###### Abstract

After exactly a century since the formulation of the general theory of
relativity, the phenomenon of gravitational lensing is still an extremely
powerful method for investigating in astrophysics and cosmology. Indeed, it is
adopted to study the distribution of the stellar component in the Milky Way,
to study dark matter and dark energy on very large scales and even to discover
exoplanets. Moreover, thanks to technological developments, it will allow the
measure of the physical parameters (mass, angular momentum and electric
charge) of supermassive black holes in the center of ours and nearby galaxies.

###### keywords:

gravitational lensing; keyword; keyword

x 10.3390/—— xx Academic Editor: Lorenzo Iorio Received: 5 January 2016;
Accepted: 7 March 2016; Published: date The Scales of Gravitational Lensing
Francesco De Paolis 1,2,†,*, Mosè Giordano 1,2,†, Gabriele Ingrosso 1,2,†,
Luigi Manni 1,2,†, Achille Nucita 1,2,† and Francesco Strafella 1,2,†
Correspondence<EMAIL_ADDRESS>Tel.: +39-0832-297493; Fax:
+39-0832-297505

##  1 Introduction

In 1911, while he was still involved in the development of the general theory
of relativity (subsequently published in 1916), Einstein made the first
calculation of light deflection by the Sun einstein1911 . He correctly
understood that a massive body may act as a gravitational lens deflecting
light rays passing close to the body surface. However, his calculation, based
on Newtonian mechanics, gave a deflection angle wrong by a factor of two. On
14 October 1913, Einstein wrote to Hale, the renowned astronomer, inquiring
whether it was possible to measure a deflection angle of about
$0.84^{\prime\prime}$ toward the Sun. The answer was negative, but Einstein
did not give up, and when, in 1915, he made the calculation again using the
general theory of relativity, he found the right value $\phi=2r_{s}/b$ (where
$r_{s}=2GM/c^{2}$ is the Schwarzschild radius and $b$ is the light rays’
impact parameter) that corresponds to an angle of about $1.75^{\prime\prime}$
in the case of the Sun. That result was resoundingly confirmed during the
Solar eclipse of 1919 dyson1919 .

In 1924, Chwolson chwolson1924  considered the particular case when the
source, the lens and the observer are aligned and noticed the possibility of
observing a luminous ring when a far source undergoes the lensing effect by a
massive star. In 1936, after the insistence of Rudi Mandl, Einstein published
a paper on science einstein1936  describing the gravitational lensing effect
of one star on another, the formation of the luminous ring, today called the
Einstein ring, and giving the expression for the source amplification.
However, Einstein considered this effect exceedingly curious and useless,
since in his opinion, there was no hope to actually observe it.

On this issue, however, Einstein was wrong: he underestimated technological
progress and did not foresee the motivations that today induce one to widely
use the gravitational lensing phenomenon. Indeed, Zwicky promptly understood
that galaxies were gravitational lenses more powerful than stars and might
give rise to images with a detectable angular separation. In two letters
published in 1937 zwicky1937a ; zwicky1937b , Zwicky noticed that the
observation of galaxy lensing, in addition to giving a further proof of the
general theory of relativity, might allow observing sources otherwise
invisible, thanks to the light gravitational amplification, thereby obtaining
a more direct and accurate estimate of the lens galaxy dynamical mass. He also
found that the probability to observe lensed galaxies was much larger than
that of star on star. This shows the foresight of this eclectic scientist,
since the first strong lensing event was discovered only in 1979: the double
quasar QSO 0957+561 a/b qso0957+561 , shortly followed by the observation of
tens of other gravitational lenses, Einstein rings and gravitational arcs. All
of that plays today an extremely relevant role for the comprehension of the
evolution of the structures and the measure of the parameters of the so-called
cosmological standard model.

Actually, there are different scales in gravitational lensing, on which we
shall briefly concentrate in the next sections, after a short introduction to
the basics of the theory of gravitational lensing (Section 2). Generally
speaking, gravitational lens images separated by more than a few tenths of
arcsecs are clearly seen as distinct images by the observer. This was the case
considered by Zwicky, and the gravitational lensing in this regime is called
strong (or macro) lensing (see Section 3), which also includes distorted
galaxy images, like Einstein rings or arcs. If instead, the distortions
induced by the gravitational fields on background objects are much smaller, we
have the weak lensing effect (see Section 4). On the other side, if one
considers the phenomenology of the star-on-star lensing (as Einstein did), the
resulting angular distance between the images is of the order of a few
$\mu$as, generally not separable by telescopes. Gravitational lensing in this
regime is called, following Paczyǹski pacz1986 , microlensing, and the
observable is an achromatic change in the brightness of the source star over
time, due to the relative motion of the lens and the source with respect to
the line of sight of the source (see Section 5). In all of these regimes, the
gravitational field can be treated in the weak field approximation. Another
scale on which gravitational lensing applies is that involving black holes. In
particular, when light rays come very close to the event horizon, they are
subject to strong gravitational field effects, and thereby, the deflection
angles are large. This effect is called retro-lensing, and we shall discuss it
in Section 6. The observation of retro-lensing events is of great importance
also because the general theory of relativity still stands practically
untested in the strong gravitational field regime (see johannsen  for a very
recent review), apart from gravitational waves gwdetection . A short final
discussion is then offered in Section 7.

##  2 Basics of Gravitational Lensing

In the general theory of relativity, light rays follow null geodesics, _i.e._
, the minimum distance paths in a curved space-time. Therefore, when a light
ray from a far source interacts with the gravitational field due to a massive
body, it is bent by an angle approximately equal to $\alpha_{S}(b)=2r_{s}/b$.
By looking at Figure 1, assuming the ideal case of a thin lens and noting that
$\alpha_{S}D_{LS}=(\theta-\theta_{S})D_{S}$, one can easily derive the so-
called lens equation:

$\theta_{S}=\theta-\frac{\theta_{E}^{2}}{\theta},$ (1)

where $\theta_{S}$ indicates the source position and:

$\theta_{E}=\left(\frac{4GM}{c^{2}}\frac{D_{LS}}{D_{S}D_{L}}\right)^{1/2}$ (2)

is the Einstein ring radius, which is the angular radius of the image when
lens and source are perfectly aligned ($\theta_{S}=0$). Therefore, one can see
that two images appear in the source plane, whose positions can be obtained by
solving Equation (1).

More generally (see for details SchneiderEhlersFalco ), the light deflection
between the two-dimensional position of the source $\boldsymbol{\theta}_{S}$
and the position of the image $\boldsymbol{\theta}$ is given by the lens
mapping equation:

$\boldsymbol{\theta}_{S}=\boldsymbol{\theta}-\boldsymbol{\nabla}\phi(\boldsymbol{\theta}),$
(3)

where $\phi=2D_{LS}\Phi_{N}^{2D}/(D_{S}c^{2})$ is the so-called lensing
potential and $\Phi_{N}^{2D}$ is the two-dimensional Newtonian projected
gravitational potential of the lens. We also note, in turn, that the ratio
$D_{LS}/{D_{S}}$ depends on the redshift of the source and the lens, as well
as on the cosmological parameters $\Omega_{M}=\rho_{M}/\rho_{c}$ and
$\Omega_{\Lambda}=\rho_{\Lambda}/\rho_{c}$, being $\rho_{c}=3H_{0}^{2}/(8\pi
G)$, $\rho_{M}$ and $\rho_{\Lambda}$ the critical, the matter and the dark
energy densities, respectively. The transformation above is thus a mapping
from the source plane to the image plane, and the Jacobian ${\cal J}$ of the
transformation is given by:

${\cal J}=\frac{d\boldsymbol{\theta}_{S}}{d\boldsymbol{\theta}}={\cal
A}^{-1}=\left(\begin{array}[]{cc}1-\phi_{,11}&-\phi_{,12}\\\
-\phi_{,12}&1-\phi_{,22}\end{array}\right)=\left(\begin{array}[]{cc}1-\kappa-\gamma_{1}&-\gamma_{2}\\\
-\gamma_{2}&1-\kappa+\gamma_{1}\end{array}\right),$ (4)

where the commas are the partial derivatives with respect to the two
components of $\boldsymbol{\theta}$. Here, $\kappa$ is the convergence, which
turns out to be equal to $\Sigma/2\Sigma_{\rm cr}$, where:

$\Sigma_{\rm cr}=c^{2}D_{S}/(4\pi GD_{L}D_{LS})$ (5)

is the critical surface density, $\boldsymbol{\gamma}=(\gamma_{1},\gamma_{2})$
is the shear and ${\cal A}$ is the magnification matrix. Thus, the previous
equations define the convergence and shear as second derivatives of the
potential, _i.e._ ,

$\kappa=\frac{1}{2}(\partial_{1}\partial_{1}+\partial_{2}\partial_{2})\phi=\frac{1}{2}\nabla^{2}\phi,\hskip
8.53581pt\gamma_{1}=(\partial_{1}\partial_{1}-\partial_{2}\partial_{2})\phi,\hskip
8.53581pt\gamma_{2}=\partial_{1}\partial_{2}\phi.$ (6)

From the above discussion, it is clear that gravitational lensing may allow
one to probe the total mass distribution within the lens system, which
reproduces the observed image configurations and distortions. This, in turn,
may allow one to constrain the cosmological parameters, although this is a
second order effect.

Figure 1: Schematics of the lensing phenomenon.

##  3 Strong Lensing

Quasars are the brightest astronomical objects, visible even at a distance of
billions of parsecs. After the identification of the first quasar in 1963
schmidt , these objects remained a mystery for quite a long time, but today,
we know that they are powered by mass accretion on a supermassive black hole,
with a mass billions of times that of the Sun. The first strong gravitational
lens, discovered in 1979, was indeed linked to a quasar (QSO 0957+561
qso0957+561 ), and although the phenomenon was expected on theoretical
grounds, it left the astronomers astonished. The existence of two objects
separated by about $6^{\prime\prime}$ and characterized by an identical
spectrum led to the conclusion that they were the doubled image of the same
quasar, clearly showing that Zwicky was perfectly right and that galaxies may
act as gravitational lenses. Afterwards, also the lens galaxy was identified,
and it was established that its dynamical mass, responsible for the light
deflection, was at least ten-times larger than the visible mass. This double
quasar was also the first object for which the time delay (about 420 days)
between the two images pelt , due to the different paths of the photons
forming the two images, has been measured. This has also allowed obtaining an
independent estimate of the lens galaxy dynamical mass. Observations can also
show four images of the same quasar, as in the case of the so-called Einstein
Cross, or when the lens and the source are closely aligned, one can observe
the Einstein ring, e.g., in the case of MG1654-1346 chen . The macroscopic
effect of multiple images’ formation is generally called strong lensing, which
also consists of the formation of arcs, as those clearly visible in the deep
sky field images by the Sloan Digital Sky Survey (SDSS; see, e.g., hennawi2008
). The sources of strong lensing events are often quasars, galaxies, galaxy
clusters and supernovae, whereas the lenses are usually galaxies or galaxy
clusters. The image separation is generally larger than a few tenths of an
arcsec, often up to a few arcsecs.

Over the years, many strong lensing events have been found in deep surveys of
the sky, such as the CLASS browne2003 , the Sloan ACS bolton2006 , the SDSS,
one of the most successful surveys in the history of astronomy (see, e.g.,
sdss  and references therein), the SQLS (the Sloan Digital Sky Survey for
Quasar Lens Search) oguri2006 , and so on.

Strong gravitational lensing is nowadays a powerful tool for investigation in
astrophysics and cosmology (see, e.g. blandford1992 ; treu2010 ). As already
mentioned in the previous section, strong lensing gives a unique opportunity
to measure the dynamical mass of the lens object using, for example, the mass
estimator $M(<R_{E})=\pi\Sigma_{\rm cr}\theta_{E}^{2}$, which directly gives
the mass within $R_{E}$, using Equation (5) in this regime. The result is that
masses obtained in this way are almost always larger than the visible mass of
the lensing object, showing that galaxy and galaxy cluster masses are
dominated by dark matter. In any case, accurately constraining the mass
distribution of the lens system (e.g., a galaxy cluster) is a generally
degenerate problem, in the sense that there are several mass distributions
that can fit the observables; thus, the best way to solve it is to use
multiple images (see, e.g., kneib1996 ).

Another important application of strong lensing is the study of dark matter
halo substructures. Indeed, sometimes flux ratio anomalies in the lensed
quasar images are detected (see, e.g., mao1998 ; metcalf2001 ), and while
smooth mass models of the lensing galaxy may generally explain the observed
image positions, the prediction of such models of the corresponding fluxes is
frequently violated. Especially in the radio band observations, since the
quasar radio emitting region is quite large, the observed radio flux anomalies
are explained as being due to the presence of substructures of about
$10^{6}-10^{8}{\rm~{}M_{\odot}}$ along the line of sight. After some
controversy regarding whether $\Lambda$CDM (cold dark matter plus Cosmological
Constant) simulations predict enough dark matter substructures to account for
the observations (for example, in xu2009 , some indication is found of an
excess of massive galaxy satellites), more recent analysis, taking also into
account the uncertainty in the lens system ellipticity, finds results
consistent with those predicted by the standard cosmological model metcalf2012
; xu2015 . However, at present, the list of multiply-imaged quasars observed
in the radio and mid-IR bands is quite short, and further observational and
theoretical work would be very helpful in this respect. Another indication of
dark matter halo substructures comes from detailed analysis of galaxy-galaxy
lensing. Although the results obtained are generally consistent with
$\Lambda$CDM simulations, more data should be analyzed in order to get strong
constraints vegetti2012 ; vegetti2014 . Strongly lensed quasars have been
observed to show a certain variability of one image with respect to the
others. This can be often attributed to microlensing (see Section 5) by the
stars throughout the lens galaxy. This effect, and in particular its variation
with respect to the wavelength, has provided an opportunity to study in detail
the central engine of the source quasar, and the magnitude of the microlensing
variability has allowed astrophysicists to constrain the stellar density in
the lens galaxy kochanek2004 ; mosquera2011 ; pooley2012 .

Strong gravitational lensing may be used as a natural telescope that magnifies
dim galaxies, making them easier to be studied in detail. For this reason,
mass concentrations, like galaxies and clusters of galaxies, can be
effectively used as cosmic telescopes to study faint sources that would not be
possible to detect in the absence of gravitational lensing (see, e.g.,
soucail1987 ; stark2008 ). At present, there is also an event of a high
magnified supernova multiply imaged and also seen exploding again, being
lensed by a galaxy in the cluster MACS J1149.6+2223 kelly2015 ; treu2016 .

The ultimate goal of strong lensing is not only to get information on the
large-scale structure of the Universe, but also to constrain the cosmological
parameters. For instance, analyzing the time delay among the lensed source
images, it is possible to estimate also the value of the Hubble constant
$H_{0}$. Indeed, the time delay is given by the difference of the light paths
from the images and is inversely proportional to $H_{0}$, as first understood
by Refsdal refsdal1964  (see also the review in jackson2007 ). At present, one
of the most accurate measurement of the Hubble constant using a gravitational
lens is provided in suyu2010 . There is also a project (COSMOGRAIL)
particularly devoted to the time delay measurements of doubly- or multiply-
lensed quasars (see cosmograil2016  and the references therein). Moreover, the
measure of both the frequency of occurrence and the redshift of multiple
images in deep sky surveys may allow one to constrain the values of
$\Omega_{M}$ and $\Omega_{\Lambda}$ in an independent way with respect to
other methods, such as those coming from SN Ia or the CMB (Cosmic Microwave
Background) power spectrum.

##  4 Weak Lensing

In addition to the macroscopic deformations discussed in the previous section,
in the deep field surveys of the sky, also arclets (i.e., single distorted
images with an elliptical shape) and weakly distorted images of galaxies, with
an almost invisible individual elongation, have been detected. This effect is
known as weak lensing and is playing an increasingly important role in
cosmology.

The weak lensing’s main feature is the shape deformation of background
galaxies, whose light crosses a mass distribution (e.g., a galaxy or a galaxy
cluster) that acts as a gravitational lens. Actually, as discussed in Section
2, gravitational lensing gives rise to two distinct effects on a source image:
convergence, which is isotropic, and shear, which is anisotropic. In the weak
lensing regime, the observer makes use of the shear, that is the image
deformation (sometimes related to the galaxy orientation), while the
convergence effect is not used, since the intrinsic luminosity and the size of
the lensed objects are unknown. For a complete and in-depth review on the
basics of weak gravitational lensing, with full mathematical details of all
the most important concepts, we refer the reader to bartelmann2001 .

The first weak lensing event was detected in 1990 as statistical tangential
alignment of galaxies behind massive clusters tyson1990 , but only in 2000,
coherent galaxy distortions were measured in blind fields, showing the
existence of the cosmic shear (see, e.g., bacon2000 ; kaiser2000 ). Here, we
remark that the weak lensing cannot be measured by a single galaxy, but its
observation relies on the statistical analysis of the shape and alignment of a
large number of galaxies in a certain direction.

Therefore, the game is to measure the galaxy ellipticities and orientations
and to relate them to the surface mass density distribution of the lens system
(generally a galaxy cluster placed in between). There are at least two major
issues in weak lensing studies, one mainly relying on the theory, the other
one on observations: the former concerns finding the best way to reconstruct
the intervening mass distribution from the shear field
$\boldsymbol{\gamma}=(\gamma_{1},\gamma_{2})$, the latter with looking for the
best way to determine the true ellipticity of a faint galaxy, which is smeared
out by the instrumental point spread function (PSF). To solve these issues,
several approaches have been proposed, which can be distinguished into two
broad families: direct and inverse methods. On the theoretical side, the
direct approaches are: the integral method, which consists of expressing the
projected mass density distribution as the convolution of
$\boldsymbol{\gamma}$ by a kernel (see, e.g., seitz1996 ), and the local
inversion method, which instead starts from the gradient of
$\boldsymbol{\gamma}$ (see, e.g., lombardi1999  and the references therein).
The inverse approaches work on the lensing potential $\phi$ (see Equation 3),
and they include the use of the maximum likelihood schneider2000 ; han2015  or
the maximum entropy methods marshall2002  to determine the most likely
projected mass distribution that reproduces the shear field. The inverse
methods are particularly useful since they make it possible to quantify the
errors in the resultant lensing mass estimates, as, for instance, errors
deriving from the assumption of a spherical mass model when fitting a non-
spherical system clowe2004 ; corless2007 .

The inverse methods allow one also to derive constraints from external
observations, such as X-ray data on galaxy clusters’ strong lensing or CMB
lensing. In particular, one can compare mass measurements from weak lensing
and X-ray observations for large samples of galaxy clusters hoekstra2007 . In
this respect, zhang2008  used a large sample of nearby clusters with good weak
lensing and X-ray measurements to investigate the agreement between mass
estimates based on weak lensing and X-ray data, as well as studied the
potential sources of errors in both methods. Moreover, a combination of weak
lensing and CMB data may provide powerful constraints on the cosmological
parameters, especially on the Hubble constant $H_{0}$, the amplitude of
fluctuations $\sigma_{8}$ and the matter cosmic density $\Omega_{m}$
contaldi2003 ; hollenstein2009 . We also mention, in this respect, that one
way to determine the fluid-mechanical properties of dark energy, characterized
by its sound speed and its viscosity apart from its equation of state, is to
combine Planck data with galaxy clustering and weak lensing observations by
Euclid, yielding one percent sensitivity on the dark energy sound speed and
viscosity majerotto2016  (see the end of this section).

On the observational side, the first priority is to use a telescope with a
wide field of view, appropriate to probe the large-scale structure
distribution at least of a galaxy cluster. On the other hand, it is also
necessary to minimize the source of noise in the determination of the
ellipticity of very faint galaxies, so that the best-seeing conditions for a
ground-based telescope or, better, a space-based instrument, are extremely
useful.

Very promising results have been obtained with the weak lensing technique so
far, as, for example, the best measure, until today, of the existence and
distribution of dark matter within the famous Bullet cluster clowe2006
(actually constituted by a pair of galaxy clusters observed in the act of
colliding). Astronomers found that the shocked plasma was almost entirely in
the region between the two clusters, separated from the galaxies. However,
weak lensing observations showed that the mass was largely concentrated around
the galaxies themselves, and this enabled a clear, independent measurement of
the amount of dark matter.

With the major aim to map, through the weak lensing effect, the mass
distribution in the Universe and the dark energy contribution by measuring the
shape and redshift of billions of very far away galaxies (for a review, see
amendola ), the European Space Agency (ESA) is planning to launch the Euclid
satellite in the near future. Also ground-based telescopes will allow one to
detect an enormous number of weak and strong lensing events. An example is
given by the LSST (Large Synoptic Survey Telescope) project, located on the
Cerro Pachón ridge in north-central Chile, which will become operative in
2022. Its 8.4-meter telescope uses a special three-mirror design, creating an
exceptionally wide field of view, and has the ability to survey the entire sky
in only three nights. The effective number density $n_{\rm eff}$ of weak
lensing galaxies (which is a measure of the statistical power of a weak
lensing survey) that will be discovered by LSST is, conservatively, in the
range of 18–24 arcmin-2 (see Table 4 in chang2013 ). The very large (about
$1.5\times 10^{4}$ square degrees) and deep survey of the sky that will be
performed by Euclid will allow astrophysicists to address fundamental
questions in physics and cosmology about the nature and the properties of dark
matter and dark energy, as well as in the physics of the early Universe and
the initial conditions that provided the seeds for the formation of cosmic
structure. Before closing this section, we also mention that strong
systematics may be present in weak lensing surveys. For example, the intrinsic
alignment of background sources may mimic to an extent the effects of shear
and may contaminate the weak lensing signal. However, these systematics may be
controlled if also the galaxy redshifts are acquired, and this fully removes
the unknown intrinsic alignment errors from weak lensing detections (for
further details, see Heymans2003 ; Valageas2014 ).

##  5 Microlensing

Let us consider now the microlensing scale of the lensing phenomenon that
occurs when $\theta_{E}$ is smaller than the typical telescope angular
resolution, as in the case of stars lensing the light from background stars
(for a review on gravitational microlensing and its astrophysical
applications, we refer to, e.g., mao2012 ). As is clear from the discussion in
Section 2, by solving the lens Equation (1), one can determine the angular
positions of the primary ($I_{1}$) and secondary ($I_{2}$) images. In Figure
2, these positions are shown for four different values of the impact parameter
$\theta_{S}$ in the case of a point-like source. If the source and the lens
are aligned (first panel on the left), the circular symmetry of the problem
leads to the formation of a luminous annulus having radius $\theta_{E}$ around
the lens position. Otherwise, increasing the $\theta_{S}$ value, the secondary
image gets closer to the lens position, while the primary image drifts apart
from it, and in the limit of $\theta_{S}\gg\theta_{E}$, the microlensing
phenomenon tends to disappear. However, observing multiple images during a
microlensing event is practically impossible with the present technology. For
instance, in the case in which the phenomenon is maximized, corresponding to
the perfect alignment, for a star in the galactic bulge (about $8$ kpc away),
one has $\Delta\theta=2\theta_{E}\simeq 1$ $\mu$as, which is well below the
angular resolving power, even of the Hubble Space Telescope (about $43$ mas at
500 nm); see, e.g., http://www.coseti.org/9008-065.htm.

Figure 2: Angular positions of the primary ($I_{1}$) and secondary ($I_{2}$)
images for four different values of the source impact parameter $\theta_{S}$
in the case of a point-like source.

When a source is microlensed, its images do not have the same luminosity;
therefore, the observer receives a total flux (or magnitude) different from
that of the unlensed source. The flux difference can be described very simply
in terms of the light magnification and the law of the conservation of the
specific intensity $I$, which represents the energy, with frequency in the
range $d\nu$ crossing the surface $dA$ during the time interval $dt$ in the
solid angle $d\Omega$ around the direction orthogonal to the surface. Indeed,
the light specific intensity turns out to be conserved in the absence of
absorption phenomena, interstellar scattering or Doppler shifts. This is also
a consequence of Liouville’s theorem, which claims that the density of states
in the phase space remains constant if the interacting forces are non-
collisional (and gravitation fulfills this condition due to its weak coupling
constant), and the propagating medium is approximately transparent (as is the
case for interstellar space). This effect can produce a magnification or a de-
magnification of the images of an extended light source (see Figure 3). If the
image is magnified, it means that it certainly subtends a wider angle with
respect to that subtended by the source in the absence of the lens. In
microlensing, the source disk size should not be neglected in general. Within
the framework of the finite source approximation for a source with flux
$F_{S}$ and assuming ${\theta}_{E}\leq{\theta}_{S}$, one can show that the
magnification $A$ of an image at angular position $\theta$ is given by
$(1-\theta_{E}^{4}/\theta^{4})^{-1}$. As a consequence, the observed flux
corresponds to $F=AF_{S}$. Of course, when the source star disk gradually
moves away from the line of sight, the magnification decreases, and the
unlensed $F_{S}$ flux is then recovered.

Figure 3: As in Figure 2, but with a conformal transformation of the source
boundary, considered extended and with radius $\rho_{S}$, by a point-like
lens. Each point of the source disk behaves as a point-like source. The black
circle represents the source disk, while the red and yellow arcs are the
deformed primary and secondary images.

As already anticipated, the observer cannot see, in the microlensing case,
well-separated images, but, instead, detects a single image made by the
overlapping of the primary and the secondary images. In this case, one can
easily obtain the classical magnification factor $A$ by summing up the
individual magnifications, _i.e._ , $A=(u^{2}+2)/\sqrt{u^{2}(u^{2}+4)}$, where
$u={\theta}_{S}/{\theta}_{E}$ is the impact factor. If there is a relative
movement between the lens and the source, $u$ changes with time, and a
standard Paczyǹski curve pacz1986  does emerge.

An important role in gravitational microlensing is played by the caustics, the
geometric loci of the points belonging to the lens plane where the light
magnification of a point-like source becomes infinite, and by the
corresponding critical curves in the source plane. In the case of a single
lens, the caustic is a point coinciding with the lens position; therefore, the
magnification diverges when the impact parameter approaches zero. However,
real sources are not point-like, so we always have finite magnifications that
can be calculated by an average procedure:

$\langle A\rangle=\frac{\int A({\bf y})I({\bf y})d^{2}y}{\int I({\bf
y})d^{2}y},$ (7)

where $A({\bf y})$ is the point-like source magnification, $I({\bf y})$ is the
brightness profile of the stellar disk (the limb darkening profile) and the
integral is extended over the source star disk.

Observations show that about half of all stars are in binary systems, and
moreover, thousands of exoplanets are being discovered around their host stars
by different techniques and instruments. Therefore, it is worth considering
binary and multiple systems as lenses in microlensing observations. In this
case, the lens equation, obviously, becomes more complicated, but it can still
be solved by numerical methods in order to obtain the magnification map where
caustics take on distinctive shapes depending on the specific geometry of the
system. In Figure 4, we show the magnification map and the resulting light
curve for a simulated microlensing event due to a binary lens with mass ratio
$q=M_{1}/M_{2}\simeq 0.01$ (e.g., a solar mass as the primary component and a
Jupiter-like planet as the secondary one). In these cases, the resulting light
curve may be rather different with respect to the typical Paczyǹski one,
depending on the system parameters. The study of these anomalies in the
microlensing light curves behavior is becoming more and more important
nowadays, since it allows one to estimate some of the parameters of the
lensing system (see, e.g., Perryman14 ). The main advantage of this technique,
compared to the other methods adopted by the exoplanets hunters (e.g., radial
velocity, direct imaging, transits), is the possibility to detect even very
small planets orbiting their own star at enormous distances from Earth. It
also allows one to discover the so-called free-floating planets (FFPs),
otherwise hardly detectable sumi . By studying the PA99-N2 microlensing event,
detected in 1999 by the French-British collaboration POINT-AGAPE an , Ingrosso
_et al_. Ingrosso2009  revealed in 2009 that the anomaly observed was
compatible with the presence of a super-Jupiter with a mass of $\simeq 5M_{J}$
around a star lying in the Andromeda galaxy (see Figure 5), thus finding the
first putative exoplanet in another galaxy.

Figure 4: Magnification map for a binary lens system characterized by two
objects separated by a projected distance of $1R_{E}$ and mass ratio $q=0.01$.
The green and red closed lines indicate the critical and caustic curves
obtained by solving the lens equation in Equation (1). The black line
indicates the trajectory of the source star, which has a radius of
$0.03R_{E}$. The simulated light curve is shown in the lower panel. Figure 5:
Light curve in three different bands (g, r and i) of the PA99-N2 event
detected in 1999 toward the Andromeda galaxy.

###  5.1 Astrometric Microlensing

During an ongoing microlensing event, the centroid of the multiple images and
the source star positions move in the lens plane giving rise to a phenomenon
known as astrometric microlensing (see, e.g., dominik2000 ; lee2010  and the
references therein). In the simplest case of a point-like lens, for a source
at angular distance $\theta_{S}$, the position $\theta$ of the images with
respect to the lens can be obtained by solving the lens Equation (1). Since
the Einstein radius $R_{E}=D_{L}\theta_{E}$ defines the scale length on the
lens plane, the lens equation reads:

$d^{2}-dd_{S}-R_{E}^{2}=0,$ (8)

where $d_{S}$ and $d$ are the linear distances, in the lens plane, of the
source and images from the gravitational lens, respectively. Moreover, using
the dimensionless source-lens distances $u=\theta_{S}/\theta_{E}$ and
$\tilde{u}=\theta/\theta_{E}$, the previous relation can be further simplified
as:

$\tilde{u}^{2}-u\tilde{u}-1=0.$ (9)

Denoting with $u_{+}$ and $u_{-}$ the solutions of this equation, one notes
that, in the lens plane, the $+$ image resides always outside the circular
ring centered on the lens position with radius equal to the Einstein angle,
while the $-$ image is always within the ring. As the source-lens distance
increases, the $+$ image approaches the source position, while the $-$ one
(becoming fainter) moves towards the lens location. For a source moving in the
lens plane with transverse velocity $v_{\perp}$ directed along the $\xi$ axis
($\eta$ is perpendicular to it), the projected coordinates of the source
result in being:

$\xi(t)=\displaystyle{\frac{t-t_{0}}{t_{E}}},~{}~{}~{}\eta(t)=\displaystyle{u_{0}}.$
(10)

where $t_{E}=R_{E}/v_{\perp}$ and $u_{0}$ is the impact parameter (in this
case, lying on the $\eta$ axis). Since $u^{2}=\xi^{2}+\eta^{2}$ is time
dependent, the two images move in the lens plane during the gravitational
lensing event.

By weighting the $+$ and $-$ image position with the associated magnification
walker1995 , one gets:

$\bar{u}\equiv\frac{\tilde{u}_{+}\mu_{+}+\tilde{u}_{-}\mu_{-}}{\mu_{+}+\mu_{-}}=\frac{u(u^{2}+3)}{u^{2}+2}.$
(11)

Finally, the observable is defined as the displacement of the centroid with
respect to the source,

$\Delta\equiv\bar{u}-u=\frac{u}{2+u^{2}}.$ (12)

Note that the centroid shift may be viewed as a vector:

${\bf\Delta}=\frac{{\bf u}}{2+u^{2}}$ (13)

with components along the axes:

$\Delta_{\xi}=\displaystyle{\frac{\xi(t)}{2+u^{2}}},~{}~{}~{}\Delta_{\eta}=\displaystyle{\frac{u_{0}}{2+u^{2}}}.$
(14)

Here, we remind that all of the angular quantities are given in units of the
Einstein angle $\theta_{E}$, which, for a source at distance $D_{S}\gg D_{L}$,
results in being:

$\theta_{E}\simeq
2\left(\frac{M}{0.5{\rm~{}M_{\odot}}}\right)^{1/2}\left(\frac{D_{L}}{{\rm
kpc}}\right)^{-1/2}~{}{\rm mas},$ (15)

which fixes the scale of the phenomenon.

It is straightforward to show (see walker1995 ) that during a microlensing
event, the centroid shift $\Delta$ traces (in the $\Delta_{\xi},\Delta_{\eta}$
plane) an ellipse centered in the point $(0,b)$. The ellipse semi-major axis
$a$ (along $\Delta_{\eta}$) and semi-minor axis $b$ (along $\Delta_{\xi}$)
are:

$a=\frac{1}{2}\frac{1}{\sqrt{u_{0}^{2}+2}},~{}~{}~{}b=\frac{1}{2}\frac{u_{0}}{u_{0}^{2}+2}.$
(16)

Then, for $u_{0}\rightarrow\infty$, the ellipse becomes a circle with radius
$1/(2u_{0})$, while it degenerates into a straight line of length $1/\sqrt{2}$
for $u_{0}$ approaching zero. Note also that Equation (16) implies:

$u_{0}^{2}=2(b/a)^{2}\left[1-(b/a)^{2}\right]^{-1},$ (17)

so that by measuring $a$ and $b$, one can determine the event impact parameter
$u_{0}$.

As observed in dominik2000 , ${\bf\Delta}$ falls more slowly than the
magnification, implying that the centroid shift may be an interesting
observable also for large source-lens distances, _i.e._ , far from the light
curve peak. In fact, in astrometric microlensing, the threshold impact
parameter $u_{\rm th}$ (_i.e._ , the value of the impact parameter that gives
an astrometric centroid signal larger than a certain quantity $\delta_{\rm
th}$) is given by $u_{\rm th}=\sqrt{T_{\rm obs}v_{\perp}/(\delta_{\rm
th}D_{L})}$, where $T_{\rm obs}$ is the observing time and $v_{\perp}$ the
relative velocity of the source with respect to the lens. For example, the
Gaia satellite should reach an astrometric precision $\sigma_{G}\simeq 300$
$\mu$as (for objects with visual magnitude $\simeq 20$) in five years of
observation eyer . Then, assuming a threshold centroid shift $\delta_{\rm
th}\simeq\sigma_{G}$, one has $u_{\rm th}\simeq 60$ for a lens at a distance
of $0.1$ kpc and transverse velocity $v_{\perp}\simeq 100$ km s-1. For
comparison, the threshold impact parameter for a ground-based photometric
observation is $\simeq 1$. Consequently, the cross-section for an astrometric
microlensing measurement is much larger than the photometric one, since it
scales as $u_{\rm th}^{2}$. Hence, in the absence of finite-source and
blending effects, by measuring $a$ and $b$, one can directly estimate the
impact parameter $u_{0}$.

A further advantage of the astrometric microlensing is that some events can be
predicted in advance paczinsky1996 . In fact, by studying in detail the
characteristics of stars with large proper motions, Proft  _et al_. proft2011
identified tens of candidates to measure astrometric microlensing by the Gaia
satellite, an European Space Agency (ESA) mission that will perform
photometry, spectroscopy and high precision astrometry (see eyer ).

###  5.2 Polarization and Orbital Motion Effects in Microlensing Events

Gravitational microlensing observations may also offer a unique tool to study
the atmospheres of far away stars by detecting a characteristic polarization
signal pol1 . In fact, it is well known that the light received from stars is
linearly polarized by the photon scattering occurring in the stellar
atmospheres. The mechanism is particularly effective for the hot stars (of A
or B type) that have a free electron atmosphere giving rise to a polarization
degree increasing from the center to the stellar limb (chandra, ). By a minor
extent, polarization may be also induced in main sequence F or G stars by the
scattering of star light off atoms/molecules and in evolved, cool giant stars
by photon scattering on dust grains contained in their extended envelopes.

Following the approach in chandra , the polarization $P$ in the direction
making an angle $\chi=\arccos(\mu)$ with the normal to the star surface is
$P(\mu)=[I_{r}(\mu)-I_{l}(\mu)]/[I_{r}(\mu)+I_{l}(\mu)]$, where $I_{l}(\mu)$
is the intensity in the plane containing the line of sight and the normal, and
$I_{r}(\mu)$ is the intensity in the direction perpendicular to this plane.
Here, $\mu=\sqrt{1-(r/R)^{2}}$, where $r$ is the distance of a star disk
element from the center and $R$ the star radius, and we are assuming that
light propagates in the direction ${\bf r\times l}$.

For isolated stars, a polarization signal has been measured only for the Sun
for which, due to the distance, the projected disk is spatially resolved.
Instead, when a star is significantly far away and can be considered as point-
like, only the polarization $\langle P\rangle$ averaged over the stellar disk
can be measured, and usually $\langle P\rangle=0$, since the flux from each
stellar disk element is the same. A net polarization of the light appears if a
suitable asymmetry in the stellar disk is present (caused by, e.g., eclipses,
tidal distortions, stellar spots, fast rotation, magnetic fields). In the
microlensing context, the polarization arises since different regions of the
source star disk are magnified differently during the event. Indeed, during an
ongoing microlensing event, the gravitational lens scans the disk of the
background star, giving rise not only to a time-dependent light magnification,
but also to a time-dependent polarization.

This effect (see also simmons2002 ) is particularly relevant in the
microlensing events where: (1) the magnification turns out to be significant;
(2) the source star radius and the lens impact parameter are comparable; (3)
the source star is a red giant, characterized by a rather low surface
temperature ($T\leq 3000\,K$), around which the formation of dust grains is
possible. This occurs beyond the distance $R_{h}$ from the star center at
which the gas temperature in the stellar wind becomes lower than the grain
sublimation temperature ($\simeq 1400$ K). The intensity of the expected
polarization signal relies on the dust grain optical depth $\tau$ and can
reach values of 0.1%–1%, which could be reasonably observed using, for
example, the ESO VLTtelescope (see ingrossopol2015 ). In Figure 6, we show
some typical polarization curves, expected in bypass (continuous curves) and
transit events (dashed curves), in which the lens trajectory approaches or
passes through the source regions where the dust grains are present. In Figure
7, the distribution of the peak polarization values (given in percent) as a
function of the intrinsic source star color index $(V-I)_{\rm int}$ (_i.e._ ,
the de-reddened color of the unlensed source star) is shown for a sample of
OGLE-type microlensing events generated by a synthetic stellar catalog
simulating the bulge stellar population. As one can see, red giants with
$(V-I)_{\rm int}\leq 3$, which corresponds to the events inside the regions
delimited by dashed lines, have $P_{\rm max}\leq 1$ percent values. These are
the typical events observed by the OGLE-III microlensing campaign. There are,
however, a few events with $1\leq P_{\rm max}\leq 10$ percent, characterized
by $(V-I)_{\rm int}\geq 3$, corresponding to source stars in the AGB phase.
These stars, which are rather rare in the galactic bulge, have not been
sources of microlensing events observed in the OGLE-III campaign, but they are
expected to exist in the galactic bulge. In this respect, the significant
increase in the event rate by the forthcoming generation of microlensing
surveys towards the galactic bulge, both ground-based, like KMTNet
Henderson2014 , and space-based, like EUCLIDPenny2013  and WFIRSTYee2014 ,
opens the possibility to develop an alert system able to trigger polarization
measurements in ongoing microlensing events.

Another way to study the atmosphere of the source star is to analyze the
amplification curve and look for dips and peaks, typically due to the presence
of stellar spots on the photosphere of the star 2000ApJ…529…69H ;
2002MNRAS.335..539H . These features may be easily confused, however, with the
signatures of a binary lensing system. When the source star has a relevant
rotation motion during the lensing event, there is the possibility to really
detect the stellar spots on the source’s surface and to estimate the rotation
period of the star Giordano et al. (2015). A new generation of networks of
telescopes dedicated to microlensing surveys, like KMTNet Kim et al. (2010),
will provide high-precision and high-cadence photometry that will enable us to
observe spots on the source’s surface. We remark that also multicolor
observations of the event would help to disentangle the aforementioned
degeneracy, as the ratio between the brightness of the spot and the
surrounding photosphere strongly depends on the frequency of the observation.
It has been shown that stellar spots can be detected also through polarimetric
observations of microlensing caustic-crossing events Sajadian (2015).

Under certain circumstances, binary lens systems are characterized by the
_close-wide degeneracy_ : if the two objects are separated by a projected
distance $s$ or $1/s$, the resulting caustics have the same structure, and
also, the observed light curves will appear the same Dominik (1999); An
(2005). This happens, for example, in systems with small mass ratio $q$, like
planetary systems Griest & Safizadeh (1998). It is possible to resolve this
degeneracy in the case of short-period binary lenses, the so-called _rapidly
rotating lenses_ , as the orbital motion induces repeating features in the
amplification curve that can be exploited to estimate important physical
parameters of the lensing systems, including the orbital period, the projected
separation and the mass Penny et al. (2011); Nucita et al. (2014).

Figure 6: Assuming parameter values $u_{0}=0.09$, $t_{E}=60$ day and
$R_{h}/R_{S}=5$, the $P(t)/\tau$ polarization curves are shown as a function
of $(t-t_{0})/t_{E}$, for increasing values of $R_{h}/u_{0}=0.35,\ 0.75,\
1.5,\ 2.5$, corresponding to the curves labeled (a), (b) (c) and (d),
respectively. Continuous curves, (a) and (b), are bypass events; dotted lines,
(c) and (d), are transit events. Here, $R_{h}$ is the minimum distance for the
formation of dust grains, and $R_{S}$ is the source star radius. Figure 7:
Distribution of the peak value of the polarization signal as a function of the
intrinsic color index $(V-I)_{\rm int}$ of the source star for simulated
transit (triangles) and bypass (purple squares) events. The dashed lines
indicate the region in which the events observed by the OGLECollaboration are
expected to lie.

##  6 Retro-Lensing: Measuring the Black Hole Features

Gravitational lensing at the scales considered in the previous sections can be
treated in the weak gravitational field approximation of the general theory of
relativity, since in those cases, photons are deflected by very small angles.
This is not the case when one considers black holes, for which it may happen
that photons get very close to the event horizon of these compact objects.

Black holes are relatively simple objects. The _no-hair theorem_ postulates
that they are completely described by only three parameters: mass, angular
momentum (generally indicated by the spin parameter $a$) and electric charge;
any other information (for which hair is a metaphor) disappears behind the
event horizon, and it is therefore inaccessible to external observers.
Depending on the values of these parameters, black holes can be classified
into Schwarzschild black holes (non-rotating and non-charged), Kerr black
holes (rotating and non-charged), Reissner–Nordström black holes (non-rotating
and charged) and Kerr–Newman black holes (rotating and charged).

Even though they appear so simple, black holes are mathematically complicated
to describe (see, e.g., chandraBH ). Nowadays, we know that black holes are
placed at the center of the majority of galaxies, active or not, and in many
binary systems emitting X-rays. Moreover, they are the engine of gamma-ray
bursts (GRBs) and play an essential role in better understanding stellar
evolution, galaxy formation and evolution, jets and, in the end, the nature of
space and time. One goal astrophysicists have been pursuing for a long time is
to probe the immediate vicinity of a black hole with an angular resolution as
close as possible to the size of the event horizon. This kind of observations
would give a new opportunity to study strong gravitational fields, and as we
will see at the end of this section, we think we are very close to reaching
this goal.

How do we measure the mass, angular momentum and electric charge of a black
hole? One possibility, rich with interesting consequences, was suggested by
Holz and Wheeler holzwheeler , who considered a phenomenon that was already
known to be possible around black holes. They used the Sun as the source of
light rays and a black hole far from the solar system. As shown in Figure 8,
some photons would have the right impact parameter to turn around the black
hole and come back to Earth. Other photons, with a slightly smaller impact
parameter, can even rotate twice around the black hole, and so on. A series of
concentric rings should then appear if the observer, the Sun and the black
hole are perfectly aligned. The two authors also suggested to do a survey and
look for concentric rings in the sky in order to discover black holes.
Unfortunately, there are two problems with this idea. First, it is unlikely
that the Sun, Earth and a black hole are perfectly aligned, and in any case,
Earth moves around the Sun, so that the alignment can occur only for a short
time interval. The second and most important problem is that the retro-image
of the Sun is so dim, that even using the Hubble Space Telescope (HST), only a
black hole with a mass larger than $10~{}M_{\odot}$ within $0.01$ pc from the
Earth could be observed with the proposed technique. Moreover, we already know
that such an object cannot be so close to the solar system without causing
observable perturbations in the planet orbits.

Figure 8: Retro-lensing of the Sun light by a black hole as seen from Earth.
On the right-hand side, the series of rings around the black hole’s event
horizon an observer would observe in the case of perfect alignment. For
clarity, only two rings are shown.

A better approach to test the idea proposed by Holz and Wheeler is to consider
a well-known supermassive black hole and a bright star around it. Of course,
the brighter the source star, the brighter will be the retro-image. Some of us
depaolisS2  soon proposed to consider retro-lensing around the black hole at
the galactic center, and in particular, the retro-lensing image of the closest
star orbiting around it. Indeed, it is known that at the center of our galaxy,
there is a supermassive black hole, with mass about $(4.2\pm 0.2)\times
10^{6}~{}M_{\odot}$, identified by studying the orbits of several bright stars
orbiting around it (see gillessen ; doku2014  and the references therein). A
method to determine the mass and the angular momentum of this black hole could
then be to measure the periastron or apoastron shifts of some of the stars
orbiting around it. Another method to estimate the black hole spin $a$ is
based on the analysis of the quasi-periodical oscillations towards Sgr A∗.
Recently, the analysis of the data in the X-ray and IR bands have allowed some
astrophysicists to find that $a=0.65\pm 0.05$ doku2014 ; doku2015 . However,
there is a drawback in this approach: periastron and apoastron shift of orbits
depend not only on the black hole parameters, but also on how stars are
distributed around the black hole and on the mass density profile of the dark
matter possibly present in the region surrounding the black hole. It is
possible to understand the difficulty of the measure by noting that the
difference of the periastron shift of the S2 star (the closest one to the
black hole at the center of our galaxy) induced by a Schwarzschild black hole
or a Kerr black hole with spin parameter $a=1$ (and the same mass of the
Schwarzschild one) is only of $\simeq 10~{}\mu$as (for the dependance of the
periastron shift on the black hole spin orientation see iorio2012 ). Then,
even if one had succeeded in measuring the periastron shift of the closest
star to the central black hole, it would be unlikely to derive the amount of
the black hole angular momentum. Our goal could be achieved anyway by
measuring the periastron shift of many stars orbiting around the center of the
Galaxy. The measure of the periastron shift could give, in turn, also an
estimate of the parameters of the dark matter concentration expected to lie
towards the Sgr $A^{*}$ region zak2007 , as well as to test different
modifications of the general theory of relativity zak2012 ; falcke2013 ;
zak2014  (see also zak  for the constraints on $R^{n}$ theories by Solar
System data)). However, this is anything but easy pasp2007 . An important step
forward in this direction has been provided recently by near-infrared
astrometric observations of many stars around Sgr $A^{*}$ with a precision of
about 170 $\mu$as in position and $\simeq 0.07$ mas$\cdot$ yr-1 in velocity
plewa2015 . A further improvement, hopefully in the near future, would make
possible the direct detection of relativistic effects in the orbits of stars
orbiting the central black hole.

Retro-lensing images of bright stars retro-lensed by the black hole at the
galactic center might give an alternative method to estimate the Sgr A∗ black
hole parameters. Even though in general it is difficult to calculate the
retro-lensing images, since this requires integrating with high precision the
trajectories followed by the light, it is possible to numerically do these
calculations not only for a Schwarzschild black hole, but also for Kerr and
Reissner–Nordström black holes. As discussed in several papers (see, e.g.,
depaolis2011  and the references therein), one finds that the shape of the
retro-lensing image depends on the black hole spin (see Figure 9), and then,
in principle, a single precise enough observation of the retro-lensing image
of a star could allow one to unambiguously estimate the parameters of the
black hole in Sgr A∗. It is possible to show that also the electric charge of
a Reissner–Nordström black hole can be obtained zak2005 . In fact, although
the formation of a Reissner–Nordström black hole may be problematic, charged
black holes are objects of intensive investigations, and the black hole charge
can be estimated by using the size of the retro-lensing images that can be
revealed by future astrometrical missions. The shape of the retro-lensing (or
shadow) image depends in fact also on the electric charge of the black hole,
and it becomes smaller as the electric charge increases. The mirage size
difference between the extreme charged black hole and the Schwarzschild black
hole case is about 30%, and in the case of the black hole in Sgr A∗, the
shadow typical angular sizes are about 52 $\mu$as for the Schwarzschild case
and about 40 $\mu$as for a maximally charged Reissner–Nordström black hole.
Therefore, a charged black hole could be, in principle, distinguished by a
Schwarzschild black hole with RADIOASTRON, at least if its charge is close to
the maximal value. We also mention that the black hole spin gives rise also to
chromatic effects (while for non-rotating lenses, the gravitational lensing
effect is always achromatic), making one side of the image bluer than the
other side depaolis2011 .

Can we really hope to observe these retro-lensing images towards Sgr A∗?
Despite what one could think, we are not so far from this goal. The successor
of the Hubble Space Telescope, the James Webb Space Telescope (JWST),
scheduled for launch in October 2018, has the sensitivity to observe the
retro-lensing image of the S2 star produced by the black hole at the galactic
center with an exposure time of about thirty hours. In Figure 10, we show the
magnification (upper panel) and the magnitude (bottom panel) light curves (in
K band) of the retro-lensing image of the S2 star produced by the black hole
at the galactic center (see also bozzamancini2004 ). Unfortunately, JWST has
not the angular resolution necessary to provide information about the shape of
the retro-lensing image. The right angular resolution could be gained with the
next generation of radio interferometers. In fact, the diameter of the retro-
lensing image around the central black hole should be of about $30$ $\mu$as,
and already in 2008, Doeleman and his collaborators doeleman  managed to
achieve an angular resolution of about $37$ $\mu$as, very close to the
required one, by using interferometrically different radio telescopes with a
baseline of about 4500 km. Progress in this field is so fast, that it is not
hard to think we can eventually reach this aim in the near future by, e.g.,
the EHT (Event Horizon Telescope) project, or by the planned Russian space
observatory, Millimetron (the spectrum-M project), or by combined observations
with different interferometers, such as the Very Large Array (VLA) and ALMA
(Atacama Large Millimeter Array).

Figure 9: Retro-lensing images of a source by a Schwarzschild black hole
(dotted circle), a Kerr black hole with spin parameter $a=0.5$ (dashed line)
and a maximally spinning black hole with $a=1$ (continuous line). The line of
sight of the observer is perpendicular to the spin axis of the black hole.
Figure 10: Upper panel: amplification as a function of time for the primary
(upper curve) and secondary (lower curve) retro-lensing images of the S2 star
by the black hole in the Galaxy center. Lower panel: light curve in K-band
magnitude of the two retro-lensing images (adapted from bozzamancini2004 ).
The standard interstellar absorption coefficient towards the Galaxy center has
been assumed.

##  7 Conclusions

In the paper, we have discussed the various scales in which gravitational
lensing manifests itself and that may lead us to obtain valuable information
about a great variety of astronomical issues ranging from the star
distribution in the Milky Way, the study of stellar atmospheres, the discovery
of exoplanets in the Milky Way and also in nearby galaxies, the study of far
away galaxies, galaxy clusters and black holes. Gravitational lensing, in
particular in the strong and weak lensing regime, may also allow scientists to
answer, in the near future, fundamental questions in cosmology related to the
nature of dark matter, why the Universe is accelerating and what is the nature
of the source responsible for the acceleration, which physicists refer to as
dark energy.

###### Acknowledgements.

Acknowledgments: The authors acknowledge the support of the TAsP (Theoretical
Astroparticle Physics) Project funded by INFN. Mosè Giordano acknowledges the
support of the Max Planck Institute for Astronomy, Heidelberg, where part of
this work has been done. We thank Philippe Jetzer, Asghar Qadir and Alexander
F. Zakharov for the valuable discussions on the paper subject during many
years. Author Contributions: All authors have equally contributed to this
paper and have read and approved the final version. Conflicts of Interest: The
authors declare no conflict of interest.

## References

  * (1) Einstein, A. On the Influence of Gravitation on the Propagation of Light. Ann. Phys. 1911, 340, 898–908.
  * (2) Dyson, F.W.; Eddington, A.; Davidson, C. A determination of the deflection of light by the sun’s gravitational field from observations made at the total eclipse of May 29, 1919. Phil. Trans. Roy. Soc. A 1920, 220, 291–333.
  * (3) Chwolson, O. Über eine mögliche Form fiktiver Doppelsterne. Astron. Nachr. 1924, 221, 329–330.
  * (4) Einstein, A. Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field. Science 1936, 84, 506–507
  * (5) Zwicky, F. Nebulae as Gravitational Lenses. Phys. Rev. 1937, 51, 290–290.
  * (6) Zwicky, F. On the Probability of Detecting Nebulae which Act as Gravitational lenses. Phys. Rev. 1937, 51, 679–679.
  * (7) Walsh, D.; Carswell, R.F.; Weymann, R.J. 0957 + 561 A,B: Twin quasistellar objects or gravitational lens? Nature 1979, 279, 381–384.
  * (8) Paczyǹski, B. Gravitational microlensing by the galactic halo. Astrophys. J. 1986, 304, 1–5.
  * (9) Johannsen, T. Sgr A∗ and general relativity. 2015, arXiv:1512.03818.
  * (10) Abbott, B.P.; Abbott, R.; Abbott, T.D.; Abernathy, M.R.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.X.; et al. Observation of Gravitational Waves from a Binary Black Hole Merger. Phys. Rev. Lett. 2016, 116, 061102.
  * (11) Schneider, P.; Ehlers, G.; Falco, E.E. Gravitational Lenses. Springer Verlag: Berlin, Germany, 1992.
  * (12) Schmidt, M. 3C 273: A star-like object with large red-shift. Nature 1963, 197, 1040–1040.
  * (13) Pelt, J.; Kayser, R.; Refsdal, S.; Schramm, T. The light curve and the time delay of QSO 0957+561. Astron. Astrophys. 1996, 306, 97–106.
  * (14) Chen, G.H.; Kochanek, C.S.; Hewitt, J.N. The Mass Distribution of the Lens Galaxy in MG 1131+0456. Astrophys. J. 1995, 447, 62–81.
  * (15) Hennawi, J.F.; Gladders, M.D.; Oguri, M.; Dalal, N.; Koester, B.; Natarajan, P.; Strauss, M.A.; Inada, N.; Kayo, I.; Lin, H. A New Survey for Giant Arcs. Astron. J. 2008, 135, 664–681.
  * (16) Browne, I.W.A.; Wilkinson, P.N.; Jackson, N.J.F.; Myers, S.T.; Fassnacht, C.D.; Koopmans, L.V.E.; Marlow, D.R.; Norbury, M.; Rusin, D.; Sykes, C.M.; _et al_. The Cosmic Lens All-Sky Survey - II. Gravitational lens candidate selection and follow-up. Mon. Not. R. Astron. Soc. 2003, 341, 13–32.
  * (17) Bolton, A.S.; Burles, S.; Koopmans, L.V.E.; Treu, T.; Moustakas, L.A. The Sloan Lens ACS Survey. I. A Large Spectroscopically Selected Sample of Massive Early-Type Lens Galaxies. Astrophys. J. 2006, 638, 703–724.
  * (18) Alam, S.; Albareti, F.D.; Allende, P.; Anders, F.; Anderson, S.F.; Anderton, T.; Andrews, B.H.; Armengaud, E.; Aubourg, E.; Bailey, S.; et al. The Eleventh and Twelfth Data Releases of the Sloan Digital Sky Survey: Final Data from SDSS-III. Astrophys. J. Suppl. Ser. 2015, 219, 12.
  * (19) Oguri, M.; Inada, N.; Pindor, B.; Strauss, M.A.; Richards, G.T.; Hennawi, J.F.; Turner, E.L.; Lupton, R.H.; Schneider, D.P.; Fukugita, M. The Sloan Digital Sky Survey Quasar Lens Search. I. Candidate Selection Algorithm. Astrophys. J. 2006, 132, 999–1013.
  * (20)

Blandford, R.D.; Narajan, R. Cosmological applications of gravitational
lensing. Annu. Rev. Astron. Astrophys.

1992, 30, 311–358.

  * (21) Treu, T. Strong lensing by galaxies. Annu. Rev. Astron. Astrophys. 2010, 48, 87–125.
  * (22) Kneib, J.-P.; Ellis, R.S.; Smail, I.; Couch, W. J.; Sharples, R.M. Hubble Space Telescope Observations of the Lensing Cluster Abell 2218. Astrophys. J. 1996, 471, 643–656.
  * (23) Mao, S.; Schneider, P. Evidence for substructure in lens galaxies? Mon. Not. R. Astron. Soc. 1998, 95, 587–594.
  * (24) Metcalf, R.B.; Madau, P. Compound Gravitational Lensing as a Probe of Dark Matter Substructure within Galaxy Halos. Astrophys. J. 2001, 563, 9–20.
  * (25) Xu, D.D.; Mao, S.; Wang, J.; Springel, V.; Gao, L.; White, S.D.M.; Frenk, C.S.; Jenkins, A.; Li, G.; Navarro, J.F. Effects of dark matter substructures on gravitational lensing: results from the Aquarius simulations. Mon. Not. R. Astron. Soc. 2009, 408, 1235–1253.
  * (26) Metcalf, R. B.; Amara, A. Small-scale structures of dark matter and flux anomalies in quasar gravitational lenses. Mon. Not. R. Astron. Soc. 2012, 419, 3414–3425.
  * (27) Xu, D.D.; Sluse, D.; Gao, L. Wang, J.; Frenk, C.; Mao, S.; Schneider, P.; Springel, V. How well can cold dark matter substructures account for the observed radio flux-ratio anomalies. Mon. Not. R. Astron. Soc. 2015, 447, 3189–3206.
  * (28) Vegetti, S.; Lagattuta, D.J.; McKean, J.P.; Auger, M. W.; Fassnacht, C.D.; Koopmans, L.V.E. Gravitational detection of a low-mass dark satellite galaxy at cosmological distance. Nature 2012, 481, 341–343.
  * (29) Vegetti, S.; Koopmans, L.V.E. Auger, M.W. Inference of the cold dark matter substructure mass function at z = 0.2 using strong gravitational lenses. Mon. Not. R. Astron. Soc. 2014, 442, 2017–2035.
  * (30) Kochanek, C.S. Quantitative Interpretation of Quasar Microlensing Light Curves. Astrophys. J. 2004, 605, 58–77.
  * (31) Mosquera, A. M., Kochanek, C.S. The Microlensing Properties of a Sample of 87 Lensed Quasars. Astrophys. J. 2011, 738, 96.
  * (32) Pooley, D.; Rappaport, S.; Blackburne, J.A.; Schechter, P.L.; Wambsganss, J. X-Ray and Optical Flux Ratio Anomalies in Quadruply Lensed Quasars. II. Mapping the Dark Matter Content in Elliptical Galaxies. Astrophys. J. 2012, 744, 111.
  * (33) Soucail, G.; Fort, B.; Mellier, Y.; Picat, J.P. A blue ring-like structure, in the center of the A 370 cluster of galaxies. Astron. Astrophys. 1987, 172, L14–L16.
  * (34) Stark, D.P.; Swinbank, A.M.; Ellis, R.S. Dye, S.; Smail, I.R; Richard, J. The formation and assembly of a typical star-forming galaxy at redshift z$\sim$3\. Nature 2008, 455, 775–777.
  * (35) Kelly, P.L.; Rodney, S.A.; Treu, T.; Foley, R.J.; Brammer, G.; Schmidt, K.B.; Zitrin, A.; Sonnenfeld, A.; Strolger, L.-G.; Graur, O.; _et al_. Multiple images of a highly magnified supernova formed by an early-type cluster galaxy lens. Science 2015, 347, 1123–1126.
  * (36) Treu, T.; Brammer, G.; Diego, J.M.; Grillo, C.; Kelly, P.L.; Oguri, M.; Rodney, A.; Rosati, P.; Sharon, K.; Zitrin, A.“Refsdal” Meets Popper: Comparing Predictions of the Re-appearance of the Multiply Imaged Supernova Behind MACSJ1149.5+2223. Astrophys. J. 2016, 817, 60.
  * (37) Refsdal, S. On the possibility of determining Hubble’s parameter and the masses of galaxies from the gravitational lens effect. Mon. Not. R. Astron. Soc. 1964, 128, 307–310.
  * (38) Jackson, N. The Hubble Constant. Living Rev. Relativ. 2007, 10, 4.
  * (39) Suyu, S.H.; Marshall, P.J.; Auger, M.W.; Hilbert, S.; Blandford, R.D.; Koopmans, L.V.E.; Fassnacht, C.D.; Treu, T.; et al. Dissecting the Gravitational lens B1608+656. II. Precision Measurements of the Hubble Constant, Spatial Curvature, and the Dark Energy Equation of State. Astrophys. J. 2010, 711, 201–221.
  * (40) Bonvin, V.; Tewes, M.; Courbin, F.; Kuntzer, T.; Sluse, D.; Meylan, G. COSMOGRAIL: the COSmological MOnitoring of GRAvItational Lenses. XV. Assessing the achievability and precision of time-delay measurements. Astron. Astrophys. 2016, 585 id.A88.
  * (41) Bartelmann, M.; Schneider, P. Weak gravitational lensing. Phys. Rep. 2001, 340, 291–472.
  * (42) Tyson, J.A.; Wenk, R.A.; Valdes, F. Detection of systematic gravitational lens galaxy image alignments—Mapping dark matter in galaxy clusters. Astrophys. J. 1990, 349, L1–L4.
  * (43) Bacon, D.J.; Refregier, A.R.; Ellis, R.S. Detection of weak gravitational lensing by large-scale structure. Mon. Not. R. Astron. Soc. 2000, 318, 625–640.
  * (44) Kaiser, N. A New Shear Estimator for Weak-Lensing Observations. Astrophys. J. 2000, 537, 555–577.
  * (45) Seitz, S.; Schneider, P. Cluster lens reconstruction using only observed local data: An improved finite-field inversion technique. Astron. Astrophys. 1996, 305, 383–401.
  * (46)

Lombardi, M.; Bertin, G. A fast direct method of mass reconstruction for
gravitational lenses. Astron. Astrophys.

1999, 348, 38–42.

  * (47) Schneider, P.; King, L.; Erben, T. Cluster mass profiles from weak lensing: Constraints from shear and magnification information. Astron. Astrophys. 2000, 353, 41–56.
  * (48) Han, J.; Eke, V.R.; Frenk, C.S.; Mandelbaum, R.; Norberg, P.; Schneider, M.D.; Peacock, J.A.; Jing, Y.; Baldry, I.; Bland-Hawthorn, J.; _et al_. Galaxy And Mass Assembly (GAMA): the halo mass of galaxy groups from maximum-likelihood weak lensing. Mon. Not. R. Astron. Soc. 2015, 446, 1356–1379.
  * (49) Marshall, P.J.; Hobson, M.P.; Gull, S.F.; Bridle, S.L. Maximum-entropy weak lens reconstruction: improved methods and application to data. Mon. Not. R. Astron. Soc. 2002, 335, 1037–1048.
  * (50) Clowe, D.; De Lucia, G.; King, L. Effects of asphericity and substructure on the determination of cluster mass with weak gravitational lensing. Mon. Not. R. Astron. Soc. 2004, 350, 1038–1048.
  * (51) Corless, V.L.; King, L.J. A statistical study of weak lensing by triaxial dark matter haloes: consequences for parameter estimation. Mon. Not. R. Astron. Soc. 2007, 380, 149–161.
  * (52) Hoekstra, H. A comparison of weak-lensing masses and X-ray properties of galaxy clusters. Mon. Not. R. Astron. Soc. 2007, 379, 317–330.
  * (53) Zhang, Y.-Y.; Finoguenov, A.; Böhringer, H.; Kneib, J.-P.; Smith, G.P.; Kneissl, R.; Okabe, N.; Dahle, H. LoCuSS: Comparison of observed X-ray and lensing galaxy cluster scaling relations with simulations. Astron. Astrophys. 2008, 482, 451–472.
  * (54) Contaldi, C.R.; Hoekstra, H.; Lewis, A. Joint Cosmic Microwave Background and Weak Lensing Analysis: Constraints on Cosmological Parameters. Phys. Rev. Lett. 2003, 22, 221303.
  * (55) Hollenstein, L.; Sapone, D.; Crittenden, R.; Schäfer, B.M. Constraints on early dark energy from CMB lensing and weak lensing tomography. J. Cosmol. Astropart. Physucs 2009, 04, id. 012.
  * (56) Majerotto, E.; Sapone, D.; Schäfer, B,M. Combined constraints on deviations of dark energy from an ideal fluid from Euclid and Planck. Mon. Not. R. Astron. Soc. 2016, 456, 109–118.
  * (57) Clowe, D.; Bradaĉ, M.; Gonzalez, A.H.; Markevitch, M. Markevitch, M.; Randall, S.W.; Jones, C.; Zaritsky, D. A Direct Empirical Proof of the Existence of Dark Matte. Astrophys. J. 2006, 648, L109–L113.
  * (58) Amendola, L.; Appleby, S.; Bacon, D. Baker, T. Baldi, M. Bartolo, N. Blanchard, A. Bonvin, C. Borgani, S. Branchini, E.; et al. Cosmology and Fundamental Physics with the Euclid Satellite. Living Rev. Relat. 2013, 16, 6.
  * (59) Chang, C.; Jarvis, M.; Jain, B. Kahn, S.M.; Kirkby, D.; Connolly, A.; Krughoff, S.; Peng, E.-H.; Peterson, J.R. The effective number density of galaxies for weak lensing measurements in the LSST project. Mon. Not. R. Astron. Soc. 2013, 434, 2121–2135.
  * (60) Heymans, C.; Heavens, A. Weak gravitational lensing: reducing the contamination by intrinsic alignments. Mon. Not. R. Astron. Soc. 2003, 339, 711–720.
  * (61)

Valageas, P. Source-lens clustering and intrinsic-alignment bias of weak-
lensing estimators. Astron. Astrophys.

2014, 561, A53.

  * (62) Mao, S. Astrophysical applications of gravitational microlensing. Res. Astron. Astrophys. 2012, 12, 947–972.
  * (63) Perryman, M. _The Exoplanet Handbook_ ; University Press: Cambridge, UK, 2014.
  * (64) Sumi, T.; Kamiya, K.; Udalski, A.; Bennett, D.P.; Bond, I.A.; Abe, F.; Botzler, C.S.; Fukui, A.; Furusawa, K.; Hearnshaw, J.B.; Itow, Y.; _et al_. Unbound or distant planetary mass population detected by gravitational microlensing. Nature 2011, 473, 349–352.
  * (65) Ingrosso, G.; Calchi Novati, S.; De Paolis, F.; Jetzer, P.; Nucita, A.A.; Strafella, F. Pixel lensing as a way to detect extrasolar planets in M31. Mon. Not. R. Astron. Soc. 2009, 399, 219–228.
  * (66) An, J.H.; Evans, N.W.; Kerins, E.; Baillon, P.; Calchi-Novati, S.; Carr, B.J.; Creze, M.; Giraud-Heraud, Y.; Gould, A.; Hewett, P.; _et al_. The Anomaly in the Candidate Microlensing Event PA-99-N2. Astrophys. J. 2004, 601, 845–857.
  * (67) Dominik, M.; Sahu, K.C. Astrometric Microlensing of Stars. Astrophys. J. 2000, 534, 213–226.
  * (68) Lee, C.-H.; Seitz, S.; Riffeser, A.; Bender, R. Finite-source and finite-lens effects in astrometric microlensing. Mon. Not. R. Astron. Soc. 2010, 407, 1597–1608.
  * (69) Walker, M.A. Microlensed Image Motions. Astrophys. J. 1995, 453, 37–39.
  * (70) Eyer, L.; Holl, B.; Pourbaix, D.; Mowlavi, N.; Siopis, C.; Barblan, F.; Evans, D.W.; North, P. The Gaia Mission. Cent. Eur. Astrophys. Bull. (CEAB) 2013, 37, 115–126.
  * (71) Paczyński, B. The Masses of Nearby Dwarfs and Brown Dwarfs with the HST. Acta Astron. 1996, 46, 291–296.
  * (72) Proft, S.; Demleitner, M.; Wambsganss, J. Prediction of astrometric microlensing events during the Gaia mission. Astron. Astrophys. 2011, 536, A50.
  * (73) Ingrosso, G.; Calchi Novati, S.; De Paolis, F.; Jetzer, P.; Nucita, A.A.; Strafella, F.; Zakharov, A.F. Polarization in microlensing events towards the Galactic bulge. Mon. Not. R. Astron. Soc. 2012, 426, 1496–1506.
  * (74) Chandrasekhar, S. _Radiative Transfer_ ; Clarendon Press: Oxford, UK, 1950.
  * (75) Simmons, J.F.L.; Bjorkman, J.E.; Ignace, R.; Coleman, I.J. Polarization from microlensing of spherical circumstellar envelopes by a point lens. Mon. Not. R. Astron. Soc. 2002, 336, 501–510.
  * (76) Ingrosso, G.; Calchi Novati, S.; De Paolis, F.; Jetzer, P.; Nucita, A.A.; Strafella, F. Measuring polarization in microlensing events. Mon. Not. R. Astron. Soc. 2015, 446, 1090–1097.
  * (77) Henderson, C.B.; Gaudi, B.S.; Han, C.; Skowron, J.; Penny, M.T.; Nataf, D.; Gould, A.P. Optimal Survey Strategies and Predicted Planet Yields for the Korean Microlensing Telescope Network. Astrophys. J. 2014, 794, 52.
  * (78) Penny, M.T.; Kerins, E.; Rattenbury, N.; Beaulieu, J.-P.; Robin, A.C.; Mao, S.; Batista, V.; Calchi Novati, S.; Cassan, A.; Fouque, P.; _et al_. Zapatero Osorio ExELS: An exoplanet legacy science proposal for the ESA Euclid mission - I. Cold exoplanets. Mon. Not. R. Astron. Soc. 2013, 434, 2–22.
  * (79) Yee, J. C.; Albrow, M.; Barry, R.K.; Bennett, D.; Bryden, G.; Chung, S.-J.; Gaudi, B.S.; Gehrels, N.; Gould, A.;  Penny, M.T.; _et al_. Takahiro SumiNASA ExoPAG Study Analysis Group 11: Preparing for the WFIRST Microlensing Survey. 2014, arXiv:1409.2759.
  * (80) Heyrovský, D.; Sasselov, D. Detecting Stellar Spots by Gravitational Microlensing. _Astrophys. J._ 2000, 529, 69–76.
  * (81) Hendry, M.A.; Bryce, H.M.; Valls-Gabaud, D. The microlensing signatures of photospheric starspots. _Mon. Not. R. Astron. Soc._ 2002, 335, 539–549.
  * Giordano et al. (2015) Giordano, M.; Nucita, A.A.; De Paolis, F.; Ingrosso, G. Starspot induced effects in microlensing events with rotating source star. _Mon. Not. R. Astron. Soc._ 2015, 453, 2017–2021.
  * Kim et al. (2010) Kim, S.-L.; Park, B.-G.; Lee, C.-U.; Yuk, I.-S.; Han, C.; O’Brien, T.; Gould, A.; Lee, J.W.; Kimet, D.-J. Technical specifications of the KMTNet observation system. _Proc. SPIE_ 2010, doi:10.1117/12.856833.
  * Sajadian (2015) Sajadian, S. Detecting stellar spots through polarimetric observations of microlensing events in caustic-crossing. _Mon. Not. R. Astron. Soc._ 2015, 452, 2587–2596.
  * Dominik (1999) Dominik, M. The binary gravitational lens and its extreme cases. _Astron. Astrophys._ 1999, 349, 108–125.
  * An (2005) An, J. H. Gravitational lens under perturbations: symmetry of perturbing potentials with invariant caustics. _Mon. Not. R. Astron. Soc._ 2005, 356, 1409–1428.
  * Griest & Safizadeh (1998) Griest, K.; Safizadeh, N. The Use of High-Magnification Microlensing Events in Discovering Extrasolar Planets. _Astrophys. J._ 1998, 500, 37–50.
  * Penny et al. (2011) Penny, M. T.; Kerins, E.; Mao, S. Rapidly rotating lenses: Repeating features in the light curves of short-period binary microlenses. _Mon. Not. R. Astron. Soc._ 2011, 417, 2216–2229.
  * Nucita et al. (2014) Nucita, A. A.; Giordano, M.; De Paolis, F.; Ingrosso, G. Signatures of rotating binaries in microlensing experiments. _Mon. Not. R. Astron. Soc._ 2014, 438, 2466–2473.
  * (90) Chandrasekhar, S. _The Mathematical Theory of Black Holes_ ; Oxford University Press: Oxford, UK, 1983.
  * (91) Holz, D.E.; Wheeler, J.A. Retro-MACHOs: pi in the Sky. Astrophys. J. 2002, 57, 330–334.
  * (92) De Paolis, F.; Geralico, A.; Ingrosso, G.; Nucita, A.A. The black hole at the galactic center as a possible retro-lens for the S2 orbiting star. Astron. Astrophys. 2003, 409, 809–812.
  * (93) Gillessen, S.; Eisenhauer, F.; Trippe, S.; Alexander, T.; Genzel, R.; Martins, F.; Ott, T. Monitoring Stellar Orbits Around the Massive Black Hole in the Galactic Center. Astrophys. J. 2009, 692, 1075–1109.
  * (94) Dokuchaev, V.I. Spin and mass of the nearest supermassive black hole. Gen. Rel. Gravit. 2014, 46, 1832.
  * (95) Dokuchaev, V.I.; Eroshenko, Y.N. Physical Laboratory at the center of the Galaxy. Phys. Uspekhi 2015, 58, 772–784.
  * (96) Iorio, L. Perturbed stellar motions around the rotating black hole in Sgr A* for a generic orientation of its spin axis. Phys. Rev. D 2012, 84, 124001.
  * (97) Zakharov, A.F.; Nucita, A.A.; De Paolis, F.; Ingrosso, G. Apoastron shift constraints on dark matter distribution at the Galactic Center. Phys. Rev. D 2007, 76, 062001.
  * (98) Zakharov, A.F.; de Paolis, F.; Ingrosso, G.; Nucita, A.A. Shadows as a tool to evaluate the black hole parameters and a dimension of spacetime. New Astron. Rev. 2012, 56, 64–73.
  * (99) Falcke, H.; Markoff, S.B. Toward the event horizon—The supermassive black hole in the galactic center. Classicala Quantum Gravit. 2013, 30, 244003.
  * (100) Zakharov, A.F.; Borka, D.; Borka Jovanović, V.; Jovanović, P. Constraints on $R^{n}$ gravity from precession of orbits of S2-like stars: A case of a bulk distribution of mass. Adv. Space Res. 2014, 54, 1108–1112.
  * (101) Zakharov, A.F.; Nucita, A.A.; De Paolis, F.; Ingrosso, G. Solar system constraints on $R^{n}$ gravity. Phys. Rev. D 2006, 74, 107101.
  * (102) Nucita, A.A.; De Paolis, F.; Ingrosso, G.; Qadir, A.; Zakharov, A.F. Sgr A*: A Laboratory to Measure the Central Black Hole and Stellar Cluster Parameters. Publ. Astron. Soc. Pac. 2007, 119, 349–359.
  * (103) Plewa, P.M. et al. Pinpointing the near-infrared location of Sgr A∗ by correcting optical distortion in the NACO imager 2015, arXiv: 1509.01941.
  * (104) De Paolis, F.; Ingrosso, G.; Nucita, A.A.; Qadir, A.; Zakharov, A.F. Estimating the parameters of the Sgr A∗ black hole. Gen. Rel. Gravit. 2011, 43, 977–988.
  * (105) Zakharov, A.F.; De Paolis, F.; Ingrosso, G.; Nucita, A. Direct measurements of black hole charge with future astrometrical missions. Astron. Astrophys. 2005, 442, 795–799.
  * (106) Bozza, V.; Mancini, L. Gravitational Lensing by Black Holes: A Comprehensive Treatment and the Case of the Star S2. Astrophys. J. 2005, 611, 1045–1053.
  * (107) Doeleman, S.S.; Weintroub, J.; Rogers, A.E.E.; Plambeck, R.; Freund, R.; Tilanus, R.P.J.; Friberg, P.; Ziurys, L.M.; Moran, J.M.; Corey, B.; _et al_. Event-horizon-scale structure in the supermassive black hole candidate at the Galactic Centre. Nature 2008, 455, 78–80.