11institutetext: 1 School of Natural Sciences, University of Tasmania, Private
Bag 37 Hobart, Tasmania, 7001, Australia
2 Sorbonne Université, CNRS, Institut d’Astrophysique de Paris, IAP, F-75014,
Paris, France
11email<EMAIL_ADDRESS>
# Finding planets via gravitational microlensing

Natalia E. Rektsini & Virginie Batista 112222

###### Abstract

Since the first microlensing planet discovery in 2003, more than 200 planets
have been detected with gravitational microlensing, in addition to several
free-floating planet and black hole candidates. In this chapter the
microlensing theory is presented by introducing the numerical methods used to
solve binary and triple lens problems and how these lead to the
characterisation of the planetary systems. Then the microlensing planetary
detection efficiency is discussed, with an emphasis on cold planets beyond the
snow line. Furthermore, it will be explained how the planetary
characterisation can be facilitated when the microlensing light curves exhibit
distortions due to second order effects such as parallax, planetary orbital
motion, and extended source. These second order effects can be turned to our
advantage, and become useful to ultimately better characterise the planetary
systems, but they can also introduce degeneracies in the light curve models.
It will be explained how the use of modern observational and computational
techniques enables microlensers to solve these degeneracies and estimate the
planetary system parameters with very high accuracy. Then a review of the main
discoveries to date will be presented while exploring the recent statistical
results from high-cadence ground-based surveys and space-based observations,
especially on the planet mass function. Finally, future prospects are
discussed, with the expected advances from dedicated space missions, extending
the planet sensitivity range down to Mars mass.

##  1 Introduction

Gravitational microlensing is a unique technique to discover exoplanets
because it doesn’t rely on the host star brightness or the orbital motion of
the planetary system. In contrast to other methods, it relies on time-variable
magnification of the light of an unrelated background star due to deflection
by the gravitational potential of the host star$+$planet system. Detectable
magnification requires very precise alignment of the planetary system with a
background source star and the observer. When this alignment occurs, the mass
of the intermediate planetary system will act as a lens, deflecting the light
emitted by the background source star. As a result, the observer receives
light from multiple projected images of the source star, each of which has its
own time-variable angular size, which can be interpreted as a time-variable
light magnification.

Predicting exactly where and when a microlensing event will occur as well as
how significant the magnification will be is impossible, and the duration of
the event is limited and unique. Therefore, the expectation of observing this
phenomenon received a rather pessimistic reaction from the scientific
community, including Einstein, who was the first to bring attention to it
(Einstein 1936), after receiving multiple requests from R. W. Mandl.

Indeed, millions of stars need to be monitored in order to catch a handful of
magnified ones. In the following decades, several scientists considered
experiments of microlensing observations (e.g Refsdal 1964; Chang & Refsdal
1979), and finally Paczynski (1986) managed to convince the international
community to develop the MACHO, EROS and OGLE experiments to search for
compact objects in the galactic halo (see acronyms at the end of the chapter).
Hunting for microlensing events towards the galactic center with the intention
to discover planets appeared a few years later, under the suggestion of Mao &
Paczynski (1991) and Gould and Loeb (1992), who gave a detailed description of
microlensing planetary perturbations. The development of this technique over
the last 30 years required tremendous efforts and steady determination from
the scientific community.

Nowadays, thousands of microlensing events are observed per year, leading to
the detection of tens of exoplanets for each season, accounting for more than
200 planetary systems published to date and many more under analysis. This
number is quite modest compared to the total amount of discovered exoplanets
but this will change with the launch of the Nancy Grace Roman Space Telescope
(Spergel et al. 2015) scheduled in late 2026. Roman will observe 2 square
degrees close to the Galactic center with high optical depth to microlensing
during 5 observing runs of 72 days each. It is estimated that it will detect
more than 1400 bound microlensing exoplanets down to the mass of Mars (Penny
et al. 2019), with orbits ranging from 0.5 AU to wide orbits. Given the recent
results of Sumi et al. (2023), we can also expect thousand or more unbound
planets down to Mars-masses.

Microlensing offers a considerable advantage in probing a parameter space that
is not accessible to other methods. It can detect planets down to Earth
masses(Bennett & Rhie 1996), within a range of separations to the host star
that is complementary to other techniques with maximum sensitivity to planets
orbiting close to or beyond the snow line. According to the core accretion
theory (Ida & Lin 2004), this is where planetary formation is predicted to be
the most efficient. Moreover, as the only effect measured by microlensing is
that of deflecting light by gravitational distortions of a given field, it is
independent of the emitted light of the lens. This makes it unbiased by the
lens properties and opens a domain of exploration down to very faint, even
invisible objects, such as brown dwarfs, black holes, neutron stars and free-
floating planets while also making possible the detection of exoplanets in
very large distances throughout our Galaxy.

Detecting distant exoplanets means that they are hard or impossible to re-
observe so studying their atmospheric and interior properties is not likely.
The importance of these detections lies in their implications for theoretical
and statistical exoplanet studies. Generally, the formation processes are
influenced by many environmental factors, such as stellar metallicity (Wang &
Fischer 2015; Zhu et al 2016), as well as the mass of the host star (Johnson
et al 2010), the star multiplicity (Kaib et al. 2013), and larger scale
conditions like higher ambient radiation and stellar density in the galactic
bulge (Thompson 2013). Thus, the distant population of planets unveiled by
microlensing provides critical constraints for planetary formation theories
and population synthesis models.

##  2 Observations

The first microlensing surveys debuted in the early 90’s, searching for the
existence of dark matter in the galactic halo (Alcock et al. 2000; Tisserand
et al. 2007; Wyrzykowski et al. 2011). The idea to use this technique to
detect exoplanets was introduced later by Mao & Paczynski (1991) and Gould and
Loeb (1992). The first generation survey telescopes were able to observe only
a small fraction of the sky with a limited observing cadence which restricted
their ability to detect the planetary perturbation. A solution to this problem
came when the OGLE-early-warning system was introduced announcing alerts of
potential microlensing events in real time. This permitted other surveys with
multiple smaller telescopes to follow-up on the events. Using numerous
observatories at all longitudes and different wavebands ensured a round-the-
clock coverage of the planetary anomalies.

The new millennium marked a real advance in the field, with two microlensing
collaborations: MOA (Bond et al. 2001; Sumi et al. 2013) and OGLE (Udalski
2003) in addition to the follow-up collaborations: PLANET (Albrow et al.
1998), $\mu$FUN (Gould et al. 2010), RobotNet (Tsapras et al. 2009) and
MiNDSTEp (Dominik et al. 2010). The Wise observatory in Israel also conducted
a survey (Gorbikov et al. 2010), despite its less favorable northern latitude.
All this hard effort and team work lead to the first microlens exoplanet
discovery (Bond et al. 2004) and to hundred more detections ever since,
establishing microlensing as a successful exoplanet detection technique.

Almost a decade later, the microlensing community made a transition towards a
network of wide field imagers. The second generation surveys conduct high
cadence observations from different longitudes of the southern hemisphere. In
2006, the MOA collaboration started using a 2.2-deg2 FOV CCD camera (Sako et
al. 2008) mounted on a 1.8m MOA-II telescope in New-Zealand. The OGLE
collaboration upgraded its camera in 2010 to a 1.4-deg2 FOV on a 1.3-m
telescope in Chile. Additionally, KMTNet (Henderson al 2014; Kim et al. 2016)
became operational in 2015, with 3 telescopes equipped with a 4 deg2 camera,
in Chile, South Africa and Australia. The addition of this microlensing survey
had a significant impact on the efficiency of microlensing planet detections
by discovering or contributing to the discovery of 204 out of 278 microlensing
planets until today according to their site (accessed
24/3/2024):(https://kmtnet.kasi.re.kr/ulens/data/KMTNet_exoplanet_list.pdf).
These three surveys are continuously monitoring hundreds of fields mostly
towards the bulge. Today most of the follow-up groups have stopped operating
but some of the observatories (e.g., LCO, UTGO ) are still following alerts
from surveys dedicated and not dedicated to microlensing (e.g., ASAS, ZTF,
GAIA).

In 2014-2017, the microlensing campaigns from the ground were also combined
with space-based observations using the telescopes Spitzer (Werner et al.
2004; Gould & Horne 2013a) and Kepler (K2-C9) (Howell et al. 2014), as a
pathway towards the future space missions Roman (Spergel et al. 2015; Penny et
al. 2019) and Euclid (Penny et al. 2013).

The PRime-focus Infrared Microlensing Experiment (PRIME) started operating in
late 2023 (Kondo et al. 2023). It is the first near-infrared microlensing
survey, using a 1.8m telescope with a wide field of view of 1.45 deg2,
positioned at the South African Astronomical Observatory. The main goal of
this survey, amongst others, is to explore and derive optimised observational
strategies for Roman ensuring the maximum possible microlensing exoplanet
detections towards the Galactic Bulge.

Since 2016 GAIA (Gaia Collaboration et al. 2016) is using astrometric
techniques covering the whole galaxy, the galactic bulge, the plane and the
disk at a wide range of magnitudes. Their photometric alert system has proven
very valuable for detecting and following-up on microlensing events on any
kind of magnitudes, using both photometric and astrometric microlensing
(Wyrzykowski et al. 2023). Ultimately, Gaia will include astrometric time
series in their fourth data release (DR4), that will be extremely valuable to
joined photometic and astrometric fits.

Interferometric methods have also been used for microlensing events, with the
first resolved microlensing image by VLTI GRAVITY (Dong et al. 2019) which was
followed by the first images of rotating gravitational microlensing arcs,
using the VLTI PIONER instrument (Cassan et al. 2022). These successful
observations have opened the path to interferometric microlensing which can
potentially lead to high precision mass measurements for future microlensing
events.

In addition, microlensing will also be included in the scientific goals of the
Legacy Survey of Space and Time (LSST) survey of the Southern Hemisphere sky
(Abrams et al. 2023; Street et al. 2023b). The survey will be conducted with
the Vera C. Rubin Observatory using an 8.4-meter telescope situated at the top
of Cerro Pachon ridge in Chile. In order to prepare for this survey the LCO
team have been running the OMEGA (2020-2023) and OMEGA 2 key projects, running
all-sky photometric and spectroscopic follow-up for microlensing alerts from
MOA, OGLE , GAIA and ZTF.

Finally, the Nancy Grace Roman Space Telescope will be the first space-based
telescope with a survey (RGBTDS) dedicated to the microlensing technique.
Roman is a NASA flagship mission scheduled for launch in October 2026 towards
the Lagrange point L2. The concept was initiated with a dedicated mission
(Bennett et al. 2002) called the Microlensing Planet Finder (MPF), which had
been proposed to NASA’s Discovery program but not selected. The objective was
to be able to monitor turn off stars in the galactic bulge as microlensing
sources. The Roman Wide Field Instrument (WFI) consists of 18 4k$\times$4k
Hawaii H4RG-10 HgCdTe detectors with a pixel scale of 0.11” with a 0.28 deg2
field of view (Spergel et al. 2015). It will also contain 6 broad band filters
(R062, Z087, Y106, J129, H158, F184, K213) from 0.62 $\mu$m to 2.13 $\mu$m and
an extremely wide filter from 0.92 $\mu$m to 2.13 $\mu$m (W146). Roman will
observe 7 fields covering 1.97 square degrees, with a cadence of 15 minutes in
W146 for 6 periods of 72 days.

Euclid is an ESA mission, launched in July 2023, with primary goals to measure
parameters of dark energy using weak gravitational lensing and baryonic
acoustic oscillation, test the general relativity and the Cold Dark Matter
paradigm for structure formation. Since its original submission in 2007 to
ESA, a microlensing planet hunting program has been listed as part of the
Legacy science (Beaulieu et al. 2008). The vision adopted by the Europeans of
a joint mission with Dark Energy probes and microlensing has been promoted and
adopted by the Astro 2010 Decadal Survey when it created and ranked as top
priority the Roman mission.

A baseline for the design of the Euclid microlensing survey was firstly
described by Penny et al. (2013) with a detailed simulation demonstrating the
capabilities and the expected scientific outcomes. (Bachelet and Penny 2019;
Bachelet et al. 2022) conducted the same simulations for the Roman survey and
predicted that precursor Euclid observations of all the currently envisioned
Roman fields would be very valuable in defining exactly which fields will be
observed, would help to understand the structure and extinction of the line of
sight, and provide a reference for relative source-lens proper motion of
forecoming Roman microlensing events. The objective is to perform 4 Euclid
observations at 9 pointing positions covering 4.5 square degree. The procedure
will take 42 hours, which is very a valuable use of Euclid time. Both VIS and
IR data will be used. The VIS instrument contains 36 4k$\times$4k e2v
CCD273-84 CCD detectors with 0.10” pixels, giving a 0.47 deg2 field of view
while the NISP instrument has the same field of view with 16 H2RG-18 HgCdTe
detectors with a pixel scale of 0.30”.

##  3 Microlensing basics

The basic principle of a microlensing effect is illustrated in Fig. 1, showing
a light ray coming from a source S being deflected by a point-like mass L,
called ‘lens’. The light is deviated by an angle $\alpha$, creating the
illusion for the observer (O) that the source is at the position I (‘image’)
in the lens plane. The three angles of this figure are linked by the lens
equation, $\beta=\theta-\alpha$, where $\alpha=4GM/(c^{2}D_{L}\theta)$, with
$M$ the lens mass, $G$ the gravitational constant and $c$ the speed of light.
Under the assumption of small angles, one can write $\alpha(D_{S}-D_{L})=\beta
D_{S}$.

The deviation of light does not affect only one ray but an ensemble of rays
coming from the source, so when the source and the lens are perfectly aligned
on the observer’s line of sight, $\beta=0$ and the source image in the lens
plan appears as a ring, called the ‘Einstein ring’, whose angular size
$\theta_{\rm{E}}$ is expressed as

$\theta_{\rm{E}}=\sqrt{\kappa
M\frac{D_{S}-D_{L}}{D_{S}D_{L}}},\quad{\rm{where}}\,\,\kappa=4G/c^{2}\simeq
8.14\,\rm{kpc.M}^{-1}_{\odot}.$ (1)

The lens equation is thus commonly expressed as
$\beta=\theta-\theta_{\rm{E}}^{2}/\theta$. The Einstein angular radius
$\theta_{\rm{E}}$ is used as a scale standard in microlensing units and as
long as its value remains unknown, most microlensing parameters will be
expressed in units of $\theta_{\rm{E}}$.

Figure 1: Deviation of a light ray from a source S due to the gravitational
field of a microlens L.

###  3.1 Single lens

By normalizing the angles by $\theta_{\rm{E}}$ and define the parameters
$u=\beta/\theta_{\rm{E}}$ and $z=\theta/\theta_{\rm{E}}$, we can express the
lens-source separation $u$ as a function of the separation $z$ between the
image of the source and the lens, both being projected in the lens plane,
$u=z-1/z$. If the lens and the source are not perfectly aligned ($u\neq 0$),
this equation has two solutions: $z_{\pm}=\pm(\sqrt{u^{2}+4}\pm u)/2$. The
positive solution creates the major image, and the negative one the minor
image, outside and inside of the Einstein ring, respectively.

The magnification of the source by the lens is then estimated as the area
ratio of the images over the source. Since the images are tangentially
stretched by a factor $z_{\pm}/u$ compared to the source and radially shrunk
by a factor $dz_{\pm}/du$, the magnification is then expressed as

$A_{\pm}=\left|{\frac{z_{\pm}}{u}\frac{dz_{\pm}}{du}}\right|=\frac{1}{2}\left[\frac{u^{2}+2}{u\sqrt{u^{2}+4}}\pm
1\right],\quad{\rm{thus}\,\,\rm{in}\,\,\rm{total,}}\quad
A(u)=\frac{u^{2}+2}{u\sqrt{u^{2}+4}}.$ (2)

In its simplest form, the lens-source relative movement is rectilinear and $u$
can be expressed as a function of the impact parameter $u_{0}$ (in units of
$\theta_{\rm E}$), the time of maximum magnification $t_{0}$, and the Einstein
crossing time $t_{\rm E}$

$u(t)=\left[u^{2}_{0}+\left(\frac{t-t_{0}}{t_{\rm
E}}\right)^{2}\right]^{1/2}.$ (3)

The observed flux is a function of time, $F(t)=A(t)F_{s}+F_{b}$, where $F_{s}$
is the source flux and $F_{b}$ the blended light (any unresolved light) that
is not magnified. It reaches its maximum at $t_{0}$, when the impact parameter
is at its minimum $u_{0}$. The smaller $u_{0}$ is, the higher the maximum of
magnification will be. For a Point-Source-Point-Lens (PSPL) event, the light
curve can be fit by five parameters: $t_{0}$, $u_{0}$, $t_{\rm{E}}$, $F_{s}$
and $F_{b}$. In practice, each telescope has its specific $F_{s}$ and $F_{b}$,
since different telescopes may have different filter bandpasses and
resolutions.

###  3.2 Multiple lens

Let’s consider now a lens composed of $N_{L}$ objects, which the total mass
will be $M=\Sigma^{N_{L}}_{i=1}m_{i}$. Let $\mathbf{\theta}_{i}$ be the
2-dimension coordinates of these individual masses in the lens plane. The
deflection angle of the source is then expressed by

$\mathbf{\alpha}(\mathbf{\theta})=\frac{4G}{c^{2}}\left(\frac{1}{D_{L}}-\frac{1}{D_{S}}\right)\sum_{i}^{N_{L}}m_{i}\frac{\mathbf{\theta}-\mathbf{\theta}_{i}}{|\mathbf{\theta}-\mathbf{\theta}_{i}|^{2}},$
(4)

where $\mathbf{\theta}$ are the angular positions of the images. It is common
to consider the lens equation in complex coordinates (Witt 1990; Rhie 1997).
The source position $\mathbf{u}=(\xi,\eta)$ and the images position
$\mathbf{z}=(x,y)$ can then be defined in the complex form, $u=\xi+i\eta$ and
$z=x+iy$ respectively. The lens equation thus becomes

$u=z-\sum_{i}^{N_{L}}\frac{m_{i}/M}{\bar{z}-\bar{z}_{i}},$ (5)

where $\bar{z}$ is the complex conjugate of $z$. Eq.(5) can be solved
numerically to find the images position $z_{j}$, where $j$ is the index for
the source images and $i$ those of the individual lens masses. The
magnification can be greater or less than 1, depending on the images area, and
is equal to the inverse of the Jacobian matrix determinant,

$A_{j}=\frac{1}{\rm{detJ}}\Big{|}_{z=z_{j}},\quad{\rm{detJ}}\equiv\frac{\partial(\xi,\eta)}{\partial(x,y)}=1-\frac{\partial
u}{\partial z}\frac{\overline{\partial u}}{\partial z}.$ (6)

The total magnification is the sum of individual ones: $A=\Sigma_{j}|A_{j}|$.
Singularities occur for positions of the source where ${\rm{detJ}}=0$,
theoretically inducing an infinite magnification in the case of a point
source. In practise, the source cannot be considered as a point in most cases
and the singularities appear as very high but finite magnifications.

From Eq.(5), $\,\partial
u/\partial\bar{z}=\sum\left[\epsilon_{i}/(\bar{z_{i}}-\bar{z})^{2}\right]$,
where $\epsilon_{i}=m_{i}/M$ is the mass fraction of individual masses
$m_{i}$. According to Eq.(6), the image positions for which ${\rm{detJ}}=0$
are given by

$\left|\sum_{i}^{N_{L}}\frac{\epsilon_{i}}{(\bar{z_{i}}-\bar{z})^{2}}\right|^{2}=1.$
(7)

The solutions of Eq.(7) shape as closed contours in the lens plane, called
‘critical curves’. To these curves correspond an ensemble of source positions
in the source plane, called ‘caustics’.

When the lens is composed of two masses, $N_{L}=2$, the lens equation is
expressed by

$u=z+\frac{\epsilon_{1}}{\bar{z_{1}}-\bar{z}}+\frac{\epsilon_{2}}{\bar{z_{2}}-\bar{z}}.$
(8)

The Eq.(8) can be written as a 5-degree polynomial, whose coefficients are
given in Witt & Mao (1995). The solutions to this polynomial are not
necessarily solutions to the Eq.(8), because when the source is outside of the
caustics, two of the five solutions yield to an imaginary magnification. Thus,
the number of images in the lens plane varies from three to five when the
source crosses a caustic.

Schneider and Weiss (1986) showed that there are three different topologies of
caustics for a given value of mass ratio $q$ between the two components of the
lens. These three topologies present either one, two or three caustics. As
shown on Fig. 2, in the case of small separations between the two lenses,
$d\ll 1$ (in Einstein angular radius units), there are three caustics, one
being the central caustic (with 4 cusps) and two secondary ones (with 3
cusps). The transition towards big separations goes through an intermediary
phase that creates a single caustic with 6 cusps, of bigger size, called
‘resonant caustic’. This configuration happens when the separation is close to
the Einstein radius. Finally, for large separations, this caustic splits into
two caustics, with 4 cusps each. Naturally these three domains are called
‘close’, ’resonant’ and ‘wide’ separations.

For triple lens modeling, Rhie (2002) gives the lens equation for point source
calculations. It is solved the same way as the binary lens equation, but the
polynomial is of 10th order instead of 5th.

Figure 2: Topology of caustics for three regimes of separation, ‘close’ (top),
‘resonant’ (middle) and ‘wide’ (bottom).

#### Lensing degeneracies

For mass ratios $q\ll 1$ and a given separation $d$, the shape and size of the
central caustic is invariant under the $d\longleftrightarrow d^{-1}$
transformation. This duality is often called ‘close-wide degeneracy’ (Griest &
Safizadeh 1998; Dominik 1999; Albrow et al. 1999; An 2005). It comes from the
fact that the Taylor development of the lens equation to the second degree is
identical for $d\longleftrightarrow d^{-1}(1+q)^{1/2}$, i.e.
$d\longleftrightarrow d^{-1}$ when $q\ll 1$. Fig. 3 gives an illustration of
this degeneracy, where the source passes very close to the central caustic of
a system with a $q=0.006$ mass ratio. Two ($d,1/d$) models provide the same
light curve. This figure also points out the importance of high magnification
events, for which $u_{0}\ll 1$, because when the source passes extremely close
to the central caustic, it probes the effect of the planetary companion on the
caustic shape.

The second known degeneracy occurs when the source passes close to the
planetary caustic. The Chang–Refsdal (Chang & Refsdal 1979) approximation
which is used to describe the planetary caustics (Gaudi and Gould 1997;
Dominik 1998) describes a point-mass lens with uniform shear. This generates a
symmetry along the star-planet axis and the line perpendicular to this axis,
causing identical light-curves for inner-position or outer-position of the
planet, named as the inner-outer degeneracy.

First, Yee et al. (2021) discovered inconsistencies in the formalism of the
close-wide degeneracy suggesting a connection between this two-fold degenerate
solution (close-wide and inner-outer degeneracies). On this thought, Zhang et
al. (2022a) used machine learning techniques to explore in great length these
lensing degeneracies, testing various separations and models in a fast and
rigorous way. They used a Neural Density Estimator (NDE) capable to learn how
to produce posteriors of the lens parameters from a large and complex sample
of microlensing light curves. They trained the network using 691.257 simulated
microlensing events for Roman. The use of NDE network permits them to retrieve
fast and automatically the posterior distributions of a large and rich sample
of binary-lens events. The study of this huge variety of light curves showed
that there is no real caustic degeneracy and that the close-wide and inner-
outer degenerate solutions can be seen as two cases of a more generalised,
unified magnification degeneracy. They call this the offset degeneracy and
they introduce an analytical formalism to demonstrate this magnification
degeneracy (Zhang and Gaudi 2022b). They also revisited 23 published
microlensing events that produced this two-fold degenerate solution and prove
that their unified offset degeneracy is valid and can be applied in all the
events. Finally, Zhang and Gaudi (2022b) use the superposition principle
(Bozza 1999; Han et al. 2001) and deduce a generalised N-body offset
degeneracy of a 2N-1 number of degenerate configurations.

Figure 3: Example of $d\longleftrightarrow d^{-1}$ degeneracy.

A third form of lensing degeneracy, defined as “central caustic cusp approach”
degeneracy (Terry et al. 2022) was detected on the OGLE-2011-BLG-0950 event
(Choi et al. 2012), where the light curve perturbation can be interpreted both
as a planetary or stellar binary anomaly. Terry et al. (2022) conducted a
follow-up analysis of this event using AO images of the target that permitted
them to resolve this sort of degeneracy by constraining the source-lens
relative proper motion and finally, to prove the target to be a stellar binary
system. Breaking this form of degeneracy is quite critical because, contrary
to the previously discussed degeneracies, even a well sampled dataset of the
perturbation will probably not help to determine this double interpretation.

###  3.3 Finite source effects and computation

As previously mentioned, in reality the source is a disk and the point-like
approximation is only valid when the source is far enough from the lens
(projected in the lens plane). This means that when the impact parameter is
very small, $u_{0}\ll 1$, finite source effects have to be taken into account
at the peak of the microlensing event, and require specific numerical
treatments. It is also true when the source approaches or crosses a caustic in
the case of multiple lens systems. In the first case, the corresponding effect
on the light curve peak is very noticeable, with a rounded shape and a damped
magnification. This effect was first analyzed by Gould (1994) and Nemiroff &
Wickramasinghe (1994). They showed that the deviation from the standard PSPL
light curve provides information on the source proper motion, under
assumptions on its brightness and color. Indeed, one can determine
$\theta_{\rm{E}}=\theta_{*}/\rho$, where $\rho$ is the source radius in
Einstein units, and $\theta_{*}$ the physical source size in $\mu$as. The
first parameter is derived from the light curve fit, and the latter from
available information on the source star color (Bensby et al. 2013) and
brightness (Nataf et al. 2013), and empirical color/brightness relations
(Kervella et al. 2004). Thus, the lens-source relative proper motion is
derived by $\mu=\theta_{\rm{E}}/t_{\rm{E}}$.

When the source crosses a caustic, the finite source effects start to be
noticeable as the source projected distance from the caustic is of a few
source radii (Pejcha & Heyrovsky 2009). Fig. 4 gives an illustration of this
effect.

Figure 4: Zoom on microlensing light curves showing a caustic crossing (see
the inset) with different source sizes: $\rho=0.0001$, 0.0002, 0.0003, 0.0004,
0.0008, 0.002. The lens system here has a mass ratio $q=0.013$ and a
separation $d=0.914$.

Similarly, caustic crossings enable us to determine the source radius in
Einstein units, as well as its luminosity profile, i.e. limb-darkening effects
on its disk.

Formally, the source magnification can be calculated by integration over its
entire surface, each point having its own magnification. Concretely, this
approach is hard to implement due to divergences near the caustics. Several
algorithms and techniques have been developed to solve the extended source
case (Dominik 1995; Bennett & Rhie 1996; Wambsganss 1997; Gould & Gaucherel
1997; Gould 2008; Pejcha & Heyrovsky 2009; Bennett 2010a). The most efficient
approach seems to be the one that consists in cutting the light curve in
segments as a function of the source distance to the caustics.

Three different methods are often applied, one when the source is relatively
far from the caustics, and the other for close approaches or crossings of
caustics. The first method is based on an analytical approximation, such as
the ‘hexadecapole’ approximation (Gould 2008; Pejcha & Heyrovsky 2009) that
evaluates the magnification of 13 points well distributed on the surface of
the source.

The second one is a more complex numerical evaluation of the source
magnification that integrates the images in the lens plane. One possibility is
to compute an ‘inverse ray shooting’, i.e. to throw rays from the lens plane
to the source plane, in order to calculate the images/source areas ratio.
Indeed, if computing the source-to-lens path of light rays is difficult, the
lens-to-source path is straightforward. One just needs to apply the equations
of light deflection for a given lens model that is tested, and then count the
amount of rays falling in the source disk. The ratio of images/source areas
gives the total magnification. By Liouville’s theorem, the surface brightness
is conserved, thus the density of light rays in the image plane is uniform.
This method requires a dense sampling of the image plane, a grid search on a
large $(d,q)$ parameter space, and an efficient algorithm to fit the other
parameters $(t_{0},u_{0},t_{\rm{E}},\rho,\alpha)$, where $\alpha$ is the angle
between the source trajectory and the lens system axis. Furthermore, triple
lens modeling is often required and has been implemented in several codes, in
particular those using a faster ray shooting technique, the image centered ray
shooting (Bennett & Rhie 1996; Bennett 2010a).

The third method consists in using Green’s theorem to compute a 1-D mapping of
the image contours, instead of 2-D (Dominik 1995; Gould & Gaucherel 1997;
Dominik 1998). Each point requires to solve a fifth order polynomial, which
makes individual computations time consuming, but this disadvantage is largely
compensated by the faster 1-D integration. However, the task of connecting
points in the lens plane in order to obtain closed contours is sometimes
confusing, in particular when the source passes near a cusp. Such a code has
been released to the public by Valerio Bozza in 2017, whose algorithm has been
developed over the years (Bozza 2010, 2018; Bozza, Khalouei, & Bachelet 2021)
and is publicly available at VBBinaryLensing (2024). The algorithm has
inspired great interest in the microlensing community and has initiated the
development of more open source microlensing light curve modelling software
(Bachelet et al. 2017; Poleski and Yee 2018; Ranc and Cassan 2018), that use
the algorithm as their core.

###  3.4 Microlens parallax, $\pi_{\rm E}$

The parallax effects count between the most remarkable and useful to solve
microlensing problems. They appear as a distorsion in the light curve, due to
a non uniform motion of the observer. For fairly long microlensing events
($t_{\rm{E}}\geq 1{\rm{year}}/2\pi$), the velocity of the Earth can no longer
be considered as constant and rectilinear, and this induces asymmetries in the
light curve called ‘orbital parallax’ (e.g Gould and Loeb 1992; Alcock et al.
1995; Mao 1999; Smith et al. 2002). Parallax effects can also be measured when
the same event is observed from different observatories. Indeed, ‘terrestrial’
parallax can result from simulatenous ground-based observations at different
longitudes, for high-magnification events monitored at very high cadence. And
‘satellite’ parallax can be measured when ground-based observations are
combined with space-based observations.

Detection of such effects yields the Einstein radius determination, projected
in the observer plane,
$\tilde{r_{\rm{E}}}\equiv(2R_{\rm{Sch}}D_{\rm{rel}})^{1/2}$, where
$R_{\rm{Sch}}\equiv 2GMc^{-2}$ is the lens Schwarzschild radius and
$D_{\rm{rel}}^{-1}\equiv D_{L}^{-1}-D_{S}^{-1}$. If in addition the Einstein
angular radius $\theta_{\rm{E}}=(2R_{\rm{Sch}}/D_{\rm{rel}})^{1/2}$ is
measured from finite source effects, then the lens mass can be deduced (Gould
& Loeb 1992)

$M=\frac{c^{2}}{4G}\tilde{r_{\rm{E}}}\theta_{\rm{E}}=0.1227M_{\odot}\left(\frac{\tilde{r_{\rm{E}}}}{1{\rm{AU}}}\right)\left(\frac{\theta_{\rm{E}}}{1{\rm{mas}}}\right).$
(9)

Parallax effects are generally defined by the vector $\mathbf{\pi_{\rm{E}}}$,
called microlens parallax, which magnitude gives the Einstein radius projected
in the observer plane,

$\pi_{\rm{E}}\equiv\frac{\pi_{L}-\pi_{S}}{\theta_{\rm{E}}}:\frac{1{\rm{kpc}}/D_{\rm{rel}}}{\theta_{\rm{E}}/1{\rm{mas}}}=\frac{1{\rm{AU}}}{\tilde{r_{\rm{E}}}},$
(10)

where $\pi_{L}/1{\rm{mas}}=1{\rm{kpc}}/D_{L}$ and
$\pi_{S}/1{\rm{mas}}=1{\rm{kpc}}/D_{S}$. The orientation of
$\mathbf{\pi_{\rm{E}}}$ is that of the lens-source relative proper motion.
Knowing $\theta_{\rm{E}}$ and $\pi_{\rm{E}}$ gives a relation between the lens
and source distances, $D_{L}=(\pi_{\rm{E}}\theta_{\rm{E}}+1/D_{S})^{-1}$.

Measuring the microlensing parallax $\pi_{\rm{E}}$ and the Einstein radius
$\theta_{\rm{E}}$ represents the optimal way to derive physical parameters of
the lens system. As orbital and terrestrial parallaxes are only detectable for
rare high magnification or long microlensing events, the idea of measuring a
satellite parallax has increased importance within the microlensing community,
as was first suggested by Refsdal (1966) and Gould (1999). The first space-
based microlensing observations were done with Spitzer for a Small Magellanic
Cloud event (OGLE-2005-SMC-0001 Dong et al 2007). Such observations were also
conducted for a microlensing event towards the bulge (MOA-2009-BLG-266 Muraki
et al. 2013) using the Deep Impact spacecraft.

Spitzer appeared to be an excellent microlensing parallax satellite being on a
solar orbit at $\sim$ 1 AU from the Earth, and the microlensing community,
under a pilot program led by A. Gould (Gould & Horne 2013a; Gould et al.
2014a, 2015), carried out Spitzer observations in 2014, 2015 and 2016. These
campaigns resulted in a fair amount of highly precise Earth-Spitzer microlens
parallaxes, e.g. the case of an isolated star (Yee et al 2015), a binary star
(Han et al. 2016a), and several exoplanets (e.g Udalski et al. 2015; Street et
al. 2016). However, a recurrent obstacle encountered with Spitzer microlensing
light curves has been the faintness of the observed targets leading to a low
S/N and the fact that only fragments are being observed, with limited baseline
data.

Fig. 5 shows an example of a microlensing event, OGLE-2014-BLG-0124Lb,
observed simultaneously with the OGLE telescope and Spitzer. The planetary
perturbation is due to a $\sim 0.5\,M_{\rm{jup}}$ planet orbiting a
$0.7\,M_{\odot}$ star. The two distinct light curves provided a parallax
measurement, to a precision $\leq 7$ %. Unfortunately, the error on the mass
determination turned out to be much higher (30%) due to poor constraints from
finite source effects (Udalski et al. 2015).

Figure 5: OGLE-2014-BLG-0124Lb: a $\sim 0.5\,M_{\rm{jup}}$ planet orbiting
$\sim 0.7\,M_{\odot}$ star, at $a_{\perp}\sim 3.1$ AU. The planetary anomaly
and the peak of the event have been observed by Spitzer simultaneously to
ground-based observations by OGLE. Figure from Udalski et al. (2015).

Space-based microlensing observations have also been conducted in 2016 with
the Kepler telescope in its K2-C9 campaign (Howell et al. 2014), with the aim
of detecting free-floating planets. These short-term campaigns have provided
important insight to this mass measurement method, which will play a major
role in the context of microlensing dedicated space missions like Roman
(Spergel et al. 2015).

To fully exploit the potential of this effect, further investigations have
been carried out to understand an underlying four-fold discrete degeneracy
associated to the Earth-space microlensing parallax. This degeneracy was
already discussed in the original paper by Refsdal (1966). Since the same
microlensing event is seen from two different observatories, the two observers
obtain distinct light curves with different $t_{0}$ and $u_{0}$, and the
parallax is given by the difference in these quantities (Gould 1994),

${\mathbf{\pi_{\rm{E}}}}=\frac{AU}{{\mathbf{D_{\perp}}}}\left(\frac{\Delta
t_{0}}{t_{\rm{E}}},\Delta u_{0}\right),$ (11)

where $\Delta t_{0}=t_{0,{\rm{sat}}}-t_{0,\oplus}$, $\Delta
u_{0}=u_{0,{\rm{sat}}}-u_{0,\oplus}$, and ${\mathbf{D_{\perp}}}$ is the
physical vector between the two observers projected in the lens plane. The
direction of ${\mathbf{D_{\perp}}}$ gives the one of $\mathbf{\pi_{\rm{E}}}$.
Since $u_{0}$ is a signed quantity which magnitude can be derived from the
single lens light curve, but not the sign, there are actually four solutions
$\Delta u_{0,\pm,\pm}=\pm|u_{0,{\rm{sat}}}|\pm|u_{0,\oplus}|$. Hence, one
degeneracy creates four different orientations of the parallax vector, and a
second, two different magnitudes of this vector. Several attempts have been
made to break these degeneracies under special circumstances (e.g Calchi
Novati et al 2015; Yee et al 2015), but it might actually require a revision
of the parallax formalism itself. Calchi Novati & Scarpetta (2016) revisited
the analysis of the microlensing parallax from an heliocentric reference frame
instead of the commonly adopted geocentric frame, and extended the Gould
(1994) expression of the parallax to observers in motion, as opposed to
observers at rest. They showed that this new approach provides a clearer
understanding of the four-fold degeneracy and interesting leverages to break
it.

In a similar way to parallax effects, if the source is animated by a
significant acceleration and non rectilinear motion during the microlensing
event, in other words if the source is a binary, the light curve can be
affected by additional distorsions. These effects are called ‘xallarap’ by
symmetry with the parallax phenomenon. Unless the source motion coincidentally
mimics the Earth’s motion, these two effects are generally well
distinguishable (Poindexter et al. 2005). Such effects have been detected in
the light curve of OGLE-2007-BLG-368Lb, a cold Neptune around a K-star (Sumi
et al. 2010).

###  3.5 Orbital motion

In the case of multiple lenses, the orbital motion of the lens components can
create deviations in the light curve (Dominik 1998) due to variations in the
shape and orientation of the caustics. These effects are generally negligible
as microlensing events mostly involve companions with long periods. Although a
Keplerian orbit would require five parameters to be characterized,
microlensing light curve perturbations can only constrain two additional
parameters. For a binary system, one reflects the variation of the separation
$d$ between the two bodies, called $s$ here for clarity, and the second being
the variation $\omega$ of the angle $\alpha$ between the source trajectory and
the binary lens axis

$s=s_{0}+{\rm{d}}s/{\rm{d}}t(t-t_{0}),\quad\alpha=\alpha_{0}+\omega(t-t_{0}).$
(12)

Considering $r_{\perp}=sD_{L}\theta_{\rm{E}}$ the projected separation of the
lens system, and evaluate the instantaneous projected velocity of the lens
components as the quadratic sum of $r_{\perp}\gamma_{\perp}=r_{\perp}\omega$
and $r_{\perp}\gamma_{||}=r_{\perp}({\rm{d}}s/{\rm{d}}t)/s$, perpendicular and
parallel to the projected binary axis, respectively. Unfortunately, there is
an important degeneracy in the microlensing models between $\pi_{\rm E,\perp}$
and $\gamma_{\perp}$ (Batista et al. 2011; Skowron et al. 2011), where
$\pi_{\rm E,\perp}$ is the component of $\mathbf{\pi_{\rm E}}$ that is
perpendicular to the instantaneous direction of the Earth’s acceleration.

For bright lenses, radial velocity (RV) measurements can be a good test of
microlensing orbital models, as it was done by Yee et al (2016) for
OGLE-2009-BLG-020, confirming the model published by Skowron et al. (2011).

###  3.6 Mass and distance estimates

The lens mass and distance have been directly determined in some microlensing
events, thanks to microlens parallax measurements combined with Einstein
angular radius and lens magnitude estimates (e.g Bennett et al. 2008a; Gaudi
et al 2008; Batista et al. 2009; Muraki et al. 2013; Kains et al. 2013;
Furusawa et al 2013; Bennett et al. 2016). However, this is not the case for
many microlensing events, and most of the time, mass and distance estimates
require constraints in addition to the light curve planet-host star mass-ratio
and separation.

#### Galactic models

A galactic model should represent the combination of the initial mass
functions, stellar densities and abundances, and proper motion distribution
for all types of stars (Duquennoy & Mayor 1991; Raghavan et al. 2002; Duchene
& Kraus 2013) of the galactic disk and bulge (Han & Gould 1995). Modelling the
stellar properties throughout our galaxy offers a probability distribution of
the microlensing events that can occur in it. This can sometimes lead to
misleading lens parameter estimates, depending on the choice of model and the
simplified features it may contain. Most of the Galactic models used in
microlensing studies were based on (Han & Gould 1995; Han and Gould 2003)
whose consistency with recent ground and space observations toward the bulge
has not been confirmed. Koshimoto et al. (2021) developed Galactic models,
towards the galactic bulge, using constraints from the spatial distributions
of the median velocity and velocity dispersion from Gaia DR2 (Gaia
Collaboration et al. 2018), OGLE-III RC star count data (Nataf et al. 2013),
VIRAC proper motion data (Smith et al. 2018; Clarke et al. 2019), BRAVA radial
velocity data (Rich et al. 2007), and OGLE-IV star count and microlensing
event data (Mróz et al. 2017, 2019b). Models like this can be used for all
types of microlensing events and can provide valid constraints for the
microlens parallax and the distance to source.

#### High angular resolution imaging

Additional observations with high angular resolution imaging are strongly
advocated, even for events where the microlens parallax and/or
$\theta_{\rm{E}}$ were detected. Observations made simultaneously or months
following the peak of the microlensing event derive constraints on the lens
brightness and the baseline and thus on the lens mass with mass-luminosity
relations. It has been applied on many microlensing systems using the VLT
(Janczak et al 2010; Batista et al. 2011; Kubas et al. 2012), Keck (Sumi et
al. 2010; Bennett et al. 2014; Batista et al. 2014, 2015; Koshimoto et al.
2017) and Subaru (Fukui et al 2015; Beaulieu et al. 2016) telescopes equipped
with adaptive optics (AO), or the Hubble Space Telescope (HST) (Dong et al
2009; Bennett et al. 2016). These follow-up observations reduced drastically
the uncertainties on the physical properties of the planetary systems.

It has been shown that (AO) observations made in the decade that follows the
microlensing event are able to constrain the lens mass and distance
measurements with up to ten percent accuracy (Batista et al. 2015; Bennett et
al. 2015). By resolving the source and lens stars, their flux ratio and
relative proper motion can be deduced in addition to mass and distance
constraints (Beaulieu et al. 2018).

Recently a modified version of the imaged centered ray shooting light curve
modeling code of (Bennett & Rhie 1996; Bennett et al. 2010b) has been
released, where they take into account the (AO) flux of the lens and relative
proper motion ensuring that the light curve fit is consistent with these
constraints (Bennett et al. 2023). This allows them to resolve degeneracy of
the microlens parallax or measure it in case of absence (Rektsini et al. 2024;
Terry et al. 2024). The code has been named eesunhong, in honor of the
original coauthor of the code (Bennett and Khavinson 2014; Bennett 2014) and
is now publicly available.

##  4 Detections overview

More than 200 exoplanets have been discovered via microlensing since the first
discovery 20 years ago (Bond et al. 2004). Although this number might seem
relatively modest compared to transit and RV methods, the microlensing
technique explores a population of planets that is hardly accessible, in the
medium term, to other techniques. It is sensitive to a large range of
planetary masses, from gas giants down to Earth and Mars-mass planets, to
wide-orbiting planets and especially planets orbiting beyond or on the snow
line and is not limited by the host-star properties (Gaudi 2012). Furthermore,
in the right conditions, microlensing permits us to observe the influence of
the gravitational field caused by the faintest objects, such as populations of
free-floating planets down to Terrestial-mass (Mróz et al. 2018; Koshimoto et
al. 2023) and compact objects distributed around our galaxy.

The unbiased sample provided by microlensing offers a large range of planetary
systems, located towards the galactic bulge or the galactic disk, with host-
stars varying from main-sequence stars, to stellar remnants. This proves that
this technique is ideal for constraining the demographics of both planetary
and stellar objects throughout our galaxy and for providing important insights
to the planetary formation theories (e.g. Ida & Lin 2004; Mordasini et al.
2009).

###  4.1 Cold super-Earths and Neptunes

Although the microlensing method is, like any other technique, more sensitive
to massive planets, with a sensitivity decreasing with mass as
$\sim\sqrt{m_{p}}$, the third and fourth microlensing detections were super-
Earth and Neptune-like planets, suggesting that these planets are more common
than gas giants. OGLE-2005-BLG-390Lb was the first super-Earth detected via
microlensing (Beaulieu et al. 2006), with light curve shown in Fig. 6. A
Bayesian analysis using a galactic model, combined with a $\theta_{\rm{E}}$
measurement indicated that the planet is likely to be a $\sim 5.5\,M_{\oplus}$
super-Earth orbiting a $M\sim 0.22\,M_{\odot}$ star, at a projected separation
of $\sim 2.6$ AU. The equilibrium temperature of the planet is $\sim 50$ K,
making it the first cold super-Earth discovered at that time.

Figure 6: OGLE-2005-BLG-390Lb light curve (Beaulieu et al. 2006).

This discovery was immediately followed by the detection of a Neptune/Uranus-
like planet, OGLE-2005-BLG-169Lb (Gould et al. 2006), confirming that such low
mass planets should be common. The mass and distance estimates of this system
were confirmed and refined down to a 10% precision ($m_{p}=13.2\pm
1.3\,M_{\oplus}$) with high angular resolution observations, taken 6 and 8
years after the event using HST (Bennett et al. 2015), and the Keck telescope
in Hawaii (Batista et al. 2015), respectively. It is the first case of a
planetary microlensing event for which the source and the lens are resolved,
being separated by $\sim$ 62 mas on the Keck images. It allowed the detection
of the lens brightness and the lens source relative proper motion. The latter
provides an additional mass-distance relation of the lens system.

More than 20 cold super-Earths, Neptunes and very low planets have been
detected until today, a number that has increased significantly with the
beginning of the KMTNet survey and it will rise to thousands after the
launching of Roman (Penny et al. 2019).

###  4.2 Triple lens systems

The first triple lens system discovered by microlensing is the 2-planet 1-star
system, OGLE-2006-BLG-109Lb,c, observed in 2006 (Gaudi et al 2008; Bennett et
al. 2010b). This system presents a remarkable similarity to a scaled version
of Jupiter and Saturn. It is also one of the most complex microlensing light
curves observed to date, exhibiting five distinct features from the two
planets (Fig. 7). Finite source and parallax effects were detected for this
event, as well as orbital motion of the Saturn-like planet. This led to a
complete solution to the lens system and provided the physical characteristics
of the two planets.

Figure 7: OGLE-2006-BLG-109Lb,c: a Jupiter/Saturn analog. Figure from Gaudi et
al (2008).

The next published multiple planet event was OGLE-2012-BLG-0026Lb,c Han et al.
(2013), composed of a Jupiter-like planet and a sub-Saturn orbiting a solar-
mass star. This result was confirmed and refined by Beaulieu et al. (2016) who
detected the flux coming from the lens using the Keck telescope.

OGLE-2007-BLG-349(AB)c is the first circumbinary planet found by microlensing
Bennett et al. (2016). The main features of the light curve are two cusp
approaches that reveal the presence of a companion with a Saturn-to-Sun mass
ratio, but only a third body could explain the slope between the two bumps. At
that point of the analysis, it is not possible to disentangle the scenario of
an additional planet, or an additional star. HST high resolution observations
ruled out the 2-planet scenario because the lens flux could not explain the
brightness of the host star in such a configuration. The system is then likely
to be a Saturn-like planet orbiting two tight M-dwarfs, separated by only
$\sim$0.08 AU, at $\sim$2.6 AU. Fig. 8 compares this system to the known
circumbinary systems, making it the most compact regarding the binary stars,
and the largest regarding the planetary orbit.

Figure 8: OGLE-2007-BLG-349(AB)c: the first circumbinary planet found via
microlensing compared to known circumbinary planet systems. The black dots
show the orbital separation of the stars, and the crosses represent the planet
separation from the stars center of mass (except for the microlensing one that
shows a black dot with error bars). Figure from Bennett et al. (2016).

OGLE-2013-BLG-0341Lb,c is a triple lens system with a very small planetary
signal, and a large signature from the crossing of a resonant caustic due to a
binary star (Gould et al. 2014b). A low amplitude bump in a very early phase
of the event enabled the authors to identify the binary as being in wide-
orbit, separated by $\sim$15 AU. The planet orbits one of them at $\sim$1 AU.
It is also one of the smallest microlensing planets discovered to date ($\sim
1.66\,M_{\oplus}$). A similar system has been published by Poleski et al.
(2014), OGLE-2008-BLG-092, i.e. a wide stellar binary in which the primary
star is orbited by a Uranus-like planet ($\sim 4\,M_{\rm{Uranus}}$).

All these triple lens events (except the last one) exhibited finite source and
parallax effects, allowing to derive precise estimates of their physical
properties from the best models. For the last 5 years KMTNet has been
conducting a project revisiting all microlensing events that contain anomalous
light curve behaviour that could not be explained by the standard 2L1S or 1L2S
models. So far they have discovered six microlensing planets orbiting binary
systems along with other 14 events (Han et al. 2024, and references therein)
that could be explained by 2L2S, 3L1S and a potential five-body model (Han et
al. 2021).

###  4.3 Massive planets around very low mass stars/BDs

Determining the frequency of planets orbiting low mass stars is of interest
because these systems provide important tests to planet formation theories. In
particular, the core accretion theory predicts that gas giants should be less
common around low mass stars (Laughlin et al. 2004; Ida & Lin 2004). Indeed,
in smaller protoplanetary disks, hydrogen and helium might dissipate before
the $\sim 10\,M_{\oplus}$ planet cores grow enough to trigger rapid accretion.
This lack of efficiency in the accretion process would be responsible for a
larger population of failed gas-giants around low mass stars. Numerous
microlensing discoveries seem to confirm these expectations, being cold
10-40$\,M_{\oplus}$ planets around M-dwarfs (e.g Gould et al. 2006; Muraki et
al. 2013; Furusawa et al 2013; Sumi et al. 2016). However, microlensing
discoveries also revealed a population of giant planets $>1M_{J}$ around very
low mass stars ($<0.2\,M_{\odot}$) (e.g Batista et al. 2011; Jung et al 2015;
Han et al. 2016b) and even around brown dwarfs (BDs) (Han et al. 2013; Choi et
al. 2013). For these particular systems, it is still not clear if they have
been formed through gravitational instabilities (Boss 2006) or core accretion.
Moreover, the microlensing BD systems present an unusual morphology compared
to most of those detected with other methods, since they are very tight binary
systems ($<1$ AU versus $\sim$10-45 AU). Here again, it is not clear if they
result from the same formation mechanisms as the wide systems, that are
believed to come from the same process as gravitational fragmentation for
stellar binaries (Lodato et al. 2005).

###  4.4 A jovian analogue orbiting a white dwarf star

One of the characteristics that make the gravitational microlensing unique is
its lack of host-star luminosity dependence, which permits observers to
discover exoplanets orbiting any type of star, even non-luminous objects (eg.
white dwarfs). Blackman et al. (2021) reported the non-detection of a lens
with most probable at least 0.5 solar masses hosting a jovian analogue planet.
This is the case of MOA-2010-BLG-477Lb (Bachelet et al. 2012). AO follow-up
observations were made for this target as part of the NASA Keck Key Strategic
Mission Support (KKSMS) program (Bhattacharya et al. 2018) which is a
pathfinder project for the Nancy Grace Roman Space Telescope.

Both epochs made with the (NIRC2) instrument on the Keck-II telescope showed
no sign of the lens, thus the lens magnitude should be lower than the Keck
detection limit in H-band (H$\sim$21.1) as shown in Fig.9. They estimate that
the system contains a white dwarf or a rare low-mass neutron star of M = 0.53
$\pm$ 0.11 $M_{\odot}$ hosting a Jovian planet with Mp = 1.43 $\pm$ 0.30 MJup
at a distance DL = 1.99 $\pm$ 0.35 kpc. If this non-detection proves to be
valid, this would be a good example of what could be the long future of our
solar system.

Figure 9: H-band brightness of possible main-sequence host lenses. In white is
the Einstein ring radius constraint, $\theta_{E}$, derived from finite source
effects in the event light curve. This constraint indicates that if the planet
host star was main sequence, it should be visible with Keck adaptive optics as
the entire area lies above the detection threshold of the H-band images.
Figure from Blackman et al. (2021).

###  4.5 Images of rotating gravitational arcs with interferometry

A benefit of interferometry method is that it is not constrained by the
luminosity of the microlens or its passage from a caustic. In practice, when a
microlensing event occurs the source image is split into several images
separated by an order of microarcsecons to milliarcseconds, forming the
Einstein ring (in perfect alignment). The resolution of images with such a
small separation is impossible with classical telescopes, but it can be
achieved by long-baseline interferometers. Delplancke et al. (2001) first
showed that the microlensed images can cause fringe visibility patterns in
interferometric observations. Several studies have been made ever since to
better understand this type of observations, the likelyhood of caching an
event and how to combine these observations with the microlensing models and
lens parameters (Dalal and Lane 2003; Rattenbury and Mao 2006). Cassan and
Ranc (2016) established a new method to connect the microlensing images
positions and the fringe visibility patterns observed by long-baseline
interferometers by defining a microlensing interferometric plane. Further
mathematical details regarding this formalism can be found in C. Ranc’s PhD
thesis Ranc (2015).

This study permitted them to build observing strategies that lead to a
successful real-time observation of rotating gravitationally lensed arcs of
the event Gaia19bld (Cassan et al. 2022; Rybicki et al. 2022). The
observations were made with the PIONIER instrument at the Very Large Telescope
Interferometer. They collected five sets of interferometric data in total, one
during the first epoch, before the peak and another two sets for each epoch
after the peak. They used the four 1.8-m Auxiliary Telescopes at medium
baseline configurations before the peak and at large baseline configurations
following the peak. Using SQUEEZE (Baron et al. 2010) to reconstruct the
source images they discovered that Gaia19bld’s arcs were captured in rotation
around the lens (Fig.10), the first image of this phenomenon ever observed.

Each of the three epochs provided independent measurements of
${\it{\theta_{\rm{E}}}}$ and its direction and thus deduced a microlensing
mass, M = 1.147 $\pm$ 0.029 M⊙. This measurement is also in agreement with the
mass derived by Rybicki et al. (2022) where they perform a photometric
analysis of the event using Gaia and Spitzer data.

This remarkable observation, in addition to the interferometric resolved
images of the microlensing event Kojima-1Lb (TCP J05074264+2447555) (Dong et
al. 2019) proves that interferometric microlensing has an important role to
play in exoplanet demographics and the existence of stellar-mass black holes
in our Galaxy.

Figure 10: Gravitationally lensed arcs in rotation around Gaia19bld’s
microlens. From left to right, the red-scale patterns show the model-
independent interferometric reconstructions of the microlensed images for each
of the three epochs. The white dot mark the position of the (unseen)
microlens, and the dotted lines join the centre of the arcs. Figure from
(Cassan et al. 2022).

###  4.6 Stellar remnants

Aside from searching for exoplanets, microlensing is also one of the rare ways
to explore the population of compact objects in our galaxy. Three black hole
candidates had been found in the past, MACHO-98-6, MACHO-96-5, MACHO-99-22,
but they could not be confirmed because there was no measurement of
$\theta_{\rm{E}}$ (Bennett et al. 2002; Poindexter et al. 2005; Mao et al.
2002).

Shvartzvald et al. (2015) analyzed a binary microlensing event,
OGLE-2015-BLG-1285La,b, that was observed simulatenously from space and Earth,
during the 2015 Spitzer campaign. They found that the lens system is likely to
lie in the galactic bulge, with the more massive component being
$>1.35\,M_{\odot}$, either a black hole or a neutron star. The authors
advocate future high resolution observations of this system, in order to
measure the proper motion of the bright companion.

From the OGLE-III database of 150 million objects observed from 2001 to 2009,
Wyrzykowski et al. (2016) published a list of 15 stellar remnants candidates,
combining parallax and brightness measurements from microlensing light curves.
The most massive black hole of their sample is estimated to be
$8.3\,M_{\odot}$ at a distance of 2.4 kpc.

Recently, (Mróz et al. 2024a, b) combined the data from the OGLE-III and OGLE-
IV phases from 2001 to 2020 collecting in total nearly 20 years of OGLE
observations of 62 million stars and searched for microlensing events towards
the LMC. Their goal was to use a long enough dataset that will enable them to
discover massive lenses, such as primordial black holes, that cause magnified
events with long timescales of up to 10 years. They discovered 13 microlensing
events, from which the longest timescale was 165.941${}^{+87.842}_{-49.125}$
days. The absence of light curves with large timelength leads them to the
conclusion that all the events can be described as stellar objects in the
Milky Way disk or inside the LMC and find no evidence of dark matter in the
form of compact objects. The authors advocate future high resolution
observations could constrain the physical parameters of these lenses and
reveal their precise location. In addition, their measurement of the optical
depth towards the LMC is in good agreement with that from Wyrzykowski et al.
(2011) and confirm the low value of the optical depth from EROS (Tisserand et
al. 2007).

In addition to photometric searches, recent projects have also sought to
detect stellar remnants, in particular single black holes and neutron stars
and even probe for the existence of intermediate-mass black holes in globular
clusters (Kains et al. 2016), using astrometric microlensing with GAIA and
HST. Indeed, in addition to the well-known photometric effect, a microlensing
event also leads to a small apparent change in the position of the source (Hog
et al. 1995; Dominik & Sahu 2000). This effect is small, but if detected,
leads to direct constraint on $\theta_{\rm{E}}$, facilitating direct mass
measurements via microlensing.

Wyrzykowski et al. (2023) reported and described the properties of the 363
microlensing events included in the third Gaia Data Release. The majority of
these events is located towards the Galactic bulge, with the density of events
following the stellar density in the surrounding Galactic regions. They also
found a low rate of individual events detected in very high Galactic
longitudes as far as the Galactic anticentre. After cross-matching these
events with previous studies from the OGLE database they confirmed the
detection of 90 new events.

In the future the combination of ground-based and space-based surveys (e.g.,
GAIA, Euclid and Roman) will offer enhanced astrometric precision, allowing us
to constrain the demographics of stellar remnants in the Milky Way.

###  4.7 First confirmation of an isolated stellar-mass black hole

Our galaxy is expected to host around $10^{8}$ isolated black holes but their
observation can’t be made by classical optical methods. Gravitational
microlensing is able to undertake such an observation but its confirmation can
present unfamiliar systematic errors. OGLE-2011-BLG-0462/MOA-2011-BLG-191 is a
magnification event caused by a stellar black hole that proved to be a big
challenge for the microlensing community. In 2022 two independent teams
released two different papers announcing the results of the photometric and
astrometric treatment of the event.

Sahu et al. (2022) used both OGLE and MOA photometric in addition to HST
observations made during 2011-2O17. Lam et al. (2022) used only the OGLE data
and the same astrometric data with additional HST data from 2021. The two
teams realised the data treatment and modelling using different methods and
concluded in two very different mass lens values, M = $7.1\pm 1.3$ M⊙ for Sahu
et al. (2022), indicating a stellar-mass BH and M = 2.15${}^{+0.67}_{-0.54}$
M⊙ for Lam et al. (2022), indicating a neutron star remnant.

Later in the same year Mróz et al. (2022) released a study where they resolve
the systematic errors causing this mass difference and after deducting an
extended data treatment conclude to a mass of M = $7.88\pm 0.82$ M⊙. Finally,
a re-analysis of the event was conducted by Lam and Lu (2023) , using the re-
reducted OGLE data and an updated HST data treatment with additional
observations from 2022 and confirm that the lens is a stellar BH of M =
6${}^{+1.2}_{-1.0}$ M⊙.

##  5 Exoplanet mass-ratio function

More than 5587 planets have been discovered to date, in about 4150 planetary
systems. The vast majority of them were detected by the transit, RV surveys
and direct imaging i.e. detection methods that are most sensitive to close-in
planets. From these surveys, Winn & Fabrycky (2015) estimated that half of the
Sun-like stars should host a small planet (Earth to Neptune) within 1 AU, and
10% of them a giant planet within a few AU. Although only about 200
microlensing planets have been discovered, this technique is complementary to
others because it probes a different parameter space, being sensitive to cold
low-mass planets beyond the snow line, at $\sim$1 to 10 AU from their host
star (Bennett 2008b; Gaudi 2012).

Several statistical microlensing studies have been published, before and after
the survey groups had updated their observational strategy and moved to high-
cadence monitoring with wide field imagers. The first studies were done even
before the first planet was discovered, computing detection efficiency on
microlensing events, to derive upper limits on the frequency of giant cold
planets (Gaudi et al 2002; Snodgrass et al. 2004). Then, from the ten first
planetary detections, Sumi et al. (2010) analyzed the distribution of planet
mass ratios beyond the snow line, and found that it could follow the power law
$dN/d\log q=q^{-0.68\pm 0.20}$, implying that Neptunes would be much more
common than Jupiters.

Gould et al. (2010) used the argument of a non-biased systematic monitoring of
high-magnification events (HME), to derive statistics from a sample of the 13
observed HME at that time, that involved 6 planets. They found a $\sim$36%
absolute frequency of planets, for a mass-ratio centered on $q=5\times
10^{-4}$ and a projected separation beyond the snow line ($\sim 3a_{{\rm
snow}}$).

Cassan et al (2012) published an analysis based on the PLANET follow-up data
during 6 years, and combined their estimate with the previous results. They
found a planet frequency that can be described by the power law $d^{2}N/d\log
M\,d\log a\sim 0.80(M/M_{\oplus})^{-0.65\pm 0.34}$, being consistent with
every star hosting a super-Earth on average.

Clanton & Gaudi (2016) searched for a common power law to connect the results
from five independent surveys, microlensing (Gould et al. 2010; Sumi et al.
2010), RV (Montet et al. 2014) and direct imaging (Lafrenière et al. 2004;
Bowler et al. 2015). They derived a planet distribution of the form
$d^{2}N_{\rm{pl}}/(d\log m_{p}d\log
a)=\mathcal{A}(m_{p}/M_{\rm{Sat}})^{\alpha}(a/2.5\,AU)^{\beta}$, where
$\alpha\sim-0.57$, $\beta\sim 0.49$, $\mathcal{A}\sim 0.26\,{\rm{dex}}^{-2}$,
and with an outer cutoff at $a_{\rm{aout}}\sim 9.7$.

All the statistical results presented above were based on a relatively small
number of microlensing events, and $\leq$10 planets. Within the last decade,
the number of microlensing events has substantially increased due to the new
generation of high-cadence wide-field surveys (OGLE, MOA, KMTNet), and these
planet abundances have been updated recently. Shvartzvald et al. (2016)
estimated the planet mass ratio function from a 4-year data survey of a 8 deg2
galactic field by OGLE, MOA, and Wise. Their results show that 55% of the
stars host at least one planet beyond the snow line, with a mass-ratio
$-4.9<\log(q)<-1.4$, with Neptunes being $\sim$ 10 times more common than
Jupiters.

Suzuki et al. (2016) conducted a statistical analysis over 6 years of the MOA-
II microlensing survey, being the study that covered the biggest amount of
microlensing events (1474) and planetary detections (23) to that time. They
find that the power law describing the planetary mass-ratio function is no
longer a single power law as previoulsy published, but a broken power law of
the form $d^{2}N/(d\log q\,d\log s)=\mathcal{A}(q/1.7\times
10^{-4})^{n}s^{m}{\rm{dex}}^{-2}$, where $\mathcal{A}=0.61^{+0.21}_{-0.16}$,
and $m=0.49^{+0.47}_{-0.49}$. For the low mass-ratio regime, $q<1.7\times
10^{-4}$, $n=0.6^{+0.5}_{-0.4}$, and for the bigger mass ratios, $q>1.7\times
10^{-4}$, $n=-0.93\pm 0.13$. Their distribution thus reaches a peak around the
Neptune mass ratio, as shown in Fig. 11, with a slightly lower abundance of
Earth-like planets than Neptunes, beyond the snow line. They also compared
their microlensing mass ratio function to previous RV results. For low mass
ratios, $q\sim 3\times 10^{-5}$, it is consistent with Mayor et al. (2009) and
Howard et al. (2010), the latter being for period orbits $<$ 50 days. For
bigger mass ratios, $\lesssim 10^{-3}$, their values are higher than Cumming
et al. (2008), and their function rises more steeply towards Neptune mass
ratios. That might indicate that there are more Neptunes beyond the snow line
than at $\sim$1 AU, and that cold Neptunes would not be highly subject to
migration. Recent studies (Ishitani Silva et al. 2022) seem to reinforce the
findings of Suzuki et al. (2016) and will be included in the upcoming
statistical analysis of cold exoplanets detected by the MOA-II survey.

Suzuki et al. (2018) simulated a population of planets around all the events
studied by Suzuki et al. (2016) using the (IL) model (Ida & Lin 2004) and the
Bern model Mordasini et al. (2009). The comparison between the planet-to-star
mass-ratio distribution measured by microlensing and the theoretical
predictions showed a discrepancy of exoplanet population for mass-ratios of
$10^{-4}$ $\leq$ q $\leq$ 4 $\times$ $10^{-4}$. The Suzuki et al. (2016)
observations predict $\sim$ 10 $\times$ more planets in these mass-ratio than
the population synthesis models, with or without migration. However, a more
complete comparison would require the planet and host star masses instead of
the mass ratios, which can be derived by high angular resolution images. As
part of the NASA (KKSMS) program AO follow-up observations with Keck and/or
HST (Bhattacharya et al. 2018) have been conducted for all the Suzuki et al.
(2016) planetary detections. The analysis of the Suzuki et al. (2016) fully
sample will provide a very precise and complete planetary mass-function and
will lead the way for the Roman mass and distance measurement techniques.

Finally, Udalski et al. (2018) conducted an analysis of seven events with mass
ratios q $\lesssim$ $10^{-4}$ from multiple photometric surveys (OGLE, KMTNet,
MiNDSTEp) and they confirm a sharp turnover in the mass-ratio function
relative to that found by Sumi et al. (2010) and Suzuki et al. (2016) but at
higher mass ratios. Jung et al. (2019) using 15 detected planets with KMTNet
deduce a break at qbr $\simeq$ 0.55 $\times$ $10^{-4}$, with an even more
severe power-law slope but argue whether this could be caused by a drop of
detection sensitivity. A new statistical analysis of more than 50 microlens
planets discovered by KMTNet is being conducted, the results of which were not
published yet while this chapter was being finalised. This will be the larger
sampled analysis conducted to date, whose conclusions might challenge all the
previous statistical results presented in this section.

Figure 11: Planetary mass ratio distribution from the 2007-2012 MOA-II
microlensing survey, compared to previous microlensing and RV results (Mayor
et al. 2009; Howard et al. 2010; Cumming et al. 2008; Johnson et al 2010;
Bonfils et al. 2013; Montet et al. 2014). Figure from Suzuki et al. (2016).

##  6 Free-floating planets

From the 2006-2007 MOA-II survey, Sumi et al. (2011) (S11 hereafter)
identified an overabundance of short timescale microlensing events
($t_{\rm{E}}<2$ days) that could not be explained by known stellar and brown
dwarf populations. They interpreted these light curves as signatures of either
free-floating planets (FFP) or wide-orbit Jupiters (separation $\gtrsim 10$ AU
from their host star). Bennett et al. (2012) estimated that if all of this
population was due to planets bound in wide orbits, their semimajor axis would
likely be $>30$ AU.

Clanton & Gaudi (2017) generated a population of distant bound planets with
distributions of mass and distance consistent with microlensing, RV and direct
imaging surveys (Clanton & Gaudi 2016), and estimated which fraction of them
would not show any signature of the host star in a microlensing light curve.
They tried to fit the microlensing event timescale distribution measured by
S11 with a lens mass function composed of brown dwarfs, main-sequence stars,
and remnants, augmented by the distribution of these bound planets with no
star signature. They found that this fraction does not entirely explain the
excess of short timescale events from S11, and concluded that $\sim 60$% of it
should be due to FFP. This would imply that FFP are $\sim 1.3$ times more
abundant than main sequence stars in the galaxy.

Mróz et al. (2017) did a more extensive analysis on the 2010-2015 OGLE-IV
dataset, analysing a sample of 2617 events and found contradictory results to
the S11 Jupiter-mass distribution. Their observed distribution deduces 43 less
events with 0.3d $<$ $t_{E}$ $<$ 1.8d and 36 less events for 0.3d $<$ $t_{E}$
$<$ 1.3d. They did discover 6 potential candidates on shorter $t_{E}$
timescales ($t_{E}$ $<$ 0.5) indicating the existence of Earth-mass unbounded
planets which is much more possible due to dynamical interactions in young
systems (Ma et al. 2016).

The main challenges of detecting FFP candidates is the short timelength of the
event due to the small lens mass and the potential absence of finite source
effects observations. These observations are required in order to measure the
angular Einstein radius $\theta_{E}$, which is extremely crucial in this type
of detections because, in contrast to $t_{E}$, $\theta_{E}$ is independent
from the source-lens relative proper motion, so it immediately offers
information on the mass of the lens.

The first report of finite source effects on FFPs was made by Mróz et al.
(2018), where they used OGLE and KMTNet databases to detect a Neptune-mass
free-floating planet candidate and managed to run successfully a finite-source
point-lens model (FSPL) on it. More FSPL candidates were detected the years
that followed (Mróz et al. 2019a, 2020a, 2020b; Ryu et al. 2021; Kim et al.
2021; Koshimoto et al. 2023), most of them in the regime of Terrestial-Neptune
masses, which confirms the mass-function distribution estimated by Mróz et al.
(2017).

Ryu et al. (2021) and Kim et al. (2021) were the first to study the
distribution of the angular Einstein radius of FFPs using the KMTNet dataset.
They discovered a gap in the distribution for 10$\mu$as $\lesssim$
$\theta_{E}$ $\lesssim$ 30$\mu$as, described as the ”Einstein desert”. This
separation can also be seen in the Mróz et al. (2017) distribution. Gould et
al. (2022) worked in details on explaining and demonstrating in theory the
existence of this desert using the 2016–2019 KMTNet microlensing data set.

Koshimoto et al. (2023) systematically analysed the MOA 2006-2014 database and
selected a statistical sample of 3535 single-lens events, including six
potential FFP candidates with tE $<$ 0.5 days. Out of these six candidates
they found two FSPL candidates with one of them being the second Terrestial
FFP candidate so far. Sumi et al. (2023) used the statistical sample selected
by Koshimoto et al. (2023) and measured the FFP mass-function distribution on
both $t_{E}$ and $\theta_{E}$. They confirm the non-existence of Jupiter
analogues claimed by S11 and find a massive amount of lower mass FFP
candidates with power law dN/dlogM = 2.18 ${}^{+0.52}_{-1.40}$ M⊙.
Furthermore, they compare the FFP mass-function distribution with the mass-
function of bound planets, as shown in Fig. 12 and found that, at Earth-
masses, FFPs are an order of magnitude higher than the bound planets but much
fewer in Jupiter-masses. This implies that low-mass planets are more likely to
be ejected from a planetary system, which confirms the findings of Mróz et al.
(2017) and the predictions from the core accretion theory (Ma et al. 2016).

The three ground-based microlensing surveys have discovered in total 10
microlensing events identified as FFP candidates, with 6 of them described by
Neptune-Earth mass planets. Meanwhile, Sumi et al. (2023) suggest that Roman
will be able to observe 988${}^{+1848}_{-424}$ FFPs down to Mars-mass.
Confirming whether these candidates are free-floating-planets or wide-orbit
planets is the next big challenge. AO follow-up observations have been used to
explore the possibility of a distant host-star existence for some of these
candidates Mróz et al. (2024a) but the limiting magnitude of the observations
doesn’t permit a certain conclusion. Further work and deeper imaging
observations will be needed to confirm all the present and future FFP
candidate detections. Space missions like JWST and Euclid will most probably
play a major role in this endeavor. Bachelet et al. (2022) has suggested that
a joint mission between Euclidd and Roman can increase the number of FFP
detections and provide immediate accurate mass measurements down to Mars-
masses via the microlensing parallax.

Figure 12:  Timescale tE distribution based on the sample selected by
Koshimoto et al. (2023). The blue dotted line represents the known population
of stars, brown dwarfs and stellar remnants, the green dashed line represents
the planetary mass-population and the red line represents the best-fit model
for all the populations. Figure from Sumi et al. (2023).

##  7 Conclusion and future prospects

Gravitational microlensing has proven to be an excellent technique to explore
the demographics of our galaxy. Its independence of the lens brightness allows
it to observe any massive object residing at the darkest corners of the Milky
Way, as long as sufficiently bright background sources are sampled. In the
last two decades microlensing has presented a moderate but extremely diverse
sample of stellar objects. It began with scientific efforts to observe dark
matter in the galactic halo but along the way so much more has beed
discovered. From Terrestial and Neptunian-mass planets to gas giants around
extremely low mass M-dwarf stars and from N-body planetary systems, down to
free-floating planets, to the possible observation of our solar-system’s
destiny and the confirmation of stellar-mass black holes and finally back to
the search for evidence (or non-evidence) of dark matter in form of compact
objects. The discovery of low mass planets beyond the snow line, with unbiased
host-star sensitivity in addition to the numerous unbound planet discoveries,
create complementary constraints to planet formation theories. Meanwhile,
stellar remnant observations can help us understand better the stellar
composition of our galaxy.

Every new discovery forced the microlensing community to exceed previous
limitations and challenge pre-existing methods and modelling techniques. The
need to increase the number of exoplanet detections led to a new generation of
high-cadence wide-field surveys, improved data treatment pipelines and more
telescope collaborations, connecting the whole globe. It also required the
possibility of real-time light-curve modelling to permit predictions of a
possible anomaly appearance revealing the existence of another lens. The
demand for fast and rigorous light-curve fitting triggered the interest of the
community resulting in multiple modelling methods and extended light-curve
studies. The use of second order effects, in particular the microlens parallax
and the finite source effects lead to a breakthrough for confirming exoplanet
detections and measuring their precise physical parameters. Nevertheless, the
limited observed signal quality didn’t always permit to deduce accurate
results and further observations were needed. High angular resolution follow-
up observations deducted in the decade that followed the source-lens alignment
proved a strong constraint for mass and distance measurements. Furthermore,
the use of AI techniques allowed the community to explore a vast number of
complex and diverse light-curves and to understand better lensing degeneracies
and ways to solve them.

Starting from what seemed like an irrational idea of an improbable detection
method, to the first few detections almost a decade later followed by more
than twenty years of dedicated effort and hard work, microlensing is now
heading to a new observational era. Roman is the first space mission with a
dedicated survey to this technique, promising a really bright future for the
microlensing community. One of the main goals of this mission is comprehending
demographics of planets down to Mars mass at separations ranging from the
habitable zone to hundred AU throughout the galaxy with maximum sensitivity to
planets orbiting at roughly the Einstein ring radius. For solar-type stars in
the Galactic disk and source-stars in the Galactic bulge, this radius is
positioned at $\sim$1-10 AU, i.e. close to or beyond the snow line. Moreover,
it will be possible to detect masses lower than Mercury Fig.13, and even down
to the mass of Ganymede at this separation range (Penny et al. 2019). This
means that in 5 years Roman will manage to exceed the low-mass sensitivity
limit of the ground-based surveys while discovering thousand of exoplanets in
a vast variety of orbital separations, something that would require decades of
observations for space-based transit missions. Furthermore, it is predicted to
detect hundreds of free-floating planets (Penny et al. 2019; Johnson et al.
2020; Sumi et al. 2023) adding more constraints to formation theories.

As discussed in previous sections, accurate measurement of the physical
parameters of the planetary system are required for comparing the statistical
results of microlensing surveys with the population synthesis models and
formation theories. Roman will be able to conduct high-angular resolution
follow-up observations regularly on the events previously discovered but it
takes time for the source and lens to be resolved. The Euclid precursor
observations of the Roman fields can guarantee a maximum number of source-lens
resolved images and thus, accurate mass measurements for all these events
(Bachelet et al. 2022, Rektsini et al in prep.). Furthermore, Bachelet et al.
(2022) has shown that a joint mission between Euclid and Roman will ensure an
even higher number of immediate accurate mass measurements for bound and
unbound planets via the microlens parallax. In addition to space-based
surveys, ground-based telescopes (e.g. Rubin Street et al. 2023a) will also
join the ”hunt” with Roman, either as follow-up or conducting previous
observations of the Roman fields or with simultaneous observations for
satellite microlens parallax detections.

Moreover, data collected during the RGBTDS will also be valuable for detecting
transiting exoplanets. It is estimated that the number of planets discovered
via transit method in the Roman fields could reach the 200.000 (Wilson et al.
2023), the majority of them being giant planets in close-orbits but it is
expected to discover a few thousands of smaller planets as well. Finally, this
survey will also provide essential and groundbreaking knowledge regarding the
stellar black hole demographics in our galaxy (Sajadian and Sahu 2023) with
further investigation on the existence of primordial black holes (Fardeen et
al. 2024) using astrometric microlensing and even explore the black hole
populations via astrometric gravitational wave detections (Wang et al. 2022).

Reflecting on all the discoveries of the last three decades and all the
fascinating and somehow curious discoveries to come, this is without a doubt a
very exciting moment for the microlensing and the astrophysical community.

Figure 13: Sensitivity range of the Roman mission in a planet-mass versus
semi-major axis diagram (Penny et al. 2019), complementary to the parameter
space probed by the Kepler mission. Figure credits: This research has made use
of the NASA Exoplanet Archive, which is operated by the California Institute
of Technology, under contract with the National Aeronautics and Space
Administration under the Exoplanet Exploration Program (Akeson et al. 2013)

.

##  8 Cross-References

  * •

Exoplanet Occurrence Rates from Microlensing Surveys

  * •

Microlensing Surveys for Exoplanet Research (OGLE Survey Perspective)

  * •

Microlensing Surveys for Exoplanet Research (MOA)

  * •

Korea Microlensing Telescope Network

##  9 Acronyms

MACHO: MAssive Compact Halo Objects
EROS: Expé rience de Recherche d’Objets Sombres
OGLE: Optical Gravitational Lens Experiment
MOA: Microlensing Observations in Astrophysics
PLANET: Probing Lensing Anomalies NETwork
$\mu$FUN: $\mu$lensing Follow-up Network
MiNDSTEp: Mincrolensing Network for the Detection of Small Terrestrial
Exoplanets
KMTNet: Korean Microlensing Telescope Network
UTGO: University of Tasmania Greenhill Observatory
LCO: Las Cumbres Observatory
RGBTDS: Roman Galactic Bulge Time Domain Survey
ZTF: Zwicky Transient Facility
GAIA: Global Astrometric Interferometer for Astrophysics
AO: Adaptive Optics

###### Acknowledgements.

NER would like to thank Naoki Koshimoto, Etienne Bachelet, Jean-Philippe
Beaulieu and P. Scott Gaudi for their comments as well as Andrew Cole and
Thomas Plunkett for proof-reading this manuscript. V.B. would like to thank
Clément Ranc for valuable discussions and Noé Kains for an input on black
holes.
Work by NER was supported by the University of Tasmania through the UTAS
Foundation and the endowed Warren Chair in Astronomy and the ANR COLD-WORLDS
(ANR-18-CE31-0002). This research was also supported by the Australian
Government through the Australian Research Council Discovery Program (project
number 200101909) grant awarded to Andrew Cole and Jean-Philippe Beaulieu.

## References

  * Abrams et al. (2023) Abrams, N. S. , Hundertmark, M. P. G., Khakpash, S. et al. 2023. Microlensing Discovery and Characterization Efficiency in the Vera C. Rubin Legacy Survey of Space and Time. arXiv e-prints. doi:10.48550/arXiv.2309.15310
  * Albrow et al. (1998) Albrow, M., Beaulieu, J.-P., Birch, P., et al. 1998. The 1995 Pilot Campaign of PLANET: Searching for Microlensing Anomalies through Precise, Rapid, Round-the-Clock Monitoring. The Astrophysical Journal 509, 687–702. doi:10.1086/306513
  * Albrow et al. (1999) Albrow, M. D., Beaulieu, J.-P., Caldwell, J. A. R. et al. (1999) A Complete Set of Solutions for Caustic Crossing Binary Microlensing Events. ApJ, 522, 1022
  * Alcock et al. (1995) Alcock, C., Allsman, R. A., Alves, D. et al. (1995). ApJ, 454, L125+
  * Alcock et al. (2000) Alcock, C., Allsman, R. A., Alves, D. R., et al. 2000. The MACHO Project: Microlensing Results from 5.7 Years of Large Magellanic Cloud Observations. The Astrophysical Journal 542, 281–307. doi:10.1086/309512
  * Akeson et al. (2013) Akeson, R. L. , Chen, X., Ciardi, D., et al. 2013. The NASA Exoplanet Archive: Data and Tools for Exoplanet Research. Publications of the Astronomical Society of the Pacific 125, 989. doi:10.1086/672273
  * An (2005) An, J. H (1999) The binary gravitational lens and its extreme cases. A&A, 349, 108
  * Baron et al. (2010) Baron, F., Monnier, J. D., Kloppenborg, B. 2010. A novel image reconstruction software for optical/infrared interferometry. Optical and Infrared Interferometry II 7734. doi:10.1117/12.857364
  * Bachelet et al. (2012) Bachelet, E. , Shin, I.-G., Han, C., et al. 2012. MOA 2010-BLG-477Lb: Constraining the Mass of a Microlensing Planet from Microlensing Parallax, Orbital Motion, and Detection of Blended Light. The Astrophysical Journal 754. doi:10.1088/0004-637X/754/1/73
  * Bachelet et al. (2017) Bachelet, E., Norbury, M., Bozza, V., Street, R. 2017. pyLIMA: An Open-source Package for Microlensing Modeling. I. Presentation of the Software and Analysis of Single-lens Models. The Astronomical Journal 154. doi:10.3847/1538-3881/aa911c
  * Bachelet and Penny (2019) Bachelet, E., Penny, M. 2019. WFIRST and EUCLID: Enabling the Microlensing Parallax Measurement from Space. The Astrophysical Journal 880. doi:10.3847/2041-8213/ab2da5
  * Bachelet et al. (2022) Bachelet, E., Specht, D., Penny, M., et al. 2022. Euclid-Roman joint microlensing survey: Early mass measurement, free floating planets, and exomoons. Astronomy and Astrophysics 664. doi:10.1051/0004-6361/202140351
  * Batista et al. (2009) Batista, V., Dong, S., Gould, A. et al. (2009) Mass measurement of a single unseen star and planetary detection efficiency for OGLE 2007-BLG-050. A&A, 508, 467
  * Batista et al. (2011) Batista, V., Gould, A., Dieters, S. et al. (2011) MOA-2009-BLG-387Lb: a massive planet orbiting an M dwarf. A&A, 529, 102
  * Batista et al. (2014) Batista, V., Beaulieu, J.-P., Gould, A. et al. (2014) MOA-2011-BLG-293Lb: First Microlensing Planet Possibly in the Habitable Zone. ApJ, 780, 54
  * Batista et al. (2015) Batista, V., Beaulieu, J.-P., Bennett, D. P. et al. (2015) Confirmation of the OGLE-2005-BLG-169 Planet Signature and Its Characteristics with Lens-Source Proper Motion Detection. ApJ, 808, 170
  * Beaulieu et al. (2006) Beaulieu, J.-P., Bennett, D. P., Fouqué, P. et al. (2006) Discovery of a cool planet of 5.5 Earth masses through gravitational microlensing. Nature, 439, 437
  * Beaulieu et al. (2008) Beaulieu, J.-P., Kerins, E., Mao, S. et al. (2008) Towards A Census of Earth-mass Exo-planets with Gravitational Microlensing. arXiv:0808.0005
  * Beaulieu et al. (2016) Beaulieu, J.-P., Bennett, D. P., Batista, V. et al. (2016) Revisiting the Microlensing Event OGLE 2012-BLG-0026: A Solar Mass Star with Two Cold Giant Planets. ApJ, 824, 83
  * Beaulieu et al. (2018) Beaulieu, J.-P. , J.-P., Batista, V., Bennett, D. P., et al. 2018. Combining Spitzer Parallax and Keck II Adaptive Optics Imaging to Measure the Mass of a Solar-like Star Orbited by a Cold Gaseous Planet Discovered by Microlensing. The Astronomical Journal 155. doi:10.3847/1538-3881/aaa293
  * Bennett (2008b) Bennett, D. P. (2008) Detection of Extrasolar Planets by Gravitational Microlensing. Exoplanets, Springer Praxis Books, ISBN 978-3-540-74007-0. Praxis Publishing Ltd, Chichester, UK, 2008, p.47
  * Bennett (2010a) Bennett, D. P. (2010) An Efficient Method for Modeling High-magnification Planetary Microlensing Events, ApJ, 716, 1408
  * Bennett & Rhie (1996) Bennett, D. P. & Rhie, S. H. (1996) Detecting Earth-Mass Planets with Gravitational Microlensing. ApJ, 472, 660
  * Bennett & Rhie (2002) Bennett, D. P. & Rhie, S. H. (2002) Simulation of a Space-based Microlensing Survey for Terrestrial Extrasolar Planets. ApJ, 574, 985
  * Bennett et al. (2002) Bennett, D. P., Becker, A. C., Calitz, J. J. et al. (2002) The Microlensing Event MACHO-99-BLG-22/OGLE-1999-BUL-32: An Intermediate Mass Black Hole, or a Lens in the Bulge. astro.ph, 7006
  * Bennett et al. (2006) Bennett, D. P., Anderson, J., Gaudi, B.S. et al. (2006) Characterization of Gravitational Microlensing Planetary Host Stars. DPS meeting 38, id.45.14; Bulletin of the American Astronomical Society, Vol.38
  * Bennett et al. (2008a) Bennett, D. P., Bond, I. A., Udalski, A. et al. (2008) A Low-Mass Planet with a Possible Sub-Stellar-Mass Host in Microlensing Event MOA-2007-BLG-192. ApJ, 684, 663
  * Bennett et al. (2010b) Bennett, D. P., Rhie, S. H, Nikolaev, S. et al. (2010) Masses and Orbital Constraints for the OGLE-2006-BLG-109Lb,c Jupiter/Saturn Analog Planetary System. ApJ, 713, 837
  * Bennett et al. (2012) Bennett, D. P., Sumi, T., Bond, I. et al. (2012) Planetary and Other Short Binary Microlensing Events from the MOA Short-event Analysis. ApJ, 757, 119
  * Bennett and Khavinson (2014) Bennett, D. P., Khavinson, D. 2014. Sun Hong Rhie. Physics Today 67, 64–64. doi:10.1063/PT.3.2318
  * Bennett (2014) Bennett, D. P. 2014. Obituary: Sun Hong Rhie (1955-2013). Bulletin of the American Astronomical Society 46.
  * Bennett et al. (2014) Bennett, D. P., Batista, V., Bond, I. et al. (2014) MOA-2011-BLG-262Lb: A Sub-Earth-Mass Moon Orbiting a Gas Giant Primary or a High Velocity Planetary System in the Galactic Bulge. ApJ, 785, 155
  * Bennett et al. (2015) Bennett, D. P., Bhattacharya, A., Anderson, J. et al. (2015) Confirmation of the Planetary Microlensing Signal and Star and Planet Mass Determinations for Event OGLE-2005-BLG-169. ApJ, 808, 169
  * Bennett et al. (2016) Bennett, D. P., Rhie, S. H., Udalski, A. et al. (2016) The First Circumbinary Planet Found by Microlensing: OGLE-2007-BLG-349L(AB)c. AJ, 152, 125
  * Bennett et al. (2023) Bennett, D. P., Bhattacharya, A., Beaulieu, J.-P., et al. 2023. Keck and Hubble Observations Show That MOA-2008-BLG-379Lb Is a Super-Jupiter Orbiting an M Dwarf. arXiv e-prints. doi:10.48550/arXiv.2311.00627
  * Bensby et al. (2013) Bensby, T., Yee, Y. C., Feltzing, S. et al. (2013) Chemical evolution of the Galactic bulge as traced by microlensed dwarf and subgiant stars. V. Evidence for a wide age distribution and a complex MDF. A&A, 549, 147
  * Bhattacharya et al. (2018) Bhattacharya, A., Beaulieu, J.-P., Bennett, D. P. et al. 2018. WFIRST Exoplanet Mass-measurement Method Finds a Planetary Mass of 39 $\pm$ 8 M ⊕ for OGLE-2012-BLG-0950Lb. The Astronomical Journal 156. doi:10.3847/1538-3881/aaed46
  * Blackman et al. (2021) Blackman, J. W., Beaulieu, J. P., Bennett, D. P. et al. 2021. A Jovian analogue orbiting a white dwarf star. Nature 598, 272–275. doi:10.1038/s41586-021-03869-6
  * Bond et al. (2001) Bond, I. A., Abe, F., Dodd, R. J. et al. (2001) Real-time difference imaging analysis of MOA Galactic bulge observations during 2000. MNRAS, 327, 868
  * Bond et al. (2004) Bond, I. A., Udalski, A., Jaroszynski, M. et al. (2004) OGLE 2003-BLG-235/MOA 2003-BLG-53: A Planetary Microlensing Event. ApJ, 606, 155
  * Bonfils et al. (2013) Bonfils, X. Delfosse, X., Udry, S. et al. (2013) The HARPS search for southern extra-solar planets. XXXI. The M-dwarf sample. A&A, 549, 109
  * Borucki et al. (2010) Borucki, W. J., Koch, D., Basri, G. et al. (2013) Kepler Planet-Detection Mission: Introduction and First Results. Science, 327, 977
  * Boss (2006) Boss, A. (2006) Rapid Formation of Gas Giant Planets around M Dwarf Stars. ApJ, 643, 501
  * Bowler et al. (2015) Bowler, B. P., Liu, M. C., Shkolnik, E. et al. (2015) Planets around Low-mass Stars (PALMS). IV. The Outer Architecture of M Dwarf Planetary Systems. ApJS, 216, 7
  * Bozza (1999) Bozza, V. 1999. Perturbative analysis in planetary gravitational lensing. Astronomy and Astrophysics 348, 311–326. doi:10.48550/arXiv.astro-ph/9904297
  * Bozza (2010) Bozza, V. (2010) Microlensing with an advanced contour integration algorithm: Green’s theorem to third order, error control, optimal sampling and limb darkening. MNRAS, 408, 2188
  * Bozza (2018) Bozza V., Bachelet E., Bartolić F., Heintz T. M., Hoag A. R., Hundertmark M. (2018) VBBINARYLENSING: a public package for microlensing light-curve computation. MNRAS, 479, 5157. doi:10.1093/mnras/sty1791
  * Bozza, Khalouei, & Bachelet (2021) Bozza V., Khalouei E., Bachelet E. (2021) A public code for astrometric microlensing with contour integration. MNRAS, 505, 126. doi:10.1093/mnras/stab1376
  * Calchi Novati et al (2015) Calchi Novati, S., Gould, A., Udalski, A. et al. (2015) Pathway to the Galactic Distribution of Planets: Combined Spitzer and Ground-Based Microlens Parallax Measurements of 21 Single-Lens Events. ApJ, 804, 20
  * Calchi Novati & Scarpetta (2016) Calchi Novati, S. & Scarpetta, G. (2016) Microlensing Parallax for Observers in Heliocentric Motion. ApJ, 824, 109
  * Cassan et al (2012) Cassan, A., Kubas, D., Beaulieu, J.-P. et al. (2012) One or more bound planets per Milky Way star from microlensing observations. Nature, 481, 167
  * Cassan and Ranc (2016) Cassan, A., Ranc, C. 2016. Interferometric observation of microlensing events. Monthly Notices of the Royal Astronomical Society 458, 2074–2079. doi:10.1093/mnras/stw372
  * Cassan et al. (2022) Cassan, A., Ranc, C., Absil, O. et al. 2022. Microlensing mass measurement from images of rotating gravitational arcs. Nature Astronomy 6, 121–128. doi:10.1038/s41550-021-01514-w
  * Chang & Refsdal (1979) Chang, K. & Refsdal, S. (1979) Flux variations of QSO 0957+561 A, B and image splitting by stars near the light path. Nature, 282, 561
  * Choi et al. (2012) Choi, J.-Y. , Shin, I.-G., Han, C., et al. 2012. A New Type of Ambiguity in the Planet and Binary Interpretations of Central Perturbations of High-magnification Gravitational Microlensing Events. The Astrophysical Journal 756. doi:10.1088/0004-637X/756/1/48
  * Choi et al. (2013) Choi, J.-Y., Han, C., Udalski, A. et al. (2013) Microlensing Discovery of a Population of Very Tight, Very Low Mass Binary Brown Dwarfs. ApJ, 768, 129
  * Clanton & Gaudi (2016) Clanton, C. & Gaudi, B. S. (2016) Synthesizing Exoplanet Demographics: A Single Population of Long-period Planetary Companions to M Dwarfs Consistent with Microlensing, Radial Velocity, and Direct Imaging Surveys. ApJ, 819, 125
  * Clanton & Gaudi (2017) Clanton, C. & Gaudi, B. S. (2017) Constraining the Frequency of Free-floating Planets from a Synthesis of Microlensing, Radial Velocity, and Direct Imaging Survey Results. ApJ, 834, 46
  * Clarke et al. (2019) Clarke, J. P., Wegg, C., Gerhard, O., Smith, L. C., Lucas, P. W., Wylie, S. M. 2019. The Milky Way bar/bulge in proper motions: a 3D view from VIRAC and Gaia. Monthly Notices of the Royal Astronomical Society 489, 3519–3538. doi:10.1093/mnras/stz2382
  * Cumming et al. (2008) Cumming, A., Butler, R. P., Marcy, G. W. et al. (2008) The Keck Planet Search: Detectability and the Minimum Mass and Orbital Period Distribution of Extrasolar Planets. PASP, 120, 531
  * Dalal and Lane (2003) Dalal, N., Lane, B. F. 2003. Bringing Closure to Microlensing Mass Measurement. The Astrophysical Journal 589, 199–209. doi:10.1086/374549
  * Delplancke et al. (2001) Delplancke, F., Górski, K. M., Richichi, A. 2001. Resolving gravitational microlensing events with long-baseline optical interferometry. Prospects for the ESO Very Large Telescope Interferometer. Astronomy and Astrophysics 375, 701–710. doi:10.1051/0004-6361:20010783
  * Dominik (1995) Dominik, M. (1995) Improved Routines for the Inversion of the Gravitational Lens Equation for a Set of Source Points. A&A, 109, 597
  * Dominik (1998) Dominik, M. (1998) A robust and efficient method for calculating the magnification of extended sources caused by gravitational lenses. A&A, 333, 79
  * Dominik (1999) Dominik, M. (1999) The binary gravitational lens and its extreme cases. A&A, 349, 108
  * Dominik & Sahu (2000) Dominik, M. & Sahu, K. C.(2000) Astrometric Microlensing of Stars. ApJ, 534, 213
  * Dominik et al. (2010) Dominik, M., Jørgensen, U. G., Rattenbury, N. J. et al. 2010. Realisation of a fully-deterministic microlensing observing strategy for inferring planet populations. Astronomische Nachrichten 331, 671. doi:10.1002/asna.201011400
  * Dong et al (2006) Dong, S., DePoy, D. L., Gaudi, B. S. et al. (2006) Planetary Detection Efficiency of the Magnification 3000 Microlensing Event OGLE-2004-BLG-343. ApJ, 642, 842
  * Dong et al (2007) Dong, S., Udalski, A, Gould, A. et al. (2007) First Space-Based Microlens Parallax Measurement: Spitzer Observations of OGLE-2005-SMC-001. ApJ, 664, 862
  * Dong et al (2009) Dong, S., Gould, A., Udalski, A, et al. (2009) OGLE-2005-BLG-071Lb, the Most Massive M Dwarf Planetary Companion? ApJ, 695,
  * Dong et al. (2019) Dong, S., Mérand, A., Delplancke-Ströbele, F. et al. 2019 First Resolution of Microlensed Images. The Astrophysical Journal 871. doi:10.3847/1538-4357/aaeffb
  * Duchene & Kraus (2013) Duchene, G. & Kraus, A.(2013) Stellar Multiplicity. A&A, 51, 269
  * Duquennoy & Mayor (1991) Duquennoy, A. & Mayor, M.(1991) Multiplicity among solar-type stars in the solar neighbourhood. II - Distribution of the orbital elements in an unbiased sample. A&A, 248, 485
  * Einstein (1936) Einstein A.(1936) Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field. Science, 84, 506
  * Fardeen et al. (2024) Fardeen, J., McGill, P., Perkins, S. E. et al. 2024. Astrometric Microlensing by Primordial Black Holes with the Roman Space Telescope. The Astrophysical Journal 965. doi:10.3847/1538-4357/ad3243
  * Fukui et al (2015) Fukui, A., Gould, A., Sumi, T. et al. (2015) OGLE-2012-BLG-0563Lb: A Saturn-mass Planet around an M Dwarf with the Mass Constrained by Subaru AO Imaging. ApJ, 809, 74
  * Furusawa et al (2013) Furusawa, K., Udalski, A., Sumi, T. et al. (2013) MOA-2010-BLG-328Lb: A Sub-Neptune Orbiting very Late M Dwarf? ApJ, 779, 91
  * Gaia Collaboration et al. (2016) Gaia Collaboration , Prusti, T., de Bruijne, J. H. J. et al. 2016. The Gaia mission. Astronomy and Astrophysics 595. doi:10.1051/0004-6361/201629272
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Katz, D., Antoja, T., et al. 2018. Gaia Data Release 2. Mapping the Milky Way disc kinematics. Astronomy and Astrophysics 616. doi:10.1051/0004-6361/201832865
  * Gaudi and Gould (1997) Gaudi, B. S., Gould, A. 1997. Planet Parameters in Microlensing Events. The Astrophysical Journal 486, 85–99. doi:10.1086/304491
  * Gaudi (2012) Gaudi, B. S. (2012) Microlensing Surveys for Exoplanets. ARA&A, 50, 411
  * Gaudi et al (2002) Gaudi, B. S., Albrow, M. D., An, J. et al. (2002) Microlensing Constraints on the Frequency of Jupiter-Mass Companions: Analysis of 5 Years of PLANET Photometry. ApJ, 566, 463
  * Gaudi et al (2008) Gaudi, B. S., Bennett, D. P., Udalski, A. et al. (2008) Discovery of a Jupiter/Saturn Analog with Gravitational Microlensing. Science, 319, 927
  * Gorbikov et al. (2010) Gorbikov, E., Brosch, N., and Afonso, C. (2010) A two-color CCD survey of the North Celestial Cap: I. The method. Ap&SS, 326, 203
  * Gould (1992) Gould, A. (1992) Extending the MACHO search to about 10 exp 6 solar masses. ApJ, 392, 442
  * Gould (1994) Gould, A. (1994) MACHO velocities from satellite-based parallaxes. ApJ, 421, 75
  * Gould (1999) Gould, A. (1999) Microlens Parallaxes with SIRTF. ApJ, 514, 869
  * Gould (2008) Gould, A. (2008) Hexadecapole Approximation in Planetary Microlensing. ApJ, 681, 1593
  * Gould and Loeb (1992) Gould, A. and Loeb, A. (1992) Discovering planetary systems through gravitational microlenses. ApJ, 396, 104
  * Gould & Gaucherel (1997) Gould, A. & Gaucherel, C. (1997) Stokes’s Theorem Applied to Microlensing of Finite Sources. ApJ, 477, 580
  * Gould & Horne (2013a) Gould, A. & Horne, K. (2013) Kepler-like Multi-plexing for Mass Production of Microlens Parallaxes. ApJ, 779, 28
  * Gould et al. (2010) Gould, A., Dong, S., Gaudi, B. S. (2010) Frequency of Solar-like Systems and of Ice and Gas Giants Beyond the Snow Line from High-magnification Microlensing Events in 2005-2008. ApJ, 720, 1073
  * Gould et al. (2006) Gould, A., Udalski, A., An, D. et al. (2006) Microlens OGLE-2005-BLG-169 Implies That Cool Neptune-like Planets Are Common. ApJ, 644, 37
  * Gould et al. (2010) Gould, A., Dong, S., Gaudi, B. S., et al. 2010. Frequency of Solar-like Systems and of Ice and Gas Giants Beyond the Snow Line from High-magnification Microlensing Events in 2005-2008. The Astrophysical Journal 720, 1073–1089. doi:10.1088/0004-637X/720/2/1073
  * Gould et al. (2013b) Gould, A., Carey, S., & Yee, J. (2013) Spitzer Microlens Planets and Parallaxes. sptz.prop10036G
  * Gould et al. (2014a) Gould, A., Carey, S., & Yee, J. (2014) Galactic Distribution of Planets from Spitzer Microlens Parallaxes. sptz.prop11006G
  * Gould et al. (2014b) Gould, A., Udalski, A., Shin, I.-G. et al. (2014) A terrestrial planet in a $\sim$1-AU orbit around one member of a $\sim$15-AU binary. Science, 345, 46
  * Gould et al. (2015) Gould, A., Yee, J., & Carey, S. (2015) Galactic Distribution of Planets From High-Magnification Microlensing Events. sptz.prop12013G
  * Gould et al. (2022) Gould, A., Jung, Y. K., Hwang, K.-H., et al. 2022. Free-Floating Planets, the Einstein Desert, and ’OUMUAMUA. Journal of Korean Astronomical Society 55, 173–194. doi:10.5303/JKAS.2022.55.5.173
  * Griest & Safizadeh (1998) Griest, K. & Safizadeh, N. (1998) The Use of High-Magnification Microlensing Events in Discovering Extrasolar Planets. ApJ, 500, 37
  * Han et al. (2001) Han, C., Chang, H.-Y., An, J. H., Chang, K. 2001. Properties of microlensing light curve anomalies induced by multiple planets. Monthly Notices of the Royal Astronomical Society 328, 986–992. doi:10.1046/j.1365-8711.2001.04973.x
  * Han & Gould (1995) Han, C. & Gould, A.(1995) Statistics of Microlensing Optical Depth. ApJ, 449, 521
  * Han & Gould (2003) Han, C. & Gould, A.(2003) Stellar Contribution to the Galactic Bulge Microlensing Optical Depth. ApJ, 592, 172
  * Han and Gould (2003) Han, C., Gould, A. 2003. Stellar Contribution to the Galactic Bulge Microlensing Optical Depth. The Astrophysical Journal 592, 172–175. doi:10.1086/375706
  * Han et al. (2013) Han, C., Jung, Y. K., Udalski, A. et al. (2016) Microlensing Discovery of a Tight, Low-mass-ratio Planetary-mass Object around an Old Field Brown Dwarf. ApJ, 778, 38
  * Han et al. (2016a) Han, C., Udalski, A., Gould, A. et al. (2016) OGLE-2015-BLG-0479LA,B: Binary Gravitational Microlens Characterized by Simultaneous Ground-based and Space-based Observations. ApJ, 828, 53
  * Han et al. (2016b) Han, C., Udalski, A., Gould, A. et al. (2016) OGLE-2015-BLG-0051/KMT-2015-BLG-0048Lb: A Giant Planet Orbiting a Low-mass Bulge Star Discovered by High-cadence Microlensing Surveys. AJ, 152, 95
  * Han et al. (2021) Han, C. , Udalski, A., Kim, D., et al. 2021. KMT-2019-BLG-1715: Planetary Microlensing Event with Three Lens Masses and Two Source Stars. The Astronomical Journal 161. doi:10.3847/1538-3881/abf4d0
  * Han et al. (2024) Han, C., Udalski, A., Jung, Y. K., et al. 2024. OGLE-2023-BLG-0836L: The sixth microlensing planet in a binary stellar system. arXiv e-prints. doi:10.48550/arXiv.2402.08116
  * Henderson et al. (2014) Henderson, C. B., Park, H., Sumi, T. et al. (2014) Candidate Gravitational Microlensing Events for Future Direct Lens Imaging. ApJ, 794, 71
  * Henderson et al. (2016) Henderson, C. B., Poleski, R., Penny, M. et al. (2016) Campaign 9 of the K2 Mission: Observational Parameters, Scientific Drivers, and Community Involvement for a Simultaneous Space- and Ground-based Microlensing Survey. PASP, 12814401
  * Hog et al. (1995) Hog, E., Novikov, I. D., Polnarev, A. G. et al. (1995) MACHO photometry and astrometry. A&A, 294, 287
  * Howard et al. (2010) Howard, A. W., Marcy, G. W., Johnson, J. A. et al. (2010) The Occurrence and Mass Distribution of Close-in Super-Earths, Neptunes, and Jupiters. Science, 330, 653
  * Howell et al. (2014) Howell, S. B., Sobeck, C., Haas, M. et al. (2014) The K2 Mission: Characterization and Early Results. PASP, 126, 398
  * Ida & Lin (2004) Ida, S. & Lin, D. N. C. (2004) Toward a Deterministic Model of Planetary Formation. I. A Desert in the Mass and Semimajor Axis Distributions of Extrasolar Planets. ApJ, 604, 388
  * Ishitani Silva et al. (2022) Ishitani Silva, S., Ranc, C., Bennett, D. P., et al. 2022 MOA-2020-BLG-135Lb: A New Neptune-class Planet for the Extended MOA-II Exoplanet Microlens Statistical Analysis. The Astronomical Journal 164. doi:10.3847/1538-3881/ac82b8
  * Janczak et al (2010) Janczak, J., Fukui, A., Dong, S. et al. (2010) Sub-Saturn Planet MOA-2008-BLG-310Lb: Likely to be in the Galactic Bulge. ApJ, 711, 731
  * Johnson et al (2010) Johnson, J. A., Aller, K. M., Howard, A. W. et al. (2010) Giant Planet Occurrence in the Stellar Mass-Metallicity Plane. PASP, 122, 905
  * Jung et al (2015) Jung, Y. K., Udalski, A., Sumi, T. et al. (2015) OGLE-2013-BLG-0102LA,B: Microlensing Binary with Components at Star/Brown Dwarf and Brown Dwarf/Planet Boundaries. ApJ, 798, 123
  * Kaib et al. (2013) Kaib, N. A., Raymond, S. N. and Duncan, M. (2013) Planetary system disruption by Galactic perturbations to wide binary stars. Nature, 493, 381
  * Johnson et al. (2020) Johnson, S. A. , Penny, M., Gaudi, B. S., et al. 2020. Predictions of the Nancy Grace Roman Space Telescope Galactic Exoplanet Survey. II. Free-floating Planet Detection Rates. The Astronomical Journal 160. doi:10.3847/1538-3881/aba75b
  * Jung et al. (2019) Jung, Y. K. , Gould, A., Zang, W., et al. 2019. KMT-2017-BLG-0165Lb: A Super-Neptune-mass Planet Orbiting a Sun-like Host Star. The Astronomical Journal 157. doi:10.3847/1538-3881/aaf87f
  * Kains et al. (2016) Kains, N., Bramich, D. M., Sahu, K. C. et al. (2016) Searching for intermediate-mass black holes in globular clusters with gravitational microlensing. MNRAS, 460, 2025
  * Kains et al. (2013) Kains, N.,Street, R. A., Choi, J.-Y. et al. (2013) A giant planet beyond the snow line in microlensing event OGLE-2011-BLG-0251. A&A, 552, 70
  * Kervella et al. (2004) Kervella, P., Thévenin, F., Di Folco, E. (2004) The angular sizes of dwarf stars and subgiants. Surface brightness relations calibrated by interferometry. A&A, 426, 297
  * Kim et al. (2016) Kim, S.-L., Lee, C.-U., Park, B.-G et al. (2016) KMTNET: A Network of 1.6 m Wide-Field Optical Telescopes Installed at Three Southern Observatories. JKAS, 49, 37
  * Kim et al. (2021) Kim, H.-W., Hwang, K.-H., Gould, A., et al. 2021. KMT-2019-BLG-2073: Fourth Free-floating Planet Candidate with $\theta$E ¡ 10 $\mu$as. The Astronomical Journal 162. doi:10.3847/1538-3881/abfc4a
  * Kondo et al. (2023) Kondo, I., Sumi, T., Koshimoto, N., Rattenbury, N. J., Suzuki, D., Bennett, D. P. 2023. Prediction of Planet Yields by the PRime-focus Infrared Microlensing Experiment Microlensing Survey. The Astronomical Journal 165. doi:10.3847/1538-3881/acccf9
  * Koshimoto et al. (2017) Koshimoto, N., Udalski, A., Beaulieu, J.-P. et al. (2016) OGLE-2012-BLG-0950Lb: The First Planet Mass Measurement from Only Microlens Parallax and Lens Flux. AJ, 153, 1
  * Koshimoto et al. (2021) Koshimoto, N., Baba, J., Bennett, D. P. 2021. A Parametric Galactic Model toward the Galactic Bulge Based on Gaia and Microlensing Data. The Astrophysical Journal 917. doi:10.3847/1538-4357/ac07a8
  * Koshimoto et al. (2023) Koshimoto, N., Sumi, T., Bennett, D. P. et al. 2023. Terrestrial- and Neptune-mass Free-Floating Planet Candidates from the MOA-II 9 yr Galactic Bulge Survey. The Astronomical Journal 166. doi:10.3847/1538-3881/ace689
  * Kubas et al. (2012) Kubas, D., Beaulieu, J.-P., Bennett, D. P. et al. (2012) A frozen super-Earth orbiting a star at the bottom of the main sequence. A&A, 540, 78
  * Lafrenière et al. (2004) Lafrenière, D., Doyon, R., Marois, C. et al. (2007) The Gemini Deep Planet Survey. ApJ, 670, 1367
  * Lam et al. (2022) Lam, C. Y. , Lu, J. R., Udalski, A., et al. 2022. An Isolated Mass-gap Black Hole or Neutron Star Detected with Astrometric Microlensing. The Astrophysical Journal 933. doi:10.3847/2041-8213/ac7442
  * Lam and Lu (2023) Lam, C. Y., Lu, J. R. 2023. A Reanalysis of the Isolated Black Hole Candidate OGLE-2011-BLG-0462/MOA-2011-BLG-191. The Astrophysical Journal 955. doi:10.3847/1538-4357/aced4a
  * Laughlin et al. (2004) Laughlin, G., Bodenheimer, P. & Adams, F. C. (2004) The Core Accretion Model Predicts Few Jovian-Mass Planets Orbiting Red Dwarfs. ApJ, 612, 73
  * Lodato et al. (2005) Lodato, G., Delgado-Donate, E. & Clarke, C. J. (2005) Constraints on the formation mechanism of the planetary mass companion of 2MASS 1207334-393254. MNRAS, 364, 91
  * Ma et al. (2016) Ma, S., Mao, S., Ida, S., Zhu, W., Lin, D. N. C. 2016. Free-floating planets from core accretion theory: microlensing predictions. Monthly Notices of the Royal Astronomical Society 461, L107–L111. doi:10.1093/mnrasl/slw110
  * Mao & Paczynski (1991) Mao, S. & Paczynski, B. (1991) Gravitational microlensing by double stars and planetary systems. ApJ, 374, 37
  * Mao (1999) Mao, S. (1999) An ongoing parallax microlensing event OGLE-1999-CAR-1 toward Carina. A&A, 350, L19
  * Mao et al. (2002) Mao, S., Smith, M. C., Wazniak, P. et al. (2002) Optical Gravitational Lensing Experiment OGLE-1999-BUL-32: the longest ever microlensing event - evidence for a stellar mass black hole? MNRAS, 329, 349
  * Mayor et al. (2009) Mayor, M., Bonfils, X., Forveille, T. et al. (2009) The HARPS search for southern extra-solar planets. XVIII. An Earth-mass planet in the GJ 581 planetary system. A&A, 507, 487
  * Montet et al. (2014) Montet, B. T., Crepp, J. R., Johnson, J. A. (2014) The TRENDS High-contrast Imaging Survey. IV. The Occurrence Rate of Giant Planets around M Dwarfs. ApJ, 781, 28
  * Mordasini et al. (2009) Mordasini, C., Alibert, Y., Benz, W. 2009. Extrasolar planet population synthesis. I. Method, formation tracks, and mass-distance distribution. Astronomy and Astrophysics 501, 1139–1160. doi:10.1051/0004-6361/200810301
  * Mróz et al. (2017) Mróz, P., Udalski, A., Skowron, J., et al. 2017. No large population of unbound or wide-orbit Jupiter-mass planets. Nature 548, 183–186. doi:10.1038/nature23276
  * Mróz et al. (2018) Mróz, P. , Ryu, Y.-H., Skowron, J. et al. 2018. A Neptune-mass Free-floating Planet Candidate Discovered by Microlensing Surveys. The Astronomical Journal 155. doi:10.3847/1538-3881/aaaae9
  * Mróz et al. (2019a) Mróz, P., Udalski, A., Bennett, D. P. et al. 2019. Two new free-floating or wide-orbit planets from microlensing. Astronomy and Astrophysics 622. doi:10.1051/0004-6361/201834557
  * Mróz et al. (2019b) Mróz, P., Udalski, A., Skowron, J., et al. 2019. Microlensing Optical Depth and Event Rate toward the Galactic Bulge from 8 yr of OGLE-IV Observations. The Astrophysical Journal Supplement Series 244. doi:10.3847/1538-4365/ab426b
  * Mróz et al. (2020a) Mróz, P. , Poleski, R., Han, C. et al. 2020. A Free-floating or Wide-orbit Planet in the Microlensing Event OGLE-2019-BLG-0551. The Astronomical Journal 159. doi:10.3847/1538-3881/ab8aeb
  * Mróz et al. (2020b) Mróz, P. , Poleski, R., Gould, A., et al. 2020. A Terrestrial-mass Rogue Planet Candidate Detected in the Shortest-timescale Microlensing Event. The Astrophysical Journal 903. doi:10.3847/2041-8213/abbfad
  * Mróz et al. (2022) Mróz, P., Udalski, A., Gould, A. 2022. Systematic Errors as a Source of Mass Discrepancy in Black Hole Microlensing Event OGLE-2011-BLG-0462. The Astrophysical Journal 937. doi:10.3847/2041-8213/ac90bb
  * Mróz et al. (2024a) Mróz, P., Ban, M., Marty, P., Poleski, R. 2024. Free-floating or Wide-orbit? Keck Adaptive-optics Observations of Free-floating Planet Candidates Detected with Gravitational Microlensing. The Astronomical Journal 167. doi:10.3847/1538-3881/ad1106
  * Mróz et al. (2024c) Mroz, P., Udalski, A., Szymanski, M. K. et al. 2024. Microlensing optical depth and event rate toward the Large Magellanic Cloud based on 20 years of OGLE observations. arXiv e-prints. doi:10.48550/arXiv.2403.02398
  * Mróz et al. (2024b) Mroz, P. , Udalski, A., Szymanski, M. K. et al. 2024. No massive black holes in the Milky Way halo. arXiv e-prints. doi:10.48550/arXiv.2403.02386
  * Muraki et al. (2013) Muraki, Y., Han, C., Bennett, D. P. et al. (2011) Discovery and Mass Measurements of a Cold, 10 Earth Mass Planet and Its Host Star. ApJ, 741, 22
  * Nataf et al. (2013) Nataf, D. M., Gould, A., Fouqué, P. et al. (2013) Reddening and Extinction toward the Galactic Bulge from OGLE-III: The Inner Milky Way’s Rv 2.5 Extinction Curve. ApJ, 769, 88
  * Nemiroff & Wickramasinghe (1994) Nemiroff, R. J. & Wickramasinghe (1994) Finite source sizes and the information content of macho-type lens search light curves. ApJ, 424, 21
  * Paczynski (1986) Paczynski, B. (1986) Gravitational microlensing by the galactic halo. ApJ, 304, 1
  * Pejcha & Heyrovsky (2009) Pejcha, O. & Heyrovsky, D. (2009) Extended-Source Effect and Chromaticity in Two-Point-Mass Microlensing. ApJ, 690, 1772
  * Penny et al. (2013) Penny, M. T., Kerins, E., Rattenbury, N. et al. (2013) ExELS: an exoplanet legacy science proposal for the ESA Euclid mission - I. Cold exoplanets. MNRAS, 434, 2
  * Penny et al. (2016a) Penny, M. T., Henderson, C. B. & Clanton, C. (2016) Is the Galactic Bulge Devoid of Planets? ApJ, 830, 150
  * Penny et al. (2016) Penny, M. T., Rattenbury, N. J., Gaudi, B. S. et al. (2016) Predictions for the Detection and Characterization of a Population of Free-Floating Planets with K2 Campaign 9. arXiv:1605.01059
  * Penny et al. (2019) Penny, M. T., Gaudi, B. S., Kerins, E. et al. 2019. Predictions of the WFIRST Microlensing Survey. I. Bound Planet Detection Rates. The Astrophysical Journal Supplement Series 241. doi:10.3847/1538-4365/aafb69
  * Poindexter et al. (2005) Poindexter, S., Afonso, C., Bennett, D. P. et al. (2005) Systematic Analysis of 22 Microlensing Parallax Candidates. ApJ, 633, 914
  * Poleski et al. (2014) Poleski, R., Skowron, J., Udalski, A. et al. (2014) Triple Microlens OGLE-2008-BLG-092L: Binary Stellar System with a Circumprimary Uranus-type Planet. ApJ, 795, 42
  * Poleski and Yee (2018) Poleski, R., Yee, J. 2018. MulensModel: Microlensing light curves modeling. Astrophysics Source Code Library. ascl:1803.006
  * Ranc (2015) Ranc, C. 2015. Exoplanets and brown dwarfs detections through gravitational microlensing. Study of interferometric observations.. Ph.D. Thesis.
  * Ranc and Cassan (2018) Ranc, C., Cassan, A. 2018. muLAn: gravitational MICROlensing Analysis Software. Astrophysics Source Code Library. ascl:1811.012
  * Rattenbury and Mao (2006) Rattenbury, N. J., Mao, S. 2006. Interferometric visibility and closure phase of microlensing events with finite source size. Monthly Notices of the Royal Astronomical Society 365, 792–800. doi:10.1111/j.1365-2966.2005.09769.x
  * Rattenbury et al. (2002) Rattenbury, N. J., Bond, I. A., Skuljan, J. et al. (2002) Planetary microlensing at high magnification. MNRAS, 335, 159
  * Raghavan et al. (2002) Raghavan, D., McAlister, H. A., Henry, T. J et al. (2002) A Survey of Stellar Families: Multiplicity of Solar-type Stars. ApJ, 190, 1
  * Refsdal (1964) Refsdal, S.(1964) The gravitational lens effect. MNRAS, 128, 295
  * Refsdal (1966) Refsdal, S.(1966) On the possibility of determining the distances and masses of stars from the gravitational lens effect. MNRAS, 134, 315
  * Rektsini et al. (2024) Rektsini, N. E., Batista, V., Ranc, C. et al. 2024. Precise Mass Measurement of OGLE-2013-BLG-0132/MOA-2013-BLG-148: A Saturn-mass Planet Orbiting an M Dwarf. The Astronomical Journal 167. doi:10.3847/1538-3881/ad2514
  * Rhie (1997) Rhie, S. H. (1997) Infimum Microlensing Amplification of the Maximum Number of Images of n-Point Lens Systems. ApJ, 484, 63
  * Rhie (2002) Rhie, S. H. (2002) How Cumbersome is a Tenth Order Polynomial?: The Case of Gravitational Triple Lens Equation. astroph, 2294R
  * Rhie et al. (2000) Rhie, S. H., Bennett, D. P., Becker, A. C. et al. (2000) On Planetary Companions to the MACHO 98-BLG-35 Microlens Star. ApJ, 533, 378
  * Rich et al. (2007) Rich, R. M., Reitzel, D. B., Howard, C. D., Zhao, H. 2007. The Bulge Radial Velocity Assay: Techniques and a Rotation Curve. The Astrophysical Journal 658, L29–L32. doi:10.1086/513509
  * Rybicki et al. (2022) Rybicki, K. A., Wyrzykowski, Ł., Bachelet, E., et al. 2022. Single-lens mass measurement in the high-magnification microlensing event Gaia19bld located in the Galactic disc. Astronomy and Astrophysics 657. doi:10.1051/0004-6361/202039542
  * Ryu et al. (2021) Ryu, Y.-H., Mróz, P., Gould, A., et al. 2021. KMT-2017-BLG-2820 and the Nature of the Free-floating Planet Population. The Astronomical Journal 161. doi:10.3847/1538-3881/abd55f
  * Sahu et al. (2022) Sahu, K. C. , Anderson, J., Casertano, S., et al. 2022. An Isolated Stellar-mass Black Hole Detected through Astrometric Microlensing. The Astrophysical Journal 933. doi:10.3847/1538-4357/ac739e
  * Sajadian and Sahu (2023) Sajadian, S., Sahu, K. C. 2023. Detecting Isolated Stellar-mass Black Holes with the Roman Telescope. The Astronomical Journal 165. doi:10.3847/1538-3881/acb20f
  * Sako et al. (2008) Sako, T., Sekiguchi, T., Sasaki, M. et al. (2008) MOA-cam3: a wide-field mosaic CCD camera for a gravitational microlensing survey in New Zealand. ExA, 22, 51
  * Schneider and Weiss (1986) Schneider, D. P. and Weiss, A. (1986) The two-point-mass lens - Detailed investigation of a special asymmetric gravitational lens. A&A, 164, 237
  * Shvartzvald et al. (2015) Shvartzvald, Y., Udalski, A., Gould, A. et al. (2015) Spitzer Microlens Measurement of a Massive Remnant in a Well-separated Binary. ApJ, 814, 111
  * Shvartzvald et al. (2016) Shvartzvald, Y., Maoz, D., Udalski, A. et al. (2016) The frequency of snowline-region planets from four years of OGLE-MOA-Wise second-generation microlensing. MNRAS, 457, 4089
  * Skowron et al. (2011) Skowron, J, Udalski, A., Gould, A. et al. (2011) Binary Microlensing Event OGLE-2009-BLG-020 Gives Verifiable Mass, Distance, and Orbit Predictions. ApJ, 738, 87
  * Smith et al. (2002) Smith, C., Rest, A., Hiriart, R. et al. (2002) in Presented at the Society of Photo-Optical Instrumentation Engineers (SPIE) Conference, Vol.4836, ed. J.Ã. Tyson & Woll, 395-405
  * Smith et al. (2018) Smith, L. C., Lucas, P. W., Kurtev, R., et al. 2018. VIRAC: the VVV Infrared Astrometric Catalogue. Monthly Notices of the Royal Astronomical Society 474, 1826–1849. doi:10.1093/mnras/stx2789
  * Snodgrass et al. (2004) Snodgrass, C., Horne, K. & Tsapras, Y. (2004) The abundance of Galactic planets from OGLE-III 2002 microlensing data. MNRAS, 351, 967
  * Spergel et al. (2015) Spergel, D., Gehrels, N., Baltay, C. et al. (2015) Wide-Field InfrarRed Survey Telescope-Astrophysics Focused Telescope Assets WFIRST-AFTA 2015 Report. eprint arXiv:1503.03757
  * Street et al. (2016) Street, R. A., Udalski, A., Calchi Novati, S. et al. (2015) Spitzer Parallax of OGLE-2015-BLG-0966: A Cold Neptune in the Galactic Disk. ApJ, 819, 93
  * Street et al. (2023b) Street, R. A., Li, X., Khakpash, S., et al. 2023. LSST Survey Strategy in the Galactic Plane and Magellanic Clouds. The Astrophysical Journal Supplement Series 267. doi:10.3847/1538-4365/acd6f4
  * Street et al. (2023a) Street, R. A. , Gough-Kelly, S., Lam, C. et al. 2023. Maximizing science return by coordinating the survey strategies of Roman with Rubin, and other major facilities. arXiv e-prints. doi:10.48550/arXiv.2306.13792
  * Sumi et al. (2010) Sumi, T., Bennett, D. P., Bond, I. A. et al. (2010) A Cold Neptune-Mass Planet OGLE-2007-BLG-368Lb: Cold Neptunes Are Common. ApJ, 710, 1641
  * Sumi et al. (2011) Sumi, T., Kamiya, K., Bennett, D. P. et al. (2011) Unbound or distant planetary mass population detected by gravitational microlensing. Nature, 473, 349
  * Sumi et al. (2013) Sumi, T., Bennett, D. P., Bond, I. A. et al. (2013) The Microlensing Event Rate and Optical Depth toward the Galactic Bulge from MOA-II. ApJ, 778, 150
  * Sumi et al. (2016) Sumi, T., Udalski, A., Bennett, D. P. et al. (2016) The First Neptune Analog or Super-Earth with a Neptune-like Orbit: MOA-2013-BLG-605Lb. ApJ, 825, 112
  * Sumi et al. (2023) Sumi, T. , Koshimoto, N., Bennett, D. P., et al. 2023. Free-floating Planet Mass Function from MOA-II 9 yr Survey toward the Galactic Bulge. The Astronomical Journal 166. doi:10.3847/1538-3881/ace688
  * Suzuki et al. (2016) Suzuki, D., Bennett, D. P., Sumi, T. et al. (2016) The Exoplanet Mass-ratio Function from the MOA-II Survey: Discovery of a Break and Likely Peak at a Neptune Mass. ApJ, 833, 145
  * Suzuki et al. (2018) Suzuki, D. , Bennett, D. P., Ida, S. et al. 2018. Microlensing Results Challenge the Core Accretion Runaway Growth Scenario for Gas Giants. The Astrophysical Journal 869. doi:10.3847/2041-8213/aaf577
  * Terry et al. (2022) Terry, S. K. , Bennett, D. P., Bhattacharya, A., et al. 2022. Adaptive Optics Imaging Can Break the Central Caustic Cusp Approach Degeneracy in High-magnification Microlensing Events. The Astronomical Journal 164. doi:10.3847/1538-3881/ac9518
  * Terry et al. (2024) Terry, S. K. Beaulieu, J.-P., Bennett, D. P. et al. 2024. Unveiling MOA-2007-BLG-192: An M Dwarf Hosting a Likely Super-Earth. arXiv e-prints. doi:10.48550/arXiv.2403.12118
  * Thompson (2013) Thompson, T. A. (2013) Gas giants in hot water: inhibiting giant planet formation and planet habitability in dense star clusters through cosmic time. MNRAS, 431, 63
  * Tisserand et al. (2007) Tisserand, P. , Le Guillou, L., Afonso, C., et al. 2007. Limits on the Macho content of the Galactic Halo from the EROS-2 Survey of the Magellanic Clouds. Astronomy and Astrophysics 469, 387–404. doi:10.1051/0004-6361:20066017
  * Tsapras et al. (2009) Tsapras, Y. , Street, R., Horne, K., et al. 2009. RoboNet-II: Follow-up observations of microlensing events with a robotic network of telescopes. Astronomische Nachrichten 330, 4. doi:10.1002/asna.200811130
  * Udalski (2003) Udalski, A. (2003) The Optical Gravitational Lensing Experiment. Real Time Data Analysis Systems in the OGLE-III Survey. Acta Astron., 53, 291
  * Udalski et al. (2015) Udalski, A., Yee, J. C., Gould, A. et al. (2015) Spitzer as a Microlens Parallax Satellite: Mass Measurement for the OGLE-2014-BLG-0124L Planet and its Host Star. ApJ, 799, 237
  * Udalski et al. (2018) Udalski, A. , Ryu, Y.-H., Sajadian, S. et al. 2018. OGLE-2017-BLG-1434Lb: Eighth q¡1×10-4 Mass-Ratio Microlens Planet Confirms Turnover in Planet Mass-Ratio Function. Acta Astronomica 68, 1–42. doi:10.32023/0001-5237/68.1.1
  * Vandorou et al. (2023) Vandorou, A. , Dang, L., Bennett, D. P., et al. 2023. OGLE-2016-BLG-1195Lb: A Sub-Neptune Beyond the Snow Line of an M-dwarf Confirmed by Keck AO. arXiv e-prints. doi:10.48550/arXiv.2302.01168
  * Vermaak (2000) Vermaak, P. (2000) The effects of resolved sources and blending on the detection of planets via gravitational microlensing, MNRAS, 319, 1011
  * Wambsganss (1997) Wambsganss, J. (1997) Discovering Galactic planets by gravitational microlensing: magnification patterns and light curves. MNRAS, 284, 172
  * Wang & Fischer (2015) Wang, J. & Fischer, D. A. (2015) Revealing a Universal Planet-Metallicity Correlation for Planets of Different Sizes Around Solar-type Stars. AJ, 149, 14
  * Werner et al. (2004) Werner, M. W., Roellig, T. L., Low, F. J. et al. (2004) The Spitzer Space Telescope Mission. ApJ, 154, 1
  * Winn & Fabrycky (2015) Winn, J. N. & Fabrycky, D. C. (2015) The Occurrence and Architecture of Exoplanetary Systems. ARA&A, 53, 409
  * Wang et al. (2022) Wang, Y., Pardo, K., Chang, T.-C., Doré, O. 2022. Constraining the stochastic gravitational wave background with photometric surveys. Physical Review D 106. doi:10.1103/PhysRevD.106.084006
  * Witt (1990) Witt, H. J. (1990) in Lecture Notes in Physics, Berlin Springer Verlag, Vol. 360, Gravitational Lensing, ed. Y. Mellier, B. Fort & G. Saucail, 192-+
  * Witt & Mao (1995) Witt, H. J. & Mao, S. (1995) On the Minimum Magnification Between Caustic Crossings for Microlensing by Binary and Multiple Stars. ApJ, 447, L105+
  * Wilson et al. (2023) Wilson, R. F., Barclay, T., Powell, B. P. et al. 2023. Transiting Exoplanet Yields for the Roman Galactic Bulge Time Domain Survey Predicted from Pixel-level Simulations. The Astrophysical Journal Supplement Series 269. doi:10.3847/1538-4365/acf3df
  * Wyrzykowski et al. (2011) Wyrzykowski, Ł., Kozłowski, S., Skowron, J. et al. 2011. The OGLE view of microlensing towards the Magellanic Clouds - III. Ruling out subsolar MACHOs with the OGLE-III LMC data. Monthly Notices of the Royal Astronomical Society 413, 493–508. doi:10.1111/j.1365-2966.2010.18150.x
  * Wyrzykowski et al. (2016) Wyrzykowski, L., Kostrzewa-Rutkowska, Z., Skowron, J. et al. (2016) Black hole, neutron star and white dwarf candidates from microlensing with OGLE-III. MNRAS, 458, 3012