# Optical Lensing by Axion Stars:
Observational Prospects with Radio Astrometry

Anirudh Prabhu<EMAIL_ADDRESS>Stanford Institute for Theoretical
Physics, Stanford University, Stanford, California 94305, USA

###### Abstract

Axions and axion-like particles (ALPs) are some of the best-motivated dark
matter (DM) candidates. Under certain circumstances, large axion fluctuations
in the early universe can collapse to form dense configurations called _axion
clumps_. The densest axion clumps are metastable states known as _oscillons_.
In this paper we propose a new class of observables that exploit the axion’s
coupling to photons. As a result of this coupling an axion clump acts like an
inhomogeneous refractive optical medium— a lens— that causes anomalous
dispersion of incident electromagnetic waves. The dispersion of
electromagnetic waves by axion clumps clearly distinguishes this lensing
effect from gravitational lensing. Axion clumps passing in front of background
radio sources act as lenses and lead to apparent positional shifts that can
potentially be discovered by high-precision radio astrometry missions with the
forthcoming Square Kilometer Array (SKA). We discuss the sensitivity of SKA to
lensing effects in a variety of axion halo models. While gravitational
microlensing surveys have placed strong constraints on the amount of DM that
exists in the form of non-luminous astrophysical objects they have been unable
to do so for objects in the mass range $[10^{-14},10^{-11}]M_{\odot}$. We find
that, over a wide range of parameter space, SKA will be sensitive to optical
lensing by oscillons in the mass range that is currently unconstrained by
microlensing surveys.

††preprint: APS/123-QED

##  I Introduction

There is overwhelming evidence that a majority of the matter density of the
universe exists in a dark sector that interacts very weakly with the Standard
Model (SM). The existence of such dark matter (DM) has been supported by
observations including the orbital motion of stars and gas clouds in disc
galaxies, temperature anisotropies in the cosmic microwave background (CMB),
strong gravitational lensing by galaxy clusters, and structure formation. All
of the observational evidence to date for the existence of DM relies solely on
its gravitational interactions with itself and ordinary matter. Determining
the non-gravitational interactions of DM is one of the most important
outstanding problems in physics. A general class of DM candidates includes
non-luminous, compact massive objects. Strong exclusions have been placed by
MACHO [1], EROS [2], OGLE [3], and HSC/Subaru [4], on objects with mass
exceeding $10^{-11}\ M_{\odot}$. For lower masses, constraints exist assuming
the compact objects are primordial black holes (PBHs) [5, 5, 6, 7, 8, 9, 10,
11]. While in the low-mass regime, the PBH scenario is strongly constrained,
bounds on general compact massive objects are scarce. Femtolensing of gamma
ray bursts provides the possibility to probe compact objects with mass less
than $10^{-14}M_{\odot}$, assuming these objects make up more than 10% of DM
[12]. As in the case of PBHs, exploiting the microphysical properties of a
given compact object may lead to stronger constraints than those from
gravitational lensing.

In this paper, we focus on the possibility that some DM could be in the form
of compact massive objects composed of axions. The QCD axion, which arises as
a pseudo Nambu-Goldstone boson of a global $U(1)$ Peccei-Quinn (PQ) symmetry
spontaneously broken at high energy scale $f_{a}$, was originally introduced
as part of a dynamical solution to the strong CP problem [13, 14, 15, 16]. The
QCD axion can also be produced, via the misalignment mechanism, in the correct
abundance to explain DM [17, 18, 19]. Its prospect as a DM candidate is
further strengthened by its feeble interactions with the Standard Model. Many
theories beyond the Standard Model, most prominently string theory, predict a
plenitude of axion-like particles (ALPs) over a wide range of masses and
couplings [20]. These particles can similarly provide good DM candidates.
Under certain circumstances, axions can form dense clumps. These clumps form
due to the gravitational collapse of large density fluctuations of the axion
field. Large fluctuations can be formed if PQ symmetry is broken after
inflation, leading to ultradense _axion miniclusters_ , or through the _large-
misalignment mechanism_ [21]. Through mergers and/or accretion, these objects
can form even denser objects called _solitons_ and _oscillons_. The
possibility that an $\mathcal{O}(1)$ fraction of dark matter is bound into
such structure can have serious implications for direct detection experiments.
Associated with the axion clump scenario is a rich set of astrophysical
signatures. Among these signatures are: bright radio flashes sourced by axion-
photon conversion in axion star-neutron star collisions [22, 23, 24, 25,
26]111Some proposals suggest that these are responsible for Fast Radio Bursts.
and gravitational microlensing, picolensing, and femtolensing [27, 28].
Another fascinating possibility is that, in certain regions of parameter
space, axion clumps may frequently collide with the Earth giving large
transient boosts in the local DM density, which could potentially be observed
in existing direct detection experiments [21]. Recently, data from searches
for compact massive objects has been used to place constraints on the
fractional abundance of axion miniclusters [29, 27].

In this article we propose a new class of observables that exploit unique
features of the axion-photon coupling. Due to this coupling, an axion clump
acts as an inhomogeneous refractive medium, which can lead to effects such as
time delays, spectral distortions, and refraction [30, 31]. These effects are
dispersive in nature, allowing them to be distinguished from purely
gravitational effects. The main result of this paper is that the refraction of
light by inhomogeneous axion configurations due to axion-photon interactions
can have unique observable lensing signatures. We refer to such lensing as
_optical lensing by axion stars_ (OLAS)222We call this effect _optical
lensing_ to specify that the lensing object—the axion star— is an _optical
medium_ that allows electromagnetic waves across the entire spectrum to pass
through it. In this paper we are interested in optical lensing primarily in
the radio band of the electromagnetic spectrum. , to distinguish it from
canonical gravitational lensing. One consequence of OLAS is that it can lead
to frequency-dependent shifts in the apparent positions of background radio
sources, that can potentially be observed by astrometric radio missions such
as the Square Kilometer Array (SKA).

We outline a program to detect such positional shifts using SKA and evaluate
its sensitivity to various axion clump models. We find that OLAS is observable
for only axion configurations with very high central amplitude— solitons and
oscillons. Over a wide range of parameter space, we find that the OLAS program
is sensitive to oscillon masses covering the unconstrained mass range
$M_{\text{osc}}\in\left[10^{-14}M_{\odot},10^{-11}M_{\odot}\right]$, even for
very small fractional abundances of oscillons. This could provide one of the
first constraints on compact massive objects in the above mass range.

The paper is organized as follows. In Sec. III, we discuss the general
properties of the axion-photon interaction and derive the effect of light
deflection by axion clumps. In Sec. IV, we review some well-motivated models
of axion clumps and discuss their formation mechanisms and viability as dark
matter candidates. In Sec. V we outline some of the observable signatures of
axion clumps and present the sensitivity of surveys like the Square-Kilometer
Array (SKA) for detection. In Sec. VI we provide some concluding remarks.

##  II Notation and Conventions

Unless otherwise stated, we work in natural units $\hbar=c=1$ and use the
Planck mass $M_{\text{pl}}=\sqrt{\frac{\hbar c}{G_{N}}}$, where $G_{N}$ is the
gravitational constant. We assume an average dark matter density of
$\rho_{{}_{\text{DM},0}}=1.1\times 10^{-6}$ GeV/cm3 and a local dark matter
density of $\rho_{{}_{\text{DM},\odot}}=0.4$ GeV/cm3. We also adopt the
$(-+++)$ metric convention.

##  III Light Refraction in an Axion Background

Interactions between the axion field, $a(x)$, and the Maxwell field are
described by the Lagrangian

$\displaystyle\mathcal{L}\supset-{g_{a\gamma\gamma}\over
4}a\tilde{F}^{\mu\nu}F_{\mu\nu}$ (1)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$,
$\tilde{F}^{\mu\nu}=\varepsilon^{\mu\nu\alpha\beta}F_{\alpha\beta}/2$. The
axion photon coupling $g_{a\gamma\gamma}$ is usually related to $f_{a}$ by
$g_{a\gamma\gamma}=C_{\gamma}\alpha_{{}_{\text{EM}}}/f_{a}$ where
$\alpha_{{}_{\text{EM}}}$ is the electromagnetic fine-structure constant and
$C_{\gamma}$ is an $\mathcal{O}(1)$ constant, although certain models allow
for enhanced axion-photon couplings [32], in which case $C_{\gamma}$ could be
arbitrarily large. Maxwell’s equation in an axion background are modified as
follows.

$\displaystyle\quad\partial_{\mu}F^{\mu\nu}+(\partial_{\mu}\theta)\tilde{F}^{\mu\nu}=0,\quad
F_{[\mu\nu,\rho]}=0$ (2)

where we have defined $\theta(x)\equiv g_{a\gamma\gamma}a(x)$ for notational
convenience and we have used the fact that the second equation in (2) is
equivalent to $\partial_{\mu}\tilde{F}^{\mu\nu}=0$. Approximate plane wave
solutions exist in the limit that the axion field is slowly varying in space
and time compared to the photon field. Formally, this WKB approximation
amounts to the following simplification:
$\partial_{\mu}\partial_{\nu}\theta/\partial\theta\ll\partial_{\mu}F/F$. This
motivates the ansatz $F_{\mu\nu}=\mathcal{R}\left\\{A_{\mu\nu}e^{iS}\right\\}$
where $A_{\mu\nu}$ is the slowly varying amplitude and $S$ the rapidly varying
phase of $F_{\mu\nu}$. In analogy with the plane wave, we may define a
frequency and wavenumber

$\displaystyle\omega\equiv-{\partial S\over\partial
t},\quad{\bf{k}}\equiv\nabla S$

Using this ansatz and working with the electric and magnetic fields, (2) takes
the form ${\bf{D}}(\omega,{\bf{k}})({\bf{E}},{\bf{B}})^{T}=0$. The dispersion
relation is determined by setting det$({\bf{D}})=0$. The eigenvalues of
${\bf{D}}$ are given by

$\displaystyle D^{\pm}=-\omega^{2}+{\bf{k}}^{2}\pm$
$\displaystyle\left[\left(\omega\dot{\theta}+{\bf{k}}\cdot\nabla\theta\right)^{2}+\left(\omega^{2}-{\bf{k}}^{2}\right)(-\dot{\theta}^{2}+(\nabla\theta)^{2})\right]^{1/2}$
(3)

where the $\pm$ refer to left-handed and right-handed helicity states,
respectively. This _birefringence_ is a natural consequence of the
pseudoscalar nature of the axion field. It is worth emphasizing that the
dispersion relation above does not assume that $\theta$ is a small parameter,
only that its gradients are small in comparison to those of the Maxwell field.
A large axion amplitude can lead to a tachyonic instability in one of the
helicity modes [33]. Assuming spatial gradients of the axion field are
subdominant compared to temporal gradients, the tachyonic instability occurs
when $\dot{\theta}>|{\bf{k}}|$ which requires a central amplitude
$a_{0}/f_{a}\gg{|{\bf{k}}|/(m_{a}\alpha_{{}_{\text{EM}}})}\gg 1$. Axion
configurations with arbitrarily high central amplitudes can form in models
with non-periodic potentials such as the ones considered in [34]. However we
restrict the following analysis to axion configurations wherein
$\theta_{0}<1$. For more discussion of tachyonic instabilities in theories
with pseudoscalar couplings to the Maxwell field, see [33]. Another peculiar
feature of this dispersion relation is that it leads to a superluminal group
velocity, which can raise questions of causality violation. Special relativity
demands that no _signal_ can propagate faster than the speed of light in
vacuum and in many situations group velocity is a poor proxy for signal
velocity [35], though there does not appear to be a generally accepted
definition of the latter [36]. The existence of superluminal group velocity
has been demonstrated in a myriad of experiments. For an exhaustive list of
these experiments and associated discussion, see [37].

For light propagating in an axion background, the position, frequency, and
wave number adjust so that (3) is satisfied at each point along the path,
defining geodesic trajectories [38]. For a trajectory parameterized by an
affine parameter $\ell$, this leads to the condition

$\displaystyle{dD^{\pm}\over d\ell}={\partial D^{\pm}\over\partial
x^{\mu}}{dx^{\mu}\over d\ell}+{\partial D^{\pm}\over\partial
k^{\mu}}{dk^{\mu}\over d\ell}=0$ (4)

In analogy with Hamiltonian optics we then define trajectories such that
$dx^{\mu}/d\ell=\partial D^{\pm}/\partial k_{\mu},dk^{\mu}/d\ell=-\partial
D^{\pm}/\partial x_{\mu}$. Using these definitions, we may use $t$ as our
affine parameter, leading to the equations for the evolution of the position,
momentum, and frequency along a trajectory.

$\displaystyle{d{\bf{x}}\over dt}$ $\displaystyle={\partial
D^{\pm}\over\partial{\bf{k}}}\left({\partial
D^{\pm}\over\partial\omega}\right)^{-1}$ (5a) $\displaystyle{d{\bf{k}}\over
dt}$ $\displaystyle=-{\partial D^{\pm}\over\partial{\bf{x}}}\left({\partial
D^{\pm}\over\partial\omega}\right)^{-1}$ (5b) $\displaystyle{d\omega\over dt}$
$\displaystyle={\partial D^{\pm}\over\partial t}\left({\partial
D^{\pm}\over\partial\omega}\right)^{-1}$ (5c)

It appears from (5) that an intervening axion clump can imprint birefringent
spectral distortions and momentum shifts (refraction) on background sources.
Claims have been made that such effects are present at linear order in
$\theta$ [39]. However, it was recently pointed out that in the presence of
only an axion background, any spectral shifts are only sensitive to the axion
configuration at the points of emission and observation [40, 30]. The authors
of [30, 31] claimed that the linear-order birefringent effect can be restored
if “medium” effects were included. As an example, the authors discussed a cold
plasma, though they claim any medium that endows the photon with a refractive
index can restore the effect. We now evaluate this claim in the context of
“realistic” axion clump models. As in Fig. 1, we consider a light ray emitted
from a source $S$ and passing by an axion clump with impact parameter
${\bm{\xi}}$ and refracting towards the observer. To leading order, the
deflection angle can be evaluated by integrating (5b) along the unperturbed
ray path. This leads to a deflection angle, defined as
${\bm{\alpha}}\equiv\delta{\bf{k}}/|{\bf{k}}|$ [30]

${\bm{\alpha}}({\bm{\xi}})=\pm{1\over
2|{\bf{k}}_{0}|}\displaystyle\int_{t_{\text{em}}}^{t_{\text{obs}}}\left[n_{0}\nabla_{\perp}\dot{\theta}(t^{\prime},{\bf{x}}_{0}(t^{\prime};{\bm{\xi}}))+(\hat{{\bf{k}}}_{0}\cdot\nabla)\nabla_{\perp}\theta(t^{\prime},{\bf{x}}_{0}(t^{\prime};{\bm{\xi}}))\right]dt^{\prime}$
(6)

where $t_{\text{em}}$ and $t_{\text{obs}}$ are the times of emission and
observation, $n_{0}$ is the refractive index of the background medium,
$\nabla_{\perp}$ is the gradient transverse to the direction of propagation,
${\bf{x}}_{0}(t,{\bm{\xi}})$ is the unperturbed photon trajectory passing
through an impact vector ${\bm{\xi}}$ in the lens plane, and ${{\bf{k}}}_{0}$
is the wavenumber of the undeflected light ray. In the case of a cold plasma
medium considered in [30, 31], $n_{0}\approx
1-{\omega_{p}^{2}/2|{\bf{k}}_{0}|^{2}}$ where $\omega_{p}$ is the plasma
frequency and $\omega_{0}$ is the unperturbed photon frequency. As first noted
by [40], in the absence of a background medium ($n_{0}=1$), the integrand of
(6) is a total derivative, which makes the bending angle only sensitive to the
boundary conditions. However, even in the presence of a medium, the effect may
be washed out by the time oscillation of the axion field. Stable axion stars
are usually non-relativistic (or semi-relativistic), as relativistic
configurations enter the unstable region in the axion star phase space [41].
Stable, non-relativistic axion configurations are well-approximated by single
harmonic with frequency of the axion mass. During a photon traversal, the
axion undergoes a large number of oscillations per coherence length,
effectively washing out the effect at linear order. The precise form of the
suppression depends on the spatial profile of the axion clump. We emphasize
that the arguments above regarding the bending of light apply to other effects
such as spectral distortion as well. At quadratic order, however, light
deflection is not averaged out as it traverses the axion configuration, though
the effect is no longer birefringent.

To compute the bending angle, we integrate terms in (5b) that are quadratic in
$\theta$ over an unperturbed ray path. We will consider axion configurations
with temporal gradients dominating spatial gradients,
$\dot{\theta}\gg(\partial_{i}\theta)$. With this simplification, the
deflection angle at quadratic order is approximately

$\displaystyle{\bm{\alpha}}({\bm{\xi}})={1\over
8|{\bf{k}}_{0}|^{2}}\displaystyle\int_{t_{\text{em}}}^{t_{\text{obs}}}\nabla_{\perp}\dot{\theta}^{2}(t^{\prime},{\bf{x}}_{0}(t^{\prime};{\bm{\xi}}))dt^{\prime}$
(7)

One feature of axion lensing that distinguishes it from gravitational lensing
is the frequency dependence of the signal. By the equivalence principle, the
gravitational bending angle is independent of frequency while, due to the
dispersive nature of the axion medium, the optical lensing effect scales as
the inverse square of the frequency. Thus a smoking gun signal of OLAS would
be the presence of a lensing signal at low frequencies and the absence of a
corresponding signal at higher frequencies.333Lensing effects by axion stars
are very similar in nature to _extreme scattering events_ , thought to be
caused by plasma inhomogeneities. Axion overdensities mimic the effects of
plasma underdensities. However, the absence of coherent time oscillation of
plasma lenses ensures that their effects are birefringent, distinguishing them
from axion overdensities.

Figure 1: Geometry of light bending by axion stars. In this scenario, a source
at distance $D_{S}$ and angular position on the sky (with respect to the
observer) ${\bm{\theta}}_{0}$ emits a light ray with momentum ${\bf{k}}_{0}$.
The light ray intersects the lens plane at an impact parameter ${\bm{\xi}}$
and gets deflected by an angle ${\bm{\alpha}}$ towards the observer. We assume
the lens is at a distance $D_{{}_{\ell}}\ll D_{S}$ within the Milky Way. The
angular separation between the image and the source is $\Delta{\bm{\theta}}$.

##  IV Axion Clump Formation

We use the term “axion clump” to describe any inhomogeneous axion
configuration with density larger than that predicted by $\Lambda$CDM. In this
section, we discuss a few well-motivated axion clump models, classifying them
by their formation mechanism.

###  IV.1 Axion Halos from Gravitational Collapse

Ultradense axion structures rely on the existence of large fluctuations in the
axion field. These can form in one of two ways. If PQ symmetry is broken after
inflation, causality demands that the axion field takes on different values in
different Hubble patches. These values are drawn from a uniform distribution
on the interval $[0,2\pi]$, leading to $\mathcal{O}(1)$ fluctuations in the
axion field on scales of order the horizon size at PQ symmetry breaking. At a
scale $\Lambda_{\text{NP}}$, non-perturbative effects (e.g. instantons)
generate a potential and hence a mass for the axion [42]. In each Hubble patch
the axion remains dynamically frozen at the value generated at PQ symmetry
breaking until $H\sim m_{a}$ at which point the axion starts oscillating and
generates $\mathcal{O}(1)$ isocurvature density fluctuations.444In many
models, including the QCD axion, the axion mass has a nontrivial temperature
dependence of the form $m_{a}=m_{a,0}(T/\Lambda_{\text{NP}})^{-n}$ where $n$
is a model-dependent index [42]. For simplicity we consider $n=0$, but a more
detailed analysis of QCD axions requires restoring this index. These
fluctuations will undergo gravitational collapse during radiation domination
and subsequently form _axion miniclusters_ [43, 44, 45, 46]. Observational
constraints on the tensor-to-scalar ratio provide an upper bound on the scale
of inflation $H_{I}\lesssim 10^{14}$ GeV and hence on the axion decay constant
in the minicluster scenario.555An intriguing scenario that circumvents this
constraint occurs if the axion gets its mass from a strongly coupled hidden
sector gauge theory with a first-order phase transition that takes
approximately Hubble time to complete [47]. The minicluster mass is given by
the total amount of mass enclosed within a spherical volume of comoving radius
$\pi/k_{0}$, where $k_{0}$ is the comoving horizon size when the axion starts
oscillating. This leads to a minicluster mass

$\displaystyle M_{\text{mc}}\approx 2.3\times
10^{-13}M_{\odot}\left({10^{-6}\text{ eV }\over m_{a}}\right)^{3/2}$ (8)

The typical radius of a minicluster is set by the (8) and the characteristic
density

$\displaystyle\rho_{\text{mc}}\simeq 140\Phi^{3}(1+\Phi)\rho_{a,\text{eq}}$
(9)

where $\rho_{a,\text{eq}}$ is the average axion density at matter-radiation
equality and $\Phi$ is the initial (dimensionless ) axion overdensity.
Simulations give the best-fit cumulative mass fraction of miniclusters
$f(\Phi>\Phi_{0})=\left(1+{\Phi_{0}/a_{1}}\right)^{a_{2}}$ where $a_{1}=1.023$
and $a_{2}=-0.462$ [28, 29]. A sizeable fraction of miniclusters are then
predicted to have overdensities exceeding $\Phi=100$. The corresponding
minicluster radius is given by

$\displaystyle R_{\text{mc}}={1.8\times 10^{7}\text{
km}\over\Phi^{3}(1+\Phi)}\left({{10^{-6}}\text{ eV}\over m_{a}}\right)^{1/2}$
(10)

Another progenitor theory for large axion density perturbations is the _large-
misalignment mechanism_ [21]. If the axion is initially very close to the top
of its potential, attractive self-interactions may delay the onset of
oscillations. Once the axion starts oscillating, certain modes undergo
parametric resonant growth sourced by the axion zero-mode and standard
adiabatic curvature fluctuations, leading to denser, more numerous halos than
predicted by $\Lambda$CDM. For a cosine potential, which is generic in
theories with instanton contributions, such as the QCD axion, this requires an
apparent tuning of the initial misalignment angle $a/f_{a}\approx\pi$. This
apparent tuning can be circumvented by models that dynamically drive the
misalignment angle to $\pi$ [48, 49]. The aforementioned parametric resonant
growth is generic to potentials in which the axion has attractive self-
interactions at large amplitude.

For such large misalignment angles, modes with physical wavenumber
$k_{\text{phys}}\sim H$ will go nonlinear earlier than in the $\Lambda$CDM
scenario, leading to denser, _large-misalignment halos_. The large-
misalignment mechanism produces dense axion halos with mass peaked around [21]

$\displaystyle M_{*}={4\pi\over 3}\left({\lambda_{*}\over
2}\right)^{3}\rho_{{}_{\text{DM},0}}=5\times
10^{-15}M_{\odot}\left({10^{-6}\text{ eV}\over m_{a}}\right)^{3/2}$ (11)

where $\lambda_{*}=2\pi/\sqrt{2m_{a}H_{\text{eq}}a_{\text{eq}}^{2}}$ (here
“eq” corresponds to values at matter-radiation equality) and
$\rho_{{}_{\text{DM},0}}$ is the average dark matter density at present. The
densities of these objects are parametrically larger than those of CDM halos
by a factor $\mathcal{B}$ which is taken to be in the range
$10^{3}\lesssim\mathcal{B}\lesssim 10^{7}$. The typical size of these objects
is determined by the scale radius (cf. the discussion after (21))

$\displaystyle R_{s}\approx 4\times 10^{-6}\text{ pc}\left({10^{-6}\text{
eV}\over m_{a}}\right)^{1/2}\left({10^{3}\over\mathcal{B}}\right)^{1/3}$ (12)

In addition to the modes discussed above, certain modes can go nonlinear deep
into radiation-domination and collapse due to self-interactions, leading to
extremely dense objects called _oscillons_. We will discuss these objects in
the following section.

###  IV.2 Solitons and Oscillons

Gravitational collapse of axion halos is eventually halted by the repulsive
kinetic (uncertainty) pressure. This occurs when the size of the halo becomes
comparable to the de Broglie wavelength of the axion field. The resulting
ground state configuration is called a _soliton_ [50, 51, 52, 53, 54].
Solitons are similar to the _bose stars_ described in [55, 56]. Stars composed
of bosons of mass $m$ were shown to be unstable to collapse above a critical
mass $M^{*}_{\text{bose}}\simeq 0.6M_{\text{pl}}^{2}/m$ [57, 58, 59]. When the
constituent bosons are axions, the resultant objects are called _axion stars_.
At masses much less than the critical mass the properties of bose stars and
axion stars coincide. As the mass approaches the critical value, weak,
attractive axion self-interactions can play a role in the balance of forces.
Axion stars have a critical mass (and corresponding critical central
amplitude) given by

$\displaystyle M_{\text{AS}}^{*}$ $\displaystyle\simeq
10{f_{a}M_{\text{pl}}\over m_{a}}$ (13) $\displaystyle\phi_{0}^{*}$
$\displaystyle\simeq{10.5\over|\lambda_{4}|}\left({f_{a}\over
M_{\text{pl}}}\right)\simeq{8.6\times
10^{-7}\over|\lambda_{4}|}\left({f_{a}\over 10^{12}\text{ GeV}}\right)$ (14)

where $\phi_{0}^{*}=a_{0}^{*}/f_{a}$ and $\lambda_{4}$ is an $\mathcal{O}(1)$
dimensionless coupling constant associated with the quartic term in the
potential. For the QCD axion with a chiral potential, $\lambda_{4}=-0.34$.
Axions stars with mass less than this value are called _dilute axion stars_
[60]. They have a well-defined mass-radius relationship [61]

$\displaystyle R_{99}$ $\displaystyle\simeq{9.95M_{\text{pl}}^{2}\over
Mm_{a}^{2}}$ (15)

where $M$ is the star mass, $R_{99}$ is the radius containing 99% of the mass.
The spatial profile will be important for understanding the lifetime and
lensing effects, however, there is no consensus on its form in the literature.
The profile is often assumed to be well-approximated by a Gaussian in, for
example, [52, 53, 21] and an exponential ($\rho(r)\propto\exp(-r/R)$) in [22,
23, 25]. Although dedicated numerical studies of the spatial profiles of
bosons stars presented in [62, 63] favor an exponential profile over a
Gaussian, however we conservatively adopt the latter.

Solitons with mass parametrically lower than the critical mass by a factor of
roughly $(H_{\text{eq}}/m_{a})^{1/4}$ can be formed directly from the large-
misalignment mechanism [21]. These solitons can either merge with each other
or accrete mass from their surrounding halos through the gravitational cooling
mechanism, resulting in a present-day abundance of near-critical solitons.
Gravitational cooling can also lead to the formation of solitons in the
minicluster scenario as shown in [41].

When the axion amplitude is greater than $a_{0}^{*}$ but less than unity, the
resulting _critical axion stars_ are unstable to any perturbations that
increase the central amplitude [53]. As the amplitude becomes
$\mathcal{O}(1)$, quartic self-interactions are no longer sufficient to
describe the balance of forces. Considering the full potential leads to
metastable objects known as _oscillons_ or _dense axion stars_ [64, 60, 65].
Oscillons may be formed as the by-product of collapse of super-critical
solitons. Solitons may become supercritical through mergers of or accretion by
near-critical solitons [66]. As mentioned earlier, oscillons may also form
directly from the large-misalignment mechanism when axion fluctuations
collapse due to self-interactions deep in radiation-domination [21].

As mentioned earlier, oscillons are metastable states wherein the outward
kinetic pressure is balanced by self-interactions described by the full
potential. We quote a few oscillon features present in a wide variety of
initial conditions of simulations performed in [21]. Firstly, the central
density is of order the natural scale, $m_{a}^{2}f_{a}^{2}$, present in the
cosine potential and in more exotic potentials like the ones in [34, 67]. In
addition, the size of these objects is roughly $R=n/m_{a}$ where
$n\sim\mathcal{O}(10)$. Assuming a roughly flat internal density profile, we
can estimate the oscillon mass to be $M_{\text{osc}}\sim
n^{3}f_{a}^{2}/m_{a}$.

Another property of oscillons is that general solutions involve infinitely
many harmonics $\omega_{0},3\omega_{0},5\omega_{0},\dots$. Here $\omega_{0}$
is the fundamental frequency, which is roughly $\omega\approx 0.9m_{a}$,
because of the small binding energy per axion. Numerical studies [60, 21] have
shown that the fundamental harmonic is dominant. Motivated by these studies,
we work in the single-harmonic approximation. The spectrum of emitted axion
radiation indicates that oscillons may be metastable due to number-changing
processes. The metastability of oscillons plays an important role in their
viability as a DM candidate.

###  IV.3 Oscillons as Dark Matter Candidates

Crucial to the observation of oscillons is their present-day abundance. This
quantity depends on the formation mechanism and lifetime. While solitons are
expected to have lifetimes greater than the age of the universe [68, 69], the
oscillon lifetime is not well understood. Since oscillons probe the full
potential they are metastable to decay through number-changing processes such
as $3a\rightarrow a$, $4a\rightarrow 2a$, etc. By conservation of energy, the
outgoing axions will be relativistic for most number-changing processes. Thus,
the oscillon lifetime is very sensitive to its size and spatial profile as
those determine the suppression of the rate of production of high-momentum
axion modes.

For the cosine potential, oscillons decay in time $\tau_{\text{osc}}\sim
10^{3}/{m_{a}}$, which means that they are unlikely to survive past matter-
radiation equality assuming they were formed via the large-misalignment
mechanism. For flatter potentials oscillons have been observed, in numerical
simulations, to have lifetimes $\tau\gtrsim 10^{8}/m_{a}$ [67], with no
apparent upper bound. Similarly, arguments by [68] suggest that for large
oscillons with certain spatial profiles the rates of number-changing processes
can be exponentially suppressed leading to large lifetimes. Mechanism to
naturally generate flat potentials were proposed in [34, 70]. We consider a
general class of well-motivated potentials [71],

$\displaystyle V(a)$ $\displaystyle={m^{2}f_{a}^{2}\over
2p}\left[\left(1+{a^{2}\over f_{a}^{2}}\right)^{p}-1\right]$ (16)

where $0\leq p\leq 1$ gives axion monodromy-like potentials [72, 73, 74] and
$p<0$ gives potentials that flatten at large field values. So-called _plateau_
potentials have been discussed in an inflationary context in [75, 76].
Potentials with $p<1$ can form potentially long-lived oscillons [67, 77].
Monodromy-like potentials were shown to lead to lifetimes in the range
$(10^{8}-10^{9})/m_{a}$, which are considerable for very low-mass axions.
However, the mass range probed by OLAS is $10^{-9}$ eV $\lesssim m_{a}\lesssim
10^{-4}$ eV. The mass range is set by the frequencies observed by radio
telescopes. Thus it seems unlikely that these potentials can lead to a
present-day abundance of oscillons. For $p<0$, where potentials become
asymptotically flat, oscillons were shown to live for $\tau>10^{9}/m_{a}$ with
no sign of decay. It is possible that oscillons formed from such potentials
can be present today and form an $\mathcal{O}(1)$ fraction of DM. Therefore in
the following discussion of observational signatures, we assume oscillons
formed not from QCD axions, but from axions with potential (16), with
$p=-1/2$. Detailed simulations of oscillon properties may be found in [78,
79].

##  V Observational Prospects

In Sec. III we showed that inhomogeneous axion configurations can lead to
deflection of light rays. This is analogous to the deflection of light due to
massive objects in general relativity, with some key differences. We thus
expect much of the gravitational lensing formalism to apply to the case of
optical lensing by axion stars.666We emphasize that this effect is analogous
to but distinct from gravitational lensing. Proposals to search for axion
clumps using gravitational lensing include [29, 27, 21, 28, 80] OLAS can
produce multiple images as well as magnification of background sources that
could potentially be detected by strong lensing and microlensing searches. We
will investigate these signatures in [81]. Even in the absence of such
dramatic effects, axion clumps may induce apparent anomalous motion of
background sources during close line-of-sight approaches. Astrometric missions
such as _Gaia_ [82] seek to measure the positions, velocities, and distances
of background sources with unprecedented precision. At lower frequencies, the
upcoming Square Kilometer Array (SKA) is slated to be the largest and most
sensitive radio interferometer to date. The array will be built in two phases.
The first phase, called SKA1, which will use only about 10% of the collecting
area of SKA2, is expected to be deployed in the early 2020s. SKA is projected
to continuously cover frequencies between 50 MHz and 14 GHz in its first two
phases and frequencies above 30 GHz in its third phase. One of the many
astrometric capabilities of SKA is its ability to measure distances and proper
motions of background radio sources with unprecedented angular precision [83].
The angular displacement of a source is often difficult to determine from a
single measurement due to the intrinsic uncertainty in the unlensed position
of the source. In this section, we calculate the sensitivity of certain time-
domain astrometric observables outlined in [80] to OLAS. In order to confirm
that a lensing event is due to OLAS and not gravitational lensing simultaneous
measurements must be made at different frequencies. If two detectors observing
in different frequency ranges record different positional shifts of the same
background source then the two observations could constitute a signal of OLAS.
For example, if Gaia, operating in the optical range, and SKA, operating in
the radio range, record different positional shifts of the same background
source, with the frequency-dependence of the positional shifts described by
(7) then we can conclude that the differential positional shift is due to OLAS
and not gravitational lensing. We note that multi-wavelength observations of
hundreds of quasars have already been made with Gaia and Very Large Baseline
Interferometry (VLBI) [84].

For a wide range of axion masses, the corresponding solitons and oscillons are
approximately point-like and traverse large arcs on the sky during the
duration of the mission (assumed to be $\tau=5$ years). The observable effects
of the axion star can be divided into two categories 777When the proper motion
of a lens is smaller than the typical minimum impact parameter in a source-
lens pair, there will be anomalously high velocities and accelerations of the
sources. These are called _outliers_. Since the axion-photon interaction is
short-range in comparison to gravity these observables are not well-suited to
oscillon searches using optical lensing:

  * •

Rare Single Events: Over the duration of the mission, we expect there to be a
very close approach (in the lens plane) between a lens and a background
source. In this close approach the source will appear to traverse a nonlinear
trajectory before returning to its original position. This observable is
called a _mono-blip_.

  * •

Multi-Source Observables: If the angular distance traveled by the lens is
larger than the angular separation between sources, there will be correlated
anomalous motion of stars, called a _multi-blip_ signal.

###  V.1 Blip Sensitivity

Blips are the most suitable observables when the distance traversed by the
lens during the mission is larger than the size of the lens and the minimum
expected approach between a source-lens pair. For the axion clump models and
masses of interest, the above condition will usually apply. To compute the
sensitivity of an SKA astrometric mission to blips caused by axion stars, we
propose making high-cadence, all-sky measurements of the positions of many
background radio sources. In the absence of any lenses, the background sources
remain approximately fixed on the sky. This is expected since the proper
angular motion of radio sources at cosmological distances over the course of
the 5 year mission is much less than the angular precision expected for SKA,
$\Delta\theta\simeq 10\ \mu$as [83]. In the presence of a lens with trajectory
${\bf{x}}_{{}_{\ell}}(t)$ (in the lens plane), OLAS predicts that a source at
angular position ${\bm{\theta}}_{i,0}$ will undergo an apparent non-linear
lensing trajectory, ${\bm{\theta}}_{i}(t)$, given by

$\displaystyle{\bm{\theta}}_{i}(t)={\bm{\theta}}_{i,0}+\left(1-{D_{{}_{\ell}}\over
D_{S}}\right){\bm{\alpha}}\left[D_{{}_{\ell}}{\bm{\theta}}_{i,0}-{\bf{x}}_{{}_{\ell}}(t)\right]$
(17)

where $D_{{}_{\ell}}$ and $D_{S}$ are the distance to the lens and source,
respectively and ${\bm{\alpha}}$ is the deflection angle, computed in (7). The
term in square brackets is the impact parameter at which light from the source
passes by the lens. We assume galactic lenses and radio sources that are at
cosmological distances. Therefore, we can neglect the term
$D_{{}_{\ell}}/D_{S}$. To test whether the presence of an axion star lens is
consistent with the data, we construct a test statistic, $\mathcal{B}$ , that
encodes the overlap of the observed angular position residuals with the
angular position shift predicted by OLAS (17), weighted by the angular
precision for each measurement, $\sigma_{\delta\theta_{i}}$ [80]. If the lens
trajectory used to compute ${\bm{\theta}}_{i}(t)$ is correct, the expected
angular position residuals should coincide with the predictions made by OLAS.
The expectation value of the test-statistic can then be calculated to be [80]

$\displaystyle\left\langle\mathcal{B}[{\bf{x}}_{{}_{\ell}}(t)]\right\rangle$
$\displaystyle=\left\langle\displaystyle\sum_{i=1}^{N}{1\over\sigma^{2}_{\delta\theta_{i}}}\displaystyle\sum_{j=1}^{n}\alpha^{2}[{\bf{x}}_{\ell}(t_{n})]\right\rangle\simeq\left({g_{a\gamma\gamma}^{2}m_{a}^{2}a_{0}^{2}\over
8\sigma_{{}_{\delta\theta,\text{eff}}}\omega^{2}}\right)^{2}\left\langle\displaystyle\sum_{i=1}^{N}{f_{\text{rep}}\over
v_{{}_{\ell}}}\displaystyle\int_{-\infty}^{\infty}f^{2}\left(\sqrt{b_{i\ell}^{2}+x^{2}}\right)dx\right\rangle$
(18)

where the first sum is performed over the $N$ sources and the second sum over
the $n$ observation times, $\omega$ is the frequency of light, $v_{{}_{\ell}}$
is the velocity of the axion lens, $\sigma^{2}_{\delta\theta_{i}}$ is the
angular position noise per observation, $b_{i\ell}$ is the minimum expected
impact parameter between a given source $i$ and lens $\ell$, and $f(x)$ is the
profile-dependent form factor defined as

$\displaystyle f({\bm{\xi}})$
$\displaystyle\simeq\displaystyle\int_{-\infty}^{\infty}\left.(\nabla_{\perp}\bar{\theta}^{2}({\bf{x}},t^{\prime}))\right|_{{\bf{x}}={\bf{x}}_{0}(t^{\prime};{\bm{\xi}})}dt^{\prime}$
(19)

where $\bar{\theta}({\bf{x}},t)=\theta({\bf{x}},t)/\theta_{0}$ and
${\bf{x}}_{0}(t;{\bm{\xi}})$ is the unperturbed path of the light ray with
impact parameter ${\bm{\xi}}$, and $\nabla_{\perp}$ refers to the gradient
normal to the direction of propagation. The second step in (18) involved
approximating the sum over observation times as a continuous integral with
integration variable $x$. For a given lens path, the signal-to-noise ratio
will be given by
$\sqrt{\left\langle\mathcal{B}[{\bf{x}}_{\ell}(t)]\right\rangle}$. This can be
large due to a single lensing event with very low impact parameter (mono-blip)
or a lens path that passes close to several sources $\sum_{\text{sources}}\gg
1$ (multi-blip). We will find that for the axion clump models of interest, the
former possibility is best-suited for detection.

For mono-blip signals, we will be interested in the closest expected approach,
$b_{i\ell}$, between any galactic lens and photons from any source on the sky
during the duration of the mission. This can be approximated as

$\displaystyle\left\langle\min\limits_{i,\ell}b_{i\ell}\right\rangle$
$\displaystyle={M_{{}_{\ell}}\over
v_{{}_{\ell}}\tau\rho_{{}_{\ell}}D_{S}\Sigma_{0}\Delta\Omega}$ (20)

where $M_{{}_{\ell}}$ and $v_{{}_{\ell}}$ are the mass and velocity of the
lens, $\tau$ is the duration of the mission, $\rho_{{}_{\ell}}$ is the average
density of lenses in the region of interest (not to be confused with the
density of a single lens), $D_{S}$ is the distance to the sources,
$\Sigma_{0}$ is the angular number density of sources, and $\Delta\Omega$ is
the angular area covered by the survey. We now discuss the astrometric
sensitivity of SKA to various axion clump models in the OLAS scenario. We
split the consideration into two cases: (1) “diffuse” objects (miniclusters
and large-misalignment halos) and (2) compact objects (solitons and
oscillons).

###  V.2 Minicluster and Large-Misalignment Halos

The formation of axion miniclusters and large-misalignment halos is analogous
to that of $\Lambda$CDM halos. We therefore assume they have the same density
profile. In $\Lambda$CDM, $N$-body simulations of gravitational collapse of
overdensities show the formation of virialized halos that are well-fitted by a
Navarro-Frenk-White (NFW) profile over a broad range of initial conditions
[85]

$\displaystyle\rho_{{}_{\text{NFW}}}(r)$ $\displaystyle={4\rho_{s}\over{r\over
R_{s}}\left(1+{r\over R_{s}}\right)^{2}}$ (21)

where $\rho_{s}$ is the scale density, the characteristic density of the halo,
and $R_{s}$ is the scale radius, the distance at which the halo density is the
scale density. The amount of mass enclosed in the scale radius is called the
scale mass. An important property of this profile is the apparent “cuspy”
behavior at small radii. This is in conflict with the “cored” DM density
profile inferred from rotation curves of dwarf galaxies [86, 87]. This
conflict is known as the _core-cusp problem_ , which may be resolved by
including baryonic feedback or the formation of solitonic cores at the center
of CDM halos. This problem highlights the fact that the NFW profile may not
apply when $r\ll R_{s}$.

The mono-blip test statistic for an NFW profile is well-approximated by

$\displaystyle\left\langle\mathcal{B}_{\text{NFW}}[{\bf{x}}_{{}_{\ell}}(t)]\right\rangle\simeq\left({g_{a\gamma\gamma}^{2}\rho_{s}\over
4\sigma_{{}_{\delta\theta,\text{eff}}}\omega^{2}}\right)\sqrt{{\pi
f_{\text{rep}}R_{s}\over
v_{{}_{\ell}}}\left(\left\langle\min\limits_{i,\ell}b_{i\ell}\right\rangle\over
R_{s}\right)^{-1}}$ (22)

Based on (8)-(12), we find that for blip searches, sensitivity to miniclusters
and large-misalignment halos relies on impact parameters much smaller than the
scale radius. The radii involved are lower than the spatial resolution of
$N$-body simulations and thus it is unclear if the NFW profile can be trusted
in this regime. The situation in which a solitonic core is formed at the
center of the halo is considered in the following subsection. The argument
above also applies to the case of multi-blip searches. Even the $\sqrt{N}$
enhancement from correlated anomalous motion of many stars will be
insufficient to make OLAS observable. Searches for blip signals sourced by
gravitational lensing is a more promising avenue for these objects [80].

###  V.3 Solitons and Oscillons

For various profiles, the deflection of light by axion stars is maximized when
the impact parameter is of order of the radius of the object. Since solitons
and oscillons are very small and compact, we expect mono-blip searches to be
the most promising. For a Gaussian profile, the mono-blip SNR is given by

$\displaystyle\text{SNR}^{\text{mono}}_{\text{Gauss}}\equiv\left\langle\mathcal{B}\left[{\bf{x}}_{{}_{\ell}}(t)\right]\right\rangle^{1/2}\simeq\left({g_{a\gamma\gamma}^{2}m_{a}^{2}a_{0}^{2}\over
8\sigma_{{}_{\delta\theta,\text{eff}}}\omega^{2}}\right)e^{-{\left\langle\min\limits_{i,\ell}b_{i\ell}\right\rangle^{2}/R^{2}}}\left[{f_{\text{rep}}R\over
v_{{}_{\ell}}}\times\left(1+{4\left\langle\min\limits_{i,\ell}b_{i\ell}\right\rangle^{2}\over
R^{2}}\right)\right]^{1/2}$ (23)

The radius containing 99% of the energy of the star, $R_{99}$ is related to
$R$ by a potential and profile-dependent $\mathcal{O}(1)$ constant of
proportionality. We compare (23) to the mono-blip SNR from gravitational
lenses [80]

$\displaystyle\text{SNR}^{\text{mono}}_{G}\simeq{4GM_{{}_{\ell}}\over\sigma_{{}_{\delta\theta,\text{eff}}}}\sqrt{{\pi
f_{\text{rep}}\over
v_{{}_{\ell}}}{1\over\left\langle\min\limits_{i,\ell}b_{i\ell}\right\rangle}}$
(24)

where $M_{{}_{\ell}}$ is the mass of the lens, and
$\left\langle\min\limits_{i,\ell}b_{i\ell}\right\rangle$ is given in (20). In
Fig. 3 the mono-blip sensitivity projections of OLAS and gravitational lensing
are shown. Since gravitational lensing is not frequency dependent, we use the
parameters expected for _Gaia_ sampling of disk stars:
$\sigma_{{}_{\delta\theta,\text{eff}}}=100\ \mu$as, $f_{\text{rep}}=14/$day,
$D_{S}=10$ kpc, $\tau=5$ years, $\Sigma_{0}=4.6\times 10^{9}$ rad-2, and
$\Delta\Omega=0.2$ in Fig. 3.

####  V.3.1 Solitons

In this section we discuss the projected sensitivity of our proposal to
solitons. Since the stability of solitons is not sensitive to the choice of
axion potential we consider solitons composed of QCD axions. In conventional
models of the QCD axion the coupling of axions to photons is assumed to lie in
a narrow band in parameter space. However, there exist several mechanisms to
expand the allowed range of couplings. In particular, many models predict
axion-photon couplings much larger than those in standard KSVZ [88, 89] and
DFSZ [90, 91] realizations of the QCD axion [32, 92, 93]. For a list of
mechanisms that can generate large axion-photon couplings, see [94]. Based on
the arguments above, we assume the axion-photon coupling is a free parameter
that is independent of $f_{a}$.

Constraining the fractional abundance of soliton DM requires knowing the mass
function for these objects. Using the Press-Schechter formalism [95], the
authors of [27] calculate the mass function for axion miniclusters. Such a
computation for solitons requires determining the formation mechanism and the
merger and accretion rates, which is beyond the scope of this work. We
investigate the possibility of detecting solitons with OLAS by assuming a
Dirac-delta function mass function, centered at the critical mass (13). The
projected sensitivity is shown in Fig. 2, assuming the solitons are described
by a Gaussian density profile. At low axion mass gradients of the axion field
are suppressed, leading to a smaller deflection angle.

Figure 2: Projected sensitivity of an SKA mono-blip search to critical-mass
solitons with fractional abundance $f_{\text{sol}}=1$ (blue),
$f_{\text{sol}}=10^{-2}$ (red), $f_{\text{sol}}=10^{-4}$ (black) where
$f_{\text{sol}}\equiv\rho_{\text{sol}}/\rho_{{}_{\text{DM}}}$. We have assumed
an angular number density $\Sigma_{0}=10^{7}$ rad-2 of background radio
sources covering the entire sky, a mission time of $\tau=5$ years, lens
velocity $v_{{}_{\ell}}=10^{-3}$, an observing cadence of
$f_{\text{rep}}=1/$day, and an angular precision of
$\sigma_{{}_{\delta\theta,\text{eff}}}=10\ \mu$as.

For critical solitons, we only have sensitivity if the axion model has
significant enhancements in the axion-photon coupling compared to conventional
QCD axion models. This may be realized in the models discussed at the
beginning of the section. However, at such high couplings critical solitons
may be unstable to parametric resonant decay into photons [96, 97, 98, 99,
100].

####  V.3.2 Oscillons

We also consider the possible lensing signatures of oscillons. As discussed in
Sec. IV.3, such signatures are contingent upon oscillons being long-lived on
cosmological timescales. Thus the following discussion does not apply to
oscillons formed from a cosine potential. For non-periodic potentials, the
central amplitude can theoretically be arbitrarily large, although the results
of Sec. III rely on the assumption that $g_{a\gamma\gamma}a_{0}<1$. Assuming
$g_{a\gamma\gamma}\sim\alpha/f_{a}$ yields the constraint $a_{0}/f_{a}\lesssim
10^{2}$. The reach of SKA mono-blip searches for optical lensing by oscillons
is shown in Fig. 3. For fixed $f_{a}$, the oscillon mass range probed is
determined by the axion mass. Since light-bending is proportional to gradients
of the axion field, it is suppressed at low axion mass (high oscillon mass).
The abrupt loss of sensitivity at high axion mass (low oscillon mass) is an
artifact of the breakdown of the eikonal approximation that occurs when the
axion mass becomes comparable to the highest frequency observed by SKA.
Extending the sensitivity to lower masses can be achieved numerically in the
regime where the eikonal approximation is no longer valid. We find that for
$f_{a}\lesssim 10^{15}$ GeV, this search will be sensitive to oscillon masses
in a range unconstrained by existing observational constraints. This appears
to be true over a variety of well-motivated spatial profiles and central axion
densities.

Existing and proposed constraints on the abundance of general astrophysical
objects come from gravitational microlensing surveys performed by MACHO [1],
EROS [2], OGLE [3], HSC/Subaru [4], on objects with mass exceeding $10^{-11}\
M_{\odot}$. As mentioned in Sec. I, constraints on general objects of mass
less then $10^{-11}M_{\odot}$ are scarce. One proposal to probe masses in this
range was femtolensing of gamma-ray bursts (GRBs); the proposal has, however,
been consequently weakened by the inclusion of finite source size and wave
effects [12]. In Fig. 3, we have also included constraints on PBH masses,
although they do not apply to oscillons or other axion stars. Constraints on
PBHs arise as a requirement that the time over which they dissipate to Hawking
radiation be greater than the age of the Universe and the requirement that the
Hawking radiation does not result in an excess in the gamma-ray background
[101]. Finally, PBHs with masses between $5\times 10^{-15}M_{\odot}$ and
$5\times 10^{-14}M_{\odot}$ can heat white dwarfs (WD) through dynamical
friction. This can lead to runaway thermonuclear fusion in and subsequent
explosion of the WDs. The observed distribution of WDs constrains PBHs in the
aforementioned mass range [9].

Figure 3: Estimated sensitivity of SKA mono-blip searches to optical lensing
by oscillons. The plot shows the projected sensitivity to oscillons with
$f_{a}=10^{13}$ GeV (blue) and $10^{14}$ GeV (red). Estimates are calculated
for central axion density $\phi_{c}=5f_{a}$, and axion-photon coupling
$g_{a\gamma\gamma}=\alpha/f_{a}$. For both plots, we have assumed an angular
number density $\Sigma_{0}=10^{7}$ rad-2 of background radio sources covering
the entire sky, a mission time of $\tau=5$ years, lens velocity
$v_{{}_{\ell}}=10^{-3}$, an observing cadence of $f_{\text{rep}}=1/$day, and
an angular precision of $\sigma_{{}_{\delta\theta,\text{eff}}}=10\ \mu$as.
Existing observational constraints come from the absence of extragalactic
$\gamma$-rays from PBH evaporation [5], femtolensing of $\gamma$-ray bursts
[102], the present-day abundance of white dwarfs [9], and microlensing from
the Kepler mission [103], the EROS/MACHO mission [2], and Subaru/HSC Andromeda
observations [4]. The thick orange lines represents the projected sensitivity
of _Gaia_ monoblip searches of the galactic disk using the gravitational
deflection of light [80].

##  VI Conclusions

In this paper, we have shown that light rays can be deflected by inhomogeneous
axion configurations due to axion-photon interactions. We emphasize that this
effect is distinct from and in some cases more significant than the canonical
bending of light due to gravity. We addressed some of the previous claims of
the possibility of birefringent light bending that occurs at linear order in
the axion-photon coupling and argued that in realistic axion clump models, an
effect is only present at quadratic order, even in the presence of a
background refractive medium. We discussed the sensitivity of SKA to the
detection of anomalous shifts in the apparent positions of background radio
sources due to optical lensing by axion stars. This complements existing
proposals to observe the astrometric effects of dark lenses through
gravitational lensing, but probes a completely different region of parameter
space. For near critical-mass solitons, the absence of frequency-dependent
blip signals in SKA can place constraints on the axion-photon coupling in the
mass range $10^{-9}\text{ eV}\lesssim m_{a}\lesssim 10^{-4}\text{ eV}$, even
assuming solitons make up a relatively small fraction of DM. However, we only
find sensitivity to QCD axion models with enhanced axion-photon couplings,
where instability to photon perturbations may dominate. We expect sensitivity
to oscillons over a wide region of parameter space. It is interesting to note
that the oscillon masses probed have significant overlap with the region of
parameter space $M\in[10^{-14}M_{\odot},10^{-11}M_{\odot}]$, where there exist
no robust bounds from gravitational microlensing. Optical lensing by axion
stars could provide the first model-dependent bounds on compact objects in
this mass range. These sensitivity projections rely on oscillons being long-
lived over cosmological time-scales. While this is unlikely to be the case for
the QCD axion, it may be true in well-motivated models with flat potentials.
For sufficiently dense axion clumps, optical lensing may give rise to dramatic
lensing signatures such as significant magnification and the production of
multiple images. These signatures may be enhanced when the clump is in the
vicinity of a critical curve of another massive object. These effects and
their observational prospects will be discussed in future work.

###### Acknowledgements.

The author is grateful to Savas Dimopoulos, Davide Racco, and Mustafa Amin for
enlightening discussions during various stages of this work and for useful
comments on preliminary drafts of this manuscript. This work was supported by
the National Science Foundation under Grant No. PHYS- 1720397 and the Gordon
and Betty Moore Foundation Grant GBMF7946. The author acknowledges the support
of the Fletcher Jones Foundation and the National Science Foundation (NSF)
Graduate Research Fellowship Program.

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