11institutetext: Department of ICT and Natural Sciences, Norwegian University
of Science and Technology, Larsgårdsvegen 2, 6009 Ålesund
11email<EMAIL_ADDRESS>22institutetext: The University Library,
Norwegian University of Science and Technology, Larsgårdsvegen 2, 6009 Ålesund

# A short review on joint weak and strong cluster lens-mass reconstruction

Ben D. Normann 11 Kenny Solevåg-Hoti 22 Hans Georg Schaathun 11

(Received after we’ve sent it; accepted at some event E on our future light-
cone)

The divide between weak and strong lensing is of course artificial, in that
both regimes are manifestations of the same physical phenomenon: gravity
bending the path of light. Nevertheless, these two regimes have to a large
extent been treated separately, with different methods. This review traces the
development of methods combining weak-lensing and strong-lensing data for
joint lens-mass reconstruction, with a particular emphasis on cluster lenses,
where both effects occur. We conclude that so-called inverse methods have been
successful in merging the two regimes insofar data analysis is concerned.
However, a number of improvements seem to be needed. First, not many studies
include weak lensing data beyond shear. In light of the unprecedented quality
of the data of JWST and future surveys, this is a clear point of improvement.
Especially so, since flexion terms have proven useful in determining sub-
structures. Second, considering the amount of data available, and the
complexity of non-parametric lenses, automating the processes of lens-mass
reconstruction would be beneficial. Towards this end, invoking machine
learning seems like a promising way forward. The silence of the literature on
this latter point is in fact somewhat surprising.

###### Key Words.:

Cosmology – gravitational lensing – cluster lensing – weak lensing – strong
lensing – roulette formalism – roulette expansion – flexion – beyond shear

##  1 Introduction

The idea of light being bent by gravity, goes back at least to Newton (1804)
himself, and later Mitchell (1784) and Laplace (1795). But it was not until
the 20th century that it was actually put to use as an observational probe on
the universe. Under a solar eclipse in 1905, Eddington and others showed that
Einstein’s theory of gravity predicted the correct lensing of light, whereas
Newton’s theory gets it wrong (misses a factor 2). Today, well over a century
later, gravitational lensing is recognised as an important cosmological probe,
and a rich source of information. One of its uses is the mapping of the
(mostly dark) matter in the universe. Mapping the dark matter through
gravitational lensing, invokes (cluster) lens-mass reconstruction, which will
be the focus of this review.

The effect of gravitational lensing is typically divided into two regimes;
weak and strong, both of which are important in understanding complicated
lensing structures, such as cluster lenses. Since they are both observational
aspects of the same phenomenon, it is natural to wonder to what extent they
are still treated separately. This paper reviews the literature on different
approaches to merging the two regimes into one unified framework for cluster
111We focus here on cluster lenses, where both effects occur. lens-mass
reconstruction.

We take a hybrid approach, combining a structured literature search with a
historical approach using prior knowledge and snowball searches. On the one
side, we use stringent search criteria, winding up with a total of 29 papers
that we examine in some detail. On the other side, we use previous knowledge
and snow-ball searches to complement the historical development. The result is
a quasi-historical development.

The rest of the paper is structured as follows. In Section 2 we describe our
literature search strategy and study selection. In Section 3, we go on to
describe lensing theory in general, before we focus in on the development of
(cluster) lens-mass reconstruction in Section 4. This is where the bulk of
papers included in the systematic literature review are presented, describing
the development up until the state-of-the-art. Finally, in Section 5 we
present our concluding remarks.

##  2 Combining weak and strong: Search method

This review considers the joint use of weak- and strong lensing data for lens-
mass reconstruction. To this end a structured search was conducted, as
described in this section.

###  2.1 Search strategy

Table 1: Preliminary searches in the topic field of the Web of Science
database 28th of October 2022

Search | String | Hits
---|---|---
1 | weak AND strong AND lens* | 1471
2 | weak AND strong AND lens* AND reconstruct* | 171
3 | (combin* OR unit*) AND weak AND strong AND lens* | 37
| AND reconstruct* |
4 | (combin* OR unit* OR unify* OR join* OR merg* | 95
| OR bind* OR link* OR expand* OR extend*) AND weak |
| AND strong AND lens* AND reconstruct* |

94-9798-0102-0506-0910-1314-1718-2122-24$0$$1$$2$$3$$4$$5$$6$YearNumber of
papers Figure 1: The histogram shows the number of papers from the shortlist
appearing each year, starting in 1994, and with a bin-size of 4 years, except
the last bin, which spans 3 years.

To create a search string that would effectively capture results relevant to
this review we went through several iterations. Searches were made in the Web
of Science database, chosen for its breadth and reliability. Attention was
restricted to title, abstract and keywords, as it was found unlikely that the
relevant literature would omit our search words in all these fields. Starting
from a broad search we modified the search string to enforce the right
context.

The initial search returned articles from a range of unrelated scientific
fields. The next iteration narrowed the hits to our field of interest, but
with noise still apparent. The main issue was articles where both weak and
strong formalisms were used, but handled independently of each other.
Correcting for this we made use of words present in highly relevant
publications in our next search. These hits were relevant, yet limited our
scope. Hence, we readjusted for the possibility of authors using a range of
synonyms. This lead us to our final query:

> (combin* OR unit* OR unify* OR join* OR merg* OR bind* OR link* OR expand*
> OR extend*) AND weak AND strong AND lens* AND reconstruct*

This search was then applied to the databases Web of Science, Scopus,
Compendex and ArXiv providing comprehensive coverage of the field of physics.
This resulted in 111 unique papers, when cross-checking the results of the
different database searches. The breakdown is given in Table 2.

Table 2: Number of papers found in each database in the search performed onthe 28th of October 2022. There are 111 unique papers. Database | Search field | Hits
---|---|---
Web of Science | Topic | 95
Scopus | Title, abstract, keywords | 67
ArXiv | Abstract | 55
Compendex | Subject, title, abstract | 53

Going through the abstracts, we selected those papers that concentrated on the
theoretical framework for weak and strong lensing. Papers that (judging from
the abstracts) seemed to be mere observational ones, or papers that
concentrated on theoretical treatment of other observables, such as X-rays,
were thus discarded. This discarding-by-abstract step was as such a step of
calculated risk, as it might sometimes be hard to judge the novelty of the
theoretical methodology underlying an observational study. Especially so,
since a successful theoretical method for observational astrophysics is highly
dependent on each individual observational situation. Nevertheless, if a
methodology is particularly novel, one would expect it to be presented in a
theoretical paper on its own. Consequently, and performed with some care, this
step was taken in order to narrow down the scope of the material.

###  2.2 Study selection

As a result a total of 29 papers (two of which appeared after the original
search was done) were left for deeper scrutiny. This list contained all the
papers that were found in an initiation search222Actually, the most
interesting papers were already found in our initial search!, before a
structured approach was adopted. The final list of 29 papers is presented in
Appendix A. Over the course of the next couple of sections we discuss all the
29 papers from the list, in a quasi-historical manner. Snowball searches
greatly expanding the list has also been adopted, so that the total number of
papers referenced is about five-fold that of the shortlist (29).

It is noteworthy that a whole strand of papers relevant for the topic at hand
was, with a couple of exceptions, left out. These were the direct methods
extending the weak lensing regime towards the strong regime, by including
higher-order effects, such as flexion. Allthough we have not performed a
systematic literature search to include such papers, they are discussed
generally in Section 3.4 and in the context of cluster lenses in Sec. 4.7.

##  3 Lensing theory

In the following, we present the basic mathematical framework used in state-
of-the-art gravitational lensing. The interested reader is referred to
standard texts such as Schneider et al. (1999) (Often denoted SEF) or
Schneider et al. (2006) for details, or more recent treatments, like Congdon &
C.R. (2018).

Figure 2: Lensing geomtry. $D\textsubscript{L}$ and $D\textsubscript{S}$ are
angular diameter distances from observer to lens and observer to source,
whilst $D\textsubscript{LS}$ is lens to source. $\beta$ and $\theta$ are
angular separations of source and image from the optical axis. The light-ray
deflection is denoted by $\hat{\alpha}$, whilst $\alpha$ is the reduced
deflection angle.

The starting point of gravitational lensing is the so-called ray-trace
equation

$\bm{\theta}D_{\mathrm{S}}=\bm{\beta}D_{\mathrm{S}}+\hat{\bm{\alpha}}D_{\mathrm{LS}},$
(1)

which relates angular position $\bm{\theta}$ in the lens plane, at distance
$D_{\textrm{L}}$ from the observer, to corresponding positions $\bm{\beta}$ in
the source plane, at distance $D_{\textrm{S}}$ from the observer. This
equation follows directly from the geometry assumed in Fig. 2, when equipped
with an additional assumption of small angels. The geometry typically333Not so
with the Schwarzschild lens, where exact results are available. follows due to
a thin-lens approximation, which means that all the mass has been projected
down to a two-dimensional plane orthogonal to the optical axis. This is often
warranted as the region where the lens affects the light beam is relatively
localized, compared to the distances $D_{\rm L}$ and $D_{\rm S}$. In fact,
most lens-mass reconstruction techniques seeks to reconstruct the
2-dimensional, lens-plane projected mass density. This reveals a fundamental
shortcoming of the current capacity of GL. Albeit being standard procedure,
the thin-lens approximation cannot reveal three-dimensional structures of a
single lens — at least not in a model-free manner. In reality lenses are
however three-dimensional, and could be of importance. Take Morandi et al.
(2011) as an example, where A1689 is modelled with a triaxial structure. This
allows to explain discrepancies between x-ray and strong-lensing data as a
result of the internal structure of the cluster. As such this demonstrates the
importance of taking the 3-dimensional structure of the lens into account in
explaining the data.

Defining the reduced deflection angle by

$\bm{\alpha}(\bm{\theta})\equiv\frac{D_{\mathrm{LS}}}{D_{\mathrm{S}}}\bm{\hat{\alpha}}(\bm{\theta}),$
(2)

the lens equation is obtained as

$\bm{\beta}=\bm{\theta}-\bm{\alpha}(\bm{\theta}).$ (3)

In the thin-lens approximation, one defines the dimensionless surface-mass
density $\kappa$ (also called convergence), given by

$\kappa(\bm{\theta})=\frac{\Sigma(\bm{\theta})}{\Sigma\textsubscript{cr}}.$
(4)

Here $\Sigma$ is the surface-mass density, and $\Sigma_{\textrm{cr}}$ is a
critical value defined such that

$\Sigma\textsubscript{cr}=\frac{c^{2}}{4\pi
G}\frac{D\textsubscript{s}}{D\textsubscript{d}D\textsubscript{ds}}.$ (5)

It is now customary and convenient to define the lensing potential $\psi$
(also called the deflection potential) from Poisson’s equation for gravity in
two dimensions444The lens plane., which takes the form

$\nabla^{2}\psi=2\kappa.$ (6)

One may use Green’s functions to show that an integral solution to this
equation is given by

$\psi(\bm{\theta})=\frac{1}{\pi}\int_{\mathbb{R}^{2}}\mathrm{\leavevmode\nobreak\
d}^{2}\theta^{\prime}\kappa\left(\bm{\theta}^{\prime}\right)\ln\left|\bm{\theta}-\bm{\theta}^{\prime}\right|.$
(7)

It follows that the (reduced) deflection angle $\bm{\alpha}$ may be written as

$\bm{\alpha}=\nabla\psi.$ (8)

Using this, the lens equation (3) takes the form

$\bm{\beta}=\bm{\theta}-\gradient\psi(\bm{\theta}).$ (9)

Consequently, for a given lens $\psi$, Eq (9) is a mapping
$\mathcal{L}_{\psi}:\mathcal{U}\,\rightarrow\,\mathcal{D}$, where
$\mathcal{U}$ and $\mathcal{D}$ are sets of undistorted and distorted images,
respectively. Since distorted background sources are much smaller than the
angular scale on which the lens properties change, the mapping can be locally
linearized and the lens mapping (9) may be written as

$\bm{d}\bm{\beta}=\mathcal{A}\,\bm{d}\bm{\theta}$ (10)

where the so-called amplification matrix $\mathcal{A}$ is the Jacobian of the
coordinate transformation. By way of (9) we have

$\displaystyle\mathcal{A}(\bm{\theta})$
$\displaystyle=\frac{\partial\bm{\beta}}{\partial\bm{\theta}}=\left(\delta_{ij}-\psi_{ij}\right)\quad\quad\textrm{where}\quad\quad\psi_{ij}\equiv\frac{\partial^{2}\psi}{\partial\theta^{i}\partial\theta^{j}}.$
(11)

Any Hermitian $2\times 2$ matrix can be decomposed into the identity matrix
and the Pauli spin matrices;

$\displaystyle\mathcal{I}=\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\quad,\quad\sigma_{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}\quad,\quad$ (12)
$\displaystyle\sigma_{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix}\quad,\quad\sigma_{3}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}.$ (13)

It follows from (9) that $\mathcal{A}$ takes the form

$\mathcal{A}=(1-\kappa)\mathcal{I}-\gamma_{\times}\sigma_{1}+\textrm{i}\rho\sigma_{2}-\gamma_{-}\sigma_{3},$
(14)

where standard theory yields $\rho=0$, since gravity is curl-free at the level
of approximation used (refer to Bacon & Schafer (2009) for an interesting
discussion). In the above we have defined the (complex) shear

$\gamma=\gamma_{+}+i\gamma_{\times}$ (15)

with (real) components

$\gamma_{+}=\frac{1}{2}\left(\psi_{11}-\psi_{22}\right),\quad\gamma_{\times}=\psi_{12}=\psi_{21}.$
(16)

By defining the reduced shear,

$g\equiv g_{+}+ig_{\times}$ (17)

with components

$g_{+}\equiv\frac{\gamma_{+}}{1-\kappa}\quad,\quad
g_{\times}\equiv\frac{\gamma_{\times}}{1-\kappa},$ (18)

one could alternatively write

$\mathcal{A}=(1-\kappa)\left(\begin{array}[]{cc}1-g_{+}&-g_{\times}\\\
-g_{\times}&1+g_{+}\end{array}\right).$ (19)

From the latter form of $\mathcal{A}$ it is particularly apparent that
$(1-\kappa)$ is a scaling factor, whereas the (reduced) shear alters the shape
of the image (by elongation). Consequently, based on the assumption of
randomly oriented intrinsic ellipticities in the sources 555Or otherwise a
known distribution of ellipticities., measurements of image ellipticities
yield an estimate of the reduced shear. This may in turn be used to obtain the
surface-mass density, as follows. Equation (7), (15) and (16) together give

$\gamma(\bm{\theta})=\frac{1}{\pi}\int_{\mathbf{R}^{2}}\mathrm{\leavevmode\nobreak\
d}^{2}\theta^{\prime}\mathcal{D}\left(\bm{\theta}-\bm{\theta}^{\prime}\right)\kappa\left(\bm{\theta}^{\prime}\right),$
(20)

with

$\mathcal{D}(\bm{\theta})\equiv\frac{\theta_{2}^{2}-\theta_{1}^{2}-2\mathrm{i}\theta_{1}\theta_{2}}{|\bm{\theta}|^{4}},$
(21)

That is, $\gamma$ is a convolution of $\kappa$ with kernel $\mathcal{D}$, a
relation that may be inverted to give

$\kappa(\bm{\theta})-\kappa_{0}=\frac{1}{\pi}\int_{\mathbf{R}^{2}}\mathrm{\leavevmode\nobreak\
d}^{2}\theta^{\prime}\mathcal{R}\text{e}\left[\mathcal{D}^{*}\left(\bm{\theta}-\bm{\theta}^{\prime}\right)\gamma\left(\bm{\theta}^{\prime}\right)\right].$
(22)

This relation opens up the possibility of obtaining matter maps of the
universe by measuring the shear of distantly incoming light congruences.
However, shear in the weak-lensing regime is a relatively weak distortion, and
thus relatively hard to detect. In fact, it is necessary to average over large
samples of galaxies for the effect to appear. Obtaining the surface-mass
density by way of shear measurements have been extensitvely applied since its
first measurement (Tyson et al. 1990). A more modern example is Jauzac et al.
(2012). But how is shear actually measured? In the following subsection we
provide a more detailed account of the advance of these so-called direct lens-
mass reconstructions, based on (22).

###  3.1 Direct lens-mass reconstruction

Obtaining the lens mass via other observables, such as the shear, is known as
lens-mass reconstruction. The relation (22) was used by Kaiser and Squires
(Kaiser & Squires 1993) in developing a parameter-free666Or non-parametric.
This simply means that no a-priori model for the lens is made, as further
discussed in a later section. inversion technique to obtain the surface-mass
density from shear measurements. This inversion technique — hereafter simply
referred to as the KS inversion technique — quantifies the shear using the
quadrupole-moment tensor of the surface brightness (Miralda-Escude 1991);
(Miralda-Escude 1991). Following Congdon & C.R. (2018, Chapt. 7), the
quadrupole-moment tensor is given by

$Q_{ij}=\frac{\int_{\mathbb{R}^{2}}(\theta_{i}-\bar{\theta}_{i})(\theta_{j}-\bar{\theta}_{j})I(\bm{\theta})\textrm{d}^{2}\theta}{\int_{\mathbb{R}^{2}}I(\bm{\theta})\textrm{d}^{2}\theta},$
(23)

where $\bar{\bm{\theta}}=(\bar{\theta_{1}},\bar{\theta_{2}})$ is the image
centroid position, given by

$\bar{\bm{\theta}}=\frac{\int_{\mathbb{R}^{2}}\bm{\theta}I(\bm{\theta})\textrm{d}^{2}\theta}{\int_{\mathbb{R}^{2}}I(\bm{\theta})\textrm{d}^{2}\theta}.$
(24)

Note that the KS inversion technique hinges on the assumption that the overall
orientation $\chi$ of each galaxy is randomly distributed777As a side remark:
Let it be mentioned that also Tyson et al (1990) assumed that the net
alignment of the source galaxies was zero in their detection prior to the work
by Kaiser and Squires. (ensamble average), so that there is no intrinsic
alignment; id est,

$\langle\chi^{2}\rangle=0.$ (25)

The validity of such an assumption is of course questionable, as reviewed in
Troxel & Ishak (2015). Making use of the assumption (25), one may however
compare source- and image ellipticities, by such providing a measure of the
(reduced) shear. This is in turn inverted to obtain the surface-mass density
of the lens. A problem with the method, is that it is valid only for
sufficiently weak lensing, where the (complex) distortions $\delta$ produced
by the lens are linearly related to the shear. In particular one has

$\delta\approx 2g\approx 2\gamma\quad\quad\textrm{linear regime}$ (26)

In Schneider & Seitz (1995); Seitz & Schneider (1995), the method is extended
to the nonlinear regime, relieving the assumption of weak lensing, as is
necessary to probe e.g. the center of clusters. The authers show that in the
generalized formulation, the shear is not an observable quantity. The only
local observable from image distortions is the (complex) distortion $\delta$,
given by

$\delta=\frac{2g}{1+\absolutevalue{g^{2}}},\quad\quad\textrm{nonlinear
regime}$ (27)

with $g$ given in Eq. (17). Evidently, Eq (27) reduces to (26) for
$\kappa,\gamma\,\ll 1$. Another problem with the KS inversion technique is
that the inversion formulae is exact only if one assumes observational data on
the whole lens plane. The finitude of the CCD or the data field thus causes
problems, an issue discussed in various ways in Schneider (1995); Seitz &
Schneider (1996); Kaiser et al. (1995); Bartelmann (1995). In the final paper
in the series (Seitz & Schneider 1997), redshift information is also taken
into account. An improved finite-field inversion method similar to the one
developed in the aforementioned literature is also presented in Seitz &
Schneider (2001).

Before the end of the millenium, the KS method had thus been improved to
account for

  1. a.

Strong tidal fields (non-linear regime) in cluster centers.

  2. b.

Finite data fields.

  3. c.

Redshift distribution of background galaxies.

The resulting methods were computationally relatively efficient. Also, they
were local, in the sense that the surface-mass density is obtained from
observed ellipticities of background galaxies. However, as discussed in (Seitz
et al. 1998), a number of drawbacks were also present:

  * •

While the data needs to be smoothed, objective criteria for how to set the
smoothing scale is lacking.

  * •

The quality of the reconstruction is hard to quantify.

  * •

Constraints from additional observables, such as multiple images or
arclets888Arclet seems to be the term used for sheared (stretched) images,
whereas the term arc is reserved for stronger (flexed) distortions., are not
easily included.

  * •

The magnification (which is needed to break the so-called mass-sheet
degeneracy (to be discussed), is incoorporated in a global fashion.

Finally, note also the so-called quadruple method introduced by Lombardi &
Bertin (1998b), where the isotropy requirement (25) is replaced by an isotropy
rquirement on the observable quadrupole moments instead. Note also the works
on improving the accuracy of mass reconstruction in various ways (Lombardi &
Bertin 1998a; Lombardi et al. 2002). In Lombardi & Bertin (1999), mass-
reconstruction is considered through a variational formulation. The procedure
formulated is reported to reduce the relevant execution time by a factor from
100 to 1000 with respect to the fastest methods available at the time.

###  3.2 Inverse methods for lens-mass reconstruction

The aforementioned issues with the standard KS method, or direct methods more
generally, lead to the development of so-called inverse methods, such as
maximum-likelihood methods (Squires & Kaiser 1996; Bartelmann et al. 1996). In
the inverse methods, one seeks to fit a very general lens model with the data.
One typically parametrizes the lens potential on a grid and then minimizes the
regularized log-likelihood function. Take $N_{\textrm{g}}$ to be the number of
galaxies and $\epsilon_{i}$ to denote ellipticity of the $i$-th galaxy. The
shear-likelihood function is then given by (Schneider et al. 2006, Eq. 62)

$\displaystyle-\ln{\mathcal{L}}=\sum_{i=1}^{N_{\textrm{g}}}\frac{\absolutevalue{\epsilon_{i}-g(\bm{\theta}_{i},{\psi_{n}})}^{2}}{\sigma_{i}^{2}(\bm{\theta}_{i},{\psi_{n}})}+2\ln{(\sigma_{i}(\bm{\theta}_{i},{\psi_{n}}))}+\lambda_{\textrm{e}}S({\psi_{n}}),$
(28)

where $\sigma_{i}$ are standard deviations in the ellipticities, $\bm{\theta}$
are angular positions and $\psi$ is the discretized potential. The parameter
$\lambda_{\textrm{e}}$ is a Lagrangian multiplier giving the relative weight
of the regularization and the likelihood function. In particular; note that
strong-lensing constraints may now be added as Lagrangian multipliers. Thus,
in such a formulation of the problem, other types of constraints (e.g. strong
lensing and magnification) are more readily included. As an early example,
consider Bartelmann et al. (1996) in which magnification constraints are
included (to effectively break the mass-sheet degeneracy). The
$\chi^{2}$-function

$\chi^{2}=\sum_{k,l}\left(\frac{(g_{i}(k,l)-\mathbf{g}_{i}(k,l))^{2}}{\sigma_{\textrm{g}}(k,l)}+\frac{(r_{i}(k,l)-\mathbf{r}_{i}(k,l))^{2}}{\sigma_{\textrm{r}}(k,l)}\right)$
(29)

is minimized numerically using a conjugate-gradient method. Here
$\mathbf{r}_{i}$ represents the ( inverse) magnification data. The method
allows for straight forward inclusion of measurement inaccuracies, and is
designed to work well with clusters, where the observed field is small or
irregularly shaped. This method was extended by Seitz et al. (1998) to include
regularization of the potential and use the individual galaxy ellipticities
and positions instead of averaging by dividing the image into cells. Bridle et
al. (1998) also includes magnification constraints by a maximum-likelihood
approach, here seen as a special case of the maximum-entropy method (MEM) in
the context of Bayes’ theorem. This method allows for direct reconstruction of
the mass distribution instead of going via the gravitational potential.

Other log-likelihood approaches include Marshall (2006), in which an atomic-
inference procedure is adopted. Only WL effects are considered, but
incorporation of SL effects is reported to be straight forward. Beyond that,
the reader is referred to the book-sized review Bartelmann & Schneider (2001)
on the early developments of weak gravitational lensing, and weak lensing
theory in general.

As we shall see in Section 4, inverse methods have become the standard tool
for lens-mass reconstruction. The key factor for its success is its linear
problem formulation, where both weak and strong constraints (and others) can
be added as linear constraints in a joint analysis.

####  3.2.1 Degeneracies in the Lens Equation

It was early recognized that the lensing equation suffers from a number of
degeneracies, owing to the fact that most observables of graviational lensing
are dimensionless. Early sources on this include Falco et al. (1985) and
Gorenstein et al. (1988), written in a time when cluster lenses were still
controversial. Later papers, such as Saha (2000) rederives the degeneracies,
putting them in a new observational context. Herein, so-called similarity
degeneracies are broadly categorized as either a (i) distance degeneracy or a
(ii) angular degeneracy. Another type, the so-called mass-sheet degeneracy,
also appears for sources at the same redshift999Note also the generalization
discussed in Liesenborgs et al. (2008). In this case the problem is that the
observable distortion, $g=\gamma/(1-\kappa)$ is invariant under the
transformation

$\kappa\rightarrow(1-\lambda)+\lambda\kappa,\quad\gamma\rightarrow\lambda\gamma.$
(30)

Globally, this is equivalent to the rescaling

$\mathcal{A}\rightarrow\lambda\mathcal{A}\quad\quad,\quad\lambda\,=\textrm{const.}\,\neq\,0,$
(31)

which leaves the critical curves $\det{\mathcal{A}}=0$ invariant (refer to
e.g. Saha & Williams (2006); Liesenborgs & De Rijcke (2012) for more). This
means that several source positions may give rise to the same observed
position, give and take a mass sheet of constant density. In particular, this
transformation keeps the critical curves of the lens mapping fixed, and thus
the location of the giant arcs and arclets. The mass-sheet degeneracy is an
example of a broader class of source-plane transformation issues (Schneider &
Sluse 2014) for strong lenses. Consequently, since one cannot distinguish
between lenses when using distortion data alone (for sources at the same
redshift), lens-mass reconstruction algorithms must therefore take measures to
break the degeneracy through some means or another. One way to go, would be to
use magnification, as tried by Broadhurst et al. (2005, 1994). However,
Schneider et al. (2000) points out uncertainties in unlensed source counts,
leaving the magnification method less prosperous. Bradač et al. (2004a) sets
out to use redshifts instead, and concludes that it is necessary to
simultaneously analyse weak and strong lensing to break the mass-sheet
degeneracy. Thus it is about time that we give a more formal definition of the
difference between the weak and strong lensing regimes.

###  3.3 The difference between weak and strong lensing

Strong lensing refers to conditions allowing for multiple images of the same
source (Congdon & C.R. 2018, Chapter 2). For this to happen, the lens must
posess a certain ’strength’. The divide between weak and strong lensing is
arguably somewhat arbitrary, since it is all part of the same underlying
physical phenomena. Nevertheless, from the mathematical side one may show that
a sufficient101010Sufficient and necessary in the case of axisymmetric lenses.
Only sufficient in the more general case. requirement for producing several
images of a source under the mapping (1) is $\kappa(0)>1$, a condition known
as supercriticality. Furthermore, it may be shown that the magnification
diverges whenever $\det\mathcal{A}=0$, producing the characteristic arcs
associated with GL. We can infer the following.

  * •

Strong-lensing data ($\kappa\,>\,1$): Incorporating multiple imaging and /or
critical curves ($\det\mathcal{A}=0$).

  * •

Weak-lensing data ($\kappa\,<\,1$): This means incorporating shear
measurements in the analysis, and sometimes flexion (to be discussed). Note
that the aforementioned linear regime ($\kappa,\gamma\,\ll\,1$) lies in the
weak-lensing regime.

The reason why these effects have been studied separately, is to some extent
historical, owing to the methods and sensitivity available. On the one hand
you have the rare but visually apparent effect of strong lensing. This effect
is visual, and observed as giant luminous arcs, or multiple images. The
sensitivity required of the instruments to capture this effect is therefore
low compared to that of weak lensing, which must typically be treated
statistically over large samples of lensed images. Whereas strong lensing was
used for lens-mass estimation already with Zwicky (1937a, b), weak lensing had
its dawn in the 90s, as described earlier.

###  3.4 Beyond shear: Banana or jelly-bean?

The lensing theory described in the beginning of this chapter, hinges on the
map given in Eq. (9), with the linearisation given by (10). For infinitesimal
beams, a linearisation is warranted, but if one intends to take into account
the finitude of the beam, the linearisation breaks down. Put differently: If
the shear varies over the cross-section of the image, higher-order effects,
such as flexion and second-flexion come into play. First-flexion is a shift in
position of the image centroid, whereas second-flexion creates the banana-
looking shape111111or is it a jelly bean?. The flexion is a second-
order121212If we by ‘order´ mean the number of derivatives in a Taylor-
expansion. effect, and thus weaker than shear in the weak field. On one hand,
this could cause problems in detecting it. On the other hand, however, source
galaxies are not believed to be intrinsically flexed, which provides an
advantage: the flexion generally belongs to the image. One way to quantify the
flexion of a lensed image, is by multipole moments, as for instance in the
early work Goldberg & Natarajan (2002), in which it is shown how flexion
(octopole moment) can be used as a probe of weak-lensing shear fields. Note
also later contributions such as Goldberg & Leonard (2007); Okura et al.
(2007, 2008). Another way to quantify flexion, is through the shaplets-
formalism due to Refregier. This was e.g. used in Goldberg & Bacon (2005),
where an inversion-technique based on flexion was investigated, reporting an
increased signal-to-noise ratio. In Bacon et al. (2006), the flexion-formalism
is tidily presented in complex form, also revealing a spin-3 field which has
come to be known as the aforementioned second-flexion. To summarize, one has

$\displaystyle\mathcal{F}=\absolutevalue{\mathcal{F}}{\rm e}^{{\rm
i}\phi}=\partial\kappa\quad\quad\textrm{first flexion (spin-1)},$ (32)
$\displaystyle\mathcal{G}=\absolutevalue{\mathcal{G}}{\rm e}^{{\rm
i}3\phi}=\partial\gamma\quad\quad\textrm{second flexion (spin-3)},$ (33)

where $\partial=\partial_{1}+i\partial_{2}$ is a complex derivative
incoorporating the two directions (${1,2}$) of the screen-space and star
denotes complex conjugation. Sometimes $\mathcal{F}$ is referred to as comatic
flexion and $\mathcal{G}$ is referred to as trefoil flexion. The flexion has
four components, and the relation between the spin-1 and spin-3 components are
given by

$\partial^{*}\partial\mathcal{G}=\partial\partial\mathcal{F}\quad\quad\textrm{consistancy
relation},$ (34)

which can be readily proven. The paper also calculates the flexion
coefficients for various spherical lensing profiles. In Bacon & Schafer (2009)
it is also shown how the flexion term arises when one includes the final Pauli
spin matrix n The reader should also note the seemingly independent
developement of higher-order lensing throughout the papers Irwin & Shmakova
(2003, 2005, 2006); Irwin et al. (2007), under the name sexupole
lensing131313This seems to be in contrast to Goldbergs’ notion of flexion as
an octopole term.. Allthough originally intended for galaxy-galaxy-lensing,
flexion measurements were already in 2007 performed on the cluster Abell 1689,
revealing a dark matter structure that was not apparent from shear
measurements alone Leonard et al. (2007). Flexion can also be estimated
through the so-called analytic image model (AIM) Cain et al. (2011), and a
subsequent study based on this method confirms that significant sub-structures
may remain invisible if this effect is not taken into account in cluster
lensing Cain et al. (2016). Flexion with elliptical lens profiles is
investigated by Hawken & Bridle (2009), where it is found that the constraints
on galaxy halo ellipticites are comparable or even up to two orders of
magnitude higher than those from shear.

Just like the actual observable of shear $\gamma$ is the reduced shear $g$,
the actual observable when it comes to flexion is the reduced flexion,
calculated in Schneider & Er (2008), and tested against simulations. One
result is that the presence of flexion affects the determination of the
reduced shear. In Er et al. (2010) the problem of cluster lens-mass
reconstruction is considered, combining strong-lensing constraints with weak-
lensing shear and flexion. By so, the weak-lensing analysis is extended to the
inner parts of the cluster, and the resolution of substructre improved. Also,
Lasky & Fluke (2009) calculates the flexion for a diversity of circularely
symmetric lenses, and concludes that the convergence and first-flexion are
better indicators of the Sérsic shape parameter, while for the concentraion of
NFW profiles, the shear and second-flexion terms are preferred. In a follow-up
work (Fluke & Lasky 2011) the ray-bundle-method (RBM) developed in Fluke et
al. (1999) is applied and found to accurately recover second-flexion for the
Schwarzschild lens. Also, the existence of a preferred flexion zone is
demonstrated. The RBM could be described as an attempt at using a strong-
lensing approach to weak lensing, by considering the fact that light rays come
in bundles, and not as single, infinitesimal rays. The idea of a strong-
lensing formalism for weak lensing is furthered by Fleury et al. (2017, 2019a,
2019b). Here the infinitessimal-beam approximation underlying weak lensing
approaches is found to yield problems. This approximation is shown to comprise
the following three approximations:

  * •

flat-sky-approximation: Distances are such that the lens and source planes can
be approximated by planes.

  * •

weak-field regime: The space-time metric can be considered locally flat within
the beam’s cross section.

  * •

Smooth curvature: The riemann curvature can be considered constant over the
beam’s cross section.

By rejecting this approximation and studying instead a finite beam, the
authors show that the Kaiser-Squires theorem is violated by finite-size
effects.

Another paper, Birrer et al. (2017), seeks to unite the two regimes (weak and
strong) by looking at line-of-sight (LOS) deflectors in galaxy-scale lensing.
In this formalism, the main deflector is a strong deflector, upon which small
LOS deflectors are added in a perturbative manner. Strong-lens systems are
shown to be accurate and precise probes of cosmic shear.

The converse; a weak-lensing approach to strong lensing is taken in Clarkson
(2015), where the general theory of secondary weak gravitational lensing is
developed. This approach is later extended to include all leading-order
screen-space derivatives Clarkson (2016a, b), as such extending the work by
Bacon et al (2006 and 2009) from an altogether different route. This latter
extension by Clarkson is referred to as the roulette formalism, and for
circularly symmetric lenses it is shown to reproduce strong-lensing features.
In a later paper recursion relations (/consistency relations) like (34) are
developed for the higher-order terms (Normann & Clarkson 2020) in context of
general non-symmetrical lenses. These relations have proven useful in a
simmulator developed (CosmoAI), with a visualization component Schaathun et
al. (2023a), to both show-case the roulette expansion and also to investigate
its ability for lens-mass reconstruction Schaathun et al. (2023b). It is
noteworthy that the roulette formalism, in its current standing, does not
include all terms in the full, non-linear geodesic deviation equation, as
described by Vines (2015).

Finally, note also Lanusse et al. (2016), where a mass-mapping algorithm
designed to recover small-scale information from a combination of
gravitational shear and flexion is developed, starting from the geodesic
deviation equation. The reconstruction method is tested on realistic weak-
lensing simulations corresponding to typical Hubble-space telescope (HST) /
Advanced Camera for studies (ACS) cluster observations. The inclusion of
flexion is found to increase signal-to-noise level on small scales.

It seems fair to conclude that despite difficulties in actually measuring
flexion Viola et al. (2012); Rowe et al. (2013), this higher-order effect has
proven to be of importance. Although perhaps limited by the current state-of-
the-art observational situation, it is natural to wonder if even higher-order
terms (Roulette expansion), could prove useful.

###  3.5 Cluster lenses

The observation of weak lensing by clusters were reported by Tyson, Valdes and
Wenk (Tyson et al. 1990) in a seminal paper, in some sense marking the
beginning of observational support for weak lensing. Attempts to determine
cluster lens-mass distributions, however, seem to go back to Webster (1985),
while Kochanek (1990) and Miralda-Escude (1991) discussed how parameterized
cluster-mass distributions could be constrained from weak-lensing data through
ellipticities of the source and image. The effects of cluster strong lensing
were however first unknowingly discovered when observing giant luminous arcs
(Lynds & Petrosian 1986)141414Zwicky also considered clusters, but at this
stage in the development shift in position was the discussed effect., with
subsequent interpretation of the phenomena as a lensing effect (Soucail et al.
1987).

Whereas a power-law of the form $\rho\sim 1/r^{\eta}$ seems to work well for
modelling lenses on galactic scale, this does not apply to clusters. A natural
generalization would be to allow the power law index to take on one value for
small radii and another for large radii, like the Navarro-Frenk-White (NFW)
model. Here

$\rho(r)=\frac{\rho_{\rm s}r_{\rm s}^{q}}{r^{p}(r_{\rm
s}+r)^{q-p}}\quad\quad\quad\textrm{NFW profile},$ (35)

for some scale radius $r_{\rm s}$. For $r\ll r_{\rm s}$ one finds $\rho\sim
1/r^{p}$, whereas $\rho\sim 1/r^{q}$ for $r\gg r_{\rm s}$. From cosmological
simulations of cluster-sized dark-matter halos Navarro et al. (1995, 1996,
1997) found $p=1$ and $q=3$, Later parametric models have often taken this
profile as a starting point. Now that the resolution of the image data is
increasing, there is however a need for going beyond such parametric models.

###  3.6 Parametric and non-parametric (free-form) models

For the purpose of this review it is useful to make a distinction between
parametric and non-parametric models. We employ the classification from Lefor
et al. (2013), where parametric models make assumptions on physical objects
(as e.g. point-masses, singular isothermal spheres and so on) to model the
lensing mass. While non-parametric (or free-form) models do lens inversion by
avoiding such assumptions and rely on the lensed images alone to do mass
reconstruction. In the non-parametric approach several models can fit the
observed data, and a regularization problem needs to be solved to restrict
results to plausible surface-mass densities. An example of a non-parametric
approach that created some stir is Jee et al. (2007), where a dark-matter ring
in the cluster CL0024+17 ($z=0.395$) was discovered, using both strong and
weak lensing constraints. When using non-parametric models, a large number of
constraints is needed to avoid spurious results, as discussed for instance in
Ponente & Diego (2011) and Sendra et al. (2014).

Abell1689 is a good example of a cluster where non-parametric models are
competitive with parametric ones, since there are so many constraints
(hundreds of arcs). Parametric models are typically popular in the inner parts
of clusters, where the lensing effects are strong. In practice, a combination
of both parametric and free-form modelling is used, such as e.g. in Jullo &
Kneib (2009) and in Beauchesne et al. (2021).

##  4 Cluster lens-mass reconstruction from weak- and strong lensing combined

The treatment so far has been general, giving the development of weak lensing
theory beyond shear (id est towards the strong regime) and by explaining
important concepts and issues. Furthermore, the turn to maximum-likelihood
methods (discussed in Sec. 3.2) was an important step, which the community
continued to develop from Bridle and on. With it, both weak and strong
constraints could be jointly analysed through linear constraints.

In the current section we narrow in to cluster lens-mass reconstruction, and
review the development of methods for unified analysis. The shortlist from the
systematic review (cf. discussion in Section 2) makes up the bulk of material
covered.

###  4.1 Weak lensing (arclets) combined with multiple images

The strong lensing regime permits for multiple images of the same source. Such
information may be used for lens-mass reconstruction, as investigated by
numerous sources, such as e.g. AbdelSalam et al. (1998c). This particular work
by Abdel Salam et al, however, is furthered in a series of papers to include
also weak-lensing information for cluster lens-mass reconstruction (AbdelSalam
et al. 1997, 1998a; Saha et al. 1999). The technique developed uses
constraints from multiple images and arclets. The technique bears resemblance
with that of Kaiser & Squires (1993) for weak lensing, but this time the
problem formulation is linear: The aforementioned strong-lensing constraints
are included as linear constraints on the projected mass distribution,
together with weak-lensing constraints. Quadratic programming is then used to
obtain the mass maps, and the method developed is reported to overcome the
drawbacks of non-linear methods, such as the mass-sheet degeneracy (by using
sources at different redshifts), and the technique is also applied to cluster
lens-mass reconstruction (see also AbdelSalam et al. (1998b)).

Another early example is Kneib et al. (2003), which studies the particular
cluster C1 0024+1654 ($z=0.395$). The study imcoorporates strong-lensing
constraints (multiple images) in a weak-lensing analysis, using the maximum-
entropy method developed in Bridle et al. (1998). The strong-lensing
constraints are found to narrow down the parameter space effectively, with an
analysis strongly rejecting the SIS model, whilst fitting that of an NFW
profile well. A shortcoming of the study is that it is not parameter-free.
Also, the parametrization only allows weak-lensing signals in the outer parts
($\sim 570$ kpc) and strong lensing constraints from the inner regions ($155$
kpc). In Smith et al. (2005) the software LENSTOOL is used together with
additional routines to incorporate weak-lensing constraints. Again the
algorithm used builds on parametric models, with a $\chi^{2}$-estimator
qantifying how well each trial lens model fits the data.

With these works in mind, it therefore seems fair to conclude that around the
dawn of the 21th century, the theory was sufficiently mature and observations
crisp enough for a joint analysis of both weak and strong lensing constraints.
For it is one thing to theoretically being able to perform joint analysis, and
quite another to implement it with actual observations. As that step was
overcome, however, combining the two regimes is quickly confirmed to be
beneficial. But a number of theoretical problems remained however in need of
closer attention: (i) The mass-sheet degeneracy must be broken for any study
relying on the lens equation, (ii) likelihood methods must be properly
regularized and (iii) non-parametric models should be developed.

###  4.2 Bradac̆, Umetsu and mass-sheet degeneracy

As already discussed (cf. Sec. 3.2.1), attempts had been made to overcome the
mass-sheet degeneracy, which also became an issue in terms of discriminating
between the SIS and the NFW profile — a hot topic in connection with cluster
lenses. In 2004, Bradac̆ et al published a paper on the mass-sheet degeneracy
Bradač et al. (2004a), suggesting a way to break it by using the distortion
and the redshift information of background galaxies. They conclude that the
method is effective for critical clusters only (multiple imaging).
Consequently, in order to break the degeneracy with the current (2004) data,
it was found necessary to extend the statistical lensing analysis closer to
the cluster center, performing a joint weak and strong lensing analysis. An
overview of the results is presented in a proceedings (Bradač et al. 2004b),
with reference to further literature, alongside an implementation with the
cluster-mass distribution of RX J1347-1145. The method developed is again
parametric, and in Bradač et al. (2004a) attention is called to non-parametric
models, since the method in principle should be applicable in such scenarios
as well. Consequently, in a series of papers, Bradac et al propose a non-
parametric method for uniting weak and strong lensing (Bradač et al. 2005;
Bradac 2005), accompanied by observational implementations (Bradač et al.
2005, 2006; Bradac et al. 2005a). The method relies on minimizing the
$\chi^{2}$ functional

$\chi^{2}(\psi_{k})=\chi^{2}_{\textrm{WL}}(\psi_{k})+\chi^{2}_{\textrm{MI}}(\psi_{k})+\eta
R(\psi_{k}),$ (36)

where subscripts WL and MI denote weak lensing and multiple imaging,
respectively. $\psi_{k}$ is the lensing potential on a regular grid, whereas
the final term is a regularisation term. A minimum is sought such that

$\partial\chi^{2}({\psi_{i}})/\partial\psi_{i}=0.$ (37)

The authors comment on the fact that other attempts at combining weak and
strong lensing have already been made. For instance, they point out that
Abdelsalam et al and Smith et al (Sec. 4.1) use the surface-mass density
$\kappa$, on which the shear $\gamma$ and the deflection angle $\alpha$
depends non-locally. Bradac et al uses instead the potential $\psi$, from
which both covergence, shear and the deflection angle may be locally derived
(derivatives). Where Kneib et al (2003) and Smith et al (2004) (cf. Sec. 4.1)
used a parametrized, Bayesian approach, a more general parametrization was
used by Bradac̆ et al. In 2009 they apply their techniques on the Bullet
cluster, with a strong and weak lensing analysis following the algorithm first
proposed by Bartelmann et al (1996) and discussed in Sec. 3.2.

Another interesting strand of work, based on the authors original idea of
using magnification to break the mass-sheet degeneracy Broadhurst et al.
(2005), is also further developed (cf. e.g. Umetsu & Broadhurst (2008);
Medezinski et al. (2007); Umetsu et al. (2011)). Finally, in Umetsu (2013), a
model-free reconstruction method is presented, using multiple populations of
background sources, correcting for magnification bias and incorporating
strong-lensing constraints. The method is applied to a number of clusters from
the Subaru images.

###  4.3 Perturbative approach

Another interesting approach within the framework of maximum-likelihood
approaches, is the perturbative approach undertaken in Natarajan et al.
(2009). Again weak and strong is combined as linear constraints in a maximum-
likelihood manner. Galaxy-galaxy lensing in the cluster CL 0024+16 at $z=0.39$
is used to track the fate of dark-matter subhalos. The gravitational potential
is modelled such that

$\phi_{\textrm{tot}}=\sum_{n}\phi_{s}+\sum_{i}\phi_{p_{i}},$ (38)

where the two $\phi_{s}$ ($n=1$ and $n=2$) components represent smooth large-
scale components, whereas the galaxy subhalo components $\phi_{p_{i}}$ are
treated as perturbers. The reason for having two large-scale components, is
that earlier studies reveal this to be the best-fit model for that particular
cluster. Thus this is taken as a prior in the current study. two The results
provide strong support for the tidal-stripping hypothesis.

###  4.4 Diego et al and the regularization problem

Due to scarce observations, parametric models were initially popular, and
superior to non-parametric ones, where the freedom in the models must be
regularized – a process that either requires parameteres or a large number of
constraints. As observations increased in amount and accuracy, however,
constraints from weak and strong lensing combined made such models competitive
with parametric ones. This seems to have accelerated the combinations of the
two lensing regimes (WL and SL). In addition to the work by Bradac̆ (previos
section) and also that by Broadhurst (Broadhurst et al. (2005) discussed in
Sec. 3.2), the work of combining the two regimes was furthered by Diego et al.
(2007) in a paper known as Paper III in a series. The work centers around the
idea of expanding the projected cluster-mass distribution into a set of basis
functions which are then constrained indivitually using shear and SL data.

Again a maximum-likelihood approach is taken, well suited for the task of
combining different types of constraints. Their discretized problem
formulation is as follows. Take $N_{\theta}$ to be the number of images,
$N_{\textrm{s}}$ the number of sources and let $N_{\textrm{c}}$ denote the
number of mass cells. Then the linearised problem takes the form

$\bm{\phi}=\Gamma\mathbf{x},$ (39)

where $\Gamma$ is a matrix of dimension
$4(N_{\theta})\times(N_{\textrm{c}}+2N_{\textrm{s}})$, and generally not
invertible. $\bm{\phi}$ is a vector containing position and shear information
whereas $\mathbf{x}$ is a vector containing all the unknowns. A weakness of
the work is that it assumes the source galaxies to be point-like151515The
paper includes a remark about a follow-up work on this issue, which we have
not been able to find.. This is however corrected for in an a-posteriori
manner, by allowing for residuals $\mathbf{r}$ in the lens equation;
$\mathbf{r}=\phi-\Gamma\mathbf{x}$. Aproximate solutions to the problem is
sought by maximizing the functional
$\mathcal{L}=\textrm{e}^{\frac{1}{2}\mathbf{x}^{2}},$ assuming $\mathbf{r}$ is
Gaussian. The $\chi^{2}$ is now defined as

$\chi^{2}=\mathbf{r}^{\textrm{T}}\bm{C}^{-1}\mathbf{r}$ (40)

where $\bm{C}$ is the covariance matrix of $\mathbf{r}$. As with most studies
combining weak and strong, this work only includes shear.

Since the approach is non-parametric, a regularization procedure is necessary.
In Paper I (Diego et al. 2005a) a bi-conjugate gradient method minimizing
$\chi^{2}$, stopping at a target value $\epsilon$ is found useful (implicitely
regularizing the problem). Paper III shows that combining weak and strong
lensing makes the choice of $\epsilon$ less relevant. Also, since the problem
formulation consists in minimizing a quadratic function with linear
constraints, a quadratic-programing approach (QADP) is taken. By using a
minimization algorithm that is constrained to minimize $\chi^{2}$ for positive
masses, unphysical negative masses are avoided, and the problem is more
properly regularized. An application using strong lensing is found in their
Paper II (Diego et al. 2005b). The result of their analysis is a software
package known as WSLAP161616Weak & Strong Lensing Analysis Package, which is
made available for public use. Also note Diego (2014).

The above works illustrate the capability of non-parametric methods to compete
with parametric ones, when a properly regularized maximum-likelihood approach
is taken.

###  4.5 Using critical curves

As discussed in Sec. 3.3, strong lensing observables include arcs, which form
around critical curves ($\det\mathcal{A}=0$). This is used in Cacciato et al.
(2006) to develop a cluster-potential reconstruction method relying on
critical-curve constraints imposed near arc-forming regions. Building on the
least-$\chi^{2}$ minimisation procedure developed in Bartelmann et al. (1996);
Seitz et al. (1998), this becomes a different route to combining weak and
strong lensing. Also note that the weak-lensing constraints are imposed on a
low-resolution grid, whereas the strong-lensing constraints are imposed on a
more fine-grained grid, so as to concentrate the analysis to the relevant
region.

One advantage of this method, is the elimination of the need to finding
multiple images of the same source, which requires work. Additionally, one
ought also know whether or not there are additional sources not imaged. But
inverting the lens equation to find out is of course a highly non-linear
process. Furthermore, multiple imaging is caused by sources inside or close to
caustics, which limits the applicability of the method.

In Cacciato et al (2006a) measured ellipticities is compared to $\gamma$
rather than to the reduced shear $\gamma(1-\kappa)^{-1}$. This approximation
is fixed in the observational follow-up (Merten et al. 2009), which also
improves the griding of the data, and adds a regularization term to not
overfit noice. Note also the follow-up Merten (2016), which incoorporates a
mesh-free, free-form approach to lens-mass reconstruction, through the use of
radial basis functions. When discretizing the data, a regular mesh is often
employed. But, as the authors argue, astronomical data is often quite
irregular. The work extends their own previous work, and other, like
Bartelmann et al. (1996); Seitz et al. (1998); Bradac et al. (2005b) into the
flexible, numerical mesh-free domain.

###  4.6 Particle-based lensing (PBL)

Let us also make mention of the works Deb et al. (2008, 2012), where a
particle-based approach is taken. Whereas grid-based techniques are often
optimized to measure at a single scale, reality is that the structure of a
cluster lens is hierarchical. Thus a method capable of increased resolution in
regions high on information would be desirable. Second, the smoothing and
weighting are arguable ad hoc by construction (motivated by model and theory),
whereas an ideal smoothing scale should be variable. Another point is the
abrupt switch in image parity at the critical curves, leading to a
discontinuity in the ellipticity as a function of $\kappa$ and $\gamma$. This
prevents linear minimization schemes from converging to a ”strong
lens”-solution if starting off by a ”weak lens” initial guess.

The particle-based lensing (PBL) methods has the ability to combine disparate
lensing scales in a coherent way without regularization schemes, and are
already in use in other fields of astrophysics where a wide range of physical
scales are present, such as numerical $N$-body simulations.

PBL is a new way of discretizing and describing a reconstructed field,
including a metric comparison to the observed data, but PBL is nevertheless
not a minimization scheme. PBL consists of a list of potentials (one at each
observed lensed image) and a metric to describe the goodness of fit. A
$\chi^{2}$ minimization to estimate a maximum-likelihood potential field is
used, and —for practical implementations anyway — some sort of minimization
procedure has to be chosen.

Note that the (parameter free) PBL method developed may in principle combine
both weak and strong constraints, including flexion. Applying the technique to
the much studied cluster CL-A1689 reveals a secondary mass peak in the north-
east direction, confirming previous optical observations Deb et al. (2012).
The study is reported to be the first non-parametric strong and weak (shear)
mass map with a covariance matrix. Also see Deb et al. (2010) for another
application of PBL to lensing, and for more on the covariance matrix.

###  4.7 Including flexion in the combined analysis

Including flexion, the lensing equation may be written as Cain et al. (2011)

$\hat{\beta}=\theta-g\theta^{*}-\frac{1}{4}\Psi_{1}^{*}\theta^{2}-\frac{1}{2}\Psi_{1}\theta\theta^{*}-\frac{1}{4}\Psi_{3}(\theta^{*})^{2},$
(41)

where $\theta$ is the complexified image-plane angular coordinate and ∗
denotes complex conjugation. Note that $\hat{\beta}=\beta/(1-\kappa)$ is the
reduced source position, $g$ is the reduced shear, as before (17), and that
$\Psi_{1}=\frac{\mathcal{F}}{1-\kappa}$ and
$\Psi_{3}=\frac{\mathcal{G}}{1-\kappa}$ are reduced flexion variables (cf. Eq.
(32)), all fit to taking the mass-sheet degeneracy into account.

It is in fact notable that weak lensing including flexion terms has also been
combined with strong lensing constraints in observational studies, as
described in this section. First, note the study Leonard et al. (2007), where
shaplets are used. Again the cluster Abell1689 is the target cluster. By use
of flexion-data alone, and with a parametric approach, the study reveals a
substructure offset from the central cluster members that was not visible from
shear measurements alone. With non-parametric approach it is also suggested,
by combination of weak and strong lensing data. Several studies show the same;
flexion is particularely sensitive to cluster substructures (e.g. Bacon et al.
(2010)). This is perhaps not so surprising from a theoretical point of view,
considering the fact that flexion encodes derivatives of the shear map.

The mass-reconstruction method presented by Bradavc and discussed earlier, is
extended in Er et al. (2010), where a flexion term $\chi_{f}^{2}$ is added,
reading

$\sum_{i=1}^{N_{\textrm{f}}}=\frac{\absolutevalue{t_{1i}-G_{1}(\bm{\theta}_{i},z_{i})}^{2}}{\sigma^{2}_{t1}}+\frac{\absolutevalue{t_{3i}-G_{3}(\bm{\theta}_{i},z_{i})}^{2}}{\sigma^{2}_{t3}},$
(42)

where $t_{1i}$ (or $t_{3i}$) refers to first (or third) flexion data, and
$G_{1i}$ (or $G_{3i}$) to the corresponding theoretical model flexion. $N_{f}$
is the number of images wiht flexion data. The approach is tested on
simulations, and concludes that flexion is sensitive to sub structures.

Also note the Analytic Image Model (AIM) method, developed in Cain et al.
(2011), which is distinct from shaplets- and moment-based methods. Starting
from the conservation of surface-brightness
$I_{\textrm{obs}}[\theta]=I_{\textrm{src}}[\beta(\theta)]$, it minimizes the
figure of merit given by

$\chi^{2}=\sum_{n}\frac{(I_{\textrm{obs}}(\theta^{(n)})-I_{\textrm{AIM}}(\theta^{(n)};p_{\textrm{int}},p_{\textrm{lens}})^{2})}{\sigma_{n}^{2}},$
(43)

where $\theta^{(n)}$ is the image-plane position of the $n$th pixel and
$\sigma_{n}^{2}$ is an estimate of the variance in that pixel’s value. Note
that $p_{\textrm{int}}$ and $p_{\textrm{lens}}$ are arrays containing the
parameters of the theory. In particular,
$p=\\{g_{1},g_{2},\Psi_{11},\Psi_{12},\Psi_{31},\Psi_{32}\\}$, where
$g_{1},\Psi_{11}$ and $\Psi_{31}$ are the real parts (and $g_{2},\Psi_{12}$
and $\Psi_{32}$ are the imaginary parts) of $g,\Psi_{1}$ and $\Psi_{3}$,
respectively. Instead of estimating derived quantities, such as shapelet
coefficients or surface-brightness moments, the AIM method instead fits a
mass-sheet transformation invariant model to eaach galaxy image. The method is
applied to CL—A1689, detecting mass structures. In a follow-up work Cain et
al. (2016), the AIM method is applied to clusters, revealing that galaxy-group
scale substructures may remain undetected, such as to underestimate the mass
in such regions by $\sim 25-40$ percent. With flexion included, subhalo masses
of $\sim 3\,\times\,10^{12}M_{\odot}$ are detectable at an angular resolution
smaller than 10 arcsec.

Albeit not including strong lensing events per say, we also mention Lanusse et
al. (2016), which combines weak-lensing shear and flexion data through
developing a method that avoids any binning of the irregularely sampled
fields. As a result, the mass-mapping problem becomes an ill-posed inverse
problem. This is regularized using a robust multi-scale wavelet sparsity
prior. Fast Fourier estimators makes the algorithm efficient, and the method
is tested on simulated data. The conclusion is again that flexion increases
the ability to detect substructures.

###  4.8 Combining lensing information with other sources

In this paper we have focused on gravitational lensing as a probe on galaxy
clusters. But there are other probes as well, which may be summarised as
follows:

  * •

weak and strong lensing constrain the line-of-sight projection of the cluster
potential

  * •

X-ray observations constrain the density and the temperature of the
intracluster gas

  * •

Sunyaev-Zeldovich effect constrains the gas pressure

  * •

Gradient of gravitational potential can be constrained by galaxy kinematics

  * •

Cluster kinematics. In e.g. Pizzuti et al. (2022) kinematics measurements of
galaxy cluster MACS J1206 is combined with weak and strong lensing
constraints.

A complete treatment would of course have to combine from all the above
approaches. Allthough alternative sources for information is beyond the scope
of the current review, we take the opportunity to mention that work has been
done in combining the available information. One example is Konrad et al.
(2013), where reconstruction from thermal gas (x-ray) and gravitational
lensing is considered in a non-parametric manner. The ultimate goal of this
study is reported to be a non-parametric method combining strong and weak
lensing, observations of thermal gas physics, and galaxy kinetmatics into one
consistent model for the projected cluster potential. Other examples include
Majer et al. (2016) where lensing information is combined with the thermal
Sunyaev-Zel′ dovich effect and Tchernin et al. (2015, 2018), where X-ray and
the Sunyaev-Zel′dovich effect is used to reconstruct the gravitational
potential by joint analysis.

###  4.9 Software developments

As far as software goes, the recently developed Niemiec et al. (2020), should
be noted, where the combination of weak and strong constraints are implemented
in the publically available modelling software package LENSTOOL, and called
hybrid-LENSTOOL. The name derives from the process: This package uses a
parametric approach to model the core of the cluster, and a non-parametric
(free-form) approach for the outskirts. The optimization is based on a joint
optimization of both weak and strong constraints, such that the combined
likelihood function is

$\mathcal{L}(\bm{\Theta},\bm{w})=\mathcal{L}_{\textrm{SL}}(\bm{\Theta},\bm{w})\times\mathcal{L}_{WL}(\bm{\Theta},\bm{w}).$
(44)

Here $\bm{\Theta}$ are the free parameters of the parametric part, whereas the
grid model composed of $N$ radial basis functions are ordered in a vector
$\bm{w}$. The details of the strong- and weak-lensing likelihoods are further
described in the paper. As a result, the package optimizes weak- and strong
lensing simultaneously, as opposed to successively, as does the fore-runner
Sequential-Fit. It should be noted, however, that Niemiec et al. (2023) still
ends up using Sequential fit, since the strong lensing constraints optimizes
the likelihood in the source plane, as opposed to the image plane. This might
not be accurate enough for complex clusters, given that best-fit mass
reconstruction is the goal. Testing both methods on simulated clusters are
left for future work. Another software is SWUnited, based on the Bradac-Hoag
model. This algorithm is based on the works Bartelmann et al. (1996) (Sec.
3.2) and Bradac et al. (2005) (Sec. 4.2), is non-parametric (free-form) and
reconstructs the lensing potential (as opposed to the convergence). This means
that assumptions on the mass density outside the field of observation is not
necessary. Also, the formalism includes flexion constraints.

Note also WSLAP+, which builds on work by Diego (discussed in Section 4.4) and
Lam. This method also unites strong and weak (shear) constraints.

A genetic algorithm (GA) for non-parametric strong-lensing systems (GRALE) was
also developed (Liesenborgs et al. (2007, 2010)), and later extended to take
time-domain and weak-lensing information into account (Liesenborgs et al.
2020). This method resembles a regularization procedure, but is different in
that it does not optimize the different criteria together. Instead, a GA looks
for a common optimum for the different so-called fitness measures. The results
as regards the addition of weak lensing to GRALE 171717Short for GRAvitational
LEnses., are reported to be somewhat mixed, and the procedure is plagued by
the mass-sheet degeneracy.

Finally, note also the newly developed RELENSING package (Torres-Ballesteros &
Castañeda 2023), written in Python. It is a free-form approach on an adaptive,
irregular grid, combining strong and weak (shear) constraints, and tested on
the simulatd clusters ARES and HERA.

A full analysis and comparison of (some of) the different packages available
shall not be undertaken here, but we refer the interested reader to Meneghetti
et al. (2017), where different groups have participated in the reconstruction
of simulated clusters, for comparisson purposes. Other tests include the Shear
TEsting Programme (STEP) and GRavitational LEnsing Accuracy Testing (GREAT).

####  4.9.1 The use of artificial intelligence

As a final remark, it is worthwhile stoping to ponder the future. As numerous
authors point out, the amount and quality of the lensing data has been
steadily increaing. Resultingly, one ought only expect that the need for fast,
automated analysis becomes more and more prevalant. To this end, people have
looked to machine learning, to see if this tool is fit for the job of
automating the process of lens-mass reconstruction. Novel attempts have been
made, such as that by Hezaveh et al. (2017), Galan et al. (2022) and Biggio et
al. (2023), all concerning strong lenses. To the best of knowledge, no other
attempts at combining weak and strong lensing in an automated ML approach has
been made, besides our own humble attempt at using the Roulette formalism to
such ends Schaathun et al. (2023a, b), and the recent work by Cha et al.
(2024), where a deep-learning technique is applied to navigate the extremely
large space of free parameters. The data set of this latter study is obtained
from wide-field JWST, which makes it all the more interesting. As mentioned,
however, this is beyond the scope of this review, and should be treated
elsewhere.

##  5 Conclusion and future outlook

In this review, we have traced the development of joint analysis of weak and
strong gravitational lensing data, with a particular emphasis on cluster lens-
mass reconstruction. In summary there seems to be two important strands of
development.

First, there is the development of the theory of weak lensing to account for
higher-order terms, such as flexion, second-flexion and terms beyond (so-
called roulette expansion). This is a natural extension of the early
development of non-linear methods (such as KS). By including terms beyond
shear in the lens map a bridge is made from the weak regime towards the
strong. As of today, the roulette expansion developed by Clarkson is the most
complete work to this end, but even here only leading-order screen-space terms
have been kept, whereas the rest, such as line-of-sight effects, are wittingly
swept under the carpet.

Second, with the development of so-called inverse methods (and most prominent
among them the class of maximum-likelihood approaches) around the turn of the
millennium, the two regimes have arguably been merged, as far as statistical
data analysis is concerned. However, a number of improvements are possible. In
weak studies, inclusion of flexion terms has already proven useful, but terms
beyond this order have not been investigated, except theoretically in the
aforementioned roulette formalism, or equivalently, by Fleury et al.
Foreseeing an increase in data quality, such decomposition of image
information beyond flexion could prove valuable.

### Future work

A number of improvements are possible. For one, degeneracies such as the mass-
sheet degeneracy still pleages the field, even in the presence of source red-
shift information, and the development of alternative methods thus seems
useful. Work on adaptive grids and alternative sources of information to break
the degeneracy should therefore be continued.

Flexion studies, adaptive grids and combined analysis of weak and strong
lensing have revealed sub-structures in cluster-mass distributions. The
development of three-dimensional parametric models of clusters should
therefore go on, reaching beyond the NFW. To such ends, studying the line-of-
sight effects from the internal structure of the lens seems to be a natural
step. Furthermore, the development of non-parametric methods for data fitting
must continue, adapting to the increase in both data quality and quantity.

To this end the recent developments within automated processes by use of
machine learning seems like a particularly promising route. Armed with a
formalism capable of drawing information beyond shear in the so-called weak-
lensing data, this might be just what we need.

###### Acknowledgements.

We thank Elise Lindegaard Hanssen for valuable comments.

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##  Appendix A Study selection

Table 3: The table presents the short-list of 29 papers selected for study
through the structured search. The column ‘MSD’ categorizes how the mass-sheet
degeneracy is treated in the study. ‘$z$’ means sources at different
redshifts. ‘magn’ is short for magnification. ‘ $\leq\,$ Flex’ means that both
shear and flexion data has been used.

Authors | Title | Free-form? |  $\psi$ or $\kappa$? | MSD | WL data | Method
---|---|---|---|---|---|---
AbdelSalam et al. (1998a) | Mass reconstruction of combined strong & weak lensing | Yes | $\kappa$ | z | Shear | Inverse
AbdelSalam et al. (1998b) | Nonparametric reconstruction of Abell 2218 from combined weak and strong lensing | Yes | $\kappa$ | z | Shear | Inverse
AbdelSalam et al. (1997) | Cluster mass profile from lensing | Yes | $\kappa$ | z | Shear | Inverse
Birrer et al. (2017) | Line-of-sight effects in strong lensing: putting theory into practice | No |  $\kappa$? | Discussed | Shear | Inverse?
Bradac (2005) | Strong and weak lensing united: the cluster mass distribution of the most X-ray |  |  |  |  |
| luminous cluster RX J1347-1145 | Yes | $\psi$ | z | Shear | Inverse
Bradac et al. (2005a) | Strong & weak lensing united: The cluster mass distribution of RX J1347-1145 | Yes | $\psi$ | z | Shear | Inverse
Bradač et al. (2005) | Strong and weak lensing united - I. The combined strong and weak lensing |  |  |  |  |
| cluster mass reconstruction method | Yes | $\psi$ | z | Shear | Inverse
Bradač et al. (2005) | Strong and weak lensing united - II. The cluster mass distribution of the most |  |  |  |  |
| X-ray luminous cluster RX J1347.5-1145 | Yes | $\psi$ | z | Shear | Inverse
Bradač et al. (2005) | Strong and weak lensing united. III. Measuring the mass distribution of the merging |  |  |  |  |
| galaxy cluster 1ES 0657-558 | Yes | $\psi$ | z | Shear | Inverse
Bradač et al. (2009) | Focusing cosmic telescopes: exploring redshift z 5-6 galaxies with the bullet cluster |  |  |  |  |
| 1E0657-56 | Yes | $\psi$ | z | Shear | Inverse
Cacciato et al. (2006) | Combining weak and strong lensing in cluster potential reconstruction | Yes? | $\psi$ | scale | Shear | Inverse
Cha et al. (2024) | Precision MARS Mass Reconstruction of A2744: Synergizing the Largest |  |  |  |  |
| Strong-lensing and Densest Weak-lensing Data Sets from JWST | Yes | $\kappa$ | scale | Shear | Inverse
Deb et al. (2012) | Mass reconstruction using Particle Based Lensing II: Quantifying substructure with |  |  |  |  |
| strong + weak lensing and X-rays | Yes | $\psi$ | z(+ scale?) | Shear | Inverse
Diego (2014) | An improved dark matter map from weak and strong lensing in A1689 | Yes | $\kappa$ | Discussed | Shear | Inverse
Diego et al. (2007) | Combined weak and strong lensing data with WSLAP | Yes | $\kappa$ | Discussed | Shear | Inverse
Er et al. (2010) | Mass reconstruction by gravitational shear and flexion | Yes | $\psi$ | z? |  $\leq\,$Flex. | Inverse
Leonard et al. (2007) | Gravitational shear, flexion and strong lensing in Abell 1689 | Partially | $\kappa$ | Discussed |  $\leq\,$Flex. | Direct
Liesenborgs et al. (2020) | Extended lens reconstructions with grale: exploiting time-domain, substructural and |  |  |  |  |
| weak lensing information | Yes | $\kappa$ | Discussed | Shear | Inverse
Merten et al. (2009) | Combining weak and strong cluster lensing: applications to simulations and MS 2137 | Yes | $\psi$ | Discussed | Shear | Inverse
Merten (2016) | Mesh-free free-form lensing - I. Methodology and application to mass reconstruction | Yes | $\psi$ | Discussed | Shear | Inverse
Natarajan et al. (2009) | The survival of dark matter halos in the cluster CI0024+ 16 | No | $\kappa$ | Not discussed | Shear | Inverse
Niemiec et al. (2020) | Hybrid-lenstool: a self-consistent algorithm to model galaxy clusters with strong- and |  |  |  |  |
| weak-lensing simultaneously | Partially | $\kappa$ | not discussed | Shear | Inverse
Normann & Clarkson (2020) | Recursion relations for gravitational lensing | Yes | $\psi$ | not discussed |  $n$th order | Direct
Niemiec et al. (2023) | Beyond the ultradeep frontier fields and legacy observations (BUFFALO): |  |  |  |  |
| a high-resolution strong+weak lensing view |  |  |  |  |
| of Abell 370 | Partially | $\kappa$ | Not discussed | Shear | Inverse
Ponente & Diego (2011) | Systemics in lensing reconstruction: dark matter rings in the sky? | Yes | $\kappa$ | $z$ | Shear | Inverse
Saha et al. (1999) | Cluster reconstruction from combined strong and weak lensing | Yes | $\kappa$ | z | Shear | Inverse
Schneider & Seitz (1995) | Steps towards nonlinear cluster inversion through gravitational distortions. |  |  |  |  |
| I. Basic considerations and circular clusters | Yes | $\kappa$ | Discussed | Shear | direct
Umetsu (2013) | Model-free multi-probe lensing reconstruction of cluster mass profiles | Yes | $\kappa$ |  $z$ and magn | Shear | Inverse
Umetsu & Broadhurst (2008) | Combining lens distortion and depletion to map the mass distribution of A1689 | Yes | $\kappa$ |  $z$ and magn | Shear | Inverse