few time slices of interval $\frac{1}{k_{-}^{(2)}}$ as long as the
change in frequency over one time period is much smaller than the frequency
itself, i.e. $\frac{2\pi a}{k_{-}^{(2)}}\ll k_{-}^{(2)}$:

$\displaystyle P_{\text{av}}$ $\displaystyle\approx\left(\frac{C\beta}{\pi
f_{a}}\right)^{2}\frac{\tilde{\phi}^{2}B_{0}^{2}m_{a}k_{m_{a}}R^{4}\pi^{5}}{12}\bigg{(}\frac{\tanh(\pi
Rk_{-}^{(2)}/2)}{\cosh(\pi Rk_{-}^{(2)}/2)}\bigg{)}^{2}.$ (44)

Near the resonant peak we also have $\pi Rk_{-}^{(2)}/2\sim\log(\sqrt{2}+1)$.
Thus, we have $a\ll(k_{-}^{(2)})^{2}\sim k_{-}^{(2)}/R$ where we have ignored
order $\sim 1$ numerical factors. Note that these very limits were used to
arrive at Eq 41. So, our result in Eq. 44 is self consistent. At this stage we
can also check the validity of the condition $t-|\mathbf{x}|\gg R$ near the
resonant peak. Recall that this condition was used to derive Eq. 37. The
resonant peak in radiation is at $k_{-}^{(2)}\sim 2\frac{\log(\sqrt{2}+1)}{\pi
R}$. Using the definitions in Eq. 39 we find

$\displaystyle t-|\mathbf{x}|$ $\displaystyle\sim$
$\displaystyle\frac{(m_{a}-\Omega_{0})-\frac{2(\log(\sqrt{2}+1))}{\pi R}}{2a}$
$\displaystyle\sim$ $\displaystyle\frac{(m_{a}-\Omega_{0})}{2a},$

where in the last line we have taken the limit
$(m_{a}-\Omega_{0})\gg\frac{2(\log(\sqrt{2}+1))}{\pi R}$ for large condensates
($Rm_{a}\gg 1$) and $m_{a}-\Omega_{0}\sim m_{a}\sim\Omega_{0}$. Thus, we see
that, the condition of $t-|\mathbf{x}|\gg R$ near the peaks translates to
$m_{a}-\Omega_{0}\gg 2aR$. Our analytical calculation for the resonant peak
for a time varying frequency therefore involves controlled approximations in
the limit of $\frac{(m_{a}-\Omega_{0})}{2a}\gg R\gg(m_{a}-\Omega_{0})^{-1}$.
We previously imposed the condition $a\ll(k_{-}^{(2)})^{2}$ which along with
$R\sim\frac{1}{k_{-}^{(2)}}$ implies $aR^{2}\ll 1$. Defining $aR^{2}\equiv b$,
we can rewrite the condition as $\frac{(m_{a}-\Omega_{0})R}{2b}\gg
1\gg((m_{a}-\Omega_{0})R)^{-1}$ which is clearly satisfied for any
$(m_{a}-\Omega_{0})R\sim m_{a}R\gg 1$. Note that some of the limits we assumed
were needed in order to perform a controlled analytic calculation. These
limits can be relaxed if the analysis is performed numerically.

In Fig. 4 we demonstrate how a time varying frequency of the external
alternating magnetic field can affect radiation from a large axion condensate,
$R=10m_{a}^{-1}$. In the figure we have set the rate of change of the magnetic
field frequency to be $0.001m_{a}^{2}$, which for an axion mass of $\sim
10^{-12}$eV corresponds to about $\sim 10^{3}\text{s}^{-2}$. We set the
oscillating magnetic field frequency at $t=0$, $\Omega_{0}=0.3m_{a}$. The
corresponding radiation reaches position $|\mathbf{x}|$ at time
$t=|\mathbf{x}|$. However, this radiated power is negligible due to the large
size of the condensate:
$R\gg\frac{1}{k_{\pm}^{(2)}(t-\lvert\mathbf{x}\rvert=0)}$. We see that the
radiated power increases with time as the frequency of the magnetic field
increases and resonant conversion is eventually realized for $k_{-}^{(2)}\sim
R^{-1}$. This shows that a wide range of axion masses can trigger resonant
conversion, if the frequency of the magnetic field is dependent on time.

##  3 Properties of interstellar plasma and alternating magnetic fields of
Neutron stars

In this section we will discuss the time scales associated with some
astrophysical sources of time dependent magnetic fields, e.g. pulsars and
neutron stars. We will also discuss how the plasma scales relevant for the
interstellar medium compare with these time scales. We mentioned earlier that
alternating magnetic fields can be sourced by spinning neutron stars. To
understand this, note that the spin of a neutron star can range from a few Hz
to a few kHz for pulsarsERA . A time period of $10^{-3}$s translates to a
energy scale of $10^{-12}$ eV in natural units. Since pulsars host very strong
magnetic fields which are typically at an angle with the axis of rotation,
they can be thought of as rotating magnetic dipoles. In fact, this rotating
magnetic dipole configuration of pulsars is expected to cause spin down of the
star by emitting magnetic dipole radiation. The frequency of this oscillating
magnetic dipole is set by the spin frequency of the star(typically a few kHz)
and interestingly it is close to the plasma frequency of the interstellar
medium ERA

$\displaystyle\nu_{P}$ $\displaystyle\sim$
$\displaystyle\left(\frac{e^{2}n_{e}}{\pi m_{e}}\right)^{1/2}$ (46)
$\displaystyle\approx$ $\displaystyle
8.97\,\,\text{kHz}\left(\frac{n_{e}}{\text{cm}^{-3}}\right)^{1/2}.$

For typical interstellar medium (ISM), one can take $n_{e}\sim
0.03\text{cm}^{-3}$ which results in $\nu_{P}\sim 1.5\,\,\text{kHz}$. 222In
fact, the presence of the interstellar plasma introduces significant damping
of the magnetic dipole radiation which in turn causes interstellar medium to
heat up. Thus, an axion condensate in the vicinity of a pulsar can experience
an alternating magnetic field, the frequency of which is of the order of the
plasma frequency of the interstellar medium. If the axion mass for such a
condensate is also of the same order, $\sim 10^{-12}$ eV, we can expect the
axion condensate to decay through electromagnetic radiation. Axions of mass
$10^{-12}$ eV are considered ultralight and can form axion stars depending on
the details of their interactions. They can also form superradiant axion
condensates around black holes of a few solar massesArvanitaki:2009fg ;
Brito:2015oca ; Arvanitaki:2014wva ; Arvanitaki:2016qwi . Thus, any time a
rotating neutron star interacts with an axion star or a superradiant
condensate of axion mass $10^{-12}$ eV, there can be significant
electromagnetic radiation which can deplete the axion condensate and leave its
imprints in gravitational waves.

Note that, it is not just pulsar spin which can produce alternating magnetic
field of frequency a few Hz to kHz. Similar alternating fields can result from
merging neutron stars which sweep a frequency of $10$ Hz to a kHz during
inspiral Piro:2012rq . This can give rise to alternating magnetic fields with
increasing frequency in time which were discussed in the previous section.

Let us now recall our discussion on dissipation from the subsection 2.3 where
we noted that for resonant radiation to take place, the condition in Eq. 31
needs to be satisfied. We will now verify to what extent this condition is met
for ultralight axion condensates in the ISM background and in an alternating
external field of frequency a few kHz. The collision frequency of electrons in
interstellar medium was discussed in Sen:2018cjt  $\sim
10^{-18}\left(\frac{1\text{eV}}{T}\frac{n_{e}}{1/\text{cc}}\right)$. Taking
the temperature to be $1$ eV and a density of $n_{e}\sim 0.01\text{cm}^{-3}$
we see that the inequality of Eq. 31 is indeed satisfied for $\omega_{P}\sim
m_{a}\sim 10^{-11}-10^{-12}$ eV. As an example, we can expect axion
condensates of size $R\gg\frac{1}{m_{a}}$ for $m_{a}\sim 10^{-12}$ eV to
radiate resonantly as long as
$R^{2}<\frac{\tau}{\omega_{P}}\sim\frac{10^{8}}{m_{a}^{2}}$.

##  4 Axion decay time scale

Since electromagnetic radiation takes away energy from the axion condensate,
the condensate amplitude will decay with time. Our analysis of electromagnetic
radiation from axion condensates so far ignores the effect of back reaction or
the decay of the condensate. As mentioned in the introduction, this is
justified so long as the decay time scale of the condensate is larger than the
time period of the outgoing electromagnetic waves. Here we will estimate this
decay time scale taking into account resonant enhancement in radiation for
condensates of size $R\sim\frac{1}{m_{a}}$ and large condensates
$R\gg\frac{1}{m_{a}}$. From Eq. LABEL:power4 we know that radiation can peak
for axion condensates of size
$R\sim\frac{1}{\kappa_{m_{a}+\Omega}}\sim\frac{1}{m_{a}}\sim\frac{1}{\Omega}$
owing to the first term on the RHS of Eq. LABEL:power4. Similarly, radiation
can peak for condensates of size $R\sim\frac{1}{\kappa_{m_{a}-\Omega}}$ owing
to the second term on the RHS. Note that the energy density stored in an axion
condensate is of the order of
$m_{a}^{2}\tilde{\phi}^{2}\operatorname{sech}\left(\frac{r}{R}\right)^{2}$ and
the corresponding energy is $\sim
m_{a}^{2}\tilde{\phi}^{2}\frac{\pi^{3}R^{3}}{3}$. So the decay time scale is
given by

$\displaystyle
T\sim\frac{\frac{\pi^{3}R^{3}}{3}m_{a}^{2}\tilde{\phi}^{2}}{\frac{4\pi}{3}\left(\frac{C\beta\tilde{\phi}}{\pi
f_{a}}\frac{B_{0}m_{a}}{8}\pi^{2}R^{2}\right)^{2}\left[\frac{\tanh\left(\frac{\pi\kappa_{m_{a}+\Omega}R}{2}\right)^{2}}{\cosh\left(\frac{\pi\kappa_{m_{a}+\Omega}R}{2}\right)^{2}}+\frac{\tanh\left(\frac{\pi\kappa_{m_{a}-\Omega}R}{2}\right)^{2}}{\cosh\left(\frac{\pi\kappa_{m_{a}-\Omega}R}{2}\right)^{2}}\right]}.$
(47)

For condensate of size $R\sim\frac{1}{\kappa_{m_{a}+\Omega}}$, if the scales
$\kappa_{m_{a}+\Omega}$ and $\kappa_{m_{a}-\Omega}$ are very different, the
decay time scale goes as

$\displaystyle
T\sim\frac{\pi^{3}R^{3}m_{a}^{2}\tilde{\phi}^{2}}{\frac{\pi}{16}\left(\frac{C\beta\tilde{\phi}}{\pi
f_{a}}B_{0}m_{a}\pi^{2}R^{2}\right)^{2}}\sim\frac{16}{m_{a}\pi^{2}\left(\frac{C\beta}{\pi
m_{a}f_{a}}B_{0}\right)^{2}}\left(\frac{\kappa_{m_{a}+\Omega}}{m_{a}}\right).$
(48)

Similarly, for a condensate of size $R\sim\frac{1}{\kappa_{m_{a}-\Omega}}$,
the decay time scale goes as

$\displaystyle
T\sim\frac{\pi^{3}R^{3}m_{a}^{2}\tilde{\phi}^{2}}{\frac{\pi}{16}\left(\frac{C\beta\tilde{\phi}}{\pi
f_{a}}B_{0}m_{a}\pi^{2}R^{2}\right)^{2}}\sim\frac{16}{m_{a}\pi^{2}\left(\frac{C\beta}{\pi
m_{a}f_{a}}B_{0}\right)^{2}}\left(\frac{\kappa_{m_{a}-\Omega}}{m_{a}}\right).$
(49)

The two time scales obtained in Eq. 48 and 49 can help us identify the region
in parameter space where back reaction of axion condensates can be ignored. In
order to ignore back reaction, for $R\sim\frac{1}{\kappa_{m_{a}+\Omega}}$, the
time scale of decay given in Eq. 48 should be larger than the time period of
outgoing radiation given by
$\frac{2\pi}{|m_{a}+\Omega|}\sim\frac{2\pi}{m_{a}}$. We can rewrite the time
scale in Eq. 48 substituting $\kappa_{m_{a}+\Omega}\sim
m_{a}\sim\Omega\sim\omega_{P}$ as

$\displaystyle T\sim\frac{16}{m_{a}\pi^{2}\left(\frac{C\beta}{\pi
m_{a}f_{a}}B_{0}\right)^{2}}.$ (50)

This time scale is larger than $\frac{2\pi}{m_{a}}$ so long as
$\left(\frac{C\beta}{\pi m_{a}f_{a}}B_{0}\right)^{2}$ is not much larger than
$1$. In the case of QCD axions, $m_{a}f_{a}\sim\Lambda_{\text{QCD}}^{2}$ and
even for the strongest fields in pulsars we can take
$B_{0}<\Lambda_{\text{QCD}}^{2}$. As an example, a magnetic field of $10^{14}$
Gauss found in magnetars translates to an energy scale of $\sim
1\,\text{MeV}^{2}$. As a result, at weak coupling, the time scale in Eq. 50 is
much larger than the time period of outgoing radiation.

When $R\sim\frac{1}{k_{m_{a}-\Omega}}$, the relevant decay time scale of Eq.
49 must be larger than the time period of the outgoing radiation given by
$\frac{2\pi}{|m_{a}-\Omega|}$. For $\kappa_{m_{a}-\Omega}\ll m_{a}$ and
$m_{a}\sim\Omega\sim\omega_{P}$ the time period of the outgoing wave
$\frac{2\pi}{|m_{a}-\Omega|}\sim\frac{2\pi}{m_{a}}$. Thus, in order to ignore
back reaction we need

$\displaystyle T\sim\frac{16}{m_{a}\pi^{2}\left(\frac{C\beta}{\pi
m_{a}f_{a}}B_{0}\right)^{2}}\left(\frac{\kappa_{m_{a}-\Omega}}{m_{a}}\right)>\frac{2\pi}{m_{a}}.$
(51)

This condition is satisfied as long as $\left(\frac{C\beta}{\pi
m_{a}f_{a}}B_{0}\right)^{2}\leq\left(\frac{\kappa_{m_{a}-\Omega}}{m_{a}}\right)\ll
1$. If we set $\left(\frac{C\beta}{\pi
m_{a}f_{a}}B_{0}\right)^{2}\sim\left(\frac{\kappa_{m_{a}-\Omega}}{m_{a}}\right)$
the decay time scale is of the order $\frac{1}{m_{a}}$ which can be a few
seconds or smaller for ultralight axions.

We conclude this section with an estimate for the total energy released in
electromagnetic radiation. As discussed in Visinelli:2017ooc , for moderately
dense axion condensates, $R\sim m_{a}^{-1}$ and $\tilde{\phi}\sim f_{a}$. In
this case, we can write the total radiated energy as
$m_{a}^{2}f_{a}^{2}R^{3}\sim f_{a}^{2}/m_{a}$. For an axion mass of
$10^{-10}-10^{-12}$ eV, the energy released is of the order of a solar mass.
Large condensates $R\gg m_{a}^{-1}$, which are also referred to as dilute
condensates, on the other hand can release energy of the order of
$\frac{m_{P}^{2}}{m_{a}}\frac{1}{(m_{a}R)^{2}}$, where $m_{\text{P}}$ is the
Planck mass Visinelli:2017ooc . For an axion mass of $10^{-10}-10^{-12}$ eV
and $Rm_{a}\sim 10-100$, this estimate too can be of the order of a few solar
masses. Thus we see that large axion condensates(dilute) $Rm_{a}\gg 1$ can
radiate as efficiently as smaller ones $R\sim m_{a}^{-1}$ (dense) if there is
resonant enhancement in radiation for the former. In both cases, energy of the
order of a few solar masses can be radiated away within a time scale of
seconds.

##  5 conclusion

Time dependent magnetic fields are ubiquitous in astrophysics. Strong time
dependent fields are often found in pulsars, supernovae and mergers of neutron
stars. We can expect electromagnetic radiation from axion condensates in the
background of such external magnetic fields to experience significant
modification due to this time dependence, provided the corresponding frequency
scale is comparable to the axion mass scale. In this paper we demonstrate this
effect by considering the most simple form of time dependence for the external
magnetic field: a sinusoidal variation in time. We choose such a form for the
time dependence in order to extract the physics of interest which clarifies
the effect of the two competing scales on radiation. We reserve the analysis
of a more realistic time dependence for future work. Our work however captures
the essential features of radiation from axion condensate in a time dependent
magnetic field. We find that, large axion condensates $Rm_{a}\gg 1$, which
don’t radiate efficiently in general, can do so in the presence of an
oscillating background field provided the frequency of oscillation is close to
the axion mass scale. This enhancement in radiation is somewhat analogous to
the resonant enhancement in radiation of axion condensates experienced in a
plasma background. We also analyzed the combined effects of an oscillating
magnetic field and a plasma medium on axion radiation. Not surprisingly, we
find that the condition for resonant enhancement of radiation now depends on
both the plasma frequency and the background field frequency. This enhancement
in radiation can result in a release of few solar mass worth energy in a few
seconds for ultralight QCD axion of mass $10^{-10}-10^{-12}$ eV. Of course,
this enormous release of energy can also take place for ultralight axion like
particles. We also emphasize that several astrophysical phenomena can give
rise to alternating magnetic fields with a frequency which itself varies in
time. Such time varying frequencies can result in resonant radiation for a
range of axion mass scales at a certain interval in time when the two
frequency scales (the mass of the axion and the frequncy of the time dependent
external field) become comparable. This can enable a non-radiating axion
condensate at a certain instant in time to radiate efficiently at a later
time. We demonstrate this behavior with a magnetic field frequency varying
linearly in time in the absence of a plasma background. Again, the physics of
interest is not dependent on the specific form of time dependence that the
magnetic field frequency can have. Our analysis can easily be extended to any
other slow variation for the time dependent frequency, as long as the change
in magnetic field frequency over a single time period of the outgoing
radiation is smaller than the frequency of the outgoing radiation. In future
work we plan to consider a more general form for the time dependence including
monotonic growth in the magnetic field. Such magnetic fields can be found in
supernovae and post merger dynamics of neutron stars Kiuchi:2015sga ;
Skoutnev:2021chg . We also intend to analyze the effects of spatial variation
in background fields and the plasma frequency for large axion condensates in
future work.

##  6 acknowledgment

This research was supported in part by DOE grant number GR-024204-00001.

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