# Astrometric Microlensing of Primordial Black Holes with _Gaia_

Himanshu Verma  Vikram Rentala

###### Abstract

The _Gaia_ space telescope allows for unprecedented accuracy for astrometric
measurements of stars in the Galaxy. In this work, we explore the sensitivity
of _Gaia_ to detect primordial black hole (PBH) dark matter through the
distortions that PBHs would create in the apparent trajectories of background
stars, an effect known as astrometric microlensing (AML). We present a novel
calculation of the lensing probability, and we combine this with the existing
publicly released _Gaia_ eDR3 stellar catalog to predict the expected rate of
AML events that _Gaia_ will see. We also compute the expected distribution of
a few event observables, which will be useful for reducing backgrounds.
Assuming that the astrophysical background rate of AML like events due to
other sources is negligible, we then compute the potential exclusion that
could be set on the parameter space of PBHs with a monochromatic mass
function. We find that _Gaia_ is sensitive to PBHs in the range of
$0.4~{}M_{\odot}$ \- $5\times 10^{7}~{}M_{\odot}$, and has peak sensitivity to
PBHs of $\sim 10~{}M_{\odot}$ for which it can rule out as little as a
fraction $3\times 10^{-4}$ of dark matter composed of PBHs. With this
exquisite sensitivity, _Gaia_ has the potential to rule out a PBH origin for
the gravitational wave signals seen at LIGO/Virgo. Our novel calculation of
the lensing probability includes for the first time, the effect of
intermediate duration lensing events, where the lensing event lasts for a few
years, but for a period which is still shorter than the _Gaia_ mission
lifetime. The lower end of our predicted mass exclusion is especially
sensitive to this class of lensing events. As and when time-series data for
_Gaia_ is released, and once we have a better understanding of the
astrophysical background rate to AML signals, our prediction of the lensing
rate and event observable distributions will be useful to estimate the true
exclusion/discovery of the PBH parameter space utilizing this data.

##  1 Introduction

Dark matter (DM) makes up 25% of the energy density of the universe today, yet
it is still unknown as to what constitutes the DM. One promising theory is
that DM is made up of Primordial Black Holes (PBHs) [1, 2, 3]. Unlike
astrophysical black holes which form due to the collapse of stars at the end
of their life cycle, PBHs are expected to form in the early universe from
excess density perturbations that gravitationally collapse while overcoming
resistance from pressure [4]. In particular, this could happen due to either
an excess in the primordial power spectrum on small scales [5, 6, 7], or due
to sudden drops in pressure in the matter-radiation fluid [8]. For other
formation channels for PBHs see for e.g. [9].

There are a number of observational constraints on PBHs that make up a
significant fraction of the DM (see ref. [10, 11] for overviews of these
constraints). Usually, these constraints are phrased assuming that the PBHs
have a monochromatic mass function – where a fraction $f$ of the DM density is
assumed to be made up of PBHs of mass $M$. The most stringent method of
setting constraints on PBHs varies depending on the assumed value of $M$. For
$M$ of $10^{-11}~{}M_{\odot}$ and above, a combination of constraints from
microlensing [12, 13, 11, 14, 15, 16], gravitational wave signatures [17, 18,
19], accretion [20, 21] and dynamical effects [22, 23, 24, 25], seem to rule
out a fraction $f\gtrsim 0.1$ of the DM as being made up of PBHs. However, it
is possible that PBHs in this mass range make up a smaller, sub-dominant
component of the DM.

A large number of experiments have leveraged the technique of photometric
microlensing (PML) to set observational constraints on the PBH abundance [26,
27, 16, 28, 29, 30, 14]. PML relies on the prediction of general relativity
that light bends around massive objects. If PBHs constitute the DM then they
are distributed throughout galaxies. If a star passes behind a PBH, then the
PBH acts like a gravitational lens, and the light from such a star would get
bent and focused, leading to a temporary brightening of such a star. Thus,
surveys which have precision photometric sensitivity, i.e. sensitivity to
small brightness variations, have been employed very fruitfully to set very
strong constraints on the existence of PBHs in the mass range
$10^{-11}-30~{}M_{\odot}$.

In addition to the apparent magnification of a star due to a PBH in its
foreground, another consequence of lensing is that the apparent trajectory of
the star would appear to be distorted relative to its true trajectory. Such an
effect is called astrometric microlensing (AML) [31, 32, 33, 34]. Surveys
which have precision astrometric sensitivity, i.e. sensitivity to the position
and velocity of the star, would be able to detect such an effect. In principle
AML should be more sensitive than PML to the presence of PBHs. This is because
in AML the relevant observable, which is the deflection of a star’s apparent
position from its true position, falls off as $1/\theta$, where $\theta$ is
the angle between the true position of the lens and the star on the celestial
sphere [35]. This is in contrast to PML where the relevant observable is the
change in magnification, and this falls off faster, as $1/\theta^{4}$ [35].

In the past few years, AML techniques have already started to be employed to
detect PBH and PBH-like candidates. Recently, there has been an exciting claim
of discovery of an isolated stellar-mass black hole lens using the AML
technique with HST data [36]. AML searches have also been employed to detect
signatures of intermediate mass black holes in the mass range
$10^{2}-10^{5}~{}M_{\odot}$ [37, 38]. AML observations have also been proposed
as follow ups for PBH candidates detected via PML methods in order to measure
the properties of the lens with greater precision [39, 40, 41]. Besides PBHs,
AML can be used to set constraints on exotic ultra-compact mini-clusters halos
(UCMCHs) [42, 43], and also on more conventional DM sub-halos [44, 45].

What is needed for precision astrometry is highly accurate position
measurements and also high cadence. The more accurate the position
measurements, the more easily small deflections in the trajectory of a star
can be detected. High cadence ensures that the star’s position is tracked
frequently enough that a lensing event is not missed. Both these factors
affect sensitivity to lower mass PBHs. At the other extreme, for high mass
PBHs, the lensing event durations can be extremely large, and if the mission
life time is small compared to a typical event duration, then the detection of
a distortion in the trajectory of a star can be difficult.

The _Gaia_ satellite, launched by ESA, has been surveying the sky since 2014.
_Gaia_ is an optical all-sky survey satellite with a large field-of-view $\sim
0.72^{\circ}\times 0.69^{\circ}$ [46]. The data collected by this satellite
provides an exciting opportunity for high precision astrometry. _Gaia_ is
expected to observe over 1 billion stars and obtain sub milli-arcsecond level
precision position measurements in a single pass. Already, it has provided us
with unprecedented views of the Milky Way. However, at the moment, time-series
data, which would enable us to make high precision searches for AML events,
has not been publicly released by the collaboration.

In this work, we set out to estimate the sensitivity of _Gaia_ to PBHs through
their expected AML signatures. We first make a prediction for the expected
rate of AML events that _Gaia_ will detect for a given assumption of PBH
parameters $f$, $M$. We argue that lensing events can be classified into one
of three types: short, intermediate, and long duration lensing events (SDLEs,
IDLEs and LDLEs). Our prediction for the lensing rate for these classes of
events involves a novel calculation of the lensing probability of a given
background star. We combine this probability calculation with a catalog of
stars for which _Gaia_ is expected to have high precision astrometric data, to
obtain a lensing event rate prediction. Besides the total rate, we also make
predictions for the distribution of expected event observables, such as
location on the sky, event duration, maximum deflection angle, etc. These
predictions will be useful for characterizing the space of observables for
genuine signal events which can distinguish them from backgrounds.

There are three main types of backgrounds that _Gaia_ will see. The first type
are statistical fluctuations in the centroiding of the camera image of the
background star. The second type are instrumental systematics. The third type
are astrophysical backgrounds which can mimic PBH induced AML signatures. For
example, besides AML events due to PBHs, _Gaia_ is also expected to see AML
events due to microlensing caused by foreground stars [47, 48], objects formed
from stellar collapse [49, 50, 51], brown dwarfs [52, 53], and free floating
planets [54]. Another type of astrophysical background could be due to stars
which do not follow rectilinear motion. Isolated stars are expected to
approximately follow rectilinear motion in the Galactic rest frame. However,
unresolved binary companions or local gravitational potential gradients, may
introduce distortions in the observed trajectories compared to the
trajectories expected from rectilinear motion. These distortions could also be
incorrectly interpreted as AML events, and could therefore lead to an
additional source of background to PBH searches.

If we could characterize the space of observables of these backgrounds, then
we could compare it to the predicted space of observables for genuine lensing
signals, and use it to reduce backgrounds – for example by cutting out regions
of parameter space that are unlikely to occur due to PBH induced AML signals.
The detailed backgrounds are hard to quantify precisely without full numerical
simulations to obtain both their rate and observable distribution. However, we
attempt to give some analytic estimates of the statistical background rate for
IDLEs. We also try to identify the astrophysical backgrounds that are likely
to be important for PBH searches. The degree to which such astrophysical
backgrounds can be further reduced by examining the distribution of event
observables is an important question that needs further investigation, but is
beyond the scope of the present work.

In order to predict the regions of PBH parameter space that are likely to be
discovered/excluded with time-series data from _Gaia_ , we need a computation
of both the signal rate (which we have done), and of the background rate
(which needs further study). We make the simplified assumption of a negligible
background rate to calculate the expected exclusion on the PBH parameter
space. While the time-series data for _Gaia_ has already been collected, it
has not yet been publicly released. We expect that our prediction of the
lensing signal rate and the signal event observable distributions will prove
useful to analyze this data, as and when it becomes available, in order to
compute the true exclusion bounds.

With our assumption of a negligible background rate, we find that _Gaia_ is
most sensitive between $0.4~{}M_{\odot}$ \- $5\times 10^{7}~{}M_{\odot}$, with
peak sensitivity to masses $\sim 10~{}M_{\odot}$, where fractions $f$ as low
as $3\times 10^{-4}$ of the DM relic density can be constrained. Our work is
the first attempt to make such a prediction for _Gaia_ by looking at
deflections of the trajectories of individual stars. Our work is enabled in
part by the existing _Gaia_ eDR3 data release [55], which already gives an
indication of the number of stars that _Gaia_ is able to track, and also lists
their measured properties that are relevant for calculating the AML rate.

Our work builds upon the work of Dominik and Sahu [35] who estimated lensing
probabilities for _Gaia_. However, in this work the authors did not attempt to
calculate the number of lensing events that _Gaia_ would see or set an
exclusion limit on the PBH parameter space. Moreover, in their work, the
authors only considered LDLE type events, with event durations longer than the
_Gaia_ mission lifetime. In our work, we show for the first time the
importance of IDLE type events, which have event durations of a few months to
a few years, a time-scale which is intermediate between _Gaia_ ’s sampling
time and its mission lifetime. As we will show, searches for IDLE type events
are important for sensitivity to relatively lower mass PBHs with masses as low
as $\sim 0.1~{}M_{\odot}$111SDLE type events have event durations shorter than
_Gaia_ ’s sampling time and will therefore show up as a fluctuation in at most
a single pass observation of a background star. However, as we shall argue in
sec. 5.1, SDLEs will be completely dominated by statistical background due to
centroiding uncertainty. This will therefore render SDLEs unusable in practice
for detecting PBHs..

We now provide some broader context for the impact of the work that we present
in this paper. For one, an exclusion of PBHs with
$\mathcal{O}(10)~{}M_{\odot}$ mass as making up less than a fraction $f\sim
10^{-3}$ of the DM would rule out the possibility that the mergers of binary
black holes being seen at LIGO/Virgo [56] are due to PBHs [9]. If _Gaia_ were
to discover even a fraction of the DM in the form of PBHs in the mass range of
$10^{-1}-10^{7}~{}M_{\odot}$, that we have claimed the survey is sensitive to,
then this would also have profound implications. If PBHs in this mass range
constitute even a small fraction of the DM density, it would rule out weakly
interacting massive particles (WIMPs) as a candidate for the rest of the dark
matter. Refs. [57, 58, 59, 60] have argued that annihilation of WIMP DM would
be greatly enhanced due to accretion around PBHs in this mass range, and this
would lead to a large indirect detection signal which is incompatible with
present limits, for a PBH density fraction $f>10^{-9}$. A technical
implication of our work is that our novel lensing probability calculation can
be easily adapted to predict the rate of lensing events due to UCMCHs/DM sub-
halos, or ordinary star-star lensing.

This paper is organized as follows. In sec. 2, we describe the properties of
the _Gaia_ satellite that are relevant for AML and we also discuss the _Gaia_
eDR3 data release from which we form a model of the stellar distribution in
the Galaxy. In sec. 3, we describe the expected signals due to PBH induced AML
events and we provide analytic expressions for the event durations and lensing
probabilities for stars in the Galaxy. We also introduce the distinction
between SDLE, IDLE, and LDLE types events in this section. In sec. 4, we
combine the lensing probability calculation with the Galactic stellar model to
numerically compute the expected total number of lensing events that _Gaia_
will see over its mission lifetime, and we also compute the expected
distribution of some simplified AML observables. Next, in sec. 5, we describe
various backgrounds to PBH searches. We also describe which backgrounds we
expect to be important, and some background reduction techniques. In sec. 6,
assuming negligible backgrounds, we then calculate an expected exclusion curve
on the PBH parameter space. In sec. 7, we discuss some key assumptions of our
rate estimation and how the robustness of our exclusion curve can be improved
by future numerical studies. Finally, we summarize and conclude in sec. 8. In
the appendix, we derive some analytic expressions for the event observables.

##  2 The _Gaia_ telescope and stellar catalog

_Gaia_ is a space-based optical-telescope designed to obtain precise positions
and velocities of stars in our Galaxy. _Gaia_ started surveying the sky in
July, 2014 and has already surpassed its expected 5 year mission lifetime. It
is expected that _Gaia_ will provide time-series data for astrometric
position, proper motion, and parallax for more than a billion stars [61]. With
such exquisite time-series data, _Gaia_ has the potential to be sensitive to
distortions in the trajectories of stars due to astrometric microlensing
events. _Gaia_ has not yet publicly released time-series data, although this
data is expected to be released in the future [55]. Thus, at the moment we can
not directly analyze _Gaia_ data for potential AML signals.

The goal of our work is two fold. First, we would like to estimate the
expected sensitivity of _Gaia_ to AML signals due to PBH induced lensing of
background stars. Second, we would like to predict the distribution of event
observables such as location on the sky, maximum astrometric deflection angle,
and event durations, for different assumptions of the PBH mass and abundance.
As and when time-series data of _Gaia_ is made available, we expect that our
work will prove useful when analyzing this data.

In this work, we utilize information from _Gaia_ in two distinct ways. First,
we make use of the known detector properties, such as how often _Gaia_ scans a
particular region of the sky and the uncertainty on the astrometric position
of a star in a single pass, to estimate the probability of _Gaia_ detecting a
lensing signature. Second, although the _Gaia_ collaboration has not released
time-series data, they have already made several public releases of high
precision, time-averaged astrometric positions, proper motion, and parallaxes
of nearly 1.5 billion stars [55]. This catalog already gives us the most
precise map of the Milky Way. We will therefore use this catalog to predict
the event rate for AML signals.

In sec. 2.1, we discuss the detector properties that will be relevant for
estimating the lensing probabilities. In sec. 2.2, we will then discuss the
_Gaia_ data release and how we build a stellar catalog from the data to
predict the expected AML event rate and event observable distributions.

###  2.1 _Gaia_ detector properties

_Gaia_ is located at the Lagrange point L2. The satellite has two telescopes
with independent fields-of-view, each of angular size $0.72^{\circ}\times
0.69^{\circ}$ [46], with an angular separation of $106.5^{\circ}$ between the
two fields-of-view. _Gaia_ spins about an axis inclined at an average angle of
45∘ with respect to the line joining the detector and the sun with an angular
speed of $60\textrm{ arcsec/sec}$. This rotation axis precesses at a rate of
$5$ revolutions per year. _Gaia_ also executes a complicated motion around the
Lagrange point L2. As the satellite spins about its rotation axis, the fields-
of-view sweep out different regions of the sky. The precession of the rotation
axis ensures that after a single rotation the satellite is scanning a slightly
different patch of the sky. Given the precession rate and the spinning rate,
it is expected that on average a given star would be seen once in each field-
of-view as _Gaia_ rotates, and would then drop out of the fields-of-view. The
observation of a star in a single telescope during a single rotation
constitutes one “pass”. The advantage of the two fields-of-view is that this
allows _Gaia_ to perform global astrometric measurements of the position of a
star relative to other background stars in another field-of-view. Thus, in a
single pass, _Gaia_ will be able to make one observation of the astrometric
position of a star which can later be assigned a specific global astrometric
co-ordinate (say, {$\alpha,\delta$} in Galactic co-ordinates).

_Gaia_ ’s motion results in a complicated scanning law [61] which determines
the time-interval between successive observations of the same star. The time-
interval between successive observations is also not constant. Over an
observational time scale of $t_{\textrm{obs}}=5$ years, _Gaia_ is expected to
observe each star approximately 70 times (35 times in each field-of-view).
Thus, at the end of the observational period _Gaia_ is expected to have non-
uniformly sampled time-series data with about 70 astrometric positions for
each star. For simplicity, we will treat the two successive observations of a
star in each field-of-view as effectively a single pass. We then take the
time-interval between these effective passes to be uniform, with a spacing
$t_{s}=5\textrm{ years}/35\approx 52.2\textrm{ days}$222In principle, the
observation of a star in the two different fields-of-view can be considered to
be two separate passes and could potentially be used to search for short
duration microlensing events with time scales of a few hours. However, it
seems unlikely given _Gaia_ ’s sampling, that such events would occur at a
significant enough rate to be detectable for realistic PBH distributions, so
we will not discuss this possibility further in this work..

_Gaia_ is equipped with three color bands, white light G-band (330-1050 nm),
Blue prism Photometer (BP-band: 330-680 nm), and Red prism Photometer (RP-
band: 640-1050 nm). It is sensitive to stars as faint as 21 G-band magnitude.
The uncertainty on the astrometric position of a star in a single pass depends
upon the apparent brightness. In order to model the uncertainty, we could use
use a fitted model for this as a function of $m_{G}$ (G-band magnitude) given
in ref. [48]. This model is based on the Monte Carlo centroiding simulations
done in ref. [62]. Rather than a direct measurement of the uncertainty on the
absolute position of a star in a single pass, the model reports different
uncertainties across (AC) and along (AL) the scanning direction of _Gaia_ , as
a star passes through the spinning fields-of-view of each telescope. These
fitted uncertainties are given by the formula,

$\displaystyle\widetilde{\sigma}_{a}=$
$\displaystyle\frac{1}{\sqrt{2}}\begin{cases}\begin{cases}0.200+0.483e^{0.690(m_{G}-12)}\textrm{
mas}&\textrm{\, for \,}m_{G}\leq 13,\\\ 1.140+0.420e^{0.771(m_{G}-12)}\textrm{
mas}&\textrm{\, for \,}m_{G}>13\end{cases}&;\textrm{\, AC}\\\
0.030+0.0111e^{0.701(m_{G}-12)}\textrm{ mas}&;\textrm{\, AL}.\end{cases}$
(2.1)

The uncertainties inside the curly brackets are for a single field-of-view.
Since, we are treating successive observations by both fields-of-view as a
single pass, we have divided these uncertainties by a factor of $\sqrt{2}$ to
characterize the global astrometric position error.

Figure 1: This figure shows the dependence of single pass uncertainty of the
angular position of a star with a given apparent G-band magnitude ($m_{G}$).
The red-dashed and blue-dashed curves show the simulation based uncertainty
($\widetilde{\sigma}_{a}$) for both the along (AL) and the across (AC)
scanning directions. When computing our results for the expected event rate
and event observable distributions, we use the empirically determined
uncertainty ($\sigma_{a}$) along the scanning direction, shown in black.

Alternatively, we can use the empirically observed uncertainty along the
scanning direction as reported in _Gaia_ DR2 [63]. We use a fit to this
empirically determined uncertainty using the fitting function of ref. [64]
given by,

$\displaystyle\sigma_{a}=$
$\displaystyle\frac{\sqrt{-1.631+680.766z+32.732z^{2}}\times
7.75+100}{\sqrt{N_{\textrm{CCD}}}}\,\,\mu\textrm{as},$ (2.2)

where,

$z=10^{(0.4(\textrm{max}(m_{G},14)-15))}.$ (2.3)

In this formula, we use $N_{\textrm{CCD}}=18$ as the number of CCDs in both
telescopes together in which each star is imaged along the scanning direction.

We have plotted the simulation based uncertainties (AC and AL) as well as the
empirically determined uncertainty (AL) as a function of $m_{G}$ in fig. 1.
From the figure we can see that all the uncertainties are roughly constant for
bright stars with $m_{G}\lesssim 13$. However, for fainter stars with
$m_{G}\gtrsim 13$ the astrometric uncertainty degrades rapidly with increasing
apparent magnitude. We also note that there is a slight difference between the
empirical AL uncertainty and the simulated AL uncertainty. In practice, the
typical astrometric uncertainty will lie between the uncertainties of the
along and across scanning directions.

We will compute the expected exclusion on the PBHs parameter space in sec. 6
and the expected event observable distribution in sec. 4, using a fiducial
choice of the empirically determined AL uncertainty ($\sigma_{a}$) as a
reference uncertainty. This model gives a roughly constant value of
$\sigma_{a}=0.076$ mas up to $m_{G}\sim 14$.

###  2.2 _Gaia_ eDR3 and stellar catalog

_Gaia_ has released an early version of its 3rd release of data (eDR3) [55]
which contains details of sources that it has observed between July 2014 and
May 2017 (34 months). This data release does not include time-series
information of different passes for individual stars. Rather the _Gaia_
collaboration fits the time-series data to “astrometric solutions” which are
parametric trajectories for each star which are assumed to only have
rectilinear motion relative to the sun.

There are three possible astrometric solutions which are used to fit the
trajectories of stars in the catalog. The type of solutions used to fit a
particular star’s trajectory depends upon the quality of data available for
that star [46]. In particular, good quality color information is important for
correcting chromatic effects which can lead to an offset in the image centroid
[65].

For those sources for which color information is of high quality and chromatic
effects can be well corrected for, _Gaia_ reports a 5-parameter (5-p)
solution. This solution is parameterized by the parallax ($\omega$), right
ascension (r.a.) ($\alpha$), declination (dec.) ($\delta$), angular velocity
along r.a. ($\mu_{\alpha*}$), and angular velocity along dec.
($\mu_{\delta}$)333$\mu_{\alpha*}$ is defined as
$\mu_{\alpha*}\equiv\frac{d\alpha}{dt}\cos{\delta}$, whereas
$\mu_{\delta}\equiv\frac{d\delta}{dt}$..

When there is lower quality color information _Gaia_ reports a 6-parameter
(6-p) solution, which includes all the parameters of the 5-p solution, plus an
additional parameter $\nu_{\textrm{eff}}$. This extra parameter is an
effective wavenumber, which allows for chromatic effects to be estimated [65].

Finally, when the data quality is insufficient to obtain a 5-p or a 6-p
solution, _Gaia_ reports a 2-parameter (2-p) solution. This solution is
parameterized only by the r.a. ($\alpha$) and dec. ($\delta$).

The data released by _Gaia_ in eDR3 only specifies the values of these fitted
parameters for each star rather than the full time-series data. The number of
sources which are fitted to a given type of solution, and their typical
astrometric position uncertainties are shown in tab. 1. Note that these
uncertainties are on the fitted $\alpha$, $\delta$, which are obtained for
multiple epochs of observation; they do not describe the uncertainty of the
astrometric position from a single pass.

Source type |  Number of
sources |  Uncertainty in position
---|---|---
Total |  1,811,709,771 |
5-parameter astrometry |  585,416,709 |  0.01-0.02 mas ; $m_{G}<$15,
0.05 mas ; $m_{G}$=17,
0.4 mas ; $m_{G}$=20,
1.0 mas ; $m_{G}$=21
6-parameter astrometry |  882,328,109 |  0.02–0.03 mas ; $m_{G}<$15,
0.08 mas ; $m_{G}$=17,
0.4 mas ; $m_{G}$=20,
1.0 mas ; $m_{G}$=21
2-parameter astrometry |  343,964,953 |  1-3 mas
Table 1: This table shows the number of sources present in the _Gaia_ eDR3
catalog which are brighter than $m_{G}=21$. It also summarizes the
distribution of the sources over 5-parameter, 6-parameter, and 2-parameter
solutions. The typical uncertainties on the fitted angular position of the
sources in the catalog is also shown. These uncertainties corresponds to the
data taken by _Gaia_ over 34 months.

According to ref. [55], time-series data will be available from the 4th data
release (DR4) onwards. In order to estimate the total number of candidate
lensing events expected to be seen by _Gaia_ over $t_{\textrm{obs}}=5$ years
of data collection, we attempt to build a simple mock catalog of the stars
that _Gaia_ will see, _and_ which will also have high quality time-series data
that can be used to identify potential lensing signals due to AML induced
deflections in the apparent trajectories of stars. To this end, we make use of
the catalog of stars already seen by _Gaia_ and reported in eDR3. While eDR3
only corresponds to 34 months of collected data, it is unlikely that there
will be a significant number of new stars discovered by _Gaia_ which will have
such a high quality of data so as to be sensitive to lensing signals. Thus we
take the eDR3 catalog to be representative of all the stars (with data quality
suitable for precision astrometric analysis) that _Gaia_ will see after 5
years of observation. We further select from among these stars to identify
those which are likely to be suitable for detecting an AML signature.

For the purposes of estimating the expected number of lensing candidate
events, we need to know the 3D position of each star ($\alpha$, $\delta$,
$D_{s}$), where $D_{s}$ is the distance to the star, and the apparent $G$-band
magnitude $m_{G}$ of the star. The position is needed to estimate the number
of lenses between us and the star and the $G$-band magnitude is needed to
estimate the astrometric uncertainty in the position of a star from a single
pass. $D_{s}$ can be inferred from the parallax and thus can only be obtained
for those stars for which 5-p or 6-p solutions are available. Thus, we prune
our selection of _Gaia_ eDR3 stars to include only those which have either 5-p
or 6-p solutions. Also, since for a large number of stars the observed
parallaxes have large uncertainties, we use distances from the GeDR3_dist
catalog [66]. This catalog converts parallax to distance after taking into
account priors from a stellar distribution model for the Galaxy444We are
trying to build a model for the stars in the Galaxy, the prior model of the
stellar distribution in ref. [67] is informed and updated by _Gaia_ eDR3 data.
These distances are not used as predicted measurements, but rather we use them
in later sections to estimate the AML probabilities and lensing rates..

Figure 2: We show the distribution of some properties of the approximately
1.47 billion sources in the _Gaia_ eDR3 catalog with 5-p or 6-p astrometric
solutions. We use this stellar distribution as a model for sources that _Gaia_
can track in order to look for signatures of astrometric microlensing due to
PBHs. We show the distribution of sources as a function of their Galactic
latitude ($\delta$) and longitude ($\alpha$), distance ($D_{s}$), and apparent
G-band magnitude $m_{G}$. The distance to each star is taken from the
GeDR3_dist catalog [66].

Once we apply the above selection criteria, the total number of stars in our
filtered catalog is $\sim 1.47$ billion. We have plotted the distribution of
these stars as a function of $\alpha$, $\delta$, $D_{s}$, and $m_{G}$ in fig.
2 (here, and in future sections we will switch conventions and use $\alpha$,
$\delta$ to represent Galactic longitude and latitude, respectively). We can
see from the figure that most stars are unsurprisingly concentrated near the
Galactic center at $\alpha=\delta=0^{\circ}$. We can also see from the the
$m_{G}$ distribution of the stars in the filtered catalog (bottom-right panel
in the figure), that the number of fainter objects increases up to
$m_{G}=20.7$ and then cuts-off sharply. Although, _Gaia_ may yet discover more
faint stars, omitting such stars from our mock catalog will not affect our
final lensing event rate prediction since such faint stars are expected to
have very large astrometric errors on their positions which will make
detecting a lensing signal difficult. This can be seen by comparing the
$m_{G}$ distribution of the catalog (bottom right panel of fig. 2) with fig.
1, which shows the astrometric positioning uncertainty as a function of
$m_{G}$.

The catalog that we are using contains all point-like sources that _Gaia_
sees. This would include all types of stars such as variable stars, white
dwarfs etc., but also luminous non-stellar objects such as brown dwarfs, hot-
luminous planets, etc. These objects can also be used as sources to detect AML
signatures. The catalog also might contain several extra-galactic objects such
as stars in the LMC and SMC, quasars, supernovae, etc. The distance to the
extra-galactic objects taken from GeDR3_dist would be incorrect because the
prior model would attempt to place these objects in the Galaxy [66]. However,
since all of the extra-galactic objects only make up a tiny fraction of the
catalog, we expect that this will not have a significant effect of our
predictions of the AML signal rates555In ref. [68], the authors attempted to
extract LMC and SMC candidate objects in _Gaia_ eDR3 and found only 11,156,431
and 1,728,303, respectively. This is less than 0.1% of the 1.8 billion objects
in the full catalog..

In our analysis, we will consider all objects in the _Gaia_ catalog to be
point sources (we will refer them as stars) within the Galactic DM halo for
purposes of estimating the lensing rate due to PBHs.

##  3 Lensing probability

In this section we will present analytic formulae for the probability of a
particular source to be lensed by a PBH DM candidate. We will first review
some basics of AML in sec. 3.1. Then in sec. 3.2, we will present a formula
for the duration of AML signals that are potentially visible to _Gaia_ and we
will classify lensing events into one of three types based on their event
durations: (i) short duration lensing events (SDLEs) – those which have
lensing event durations which are shorter than _Gaia_ ’s sampling time
$t_{s}=52.2\textrm{ days}$, (ii) intermediate duration lensing events (IDLEs)
– those which have event durations longer than _Gaia_ ’s sampling time but
shorter than the mission lifetime $t_{\textrm{obs}}=5$ years, (iii) long
duration lensing events (LDLEs) – those which have event duration longer than
$t_{\textrm{obs}}$. Previous literature [35], has primarily focused on LDLE
detection by _Gaia_. As we shall show, SDLEs and IDLEs will also lead to
distortions in the apparent trajectories of background stars which possibly
could be detectable by _Gaia_. Because of the short durations of SDLEs, they
will have a much smaller detection probability compared to IDLEs and in
addition, as we will argue in sec. 5.1, they will be completely dominated by
the statistical background and thus will be unimportant for constraining PBHs.
On the other hand, the study of IDLE events will be shown to be of utmost
importance for detection of relatively lower mass PBHs of mass
($0.1-100~{}M_{\odot}$). Our treatment of IDLEs is a novel feature of this
work. Then, in sec. 3.3, we will present analytic formulae for the lensing
probabilities in terms of a line-of-sight integral to a source of magnitude
$m_{G}$ with co-ordinates $(D_{s},\alpha,\delta)$. In the last two sub-
sections we will address some subtleties of our calculation.

###  3.1 AML basics

When a star and PBH lens are nearly aligned, gravitational lensing effects are
important. Since, both the stars and PBH lenses have proper motion in the sky,
lensing signals will be transient phenomena. Before discussing the transient
behaviour and durations of such lensing events, we will first discuss the
simple case of a static source and a static lens, relative to us, in the sky.
A more detailed version of the discussion that follows can be found in refs.
[69, 70, 71].

We take the source to be located at a distance $D_{s}$ from us and the lens is
assumed to be at a distance $D_{l}<D_{s}$, and is assumed to have mass $M$. We
will denote the angular separation between the source and lens as $\theta$.
Gravitational lensing results in magnified multiple images of a source due to
the presence of a massive lensing object in the foreground. We will focus on
the case of point like source and a point like lens. Lensing phenomena in such
a case are characterized by an Einstein ring defined as a circle centered at
the lens position on the celestial sphere with an angular size given by the
Einstein angle,

$\displaystyle\theta_{E}$
$\displaystyle=\sqrt{\frac{4GM}{c^{2}}\frac{D_{s}-D_{l}}{D_{l}D_{s}}},$
$\displaystyle\approx 2.85\text{ mas
}\sqrt{\frac{M}{1~{}M_{\odot}}\frac{1\textrm{
kpc}}{D_{s}}\frac{D_{s}-D_{l}}{D_{l}}}.$ (3.1)

Gravitational lensing results in two distinct images of the source which are
located on either side of the true source position, along the line joining the
source and the lens on the celestial sphere. The angular separation between
each of these images and the lens position is given by,

$\theta_{I}^{\pm}=\frac{\theta_{E}}{2}\left(u\pm\sqrt{u^{2}+4}\right),$ (3.2)

respectively, where $u\equiv\theta/\theta_{E}$ is the source position relative
to the lens, scaled by the Einstein angle. Both these images will have
different magnification in general, with the magnifications given by the
following expressions,

$m_{I}^{\pm}=\frac{u^{2}+2}{2u\sqrt{u^{2}+4}}\pm\frac{1}{2}.$ (3.3)

It is almost always the case that for the lens masses that we will be
considering, the two images will not be resolvable by _Gaia_. In such a case,
we would only detect a single image located at the “center-of-light”
(centroid) of the two images666Center-of-light is defined as magnification-
weighted average position of the two images..

In general this image is magnified and shifted relative to what we would have
seen if there were no lens. The relative magnification of this image, as
compared to the unlensed case, is given by,

$m=\frac{u^{2}+2}{u\sqrt{u^{2}+4}}.$ (3.4)

We denote the angular shift in the centroid away from the true source position
on the celestial sphere as $\delta\theta=\theta_{E}\delta u$, where $\delta u$
is the angular shift in units of $\theta_{E}$ and is given by [32],

$\delta u=\frac{u}{u^{2}+2}.$ (3.5)

This shift is along the line joining the lens to the true position of the
source, and it is directed away from the lens.

Figure 3:  This figure shows the AML observable, i.e. the shift in angular
position $\delta u$, and the PML observable, i.e. the change in relative
magnification $m-1$, as a function of $u$, the angular separation between the
star and the PBH in units of the Einstein angle, $\theta_{E}$. The AML
observable falls off as $\frac{1}{u}$ for large $u$, whereas the PML
observable falls of as $\frac{1}{u^{4}}$ for large $u$. Also note that unlike
the PML observable, the AML observable decreases for small separation between
the source and the lens. The AML observable peaks for an angular separation
$\delta\theta=\sqrt{2}\theta_{E}$.

Photometric microlensing attempts to detect the magnification change, whereas
astrometric microlensing attempts to detect the angular shift of the image. In
fig. 3, we show the astrometric shift, $\delta u$ and the change in
photometric magnification, $m-1$ as a function of the scaled relative angle
between the true position of the source and the lens, $u$.

Both the change in magnification and the angular shift are small for large
separation between the source and the lens – typically for a separation larger
than a few $\theta_{E}$. However, as we can see from the figure and the
formulas above, the change in magnification, $m-1$, falls off as $1/u^{4}$,
whereas the shift in angular position, $\delta u$, falls off as $1/u$. This
implies that, for large separation between the source and the lens,
astrometric microlensing can be more sensitive to the presence of a lens. In
practice, the relative sensitivity of these two microlensing techniques also
depends on the instrumental sensitivity to changes in magnification versus
sensitivity to astrometric position measurements. In the rest of this section,
we will focus only on the astrometric microlensing technique.

###  3.2 Event durations and classifications of AML signals

Now we consider the effect of relative motion between the star and the lens on
the apparent trajectory of the star. The relative motion of the star and the
lens will change their relative separation parameter $u$. This in turn will
lead to a change in the astrometric shift $\delta\theta$ of the apparent
source position over time.

For simplicity, we will consider the case where only the source is in motion
with a tangential speed $v_{s}$ relative to us, and the lens is assumed to be
static. We will also ignore the effects of parallax due to the earth’s motion
(similar to refs. [35, 52]). We will comment on how the computations of this
subsection can be generalized to scenarios where both the lens and the source
are moving in sec. 3.4. We will also discuss the effect of the second
assumption (the neglect of parallax) on our analysis in sec. 3.5.

We define the lens plane as a plane perpendicular to our line-of-sight which
contains the lens. The source’s position projected on to this plane moves with
an angular speed $\mu\approx v_{s}/D_{s}$, where $D_{s}$ is the distance to
the source. We will then define an effective speed parameter $v\equiv
v_{s}\frac{D_{l}}{D_{s}}$ such that $\mu=v/D_{l}$, where $D_{l}$ is the
distance from us to the lens. A further discussion and interpretation of the
parameter $v$ as a relative separation velocity between the source and the
lens will be discussed in sec. 3.4. The parameter $v$ will crucially determine
the duration of lensing events as well as their probabilities.

The astrometric lensing effect can be treated as the instantaneous
displacement of the apparent source position relative to the true position of
the source, as the source moves across the sky. Up to an overall rotation of
the source-lens configuration, a general trajectory of the source in the
presence of a foreground lens can then be characterized by the speed $v$ and
the impact parameter $u_{0}$. The impact parameter is defined by the angular
separation at the point of closest approach between the source and the lens,
when the source’s position is projected onto the lens plane.

Figure 4: The figure shows three rectilinear trajectories of stars labelled
$A$, $B$, $C$ (yellow horizontal lines), and the distortion in the apparent
trajectories (blue curves) due to a PBH lens in the foreground (black dot at
the center). The Einstein ring around the PBH is shown by the grey dashed
circle. The arrows from the yellow to the blue curves indicate the size and
direction of the astrometric shift of the apparent position, away from the
true position, due to the lensing effect of the PBH. The direction of the
shift is always radially away from the PBH. The magnitude of the shift is
above _Gaia_ ’s sensitivity threshold (reference arrow
$\frac{\sigma_{a}}{\theta_{E}}$ shown in the bottom right) in the green
annular region around the lens. The inner radius of this annulus is $u_{-}$,
and the outer radius is $u_{+}$. The image is color coded so that the
deviation vectors are in green if the shift is above threshold, and they are
shown in black otherwise. Trajectories $A$ and $B$ have some portion which
lies in the annular region, however, trajectory $C$ lies wholly outside the
annular region and will not give rise to a detectable AML signal at _Gaia_.
The event durations for trajectories $A$ and $B$ are given by the time that
the star spends in the annular region. We classify AML events based on their
average lensing event durations into either SDLEs, IDLEs, or LDLEs.

In fig. 4, we show in yellow, several rectilinear trajectories, with different
impact parameters for a background star. Here the lens, indicated by the black
dot, is located at the center of the figure. In this figure, we are showing
relative angular positions $u_{x},u_{y}$ on the celestial sphere, scaled to
the Einstein angle $\theta_{E}$. The Einstein ring ($u=1$) is indicated by a
grey dashed circle. At every position the star undergoes an AML effect which
leads to a shift in its apparent position relative to its true position. This
deflection is along the line joining the lens to the source, and directed away
from the lens. These deflections from the true trajectories are denoted with
arrows in the figure. The locus of all the arrow-tips, shown in blue,
indicates the apparent lensed trajectory that we would see. The effect of
lensing is thus to create a distortion of the apparent trajectory away from
the true simple rectilinear motion.

By examining the observed trajectory of a star and looking for deviations from
rectilinear motion, we can detect signatures of PBH lenses. Next we discuss
whether such deviations can be detected by _Gaia_.

_Gaia_ ’s sensitivity to AML distortion in the trajectory will depend on three
factors:

  1. 1.

The uncertainty in the position measurement of a star in a given pass.

  2. 2.

The rate at which _Gaia_ makes observations of a particular star’s trajectory
(cadence).

  3. 3.

_Gaia_ ’s total observation time $t_{\textrm{obs}}$.

As discussed in sec. 2, _Gaia_ scans a particular region of the sky after
every $t_{s}$, where $t_{s}\sim 52.2$ days is the sampling time. In a given
sampling, the uncertainty on the source position measurement is given by
$\sigma_{a}(m_{G})$ and depends on the G-band magnitude $m_{G}$ of the star,
see sec. 2.1. Thus, a minimum criteria for _Gaia_ to be sensitive to the
deviation in the star’s lensed trajectory relative to its true trajectory, is
that the astrometric shift $\delta\theta=\theta_{E}\delta u$ should exceed the
astrometric resolution $\sigma_{a}(m_{G})$. It is easy to show from eq. 3.5
that this condition can be satisfied for an angular separation between star
and lens (expressed in $\theta_{E}$ units) which lies between $u_{+}$ and
$u_{-}$ where,

$u_{\pm}=\frac{1}{2}\left(\frac{\theta_{E}}{\sigma_{a}(m_{G})}\pm\sqrt{\frac{\theta_{E}^{2}}{\sigma_{a}(m_{G})^{2}}-8}\right).$
(3.6)

In fig. 4, we also show in green, an annular region around the lens, where the
inner radius of the annulus is $u_{-}$ and the outer radius is $u_{+}$. A star
passing through this annular region will experience a deflection in its
trajectory which is larger than _Gaia_ ’s single pass astrometric resolution.

In the figure, we have shown a representative measure of the single pass
astrometric resolution $\sigma_{a}(m_{G})/\theta_{E}=0.3$, as a red arrow
annotation at the bottom right. The deflection arrows along each trajectory
are color coded such that the arrows are show in black if the deflection is
smaller than this threshold, and the arrows are shown in green if they are
larger than this threshold. For the portion of the trajectory that lies within
the annular region, the deflection arrows are green, indicating that the
deflection in the trajectory is above _Gaia_ ’s detection threshold.

In the figure, trajectory $A$ enters the green annular region and thus creates
an above threshold deflection, after which it enters the inner annular region
and the deflection falls below threshold. Then it once again enters the
annular region and the deflection is above threshold, before finally exiting
the annular region, and the deflection drops below threshold once again.
Trajectory $B$ enters the annular region and creates an above threshold
deflection, but it does not enter the inner annular region before exiting the
annulus. Finally, trajectories with a large impact parameter ($u_{0}>u_{+}$),
such as trajectory $C$, will lie wholly outside the annulus and will always
have deflections below threshold. Such trajectories will always yield
undetectable lensing signatures. For $\theta_{E}<2\sqrt{2}\sigma_{a}(m_{G})$,
the annulus shrinks to zero size, and the lensing signal would be too weak to
be detectable for any impact parameter. Thus, for
$\theta_{E}>2\sqrt{2}\sigma_{a}(m_{G})$, and trajectories with impact
parameter $u_{0}<u_{+}$, we will have AML deflections above _Gaia_ ’s
threshold sensitivity, and thus, a potentially detectable lensing signature.

For a source moving with angular speed $\mu$, we can define a lensing duration
$t_{e}$ for a given trajectory, during which the deflection of the apparent
trajectory from the true trajectory is above _Gaia_ ’s sensitivity threshold,
i.e. $\delta\theta>\sigma_{a}(m_{G})$777For photometric microlensing, we can
also define event duration for which the $m-1>\sigma_{p}(m_{G})$, where
$\sigma_{p}(m_{G})$ is the uncertainty in the apparent magnitude $m_{G}$.
Typically we expect that the duration of the photometric event would be much
smaller than the astrometric event. This is because photometric signal
($m-1\sim 1/u^{4}$) decays much faster than astrometric signal
($\delta\theta\sim 1/u$) with the angular separation $u$. However, an exact
comparison depends on the photometric and astrometric resolutions.. Thus,
$t_{e}$ is just the time that would be spent by the star in the annular
region. Note that this time $t_{e}$ need not be something that _Gaia_ actually
observes, rather it should be thought of as a parameter associated with a
source-lens configuration that will help us quantify whether the lensing
signature is detectable by _Gaia_.

We can divide lensing events into three categories based on the value of
$t_{e}$,

  * •

Short duration lensing events (SDLEs) – those with $t_{e}<t_{s}$. In this
case, these events will most likely be missed by _Gaia_ , as the above
threshold deflection is most likely to occur in between _Gaia_ ’s samplings of
the region of the sky containing the source and the lens. However, it is
possible that a single sample of the trajectory may show a large deviation due
to an SDLE which coincidentally occurs when _Gaia_ happens to be observing a
star while it is being lensed.

  * •

Intermediate duration lensing events (IDLEs) – those with
$t_{s}<t_{e}<t_{\text{obs}}$. In this case, most of the trajectory that _Gaia_
observes will not suffer significant distortion due to lensing. However, there
will be at least a few sample observations seen along the trajectory that show
a lensing deflection $\delta\theta>\sigma_{a}(m_{G})$. Such a trajectory can
be easily distinguished from rectilinear motion because of the blip/distortion
in the apparent trajectory.

  * •

Long duration lensing events (LDLEs) – those with $t_{\text{obs}}<t_{e}$. For
these trajectories, almost the entire trajectory suffers from a significant
distortion due to lensing, and in the extreme limit of large event duration,
_Gaia_ will only see a fraction of the lensed trajectory. Counter-intuitively,
this can make detection of such signals harder than the previous case since we
will not have a reference for rectilinear motion against which we can detect a
lensing signature.

In summary, detectable lensing events at _Gaia_ will require the following
conditions to be met for the source-lens configuration.

  1. 1.

First, the annular region of source positions around the lens, in which the
deflection of the star’s apparent position is above threshold, should exist.
This condition requires $\theta_{E}>2\sqrt{2}\sigma_{a}(m_{G})$.

  2. 2.

Second, a portion of the star’s trajectory must pass through this annular
region.

  3. 3.

The lensing event duration $t_{e}$, which is the time spent by the star in the
annular region will determine the type of event. For $t_{e}<t_{s}$ and
assuming that the above two conditions are satisfied, we will get an SDLE type
of event. However, the probability of detection of such an event will be
suppressed by the coincidence probability $t_{e}/t_{s}$ of sampling a star
while such a lensing event is in progress. For $t_{s}<t_{e}<t_{\textrm{obs}}$,
we will get an IDLE type of event, which we will assume will be detected by
_Gaia_ as long as the above two criteria are satisfied. For
$t_{e}>t_{\textrm{obs}}$, we will have LDLE type events if both of the above
criteria are satisfied. In addition, we will also assume for LDLE events a
threshold criteria for detection by _Gaia_ by requiring that the difference
between the deflections at the initial and final epochs of observation differ
by an angle greater than $\sigma_{a}(m_{G})$ (see ref. [35]).

Assuming that the annular region exists, and a star and lens come close enough
to each other that the star passes through this region, we would like to
estimate what the typical event durations would be. We compute the event
duration averaged over viable impact parameters which we denote as $\langle
t_{e}\rangle$. For a given impact parameter, the event duration is
$t_{e}=l(u_{0})/\mu$, where $l(u_{0})$ is the (angular) length of the portion
of the trajectory which lies within the annulus. Integrating this time over
$u_{0}$ and dividing by the allowed range of $u_{0}$ gives the average event
duration $\langle t_{e}\rangle$. The allowed range of impact parameters for
which the star will pass through the annular region around the lens is
$0<u_{0}<u_{+}$. The integration of $l(u_{0})$ over $u_{0}$ therefore gives
the area of the upper half of the annulus. Thus we have,

$\displaystyle\langle t_{e}\rangle=$
$\displaystyle\frac{\pi\theta_{E}(u_{+}^{2}-u_{-}^{2})}{2u_{+}\mu},$
$\displaystyle=$ $\displaystyle\frac{\pi
D_{l}\theta_{E}}{v}\frac{\frac{\theta_{E}}{\sigma_{a}(m_{G})}\sqrt{\frac{\theta_{E}^{2}}{\sigma_{a}(m_{G})^{2}}-8}}{\frac{\theta_{E}}{\sigma_{a}(m_{G})}+\sqrt{\frac{\theta_{E}^{2}}{\sigma_{a}(m_{G})^{2}}-8}},$
(3.7)

where the quantity under the square root is positive for detectable lensing
events. For a $10~{}M_{\odot}$ lens, with source and lens at a distance of
$\mathcal{O}(1)\text{ kpc}$ from us, and an astrometric precision of 1 mas, we
would find that the average event duration is $\sim 1-2$ years, giving us an
IDLE type event. For larger masses for the PBH, or better astrometric
precision (brighter sources), or larger distances to the source, we can get
event durations longer than the _Gaia_ mission time $t_{\text{obs}}$, i.e. we
would get LDLE type events.

###  3.3 AML event probability due to PBH DM

Given an assumption of the PBH parameters $(f,M)$, we will now derive an
expression for the probability $P_{\textrm{star}}$ for a source with apparent
magnitude $m_{G}$ and Galactic co-ordinates $(D_{s},\alpha,\delta)$ to undergo
a lensing event which is detectable by _Gaia_. This probability will be the
sum of the probabilities for a source to yield either an SDLE, IDLE, or an
LDLE type event.

We split this calculation into two parts, first we will calculate the
conditional probability of lensing of such a star $p_{\textrm{lensing}}^{c}$,
assuming that there is a lens located at a distance $D_{l}$ between us and the
source, such that the lens is localized within a solid angle $\Delta\Omega$.
Then we will weight this conditional probability by the probability that such
a lens is actually present, assuming that the lenses are in the form of PBHs
of mass $M$, which make up a fraction $f$ of the DM in the Galactic halo.

We will first compute $p_{\textrm{lensing}}^{c}$. Let us consider a patch of
sky that subtends a solid angle $\Delta\Omega$. We assume that this patch
contains a source located at a distance $D_{s}$ from us, and a lens located at
a distance $D_{l}$ from us. As the source moves across the sky over a time
$t_{\text{obs}}$, the source traverses an angular distance on the sky $\mu
t_{\text{obs}}$, where $\mu=v/D_{l}$ is the proper speed of the source. We
assume that $\Delta\Omega$ is sufficiently large such that the entire
trajectory of the star is well contained within this patch of the sky. If the
lens lies within an impact parameter $u_{+}\theta_{E}$ on either side of the
star’s trajectory, a portion of the trajectory will have an AML deflection
which is larger than _Gaia_ ’s threshold resolution $\sigma_{a}(m_{G})$.
However, the impact parameter $u_{0}$, and hence the event duration $t_{e}$,
will depend upon the angular orientation of the source’s velocity vector. We
take our formula in eq. 3.2 for $\langle t_{e}\rangle$, averaged over impact
parameters, to represent the typical event duration averaged over orientations
of the source velocity vector.

For IDLEs ($t_{s}<\langle t_{e}\rangle<t_{\textrm{obs}}$), the bulk of the
trajectory is sampled while the star is not undergoing any lensing signal and
therefore the apparent motion appears rectilinear. However, for a small
portion of the trajectory, during which it is sampled at least a few times,
the trajectory will show a clear deviation from rectilinear motion,
characteristic of an IDLE. In this case we assign
$p_{\textrm{lensing}}^{c}=\frac{\delta\omega_{\textrm{IDLE}}}{\Delta\Omega}$,
where $\delta\omega_{\textrm{IDLE}}=\mu t_{\textrm{obs}}\times
2u_{+}\theta_{E}$ is the angular area of an imaginary rectangle swept out by
the source as it moves across the sky, of width $2u_{+}\theta_{E}$. If the
lens lies within the rectangle, it would lead to an IDLE type event.

For SDLEs, $p_{\textrm{lensing}}^{c}$ will be same as that for IDLEs, except
for an additional coincidence probability suppression factor of $\langle
t_{e}\rangle/t_{s}$.

For LDLEs ($t_{\textrm{obs}}<\langle t_{e}\rangle$), although the event
duration is large, and the trajectory will be sampled frequently during a
period in which the deflection of the source’s apparent position from its true
position is above the _Gaia_ threshold, there is no reference rectilinear part
of the apparent trajectory against which we can compare this deviation. In
this case, a possible requirement for the threshold for detection could be to
demand that the difference between the deflections at the start and at the end
of the observational period is larger than $\sigma_{a}(m_{G})$. In ref. [35],
the authors made exactly this assumption and computed
$p_{\textrm{lensing}}^{c}=\frac{\delta\omega_{\textrm{LDLE}}}{\Delta\Omega}$,
where $\delta\omega_{\textrm{LDLE}}=\mu t_{\textrm{obs}}\times
2\sqrt{\frac{vt_{\text{obs}}}{D_{l}}\frac{1}{\sigma_{a}(m_{G})}}\theta_{E}$.

In summary we find,

$p_{\textrm{lensing}}^{c}=\begin{cases}\frac{2}{\Delta\Omega}\left(\frac{\theta_{E}}{\sigma_{a}(m_{G})}+\sqrt{\frac{\theta_{E}^{2}}{\sigma_{a}(m_{G})^{2}}-8}\right)\theta_{E}\mu
t_{\textrm{obs}}\times\frac{\langle t_{e}\rangle}{t_{s}}&;\,\langle
t_{e}\rangle<t_{s},\\\
\frac{2}{\Delta\Omega}\left(\frac{\theta_{E}}{\sigma_{a}(m_{G})}+\sqrt{\frac{\theta_{E}^{2}}{\sigma_{a}(m_{G})^{2}}-8}\right)\theta_{E}\mu
t_{\textrm{obs}}&;\,t_{s}<\langle t_{e}\rangle<t_{\textrm{obs}},\\\
\frac{2}{\Delta\Omega}\sqrt{\frac{t_{\text{obs}}v}{D_{l}}\frac{1}{\sigma_{a}(m_{G})}}\theta_{E}\mu
t_{\textrm{obs}}&;\,t_{\textrm{obs}}<\langle t_{e}\rangle.\end{cases}$ (3.8)

This computation of $p_{\textrm{lensing}}^{c}$ is valid only when the
astrometric deflection is above threshold. To be explicit, we should multiply
the right-hand-side of the above formula by the step function
$\Theta(\theta_{E}-2\sqrt{2}\sigma_{a}(m_{G}))$, in order to ensure this
condition. In computing this probability above, we have averaged over an
assumed uniform distribution of the star and the lens positions within the
angular window of size $\Delta\Omega$, and we have also averaged over
different orientations of the source velocity vector.

Now, for a particular star located at a given ($D_{s},\alpha,\delta$), we can
estimate the probability $P_{\textrm{star}}$ of such a star undergoing a
microlensing event caused by PBHs of mass $M$, that make up a fraction $f$ of
the Galactic DM halo. We make the assumption that the probability of lensing
of a particular star is small and the lens distribution is diffuse enough such
that the apparent trajectory of the star is only affected by a single lens at
a time. We call this the unilensing assumption.

$P_{\textrm{star}}$ can then be computed by first multiplying the conditional
probability $p_{\textrm{lensing}}^{c}$, weighted by the Poissonian probability
of having a single lens in the angular window $\Delta\Omega$ located at a
distance between $D_{l}$ and $D_{l}+\Delta D_{l}$, which lies between us and
the source. The Poissonian probability can just be taken to be the average
number of lenses in the aforementioned region. We then sum this over all
distances $D_{l}$ between us and the source.

Thus we obtain,

$\displaystyle P_{\textrm{star}}=$
$\displaystyle\int_{0}^{D_{s}}D_{l}^{2}dD_{l}\Delta\Omega\frac{f}{M}\rho_{\textrm{DM}}(D_{l},\alpha,\delta)p_{\textrm{lensing}}^{c},$
(3.9)

where we assume that the DM density is given by a standard spherically-
symmetric NFW profile [72] about the Galactic center,

$\rho_{\textrm{DM}}=\frac{\rho_{0}}{\frac{r}{r_{s}}\left(1+\frac{r}{r_{s}}\right)^{2}},$
(3.10)

where $\rho_{0}=1.06\times 10^{7}~{}M_{\odot}/\textrm{kpc}^{3}$ and
$r_{s}=12.5\textrm{ kpc}$ is the DM scale radius. The distance $r$ of a lens
from the center of the Galaxy can be written in terms of of the distance of
the lens from the earth’s position $D_{l}$, as
$r=\sqrt{D_{l}^{2}+r_{e}^{2}-2D_{l}r_{e}\cos{\delta}\cos{\alpha}}$. Here,
$r_{e}=8.5\textrm{ kpc}$ is the distance between the earth and the Galactic
center, and $\alpha,\delta$ give the position of the star in Galactic co-
ordinates. Note that the $\Delta\Omega$ factor cancels out with the factor
$1/\Delta\Omega$ in $p_{\textrm{lensing}}^{c}$. Thus, we are left with a
single line-of-sight integral to the star in order to compute
$P_{\textrm{star}}$.

For a given hypothesis of the PBH mass $M$ and density fraction $f$, the
probability $P_{\textrm{star}}$ is a function of (i) the location
$(D_{s},\alpha,\delta)$, (ii) the separation velocity $v$, and (iii) the
apparent magnitude $m_{G}$ (through the astrometric resolution
$\sigma_{a}(m_{G})$), of the star. The line-of-sight integral can be
numerically evaluated for a given star in the _Gaia_ catalog, under a specific
hypothesis for the PBH parameters. Care must be taken to evaluate $\langle
t_{e}\rangle$ to check which case of lensing applies, and care must also be
taken to check the value of the step function argument, to ensure that the
lensing signal is above threshold. Both these conditions depend on the value
of $D_{l}$ and this complicates the numerical integration of the expression in
eq. 3.9.

To minimize the computational time, instead of evaluating $P_{\textrm{star}}$
individually for each star in the _Gaia_ catalog, we tabulated its values.
First, we fixed the values of PBH parameters $(f,M)$. We assumed a fixed
relative separation velocity $v=200\textrm{ km/sec}$888See sec. 3.4 for a
discussion on the implications of this assumption.. Then, we numerically
evaluated the expression in eq. 3.9 on a grid of $(D_{s},\alpha,\delta,m_{G})$
values. We show the range of values of these parameters that we have scanned
over in tab. 2.

We then repeated our tabulation of $P_{\textrm{star}}$ for different
assumptions of the PBH parameters. The dependence on $M$ is complicated,
because $\theta_{E}$ depends on $M$, and $P_{\textrm{star}}$ depends on
$\theta_{E}$ in a non-trivial way. However, the dependence of the probability
on $f$ can be found by just a trivial overall rescaling.

Parameters |  min value |  max value |  grid size
---|---|---|---
$\alpha$ |  -180 deg |  180 deg |  50
$\delta$ |  -90 deg |  90 deg |  100
$D_{s}$ |  0.5 kpc |  20 kpc |  100
$m_{G}$ |  10 |  21.6 |  50
Table 2: Rather than evaluate $P_{\textrm{star}}$ individually for each star
in the _Gaia_ catalog, we tabulate $P_{\textrm{star}}$ on a grid of
($D_{s},~{}\alpha,~{}\delta,~{}m_{G}$) values. This table show the minimum and
maximum values of each parameter, and the grid size gives us the number of
samples that we take for each parameter, i.e. we are evaluating
$P_{\textrm{star}}$ on a $50\times 100\times 100\times 50$ grid. In order to
evaluate $P_{\textrm{star}}$ for a given star in the _Gaia_ catalog, we simply
read off the value from the nearest point on our grid. Figure 5:  The
probability $P_{\textrm{star}}$ for a given star located at
$(D_{s},\alpha,\delta)$ and with apparent magnitude $m_{G}$ to undergo an
astrometric microlensing event which will be detectable by _Gaia_ over an
observational time $t_{\textrm{obs}}=5~{}\textrm{years}$. This figure shows
the probability for specific PBH parameters $(f,M)$ = $(1,14~{}M_{\odot})$.
The plot depicts $P_{\textrm{star}}$ as a function of any two parameters among
$(D_{s},\alpha,\delta,m_{G})$, while averaging uniformly over the other two
parameters over their full range (see tab. 2). Also shown is the dependence of
$P_{\textrm{star}}$ on any one parameter, while averaging over the other three
parameters. An intuitive explanation for the dependence of this function on
each of the parameters is given in the text. In sec. 6, we will weight the
stars in the _Gaia_ catalog by this probability distribution to obtain the
total number of lensing events that _Gaia_ is expected to detect.

We show the results of our numerical evaluation of
$P_{\textrm{star}}(D_{s},\alpha,\delta,m_{G})$ for $f=1$ and
$M=14~{}M_{\odot}$ in fig. 5999Our plot actually shows $P_{\textrm{star}}$
only taking into account IDLE and LDLE type events. SDLE type events have a
suppressed probability distribution and will in any case have a small
contribution to $P_{\textrm{star}}$ for this PBH mass.. To visualize this
probability we have plotted its distribution as a function of two parameters
at a time, while uniformly averaging over the other parameters. We have also
shown the 1D distributions of this probability as function of each of the
individual parameters.

From the figure, we can see several interesting features. First, the
probability of lensing of a star is largest in the direction of the Galactic
center $\alpha=0^{\circ}$, $\delta=0^{\circ}$. This is expected because of the
large DM density, and hence large number of potential lenses in that
direction. Second, we see that $P_{\textrm{star}}$ increases as a function of
$D_{s}$ before eventually saturating beyond the NFW scale radius. This
saturation is expected, as there is no further significant gain in the number
of lenses along any line-of-sight at large Galactic radii. Finally, we see
that $P_{\textrm{star}}$ is almost constant as a function of $m_{G}$ and then
decreases sharply for $m_{G}\gtrsim 15$. This behaviour can be attributed to
_Gaia_ ’s astrometric position measurement uncertainty which becomes large for
$m_{G}>14$, see eq. 2.2.

We also note here that it is possible to define
$P_{\textrm{star}}^{\textrm{SDLE}}$, $P_{\textrm{star}}^{\textrm{IDLE}}$, and
$P_{\textrm{star}}^{\textrm{LDLE}}$ as the probabilities for a given star to
undergo either an SDLE, IDLE, or LDLE, respectively. These can be computed by
using eq. 3.9, with a selection of the appropriate case for
$p_{\textrm{lensing}}^{c}$ in eq. 3.8. $P_{\textrm{star}}$ is then just the
sum of these individual probabilities. If we vary the mass $M$ of the PBHs,
this changes the typical event duration. As a result, in addition to the
overall probability $P_{\textrm{star}}$ changing as a function of $M$, the
relative probabilities of these three event types will also change. We will
utilize these probabilities for specific event types in sec. 4.2, when we try
to compute the expected distribution of event observables for different types
of lensing events.

###  3.4 Modeling of the signal in the case where both the source and the lens
are moving

Our model of the AML signals in this section was based upon the assumption of
a moving source and a static lens. Given that the typical random DM velocity
in the halo is of a similar order as that of the typical stellar velocities,
and given that the source lies in the background and that the PBH DM lies in
the foreground, we would expect, on average, larger proper motion for the lens
than for the source, which is counter to our modeling assumption in this
section. One may therefore wonder how considering a motion of the lens will
alter our calculations of the event duration and the lensing probability.

The displacement of the source image relative to the true position of the
source depends only upon the relative angular separation of the source and the
lens. This is an essential input to calculating the image position, which is
valid both in the case where only the source is moving, as well as in the more
general case where both the source and the lens are moving.

Let us suppose that the source is moving with a velocity $\vec{v}_{s}$ and the
lens is moving with a velocity $\vec{v}_{l}$, relative to us. We will assume
for simplicity that these velocities are perpendicular to our line-of-sight.
Any velocity component parallel to our line-of-sight will not play any role in
our lensing calculations. The relative angular separation of the source and
the lens changes at a rate defined by the relative proper motion between the
two. The relative proper motion is given by,

$\displaystyle\vec{\mu}=\frac{\vec{v}_{s}}{D_{s}}-\frac{\vec{v}_{l}}{D_{l}}.$
(3.11)

The relative proper motion projected on to the lens plane thus yields an
effective separation velocity parameter defined as,

$\vec{v}_{\textrm{sep}}\equiv
D_{l}\times\vec{\mu}=\vec{v}_{s}\frac{D_{l}}{D_{s}}-\vec{v}_{l}.$ (3.12)

Our calculation of the event duration and the lensing probability can be
thought of as being computed in the frame of a hypothetical observer centered
at the earth (or really at the sun, since we are ignoring parallax). However,
if we think of this observer as rotating such that the lens maintains a fixed
position in their field-of-view, then such an observer would see the kind of
signal that we have assumed previously in this section, corresponding to a
static lens with $v_{l}=0$. Although this rotation can never be realized by a
true observer since we do not know the position of the PBH lens a priori, the
event duration101010Recall that the event duration is defined as the period of
time for which the source and the lens separation is such that it gives rise
to a displacement in the source position which is above _Gaia_ sensitivity
threshold. will be the same for the fictitious rotating observer and a true
non-rotating observer, provided that we replace the parameter $v$ by
$v_{\textrm{sep}}\equiv|\vec{v}_{\textrm{sep}}|$. Besides the event duration,
the lensing probability calculations also remain unchanged provided we make
this replacement of $v$ by $v_{\textrm{sep}}$.

Thus, the speed $v=200$ km/s should be thought of as representing
$v_{\textrm{sep}}$. Realistically, this replacement can not be fully correct
because $v_{\textrm{sep}}$ depends upon the ratio $D_{l}/D_{s}$ which varies
for different source-lens system. However, in lieu of a full modeling of the
source and lens velocities, our modeling should be thought of as using a
“typical value” $v_{\textrm{sep}}=200$ km/s.

While the replacement of $v$ by $v_{\textrm{sep}}$ allows us to compute the
event duration and the probability of lensing in the more general case of a
moving lens, the key difference will be the actual signals seen by an observer
who is not co-rotating with the lens as compared to the signals predicted in
our modeling (see fig. 4). We have shown for comparision three different cases
of source and lens velocities relative to us and the effect on the actual
signal that would be seen by _Gaia_.

(a)

(b)

(c)

Figure 6: This figure shows the realistic AML signals that would be seen for
different cases of source and lens relative motion. In (a) the lens is static
and the source is moving, in (b) both the source and the lens are moving, with
the lens shown as the black solid line running from the bottom to the top of
the figure, and in (c) the source is static and the lens is moving, with the
lens trajectory indicated by the black horizontal line. In all three cases,
the yellow line/yellow dot indicates the true source position, and the blue
curve indicates the apparent position of the image due to the lensing effect.
The black and green arrows indicate the displacement of the source image from
its true position, with green arrows indicating displacements above threshold,
and black arrows displacements below threshold. We can see from cases (b) and
(c) in the figure that the actual signals seen by _Gaia_ can differ from our
default modeling of moving source and static lens (case a).

In fig. 6(a), we show the case of a static lens (denoted by a black dot) with
a moving source (whose true position is shown in yellow). The image of the
source is shown in blue, with vectors indicating the displacement between the
true positions and the corresponding image positions. As before we have shown
the arrows in green when the astrometric shift is above _Gaia_ sensitivity
threshold and in black when the astrometric shift is below threshold. This
figure is identical to fig. 4.

In fig. 6(b), we show exactly the same quantities (true positions of the
source in yellow, apparent positions of the image in blue, and displacement
vectors in black and green), however in this case the lens has a rectilinear
motion of its own and its trajectory is shown by the black line which extends
from the bottom to the top of the figure.

In fig. 6(c), we show the case of a static source with a moving lens (whose
trajectory is depicted as the horizontal black line). In this case, the true
source position is static but the image traces a closed loop as the lens
passes in the vicinity of the source [32].

For all three cases, we have shown three different source trajectories, each
of which corresponds to a different choice of impact parameter. Depending upon
the event duration, for each case, a part of, or the entirety of, the
trajectory may be seen by _Gaia_. As we have argued above, each of these cases
will give rise to the same event duration if the parameter $v_{\textrm{sep}}$
is the same. Moreover, the probability of detecting a lensing signal in each
of these cases is also the same given a fixed parameter $v_{\textrm{sep}}$.
However, the actual signal that _Gaia_ will see is clearly very different in
each of these three cases. Thus, while the choice of our modeling of selecting
a static lens and a moving source does not affect our prediction of signal
event rate at _Gaia_ , it does not adequately describe the types of signals
that will be seen by the telescope. The reason why this issue is important is
because in order to predict the background rate and how well the background
can be separated from the signal, we need to know what the actual signals look
like in the data.

Given the fact that the source lies behind the lens, one might typically
expect that the lens proper motion dominates over source proper motion in
contributing to the relative proper motion. If this were true then we would
expect realistic signals that look closer to those of fig. 6(c) rather than
those of fig. 6(a). However, we caution that a full numerical study which
includes a probabilisitic distribution of source and lens velocities is
necessary to determine what the typical values of $v_{\textrm{sep}}$ are and
what the relative importance of source proper motion versus lens proper motion
are in the expected signal sample that _Gaia_ will be able to detect. Only
then one can make definitive claims about which of the cases in fig. 6 is more
representative of the typical signals that _Gaia_ will be able to see.

In sec. 5, where we discuss various sources of background, we will make the
assumption that the lensing signals are of the type in fig. 6(a). Since our
analysis of the background rate will be highly qualitative, we do not expect
this assumption to have a significant effect on our estimate of this rate,
however a more quantitative study of the background rate would actually have
to take in to account the relative fraction of the different types of lensing
signals. This study would perhaps best be performed numerically.

###  3.5 Including the effect of parallax

The analysis of this section has one major omission, which is the neglect of
parallax. When looking for a lensing signal, we would have to look for
deviations of the source motion from not just rectilinear motion but from
rectilinear plus parallactic motion. The parallax of the source itself is an
unknown parameter that needs to be fitted for. One would have to define a
parameter (such as the best fit $\chi^{2}$) which captures the difference
between the observed trajectory and an expected trajectory with rectilinear
plus parallactic motion only. A large value of this parameter would indicate a
significant deviation, which could perhaps be attributed to a lensing signal.

The types of lensing signals that would be seen by _Gaia_ would also be
different from the expectations of fig. 6 because of the omission of parallax.
Since the lens is closer to us than the source, we would expect the lens to
have a larger parallax than the source. Lens parallax would significantly
complicate the types of deviations seen in fig. 6.

A numerical study which includes the effect of parallax is needed to see the
effect on the event duration and the probability of lensing. Moreover, once we
understand the full trajectories including parallax, both with and without
lensing, we can then try to see which backgrounds can mimic lensing signals,
and to see if there are new discriminatory variables that can separate genuine
lensing signals from these backgrounds. We leave such a study to future work.

##  4 Lensing signal rate due to PBHs

Now we are ready to numerically compute the expected total number of lensing
($N_{l}$) events that _Gaia_ will see over a time period $t_{\textrm{obs}}$,
for a given hypothesis of the PBH parameters $(f,M)$. We have seen in sec. 3
that there are three kinds of AML signals that can be detected by _Gaia_.
However, the SDLE event rate is typically highly suppressed by the coincidence
probability for most PBH masses, and more importantly, as we will show in sec.
5.1 SDLEs will be completely dominated by the background due to centroiding
uncertainty. Thus, in this section we will focus only on the case of IDLEs and
LDLEs. The numerical results for the computation of $N_{l}$ will be shown in
sec. 4.1. Later, in sec. 4.2, we will also introduce some simplified event
observables for IDLE and LDLE type events, and compute the expected
distribution of these observables.

###  4.1 Computing the number of lensing events that _Gaia_ will see

Figure 7: Differential distribution of the expected number of lensing events
($N_{l}$) for PBH parameter $(f,M)=(1,~{}14~{}M_{\odot})$. The total number of
detectable lensing events is $\sim 3725$, with 55% of these being LDLE type,
and the rest IDLE type (we have once again neglected SDLE type events which
will have a highly suppressed rate due to the coincidence probability).
$N_{l}$ is obtained by weighting the distribution of stars in the _Gaia_
catalog by the probability $P_{\textrm{star}}$ (eq. 3.9) for a star to undergo
a detectable lensing event. $N_{l}$ depends on the location of stellar targets
for AML searches ($\alpha,\delta,D_{s}$) and the apparent magnitude $m_{G}$ of
these sources. The figure shows 2D as well as 1D distributions of $N_{l}$ as a
function of these parameters. From the figure we can see that the maximum
number of lensing events will be obtained for stars in the direction of the
Galactic center ($\alpha=\delta=0^{\circ}$). The most interesting part of this
figure is the 1D dependence of $N_{l}$ on the apparent brightness $m_{G}$ of
target stars (bottom right panel). This dependence arises due to an increase
in the number of stars at low apparent brightness (high $m_{G}$) weighted by a
higher probability of a detectable lensing signals from stars with higher
apparent brightness (low $m_{G}$).

We start by fixing a hypothesis for the PBH parameters $(f,M)$. In order to
calculate the expected total number of lensing events that _Gaia_ will see
over $t_{\textrm{obs}}=5$ years, we need to compute the average number of
lensing events that we expect to see for each _individual_ star in the _Gaia_
catalog, and then add up all these values.

For each star in our reduced _Gaia_ catalog, described in sec. 2.2, we have
$D_{s}$, $\alpha$, $\delta$, and $m_{G}$ values. We can then numerically
calculate the probability $P_{\textrm{star}}$ (eq. 3.9) that a lensing event
will occur and will be detected by _Gaia_ in $t_{\textrm{obs}}=5$ years. This
probability can also be interpreted as the average number of lensing events
associated with a given star. Since, the computation for $P_{\textrm{star}}$
was numerically challenging, we take the nearest grid point in $D_{s}$,
$\alpha$, $\delta$, and $m_{G}$ that we have tabulated $P_{\textrm{star}}$
values for, to determine its value. We then sum over all the stars in the
_Gaia_ catalog to determine the average expected number of detectable lensing
events. This is equivalent to a weighting of $P_{\textrm{star}}$ by the
distribution of stellar locations and apparent magnitudes in the Galaxy, or
equivalently, for the choice of $f=1$ and $M=14~{}M_{\odot}$, it is a
weighting of fig. 5 by fig. 2. The result of this procedure is shown in fig.
7, in which we plot the total number of detectable lensing events as a
function of $D_{s}$, $\alpha$, $\delta$, and $m_{G}$.

From fig. 7, we can see that the location on the sky where the maximal number
of lensing events will occur is in the direction towards the Galactic center.
This is because of a combination of a large number of stars along with a large
DM density integrated along the line-of-sight in this direction. A more
interesting trend is the dependence of the number of lensing events on
$m_{G}$. As $m_{G}$ increases, _Gaia_ ’s astrometric positioning errors also
increase (see eq. 2.2), and hence the lensing probability $P_{\textrm{star}}$
decreases. However, there are a larger number of stars with higher values of
$m_{G}$ in the _Gaia_ catalog (see fig. 2). The competition between these two
opposing effects is what leads to the non-trivial dependence on $m_{G}$,
leading to a peak in the distribution of number of lensing events near
$m_{G}\simeq 19$ mag.

We can also try to understand how the distribution of event numbers will
change as a function of the PBH mass $M$. Since the number of events is still
expected to peak in the direction of the Galactic center, we will focus our
attention on the $m_{G}$ dependence of the expected number of events. In fig.
8, we show for different PBH masses $M$, the 1D dependence of the expected
number of events on $m_{G}$, after marginalizing over all $D_{s}$, $\alpha$,
and $\delta$ values. At low values of $M\lesssim~{}10^{-2}M_{\odot}$, the
expected number of events drops rapidly to zero. This is because of a
combination of several factors: (i) the typical AML event duration drops below
_Gaia_ ’s sampling time $t_{s}$ and (ii) the typical AML deflection drops
below _Gaia_ ’s sensitivity threshold $\sigma_{a}(m_{G})$. The combination of
these two factors makes the AML signals unobservable at _Gaia_ for low PBH
masses. For large masses the expected number of events drops more smoothly, as
can be seen in the figure. This drop in the number of lensing events at large
masses is because there are fewer lenses for a fixed PBH mass density, and
hence the probability of AML lensing decreases.

Figure 8:  1D distribution of the expected number of lensing events for stars
in the _Gaia_ catalog as a function of the target star’s apparent brightness
($m_{G}$). We have shown this distribution for several values of the assumed
PBH mass $M$, while assuming that they make up a fraction $f=1$ of the
Galactic DM. For PBH masses lower than $10^{-2}~{}M_{\odot}$, the number of
expected events drops rapidly to zero as both the deflection and event
duration become small. For larger PBH masses the number of potential lenses
decreases, which leads to fewer lensing events with increasing PBH mass.

###  4.2 Expected distribution of observables for AML events

In the case of IDLEs one would observe a rectilinear trajectory with a
distortion or deviation from apparent rectilinear motion for a short time,
i.e. for the duration of the event $t_{e}$. Two simplified observables can
characterize such trajectories: the event duration $t_{e}$ during which the
deflection from rectilinear motion is above _Gaia_ ’s threshold, and the
maximum deflection $(\delta\theta)_{\textrm{max}}$ of the trajectory away from
rectilinear motion. For LDLE type events it is harder to find a simplified
observable that would truly characterize such events. However, we can try to
examine the relative deflection difference between the start point and the end
point of the trajectory away from the true rectilinear trajectory. Note that
for LDLE type events, the event duration $t_{e}$ will not be an easily
reconstructable observable. Given our computation of $P_{\textrm{star}}$ in
the previous section, and more specifically, the computation of
$P_{\textrm{star}}^{\textrm{IDLE}}$ and $P_{\textrm{star}}^{\textrm{LDLE}}$,
we can plot the expected distribution of these simplified observables, for a
given hypothesis of PBH parameters.

####  4.2.1 IDLE observables

First, we need to compute, for a star located at $D_{s},\alpha,\delta$, and
with apparent brightness $m_{G}$, the average expected values of the
observables $t_{e}$ and $(\delta\theta)_{\textrm{max}}$.

We had earlier evaluated the average event duration $\langle t_{e}\rangle$
over impact parameters in eq. 3.2. We further average this over all candidate
IDLE lensing events that such a star will undergo, due to lenses at various
distances $D_{l}$ between us and the star along the line-of-sight. This gives
us,

$\displaystyle\langle\langle t_{e}\rangle\rangle$
$\displaystyle=\frac{\int_{0}^{D_{s}}\langle
t_{e}\rangle\frac{dP_{\textrm{star}}^{\textrm{IDLE}}}{dD_{l}}dD_{l}}{\int_{0}^{D_{s}}\frac{dP_{\textrm{star}}^{\textrm{IDLE}}}{dD_{l}}dD_{l}},$
(4.1)

where the double angular brackets around $t_{e}$ denote averaging with respect
to both – impact parameters as well as candidate lens distances.

We then compute a similar double average for $(\delta\theta)_{\textrm{max}}$.
The averaging over impact parameters can be performed analytically, and is
given by,

$\displaystyle\langle(\delta\theta)_{\textrm{max}}\rangle$
$\displaystyle=\theta_{E}\frac{1+\log\left[\frac{1}{8}K(K+\sqrt{K^{2}-8})\right]}{K+\sqrt{K^{2}-8}},$
(4.2)

where $K=\frac{\theta_{E}}{\sigma_{a}(m_{G})}$. The derivation of this
equation is given in appendix A. We can then perform a similar further
averaging as we did for $t_{e}$, for this observable, over lens distances
$D_{l}$.

We now compute these double average observables for each star in the _Gaia_
catalog and construct a 2-D histogram of $\langle\langle t_{e}\rangle\rangle$
and $\langle\langle(\delta\theta)_{\textrm{max}}\rangle\rangle$. While
populating the histogram, we weight each star’s contribution by
$P_{\textrm{star}}^{\textrm{IDLE}}$, i.e. the probability that such a star
will undergo an IDLE type event. The resulting histogram shows the expected
distribution of the observables for IDLE type events. This histogram is shown
in fig. 9(a) for $M=14~{}M_{\odot}$. We have also shown with vertical dashed
lines the sampling time $t_{s}=52.2$ days and the observational time
$t_{\textrm{obs}}=5$ years.

From the figure we can see that the distribution of typical IDLE type event
durations are between 1.5 - 4 years at this mass. These event durations lie
between $t_{s}$ and $t_{\textrm{obs}}$. However, for lower choices of mass for
the PBH, the distribution of event durations will drift to lower values, and
eventually for sufficiently small masses, $M\lesssim 10^{-2}~{}M_{\odot}$,
would fall below the sampling time for _Gaia_. Thus, such low mass PBHs would
typically not induce an IDLE lensing signal (although they might induce SDLE
type events). As we shall see in the next section, the statistical backgrounds
are significant for event durations $t_{e}\lesssim 2$ years. We will demand
that the IDLE signals we are looking for have event durations greater than 2
years so that this background is negligible, and this will result in _Gaia_
only being sensitive to PBHs with masses $M\gtrsim 0.1~{}M_{\odot}$.

(a) IDLE observables

(b) LDLE observables

Figure 9: The two histograms above show the simplified distribution of IDLE
and LDLE observables. For IDLEs, we have plotted the distribution for a PBH
mass of $14~{}M_{\odot}$, whereas we have shown the LDLE observable
distribution for several different PBH masses. IDLE event durations lie
between _Gaia_ ’s sampling time $t_{s}$ and observational time
$t_{\textrm{obs}}$. LDLE event durations are larger than $t_{\textrm{obs}}$
and are thus not easy to extract from an observed trajectory.

####  4.2.2 LDLE observables

In the case of LDLE type events, the relative deflection difference
observable, averaged over impact parameters, is computed in appendix A, and is
given by,

$\langle\Delta_{\textrm{LDLE}}\rangle=\frac{vt_{\textrm{obs}}}{D_{l}}\frac{2\textrm{log}(u_{T}/u_{\textrm{min}})}{u^{2}_{T}-u^{2}_{\textrm{min}}},$
(4.3)

where $u_{T}=\sqrt{\frac{vt_{\text{obs}}}{D_{l}\sigma_{a}(m_{G})}}$ and
$u_{\textrm{min}}=\sqrt{2}$. We can then further average this over $D_{l}$
values, using a formula similar to 4.1, however this time with
$P_{\textrm{star}}^{\textrm{IDLE}}$ replaced by
$P_{\textrm{star}}^{\textrm{LDLE}}$.

We could now build a 1D histogram for this observable by populating it with
all the stars in the _Gaia_ catalog, appropriately weighting each star’s
contribution by $P_{\textrm{star}}^{\textrm{LDLE}}$, i.e. the probability for
that star to undergo an LDLE. However, instead of simply computing the 1-D
distribution of $\Delta_{\textrm{LDLE}}$, we compute the 2-D distribution of
$\Delta_{\textrm{LDLE}}$ and $t_{e}$. Although $t_{e}$ will not be a genuine
direct observable for LDLEs, it can still be thought of as a parameter
characterizing the trajectory, albeit one that is much harder to extract.
Furthermore, it has a straightforward comparison to $t_{e}$ for IDLE type
events. The 2-D histogram for these observables for LDLE type events is shown
in fig. 9(b) for several values of the PBH mass $M$ ranging from
$14~{}M_{\odot}$ to $5.9\times 10^{5}~{}M_{\odot}$.

Higher values of the PBH mass lead to larger event durations, but they give a
similar distribution of the deflection observable. However, the total number
of events decreases at large PBH masses because of the decrease in number of
PBHs in the Galaxy for a fixed DM density. This decrease will limit the
sensitivity of _Gaia_ to very high mass PBHs.

Note that for a given PBH mass, such as for $14~{}M_{\odot}$ as shown in the
figure, it is possible to have both IDLE and LDLE type events. Ideally the
distribution of $t_{e}$ should look continuous as we transition from IDLE to
LDLE type events in figs. 9(a) and 9(b), but the lack of smooth continuity is
because of our assumptions of very large event durations $t_{e}\gg
t_{\textrm{obs}}$, when deriving the probabilities and event rates for LDLE
type events.

##  5 Background rate

In this section we will discuss various sources of background that can mimic
the AML signal caused due to PBHs. These backgrounds can lower the statistical
significance of a potential PBH signal, and therefore careful modelling is
required to estimate the background rate. We can categorize the backgrounds
into three types: statistical, systematic, and astrophysical. The statistical
background is due to uncertainty in the centroiding of the image of a given
star. The systematic backgrounds are due to instrumental systematics. The
astrophysical backgrounds are due to i) astrophysical lenses which can mimic
the effects of PBH lensing, and ii) deviation from our assumption of
rectilinear motion, for example due to a binary companion. A detailed study of
all the backgrounds is beyond the scope of this work. We will however discuss
the statistical and astrophysical backgrounds. For the statistical backgrounds
we will estimate their rate for the case of IDLEs and compare it to our
predicted event rates from PBHs. We will use this estimate to argue that the
background rate for SDLEs is so large that it makes SDLE event detection at
_Gaia_ nearly impossible. We will not estimate the astrophysical background
rates, but we will discuss which sources of this background we expect to be
relevant, and some possible ways to reduce these.

In the previous section we showed that the lensing signals can be
characterized by a few observables. For example, for IDLEs, we would have the
event duration $t_{\textrm{e}}$ during which the deflection is above _Gaia_ ’s
threshold sensitivity, and the maximum astrometric shift
$(\delta\theta)_{\textrm{max}}$, whereas for LDLEs, we would have
$\Delta_{\textrm{LDLE}}$. In addition, for both these types of events, we
would also have the location in the sky $(D_{s},\alpha,\delta)$ of the
background star, where $D_{s}$ could be measured through parallax. Under a
specific hypothesis of the PBH parameters $(f,M)$, we have computed the
expected distribution of a few of these simplified observables for genuine AML
signals from PBHs. In principle, one could also augment these astrometric
observables with _Gaia_ ’s relatively crude photometric measurements to also
characterize the PML magnification signal and predict the expected
distribution. Each candidate lensing event can thus be thought of as a single
point in the multi-dimensional space of these observables.

In general we expect that the backgrounds that could mimic an AML signal will
have a different distribution in this space as compared to that of genuine AML
signals. Thus, by applying suitable cuts on an observed distribution in this
space, we should be able to enhance the signal-to-background ratio.

###  5.1 Statistical Backgrounds

Let us first discuss the case of statistical background for IDLEs. For an
unlensed trajectory, since the centroiding of a star in _Gaia_ ’s detectors
has an intrinsic uncertainty of $\sigma_{a}(m_{G})$, this would lead to
statistical fluctuations in the observed position of a star. If these
statistical fluctuations mimic the lensing signal due to PBHs, then they would
form a statistical background.

While estimating the lensing probability for a given star $P_{\textrm{star}}$
due to PBHs, we assumed that the instantaneous astrometric deflection of a
star’s trajectory $\delta\theta$ is greater than _Gaia_ ’s threshold
sensitivity $\sigma_{a}(m_{G})$. If the event duration is $t_{e}$, then there
would be $N_{s}\simeq t_{e}/t_{s}$ observations of the star’s position with an
above threshold lensing signal, where $t_{s}$ is _Gaia_ ’s sampling time.

We can thus estimate the rate at which the statistical fluctuations of an
unlensed trajectory generate a fake IDLE type lensing signature using the
critera above. For the statistical fluctuations to mimic a lensing signal with
event duration $t_{e}$ (where $t_{s}<t_{e}<t_{\textrm{obs}}$), we need $N_{s}$
consecutive fluctuations in the centroiding, each of which are greater than
$1$-$\sigma$ away from the true rectilinear trajectory. Here
$N_{s}=t_{e}/t_{s}$ is the number of passes that _Gaia_ would make of the star
during a time $t_{e}$. Moreover, these fluctuations all need to be in the same
direction to mimic a lensing signal. If we assume that, for a given star in
the _Gaia_ catalog, the centroiding fluctuations in a single pass are governed
by a gaussian distribution with standard deviation $\sigma_{a}(m_{G})$, then
the probability of greater than $1$-$\sigma$ single-sided deviation is
$1-\textrm{erf}(1)\simeq 0.16$, where erf is the error-function. Thus, for a
typical trajectory with $35$ samples of the star’s position as measured by
_Gaia_ over $t_{\textrm{obs}}=5$ years, we would find that the probability for
a star to undergo such a fake event due to statistical fluctuations is,

$P_{\textrm{star}}^{\textrm{stat}}=2\times(0.16)^{N_{s}}(35-N_{s}),$ (5.1)

where $0.16^{N_{s}}$ is the probability for $N_{s}$ $1$-$\sigma$ single-sided
deviations in the star’s apparent trajectory, the factor of $2$ takes into
account that the deviations can be on either side of the trajectory, and the
factor of $35-N_{s}$ gives the number of ways of choosing the starting pass
for which the fake lensing event begins. $P_{\textrm{star}}^{\textrm{stat}}$
does not depend on the specific properties of the star such as its location
and apparent magnitude. Thus, we can directly multiply it by the number of
stars in the _Gaia_ catalog to obtain the expected number of statistical
background events.

Figure 10: This plot shows the estimate of the number
$N_{\textrm{B}}^{\textrm{stat}}$ of statistical background events for IDLEs
and its dependence on the duration $t_{e}$ of AML events. Here,
$N_{\textrm{B}}^{\textrm{stat}}=P_{\textrm{star}}^{\textrm{stat}}\times
1.47\times 10^{9}$, where $P_{\textrm{star}}^{\textrm{stat}}$ is computed in
eq. 5.1. We have also shown guiding lines (dashed-red) to indicate an event
duration of 2 years for which the statistical background is $\sim$ 1 event.
For event durations larger than this, the statistical background for IDLEs is
nearly negligible.

The fake rate is obviously very small for longer event durations ($t_{s}\ll
t_{e}\lesssim t_{\textrm{obs}}$ or $1\ll N_{s}\lesssim 35$) because of the
requirement that multiple consecutive passes show large deflections, for e.g.
for $t_{e}\simeq$ 2 years, we find $N_{s}=14$, and
$P_{\textrm{star}}^{\textrm{stat}}=3\times 10^{-10}$. Upon multiplying this by
the number of stars in the _Gaia_ catalog ($1.47\times 10^{9}$), we would
obtain $\sim 0.4$ background events with a 2 year event duration. For smaller
event durations, we would find a much larger number of background events. We
have plotted the number of statistical background events for IDLEs as a
function of the event duration $t_{e}$ in fig. 10. In order to reduce the
statistical background to negligible levels, we make the simple choice to only
look at IDLEs with event duration greater than 2 years.

We can also now understand why the statistical backgrounds for SDLEs would be
large from the figure. As we can see, the number of background events
increases rapidly for small event durations. Thus, we expect to get a very
large number of statistical background events for SDLEs as well, which could
correspond to an astrometric fluctuation in the position of the source image
in a single pass through the entire trajectory. This validates the claim that
we have made earlier that SDLEs will be undetectable at _Gaia_ due to the
overwhelmingly large statistical background.

Here, we have looked at the dependence of the background rate on only one
observable, the event duration $t_{e}$ and discussed the distribution of this
observable due to statistical background. It is conceivable that if we examine
the multi-dimensional space of observables, the background could be reduced
further, thus allowing for relatively low background searches for IDLE signals
with event durations slightly shorter than 2 years.

For LDLE type events, estimating the statistical background is trickier. The
reason for this is that there is no reference rectilinear motion against which
we can study a deviation. Instead, in a numerical study we could simulate
several rectilinear trajectories with statistical fluctuations, and compute
the probability of some of these trajectories to be statistical outliers,
perhaps through quantifying their $\chi^{2}$ fit to a pure rectilinear
trajectory. Such a detailed numerical study is beyond the scope of the present
work, but we hope to take it up in a future work.

###  5.2 Astrophysical background

There are several astrophysical sources of background that can mimic the
lensing signatures due to PBHs, i) astrophysical lenses which can mimic the
effects of PBH lensing, ii) deviation from our assumption of rectilinear
motion, for example due to a binary companion. We will only briefly discuss
these here, along with some suggestions for background reduction techniques.

Besides AML events due to PBHs, _Gaia_ is also expected to see AML events due
to microlensing caused by foreground stars (including objects formed from
stellar collapse) [73, 47, 48], brown dwarfs [52, 53, 74], and free-floating
planets [54]. Such events form a background to the detection of PBHs through
their AML signatures. These objects would predominantly lie in the disk and
thus the spatial distribution of the lensing event could in principle be used
to partially reduce this background. However, since both genuine AML
signatures due to PBHs and such background events would peak in the direction
of the Galactic center, we expect the spatial distribution of such events to
provide only a modest background reduction.

Brown dwarfs and free-floating planets would have masses less than $\sim
10^{-2}~{}M_{\odot}$. In practice, these objects would be indistinguishable
from PBHs of similar mass. As we will see in the next section, our expected
exclusion on the PBH parameter space is relevant only for PBH masses
$M>10^{-2}~{}M_{\odot}$, with lower mass objects giving rise either to SDLEs,
or below threshold lensing. Thus, we do not expect brown dwarfs and free-
floating planets to be a relevant background.

Lensing signals due to foreground stars form a reducible background when the
foreground star can be resolved by _Gaia_ and thus the source of lensing can
be clearly identified. In such a case, either a portion of the trajectory or
the entire trajectory could be excluded from an analysis. However, if the
foreground star is too dim, for example if it is a black hole or a neutron
star or a white dwarf formed from stellar collapse, then lensing due to such
objects would form an irreducible background. We would need to model the rate
of such lensing events and their observable distribution to predict the
expected background rate. The rate of AML signals due to unresolved foreground
stars can be predicted in a way similar to that in which we predicted the
event rate due to PBHs. One would have to perform a similar calculation as in
sec. 3, where instead of using the DM density to characterize the lens
distribution we would use a model distribution density for the dark lenses in
eq. 3.9. In addition, we would have to modify the monochromatic mass function
assumption used for the PBHs, and replace it by a distribution based on a mass
model for these faint stellar objects. A recent study along these lines using
a population synthesis code was performed in ref. [49]. The authors of this
study constructed the distribution of a similar set of observables as those
for our IDLEs ($\delta\theta_{\textrm{max}}$ and $t_{e}$)111111The authors of
ref. [49] actually used the Einstein ring crossing time which is
$\theta_{E}/\mu$ instead of the event duration $t_{e}$ as one of the
simplified observables. In order to approximately relate the former to the
latter, we can multiply it by $\theta_{E}/\sigma_{a}(m_{G})$, see eq. 3.2.,
but for lensing due to stellar objects. An examination of their results
suggests that the main backgrounds that could give rise to observables in a
similar region of parameter space as the PBH induced observable space (compare
our fig. 9 with fig. 14 in ref. [49]), are those which would arise from
neutron stars which could mimic IDLE type events, and black holes from stellar
collapse which could mimic very long duration LDLE type events121212The
authors of this study found that microlensing due to white dwarfs would give
rise comparatively smaller astrometric shift due to their lower masses.. From
a naive comparison by eye, it appears that they expect $\sim$ 1 neutron star
event which could mimic an IDLE signal. Although this background seems
negligible, a detailed study specifically with _Gaia_ observables is needed to
estimate this background more precisely.

The second source of astrophysical background is due to a violation of our
assumption of rectilinear motion. Isolated stars are expected to approximately
follow rectilinear motion in the Galactic rest frame over the time period of
observation of _Gaia_. However, gravitational potential gradients can alter
this expected behaviour. Such gradients can arise due to localized effects,
e.g a binary companion, or localized gravitational perturbations (for e.g. due
to a nearby star cluster), or they can arise due to steeper global
gravitational gradients, such as near the center of the Galaxy.

Of these possibilities, binary companions would probably be the most
significant effect, with nearly a third of main sequence stellar systems in
the Galaxy expected to be binary systems [75]. Furthermore, when accounting
for the bias in selection due to brightness, one would expect a significantly
larger fraction of the _Gaia_ sources to be binaries. A study by Penoyre,
Belokurov, and Evans [76, 77] which used a subsample of _Gaia_ data which only
utilized nearby stars, found that roughly 10% of stars in their sample showed
significant deviation away from the trajectory expected due to rectilinear
motion (more precisely rectilinear + parallax). These deviations were expected
to be because of binary companions.

However, this number is not necessarily representative of the expected binary
background rate to AML signals. In cases where the binary can be resolved and
both companions identified, such systems can easily be identified as
backgrounds. In the case of unresolved binaries, or binaries where the
companion is too faint to be detected, the relevant astrometric observable
would be the motion of the center-of-light (centroid). While the center-of-
mass of a binary system is expected to follow a rectilinear trajectory,
asymmetric mass-to-light ratios of the binary companions will result in a
motion of the center-of-light that will in general show deviations from
rectilinear motion. In most cases such deviations will exhibit behavior that
is different from that of a genuine lensing signal. One could perform a
detailed study of simulated trajectories of the centroid of such binaries and
compute their expected event observable distributions. It might then be
possible to develop additional discriminatory variables (such as periodic
features) that could be used to further separate the binary background from
signal. It is also important to know what the true type of lensing signals
are, corresponding to figs. 6(a), 6(b), and 6(c), in order to understand the
level to which binaries can be confused with genuine AML signals. For example
in the case where the signal involves a static source and a moving lens as in
fig. 6(c), the trajectory for IDLEs would involve some period of the centroid
being static, and then moving around a closed loop, and then being static
again – this should be easily distinguishable from most types of binary
motion. Alternatively, it might also be possible to leverage the relatively
crude photometry of _Gaia_ to check if the candidate AML event also displays a
simultaneous PML signal. The development of discriminatory variables, and the
degree to which reduction in the background can be achieved requires numerical
study which we hope to take up in future work.

Thus, we expect that the true binary background rate can be estimated by a
combination of a population model to characterize the number of binary systems
which satisfy certain basic criteria to constitute a background (such as
asymmetry of the mass-to-light ratios, resolvability, etc.), and further
studying the expected reduction in the rate once suitable criteria are imposed
to identify binary trajectories from the astrometric motion of the centroid.

In case it is found that the predicted binary background is sufficiently small
after such a study, and if the number of detected candidate AML events with
_Gaia_ is also sufficiently small, follow up observation of the detected
candidates (possibly with terrestrial or other space-based telescopes) could
be further used to identify binary companions. If the binary companions are in
different spectral classes, photometric follow ups could detect intensity
peaks in two widely separated bands [78, 79, 80, 81], thus identifying it as a
binary system. Although this technique would have low efficacy, other
additional techniques such as spectroscopic measurements (either to detect
different spectral types [82] or time-domain spectroscopy to measure Doppler
shifts [83, 84, 85]) or high resolution imaging [86, 87, 88], can be used to
identify binary systems among candidate lensing events.

We can also estimate the typical time-scale for deviations in trajectories
induced by the gravitational potential in a stellar cluster, by computing the
ratio of velocity and gravitational acceleration of a star in a typical
cluster. Assuming that $v=200$ km/s as before, and a cluster mass of
$10^{6}~{}M_{\odot}$, with a distance of 100 pc between the star and the
cluster, we would find a time scale of $\sim 10^{8}$ years. Therefore, such
gravitational gradients would typically only mimic very long duration LDLE
type events. Such anomalous accelerations will not exhibit any photometric
brightening signals, and this could perhaps be one way to separate them from
genuine AML signals.

We can briefly summarize the preceding discussion on astrophysical backgrounds
as follows. The potentially largest source of background is due to binary
motion. However, a population synthesis modelling, plus a study of the
detailed trajectories is needed to estimate the true background rate due to
binaries. This background could potentially be further reduced by follow up
observations of candidate lensing events. In addition, for very long duration
LDLEs we also expect some background from lensing due to black holes from
stellar collapse, as well as possible effects of local gravitational gradients
on stellar trajectories.

##  6 _Gaia_ ’s expected exclusion on the PBH parameter space

For a given value of $(f,M)$, we have shown that we can calculate the expected
number of IDLE and LDLE type events that _Gaia_ will be able to detect. We
have also given a procedure to calculate the distributions of some simplified
event observables. As discussed in sec. 5, there are various types of
backgrounds which can mimic genuine PBH induced lensing signals. Given the
distribution of expected observables for both signal and background, one could
set cuts on the observable parameter space to reduce the amount of background,
and maximize the signal-to-background ratio. Then, assuming that _Gaia_ takes
$t_{\textrm{obs}}$ worth of time-series data and sees events consistent with a
background-only rate, we could then find the expected exclusion on the regions
of PBH parameter space that _Gaia_ can set due to non-observation of any
excess event rate over background.

In this work we have only calculated the signal rate and the observable
distribution from signal, however we have not computed the full background
rates and the observable distributions for all the backgrounds. Thus, we can
not give a fully accurate projected exclusion on the PBH parameters that can
be set by _Gaia_.

In order to get some indication of _Gaia_ ’s expected sensitivity in the
absence of a full background rate calculation, we can make some simple
assumptions on the background rate and calculate the projected exclusion. We
will assume that we only study AML events with event durations larger than 2
years in order to reject the IDLE statistical background (see sec. 5.1). We
will further make the simplified assumption that after setting this cut, the
total background from all sources is negligible in the region of observable
space where we expect genuine PBH induced AML signatures.

Thus, with the above assumptions, assuming that we see zero candidate lensing
events, consistent with background only, we expect to rule out regions of the
PBH parameter space that predict $N_{l}>2.3$ events at the 90% confidence
level, assuming Poissonian statistics.

Figure 11: This plot shows excluded regions of the PBH parameter space
$(f,M)$. Dashed curves indicate the exclusions that have already been set
using PML techniques (HSC [26, 27], MACHO [14], EROS [30], OGLE [16]) and
dotted curves indicate constraints from a variety of other observations, such
as PML signals in giant arcs [12], lensed SNe1a [15], distortions of the CMB
[89], and effect on large-scale-structure (LSS) [90]. The green curve is our
result for the expected $90\%$ C.L. exclusion curve that _Gaia_ will set using
astrometric microlensing with $t_{\textrm{obs}}=5$ years of data. This
exclusion curve is computed assuming that _Gaia_ will see no events,
consistent with our simplified assumption of a negligible number of expected
background events. The total exclusion expected from _Gaia_ can be decomposed
into an exclusion based on IDLE type events (blue curve) with event durations
between 2 - 5 years, and LDLE type events which have longer event durations.
At masses lower than $\sim 0.4~{}M_{\odot}$, _Gaia_ loses sensitivity to PBHs
because such low mass PBHs would yield small deflections and event durations
below _Gaia_ ’s sensitivity thresholds. At large masses, the number of lenses
in the Galaxy decreases and hence the lensing probability also decreases, once
again yielding a loss in sensitivity near $\sim 10^{8}~{}M_{\odot}$. From the
curve we see that _Gaia_ is expected to be most sensitive to PBHs with mass
$\sim 10~{}M_{\odot}$. Our projection indicates that _Gaia_ is sensitive
enough to rule out a PBH origin for the LIGO/Virgo gravitational wave (G.W.)
signals (indicated by the 2-$\sigma$ best-fit region shown by the cyan contour
[91]).

In fig. 11, we show the PBH parameter space $(f,M)$ and shaded contours which
indicate the average total number of lensing events $N_{l}$ due to PBHs, for
both IDLE and LDLE types, for each point in the parameter space. The boundary
of these contours is the green curve, which corresponds to $N_{l}=2.3$. The
region of shaded parameter space interior to the green curve can be excluded
at the 90% confidence level.

We have also shown on the same plot, the curves corresponding to regions of
parameter space that give rise to 2.3 IDLE type events (blue curve), or 2.3
LDLE type events (orange curve). The regions above these curves can be
interpreted as expected exclusions purely due to null search results for
either one of these classes of lensing events.

We can see from the figure that _Gaia_ is sensitive to PBHs with masses
between $0.4~{}M_{\odot}$ \- $5\times 10^{7}~{}M_{\odot}$. _Gaia_ is most
sensitive to PBH masses near $10~{}M_{\odot}$ where a fraction $f$ as low as
$3\times 10^{-4}$ of the DM relic density can be ruled out. For PBH masses
lower than $10^{3}$ $M_{\odot}$, most of the detectable lensing events would
correspond to IDLEs with average event durations that lie between 2 - 5 years.
For sufficiently low PBH masses, two conditions cause the number of detectable
lensing events to drop rapidly to zero. Firstly, the Einstein angle
$\theta_{E}$ decreases to such an extent that many of the lensing signals
would fall below _Gaia_ ’s detection threshold. Secondly, the average event
duration $t_{e}$ would decrease below 2 years, for which the statistical
backgrounds to IDLEs would be large. This makes the exclusion bound weaker for
low PBH masses. For PBH masses greater than $10^{3}M_{\odot}$, most of the
detectable lensing events would correspond to LDLE lensing with average event
duration much greater than observation time $t_{\textrm{obs}}<t_{e}$. For
larger PBH masses, the number of lenses decreases with increasing PBH mass,
and thus relatively fewer signal events are expected. This makes the bound
weaker for larger values of the PBH mass $M$.

We have shown several other exclusion bounds from the literature in fig. 11,
which can be compared against our projected exclusion bound. Among the
existing constraints, lensing constraints are the most robust, since they rely
on fewer assumptions. We find that the AML bound on PBH density fraction $f$
that can be set by _Gaia_ for masses $\gtrsim~{}0.4~{}M_{\odot}$ is stronger
than that of the current best bounds from PML surveys, possibly by several
orders of magnitude. We have also shown in the figure, a best-fit region of
the PBH parameter space corresponding to PBH mergers giving rise to all the
observed gravitational wave signals seen at LIGO/Virgo [91]. We find that our
projected exclusion could rule out this region, thus ruling out PBHs as the
origin for the observed gravitational wave events.

A more realistic study of a possible PBH origin for the LIGO/Virgo signals was
performed in refs. [92, 93]. In these works, the authors studied mixed models
with some fraction of the LIGO/Virgo signals arising from astrophysical black
hole (ABH) mergers, and some from PBHs. Using a Bayesian analysis, they found
that only 20% of the LIGO/Virgo signals are likely to have a PBH origin. This
would imply a best fit fraction $f$ of PBH dark matter which is lower than
that of [91] by a factor of $\sim$ 5 [92]. However, the exact fraction would
depend on the ABH model which is highly uncertain. We note that the expected
_Gaia_ exclusion that we have found is strong enough to rule out even this
smaller value of $f$, thus potentially ruling out even the best-fit prediction
of [92]131313Although no other present day constraints exist on such a low
fraction of PBHs in this mass range, future gravitational wave searches for
PBH mergers at higher redshift $z\gtrsim 30$ with the Cosmic explorer and
Einstein telescopes could rule out a fraction $f\gtrsim 10^{-5}$ of PBHs at
$10~{}M_{\odot}$ in the next decade [94]..

##  7 Discussion

The computation of the expected exclusion curve that _Gaia_ will set based on
the AML technique crucially depended on the calculation of the lensing
probability $P_{\textrm{star}}$. There are a number of assumptions that we
made while calculating $P_{\textrm{star}}$, some of which were made for
convenience of calculation. We list some ways in which the calculation can be
improved.

  * •

We made an assumption of a constant relative tangential separation velocity
$v=200$ km/s. This can be further improved by taking into account the stellar
velocities and DM velocities. Stellar velocities can potentially be obtained
from the _Gaia_ catalog itself, whereas the DM velocities can be obtained
perhaps from Eddington inversion techniques of the Galactic rotation curve
(see for e.g. [95]). A more accurate prediction of relative velocities is
needed because our prediction of the typical event durations (and hence
separability from background) has a direct inverse dependence on the
separation velocity.

  * •

We made the assumption of the rectilinear motion of target stars. However, the
parallax motion of these stars (and PBHs) would result in non-rectilinear
motion. Moreover, a large fraction of stars in our Galaxy are in binary
systems and this would result in wobbles in the star’s trajectory due to its
bound motion. In order to more carefully take into account these effects in
the computation of $P_{\textrm{star}}$, we would need to perform a numerical
simulation of trajectories to study the effects of lensing and how it can be
distinguished from other kinds of motion.

  * •

We have also assumed uniformly spaced sampling in time by _Gaia_. However, in
a numerical study one could also take into account _Gaia_ ’s scanning law
which would give us the correct time sampling rate for stars in a given region
of sky.

  * •

For simplicity, we have taken the astrometric error $\sigma_{a}(m_{G})$ to be
the error along the scanning direction of _Gaia_. In a numerical study, a
better estimate of $P_{\textrm{star}}$ could be obtained using both the across
and along astrometric errors depending upon _Gaia_ ’s scanning strategy.

  * •

The probability $P_{\textrm{star}}$ depends upon the assumed DM density
profile. We computed $P_{\textrm{star}}$ for the NFW profile, but we could
also try other profiles such as Burkert [96], Einasto [97, 98], isothermal
[99, 100] etc. These DM density profiles differ from one another near the
Galactic center, and in general are less cuspier than the NFW profile. Since
the Galactic center region was where we have found the maximum probability for
detecting lensing events, we therefore expect that these alternate profiles
could have a sizeable effect on our estimate of the expected number of AML
events.

  * •

We have ignored blending effects due to foreground stars which could affect
the centroiding precision when detecting AML events. A realistic simulation is
needed to estimate the size of such effects.

  * •

We have assumed that our exclusion is set by using only the AML signature
detected by _Gaia_. However, _Gaia_ ’s relatively crude photometry could also
be used in conjunction with the AML measurements to measure a PML signal which
could potentially improve the sensitivity of the search.

In addition to improvements in the signal rate calculation, a detailed
background study is needed to obtain an accurate exclusion curve. In
particular, numerical studies of the LDLE statistical background are important
as this could potentially affect the expected bound at high PBH masses where
we expect to have the longest event durations. Also, as outlined in sec. 5.2,
we expect that the astrophysical backgrounds that need a more careful study
are due to i) wobbles in a stellar trajectory due to an unresolved binary
companion or local gravitational potential gradients and ii) lensing due to
dark stellar objects.

##  8 Summary: Main takeaway

_Gaia_ ’s unprecedented astrometric precision has ushered in a new era in our
understanding of the Galaxy. With its milliarcsecond level precision in a
single pass, _Gaia_ has the capability to be extremely sensitive to any
unusual proper motion of stars. Primordial black holes are expected to form a
significant fraction of the dark matter density of the universe in many
cosmological models. PBH dark matter in our own Galaxy could lead to
astrometric microlensing signals of background stars which could be detected
by _Gaia_. In this work, we have attempted to make a prediction for the
expected sensitivity of _Gaia_ to the PBH parameter space ($f,M$), and to
compute the expected AML event observable distributions from PBH induced
lensing.

In order to estimate the potential exclusion limit that _Gaia_ can set on the
PBH parameter space with 5 years of observational data, we needed to estimate
both the rate of genuine PBH induced AML signals, as well as the rate of
background events that could mimic such signals.

The present work has primarily focused on a precise prediction of the signal
rate and associated observables. In order to compute the signal rate, we used
the existing _Gaia_ eDR3 catalog as a model of the stars in the Galaxy which
are potential AML targets. We then combined this with a novel probability
calculation to estimate the likelihood that a given star in the _Gaia_ catalog
would undergo an AML event which is detectable at _Gaia_. While estimating
this probability we argued that there would be three different classes of
detectable lensing events, Short Duration Lensing Events (SDLEs), Intermediate
Duration Lensing Events (IDLEs), and Long Duration Lensing Events (LDLEs). We
also suggested appropriate simplified observables that can be extracted from
the apparent trajectory, which characterize the signatures of AML events. For
a given set of PBH parameters, our calculations showed a) the regions of the
sky which are likely to yield a large event rate and b) the distribution of
the simplified observables for both IDLE and LDLE types of lensing events.

Although a full background study was beyond the scope of this work, we have
also discussed various sources of background, and highlighted which
backgrounds we expect to be important and deserving of further investigation.
We did perform a slightly more quantitative estimate of the background for
IDLE type events due to statistical fluctuations induced by centroiding
uncertainties. We found that event durations greater than $\sim 2$ years,
would show negligible background of this type. Thus only IDLEs of event
durations between 2 - 5 years can be used to search for PBHs, whereas shorter
duration IDLEs would not be usefull for PBH searches. We used this to also
argue that similarly SDLEs would also be completely swamped by the same
statistical background and thus irrelevant for PBH searches. We also
highlighted the major sources of expected astrophysical backgrounds and
possible techniques that could be used to reduce these backgrounds.

We then computed the expected exclusion curve for _Gaia_ using our predicted
signal rate, and the simplified assumption of a negligible background rate.
With this assumption, we found that _Gaia_ is sensitive to PBHs with mass
between $0.4~{}M_{\odot}$ to $5\times 10^{7}~{}M_{\odot}$ with peak
sensitivity to PBH masses of $10~{}M_{\odot}$, for which we can rule out such
PBHs as making up as little as a fraction $f=3\times 10^{-4}$ of the DM
density. The lower end of the mass sensitivity window is due to IDLE type
events which yield event durations close to 2 years, near where the
statistical background becomes a significant effect. At the higher end of the
mass range of interest, _Gaia_ is sensitive to LDLE type events, but loses
sensitivity at high masses because the number of lenses decreases, hence
decreasing the expected event rate.

As compared to other existing bounds on the PBH parameter space, we have found
that _Gaia_ will potentially have the best sensitivity to PBHs with masses
near $10~{}M_{\odot}$. This region of PBH masses is particularly intriguing
since it overlaps with the range of PBH masses that current gravitational wave
detectors are sensitive to. An exciting implication of our work is that we
expect that _Gaia_ can potentially exclude a PBH origin for the LIGO/Virgo
black hole merger events.

Our work is the first attempt to combine a detailed signal rate estimation
with a rudimentary background assumption to calculate the expected sensitivity
of _Gaia_ to the PBH parameter space. Previous works in the literature have
estimated the lensing probabilities, but not the total expected event rate or
the exclusion curve. Moreover, these studies have only focused on the case of
LDLEs. Our work also shows for the first time that IDLE type events can also
play a significant role in excluding or discovering regions of the PBH
parameter space with _Gaia_.

We have also discussed a number of ways in which our calculation of the
expected exclusion could be further improved. Most of these improvements
require detailed numerical studies which we plan to pursue in the future. Once
time-series data of _Gaia_ is publicly available, we expect to be able to
analyze this data for potential lensing signals, and detection/non-detection
of such signals can then be used to place constraints on the PBH parameter
space using our expected event rate calculations.

## Acknowledgments

We thank the anonymous referee for many useful comments on the draft. We
acknowledge helpful discussions and correspondence with Jeffery J. Andrews,
Varun Bhalerao, Surhud More, Jan Rybizki, and Sourav Chatterjee. The work of
VR was supported by a DST-SERB Early Career Research Award (ECR/2017/000040).

##  Appendix A Derivation of some expressions for averaged astrometric lensing
observables

In this appendix, we derive expressions for the simplified event observables
defined in sec. 4.2, averaged over impact parameters.

For IDLEs, the simplified observable is given by the maximum astrometric
deflection $(\delta\theta)_{\textrm{max}}$. We first find the maximum
astrometric shift $\delta u$, in $\theta_{E}$ units, using eq. 3.5. For a
rectilinear trajectory, we can take the lens-source separation to be of the
form, $u=\sqrt{u_{0}^{2}+\xi^{2}(t)}$, where $u_{0}$ is the impact parameter