###### Abstract

Probing the existence of hypothetical particles beyond the Standard model
often deals with extreme parameters: large energies, tiny cross-sections,
large time scales, etc. Sometimes laboratory experiments can test required
regions of parameter space, but more often natural limitations leads to poorly
restrictive upper limits. In such cases astrophysical studies can help to
expand the range of values significantly. Among astronomical sources, used in
interests of fundamental physics, compact objects — neutron stars and white
dwarfs, — play a leading role. We review several aspects of astroparticle
physics studies related to observations and properties of these celestial
bodies. Dark matter particles can be collected inside compact objects
resulting in additional heating or collapse. We summarize regimes and rates of
particle capturing as well as possible astrophysical consequences. Then we
focus on a particular type of hypothetical particles — axions. Their existence
can be uncovered due to observations of emission originated due to Primakoff
process in magnetospheres of neutron stars or white dwarfs. Alternatively,
they can contribute to cooling of these compact objects. We present results in
these areas, including upper limits based on recent observations.

###### keywords:

dark matter; axions; neutron stars; white dwarfs

1 1 0 https://doi.org/ Astroparticle physics with compact objects
Astroparticle physics with compact objects Peter Tinyakov 1, Maxim Pshirkov
2,3,4 and Sergei Popov 2* Peter Tinyakov, Maxim Pshirkov and Sergei Popov
Tinyakov, P.; Pshirkov, M.; Popov, S. Correspondence<EMAIL_ADDRESS>Tel.: +7-495-939-5006 (S.P.)

##  1 Introduction

It is a commonplace to state that compact objects — white dwarfs, neutron
stars, and black holes, — are unique natural laboratories which allows us to
study matter and various processes under extreme conditions. However, this is
indeed true and quite often best limits on different hypothesis in fundamental
physics beyond standard scenarios are given by observations of these
astrophysical sources.

Leaving black holes (BHs) aside, neutron stars (NSs) and white dwarfs (WDs)
with their high density and temperature of interiors and large external and
internal magnetic fields in many respects are well-suited for studies of
interactions with even very elusive particles (like dark matter candidates,
axions and axion-like particles, etc.) and for their production.

Internal densities of NSs can reach up to $\lesssim 10$ nuclear saturation
density Haensel et al. (2007). This allows to obtain a reasonable rate of
interaction even with particles with very small cross-section reducing their
energy in the process and eventually trapping them inside compact objects.
Despite WDs have about seven orders of magnitude lower density, they also can
capture weakly interacting particles.

High densities combined with high temperatures result in non-trivial cooling
processes of compact objects which are born very hot. The main cooling channel
for many of them is related to neutrino emission Yakovlev and Pethick (2004).
However, it was also proposed that axions can play a role in thermal energy
losses in both NSs and WDs (see , e.g. Isern et al. (1992); Sedrakian (2016)).

In the case of magnetars Turolla et al. (2015) and magnetic WDs (MWDs)
Ferrario et al. (2015) magnetic moments can be as high as $10^{33}$ G cm3.
Magnetospheres of such compact objects can interact with axions via the
Primakoff process, thus working as a kind of a haloscope Raffelt (1996).
Typically, magnetic fields of other astronomical objects (or in realistic
medium) are not that high to produce significant effect for discussed
parameters of particles. Thus, NSs and WDs can give us a unique opportunity to
discover axions with purely astrophysical methods.

In this review BHs generally are not discussed, together with different exotic
solutions like boson stars Colpi et al. (1986), fermion stars Lee and Pang
(1987), pion stars Brandt et al. (2018), etc. Instead we focus on three
abovementioned aspects linking particle physics beyond the Standard model with
properties of NSs and WDs.

At first, in Sec. 2, we discuss how dark matter (DM) particles can appear
inside NSs and WDs due to capturing by progenitors or already after the
compact object formation. Then, studies of the main possible consequences —
heating and collapse, — of particles capture are reviewed.

In the next section we turn to a specific type of hypothetical particles —
axions (including so-called axion-like particles, ALPs). We present theory of
axion interaction with magnetospheres of NSs. We discuss interactions with
axions of cosmological origin that could constitute dark matter and those
produced in NSs. Interaction of axion clusters with magnetospheres is also
discussed in the context of models of fast radio bursts (FRBs) origin.

Finally, in Sec. 4 we come to WDs and review possible role of axions in
cooling of these compact objects after which present several upper limits on
properties of these particles based on observations of WDs.

##  2 Compact objects and dark matter

According to the theory of structure formation, at galactic and larger scales
the distributions of baryons and dark matter are correlated. Stars are
therefore formed in regions of DM overdensities. As any gravitating objects,
they have some amount of DM gravitationally bound to them. They may also
accumulate DM during their lifetime. While the total amount of the accumulated
DM stays tiny, it may nevertheless have observable effects. In this section we
review the DM capture mechanisms and estimate the amount of DM that may be
accumulated by a star in different conditions. We then discuss possible
observational signatures that may result.

###  2.1 Capture of DM by stars

Stars capture DM at formation and during their lifetime by two different
mechanisms. The second one — accumulation during the star lifetime — requires
non-zero DM-to-nucleon interactions for capture; the first mechanism is
gravitational at the stage of capture, but still requires DM interactions with
nucleons for the thermalization of the captured DM. The two mechanisms
generally give comparable contributions to the total amount of DM accumulated
by a star. We start with the second one as it is more extensively studied in
the literature and allows for simple analytical estimates.

####  2.1.1 Capture during star lifetime

To gain a qualitative understanding of the capture mechanism, consider an
isolated star111An enhancement of capture rate may be expected in binary
systems, but only by a factor of a few Brayeur and Tinyakov (2012). embedded
in a gas of non-interacting DM particles, as would be the case for stars in
galactic halos. Assume the DM distribution in velocities is Maxwellian,
characterised by a velocity dispersion $\bar{v}$. Assume as well that the star
is at rest with respect to the DM distribution. This is not the case for most
of the stars in halos as their typical velocities are the same as those of DM
particles, that is, of order $\bar{v}$. Also, the DM distribution in
velocities may not be Gaussian in a realistic halo. We will discuss later the
effect of these complications on the capture rate.

The star forms a gravitational potential well where DM particles are confined
provided they have negative total energy. A particle from the DM halo that
originally has a positive energy can lose it in collisions with the stellar
material and become gravitationally bound. Since the collisions happen while a
particle is inside the star, it settles on a star-crossing orbit and will
eventually come back and cross the star again. The energy losses will thus
continue until the particle reaches an equilibrium with the matter inside the
star. The rate of capture is determined by the number of particles that get
gravitationally bound after the first star crossing.

It follows immediately from this picture that there is an upper bound on the
capture rate $F$ independent of the energy loss mechanism. It is given by the
rate of star crossings by the DM particles. For a given particle with
asymptotic velocity, $v$, the cross section of the star crossing is

$\sigma_{\rm cross}=\pi R_{*}^{2}\left(1+{2GM_{*}\over R_{*}v^{2}}\right)=\pi
R_{*}^{2}\left(1+{v_{\rm esc}^{2}\over v^{2}}\right),$ (1)

where $R_{*}$ and $M_{*}$ are the star radius and mass, $G$ is the
gravitational constant, and $v_{\rm esc}$ the escape velocity from the surface
of the star. Here for simplicity the Newtonian gravity has been assumed, which
gives a reasonable approximation even for a NS. The account for General
Relativity corrections changes the capture rate by a factor of order 1
Kouvaris (2008). The second term in the parenthesis is the result of
gravitational focusing; this term is dominant in most cases. Averaging this
expression over the Maxwellian distribution of the velocity dispersion
$\bar{v}$ yields the maximum value of the DM capture rate $F$,

$F_{\rm max}=\left({8\pi\over 3}\right)^{1/2}nR_{*}^{2}\bar{v}\left(1+{3v_{\rm
esc}^{2}\over 2{\bar{v}}^{2}}\right)\simeq\sqrt{6\pi}nR_{*}^{2}v_{\rm
esc}^{2}/{\bar{v}}=\sqrt{6\pi}n{R_{*}R_{g}\over{\bar{v}}}c^{2},$ (2)

where $n$ is the DM number density and $R_{g}=2GM_{*}/c^{2}$ is a stellar
gravitational radius. In the final expression we have neglected the first term
in the r.h.s. of eq. (1) as compared to the second one. Detailed calculations
can be found in Refs. Press and Spergel (1985); Gould (1987, 1988). Eq. (2)
suggests that loosely bound haloes with smaller velocity dispersion give
higher capture rates.

To get an idea of how much DM can possibly be captured by this mechanism let
us substitute the numbers. For a Sun-like star in the typical Galactic
environment, assuming the lifetime of 10 Gyr eq. (2) gives for the total mass
of the accumulated DM $M_{\rm tot}\sim 5\times 10^{47}~{}{\rm GeV}\sim 5\times
10^{-10}~{}M_{\odot}$. For a NS in the same conditions $M_{\rm tot}\sim
10^{43}~{}{\rm GeV}\sim 10^{-14}~{}M_{\odot}$. The upper limits may be higher
by 2-3 orders of magnitude in an environment with higher DM density and
smaller velocity.

The capture rate, however, is usually further suppressed for two reasons: not
every particle that crosses the star may get scattered, and even if it does,
not every particle loses enough energy to become gravitationally bound. The
scattering is controlled by the cross section of DM interactions with
nucleons. There is a critical value of this cross section $\sigma_{\rm cr}$ at
which the star becomes opaque to DM particles, that is, at which a DM particle
scatters once per crossing on average. It is estimated as $\sigma_{\rm
cr}=R_{*}^{2}m_{p}/M_{*}$ and has the following value for the Sun, a WD and a
NS, respectively:

$\displaystyle\sigma_{\rm cr,~{}Sun}$ $\displaystyle\simeq 4\times
10^{-36}{\rm cm}^{2},$ $\displaystyle\sigma_{\rm cr,~{}WD}$
$\displaystyle\simeq 4\times 10^{-40}{\rm cm}^{2},$ $\displaystyle\sigma_{\rm
cr,~{}NS}$ $\displaystyle\simeq 8\times 10^{-46}{\rm cm}^{2}.$ (3)

For NSs and WDs we have neglected the effects of matter degeneracy; they will
be commented on later. If the DM-nucleon cross section is smaller than the
critical one, the capture rate gets proportionally suppressed. Note that this
is a likely case as the experimental bounds on the DM-nucleon cross section
$\sigma_{\rm DM-n}$ are becoming quite strict. In the case of the spin-
independent interactions they are at the level of $10^{-46}$ cm2 at DM mass
around $50$ GeV Aprile et al. (2018), while for the spin-dependent
interactions the limits on the cross section are at the level of
$10^{-40}-10^{-41}$ cm2 Aprile et al. (2019); Akerib et al. (2017); Xia et al.
(2019); Amole et al. (2019). The constraints become much weaker for DM masses
below $10$ GeV and higher than several TeV.

Consider now the effect of energy losses on the capture rate. A particle
crossing the star gets trapped if it loses the energy $E_{\rm loss}$ which is
larger than its asymptotic energy. If $E_{\rm loss}\gtrsim m\bar{v}^{2}/2$
most of the particles crossing the star get trapped and no additional
suppression results as compared to eq. (2). In the opposite case only a
fraction of particles that have sufficiently small asymptotic velocity
$v<\bar{v}$ such that $mv^{2}/2<E_{\rm loss}$ will become gravitationally
bound. Given the distribution of DM particles over velocities it is a matter
of a direct calculation to find the capture as a function of $E_{\rm loss}$.
Assuming all particles lose the same energy one gets the capture rate

$F=\sqrt{6\pi}{\rho_{\rm DM}R_{*}R_{g}\over
m{\bar{v}}(1-R_{g}/R_{*})}\left[1-\exp\left(-{3E_{\rm loss}\over
m\bar{v}^{2}}\right)\right]f.$ (4)

Here $\rho_{\rm DM}$ is the DM ambient density and $f$ is the fraction of
particles that scatter among all particles crossing the star

$\begin{array}[]{ll}f=\sigma/\sigma_{\rm cr}&\text{if $\sigma<\sigma_{\rm
cr}$},\\\ f=1&\text{otherwise.}\end{array}$ (5)

For completeness, we have also included in eq. (4) the relativistic correction
$1/(1-R_{g}/R_{*})$ Kouvaris (2008) which however is only noticeable in case
of NSs where it leads to a slight enhancement of the rate. We will disregard
it in what follows.

The dependence of the capture rate on the energy loss $E_{\rm loss}$ is
controlled by the factor in the square brackets. When the energy loss in a
single star crossing is of order or larger than the typical asymptotic energy
$m\bar{v}^{2}/2$, this factor is close to 1 and gives no extra suppression of
the capture rate. In this case, and assuming all particles scatter ($f=1$) one
recovers the maximum rate (2). Note that in this regime the mass capture rate
$mF$ is independent of the DM mass $m$ at a given ambient DM density
$\rho_{\rm DM}$. In the opposite limit when the energy losses are small,
$E_{\rm loss}\ll m\bar{v}^{2}$, the capture rate is further suppressed. In
this regime the mass capture rate is inversely proportional to the DM mass.
Note also that in this case the dependence on the DM asymptotic velocity is
much stronger, $F\propto 1/\bar{v}^{3}$. In summary,

$F\simeq\left\\{\begin{array}[]{ll}\vrule
height=18.0pt,depth=18.0pt,width=0.0pt\displaystyle\sqrt{6\pi}{\rho_{\rm
DM}R_{*}R_{g}\over m{\bar{v}}}\,f&\quad\text{at large $E_{\rm loss}\gg
m\bar{v}^{2}$},\\\ \vrule
height=18.0pt,depth=18.0pt,width=0.0pt\displaystyle\sqrt{6\pi}{\rho_{\rm
DM}R_{*}R_{g}\over m{\bar{v}}}{3E_{\rm loss}\over
m\bar{v}^{2}}\,f&\quad\text{at small $E_{\rm loss}\ll
m\bar{v}^{2}$}.\end{array}\right.$ (6)

The extra suppression factor $\propto E_{\rm loss}/(m\bar{v}^{2})$ at small
$E_{\rm loss}$ arises because in this case only a small fraction of the DM
particles — those that have $mv^{2}\lesssim E_{\rm loss}$ — can lose enough
energy in one star crossing and become gravitationally bound. Clearly, in this
regime only the low-velocity part of the DM distribution is important. The
velocity dependence of the capture rate (6) originates in this case from the
volume element in the velocity space $d^{3}v$ and will be the same for any
velocity distribution that, like the Maxwellian one, goes to a constant at
small $v$. Therefore, regardless of the details of the DM velocity
distribution, the capture rate is proportional to $F\propto 1/\bar{v}$ and is
given by eq. (2) times the probability of scattering $f$ for small $\bar{v}$,
and changes to $F\propto 1/\bar{v}^{3}$ as given by the small $E_{\rm loss}$
case of eq. (6) with possibly a different numerical coefficient.

When expressed in terms of the energy loss $E_{\rm loss}$ and the probability
of scattering $f$, eqs. (6) are completely general and apply to any type of DM
and/or stars. The specific features of the models determine these two
quantities. The probability $f$ is determined by the scattering cross section
through eq. (5) for particle DM, for primordial black holes $f=1$ .

Consider now the energy losses $E_{\rm loss}$. We start with the simplest case
of particle DM crossing an ordinary star. A typical kinetic energy of a DM
particle away from the star is $m\bar{v}^{2}/2$. When the particle falls onto
the star it accelerates and picks the velocity of order of the escape
velocity. For a Sun-like star the latter is $\sim 600$ km/s, which is about
factor 3 higher than the dispersion velocity in our Galaxy at the position of
the Sun. We can therefore estimate the particle velocity while crossing the
star as the escape velocity $v_{\rm esc}\propto\sqrt{R_{g}/R_{*}}$. At the
temperature of order $10^{7}$ K the protons in the core of a Sun-like star
move with the thermal velocity $\sqrt{3kT/m_{p}}\sim 500$ km/s, and with
smaller velocities away from the core. For the sake of the estimate we may,
therefore, consider the collision of DM particles with protons at rest. Assume
for simplicity that DM particles are much heavier than protons. The typical
energy transfer in the collision, in the frame of the star, is then $E_{\rm
loss}\sim m_{p}v_{\rm rel}^{2}/2$, where $v_{\rm rel}\sim v_{\rm esc}$ is the
relative velocity of particles. Depending on the environment this energy loss
may be larger or smaller than the particle asymptotic energy. In our Galaxy
and for DM particles heavier than $\sim 10$ GeV the energy loss is smaller
than typical asymptotic energy, so we are in the suppressed small $E_{\rm
loss}$ regime of eq. (6). In dwarf galaxies with typical velocity dispersion
$\sim 10$ km/s this regime occurs for DM masses $\gtrsim 500$ GeV, otherwise
the large $E_{\rm loss}$ regime of eq. (6) gives the right estimate of the
rate. Thus, in these two cases we have for the mass capture rate:

Sun-like, Milky Way: $\displaystyle\qquad\vrule
height=15.0pt,depth=15.0pt,width=0.0pt\displaystyle mF\sim 2\times
10^{30}\,{\rm GeV/s}\left({\rho_{\rm DM}\over 0.5\,{\rm
GeV/cm}^{3}}\right)\left({10\,{\rm GeV}\over m}\right)\,f,$ Sun-like, dwarf
galaxies, $m\gtrsim 500$ GeV: $\displaystyle\qquad\vrule
height=15.0pt,depth=15.0pt,width=0.0pt\displaystyle mF\sim 9\times
10^{34}\,{\rm GeV/s}\left({\rho_{\rm DM}\over 100\,{\rm
GeV/cm}^{3}}\right)\left({500\,{\rm GeV}\over m}\right)\,f,$ Sun-like, dwarf
galaxies, $m\lesssim 500$ GeV: $\displaystyle\qquad\vrule
height=15.0pt,depth=15.0pt,width=0.0pt\displaystyle mF\sim 9\times
10^{34}\,{\rm GeV/s}\left({\rho_{\rm DM}\over 100\,{\rm
GeV/cm}^{3}}\right)\,f,$ (9)

The numerical values in eqs. (2.1.1)–(9) are close to the maximum achievable
capture rates for the Sun-like star in the corresponding environment. Note,
however, that in view of eqs. (3), (5) for the Sun-like star the scattering
probability is $f\lesssim 10^{-10}$ if current experimental bounds on the
spin-independent DM-nucleon cross section are assumed.

In the case of compact objects — WDs or NSs, — the escape velocities are
substantially higher, $v_{\rm esc}\sim 6\times 10^{3}$ km/s and $v_{\rm
esc}\sim 0.5c$, respectively. In this case we are always in the large $E_{\rm
loss}$ regime of eq. (6) for DM masses below $\sim 1$ TeV. The capture rate is
then estimated as follows,

$\displaystyle\text{WD:}\quad\vrule
height=15.0pt,depth=15.0pt,width=0.0pt\displaystyle mF\sim 2\times
10^{28}\,{\rm GeV/s}\left({\rho_{\rm DM}\over 0.5\,{\rm
GeV/cm}^{3}}\right)\left({220\,{\rm km/s}\over\bar{v}}\right)\,f,$ (10)
$\displaystyle\text{NS:}\quad\vrule
height=15.0pt,depth=15.0pt,width=0.0pt\displaystyle mF\sim 2.5\times
10^{26}\,{\rm GeV/s}\left({\rho_{\rm DM}\over 0.5\,{\rm
GeV/cm}^{3}}\right)\left({220\,{\rm km/s}\over\bar{v}}\right)\,f.$ (11)

In the case of compact stars the critical cross sections are smaller, cf. eq.
(3), so the scattering probability $f$ is less suppressed than in the case of
Sun-like stars. For NSs, assuming current experimental limits on the
scattering cross section, it may reach values close to 1 even for the spin-
independent cross section. In all cases, however, the accumulation rates are
small: only a tiny fraction of $M_{\odot}$ can be accumulated over the age of
the Universe.

A number of subtleties arises in the case of capture by NSs because NS matter
is degenerate, and because the scattering happens at semi-relativistic
velocities. A detailed analysis of these subtleties and further discussion can
be found in Ref. Acevedo et al. (2021); Bell et al. (2021, 2020). Note that
the estimates presented above neglect the self-interactions of DM particles.
Accounting for such self-interactions may in some cases significantly changes
(enhances) the capture rate Güver et al. (2014).

####  2.1.2 Capture at star formation

In addition to the DM accumulated during the lifetime, stars capture some
amount of DM in the process of their formation. This capture mechanism is
different from the accumulation considered in the previous section as its
first stage is gravitational and thus is independent of the DM properties. We
summarise here this mechanism following Refs.Capela et al. (2013, 2014).

Stars are formed from the pre-stellar cores. These baryonic overdensities
create dips in the gravitational potential filled with slow DM particles bound
to the pre-stellar core. These slow particles are, of course, only a small
fraction of the ambient DM density, and their mass density is completely
negligible to that of the baryons. When the star forms, this bound DM is
dragged along by the time-dependent gravitational field of contracting
baryons. This process is slow in the sense that the star formation time, being
determined by the energy dissipation by the baryons, is much larger than the
free fall time. It can therefore be described in the adiabatic approximation.
When the star is finally formed, the space distribution of the bound DM
becomes peaked at the star position. This is the first stage of the DM
capture.

Among the DM particles that form the resulting cuspy profile, of interest to
us are those that have trajectories crossing the newly-born star. These
particles are in the same conditions as the ones captured by direct scattering
(cf. Sect. 2.1.1): they will sooner or later scatter off the star nucleons,
lose their energy and settle inside the star.

The amount of DM that is captured in this way is proportional to the density
of DM that was gravitatonally bound to the pre-stellar core, and depends on
the parameters of the DM distribution after the adiabatic contraction of
baryons. The bound DM density can be estimated in terms of the depth of the
gravitational potential of the pre-stellar core $\phi$ as follows:

$\rho_{\rm bound}=\rho_{\rm DM}{4\pi\over
3}\left({3\phi\over\pi\bar{v}^{2}}\right)^{3/2},$ (12)

where $\rho_{\rm DM}$ is the ambient DM density and its velocity distribution
is assumed to be Gaussian with the characteristic velocity $\bar{v}$. The
gravitational potential in turn can be estimated in terms of the parameters of
the pre-stellar core and is in the range $3\times 10^{-12}-3\times 10^{-11}$
for stellar masses $1-10$ $M_{\odot}$ Capela et al. (2013). In all cases one
has $\rho_{\rm bound}/\rho_{\rm DM}\ll 1$.

The distribution of the bound DM after the contraction of baryons can be
calculated numerically. The simulation is straightforward due to two
simplifications: the contribution of the DM into the gravitaional potential is
negligible, and therefore particle trajectories may be simulated one at a
time, and the process is adiabatic so that the detailes of the baryonic
contraction are irrelevant as long as it is sufficiently slow (in practice,
several times slower than the free fall time is enough). Combining the results
of Refs. Capela et al. (2013, 2014) one obtains the following estimates of the
total mass of the DM that acquires orbits crossing the star and may eventually
be captured, for two different star masses:

$\displaystyle\text{$M_{\odot}$:}\quad\vrule
height=15.0pt,depth=15.0pt,width=0.0pt\displaystyle M_{\rm tot}\sim 2\times
10^{-13}M_{\odot}\left({\rho_{\rm DM}\over 100\,{\rm
GeV/cm}^{3}}\right)\left({10\,{\rm km/s}\over\bar{v}}\right)^{3},$ (13)
$\displaystyle\text{$10~{}M_{\odot}$:}\quad\vrule
height=15.0pt,depth=15.0pt,width=0.0pt\displaystyle M_{\rm tot}\sim 6\times
10^{-11}M_{\odot}\left({\rho_{\rm DM}\over 100\,{\rm
GeV/cm}^{3}}\right)\left({10\,{\rm km/s}\over\bar{v}}\right)^{3}.$ (14)

These numbers can be compared to eq. (9). Taking $f\sim 10^{-10}$ and assuming
for a Sun-like star lifetime $\sim 10$ Gyr one has from direct capture, eq.
(9), $M_{\rm tot}\sim 3\times 10^{-15}M_{\odot}$ for the same environment of a
typical dwarf galaxy. We see that capture at the stage of star formation may
give a dominant contribution to the total amount of accumulated DM.

####  2.1.3 Thermalization of captured DM

Both mechanisms that have been described above require that DM particles which
have been captured on bound star-crossing orbits interact with the stellar
nucleons in order to eventually settle inside the star. This thermalization
process takes time that is controlled by the DM parameters: the mass $m$ and
the cross section $\sigma$ of DM with nucleons. The thermalization time can be
estimated as follows Kouvaris and Tinyakov (2011). One has to consider
separately two stages. At the first stage, the DM trajectories are mostly
outside of the star while crossing it twice per period. At the second stage
the DM particles move on orbits completely contained inside the star. In both
cases the energy loss happens over many orbits and can be described by a
differential equation in a continuous energy loss approximation. Solving this
differential equations gives the thermalization times for the two stages. For
the second stage the initial conditions are the same for both capture
mechanisms considered previously, and one has

$t_{2}={m^{3/2}\over\sqrt{3}\rho_{*}\sigma}{1\over\sqrt{kT_{*}}}\sim 64{\rm
yr}\,\left({m\over 100\,{\rm GeV}}\right)^{3/2}\left({10^{-36}\,{\rm
cm}^{2}\over\sigma}\right),$ (15)

where $T_{*}$ is the star core temperature which determines the final particle
energy.

For the first stage the initial conditions are different for the two capture
mechanisms. In the case of direct capture they are set by the energy loss in
the first collision. One has Kouvaris and Tinyakov (2011):

$t_{1}\sim{3\pi mR_{*}^{3/2}\sigma_{\rm cr}\over
4m_{p}cR_{g}^{1/2}\sigma}\sqrt{\left|{E_{*}\over E_{0}}\right|}\sim 20{\rm
yr}\,\left({m\over 100\,{\rm GeV}}\right)^{3/2}\left({10^{-36}\,{\rm
cm}^{2}\over\sigma}\right),$ (16)

where $R_{*}$ is the stellar radius, $R_{g}$ is its gravitational radius,
$m_{p}$ is proton mass, $E_{*}=GMm/R_{*}$ is the particle binding energy at
the star surface, and $E_{0}=m_{p}c^{2}R_{g}/R_{*}$ is a typical energy loss
in the first collision, so that $E_{*}/E_{0}\sim m/m_{p}$. In the case of the
direct capture by a NS the parameters of the NS should be used in these
estimates. Assuming $\sigma\sim\sigma_{\rm cr}$, eqs.(15) and (16) then give
$t_{1}\sim 0.16$ s, $t_{2}\sim 100$ s for $m=100$ GeV, i.e., in this case the
thermalization time is negligibly short.

For the capture at formation the initial conditions are determined by the size
of the pre-stellar core which sets the size of largest orbits. The ratio of
the final to initial energy becomes instead $E_{*}/E_{0}\sim R_{\rm
psc}/R_{*}$ where $R_{\rm psc}$ is the radius of the pre-stellar core. At
$m=100$ GeV this gives an extra factor 100 in the thermalization time, so that

$t_{1}\sim 2\times 10^{3}{\rm yr}\,\left({m\over 100~{}{\rm
GeV}}\right)\left({10^{-36}{\rm cm}^{2}\over\sigma}\right).$ (17)

The thermalization time becomes much larger than the age of the Universe for
allowed spin-independent cross sections and DM masses in the $100$ GeV range.

The above estimates of the thermalization time have to be modified in the case
of a direct capture by the NS. In this case the degeneracy of the nuclear
matter has to be taken into account. For heavy DM the modifications are not
dramatic Goldman and Nussinov (1989); Kouvaris and Tinyakov (2011) because in
the case of a neutron star DM particles falling onto the star have semi-
relativistic velocities, so that for particles much heavier than protons the
momentum transfer is of the order of the neutron Fermi momentum. For lighter
DM particles the modifications are large. The recent detailed calculations can
be found in Refs. Bertoni et al. (2013); Cermeño et al. (2017); Garani et al.
(2019, 2021). In case of very small interaction cross-sections, $\sigma\ll
10^{-36}~{}{\rm cm}^{2}$, and corresponding long thermalization times, the
capture process could be affected by influence of external perturbers which
would prevent some particles from moving into completely contained orbits
Montero-Camacho et al. (2019).

Once the DM particles thermalize with the star, they form a spherical cloud in
the center of a radius

$r_{\rm th}=\left({9T_{*}\over 8\pi G\rho_{*}m}\right)^{1/2}.$ (18)

For a Sun-like star with a core temperature of $T_{*}=1.5\times 10^{7}$ K,
core density $\rho_{*}=150$ g/cm3, and DM mass of $100$ GeV this cloud is of
the size of $\sim 1000$ km. For a typical old NS the thermal radius is $r_{\rm
th}\sim 20$ cm at temperature $10^{5}$ K.

###  2.2 Signatures of DM in compact stars

We have seen in the previous section that, even though the DM may accumulate
in stars, its total amount may only constitute a tiny fraction $\lesssim
10^{-10}$ of the stellar mass. It may nevertheless produce a number of
potentially observable signatures, particularly in compact stars. We discuss
below two of these signatures: the heating of NSs by DM annihilations, and
destruction of the star by DM collapse into a small BH inside the star.

####  2.2.1 Annihilation and heating

Because of a very long DM lifetime the decay of the accumulated DM can be
safely ignored. In many conventional DM models, however, the DM particles can
annihilate into the standard model ones. The annihilation rate is proportional
to the square of the DM density which is many orders of magnitude higher for
DM accumulated inside the star compared to the Galactic DM density. The
annihilation may therefore be efficient even for very low annihilation cross
sections, particularly for NSs where the accumulated DM concentrates in a very
small central region. These annihilations produce heat. In the case of a Main
sequence star this heat is smaller compared to energy release of nuclear
reactions by at least several orders of magnitude (cf. eqs. (2.1.1) – (9) with
$f\ll 1$) and is therefore negligible. On the contrary, NSs have no major
internal heat source; the heat produced by DM annihilations can in principle
be detectable Kouvaris (2008); Kouvaris and Tinyakov (2010).

In principle, depending on the structure of the dark sector, DM annihilations
amplified by its concentration in neutron stars may produce more specific
signatures. For instance, it has been pointed out in Ref. Leane et al. (2021)
that if DM annihilates into sufficiently long-lived particles that can escape
from the neutron star and decay outside into the visible sector, the existing
experiments such as Fermi and HESS may impose constraints on the DM parameters
several orders of magnitude better than those from the direct detection
experiments. Here we concentrate on the more model-independent heating
signature.

In case of annihilation, the current amount of DM in a star is determined by
the balance between the accretion and annihilation (for DM masses larger than
$\sim 10$ GeV that we consider here, the evaporation is negligible; for
smaller masses see Ref.Garani and Palomares-Ruiz (2021)). In this case the
number of DM particles in a star $N(t)$ is governed by the equaion

${dN(t)\over dt}=F-C_{A}N(t)^{2},$ (19)

where as before $F$ is the accretion rate and
$C_{A}=\langle\sigma_{A}v\rangle/V$ is the thermal averaged annihilation cross
section $\sigma_{A}$ divided by the annihilation volume $V$. The solution to
this equation asymptotes to a constant over the time scale
$\tau=1/\sqrt{FC_{A}}$. This time scale is typically very short: for a
characteristic value of $\langle\sigma_{A}v\rangle\sim 10^{-26}$cm3/s and the
rate of eq. (11) taken at $f=1$ one finds $\tau\sim 10^{-4}$ yr. For the cross
sections as low as $\langle\sigma_{A}v\rangle\gtrsim 10^{-46}$cm3/s this
timescale is still below 1 Myr Kouvaris and Tinyakov (2010). In the
equilibrium when the annihilation and capture rates exactly balance each
other, the power produced by the annihilation is given simply by the mass
accretion rate of eq. (11).

In the absence of other heat sources, the ultimate NS surface temperature is
determined by equating the mass capture rate (11) to the black body radiation
from the star surface $L=4\pi R_{*}^{2}\sigma_{B}T^{4}$ where $\sigma_{B}$ is
the Stephan-Boltzmann constant. This gives

$T=\left({mF\over 4\pi R_{*}^{2}\sigma_{B}}\right)^{1/4}\sim 5000\,{\rm K}.$
(20)

In the absence of annihilations a NS would cool down below this temperature in
about $2\times 10^{7}$ yr. Having observed an old NS with surface temperature
in or below this range would thus impose constraints on the DM annihilation
cross sections. Depending on the observed NS temperature and on the DM density
at the star location, cross sections a few orders of magnitude below current
direct detection limit may be constrained in this way.

An important observation has been made in Ref.Baryakhtar et al. (2017) where
it was noted that capture of DM particles by the NS itself provides a heat
source comparable in power to that of DM annihilations. The gravitational
potential inside the NS is $0.2-0.3$, so capturing a DM particle releases a
sizeable fraction of its rest mass in the form of thermal energy. This energy
source is therefore only a factor of a few less efficient than annihilations.
This mechanism, referred to as ’kinetic heating’, is completely free from the
assumptions about the DM annihilation cross section. The constraints on the DM
from the kinetic heating have been worked out in detail in Refs. Raj et al.
(2018); Bell et al. (2018); Acevedo et al. (2020); Joglekar et al. (2020);
Acevedo et al. (2021).

Internal heating can be important also for WDs. This was recently analysed in
Panotopoulos and Lopes (2020). These authors studied the case of relatively
light weakly interacting massive particles thermalized in compact objects and
compared theoretical predictions on additional luminosity with observations of
10 WDs in a globular cluster M4. In the mass range $\sim 0.1$ –5 GeV the limit
on the cross section of DM particles on baryons is
$\sigma_{\mathrm{DM-n}}\lesssim$ few $10^{-41}$ cm2.

A more complicated model of DM annihilation (with formation of metastable
mediators decaying into photons) in application to WDs from the same globular
cluster M4 was presented in Cermeño and Pérez-García (2018). Mediators can
decay to $\gamma$-photons outside a WD, thus this emission can be detected. If
mediators decay inside a WD, then an additional contribution to its thermal
luminosity appears. Both variants were discussed in Cermeño and Pérez-García
(2018) and compared with limits on the thermal luminosity of WDs in M4. This
allows to put some model-dependent constrains on the lifetime of mediators.

In principle, the arguments based on the NS heating by DM can be inverted and
used to search for DM. In practice, however, the discovery potential of this
method is limited by potential presence of alternative heat sources in old NSs
Gonzalez and Reisenegger (2010); Hamaguchi et al. (2019). Other DM-related
effects may help to overcome this difficulty Casanellas and Lopes (2011);
Camargo et al. (2019); Graham et al. (2015, 2018); Bell et al. (2019); Acevedo
and Bramante (2019); Janish et al. (2019).

####  2.2.2 BH formation and star destruction

Another potential signature of DM accumulation in stars is related to its
possible collapse into a small seed BH inside the star with the subsequent
accretion of the stellar matter and the star destruction. In order to
collapse, the DM has to be concentrated in a small volume. Only models with
non-annihilating DM (e.g., asymmetric DM models Petraki and Volkas (2013);
Zurek (2014)) are relevant in this context as even a tiny annihilation cross
section is enough to burn out the accumulated DM. The size of the DM cloud
inside the star depends on the stellar core density and is many orders of
magnitude smaller for compact stars than for Main sequence ones. We
concentrate here on NSs and WDs as providing most favorable conditions for the
collapse.

The arguments based on the DM collapse into a BHs inside compact stars can be
used in both directions: an observation of a NS or a WD in a given DM-rich
environment would exclude/constrain those DM models/parameters for wich a BH
should have been formed inside a WD/NS and converted it into a (sub)solar mass
BH. Alternatively, in models where this destruction could take place only in
some particularly favorable conditions one predicts the existence of
(sub)solar mass BHs which are not formed by conventional stellar evolution
mechanisms.

The general stages of the DM collapse inside a star are as follows. The
captured DM forms the cloud of radius given by eq. (18) in the center of the
star. As the DM accumulates, its mass density grows and at some point becomes
larger than the density of the stellar matter. At this point the cloud becomes
self-gravitating and starts to collapse. As the cloud becomes stronger
bounded, the released energy is evacuated through the collisions with the star
nucleons. The shrinking continues either until the extra source of pressure
arises that can stabilize the DM sphere, or until the BH is formed in the
absence of such a source.

At this last stage, the details of the collapse depend on the DM nature –
whether it is bosonic or fermionic. In case of free fermions the Fermi
pressure will halt the collapse unless the number of particles is sufficiently
large. The required number of DM particles is given by an analog of the famous
Chandrasekhar condition which in the case at hand reads:

$N_{\rm min,f}\sim\left({M_{\rm Pl}\over m}\right)^{3}=2\times
10^{51}\left({100\,{\rm GeV}\over m}\right)^{3}.$ (21)

Note that for $m\sim 1$ GeV this equation roughly reproduces the number of
neutrons in a NS. In case of free bosons this is the quantum uncertainty
principle that halts the collapse if the number of particles is smaller than

$N_{\rm min,b}\sim\left({M_{\rm Pl}\over m}\right)^{2}=2\times
10^{34}\left({100\,{\rm GeV}\over m}\right)^{2}.$ (22)

The required number of particles in this case is much smaller than in the case
of fermions. Note, however, that eq. (22) assumes that bosons are non-
interacting; interactions may significantly modify this equation as will be
discused below.

Consider first the case of free fermions. In this case the number of DM
particles required to make a BH is given by eq. (21). Let us compare it to the
number that can be accumulated according to eq. (11). Assuming the NS lifetime
of 1 Gyr, eq. (11) implies the total accumulated amount of $8\times 10^{42}$
GeV in the conditions typical for Milky Way, regardless of the DM mass.
Dividing by the mass and requiring that the resulting number is larger than
that of eq. (21) one finds the condition

$m\gtrsim 10^{7}~{}{\rm GeV},$ (23)

in agreement with the results of Ref. Goldman and Nussinov (1989). The DM
models with heavier fermionic DM would then contradict to observations of NS
in the Milky Way, provided the DM has scattering cross section on neutrons
that is larger than $\sigma\gtrsim 10^{-45}$ cm2 so that eq. (11) is valid
with $f=1$.

This constraint on the mass can be improved by excluding lower DM masses in
two ways. The minimum mass to form a BH scales like the square root of the
capture rate. The latter, in turn, scales like $\rho_{\rm DM}/\bar{v}$.
Observation of NSs in DM-rich environments thus leads to stronger constraints.
For instance, in dwarf galaxies both the DM density is higher by a factor up
to $\sim 200$, and the velocity of DM particles is lower by up to $30$ times.
Thus, in dwarf galaxies the capture rate may be higher by nearly 4 orders, and
correspondingly the constraints on the fermionic DM mass start at masses lower
by 1.5-2 orders of magnitude compared to eq. (23). The second improvement may
be achieved by taking in consideration the accumulation of DM by the NS
progenitor. The efficient accumulation of DM in the Main sequence stars
requires large DM-nucleon cross section in view of eqs. (3). The argument,
therefore, applies to models with the spin-dependent DM-nucleon cross section
that is less constrained. Depending on the value of this cross section, masses
as low as $\sim 10$ TeV can be potentially constrained Kouvaris and Tinyakov
(2011).

We have considered so far non-interacting fermions. Adding self-interactions
may change the collapse conditions. The simplest Yukawa interaction with the
scalar mediator gives the attractive interaction which may reduce the number
of DM particles required for the collapse Kouvaris (2012); Bramante et al.
(2014); Kouvaris et al. (2018). Note, however, that the non-relativistic
approximation becomes inadequate in the relativistic regime close to the
collapse Gresham and Zurek (2019), which may limit this reduction to a mere
factor of a few as compared to the non-interacting case.

Let us turn now the case of bosonic DM. In the case of bosons one more
phenomenon has to be taken into account, the formation of the Bose-Einstein
condensate (BEC) Kouvaris and Tinyakov (2011). The condition for the BEC
formation (large phase space density) can be formulated in terms of DM number
density $n$ and the star core temperature $T_{c}$ and reads

$n\gtrsim 5\times 10^{28}{\rm cm}^{-3}\left({m\over{\rm
GeV}}\right)^{3/2}\left({T_{c}\over 10^{5}{\rm K}}\right)^{3/2},$ (24)

where we assumed that DM particles are in thermal equilibrium with the star.
In view of eq. (11) this condition becomes satisfied very quickly. Once the
BEC is formed, newly captured DM particles go into the condensed state. The
size of the condensate $d$ is determined by the size of the lowest quantum
level of DM in the gravitational potential of the NS:

$d=\left({8\pi\over 3}G\rho_{c}m^{2}\hbar^{-2}\right)^{-1/4}\sim 2\times
10^{-4}{\rm cm}\left({{\rm GeV}\over m}\right)^{1/2},$ (25)

where $\rho_{c}$ is the star core density. This size is much smaller than the
size of the thermal DM cloud. Because of the small size of the BEC, the
density of DM in the BEC quickly exceeds the density of nucleons in the star.
This happens when the accumulated DM mass becomes larger than

$M\gtrsim 10^{28}{\rm GeV}\left({m\over{\rm GeV}}\right)^{-3/2},$ (26)

i.e., almost immediately after the star formation. The self-gravitating BEC
then grows until the condition of BH formation is satisfied.

In the case of zero DM self-interactions there is nothing that can stop the BH
formation once the number of DM particles exceeds that given in eq. (22). In
this case the constraints can be imposed on the DM parameters. Requiring that
the BH that is formed is heavy enough to grow by accretion faster than it
evaporates by the Hawking radiation, one may exclude the DM masses in the
range $100{\rm keV}-10$ GeV for DM-nucleon cross sections at the current
experimental limit Kouvaris and Tinyakov (2011); McDermott et al. (2012). For
heavier masses the BH formed in the DM collapse is too light and evaporates
through Hawking radiation faster than it accretes the matter of the star
Kouvaris and Tinyakov (2013) with no observable consequences apart from
heating of the star as already discussed above. Adding even a tiny
$\lambda\phi^{4}$ self-interactions shifts the exclusion region into the
multi-TeV range (see fig.2 of Kouvaris and Tinyakov (2011)). A detailed recent
analysis of these constraints can be found in Garani et al. (2019).

It has been pointed out in Ref.Bell et al. (2013) that self-interactions must
be present at a non-negligible level, the reason being as follows. Even if a
direct coupling between DM particles is not included in the Lagrangian, the DM
interactions with nucleons that are necessary for capture will induce the DM-
DM self-interactions in the higher orders of perturbation theory. There is
however, a subtlety in this argument. The DM-nucleon scattering which is
necessary for capture occurs at low energies, while the DM-DM self-
interactions present an obstacle for collapse at very high (Planckian) values
of fields. These two scales are not necessarily related. So, while it is easy
to avoid the constraints by adding an appropriate DM self-interaction, the
assumption that self-interactions are negligible at the collapse is not in
contradiction with the non-zero DM interaction with nucleons at low energies.

Having excluded some regions of the DM parameter space by NS implosions due to
formation of seed BHs automatically implies, by continuity, that for the
parameters close to those regions but not excluded, one may look for
signatures of these implosion events. These signatures may include particular
gravitational wave events Kurita and Nakano (2016); Bramante et al. (2018);
Takhistov (2018); Nelson et al. (2019), quiet kilonovae Bramante et al.
(2018), supernovae Bramante and Linden (2014); Bramante and Elahi (2015);
Bramante (2015), an impact on star population Casanellas and Lopes (2011), and
existence of (sub)Solar-mass BHs Kouvaris et al. (2018); Abbott et al. (2019)
not expected to result from conventional stellar evolution mechanisms.

##  3 Axions and neutron star magnetospheres

###  3.1 Theory

One of the unsolved issues of the Standard model (SM) of particles is strong
CP problem, i.e. absence of CP-violation in the quantum chromodynamics (QCD).
The theory contains terms that could generate CP-violation, so extreme
closeness of parameter $\theta$ that describes strength of the violation and
could take values anywhere in $[0,2\pi]$ interval, to zero seems unnatural.
One of the most elegant solutions of the strong CP problem was suggested by
Peccei and Quinn (Peccei and Quinn (1977)) — to treat $\theta$ as a new
dynamical field, rather than a fixed parameter. Dynamics of this field could
drive the value of CP-parameter to zero, effectively solving the problem.
There also would be particles corresponding to this field — axions Wilczek
(1978); Weinberg (1978). There is no unique way to add axion to the SM, and
two most popular models are KSVZ (Kim-Shifman-Vainshtein-Zakharov) and DFSZ
(Dine-Fischler-Srednicki- Zhitnitsky) Kim (1979); Shifman et al. (1980);
Zhitnitsky (1980); Dine et al. (1981) which predict couplings of axion to SM
particles of similar strengths and frequently serve as benchmarks. In these
models the axion mass $m_{a}$ is a free parameter that can change in a very
wide range and should be found from experiments. One of the general prediction
for axions is that their coupling constants, e.g. the coupling constant to
photons $g_{a\gamma\gamma}$, are proportional to mass $m_{a}$, meaning that
less massive axions become progressively more elusive for direct or indirect
detection. Still, it is at least theoretically possible to reach this line,
or, given uncertainties, band on $m_{a}-g_{a\gamma\gamma}$ plane for some
values of $m_{a}$ and therefore test model of QCD axion in a certain mass
range. In low-energy effective field theories derived from some string models
more general class of axion-like particles (ALPs) could emerge (e.g., Svrcek
and Witten (2006); Choi et al. (2009); Arvanitaki et al. (2010)). For ALPs
mass and coupling constants are independent.

Axions (and ALPs) are also regarded as a possible DM candidate Preskill et al.
(1983); Abbott and Sikivie (1983). Though they are very light by standards of
WIMPs, with masses $<1~{}$eV, still axions could contribute to cold DM (CDM).
A concise review of axion and ALPs and present status of different experiments
searching for them could be found in Particle Data Group et al. (2020).
Properties of axions and ALPs which are relevant to us are very close, so from
now on we will refer to both of them as axions. Throughout this section we use
units with $\hbar=c=1$ and $\alpha=e^{2}/4\pi\approx 1/137$.

Most of the search methods rely on the very weak axion coupling to photons.
The lagrangian of axion-photon system in vacuum without QED corrections is
described as follows:

$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}{F}^{\mu\nu}+\frac{1}{2}\left(\partial_{\mu}a\partial^{\mu}a-m_{a}^{2}a^{2}\right)+\frac{1}{4}g_{a\gamma\gamma}aF_{\mu\nu}\widetilde{F}^{\mu\nu},$
(27)

where $F_{\mu\nu}\equiv\partial_{\nu}A_{\nu}-\partial_{\mu}A_{\mu}$ is the
electromagnetic field tensor,
$\widetilde{F}^{\mu\nu}\equiv\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}$
is its dual, and $a,m_{a}$ are the axion field and mass respectively. The
coupling term could be rewritten using magnetic and electric fields strengths
$\vec{E},\vec{B}$:
$\frac{1}{4}g_{a\gamma\gamma}aF_{\mu\nu}\widetilde{F}^{\mu\nu}=\frac{1}{4}g_{a\gamma\gamma}a\vec{E}\cdot\vec{B}$.
The equations of axion electrodynamics could be derived from the Lagrangian
(27) Wilczek (1987); Battye et al. (2020):

$\displaystyle\nabla\cdot\vec{E}=-g_{a\gamma\gamma}\vec{B}\cdot\nabla a,$ (28)
$\displaystyle\nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial t},$ (29)
$\displaystyle\nabla\cdot\vec{B}=0,$ (30)
$\displaystyle\nabla\times\vec{B}=\frac{\partial\vec{E}}{\partial
t}+g_{a\gamma\gamma}\dot{a}\vec{B}+g_{a\gamma\gamma}\nabla a\times\vec{E},$
(31) $\displaystyle\Box a+m_{a}^{2}a=g_{a\gamma\gamma}\vec{E}\cdot\vec{B}.$
(32)

Let us assume that there is some background magnetic field $\vec{B}_{0}$ and
perturbations $\vec{E},\vec{B}$ propagating over it. In most astrophysical
scenarios background electric field $\vec{E}_{0}$ could be safely set equal to
zero. Furthermore, as we deal with propagating EM waves, it is sufficient to
describe $E$-component only:

$\displaystyle\Box\vec{E}+\nabla(\nabla\cdot\vec{E})=-g_{a\gamma\gamma}\ddot{a}\vec{B}_{0}.$
(33) $\displaystyle\Box
a+m_{a}^{2}a=g_{a\gamma\gamma}\vec{E}\cdot\vec{B}_{0}.$ (34)

It is illuminating to consider the simplest setup: the EM wave and axion
propagate along $z$-axis in a static and uniform background magnetic field
$\vec{B}_{0}$. As the coupling is sourced by the scalar product
$\vec{E}\cdot\vec{B}_{0}$, only transverse component of the magnetic field
$\vec{B}_{0}$ enters into the equations of motion — we use it to define the
direction of $x$-axis. Also, only one polarization $E_{||}=E_{x}$ couples to
axion and experiences conversion to axion and vice versa, as for the other one
the scalar product $\vec{E}\cdot\vec{B}$ vanishes.

Factorizing out temporal dependence $\sim e^{i\omega t}$, we arrive at the
following set of equations:

$\displaystyle\begin{pmatrix}\omega^{2}+\partial_{z}^{2}-m_{a}^{2}&\omega
g_{a\gamma\gamma}B(z)\\\ \omega
g_{a\gamma\gamma}B(z)&\omega^{2}+\partial_{z}^{2}\end{pmatrix}\begin{pmatrix}a\\\
E_{||}/\omega\end{pmatrix}=0$ (35)

These second-order equations could be solved numerically. Another approach is
to linearize them and obtain analytic results from first-order equations. The
details of the linearization procedure depends on the dispersion relation for
axion $\omega^{2}=m_{a}^{2}+k^{2}$, where $k$ is the wave vector, and slightly
differs for cases of relativistic and non-relativistic axion. The former case,
which is e.g. relevant for propagation of X-ray photons in magnetospheres of
NS, was first thoroughly studied in Raffelt and Stodolsky (1988). For
ultrarelativistic axions $k\approx\omega$ and the linearized equations have a
very simple form222For non-relativistic axions an analogous expression can be
found e.g. in Battye et al. (2020).:

$\displaystyle\begin{pmatrix}\omega-i\partial_{z}-\frac{m_{a}^{2}}{2\omega}&\frac{g_{a\gamma\gamma}B(z)}{2}\\\
\frac{g_{a\gamma\gamma}B(z)}{2}&\omega-i\partial_{z}\end{pmatrix}\begin{pmatrix}a\\\
E_{||}/\omega\end{pmatrix}=0$ (36)

These Schrodinger-like equations describe behaviour of a system of two mixed
states. We are mainly interested in a probability of conversion between them.
In case of homogeneous constant magnetic field $B$ the probability of
conversion of one particle into another depends on the travelled distance $L$
Raffelt and Stodolsky (1988); Fairbairn et al. (2011):

$\displaystyle
P(L)=\frac{g_{a\gamma\gamma}^{2}B^{2}}{m_{a}^{4}/4\omega^{2}+g_{a\gamma\gamma}^{2}B^{2}}\sin\left(\frac{L\Delta_{osc}}{2}\right),$
(37)
$\displaystyle\Delta_{osc}^{2}=m_{a}^{4}/4\omega^{2}+g_{a\gamma\gamma}^{2}B^{2}.$
(38)

Photons and axions are in a strong mixing regime, when $g_{a\gamma\gamma}B\gg
m_{a}^{2}/2\omega$. In this regime almost full conversion of photons to axions
and back is possible with a characteristic length scale of oscillations
$l_{osc}=2\pi/\Delta_{osc}=2\pi/g_{a\gamma\gamma}B$. In the opposite case,
$g_{a\gamma\gamma}B\ll m_{a}^{2}/2\omega$, the amplitude of oscillations is
greatly suppressed. This suppression comes from the fact that the phase
velocities of waves corresponding to massless photon and massive axion are
slightly different and conversion process has only limited time to build up
while these waves propagate in phase. After that the conversion process falls
out of the resonance, effectively limiting the amplitude of oscillations.

This picture of axion/EM wave propagating in vacuum could be generalized to a
more physical scenario. Now we take into account effects of medium and QED-
induced terms which changes velocity of propagation of EM waves Raffelt and
Stodolsky (1988); Fairbairn et al. (2011). Equation (36) should be rewritten
as follows:

$\displaystyle\begin{pmatrix}\omega-i\partial_{z}+\Delta_{a}&\Delta_{M}\\\
\Delta_{M}&\omega-i\partial_{z}+\Delta_{||}\end{pmatrix}\begin{pmatrix}a\\\
E_{||}/\omega\end{pmatrix}=0,$ (39)

where $\Delta_{a}=-\frac{m_{a}^{2}}{2\omega}$,
$\Delta_{M}=\frac{g_{a\gamma\gamma}B\sin\Theta}{2}$, and $\Theta$ is the angle
between the direction of axion propagation and background magnetic field. In
astrophysical sources the mixing term $\Delta_{||}$ is a sum of two parts
$\Delta_{||}=\Delta_{pl}+\Delta_{QED,||}$. The first one arises due to the
influence of plasma, where photon acquires effective mass and propagates
slower than in vacuum:
$\Delta_{pl}=-\frac{\omega_{pl}^{2}}{2\omega},~{}\omega_{pl}=\sqrt{\frac{4\pi\alpha
n_{e}}{m_{e}}}$, $\alpha$ is the fine-structure constant, $n_{e}$ is the
number concentration of electrons and $m_{e}$ is the electron charge 333If the
main charge carriers are not electrons, than the $n_{e},m_{e}$ should be
substituted for $n_{c},m_{c}$. The second term $\Delta_{QED,||}$ comes from
QED-induced corrections and depends on the strength of magnetic field $B$,
$\Delta_{QED,||}=\frac{1}{2}q(b)\omega\sin^{2}\Theta$, $q(b)$ is a function of
parameter $b\equiv\frac{B}{B_{cr}}$, $B_{cr}=m_{e}^{2}/e=4.4\times
10^{13}~{}$G, which can be approximated in the following form Potekhin et al.
(2004):

$q=\frac{7\alpha}{45}b^{2}\hat{q},\,\,\,\hat{q}=\frac{1+1.2b}{1+1.33b+0.56b^{2}}.$
(40)

Usually it is sufficient to use the small-$b$ approximation
$\Delta_{QED,||}=\frac{7\alpha}{90}\omega\left(\frac{B}{B_{cr}}\right)^{2}\sin^{2}\Theta$.
The only possible exception could arise when one considers propagation of EM
waves in the immediate vicinity of magnetars, where magnetic field could be
considerably higher than $B_{cr}$. Introduction of these new mixing terms
slightly modifies the expression (37) for probability $P(L)$ :

$\displaystyle
P(L)=\frac{4\Delta_{M}^{2}}{(\Delta_{pl}+\Delta_{QED,||}-\Delta_{a})^{2}+4\Delta_{M}^{2}}\sin\left(\frac{L\Delta_{osc}}{2}\right),$
(41)
$\displaystyle\Delta_{osc}^{2}=(\Delta_{pl}+\Delta_{QED,||}-\Delta_{a})^{2}+4\Delta_{M}^{2}.$
(42)

It can be seen that now it is possible for photons and axions to experience
very efficient conversion when the resonance condition
$(\Delta_{pl}+\Delta_{QED,||}-\Delta_{a})=0$ is met. This effect is used in
radio searches for axions both in laboratory and in observations of compact
stars (see Section 3.3 below). On the other hand, as $\Delta_{QED,||}$ and
$\Delta_{a}$ have different signs, QED-induced term can further suppress
conversion, comparing to the vacuum case. This is relevant for propagation of
X-ray photons and axions of correspondind energies in magnetospheres of MWDs
and NSs.

To conclude this section, we will present some benchmark values for the mixing
terms, using typical values for relevant parameters:

$\displaystyle\Delta_{a}=-\frac{m_{a}^{2}}{2\omega}=-5\times
10^{-16}\left(\frac{m_{a}}{1~{}\mu\mathrm{eV}}\right)^{2}\left(\frac{\omega}{1~{}\mathrm{keV}}\right)^{-1}~{}\mathrm{eV}=$
$\displaystyle=-2.538\times
10^{-11}\left(\frac{m_{a}}{1~{}\mu\mathrm{eV}}\right)^{2}\left(\frac{\omega}{1~{}\mathrm{keV}}\right)^{-1}~{}\mathrm{cm}^{-1},$
(43)
$\displaystyle\Delta_{pl}=-\frac{\omega_{pl}^{2}}{2\omega}=-\frac{2\pi\alpha
n_{e}}{m_{e}\omega}=$ $\displaystyle=-6.87\times
10^{-15}\left(\frac{n_{e}}{10^{10}~{}{\mathrm{cm}^{-3}}}\right)^{2}\left(\frac{\omega}{1~{}\mathrm{keV}}\right)^{-1}~{}\mathrm{eV}=$
$\displaystyle=-3.48\times
10^{-10}\left(\frac{n_{e}}{10^{10}~{}{\mathrm{cm}^{-3}}}\right)^{-1}\left(\frac{\omega}{1~{}\mathrm{keV}}\right)^{-1}~{}\mathrm{cm}^{-1},$
(44)
$\displaystyle\Delta_{M}=\frac{g_{a\gamma\gamma}B\sin\Theta}{2}=9.76\times
10^{-10}\left(\frac{g_{a\gamma\gamma}}{10^{-10}~{}\mathrm{GeV}^{-1}}\right)\left(\frac{B}{10^{12}~{}\mathrm{G}}\right)\sin\Theta~{}\mathrm{eV}=$
$\displaystyle=4.94\times
10^{-5}\left(\frac{g_{a\gamma\gamma}}{10^{-10}~{}\mathrm{GeV}^{-1}}\right)\left(\frac{B}{10^{12}~{}\mathrm{G}}\right)\sin\Theta~{}\mathrm{cm}^{-1},$
(45)
$\displaystyle\Delta_{QED,||}=\frac{7\alpha}{90}\omega\left(\frac{B}{B_{cr}}\right)^{2}\sin^{2}\Theta=$
$\displaystyle=9.33\times
10^{-5}\left(\frac{B}{10^{12}~{}\mathrm{G}}\right)^{2}\left(\frac{\omega}{1~{}\mathrm{keV}}\right)\sin^{2}\Theta~{}\mathrm{eV}=$
$\displaystyle=4.73\left(\frac{B}{10^{12}~{}\mathrm{G}}\right)^{2}\left(\frac{\omega}{1~{}\mathrm{keV}}\right)\sin^{2}\Theta~{}\mathrm{cm}^{-1}.$
(46)

Relative strength of these terms defines which effect dominates during
propagation. In realistic situations where the field is inhomogeneous the
probability is calculated by numerical integration of equations (39) from some
starting point to infinityRaffelt and Stodolsky (1988):

$P_{a\rightarrow\gamma}=\left|\int\limits_{R_{0}}^{\infty}dr^{\prime}\Delta_{M}(r^{\prime})e^{i\Delta_{a}r^{\prime}-i\int\limits_{R_{0}}^{r^{\prime\prime}}dr^{\prime\prime}\Delta_{QED,||}(r^{\prime\prime})}\right|^{2}.$
(47)

###  3.2 Hot axions

Extremely strong magnetic fields around NSs naturally make them obvious
targets for searches for signatures of axion-photon oscillations. First, it
was suggested to search for high-energy X-ray photons from conversion of
axions produced in cores of NSs Morris (1986); Raffelt and Stodolsky (1988).
Temperature in central parts of these objects can reach extremely high values,
$T_{c}\sim 10^{9}$ K, and the axions could be generated there in
bremsstrahlung process during interactions of nucleons, $n+n\longrightarrow
n+n+a$. These axions would propagate freely in the interiors of the NS, but
could convert in the external magnetic fields into energetic photons that
could be easily distinguished from thermal photons coming from the surface as
its temperature is much lower, $T_{s}\sim 10^{6}$ K. Inverse process of photon
to axion conversion could also lead to emergence of potentially observable
spectral and polarization features in X-ray and other energy ranges Lai and
Heyl (2006); Chelouche et al. (2009); Perna et al. (2012).

At $\mathcal{O}(\mathrm{keV})$ energies conversion takes place in a weak
mixing mode almost everywhere, as could be seen from eqs. (46), and the
dominating term is $\Delta_{QED,||}$. Particles could still experience photon-
axion resonance, when resonance condition
$\Delta_{pl}+\Delta_{QED,||}-\Delta_{a}=0$ is met. This is possible when
$\Delta_{pl}\approx-\Delta_{QED,||}$, which defines corresponding density Lai
and Heyl (2006):

$\rho_{res}=2.25\times
10^{-4}\frac{1}{Y_{e}}\left(\frac{B}{10^{12}~{}\mathrm{G}}\right)^{2}\left(\frac{\omega}{1~{}\mathrm{keV}}\right)^{2}~{}\mathrm{g~{}cm}^{-3},$
(48)

where $Y_{e}$ is the electron fraction. In resonance photon-axion system is in
the strong mixing regime and it is theoretically possible to obtain complete
conversion from one state to another. However, the needed oscillation length
$l_{osc}=\frac{\pi}{\Delta_{M}}$ is much larger than the characteristic scale
of the NS atmosphere $h_{atm}\sim\mathcal{O}(10)$ cm, let alone more narrow
region where density is close to the value which makes the resonance possible.
Because of that the contribution from the resonance conversion is subdominant
and in all calculations of conversion probability it is possible to neglect
the plasma term $\Delta_{pl}$.

Hot axions generated in the centers of NSs have a modified thermal
distribution Iwamoto (1984); Nakagawa et al. (1988) with spectrum peaking
around $E_{peak}\sim 3.3T_{c}$ and for temperatures $T_{c}\sim 10^{9}$ K the
flux is non-negligible in 10-1000 keV energy range. It is customary to
calculate probability of conversion using eq. (47) assuming a simple dipolar
model for a NS magnetic field and purely radial propagation Fortin and Sinha
(2018); Buschmann et al. (2021). This probability could be estimated as
follows Buschmann et al. (2021):

$\displaystyle P_{a\rightarrow\gamma}=6\times
10^{-3}\left(\frac{g_{a\gamma\gamma}}{10^{-10}~{}\mathrm{GeV}^{-1}}\right)^{2}\left(\frac{B}{10^{12}~{}\mathrm{G}}\right)^{2/5}$
$\displaystyle\left(\frac{\omega}{1~{}\mathrm{keV}}\right)^{-4/5}\left(\frac{R_{NS}}{10~{}\mathrm{km}}\right)^{6/5}.$
(49)

Multiplying theoretically expected axionic spectrum by the probability it is
now possible to obtain spectrum of X-rays from the conversion and compare it
to observations. Also polarization measurements could be used to constrain
properties of axions: converted photons could have only single polarization
$E_{||}$, also known as O-mode (ordinary), while many models of NS atmospheres
predict that NS emission, especially at lower energies $<1$ keV, is primarily
polarized in perpendicular X-mode (extraordinary) van Adelsberg and Lai (2006)
and an admixture of differently polarized mode could be in principle detected
Fortin and Sinha (2019).

Different classes of NSs was suggested to use for this type of searches.
First, the magnetars Fortin and Sinha (2018, 2019) — NSs with extremely high
magnetic fields exceeding $10^{14}$ G and high central temperatures, up to
$\sim 10^{9}$ K which boosts their axionic luminosities. There are obvious
downsides as well: all of them are rather distant and that greatly decreases
the flux observed at the Earth. Also, these objects demonstrate high-energy
activity which makes it non-trivial to disentangle possible contribution from
axion-photon oscillations. Nevertheless, the observations of magnetars in the
energy range 10-1000 keV allowed to put meaningful constraints on the product
of couplings $g_{a\gamma\gamma}\times g_{ann}$ Lloyd et al. (2021); Fortin et
al. (2021). The most stringent constraints come from magnetar 1E 1547.0-5408:
$g_{a\gamma\gamma}\times g_{ann}<4.4\times 10^{-20}~{}\mathrm{GeV}^{-1}$ for
$T_{c}=10^{9}$ K.

Another interesting set of candidates are X-ray dim isolated NSs (XDINSs, also
known as the Magnificent Seven) (e.g. Posselt et al. (2007)). These NSs are
detected mainly in X-rays (plus, dim optical counterparts are known for most
of them) and their emission is thought to originate at their surfaces with
spectra which are very close to blackbody with temperatures $T_{s}\sim 0.1~{}$
keV. Detected evolution of spin periods allowed to infer magnetic fields of
these objects: $B\sim 10^{13}$ G which makes possible quite effective
conversion. Their relative proximity to the Earth, with distances in 100-600
pc range also benefits searches. Deep observations in 2-8 keV range by  XMM-
Newton and Chandra revealed existence of X-ray excess for two XDINS: RX
J1856.6-3754 and RX J0420.0-5022 Dessert et al. (2020). This excess could be
explained by axion-photon conversion if $g_{a\gamma\gamma}\times g_{ann}\sim
4\times 10^{-20}~{}\mathrm{GeV}^{-1}$ (only central value is quoted,
systematic and statistical uncertainties could exceed an order of magnitude )
Buschmann et al. (2021). This value is only slightly lower than one coming
from a combination of constraints from CAST helioscope on $g_{a\gamma\gamma}$
and from SN 1987A on $g_{ann}$, so it is possible that this result would soon
be tested with new experiments such as IAXO.

Photon to axion conversion and induced changes in observed spectrum and
polarization of NS X-ray emission was extensively studied in Perna et al.
(2012). Although the possible constraints depend now only on
$g_{a\gamma\gamma}$ coupling, the theoretical uncertainties in modelling of NS
atmospheres greatly complicate the task of setting them. Usually it is assumed
that below 1-2 keV intrinsic emission of the NS is X-polarized due to an
increased opacity for O-mode. As X- and O- mode could convert into each other
in process similar to axion-photon conversion Lai and Heyl (2006); Perna et
al. (2012) the emission is mostly polarized in O-mode at higher energies —
originally produced X-polarized photons experience conversion into O-photons
in so-called vacuum resonance. Vacuum and axion-photon resonances occur at
well separated regions, so it is possible to treat them independently and
instead of three-state system (X, O, axion) study consequently two two-state
system (O, axion) and (X, O) Lai and Heyl (2006) It was shown that if the
photons come from hot spots rather than entire surface, their conversion to
axions would lead to phase-dependent suppression of high-energy tail of the
spectrum Perna et al. (2012). In the opposite case, the conversion would
affect the inferred radius of NS which could lead to contradictions with
values found by other methods.

Another possible way to use photon to axion conversion was suggested recently
in Zhuravlev et al. (2021) — optical photons emitted from the surface of XDINS
should also follow blackbody distribution with temperature that can be
inferred from X-ray observations of these objects. Conversion into axions
would lead to dimming in the optical part of the spectrum which theoretically
allows to probe coupling constraints down to values $g_{a\gamma\gamma}\sim
10^{-11}~{}\mathrm{GeV}^{-1}$.

Further observational progress would be reached with new generation of X-ray
telescopes: Athena Barret et al. (2016) would reach the level of sensitivity
which would make possible phase-resolved observations of XDINs and magnetars.
Effectiveness of the axion-photon conversion process varies with rotational
phase due to changing orientation of magnetic field, so these observations
could make crucial contribution to the field. Also polarimetric observations
with telescopes like XIPE Soffitta et al. (2013) would significantly further
our progress in understanding of NS atmospheres and propagation of X-ray
photons in them, thus allowing to put stronger constraints on axion
properties.

###  3.3 Compact object as haloscopes

Another promising method of axion detection was initially suggested in
Pshirkov and Popov (2009). If the axions are the main DM component, than their
conversion in strong magnetic fields of NS magnetosphere could be possibly
revealed by the presence of sharp spectral feature, similar to the laboratory
haloscopes used to search for axions (e.g. ADMX Du et al. (2018)).

Flux of axions through magnetosphere is enhanced by gravitational focusing and
infalling particles travelling through the inner regions of the magnetosphere
have a chance to convert into photons. In this case we are dealing with
conversion of non-relativistic axions, so the resulting signal would be almost
monochromatic at the frequency corresponding to axion mass $m_{a}$ with
certain shift and widening due to the Doppler effect.

Renewed interest to this method of axion searches lead to fast progress in our
understanding of the details of the process Huang et al. (2018); Hook et al.
(2018); Safdi et al. (2019); Battye et al. (2020); Leroy et al. (2020); Witte
et al. (2021); Battye et al. (2021). Conversion process is slightly different
for non-relativistic axions: governing linearized equations are altered Battye
et al. (2020), and conversion occurs in a resonant mode, in a stark contrast
to conversion of ’hot’ axions, as discussed above. Also plasma effects that
influence propagation of EM waves are much more important for conversion of
non-relativistic axions Witte et al. (2021); Battye et al. (2021).

One of the main properties that directly affects potential sensitivity of
radio observations is the resulting width of the radio line. It is mostly
defined by interaction of infalling axions with rotating plasma of a NS
magnetosphere Battye et al. (2020) and can reach values as high as:

$\frac{\delta f}{f}\sim\mathcal{O}\left(\frac{\Omega r_{c}}{c}\right)=6\times
10^{-4}\left(\frac{\Omega}{1~{}\mathrm{Hz}}\right)\left(\frac{r_{c}}{200~{}\mathrm{km}}\right),$
(50)

where $r_{c}$ is the critical radius where conversion takes place,
$\omega_{pl}(r_{c})=m_{a}$. This contribution due to Doppler broadening
strongly dominates over one caused by DM particle velocity dispersion
$\bar{v}$:

$\frac{\delta f_{disp}}{f}=\frac{{\bar{v}}^{2}}{2c^{2}}=8\times
10^{-7}\left(\frac{\bar{v}}{100~{}\mathrm{km~{}s^{-1}}}\right)^{2}.$

More rigorous treatment of 3D propagation in plasma, including ray-tracing
technique, lead to modification of the magnitude of the expected effect, when
compared to simple 1D calculations. In 3D non-radial trajectories of photons
unlike axions are not rectilinear and that in turn leads to de-phasing and
suppression of resonance which results in somewhat lower flux from conversion
Witte et al. (2021). Also plasma effects could lead to emergence of rich
temporal structure of a signal.

Due to their proximity and relatively high magnetic fields XDINSs are again
considered to be very good observational targets for searches in radio domain.
Observations of RX J0720.4-3125 and RX J0806.4-4123 with Green Bank radio
telescope found no excess signal and that allowed to set constraints:
$g_{a\gamma\gamma}<10^{-11}~{}\mathrm{GeV}^{-1}$ for $5.5<m_{a}<7.0~{}\mu eV$
Foster et al. (2020).

Signals from NSs could be greatly increased if they reside in regions with
enhanced DM density. Another candidate that potentially could lead to even
more stringent constraints is magnetar PSR J1745-2900 with $B\approx 1.6\times
10^{14}$ G which is only 0.1 pc away from the central supermassive black hole
Sgr A∗. The boost factor which can be defined as ratio of DM density at the NS
position relative to DM density near the Solar system
$\eta=\rho_{DM}/\rho_{DM,\odot}$ could reach $1.5\times 10^{5}$ in Navarro-
Frenk-White(NFW) model of the DM halo and could exceed $10^{9}$ in models with
a spike near the SMBH. This object was extensively studied using Effelsberg
observations Foster et al. (2020) and archival data from Karl G. Jansky Very
Large Array Darling (2020a, b); Battye et al. (2021). Constraints can be set
in several broad windows from 4 to 160 $\mu$eV, which correspond to
radiotelescope frequency bands spanning from 1 to 40 GHZ. Assuming NFW model
Navarro et al. (1997) constraints can reach
$g_{a\gamma\gamma}<10^{-11}~{}\mathrm{GeV}^{-1}$ level, while they could be
two orders of magnitude stronger if there was a DM spike around the SMBH. At
the moment the greatest uncertainty comes from our limited knowledge of DM
distribution near the Galactic center. In the future, supreme sensitivity of
the SKA would allow to test couplings as small as $g_{a\gamma\gamma}\sim
10^{-13}~{}\mathrm{GeV}^{-1}$ even assuming NFW model Witte et al. (2021).

MWDs could also be a possible targets of observations, as the conversion of DM
could analogously take place in their magnetospheres. In Wang et al. (2021) it
was shown that future observations of MWD 2010+310 with SKA phase 1 can probe
coupling as low as $g_{a\gamma\gamma}\sim 10^{-12}~{}\mathrm{GeV}^{-1}$ in
0.2-3.7 $\mu$eV mass range.

###  3.4 Axions and FRBs

Fast radio bursts (FRBs) are extragalactic radio flares with millisecond
duration discovered by Lorimer et al. (2007). The question of their origin is
one of the major astrophysical mysteries nowadays (see reviews e.g. in Popov
et al. (2018); Zhang (2020); Xiao et al. (2021)). Axion to photon conversion
in strong magnetic field was one of the first scenarios involving
astroparticle physics.

Sizable fraction of axions constituting DM could undergo process of
condensation, forming relatively dense clusters with masses
$\sim\mathcal{O}(10^{-12}~{}M_{\odot})$ and characteristic sizes of
$\sim\mathcal{O}(100~{}\mathrm{km})$. If during a passage through
magnetosphere of a NS such cluster experiences partial or full conversion and
the axion mass falls in the $\mathcal{O}(\mu\mathrm{eV})$ range, there would
be a bright short radio flare. Conversion of
$\sim\mathcal{O}(10^{-13}-10^{-12}~{}M_{\odot})$ into EM radiation could have
explained outstanding energetics of FRBs with total emitted energy reaching
$E_{tot}=10^{40}$ ergs Tkachev (2015); Iwazaki (2015).

As it was quickly demonstrated, this simplified scenario could not work as
axion miniclusters are too loosely bound and would be destroyed by tidal
forces long before they could reach the region of efficient conversion and so
duration of the resulting flare would be of order of seconds rather than
milliseconds Pshirkov (2017). However, accurate treatment of equations,
describing axion condensation shows that there is a branch of solutions, which
correspond to much more concentrated objects, dubbed dense axion stars. With
masses around $10^{-13}~{}M_{\odot}$ but characteristic sizes less than about
a meter they are immune to forces of tidal disruption and could reach region
of resonant conversion intact Prabhu and Rapidis (2020); Buckley et al.
(2021). So it is still possible that some sub-population of FRBs could be
explained by this process.

##  4 Axions and cooling of white dwarfs

Already in 1986 it was proposed that axions can be emitted by WDs due to
bremsstrahlung process when electrons scatter off nucleons (or nuclei) Raffelt
(1986). This additional energy loss might contribute to WD cooling and so can
be detected (directly or indirectly) by different methods. Since publication
of the original idea (when some constraints on the axion parameters have been
made already) many researchers advanced this approach in many ways. In this
section we present and summarize main recent results in this field of
astroparticle physics trying to mention different methods of constraining
axion parameters via astronomical observations involving WDs.

The first approach is quite similar to the one described in Sec. 3 above for
the case of NSs. Axions emitted in hot and dense WD interiors can be converted
to photons in strong magnetic fields of MWDs Dessert et al. (2019) (see
therein references to earlier similar ideas in application to other types of
astronomical sources). In MWDs surface field can reach $10^{9}$ G Ferrario et
al. (2020). In such strong field axions can turn to X-ray photons via already
discussed above Primakoff effect.444An opposite process — conversion of
thermal surface photons to axions in magnetospheres of MWDs, — was used by
Gill and Heyl (2011) to put stringent limits on $g_{a\gamma\gamma}$ by
observation of linear polarization of optical emission from MWDs. Production
of X-ray emission is related to the fact that temperatures at WDs interiors
can be of the order of $10^{7}$ K. In comparison with NSs, WDs have one
important advantage: their surface temperatures are too low to produce
significant flux of thermal X-ray photons of keV energy. Thus, all detected
photons can be related to conversion of axions.

Expected luminosity is non-negligible, in Dessert et al. (2019) the authors
provide the following estimate:

$L_{ax}=1.6\times
10^{-4}\left(\frac{g_{aee}}{10^{-13}}\right)^{2}\left(\frac{M}{1\,M_{\odot}}\right)\left(\frac{T_{\mathrm{c}}}{10^{7}\,\mathrm{K}}\right)^{4}\,L_{\odot}.$
(51)

Here $M$ is WD mass, and $T_{\mathrm{c}}$ — its central temperature. If all
axions are converted to X-ray photons in the WD’s magnetosphere, this would
correspond to flux $\approx 5\times 10^{-13}\,d_{100}^{-2}$ erg cm-2 s-1,
where $d_{100}$ is distance normalized to 100 pc (we neglect interstellar
absorption which is reasonable for near-by sources in keV range). Such fluxes
are well within reach by modern X-ray observatories like Chandra or XMM-
Newton, and in future Athena will bring much more examples. Of course,
conversion efficiency is far from ideal. Still, X-ray data on MWD RE J0317-853
obtained with Suzaku satellite allowed the authors of Dessert et al. (2019) to
obtain a strong limit on low-mass axions parameters. Later, dedicated
observations of the same source produced even better constraint.

The idea proposed in Dessert et al. (2019) was recently realized in the study
by Dessert et al. Dessert et al. (2021). These authors analysed Chandra
observations of MWD RE J0317-853 at $\sim 30$ pc from the Sun. No X-ray
emission was detected. Accurate modeling and usage of recent data allowed to
obtain a better estimate of the central temperature: $\approx 1.4$ keV.
Altogether this allowed to place an upper limit $g_{a\gamma\gamma}\lesssim
5.5\times 10^{-13}\sqrt{C_{a\gamma\gamma}/C_{aee}}$ GeV-1. For low-mass axions
($m_{a}\lesssim 10^{-5}$ eV) this constraint is much better than the one
obtained by CAST Anastassopoulos et al. (2017). Future observations with new
generation of space X-ray observatories might allow to increase this limit
substantially, or even to detect the effect.

If axions are important for WD cooling, then this effect influences not only
individual sources, but also general properties of the whole population of
these compact objects. In particular, luminosity distribution of WDs — so-
called, luminosity function, $n(L)$ Isern et al. (2018):

$n(L)=\int^{M_{up}}_{M_{low}}\Phi(M)\Psi(\Delta t)\tau_{cool}(L,M)dM.$ (52)

In this expression $L$ and $M$ are luminosity and mass of a WD, respectively.
$\Phi(M)$ is the initial mass function and $\Psi(t)$ — star formation rate.
Characteristic time scale is defined as $\tau_{cool}=dt/dM_{bol}$, where
$M_{bol}$ is absolute bolometric magnitude. Finally, $M_{up}$ and $M_{low}$
are stellar masses that define the range of WD formation in a given
population. Each stellar mass has one-to-one correspondence with a WD mass.

In eq.(52) time interval $\Delta t$ is the difference between the age of
population under consideration (can be equal to the Galactic age) and sum of
the cooling age (time necessary to reach the given luminosity $L$) and
lifetime of the progenitor: $\Delta t=t_{pop}-(t_{cool}(L,M)+t_{pro}(M))$. The
minimum progenitor mass can be defined from equation:
$t_{pop}=t_{cool}(L,M_{low})+t_{pro}(M_{low})$ (stars with lower masses had no
time to evolve up to WD formation, yet).

For a given starformation history standard model of stellar evolution allows
to calculate birth rate of WDs of different masses along the lifetime of the
analysed population. Then, using standard model of WD cooling we can obtain
the luminosity function and compare it with observational data. If they
coincide, then there is no necessity to introduce additional cooling
agents.555Axion cooling can also modify stellar evolution and, thus,
indirectly influence WD luminosity function, see e.g. Raffelt (2008).
Oppositely, deviations can be attributed, for example, to the effect of
axions.

Number of known WDs and precision of determination of their parameters rapidly
grow in our century thanks to many surveys, especially SDSS and Gaia (see,
e.g. Gentile Fusillo et al. (2019) and references therein). This allows to
determine with high confidence different features, including change of the
slope of the luminosity function due to neutrino cooling in hot WDs and
effects of crystallization in old (and cold) sources. However, some
uncertainties prevent equally solid conclusion about the role of axions, as it
is more subtle.

In the first place, the history of WD formation rate and details of their
Galactic distribution are not known well enough. In particular, recent (1-2
Gyrs) bursts of starformation can modify the luminosity function and result in
fallacious conclusions about additional cooling agents. This was recently
studied in Isern et al. (2018).

In their previous studies Isern et al. (2008) these authors already came to
the conclusion that inclusion of axion cooling helps to improve coincidence
between observational data and theoretical luminosity functions for absolute
bolometric magnitude $M_{bol}\sim 6-12$. The same range of $M_{bol}$, where
models without axion cooling slightly overpredict the number of sources, was
studied also in Isern et al. (2018). The new analysis accounting for possible
episodes of intensive starformation confirms the older result favouring axions
with masses about few meV and (electron) coupling constant $\sim$few$\times
10^{-13}$ GeV-1. Similar behaviour of luminosity functions for WDs in the
Galactic halo, thin and thick disks provides support for the authors’
conclusions, as starformation histories of these three elements of the
Galactic structure might not be strongly correlated. Still, uncertainties
remain, and new observations are necessary to improve data on the WD
luminosity function and new studies of stellar and WD properties would be
beneficial too.

As we already mentioned above, axion cooling influences stellar evolution:
stars contract due to additional channel of energy losses from the region of
nuclear reactions. Thus, temperature is increased as well as the rate of
reactions. The star evolves faster and its helium core grows more massive. On
other hand, at the asymptotic giant branch increase of the helium core mass is
smaller in the presence of massive axion-like particles. All this results on
the mass of a WD, i.e., the relation between initial stellar mass and the
compact object mass (IFMR) is modified.

Properties of the IFMR were studied recently in Dolan et al. (2021). These
authors slightly improve constraints in comparison to those obtained from
helium burning (so-called horizontal branch) stars. These objects have dense
hot cores and their lifetime can be estimated from observations. Additional
cooling due to axions can significantly modify duration of helium burning in
contradiction with observations. This excludes massive axion-like particles in
the range $\sim 1-100$ keV with axion-photon coupling constant
$g_{a\gamma\gamma}\gtrsim 10^{-10}$ GeV-1. In Dolan et al. (2021) the authors
enlarge the forbidden area in the region of so-called cosmological triangle
with axion masses about hundreds of keV and coupling constant $\sim 10^{-5}$
GeV-1 analysing helium core growth at the asymptotic giant branch stage.

In helium burning shells of asymptotic giant branch stars temperature is about
15-20 keV. At these conditions Primakoff process and photon-photon interaction
can result in production of axion-like particles even if their masses
$\gtrsim$ than few tens of keV. Additional cooling leads to faster evolution
and more active mixing of matter in outer layers. The latter process results
in smaller amount of helium available for burning in the shell and so, the
mass of carbon-oxygen core grows slower. Finally, the WD originated from this
core is lighter than the one which could be formed in the standard scenario
without additional cooling.

In Dolan et al. (2021) the authors used the IFMR based on 14 double WD wide
binary systems. This IFMR covers the range progenitor masses from 2 up to
7$M_{\odot}$. If massive ($\sim 10$– few$\times 100$ keV) axion-like particles
are considered for $g_{a\gamma\gamma}\gtrsim 10^{-10}$ Gev-1 theoretical IFMR
starts to be out of limits defined by observations in the region of progenitor
masses $\sim 4-7\,M_{\odot}$ (excluding very large values
$g_{a\gamma\gamma}\gtrsim 10^{-3}$GeV-1 where other effects become important
as axion-like particles do not leave the helium burning shell, but decay
inside it). Accounting for rotation slightly softens the constraint, but still
it increases forbidden area in comparison with limits based on analysis of
core helium-burning stars.

May be the most interesting constraint using WDs is obtained from the analysis
of a pulsating source belonging to the so-called ZZ Ceti class (on formation
of this type of WDs, see e.g. Althaus et al. (2002) and references therein).
These WDs are situated in the instability strip of the Hertzsprung-Russel
diagram. They are mid-aged ($<10^{9}$ yrs) and so still warm ($>10^{4}$ K)
objects with convective envelopes containing zones of partial ionization and
hydrogen atmospheres.

The constraint is based on observations of pulsation period evolution of
G117–B15A — a near-by ($\approx 57.5$ pc) WD with temperature $T\approx
12,000$ K and mass $\sim 0.5\,M_{\odot}$. This source is monitored since
mid-70s. ZZ Ceti stars increase pulsation period as they cool down. In the
simplest model $\dot{P}/P\propto-\dot{T}/T$ (e.g. Isern et al. (1992)).
G117–B15A has the period equal to 215.2 seconds and it is increasing with the
rate $\dot{P}=5.47\pm 0.82\times 10^{-15}$ s s-1 Kepler et al. (2021), less
than 10% of thus value can be attributed to the proper motion of the source.
Already in Isern et al. (1992) it was proposed that large value of the period
derivative ($\dot{P}=12\pm 3.5\times 10^{-15}$ s s-1 at that time) can be
explained by additional cooling due to axions. In that paper the authors
obtained the value of axion mass about a few meV.

In the new study Kepler et al. (2021) the authors estimate axion mass as
$m_{a}\cos^{2}\beta=19.9^{+2.1}_{-3.1}$ meV, where $\cos^{2}\beta$ is a free
parameter666This model-dependent parameter links electron coupling constant
$g_{ae}$ to axion mass $m_{a}$ (in units of eV) in the DFSZ model:
$g_{ae}=2.83\times 10^{-11}m_{a}/\cos^{2}\beta$ and is usually set equal to
unity.. This value might be considered as an upper limit as several
uncertainties remain.

##  5 Conclusions

Perspectives for laboratory detection of DM particles and axions in near
future are not very bright. Thus, most probably astronomical data will not
only have a chance to continue producing the best upper limits, but also are
number one candidate to provide direct evidence in favour of existence of such
particles. Various approaches related to different types of sources are used
in astroparticle physics. But continuously NSs and WDs are among best sites to
look for effects linked to new physics.

In the review we presented several lines of considerations to identify
observational tests of existence of exotic particles predicted by theories. We
expect that in near future some of them have chances to provide positive
results. To name a few:

  * •

deep surveys can identify dim old sources related to isolated NSs heated by DM
particles annihilation;

  * •

dedicated searches for radio emission related to Primakoff processes can
result in its detection;

  * •

finally, detailed studies (including polarization) of surface X-ray and
optical emission of cooling NSs also can demonstrates effects related to the
existence of axions.

Advances in technology promise persistent progress in sensitivity of
astronomical observations in next decades. All this raises the chances to
discover new particles by astronomical means

All authors contributed equally to secs. 1 and 5. Sec. 2 was mainly written by
PT, sec. 3 — by MP, and sec. 4 — by SP. All authors have read and agreed to
the published version of the manuscript.

MP and SP were supported by the Ministry of Science and Higher Education of
Russian Federation under the contract 075-15-2020-778 in the framework of the
Large Scientific Projects program within the national project “Science”.

###### Acknowledgements.

The authors are grateful to Sergey Troitsky and Alexey Zhuravlev for helpful
discussions. The authors declare no conflict of interest. The following
abbreviations are used in this manuscript:
ALP | Axion-like particle
---|---
BH | black hole
BEC | Bose-Einstein condensate
CP | Charge parity
DFSZ | Dine-Fischler-Srednicki-Zhitnitsky model
DM | Dark matter
EM | Electro-magnetic
FRB | Fast radio burst
IFMR | Initial to final mass ratio
KSVZ | Kim-Shifman-Vainshtein-Zakharov model
MWD | Magnetic white dwarf
NS | Neutron star
NFW | Navarro, Frenk and White model
QCD | Quantum chromodynamics
QED | Quantum electrodymanics
SM | Standard model
SMBH | Supermassive black hole
WD | White dwarf
WIMP | Weakly interacting massive particle
XDINS | X-ray dim isolated neutron stars
no References yes

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