# Q-balls in the sky

Arhum Ansari  Lalit Singh Bhandari  and Arun M. Thalapillil

###### Abstract

There may exist extended configurations in the dark matter sector that are
analogues of structures in the visible sector. In this work, we explore non-
topological solitonic configurations, specifically Q-balls, and study when
they may form macroscopic astrophysical structures and what their distinct
characteristics might be. We study in some detail theoretical bounds on their
sizes and constraints on the underlying parameters, based on criteria for an
astrophysical Q-ball’s existence, gravitational stability and viability of
solutions. Following this path, one is able to obtain novel limits on
astrophysical Q-ball sizes and their underlying parameters. We also explore
the gravitational lensing features of different astrophysical Q-ball profiles,
which are more general than the simple thin-wall limit. It is seen that the
magnification characteristics may be very distinct, depending on the actual
details of the solution, even for astrophysical Q-balls having the same size
and mass. Assuming that such astrophysical Q-balls may form a small component
of the dark matter in the universe, we place limits on this fraction from the
gravitational microlensing surveys EROS-2, OGLE-IV and the proposed future
survey WFIRST. Exploring various astrophysical Q-ball profiles and sizes, it
is found that while for smaller masses, the dark matter fraction comprising
astrophysical Q-balls is at most sub-percent, for larger masses, it may be
significantly higher.

##  1 Introduction

The true nature of dark matter is among the foremost questions in physics
today (See [1, 2] and references therein, for instance). The present evidence
suggests that this sector interacts only feebly with visible matter. Given the
presence of such an extraordinary dark sector, one is led to speculate if this
domain furnishes, apart from some plethora of dark elementary particles, bound
states similar to those in the visible sector. This may, for instance, include
field-theoretic, extended, solitonic objects as well as gravitationally bound
astrophysical objects that are similar to stars. Analogues in the visible
sector which are electromagnetically inert, for instance, massive
astrophysical compact halo objects (MACHOs) and dark clouds (see, for
instance, [3, 4] and related references), have long been investigated [5] with
the speculation that they may form at least some fraction of the missing
matter in the universe.

The constituents of the dark sector could also possibly form a rich set of
structures. Potential dark matter structures that have been speculated range
from galactic or sub-galactic scales all the way to almost particle-like bound
states (See, for example, [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18] ).
Particularly intriguing in recent years have been bosonic field
configurations, either forming expansive diffuse structures or exotic
astrophysical compact objects, that mimic ordinary neutron stars or white
dwarfs (for instance, see [19, 20, 21, 22, 23, 24, 25] and related
references).

There have been seminal works on various astrophysical aspects of extended
dark matter structures and star-like objects—for instance, looking at their
gravitational lensing characteristics [26, 27, 28, 29], signatures from
baryonic accretion [17, 27], gravitational wave signatures [30, 31] and so on
(For a more comprehensive survey and summary of the existing literature, in
these contexts, we kindly refer the interested reader to some of the excellent
current reviews and white papers [32, 24, 25, 33, 34]).

In this work we focus on non-topological solitions [35, 36] that could form
astrophysical structures. Specifically, we would like to study Q-balls [37] in
an astrophysical context. These are bound configurations of bosonic fields,
owing their existence to a conserved global charge ($Q$) [37], hence the ‘Q’
in their nomenclature. Their existence and stability are predicated on unique
characteristics in the theory and for large charges, their energy and volume
scale linearly with it. It is the latter scaling that makes such objects
behave almost like homogeneous lumps of exotic matter– Q-matter—with ‘Q’
playing the role of a putative particle number. Such objects have been studied
in various contexts since their conception—decay of such objects when
couplings to other fields are introduced [38], their small charge limits [39],
as dark matter candidates [40, 41], in the context of supersymmetry and
baryogenesis [42], cosmological phase transitions [43], spinning Q-balls [44],
charge-swapping Q-balls [45, 46, 47], gravitational waves during Q-ball
formation from condensate fragmentation [48] and so forth. Interest in such
objects has persisted, and they have also been subjects of many dedicated
recent studies, for instance, in the context of generating lepton asymmetry
and enhanced gravitational waves during Q-ball decay [49], Q-ball
superradiance [50], stress stability criteria in gauged Q-balls [51], fermion
scattering solutions for one-dimensional Q-balls [52], topological and non-
topological charged Q-monopole-balls [53], and many others.

There exist in the literature few analytical solutions and approximations for
particular potentials [54, 55, 56, 57, 58, 59, 60, 61], that accommodate
Q-ball solutions, but the analytic tractability of general scenarios has
largely been challenging. Recently, there has been some progress [62, 63, 64]
in obtaining better analytic approximations to Q-ball profiles and associated
quantities.

We hope to leverage this recent progress [62, 63, 64] in the analytic
approximations for Q-ball profiles by adapting it to study aspects of Q-balls
when they may form astrophysically viable objects. Particularly, in this
context, we explore a broader set of solutions than the strict thin-wall limit
[37]. We theoretically study when the flat spacetime Q-ball solutions may also
be considered as good approximations in the astrophysical context. We broach
this question by considering Jean’s instability arguments and probing the non-
gravitational limit of such macroscopic Q-balls. Similarly, for
astrophysically viable Q-balls, from a careful study of the mass density
profiles and gravitational lensing features, we point out distinct
characteristics of various solutions. Through microlensing survey constraints,
we also bound the fraction of total dark matter that may be in the form of
astrophysical Q-balls. For this latter part, we will partly modify and adapt
few of the methodologies in [26, 27]. The actual cosmological origins and
formation of astrophysical Q-balls is an intruiging topic of study (see, for
instance, [40, 43, 65, 66, 67] and related references). For the aspects we
focus on in this study, we will nevertheless be largely agnostic to any
specific formation mechanism.

In Sec. 2, we briefly review the theoretical framework for Q-balls, fixing
conventions and notations along the way. Then, in Sec. 3, we summarise aspects
of gravitational lensing, specifically microlensing, that we will subsequently
utilise to study astrophysical Q-balls. Sec. 4 contains the main results of
the study. Here, we present our theoretical discussions pertaining to the
astrophysical viability of Q-ball solutions and their unique gravitational
lensing characteristics, as well as microlensing constraints. We summarise and
conclude in Sec. 5.

##  2 Non-topological solitons with a conserved charge

Q-balls [37] are non-topological solitons that may occur in theories with a
conserved global charge. They arise as stable, localised configurations in
generic scalar field theories whose potential satisfies specific conditions.

For the purposes of our present discussion, consider a theory with a conserved
$U(1)$ charge, described by a Lagrangian density given by

$\mathcal{L}=\partial_{\mu}\Phi\partial^{\mu}\Phi^{\dagger}-U(\Phi^{\dagger}\Phi)\;.$
(2.1)

The stable vacuum is assumed to be at $\lvert\Phi\rvert=0$, and without loss
of generality, the potential is assumed to be vanishing in the vacuum. Q-ball
solutions will exist for all potentials $U(\Phi^{\dagger}\Phi)$ when
$U(\Phi^{\dagger}\Phi)/\Phi^{\dagger}\Phi$ has a minimum, without loss of
generality, at a real, positive field value $\Phi_{\text{\tiny{Q}}}$
satisfying

$0\leqslant\frac{U(\Phi_{\text{\tiny Q}})}{\Phi^{2}_{\text{\tiny
Q}}}\equiv\omega^{2}_{\text{\tiny Q}}<m^{2}\;.$ (2.2)

Here, the mass of the fundamental scalar quanta $\Phi$ is defined as
$m^{2}\equiv\frac{d^{2}U(\Phi^{\dagger}\Phi)}{d\Phi^{\dagger}d\Phi}\big{|}_{\Phi=0}$.
The theory defined by Eq. (2.1) is invariant under a global $U(1)$ symmetry
with the associated conserved charge for Q-balls in the local
$\Phi_{\text{\tiny{Q}}}$ minimum given by

$Q\equiv i\int d^{3}x(\Phi\dot{\Phi}^{\dagger}-\Phi^{\dagger}\dot{\Phi})\;.$
(2.3)

The condition for the existence of Q-balls given in Eq. (2.2) is equivalent,
for large $Q$-charges, to a requirement that the Q-ball configuration is more
energetically favourable than a collection of scalar particles with the same
total charge.

The Hamiltonian for the theory is

$H=\int
d^{3}x\left(|\dot{\Phi}|^{2}+\absolutevalue{\nabla\Phi}^{2}+U(\absolutevalue{\Phi})\right)\;,$
(2.4)

and to study the existence and possibility of objects with a fixed charge $Q$,
we may analyse the functional [39]

$\mathcal{F}=H+\omega\left(Q-i\int
d^{3}x(\Phi\dot{\Phi}^{\dagger}-\Phi^{\dagger}\dot{\Phi})\right)\;.$ (2.5)

Here, $\omega$ is a Lagrange multiplier. Minimising the functional in Eq.
(2.5), it may be shown that the minimum energy configuration, subject to a
fixed Q-charge, will have a time dependence that goes like

$\Phi(r,t)=\phi(r)e^{i\omega t}\;.$ (2.6)

The above time dependence gives a stationary solution that helps evade
Derrick’s theorem [68]. With this time dependence, the equation of motion now
becomes

$\phi^{\prime\prime}+\frac{2}{r}\phi^{\prime}=-\frac{1}{2}\frac{\partial}{\partial\phi}\left[\omega^{2}\phi^{2}-U(\phi)\right]\;,$
(2.7)

and the total charge, energy and equation of motion may be written as

$Q=8\pi\omega\int drr^{2}\phi(r)^{2}\;,$ (2.8) $\displaystyle E$
$\displaystyle=$ $\displaystyle 4\pi\int drr^{2}\left[\phi^{\prime
2}+\phi^{2}\omega^{2}+U(\phi)\right]\;,$ (2.9) $\displaystyle=$
$\displaystyle\omega Q+\frac{8\pi}{3}\int drr^{2}\phi^{\prime 2}\;.$

At this juncture, we would like to re-emphasize that, in this study, by Q-ball
solutions, we mean the solution one obtains strictly from Eq. (2.7). This
distinction is being made to distinguish Q-balls from Q-stars, where the
effects of gravity may have to be taken into account by solving a Gross-
Pitaevskii-Poisson equation. We will also comment on some of these aspects
later in Sec. 4.1.

From above, we identify the corresponding charge and energy density profiles
for a given field profile to be

$\rho_{\text{\tiny Q}}^{\text{\tiny C}}(r)=2\omega\phi(r)^{2}\;,$ (2.10)
$\rho_{\text{\tiny Q}}^{\text{\tiny
M}}(r)=2\omega^{2}\phi(r)^{2}+\frac{2}{3}\phi(r)^{\prime 2}\;.$ (2.11)

Also, differentiating Eq. (2.9) it is seen that [57]

$\derivative{E}{\omega}=\omega\derivative{Q}{\omega}\;.$ (2.12)

This is an exact relation that shows that $\omega$ may be interpreted as a
chemical potential. When $dE/dQ>0$, the Q-ball configuration is expected to be
unstable or metastable, with it being energetically favourable for the Q-ball
to shed fundamental $\Phi$ quanta.

To make the study more concrete, let us specifically look at the simplest
potential that allows for Q-ball solutions. Stable quartic potentials do not
yield Q-ball configurations. Confining to simple field representations, the
simplest potential that contains Q-ball-like solutions is the non-
renormalizable sextic potential

$U(\phi)=m^{2}\phi^{2}+\lambda\phi^{4}+\zeta\phi^{6}\;.$ (2.13)

Apart from being the simplest potential where Q-ball configurations may be
studied, thereby acting as a good proxy for investigating salient features, it
is also a potential where so far no exact solutions have been found.
Therefore, unlike few other special potentials where exact Q-ball solutions
have been found [54, 55, 56, 57, 58, 59, 60, 61], in this case, improved
analytic solutions and studies may be even more pertinent [62]. The second
term with $\lambda<0$ provides the attractive interaction, whereas the third
term is repulsive in nature and can balance the attractive interaction, which
hints towards forming stable bound states like Q-balls. When $\lambda<0$,
Q-ball solutions may be found, with the model having its global minima at
$\phi=0$ and a local minimum for $U(\Phi^{\dagger}\Phi)/\Phi^{\dagger}\Phi$ at

$\phi=\phi_{\text{\tiny Q}}\equiv\sqrt{\frac{\lvert\lambda\rvert}{2\zeta}}\;.$
(2.14)

In the latter minima, the ground state quanta may be Q-ball configurations,
subject to the fulfilment of the required conditions. From Eq. (2.2) we have
for this potential [62]

$\omega_{\text{\tiny Q}}=m\sqrt{1-\frac{\lambda^{2}}{4m^{2}\zeta}}\;.$ (2.15)

It is clear from above that $\omega_{\text{\tiny Q}}<m$ is satisfied, and the
existence of Q-balls is possible for

$0<\lambda^{2}\leqslant 4m^{2}\zeta\;.$ (2.16)

Note that these restrictions on the Lagrangian parameters, among other
features, make diffuse Q-ball structures potentially more distinct than
generic bosonic field configurations like bose stars.

Analysing the form of the effective potential
$\left(\omega^{2}\phi^{2}-U(\phi)\right)$ appearing in the equation of motion
Eq. (2.7), as a function of $\omega$, one finds that suitable Q-ball solutions
only occur for $\omega>\omega_{\text{\tiny Q}}$ and $\omega<m$. This, in
combination with the necessary condition Eq. (2.2), then implies that we must
have

$\omega_{\text{\tiny Q}}<\omega<m\;,$ (2.17)

for viable Q-ball solutions.

In the lower limit of the viable $\omega$ region,
$\omega\rightarrow\omega_{\text{\tiny Q}}$, one recovers the original Q-ball
solution of Coleman [37]. In the limit $\omega\rightarrow\omega_{\text{\tiny
Q}}$, the actual field profile obtained by solving Eq. (2.7) is very well
approximated by [37]

$\phi^{\text{\tiny TW}}(r)=\begin{cases}\phi_{\text{\tiny Q}}&r\leqslant
R_{\text{\tiny Q}}\;,\\\ 0&r>R_{\text{\tiny Q}}\;,\end{cases}$ (2.18)

where $R_{\text{\tiny Q}}$ is defined implicitly in terms of the actual field
profile through

$R_{\text{\tiny Q}}=\left(\frac{3\omega}{4\pi\omega_{\text{\tiny
Q}}\phi_{\text{\tiny
Q}}^{2}}\int_{0}^{\infty}d^{3}r\phi(r)^{2}\right)^{\frac{1}{3}}\;.$ (2.19)

We will refer to these solutions as thin-wall (TW) Q-balls.

Recently, better theoretical approximations for Q-ball solutions have been
found, enabling one to explore more of the $\omega$-parameter space regions,
encompassing $\omega_{\text{\tiny Q}}<\omega<m$. In particular, this allows
studies beyond the simplest step-function profiles of the exact thin-wall
limit Q-balls. Following methodologies developed in [62], better analytic
profiles may be obtained for the interior, exterior and transition regions of
the Q-ball. Remarkably, for large Q-balls, the full analytic profile may be
approximated by a single function [62] of the form

$\phi(r)=\frac{\phi_{*}}{\sqrt{1+2\exp\left[2\sqrt{m^{2}-\omega_{\text{\tiny
Q}}^{2}}\,(r-R_{\text{\tiny Q}})\right]~{}}}\;,$ (2.20)

where

$\phi_{*}^{2}=\frac{\phi_{\text{\tiny
Q}}^{2}}{3}\left[2+\sqrt{1+3\left(\frac{\omega^{2}-\omega_{\text{\tiny
Q}}^{2}}{m^{2}-\omega_{\text{\tiny Q}}^{2}}\right)}~{}\right]\;.$ (2.21)

The radius $R_{\text{\tiny Q}}$ in the expression above is defined implicitly
through

$\frac{d^{2}\phi(r)}{dr^{2}}\Big{|}_{r=R_{\text{\tiny Q}}}=0\;,$ (2.22)

using the full field profile approximation. When $\omega$ is very near the
$\omega_{\text{\tiny Q}}$ lower limit, the above definition starts coinciding
with the thin-wall definition Eq. (2.19) and the field profile also starts
coinciding with that given in Eq. (2.18). To leading order, Eq. (2.22) gives
large Q-ball radii to be [62]

$R_{\text{\tiny Q}}=\frac{\sqrt{m^{2}-\omega_{\text{\tiny
Q}}^{2}}}{~{}\omega^{2}-\omega_{\text{\tiny Q}}^{2}}\;.$ (2.23)

To distinguish the improved, more general field profile, given in Eq. (2.20),
from a limit of it given by Eq. (2.18) (i.e. the TW solution limit), we will
refer to the former as beyond-thin-wall (BTW) Q-balls.

For large Q-balls, the approximate BTW Q-ball profile given by Eq. (2.20) is
in agreement with the relation given by Eq. (2.12) [62]. $dE/dQ<0$ and
$E<m\,Q$ ensure the field-theoretic stability of Q-balls, against decay into
free scalar quanta. These will restrict the viable Lagrangian parameters in
turn. We will, therefore, also impose these field-theoretic stability
constraints along with the gravitational stability criteria later in Sec. 4.1,
while investigating astrophysically viable Q-balls.

If we expand the domain of symmetry, one may make the $U(1)$ invariance of the
Lagrangian, given in Eq.(2.1), local. This gives rise to gauged Q-balls [69],
which imposes some restrictions on the charge and radius of the Q-balls.
However, it has been shown that gauged Q-ball profile can still be described
by the expression given in Eq.(2.20) with the following identification of the
Lagrange multiplier [63]

$\omega_{\text{\tiny G}}=\omega g\phi_{\text{\tiny Q}}R_{\text{\tiny
Q}}\coth(e\phi_{\text{\tiny Q}}R_{\text{\tiny Q}})\;.$ (2.24)

In the above expression, $\omega_{\text{\tiny G}}$ and $\omega$ are the
Lagrange multipliers for gauged and global Q-balls, respectively, and $g$ is
the coupling constant between the scalar field and the gauge field. Owing to
this mapping, some of the analyses for global Q-balls may be translated for
gauged Q-balls. We leave a detailed analyses for future work.

One may also look at more generalised potentials given that the particular
polynomial potential given in Eq.(2.13), the simplest one which accommodates
Q-balls, may be considered as an effective potential coming from some
fundamental renormalisable theory. Most renormalisable theories in the low
energy limit give rise to polynomial potential with arbitrary exponents whose
first few terms can be of the form,

$U(\phi)=m^{2}\phi^{2}+\lambda\phi^{p}+\zeta\phi^{q}\;,$ (2.25)

with arbitrary powers $p$ and $q$. General polynomial potentials of the above
form were studied in detail recently [64]. It is seen that global properties
like total charge and total energy have only a mild dependence on the
polynomial powers $p$ and $q$ [64]. From such indications, it may be
speculated that the sextic potential in (2.13) may capture some of the more
salient features of generic Q-ball configurations while being analytically
more tractable.

In the next section, we briefly review the theoretical framework for
gravitational lensing while defining and clarifying relevant ideas and
notations that will be useful to us in our study.

##  3 Gravitational microlensing by extended structures

If there is an extended mass distribution in the path of light emanating from
astrophysical or cosmological sources, they may be deflected or distorted.
This broad effect is termed gravitational lensing [70].

Figure 1: A diagrammatic representation of a lens, observer and source lensing
system showing the various relevant quantities involved. The directed solid
line illustrates the path of a null ray. The lens plane is shown at the
centre, along with a spherical lens.

Consider Fig.1 illustrating a typical gravitational lensing system. In the
diagram, the light source (S), observer (O) and image position (I) are shown
along with the respective source angular position ($\theta_{S}$), image
angular positions ($\theta_{I}$) and deflection angle ($\hat{\theta}_{D}$).
$d_{\text{\tiny S}}$ denotes the angular diameter distance of the source from
the observer, $d_{\text{\tiny L}}$ the angular diameter distance of the
lensing mass from the observer and $d_{\text{\tiny LS}}$ is the angular
diameter distance from the source to the lensing mass. $\chi$ denotes the
transverse distance to the null ray, and z-distances are measured
perpendicular to the lens plane. In almost all cases of interest, and in
particular, in the case of interest to us, the deflection angles involved are
small, and the lens may be approximated to be a planar mass distribution by
projecting the mass of the lens on a sheet orthogonal to the line of sight.

For point-like objects, meaning their intrinsic size is negligible compared to
the other characteristic length scales in the system, the deflection angle of
light rays sufficiently far from an object is directly related to the
transverse gradient of the configuration’s Newtonian gravitational potential
($\varphi^{\text{\tiny{G}}}$). More precisely, for a total configuration mass
$M$, the point lens approximation is valid when the size of the lens is much
smaller than the corresponding Einstein radius of the system defined by [70]

$R_{\text{\tiny E}}=\left(\frac{4G_{\text{\tiny
N}}M}{c^{2}}\frac{d_{\text{\tiny L}}d_{\text{\tiny LS}}}{d_{\text{\tiny
S}}}\right)^{\frac{1}{2}}\;.$ (3.1)

Here $G_{\text{\tiny N}}$ is the Newton’s gravitational constant. This is, for
instance, relevant in many gravitational lensing studies involving compact
astrophysical objects like black holes or neutron stars. More pertinently, for
our purposes, this is also applicable to small Q-balls. For a total lens mass
$M$, the deflection angle would for instance be given by [71]

$\hat{\theta}_{\text{\tiny
D}}=\frac{2}{c^{2}}\int\nabla_{\chi}\varphi^{\text{\tiny{G}}}~{}dz=\frac{4G_{\text{\tiny
N}}M}{c^{2}|\vec{\chi}|}\hat{\chi}\;.$ (3.2)

For extended mass distributions, like the Q-ball configurations we are
interested in, one essentially adds up the contributions due to the individual
mass elements. If $\chi$ denotes the transverse distance (with respect to some
origin) of a light ray passing through an extended mass distribution and
$\chi^{\prime}$ denotes the position of an individual mass element of the
distribution, then the deflection angle $\hat{\theta}_{\text{\tiny D}}(\chi)$
is given by

$\hat{\theta}_{\text{\tiny D}}(\chi)=\frac{4G_{\text{\tiny
N}}}{c^{2}}\int\frac{(\vec{\chi}-\vec{\chi}^{\prime})dM^{\prime}}{\absolutevalue{\vec{\chi}-\vec{\chi}^{\prime}}^{2}}=\frac{4G_{\text{\tiny
N}}}{c^{2}}\int
d^{2}\chi^{\prime}\frac{(\vec{\chi}-\vec{\chi}^{\prime})\sigma(\chi^{\prime})}{\absolutevalue{\vec{\chi}-\vec{\chi}^{\prime}}^{2}}\;.$
(3.3)

Here, for a density distribution $\rho(\chi,z)$, we have defined the planar
density distribution $\sigma(\chi)=\int_{-\infty}^{\infty}\rho(\chi,z)dz$. A
special case arises for spherically symmetric lenses. In that case, we can
shift the origin to the centre, and Eq.(3.3) becomes [71]

$\displaystyle\hat{\theta}_{\text{\tiny D}}(\chi)$ $\displaystyle=$
$\displaystyle\frac{4G_{\text{\tiny
N}}}{c^{2}|\vec{\chi}|}\int_{0}^{\chi}d^{2}\chi^{\prime}\sigma(\chi^{\prime})\;,$
(3.4) $\displaystyle=$ $\displaystyle\frac{4G_{\text{\tiny
N}}\widetilde{M}(\chi)}{c^{2}\chi}\;,$

where

$\widetilde{M}(\chi)=\int_{0}^{\chi}d^{2}\chi^{\prime}\sigma(\chi^{\prime})=2\pi\int_{0}^{\chi}d\chi^{\prime}\chi^{\prime}\sigma(\chi^{\prime})\;.$
(3.5)

The angular source position and angular image positions are related via the
lens equation

$\theta_{\text{\tiny S}}=\theta_{\text{\tiny I}}-\theta_{\text{\tiny
D}}(\theta_{\text{\tiny I}})\;,$ (3.6)

where

$\theta_{\text{\tiny D}}=\frac{d_{\text{\tiny LS}}}{d_{\text{\tiny
S}}}\hat{\theta}_{\text{\tiny D}}\;.$ (3.7)

Using Eq. (3.4) with the identification $\chi=d_{\text{\tiny
L}}\theta_{\text{\tiny I}}$, the lens equation may be written as

$\theta_{\text{\tiny S}}=\theta_{\text{\tiny I}}-\frac{d_{\text{\tiny
LS}}}{d_{\text{\tiny S}}d_{\text{\tiny L}}}\frac{4G_{\text{\tiny
N}}\widetilde{M}(\theta_{\text{\tiny I}})}{c^{2}\theta_{\text{\tiny I}}}\;.$
(3.8)

For a general extended mass distribution, given a source position
$\theta_{\text{\tiny S}}$, we solve the above lens equation to find the image
positions $\theta_{\text{\tiny I}}$.

When $\theta_{\text{\tiny S}}=0$—meaning the source, lens and observer are
aligned along the line of sight—the angular image position, for a point lens
of total mass $M$, is defined to be at the point-like Einstein angle
($\theta_{\text{\tiny E}}$). This is related to Eq. (3.1) through

$\theta_{\text{\tiny E}}=\frac{R_{\text{\tiny E}}}{d_{\text{\tiny L}}}\;.$
(3.9)

We will use $R_{\text{\tiny E}}$ and $\theta_{\text{\tiny E}}$, when required,
to normalise the Q-ball radius and relevant lensing angles.

By Liouville’s theorem, the surface brightness of the source is conserved
under gravitational lensing. Under gravitational microlensing, as a Q-ball
configuration gradually moves across a light source, say a background galaxy,
there will be a waxing and waning of its measured brightness due to the
magnification. In this context, for the aforementioned reasons, the
magnification of the source would just be given by the ratio of the image area
to the source area. For each $\theta_{\text{\tiny I}}$ solution of Eq. (3.8),
this magnification ($\mathfrak{m}_{\text{\tiny I}}$) may be expressed as

$\mathfrak{m}_{\text{\tiny I}}=\frac{\theta_{\text{\tiny
I}}}{\theta_{\text{\tiny S}}}\derivative{\theta_{\text{\tiny
I}}}{\theta_{\text{\tiny S}}}\;.$ (3.10)

The total magnification ($\mathfrak{m}$) obtained in the detector is then just
the sum of magnification produced by all the images; obtained as solutions to
Eq. (3.8)

$\mathfrak{m}=\sum_{\text{\tiny I}}\absolutevalue{\mathfrak{m}_{\text{\tiny
I}}}\;.$ (3.11)

Let us now proceed to discuss the main results of the study in the context of
Q-ball configurations that may exist in the universe.

##  4 Astrophysical Q-ball structures

In this section, we discuss some of the theoretical considerations for
Q-balls, when they may form extended astrophysical structures, followed by
discussions of their gravitational lensing characteristics and constraints on
astrophysical Q-balls from microlensing surveys.

###  4.1 Theoretical bounds from existence, stability and viability

As we mentioned earlier, by Q-balls, we strictly mean here the pure field
theoretical non-topological soliton solutions, ignoring any role of gravity.
Obviously, the latter regime puts constraints on how dense these astrophysical
Q-ball structures could be and are constrained by relevant Jean’s instability
criteria. We will be considering diffuse Q-balls in the low compactness
regime.

When a Q-ball of total mass $M_{\text{\tiny{Q}}}$ is confined within a
characteristic radius $R_{\text{\tiny{Q}}}$ and is slightly compressed, there
is a propensity for gravity to compress the system, thereby attempting to
decrease the gravitational potential energy, while in contrast, the Q-matter
inside would try to resist this compression by virtue of outward internal
pressure. Assuming a homogeneous medium with almost constant density, which is
a good approximation for large Q-ball solutions, we may now formulate the
relevant Jean’s instability criteria.

The characteristic speed of the outward pressure waves is the sound speed in
the homogeneous medium ($v^{\text{\tiny s}}_{\text{\tiny Q}}$). The relevant
time scales are, therefore, the sound crossing time in Q-matter and the
corresponding gravitational free-fall time, which is given by

$t^{\text{\tiny{g}}}_{\text{\tiny{Q}}}\simeq\frac{1}{\sqrt{G_{\text{\tiny
N}}\,\rho_{\text{\tiny Q}}^{\text{\tiny M}}}}\;.$ (4.1)

Here, $G_{\text{\tiny N}}$ is Newton’s gravitational constant and
$\rho_{\text{\tiny Q}}^{\text{\tiny M}}$, as before, is the mass density of
the Q-ball. Within this characteristic free-fall time
$t^{\text{\tiny{g}}}_{\text{\tiny{Q}}}$ the pressure waves would travel a
characteristic length

$d^{\text{\tiny J}}_{\text{\tiny Q}}=v^{\text{\tiny s}}_{\text{\tiny
Q}}\,t^{\text{\tiny{g}}}_{\text{\tiny{Q}}}\simeq\frac{v^{\text{\tiny
s}}_{\text{\tiny Q}}}{\sqrt{G_{\text{\tiny N}}\,\rho_{\text{\tiny
Q}}^{\text{\tiny M}}}}\;.$ (4.2)

This characteristic length scale $d^{\text{\tiny J}}_{\text{\tiny Q}}$ is the
Jean’s length for the Q-ball configuration.

The basic idea is that if Q-matter is extended beyond $d^{\text{\tiny
J}}_{\text{\tiny Q}}$ then the pressure waves resisting gravity will not have
enough time to travel the length scale and counter the external compressive
disturbance due to gravity. In terms of the Lagrangian parameters, assuming to
a good approximation almost homogeneous internal densities, we have

$v^{\text{\tiny s}\,2}_{\text{\tiny Q}}\equiv\frac{dP_{\text{\tiny
Q}}}{d\rho_{\text{\tiny Q}}^{\text{\tiny
M}}}=\frac{\lambda^{2}}{4m^{2}\zeta}\;.$ (4.3)

We note that as a direct consequence of the existence condition for Q-balls
given in Eq. (2.16), one automatically satisfies the causality criterion
$v^{\text{\tiny s}}_{\text{\tiny Q}}\leqslant 1$ in the Q-matter.

From Eq.(2.23), we have for the radius of a Q-ball

$R_{\text{\tiny{Q}}}=\frac{\sqrt{\lambda^{2}/4\zeta}}{(\omega^{2}-\omega_{\text{\tiny
Q}}^{2})}\;.$ (4.4)

Since $\omega$ has an upper bound, as proscribed by Eq.(2.17), from Eqs.
(2.15) and (4.4) one obtains a minimum radius for the Q-ball solution

$R_{\text{\tiny{Q}},\,\text{\tiny
min}}^{\,\omega<m}\equiv\sqrt{\frac{4\zeta}{\lambda^{2}}}\;.$ (4.5)

The existence of such a minimum radius was also recently pointed out to be
valid even in general polynomial potentials [64]. This agrees with a recently
proposed conjecture [72] that all bound states in a theory must be greater
than the Compton wavelength $\lambda_{\phi}^{\text{\tiny C}}\equiv 1/m$. In
the case of Q-balls we note particularly that
$R_{\text{\tiny{Q}},\,\text{\tiny
min}}^{\,\omega<m}/\lambda_{\phi}^{\text{\tiny C}}=\sqrt{4\zeta
m^{2}/\lambda^{2}}\geqslant 1$ by virtue of the Q-ball existence condition Eq.
(2.16).

From Eqs. (4.2), (4.3), and (4.24), we may also calculate the corresponding
Jean’s length explicitly. This gives

$d^{\text{\tiny J}}_{\text{\tiny
Q}}=\sqrt{\frac{\lvert\lambda\rvert}{4m^{4}G_{\text{\tiny
N}}\left(1-\frac{\lambda^{2}}{4m^{2}\zeta}\right)}}\equiv
R_{\text{\tiny{Q}},\,\text{\tiny max}}^{\text{\tiny J}}\;.$ (4.6)

For the pure non-topological soliton-like Q-ball solution to be valid, the
radius of the corresponding stable Q-ball must be less than this Jeans length.

The above equation, therefore, furnishes an upper bound on the radius of
astrophysically viable Q-balls. Otherwise, the initial Q-ball will be unstable
to gravitational collapse and potentially reconfigure into a different field
configuration, whose features may now depend on the effects of gravity as
well. The criterion

$R_{\text{\tiny{Q}}}~{}<~{}d^{\text{\tiny J}}_{\text{\tiny Q}}\equiv
R_{\text{\tiny{Q}},\,\text{\tiny max}}^{\text{\tiny J}}\;,$ (4.7)

for stability with respect to gravitational collapse and continued validity of
the pure Q-ball solution, then gives a condition on the Lagrange multiplier
$\omega$ as

$\omega>\omega_{\text{\tiny
Q}}\left[1+\left\\{\frac{\absolutevalue{\lambda}G_{\text{\tiny
N}}/\zeta}{1-\frac{\lambda^{2}}{4m^{2}\zeta}}\right\\}^{1/2}\right]^{1/2}\equiv\omega^{\text{\tiny
J}}_{\text{\tiny min}}\;.$ (4.8)

At $\omega=\omega^{\text{\tiny J}}_{\text{\tiny min}}$ the Q-ball will become
unstable to collapse.

Therefore, we note that there exists both a minimum and maximum limit on the
radius of astrophysical Q-balls. Of course, for physically viable
astrophysical Q-ball solutions to actually exist, we must then obviously
require

$R_{\text{\tiny{Q}},\,\text{\tiny
min}}^{\,\omega<m}<R_{\text{\tiny{Q}},\,\text{\tiny max}}^{\text{\tiny J}}\;.$
(4.9)

As emphasised by the superscripts, the lower and upper bound respectively come
from the restriction on the Lagrangian parameter $\omega$, for the existence
of Q-ball solutions, and Jean’s stability criteria, signifying stability to
gravitational collapse and continued validity of the flat-spacetime Q-ball
solutions. From Eqs. (2.16), (4.5) and (4.6) this gives the constraint

$\left\\{\left(1-\frac{\lambda^{2}}{4m^{2}\zeta}\right)\frac{G_{\text{\tiny
N}}\absolutevalue{\lambda}}{\zeta}\right\\}^{1/2}<\frac{\lambda^{2}}{4m^{2}\zeta}\leqslant
1\;.$ (4.10)

Note that here the first part of the inequality comes from Jeans instability
criteria, and the second part of the inequality results from the existence
condition for Q-balls.

A measure of the extent to which gravity influences the global properties of
an astrophysical object is quantified by the compactness parameter, which is
defined as $\mathcal{C}\equiv\frac{G_{\text{\tiny N}}M}{R_{\text{\tiny{Q}}}}$.
For large Q-balls, we can express this in terms of the Lagrangian parameters
and Lagrange multiplier $\omega$ as follows,

$\mathcal{C}_{\text{\tiny Q}}=\frac{8\pi G_{\text{\tiny
N}}}{3}\left(\frac{\absolutevalue{\lambda}}{2\zeta}\right)\left(m^{2}-\frac{\lambda^{2}}{4\zeta}\right)\frac{\lambda^{2}/4\zeta}{\left[\omega^{2}-\left(m^{2}-\frac{\lambda^{2}}{4\zeta}\right)\right]^{2}}\;.$
(4.11)

The maximum compactness of the profiles that we work with is
$\mathcal{O}(10^{-5})$. Hence we can safely say that gravity has very little
role to play in determining the global properties of such Q-balls.

Let us examine another perspective to think about the non-gravitational limit
and the validity of the flat spacetime Q-ball solutions in an astrophysical
context. Let us start from the case where any effects due to gravity are
negligible and a scalar field potential with non-zero self-interactions (i.e.
$\lambda\neq 0$, $\zeta\neq 0$) gives us viable flat spacetime Q-ball
configurations; as a solution to Eq. (2.7). Let us now consider how slowly
incorporating any effects due to gravity may start changing the solutions.

Towards this, let us change the flat spacetime metric in the scalar field
equation of motion to include gravity. In the weak-field limit this may be
done through the substitution

$\eta_{\mu\nu}\rightarrow g_{\mu\nu}(x)=(1+2\varphi^{\text{\tiny
G}},-1,-1,-1)\;.$ (4.12)

$\varphi^{\text{\tiny G}}$, as before, is the Newtonian gravitational
potential. Let us express the scalar field $\Phi$ as

$\Phi(r,t)=(2E/N)^{-1/2}\psi(r)e^{-iEt}\;.$ (4.13)

Here, $N$ is the total number of particles in the configuration, $E\approx
m+2E_{B}$ is the energy of the ground state, and $E_{B}\ll m$ is the binding
energy. The equation of motion takes the form

$E_{B}\,\psi=-\dfrac{1}{2m}\nabla^{2}\psi+m\varphi^{\text{\tiny
G}}\psi+\dfrac{N\lambda}{2m^{2}}\psi^{3}+\dfrac{3\zeta
N^{2}}{8m^{3}}\psi^{5}\;.$ (4.14)

The second and third term on the right-hand side leads to attraction between
bosons; due to gravity and attractive self-interactions ($\lambda<0$). The
last term on the right-hand side leads to repulsive interactions among the
bosons.

After inspecting the attractive components in Eq. (4.14), we discover that the
effect of gravitational interaction is weaker than that due to the attractive
self-interaction among the fields when

$|m\varphi^{\text{\tiny
G}}\rvert<\frac{N\absolutevalue{\lambda}}{2m^{2}}\psi^{2}\simeq\frac{M_{\text{\tiny{Q}}}\absolutevalue{\lambda}}{2\omega
m^{2}}\psi^{2}\;.$ (4.15)

Here, as suggested by Eq. (2.9), $M_{\text{\tiny{Q}}}\sim N\omega$ denotes the
approximate mass of the Q-ball. Using $\varphi^{\text{\tiny
G}}\sim-\,G_{\text{\tiny N}}M_{\text{\tiny{Q}}}/R_{\text{\tiny{Q}}}$ and
$\psi^{2}\sim 1/R^{3}_{\text{\tiny{Q}}}$, we may estimate the strength of
gravitational and self-interaction terms.

From Eqs. (4.15), (4.4) and (2.15), along with the scalings already
identified, one finds that the quartic self-interaction among the fields is
the dominant attractive interaction when

$\frac{G_{\text{\tiny
N}}m^{3}\omega\absolutevalue{\lambda}}{2\zeta\left(\omega^{2}-m^{2}+\frac{\absolutevalue{\lambda}^{2}}{4\zeta}\right)^{2}}<1\;.$
(4.16)

This implies that for the Lagrangian parameters satisfying the above
condition, gravity does not play a dominant role when compared to the
attractive self-interaction.

An analogous but slightly different reasoning has been used to converge on the
non-gravitational limit for generic Boson stars [21], starting now from an
initial configuration which is non-self-interacting ($\lambda=0$), and where
the scaling, in contrast, is now $R\sim 1/(G_{\text{\tiny N}}Mm^{2})$.

The above discussions mean that with these constraints the field-theoretic
Q-ball solutions of Sec. 2 are still good approximations for the astrophysical
structures under consideration. In our study, we ensure that the Jean’s
stability and subservient gravity conditions, given by Eqs. (4.7) and (4.16)
respectively, are well satisfied. This legitimises the consideration of these
astrophysical configurations as actual Q-balls [37] and, furthermore, the use
of the theoretical expressions and profiles we work with.

For brevity, we will henceforth refer to Q-balls satisfying the criteria of
Secs. 2 and 4.1 as astrophysical Q-balls. These bosonic field configurations
may be expected to be very similar in characteristics to ideal flat-spacetime
Q-balls while satisfying all the requisite existence, stability and non-
gravitational limit qualifications. For our studies, we will require these
conditions to ensure the theoretical validity and physical viability of the
astrophysical Q-ball structures.

Let us now discuss the various gravitational lensing characteristics of
astrophysical Q-ball structures and how they may change for the various types
of profiles possible for such objects.

###  4.2 Gravitational lensing by thin-wall and beyond-thin-wall Q-balls

When an astrophysical Q-ball passes across the field of view of the observer
and progressively occludes the source, the gravitational lensing by the Q-ball
will cause a waxing and waning of the source brightness. If we have a moving
Q-ball lens, then $\theta_{\text{\tiny S}}$ may vary with time. Assuming
approximately rectilinear motion of the source for the duration of the lensing
transit, the temporal dependence will be given by

$\theta_{\text{\tiny S}}=\frac{\sqrt{d_{\text{\tiny L}}^{2}\theta_{\text{\tiny
S},\text{min}}^{2}+v^{2}t^{2}}}{d_{\text{\tiny L}}}\;.$ (4.17)

where $\theta_{\text{\tiny S},\text{min}}$ is the minimum source angle
attained during transit. The above relation is just a consequence of the
Pythagorean theorem satisfied by the angular diameter distances in the lens
plane. As the Q-ball travels across the foreground of the source,
gravitational lensing will result in the formation of images, whose positions
will be given as a function of time by

$\frac{\sqrt{d_{\text{\tiny L}}^{2}\theta_{\text{\tiny
S},\text{min}}^{2}+v^{2}t^{2}}}{d_{\text{\tiny L}}}=\theta_{\text{\tiny
I}}-\frac{d_{\text{\tiny LS}}}{d_{\text{\tiny S}}d_{\text{\tiny
L}}}\frac{4G_{\text{\tiny
N}}\widetilde{M}_{\text{\tiny{Q}}}(\theta_{\text{\tiny
I}})}{c^{2}\theta_{\text{\tiny I}}}\;,$ (4.18)

following Eq. (3.8). In the microlensing context, the images so formed at each
instance of time will contribute to the overall magnification of the source.
These, when put together for the transit duration, leads to the full
microlensing light curve.

This gravitational microlensing will depend on the density profile of the
spherical Q-ball. Specifically, the maximum magnification and the brightness
profile crucially depend on the number and contribution from the images formed
and hence on the density profile through $M(\chi)$. Let us explore these
aspects now for thin-walled and beyond-thin-walled Q-ball solutions.

Figure 2: Plot showing the charge density and energy density profiles
calculated using transition profile function (solid) and exact numerical
profile (dashed). Here, $\bar{\rho}_{\text{\tiny Q}}^{\text{\tiny
C}}\equiv\rho_{\text{\tiny Q}}^{\text{\tiny C}}/2\omega_{\text{\tiny
Q}}\phi_{\text{\tiny Q}}^{2}$, $\bar{\rho}_{\text{\tiny Q}}^{\text{\tiny
M}}\equiv\rho_{\text{\tiny Q}}^{\text{\tiny M}}/2\omega_{\text{\tiny
Q}}^{2}\phi_{\text{\tiny Q}}^{2}$ and $\bar{r}\equiv
r\sqrt{m^{2}-\omega_{\text{\tiny Q}}^{2}}$. One sees that the agreement
between the exact numerical solution and the transition profile approximation
are relatively good across most regions, as well as for the two distinct
profiles shown. The two profiles shown have been labelled by their $\omega/m$
ratios. For both cases we have $m/\omega_{\text{\tiny Q}}\sim 10^{-5}$.

From the improved large Q-ball interior, exterior and intermediate region
field profiles [62], which are in a unified form well approximated by Eq.
(2.20), the charge and mass density profiles may be computed. We find that
they have the respective forms

$\rho_{\text{\tiny Q}}^{\text{\tiny
C}}(r)=\frac{2\omega\phi_{*}^{2}}{1+2\exp\left[2\sqrt{m^{2}-\omega_{\text{\tiny
Q}}^{2}}(r-R_{\text{\tiny{Q}}})\right]}\;,$ (4.19) $\rho_{\text{\tiny
Q}}^{\text{\tiny
M}}(r)=2\phi_{*}^{2}\left[\frac{\omega^{2}}{1+2\exp\left[2\sqrt{m^{2}-\omega_{\text{\tiny
Q}}^{2}}(r-R_{\text{\tiny{Q}}})\right]}+\frac{4(m^{2}-\omega_{\text{\tiny
Q}}^{2})}{3}\left(\frac{\exp\left[4\sqrt{m^{2}-\omega_{\text{\tiny
Q}}^{2}}(r-R_{\text{\tiny{Q}}})\right]}{\left(1+2\exp\left[2\sqrt{m^{2}-\omega_{\text{\tiny
Q}}^{2}}(r-R_{\text{\tiny{Q}}})\right]\right)^{3}}\right)\right].$ (4.20)

Here, $\phi_{*}$ is as defined earlier in Eq. (2.21). Note that the quantity
$R_{\text{\tiny{Q}}}$ in the above expressions is defined implicitly through
Eq. (2.22). For our lensing studies, the mass density $\rho_{\text{\tiny
Q}}^{\text{\tiny M}}(r)$ will be of primary interest.

The transition field profile in Eq. (2.20), the density profiles of Eqs.
(4.19) and (4.20), along with the Q-ball radius defined through Eq. (2.23),
all approximate the respective exact numerical profiles very well for large
Q-balls [62]. In Fig. 2, we show a comparison of the charge and energy
densities calculated using Eqs. (4.19) and (4.20) with the corresponding
numerically computed exact density profiles. One notes that the approximation
is relatively good across the radial range and even for distinct profiles
differing in their radius.

The mass profile for a large radius Q-ball may be calculated based on Eq.
(4.20), and is given by

$\displaystyle M_{\text{\tiny Q}}(r)$ $\displaystyle=$ $\displaystyle
4\pi\int_{0}^{r}drr^{2}\rho_{\text{\tiny Q}}^{\text{\tiny M}}(r)$ (4.21)
$\displaystyle=$ $\displaystyle
8\pi\phi_{+}^{2}\left[\frac{\omega^{2}}{4\vartheta^{3}}\left\\{2\vartheta
r\text{Li}_{2}\left(-\frac{1}{2}e^{2(R_{\text{\tiny{Q}}}-r)\vartheta}\right)+\text{Li}_{3}\left(-\frac{1}{2}e^{2(R_{\text{\tiny{Q}}}-r)\vartheta}\right)\right.\right.$
$\displaystyle\left.-\text{Li}_{3}\left(-\frac{1}{2}e^{2R_{\text{\tiny{Q}}}\vartheta}\right)+\vartheta^{2}r^{2}\left(\log(4)-2\log\left(e^{2\vartheta(R_{\text{\tiny{Q}}}-r)}+2\right)\right)\right\\}$
$\displaystyle+\frac{1}{24\vartheta}\left\\{-\text{Li}_{2}\left(-2e^{2(r-R_{\text{\tiny{Q}}})\vartheta}\right)+\text{Li}_{2}\left(-2e^{-2R_{\text{\tiny{Q}}}\vartheta}\right)+\frac{4\vartheta
r\left(2e^{4\vartheta r}(\vartheta
r+1)+e^{2\vartheta(r+R_{\text{\tiny{Q}}})}\right)}{\left(2e^{2\vartheta
r}+e^{2\vartheta R_{\text{\tiny{Q}}}}\right)^{2}}\right.$
$\displaystyle\left.\left.-(2\vartheta
r+1)\log\left(2e^{2\vartheta(r-R_{\text{\tiny{Q}}})}+1\right)+\log\left(2e^{-2\vartheta
R_{\text{\tiny{Q}}}}+1\right)\right\\}\right]\;,$

where we have defined $\vartheta\equiv\sqrt{m^{2}-\omega_{\text{\tiny
Q}}^{2}}=\sqrt{\lambda^{2}/4\zeta}$, for brevity of terms, and defined

$\text{Li}_{n}(z)=\sum_{k=1}^{\infty}\frac{z^{k}}{k^{n}}\;.$ (4.22)

The total energy and charge for large Q-balls calculated using the above
densities satisfy the theoretical consistency condition Eq. (2.12) to a good
approximation. In the large radius limit, they also satisfy the corresponding
expressions in the literature [62], to the given order. For the BTW Q-ball
transition profiles and parameter values that we work with, additionally, all
relevant boundary conditions and exact Q-ball relations [62] are well-
satisfied with any conservative errors being at most $\mathcal{O}(10\%)$.

Let us note an elementary limit of Eqs. (4.19) and (4.20), the so called
simple thin-wall limit mentioned earlier. When
$\omega\rightarrow\omega_{\text{\tiny Q}}$, we have the thin-wall limit, and
the field profiles are well-approximated by Eq. (2.18), with the charge and
energy density profiles also taking a simple form given by

$\rho_{\text{\tiny Q}}^{\text{\tiny C},\text{\tiny TW}}(r)\simeq
2\omega_{\text{\tiny Q}}\phi_{\text{\tiny
Q}}^{2}\theta(R_{\text{\tiny{Q}}}-r)=2m\left(\frac{\lvert\lambda\rvert}{2\zeta}\right)\sqrt{1-\frac{\lambda^{2}}{4m^{2}\zeta}}~{}\Theta(R_{\text{\tiny{Q}}}-r)\;,$
(4.23) $\rho_{\text{\tiny Q}}^{\text{\tiny M},\text{\tiny TW}}(r)\simeq
2\omega_{\text{\tiny Q}}^{2}\phi_{\text{\tiny
Q}}^{2}\theta(R_{\text{\tiny{Q}}}-r)=2m^{2}\left(\frac{\lvert\lambda\rvert}{2\zeta}\right)\left\\{1-\frac{\lambda^{2}}{4m^{2}\zeta}\right\\}~{}\Theta(R_{\text{\tiny{Q}}}-r)\;.$
(4.24)

Here, $\Theta(x)$ is the Heaviside step function. In this strict limit of
approximation, some of the gravitational lensing features will start to
coincide with that of uniform density profiles [26, 27].

In order to solve the lens equation given in Eq.(3.8), for a given Q-ball
solution, we need to know the corresponding mass-ratio function
$\widetilde{M}_{\text{\tiny Q}}(\theta_{\text{\tiny I}})/M_{\text{\tiny Q}}$.
This may be calculated starting from Eq.(3.5) as

$\frac{\widetilde{M}_{\text{\tiny Q}}(\theta_{\text{\tiny I}})}{M_{\text{\tiny
Q}}}=\frac{\int_{0}^{\theta_{\text{\tiny I}}/\theta_{\text{\tiny
E}}}du~{}u\int_{0}^{\infty}dv~{}\rho^{\text{\tiny M}}_{\text{\tiny
Q}}\left(\sqrt{u^{2}+v^{2}}\right)}{\int_{0}^{\infty}dw~{}w^{2}\rho^{\text{\tiny
M}}_{\text{\tiny Q}}\left(w\right)}\;.$ (4.25)

Here, we have normalised all the coordinates with respect to the Einstein
radius $R_{\text{\tiny E}}=d_{\text{\tiny L}}\theta_{\text{\tiny E}}$, as
defined in Eq. (3.1), in the following way– $u\equiv\chi/R_{\text{\tiny E}}$,
$v\equiv z/R_{\text{\tiny E}}$ and $w\equiv r/R_{\text{\tiny E}}$.
$M_{\text{\tiny Q}}$ is the total mass of the Q-ball lens. Thus, it is through
this mass-ratio function that the precise details of the Q-ball density
profile will appear in the lens equation, its solutions and the total
magnification finally obtained.

Let us start by analysing astrophysical Q-ball solutions very close to the
$\omega\rightarrow\omega_{\text{\tiny Q}}$ limit, the so-called TW Q-balls.
For the case of TW Q-ball solutions, evaluating Eq. (4.25) with Eq.(4.24), one
obtains the mass-ratio function

Figure 3: Plot showing total magnification produced with changing
$\theta_{\text{\tiny S}}$ for the case of TW astrophysical Q-balls. Here the
legend indicates the ratio $R_{\text{\tiny{Q}}}/(R_{\text{\tiny
E}}\sqrt{3/2})$, where $\sqrt{3/2}$ denotes the critical ratio of
$R_{\text{\tiny{Q}}}/R_{\text{\tiny E}}$ which separates the regimes with
different number of solutions. The dashed line refers to the total
magnification produced due to a point lens.

$\frac{\widetilde{M}_{\text{\tiny Q}}(\theta_{\text{\tiny I}})}{M_{\text{\tiny
Q}}}=\begin{cases}1-\left(1-\frac{(\theta_{\text{\tiny I}}/\theta_{\text{\tiny
E}})^{2}}{(R_{\text{\tiny{Q}}}/R_{\text{\tiny E}})^{2}}\right)^{3/2}&;\hskip
14.22636pt\left(\frac{\theta_{\text{\tiny I}}}{\theta_{\text{\tiny
E}}}\right)R_{\text{\tiny E}}<R_{\text{\tiny{Q}}}\\\ 1&;\hskip
14.22636pt\left(\frac{\theta_{\text{\tiny I}}}{\theta_{\text{\tiny
E}}}\right)R_{\text{\tiny E}}\geqslant R_{\text{\tiny{Q}}}\;.\end{cases}$
(4.26)

We note from above, with suitable re-scalings, that the dependence on
$\omega_{\text{\tiny Q}}$ and $\phi_{\text{\tiny Q}}$ in the lens equation Eq.
(3.8) comes solely through $R_{\text{\tiny{Q}}}/R_{\text{\tiny E}}$ and
$\theta_{\text{\tiny E}}$ for thin-wall Q-balls. The lens equation Eq. (3.8)
may now be solved with the mass function Eq. (4.26) to find the image
positions. It is found that there are two broad regimes for the lens equation
with respect to the obtained solutions, depending on the size of the Q-ball.

As we see from Eq. (4.26), for $\theta_{\text{\tiny I}}/\theta_{\text{\tiny
E}}<R_{\text{\tiny{Q}}}/R_{\text{\tiny E}}$, the lens equation as given in
Eq.(3.8) is a quintic polynomial in $\theta_{\text{\tiny I}}$. For the special
case $\theta_{\text{\tiny S}}=0$, it can be factorised into a product of a
quartic and linear polynomial. The linear polynomial gives the trivial
solution at the origin (i.e. $\theta_{\text{\tiny I}}=0$), irrespective of the
value of $R_{\text{\tiny{Q}}}/R_{\text{\tiny E}}$. The quartic equation has
two real solutions for $0<R_{\text{\tiny{Q}}}/R_{\text{\tiny E}}<\sqrt{3/2}$.
For $R_{\text{\tiny{Q}}}/R_{\text{\tiny E}}\leq 1$ there exits two solutions
which are equally spaced from the origin—at $\theta_{\text{\tiny
I}}=\pm\theta_{\text{\tiny E}}$—on the Einstein ring. For
$1<R_{\text{\tiny{Q}}}/R_{\text{\tiny E}}<\sqrt{3/2}$, we again have equally
spaced solutions which lie at $\lvert\theta_{\text{\tiny
I}}\rvert<\theta_{\text{\tiny E}}$, i.e. inside the Einstein ring.
Specifically, they are located at,

$\theta_{\text{\tiny I}}=\pm\frac{\theta_{\text{\tiny
E}}R_{\text{\tiny{Q}}}}{R_{\text{\tiny
E}}\sqrt{2}}\left(3-(R_{\text{\tiny{Q}}}/R_{\text{\tiny
E}})^{4}-((R_{\text{\tiny{Q}}}/R_{\text{\tiny
E}})^{2}-1)^{3/2}\sqrt{(R_{\text{\tiny{Q}}}/R_{\text{\tiny
E}})^{2}+3}\right)^{1/2}\;.$ (4.27)

For $R_{\text{\tiny{Q}}}/R_{\text{\tiny E}}\geqslant\sqrt{3/2}$, all four
solutions to the quartic equation becomes imaginary and we are left with only
one solution at the origin.

For non zero $\theta_{\text{\tiny S}}$, we can use Descarte’s rule of signs
(See, for instance, [73]) to get the number of positive and negative real
roots. From Descarte’s rule of signs, we deduce that there will be two
negative real roots and one positive real root. Among them the two negative
real roots disappear when either $\theta_{\text{\tiny S}}\gg 1$ or
$R_{\text{\tiny{Q}}}/R_{\text{\tiny E}}>\sqrt{3/2}$. Below we will try to get
some analytical form of some of these solutions in particular limits. We will
be largely restricted to numerical analysis, though, due to the Abel-Ruffini
theorem (for instance, see [74]).

For a thin-walled Q-ball with a radius satisfying
$R_{\text{\tiny{Q}}}>\sqrt{3/2}R_{\text{\tiny E}}$, Eq. (3.8), with the
functional form Eq.(4.26), yields only a single viable solution. The solution
may be readily obtained numerically, but in special cases, a semi-analytical
expression may be found. If the domain of the solution is such that
$\left(\theta_{\text{\tiny I}}/\theta_{\text{\tiny E}}\right)R_{\text{\tiny
E}}\ll R_{\text{\tiny{Q}}}\simeq\sqrt{m^{2}-\omega_{\text{\tiny
Q}}^{2}}/\left(\omega^{2}-\omega_{\text{\tiny Q}}^{2}\right)$, which is the
case when $\theta_{\text{\tiny S}}/\theta_{\text{\tiny E}}\ll 1$, then in this
region we would have

$\frac{\widetilde{M}_{\text{\tiny Q}}(\theta_{\text{\tiny I}})}{M_{\text{\tiny
Q}}}\Big{|}_{\left(\frac{\theta_{\text{\tiny I}}}{\theta_{\text{\tiny
E}}}\right)R_{\text{\tiny E}}\ll
R_{\text{\tiny{Q}}}}\simeq\frac{3}{2}\left(\frac{\theta_{\text{\tiny
I}}R_{\text{\tiny E}}}{\theta_{\text{\tiny
E}}R_{\text{\tiny{Q}}}}\right)^{2}\;,$ (4.28)

and Eq. (3.8) may be solved analytically to obtain an approximate solution

$\theta_{\text{\tiny I}}^{(0)}\simeq\frac{\theta_{\text{\tiny
S}}}{1-\left(3R_{\text{\tiny E}}^{2}/2R_{\text{\tiny{Q}}}^{2}\right)}\;.$
(4.29)

One may calculate the magnification in this regime by taking the derivative of
Eq. (4.26), with the limit $\left(\theta_{\text{\tiny I}}/\theta_{\text{\tiny
E}}\right)R_{\text{\tiny E}}\ll R_{\text{\tiny{Q}}}$ and using the solution in
Eq. (4.29). In this regime, from the single viable solution, one obtains a
magnification

$\mathfrak{m}\left({\theta_{\text{\tiny
I}}^{(0)}}\right)\Big{|}_{\left(\frac{\theta_{\text{\tiny
I}}}{\theta_{\text{\tiny E}}}\right)R_{\text{\tiny E}}\ll
R_{\text{\tiny{Q}}}}=\frac{{\theta_{\text{\tiny
I}}^{(0)}}}{{\theta_{\text{\tiny S}}}}\frac{d\theta_{\text{\tiny
I}}^{(0)}}{d\theta_{\text{\tiny
S}}}=\left(1-\frac{3}{2(R_{\text{\tiny{Q}}}/R_{\text{\tiny
E}})^{2}}\right)^{-2}\left[1-\frac{3(\theta_{\text{\tiny
S}}/\theta_{\text{\tiny E}})^{2}}{2(R_{\text{\tiny{Q}}}/R_{\text{\tiny
E}})^{4}}\left(1-\frac{3}{2(R_{\text{\tiny{Q}}}/R_{\text{\tiny
E}})^{2}}\right)^{-3}\right]$ (4.30)

It is straightforward to deduce by looking at Eq. (4.30) that as
$\theta_{\text{\tiny S}}\rightarrow 0$, we get a finite maximum magnification.
Hence we conclude that for Q-balls with a radius
$R_{\text{\tiny{Q}}}>\sqrt{3/2}R_{\text{\tiny E}}$ the maximum magnification
produced is finite and only depends on the ratio
$R_{\text{\tiny{Q}}}/R_{\text{\tiny E}}$.

Let us now consider the other regime in Q-ball sizes. For Q-balls satisfying
$R_{\text{\tiny{Q}}}<\sqrt{3/2}R_{\text{\tiny E}}$, there may be three viable
solutions for small $\theta_{\text{\tiny S}}/\theta_{\text{\tiny E}}$, two of
which disappear as we move towards larger values. Again, one may numerically
solve the lens equation to find the solutions, but in special cases, semi-
analytic expressions may be obtained.

Two out of the three solutions are found in the region satisfying
$\frac{\theta_{\text{\tiny I}}}{\theta_{\text{\tiny
E}}}\geqslant\frac{R_{\text{\tiny{Q}}}}{R_{\text{\tiny E}}}$. Here, in fact,
we can analytically solve the lens equation, which is now only quadratic in
$\theta_{\text{\tiny I}}/\theta_{\text{\tiny E}}$. The solutions obtained are

$\theta_{\text{\tiny I}}^{(\pm)}=\frac{\theta_{\text{\tiny
S}}}{2}\left[1\pm\sqrt{1+\frac{4\theta_{\text{\tiny
E}}^{2}}{\theta_{\text{\tiny S}}^{2}}}\right]\;.$ (4.31)

These are observed to just coincide with image solutions for a point-like
lens. The magnification of each of these images is given by

$\mathfrak{m}\left({\theta_{\text{\tiny
I}}^{(\pm)}}\right)\Big{|}_{\left(\frac{\theta_{\text{\tiny
I}}}{\theta_{\text{\tiny E}}}\right)R_{\text{\tiny E}}\geqslant
R_{\text{\tiny{Q}}}}=\frac{{\theta_{\text{\tiny
I}}^{(\pm)}}}{{\theta_{\text{\tiny S}}}}\frac{d\theta_{\text{\tiny
I}}^{(\pm)}}{d\theta_{\text{\tiny S}}}=\pm\frac{(\theta_{\text{\tiny
S}}/\theta_{\text{\tiny E}})^{2}+2}{2\,\theta_{\text{\tiny
S}}/\theta_{\text{\tiny E}}\sqrt{(\theta_{\text{\tiny S}}/\theta_{\text{\tiny
E}})^{2}+4}}+\frac{1}{2}\;,$ (4.32)

leading to a contribution to the total magnification from these two solutions
to be

$\mathfrak{m}\Big{|}^{(\pm)}_{\left(\frac{\theta_{\text{\tiny
I}}}{\theta_{\text{\tiny E}}}\right)R_{\text{\tiny E}}\geqslant
R_{\text{\tiny{Q}}}}=\absolutevalue{\mathfrak{m}_{\text{\tiny
I}}^{(+)}}+\absolutevalue{\mathfrak{m}_{\text{\tiny
I}}^{(-)}}=\frac{(\theta_{\text{\tiny S}}/\theta_{\text{\tiny
E}})^{2}+2}{\theta_{\text{\tiny S}}/\theta_{\text{\tiny
E}}\sqrt{(\theta_{\text{\tiny S}}/\theta_{\text{\tiny E}})^{2}+4}}\;.$ (4.33)

In this regime of Q-ball sizes, the contribution from two of the solutions,
therefore, just coincides with the analogous contribution from point-like
lenses. Note that in the above expressions, the characteristics of the Q-ball
under consideration are still present, entering through the Einstein angle
$\theta_{\text{\tiny E}}$. We also note here that the maximum magnification
produced due to these images formally diverges as $\theta_{\text{\tiny
S}}\rightarrow 0$. In actuality, this apparent divergence will be mitigated by
finite source effects coming into play as we approach the limit.

An analytic form for the third solution may also be sought if it lies in the
region $\left(\theta_{\text{\tiny I}}/\theta_{\text{\tiny
E}}\right)R_{\text{\tiny E}}\ll R_{\text{\tiny{Q}}}$. It yields an expression
identical to that given in Eq. (4.29), and therefore enhances the image by the
same magnification factor as that given in Eq. (4.30). As we discussed before,
we can see from Eq. (4.30), that as we decrease $\theta_{\text{\tiny
S}}/\theta_{\text{\tiny E}}$ the total magnification increases. The maximum
magnification is produced for $\theta_{\text{\tiny S}}=0$ with a magnitude

$\mathfrak{m}_{3}^{\text{max}}\approx\left(1-\frac{3}{2(R_{\text{\tiny{Q}}}/R_{\text{\tiny
E}})^{2}}\right)^{-2}\;.$ (4.34)

Figure 4: In the above figures, we display the density profiles for three
different astrophysical Q-ball profiles with approximately equal radii and
average densities and compare their relative magnification profiles. We define
$\alpha\equiv\sqrt{(m^{2}-\omega_{\text{\tiny
Q}}^{2})/(\omega^{2}-\omega_{\text{\tiny Q}}^{2})}$ to quantify the different
Q-ball profiles. Top: Density profile for a TW and two BTW Q-balls are shown,
all of which have an equal radius, around $1\,R_{\odot}$. Here, we have
normalised the radial coordinate with respect to the solar radius,
$R_{\odot}=6.9\times 10^{8}\,\mathrm{m}$, and the density is normalised to the
average solar density, $\rho_{\odot}=1.4\times
10^{3}\,\mathrm{kg}/\mathrm{m}^{3}$. Bottom: The figure shows the dependence
of the total magnification on the source position $\theta_{\text{\tiny S}}$
for the same set of Q-ball profiles as in the adjoining figure. Though the
magnification profiles have similar shapes overall, the position of the
caustic progressively shifts towards lower $\theta_{\text{\tiny S}}$ values
for thicker transition regions. A larger magnification is also seen in the
small $\theta_{\text{\tiny S}}$ domain for profiles with a thicker transition
region.

For intermediate values of $R_{\text{\tiny{Q}}}/R_{\text{\tiny E}}$,
analytical expressions for the solutions and magnification are challenging to
obtain in general. In our subsequent studies, we will, therefore, numerically
compute all the solutions to the lens equation.

The numerically computed total magnification produced by thin-wall Q-balls are
shown in Fig. 3, for different values of $R_{\text{\tiny{Q}}}/R_{\text{\tiny
E}}$. From Fig. 3, we see that for TW Q-balls having
$R_{\text{\tiny{Q}}}>\sqrt{3/2}R_{\text{\tiny E}}$, decreasing the value of
$\theta_{\text{\tiny S}}$ to zero leads to a finite maximum value for the
magnification. This is quantified by the analytical expression given in
Eq.(4.30).

This may be contrasted with the
$R_{\text{\tiny{Q}}}\leqslant\sqrt{3/2}R_{\text{\tiny E}}$ case where there is
formally no upper bound on the total magnification, though in reality it will
be regulated when finite source sizes are correctly accounted for, as we
mentioned earlier. When the size of TW Q-balls are of the order of their
corresponding Einstein radius i.e. $R_{\text{\tiny{Q}}}\approx R_{\text{\tiny
E}}$, the magnification profile displays some interesting features—such as the
peak for the case $R_{\text{\tiny{Q}}}/R_{\text{\tiny
E}}=0.5\times\sqrt{3/2}$. Such a peak is termed a ‘caustic’, and is the value
of $\theta_{\text{\tiny S}}$ at which the number of solutions changes
discontinuously. The sudden divergence of the magnification comes from the
fact that the derivative $d\theta_{\text{\tiny S}}/d\theta_{\text{\tiny I}}$
vanishes, and by virtue of Eq.(3.10) makes the magnification formally diverge.

Intuitively, it may also be understood that Q-balls which are not in the
vicinity of the line of sight joining the source and the observer should not
affect the source brightness significantly. Therefore we see from Fig. 3 that
for large values of $\theta_{\text{\tiny S}}$, Q-balls of all sizes fail to
produce any magnification and therefore the magnification factor
$\mathfrak{m}\rightarrow 1$ irrespective of the size of the Q-balls.

Having looked at the salient features of lensing due to thin-wall Q-balls, we
now move on to investigate the gravitational lensing by BTW Q-balls, where
$\omega$ may take more general values. Here again, in order to solve the lens
equation given in Eq.(3.8), we need to calculate the mass ratio given in
Eq.(4.25), but now using the full mass density profile for BTW Q-balls given
in Eq. (4.20), with arbitrary $\omega$. This does not yield a simple
analytical form, except in very special cases. So, for the case of BTW
Q-balls, we will rely on numerical analysis for finding out solutions to lens
equation as well as calculating the total magnification. We also ensure that
all the requisite criteria for viable astrophysical Q-balls, as encapsulated
in Secs. 2 and 4.1 are satified. Note that the mass profile Eq. (4.21)
implicity appears in Eq. (3.5), for the definition of
$\widetilde{M}_{\text{\tiny Q}}(\chi^{\prime})$, through the identification
$r=\sqrt{\chi^{\prime 2}+z^{2}}$, for a fixed transverse distance
$\chi^{\prime}$.

The major difference in TW and BTW Q-balls comes from the possibility of a
distinct overall functional form and thicker transition region for the
latter’s field profiles and mass density profiles. It is natural to suspect
that such a feature in the mass density profile may have a different overall
impact on the microlensing signature, depending on the actual functional form,
even for Q-balls which have roughly the same mass and size. One of the
astrophysical observables to help contrast the distinct characteristics of TW
and BTW Q-balls is to compare their respective magnification profiles.

For this particular reason, in Fig. 4, we plot the magnification profile for a
thin wall Q-ball and two BTW Q-balls. All three Q-ball profiles have equal
radii and roughly the same average density. We can see from the figure that
qualitatively the magnification profiles look the same, but the position of
the caustic progressively shifts towards lower $\theta_{\text{\tiny S}}$
values for profiles with thicker transition regions. Also, for smaller values
of $\theta_{\text{\tiny S}}$, the profile with the thicker region of
transition produces a larger magnification. For large $\theta_{\text{\tiny
S}}$ values, all three Q-balls lens cases converge to a magnification of
unity, which is true for any generic astrophysical object.

As we commented earlier, when a lens passes through the line of sight of the
observer and the source, it magnifies the background source. Whether this is
observed or detected will depend on the microlensing survey instrument’s
sensitivity to this waxing of brightness. Usually, for categorising an event
to be a microlensing event, we define a threshold magnification value
($\mathfrak{m}^{*}$) above which an instrument is able to detect it as a
viable event. This threshold magnification ($\mathfrak{m}^{*}$) is
conventionally defined as the magnification generated by a point lens when
$\theta_{\text{\tiny S}}=\theta_{\text{\tiny E}}$. Utilising Eq. (4.33), this
gives the threshold magnification as $\mathfrak{m}^{*}=1.34$. The idea is
that, with the above prototypical magnification threshold assumption for a
survey, all point lenses which are positioned with $\theta_{\text{\tiny
S}}<\theta_{\text{\tiny S}}^{*}$ will produce magnification
$\mathfrak{m}>\mathfrak{m}^{*}$, and hence will be detected.

Figure 5: The threshold value of the source position (denoted by
$\theta_{\text{\tiny S}}^{*}$), below which one obtains a total magnification
$\mathfrak{m}\geqslant\mathfrak{m}^{*}$, is shown as a function of
$R_{\text{\tiny{Q}}}/R_{\text{\tiny E}}$ . Here, a range bounded by
$\mathfrak{m}^{*}=1.34\pm 0.01$ is the assumed threshold magnification band
(corresponding to the magnification due to a point mass lens when
$\theta_{\text{\tiny S}}\rightarrow\theta_{\text{\tiny E}}$) delimiting what
may be detectable by a microlensing survey. The dashed and dotted black curves
indicate point lens and TW Q-ball profiles, respectively. The green band is
composed of various distinct astrophysical Q-ball profiles with the
restriction $\sqrt{(m^{2}-\omega_{\text{\tiny
Q}}^{2})/(\omega^{2}-\omega_{\text{\tiny Q}}^{2})}\lesssim 0.5$, so as to
comfortably satisfy the field-theoretic stability constraints. For the
$R_{\text{\tiny{Q}}}/R_{\text{\tiny E}}$ values shown, we have verified the
existence of astrophysical Q-balls, satisfying the criteria of Secs. 2 and
4.1.

For extended distributions of matter like Q-balls, due to the non-trivial
density profiles, the value of $\theta_{\text{\tiny S}}$ that produces a
magnification of $\mathfrak{m}^{*}$ may not be the same as
$\theta_{\text{\tiny E}}$, and may even vary depending on the relative size of
the Q-ball as compared to its Einstein radius.

The threshold characteristics may be understood more clearly from Fig. 5,
where we have computed and plotted how the threshold value of
$\theta_{\text{\tiny S}}$, which gives magnification in the range
$\mathfrak{m}^{*}=1.34\pm 0.01$, changes with the ratio
$R_{\text{\tiny{Q}}}/R_{\text{\tiny E}}$. In Fig. 5, the green band depicts a
range of astrophysical Q-balls, with distinct density profiles. The band is
obtained by sifting through numerous Q-ball profiles with the requirement that
they satisfy the criteria of Secs. 2 and 4.1. The band is not an artefact of
the narrow range ($\pm 0.01$) we have assumed for $\mathfrak{m}^{*}$. From
Fig. 5, it is particularly obvious that when
$R_{\text{\tiny{Q}}}/R_{\text{\tiny E}}\rightarrow 0$, for all Q-balls, the
corresponding $\theta_{\text{\tiny S}}$ value goes to the point lens value
(i.e. $1$), as should be expected. Additionally, the band gets terminated at
$R_{\text{\tiny{Q}}}/(R_{\text{\tiny E}}\sqrt{3/2})\sim 2.7$, because above it
the magnification is always smaller than $1.34$. When
$R_{\text{\tiny{Q}}}/(R_{\text{\tiny E}}\sqrt{3/2})$ is near $0.4$ or in the
range $[1.4,\,2.5]$ the BTW Q-ball lenses are more effective in magnifying
sources than their point-lens counterparts. One also notes that there is a
peak near $R_{\text{\tiny{Q}}}/(R_{\text{\tiny E}}\sqrt{3/2})\sim 0.4$. In the
range $[0.5,\,1.3]$ the astrophysical Q-ball lenses are weaker lenses than
point lenses.

Figure 6: The threshold $\theta_{\text{\tiny S}}^{*}$ is shown now as a
function of the lens distance $d_{\text{\tiny L}}$. The lensing source objects
are assumed to be the Large Magellanic Cloud (top figure) and the Milky Way
bulge (bottom figure). The point lens and TW Q-balls are again represented by
dashed and dotted black curves, respectively. The green band, as in Fig. 5,
represents various astrophysical Q-ball configurations, all with similar mass
and radii but distinct density profiles. The plots have been made for
$M_{\text{\tiny{Q}}}\simeq 5\times 10^{-6}~{}\textup{M}_{\odot}$ and
$R_{\text{\tiny{Q}}}=1~{}\textup{R}_{\odot}$. We have again adopted
$\mathfrak{m}^{*}\in[1.33,1.35]$ and have restricted
$\sqrt{(m^{2}-\omega_{\text{\tiny Q}}^{2})/(\omega^{2}-\omega_{\text{\tiny
Q}}^{2})}\lesssim 0.5$ based on field-theoretic stability.

In Fig. 6, we further plot the variation of $\theta_{\text{\tiny S}}^{*}$ as a
function of the lens position $d_{\text{\tiny L}}$ for the Large Magellanic
Cloud (LMC) and Milky Way bulge (MW) sources; top and bottom figures,
respectively. The LMC is located at $50$ kpc and the MW at $8.5$ kpc from
earth. As we shall see in the next section, akin to the data encapsulated in
Fig. 6, a dictionary of $\theta_{\text{\tiny S}}^{*}$ for various
astrophysical Q-ball configurations is critical for determining microlensing
event rates and physically viable astrophysical Q-ball populations. Here, the
band indicates various astrophysical Q-balls with nearly identical masses and
radii but differing density profiles. For Fig. 6, as in Fig. 4, the mass and
radius of the Q-ball are taken to be about $5\times
10^{-6}\,\textup{M}_{\odot}$ and $1\,\textup{R}_{\odot}$. Clearly, for the MW
source, the Q-ball lens is more effective compared to LMC, manifesting as
higher values for $\theta_{\text{\tiny S}}^{*}$. This will reflect in the
microlensing event rates and hence in the limits on viable astrophysical
Q-ball populations.

In the next section, we place constraints on the population of astrophysical
Q-balls, assuming them to form a component of the dark matter sector, by
leveraging the observations from microlensing surveys like EROS-2, OGLE-IV and
WFIRST (proposed, future survey).

###  4.3 Contraints on astrophysical Q-balls from microlensing surveys

If astrophysical Q-balls exist and may be forming a small fraction of the
missing matter in the universe, they may be detected or at least constrained
by gravitational microlensing surveys. In this section, we wish to study
bounds on the fraction of dark matter that may, in principle, be in the form
of Q-balls.

The number of microlensing events induced by astrophysical Q-balls will be
directly proportional to the number density in the line of sight of the
observer and the source. Therefore, the idea is that depending on the number
of ‘anomalous’ events observed in a microlensing survey, we may hope to put
crude upper bounds on the population of astrophysical Q-balls in the vicinity
of our galaxy. We will adapt the methodologies expounded in [26, 27] for
studying our particular cases of interest. If the dark matter density at a
distance $d_{\text{\tiny L}}$ from observer is
$\rho_{\text{DM}}(d_{\text{\tiny L}})$, then we may write the fraction of that
density contained in Q-balls as
$\rho^{\text{\tiny{Q}}}_{\text{DM}}(d_{\text{\tiny
L}})=f_{\text{DM}}\,\rho_{\text{DM}}(d_{\text{\tiny L}})$. Here,
$f_{\text{DM}}$ is the fraction of total dark matter contained in the form of
Q-balls. This is the quantity we hope to put constraints on.

For a single background source, we may quantify the rate of events, assuming
unit exposure time, from the differential event rate expression [75]

$\frac{d^{2}\Gamma}{d\gamma d\tau}=\frac{2d_{\text{\tiny
S}}e(\tau)}{v_{\odot}^{2}M_{\text{\tiny{Q}}}}f_{\text{DM}}\rho_{\text{DM}}(\gamma)v_{\text{\tiny{Q}}}^{4}(\gamma)e^{-v_{\text{\tiny{Q}}}^{2}(\gamma)/v_{\odot}^{2}}\;.$
(4.35)

We have defined $\gamma\equiv d_{\text{\tiny L}}/d_{\text{\tiny S}}$ and
$v_{\odot}=2.2\times 10^{5}\mathrm{m\,s}^{-1}$, which is the circular speed of
the solar system around the galactic centre. In the above expression, $\tau$
is the transit time for a Q-ball to pass through the “lensing tube”. The
latter is defined as the volume of space between the observer and the source,
constructed by cylinders of infinitesimal thickness and radius
$\theta_{\text{\tiny S}}^{*}d_{\text{\tiny L}}$. This basically means that as
long as the Q-ball resides inside the lensing tube, it produces a
magnification greater than the threshold value $\mathfrak{m}^{*}$. Here,
$\theta_{\text{\tiny S}}^{*}$ is the threshold source position as a function
of $d_{\text{\tiny L}}/d_{\text{\tiny S}}$ , similar to Fig. 6. $e(\tau)$ is
the efficiency of detection, $M_{\text{\tiny{Q}}}$ is the mass of the
astrophysical Q-ball and $v_{\text{\tiny{Q}}}\equiv 2\theta_{\text{\tiny
S}}^{*}(\gamma)R_{\text{\tiny E}}(\gamma)/\theta_{\text{\tiny E}}\tau$ is the
velocity of the astrophysical Q-ball. The above expression is derived assuming
that dark matter and astrophysical Q-ball velocities follow a simple Maxwell-
Boltzmann distribution [75]. For the dark matter distribution concerned, we
assume an isothermal profile [76]

$\rho_{\text{\tiny DM}}(r)=\frac{\rho_{c}}{1+(r/r_{c})^{2}}$ (4.36)

where $\rho_{c}=1.4\,\mathrm{GeV}/\mathrm{cm}^{3}$ is the dark matter core
density and $r_{c}=4.4\,\mathrm{kpc}$ quantifies the size of the dark matter
core.

Figure 7: Bounds on the fraction of dark matter comprised of astrophysical
Q-balls are shown from various microlensing surveys. The point lens and TW
astrophysical Q-balls are shown by the dashed and dotted lines, respectively.
While the solid and dashed-dotted lines represent the two BTW Q-balls. The
three profiles have the same radius $R_{\text{\tiny Q}}$ in each of the cases.
As in Fig. 4 we have taken $\alpha=\\{10^{4},3,2\\}$. From top to bottom, the
constraints are given for astrophysical Q-balls of sizes
$0.5\,R_{\odot},1\,R_{\odot}$ and $2\,R_{\odot}$ respectively; again with the
solar radius $R_{\odot}=6.9\times 10^{8}\,\mathrm{m}$. We note that inspite of
having quantitatively disntinct individual magnification signatures, the BTW
Q-balls and TW Q-ball have almost the same $f_{\text{DM}}$ bound. The
$M/M_{\odot}$ ranges adopted are accommodating regions where we have verified
the existence of astrophysical Q-balls, following the conditions derived in
Secs. 2 and 4.1. We have taken $M_{\odot}=1.98\times 10^{30}\,\mathrm{kg}$.

We may now integrate Eq.(4.35), along the line joining the source and the
observer and between a range of astrophysical Q-ball transit times. This gives
the total number of Q-ball microlensing events registered ($\eta$) for a
single source and for unit observation time

$\eta=\int_{0}^{1}d\gamma\int_{\tau_{\text{min}}}^{\tau_{\text{max}}}d\tau\frac{d^{2}\Gamma}{d\gamma
d\tau}\;.$ (4.37)

Multiplying $\eta$ with the total number of sources in a survey ($N_{s}$) and
the total observation time ($T_{o}$), one obtains the total number of expected
astrophysical Q-ball events

$N_{\text{\tiny exp}}=\eta N_{s}T_{o}\;.$ (4.38)

Here, we emphasize that $\eta$ indirectly contains information about the
astrophysical Q-ball density profile distribution through $\theta_{\text{\tiny
S}}^{*}(\gamma)$. For an astrophysical Q-ball with fixed radius and mass,
$\theta_{\text{\tiny S}}^{*}/\theta_{\text{\tiny E}}$ still varies with
$\gamma$; as we already observed in Fig. 6, for instance. This may be partly
understood from Eq. (3.1). This information goes into Eq. (4.35) through
$v_{\text{\tiny{Q}}}$ and therefore in putting the constraints for Q-balls of
different radii and mass. Specifically, the dependence is through relations
similar to those encapsulated in Fig. 6.

We will leverage the gravitational microlensing surveys EROS-2 [77] and OGLE-
IV [78, 79] and the proposed survey WFIRST [80, 81] to put bounds on the
astrophysical Q-ball fraction ($f_{\text{DM}}$). We will place limits on the
population of astrophysical Q-balls in the vicinity of our Milky Way based on
the number of anomalous events detected (for EROS-2 and OGLE-IV) or assuming
future detection (for WFIRST).

EROS-2 [77] is pointed at the Large and Small Magellanic Clouds, which are
located at a distance of $50\,\mathrm{kpc}$ and $60\,\mathrm{kpc}$ from the
Sun, respectively. The total number of source stars in the survey is
$N_{s}=5.49\times 10^{6}$ [82]. The total observation time was
$T_{o}=2500\,\mathrm{days}$, with the accommodated transit times ranging from
a day up to a thousand days. We have taken the efficiency $e(\tau)=0.24$,
which is the time average over all transit periods. It has detected one event
that cannot be fully explained by background modelling. We have placed
constraints on the astrophysical Q-ball constituents in dark matter, taking
LMC as the source field.

The centre of the Milky Way contains a large number of source stars, around
$N_{s}=4.88\times 10^{7}$. OGLE-IV [78] takes its source field as the Milky
Way bulge, which is at a distance of $8.5\,\mathrm{kpc}$ from the Sun [82].
OGLE-IV has observed more than $2500$ events in its observation period of
$T_{o}=1826\,\mathrm{days}$. We have taken the efficiency $e(\tau)=0.1$, which
is the time average over all transit periods. Assuming these events to be due
to stellar structure, we put conservative constraints on the population of
astrophysical Q-balls in the Milky Way.

Proposed, future microlensing surveys, like WFIRST, will enable us to look for
astrophysical Q-balls in ever smaller mass ranges. The proposed WFIRST mission
expects to adopt the Milky Way Buldge as well as Magellanic clouds as its
source field. It will scan the sky in intervals spanning $72\,\mathrm{days}$
[81]. We have taken the efficiency $e(\tau)=0.5$. As we mentioned, it will be
most sensitive to very low sub-solar mass range objects. We have put
constraints on the population of low-mass Q-balls—assuming it detects one or
zero anomalous events—taking LMC as the source field.

While counting events, we assume Poisson statistics for the distribution of
microlensing events. Therefore, if the actual number of anomalous events
detected by one of these surveys is $N_{\text{obs}}$, say, then assuming
Poisson distribution for these events, we may calculate the expected number
$N_{\text{exp}}$ as

$\sum_{k=0}^{N_{\text{obs}}}P(k,N_{\text{exp}})=0.05\;.$ (4.39)

Here, $P(k,N_{\text{exp}})=N_{\text{exp}}^{k}e^{-N_{\text{exp}}}/k!$ denotes
the probability of observing $k$ events for a fixed $N_{\text{exp}}$. So if we
have already observed $N_{\text{obs}}$ events then by virtue of Eq.(4.39) we
can exclude the values of $N_{\text{exp}}$ such that
$\sum_{k=0}^{N_{\text{obs}}}P(k,N_{\text{exp}})<0.05$ with $95\%$ C.L. This
puts an upper bound on $N_{\text{exp}}$ which can be used to put the upper
bound on $f_{\text{DM}}$ as function of the lens mass through Eq.(4.38). We
have performed this analysis for the surveys mentioned above and our results
are shown in Fig. 7.

The limits on three astrophysical Q-ball profiles, the same as those displayed
in Fig. 4, are shown in Fig. 7. In each case, the astrophysical Q-ball radii
are the same. The profiles are chosen to have $\alpha=\\{10^{4},3,2\\}$, where
$\alpha\equiv\sqrt{(m^{2}-\omega_{\text{\tiny
Q}}^{2})/(\omega^{2}-\omega_{\text{\tiny Q}}^{2})}$. It is worth noting that
for sub-earth mass astrophysical Q-balls, very minimal variance exists in the
constrained $f_{\text{DM}}$, for various BTW Q-ball profiles. This is a
consequence of the fact that the area under the curve in Fig. 6 is almost the
same for the different profiles. This renders the contribution from the
lensing tube integration to the number of microlensing events to be almost
equal. There are also no significant differences in the $f_{\text{DM}}$ bounds
for higher-mass TW and BTW astrophysical Q-balls. This is because, for large-
mass Q-balls (at fixed radii), the ratio $R_{\text{\tiny{Q}}}/R_{\text{\tiny
E}}\ll 1$ and hence all Q-ball profiles start showing point lens behaviour.
This is clearly visible from Fig. 5, for instance. Therefore, one concludes
that despite their quantitatively unique individual magnification
characteristics, as far as the $f_{\text{DM}}$ bounds are concerned, the
various astrophysical Q-balls generate almost identical bounds.

We observe from Fig. 7 that the strictest limits on solar mass astrophysical
Q-balls may be placed from OGLE-IV. Also, for $R_{\text{\tiny{Q}}}\sim
R_{\odot}$, we see that for the mass range $10^{-5}-10^{-4}M_{\odot}$, the
astrophysical Q-ball population is restricted to being within about $0.1\%$ of
the total dark matter. As we move away from $10^{-5}M_{\odot}$, the
$f_{\text{DM}}$ constraints weaken. For low mass Q-balls with mass range
$10^{-7}M_{\odot}<M_{\text{\tiny{Q}}}<10^{-5}M_{\odot}$, we observe that
WFIRST and OGLE-IV put stricter bounds than EROS-2 when $R_{\text{\tiny{Q}}}$
is below $R_{\odot}$. This trend could also have been already speculated based
on the characteristics observed in Fig. 6. For
$R_{\text{\tiny{Q}}}>R_{\odot}$, in the vicinity of $10^{-7}M_{\odot}$, EROS-2
currently gives better limits than OGLE-IV, while WFIRST can potentially
constrain this region very strongly in the future. For mass range
$10^{-5}M_{\odot}-10^{-2}M_{\odot}$, OGLE-IV and EROS-2 put more strict bounds
on $f_{\text{DM}}$ as compared to WFIRST. This is expected as WFIRST is
designed to detect earth-mass astrophysical objects. We also observe that for
astrophysical Q-ball mass ranges $10^{-1}\,M_{\odot}-10\,M_{\odot}$, one
cannot put any really strong restrictive bounds on $f_{\text{DM}}$, from the
microlensing surveys considered. In these regions, therefore, astrophysical
Q-balls, if they exist, could abound.

##  5 Summary and conclusions

In this paper, we have explored a few aspects of non-topological solitonic
Q-ball solutions when such configurations may form astrophysically viable
structures in the universe. We focused on theoretical aspects related to their
viability, as well as investigated their gravitational lensing features for
different astrophysical Q-ball profiles. Speculating on what fraction of dark
matter may be comprised in Q-balls, we derived constraints from microlensing
surveys in the viable ranges. Apart from Figs. 3-7, let us summarise below
some of the main results.

In the context of the theoretical bounds, we deduced novel limits on the size
of astrophysical Q-balls in Eqs. (4.5) and (4.6). These were obtained from the
analytic form of the Q-ball radius and by considerations of gravitational
instability, respectively. In the same context, while exploring the Jeans
instability criteria for astrophysical Q-balls, we found a new lower bound in
Eq. (4.8) on the Lagrange multiplier, which is akin to a chemical potential in
the Q-ball context; as may be motivated from Eq. (2.12). Also, these limits on
the astrophysical Q-ball radii provided new inequalities for the Lagrangian
parameters, as encapsulated by Eq. (4.10).

Through the study of gravitational lensing features by different Q-ball
profiles, we noted distinct characteristics as the profiles were varied. We
observed, for instance, in Fig. 4 that thin-wall and beyond-thin-wall Q-balls,
having roughly the same mass and size, exhibit different magnification
profiles. This suggests that the microlensing light curve will also be
distinct and may help differentiate profiles, provided sufficient anomalous
events are observed in the future, if astrophysical Q-balls indeed exist in
the universe. In Figs. 5 and 6, we explored threshold angular source
positions, leading to potentially detectable magnifications in surveys, for
beyond-thin-wall Q-balls.

Finally, we derived gravitational microlensing constraints on astrophysical
Q-balls—both for the case of thin wall and beyond-thin-wall profiles—using
data from EROS-2, OGLE-IV and projections for the proposed WFIRST survey. The
limits on $f_{\text{DM}}$, the fraction of dark matter that may be in the form
of astrophysical Q-balls, are shown in Fig. 7. For lower masses, it is seen
that this fraction may at most be $\mathcal{O}(0.1\%)$, while for higher
masses, the fraction may be much higher, even in the $\mathcal{O}(10\%)$
range.

.

## Acknowledgments

We would like to thank Ranjan Laha for discussions. AT would like to thank the
organisers of the ‘Horizons in Accelerators, Particle/Nuclear Physics and
Laboratory-based Quantum Sensors for HEP/NP’ and ‘Particle Physics: Phenomena,
Puzzles, Promises’ ICTS conferences, as well as ICTS, Bengaluru, for their
kind hospitality during the completion of parts of this work. AA and LB
acknowledge support from a Junior Research Fellowship, granted by the Human
Resource Development Group, Council of Scientific and Industrial Research,
Government of India. AT acknowledges support from an Early Career Research
award, from the Department of Science and Technology, Government of India.

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