††thanks: ⋆mailto:aoife.boyle<EMAIL_ADDRESS>
# The cumulant generating function as a novel observable
to cumulate weak lensing information

Aoife Boyle1,⋆ Alexandre Barthelemy2 Sandrine Codis1 Cora Uhlemann3 Oliver
Friedrich2 1 AIM, CEA, CNRS, Université Paris-Saclay, Université Paris
Diderot, Sorbonne Paris Cité, 91191 Gif-sur-Yvette, France  2 Universitäts-
Sternwarte, Fakultät für Physik, Ludwig-Maximilians Universität München,
Scheinerstr. 1, 81679 München, Germany  3 School of Mathematics, Statistics
and Physics, Newcastle University, Herschel Building, NE1 7RU Newcastle-upon-
Tyne, U.K.

###### Abstract

Key non-Gaussian properties of cosmological fields can be captured by their
one-point statistics, providing a complement to two-point statistical
measurements from power spectra or correlation functions. Large deviation
theory can robustly predict the one-point statistics of cosmological density
fields on mildly non-linear scales from first principles. It provides a direct
prediction for the cumulant generating function (CGF) of such fields, from
which a prediction for the more commonly used probability density function
(PDF) is extracted through an inverse Laplace transform. For joint one-point
statistics of multiple fields, the inverse Laplace transform rapidly becomes
more cumbersome and computationally expensive. In this work, we demonstrate
for the first time that the weak lensing CGF itself can be used as an
observable that captures an equal amount of cosmological information to the
PDF. While we use the weak-lensing convergence field as a simplistic and
instructive example, this work is intended as a first step towards a
cosmological analysis based on large deviation theory in the context of a
nulling framework, which excludes contributions from small scales to
facilitate highly accurate theoretical predictions. In this context, the
method should be generally applicable for a multi-scale tomographic analysis
of weak lensing and galaxy clustering.

##  1 Introduction

Squeezing the maximum amount of cosmological information out of galaxy surveys
with summary statistics requires statistics that go beyond the standard two-
point correlation functions or power spectra, which only contain complete
statistical information for Gaussian random fields. Most recently this has
been exemplified by the exploitation of the Dark Energy Survey (see for
example Zürcher et al., 2022b; Secco et al., 2022; Gatti et al., 2021; Gruen
et al., 2018), and numerous papers (see for example Zürcher et al., 2021;
Zürcher et al., 2022a, and references therein) that have underlined the
importance of non-Gaussian statistics for upcoming surveys like Euclid
(Laureijs et al., 2011) or the Vera Rubin Observatory (LSST) (Ivezić et al.,
2019). One-point statistics are of particular interest because of their ease
of measurement and theoretical accessibility. In particular, the moments of
the density field have received a lot of attention from both a theoretical
(e.g Fry, 1984; Lokas et al., 1995; Bouchet et al., 1995) and observational
(see Wolk et al., 2013; Gatti et al., 2021, and references therein) point of
view. Going one step further, several works have shown that the traditional
one-point probability density function (PDF) is a particularly relevant
additional non-Gaussian probe, which contains information on the full set of
higher-order moments (Codis et al., 2016a; Uhlemann et al., 2020a; Friedrich
et al., 2021a; Boyle et al., 2021). One other notable advantage of the PDF is
that cutting out the extreme tails allows removal of the impact of rare
events, which are difficult to account for properly in a cosmological data
analysis. In this paper, we advocate for the cumulant generating function
(CGF) as a new observable analogous to the PDF, with desirable properties for
straightforward extensions to multiple fields.

The cumulant generating function is a widely used mathematical object, closely
related to the moment generating function (MGF), whose successive derivatives
in zero are the cumulants (resp. moments) of the random variable under
consideration. The CGF is actually defined as the natural logarithm of the
MGF. From a mathematical viewpoint, cumulants and the CGF are preferred over
moments and the MGF for problems that consider sums of independent (multi-
variate) random variables, because the CGF of the sum in such cases is simply
the sum of the CGFs. More concretely, cumulants represent the connected part
of the corresponding moments. This means that the $n^{\rm th}$ cumulant gives
extra information not contained in any previous cumulant, while additional
moments can exist without adding any extra information. A typical example of
the latter case is found in Wick’s theorem for Gaussian fields, which
quantitatively expresses all the higher order moments as a function of the
first two (the mean and variance). The same Gaussian field, however, only
admits two non-zero cumulants. This gives an additional advantage to
considering cumulants in cosmology. In this context, the fields we consider
are typically mildly non-Gaussian, so the higher-order cumulants are small and
become smaller as their degree increases (a natural consequence of the process
of gravitational instability on cosmological scales, as shown extensively by,
for example Peebles, 1980; Baugh et al., 1995; Bernardeau et al., 2002).

From a physical viewpoint, CGFs are ubiquitous, especially in the realm of
statistical physics where typically any extensive quantity can be related to
the cumulants of a random variable (often the energy of the system). It can be
noted, for example, that Helmholtz free energy is exactly the CGF of the
energy and as such all derivatives of the free energy, such as the internal
energy, entropy, and specific heat capacity, can be readily expressed in terms
of cumulants. It is thus no surprise that the CGF is at the heart of the
mathematical theory of large deviations (see for example Touchette (2011) for
an introduction and the link with statistical mechanics and more historical
references, or Ellis (1985) for an earlier but more complete reference), which
can be seen from a practical point of view as the mathematical formalism that
underpins statistical mechanics.

In cosmology, moment or cumulant generating functions were introduced in
theoretical papers in the late 1970s as a way to probe some part of high-order
correlation functions of clustering fields, which are hard to measure in
practical experiments (see for example White, 1979, on using the probability
of voids as a generating function for all other probabilities). Many attempts
were made to compute these clustering functions and compare the results with
numerical simulations in the following decade (for a detailed discussion of
the advances at the time, see the introduction of Balian & Schaeffer, 1989).
We highlight in particular Balian & Schaeffer (1989), who showed how the
existence of a critical point in the matter density CGF induces an exponential
cut-off in the corresponding PDF, and (Bernardeau, 1992), who made the link
between the tree-order cumulant hierarchy in Eulerian perturbation theory and
the dynamics of spherical collapse. In recent years, and as a generalisation
of the previous results using Eulerian perturbation theory, it has been shown
that large-deviation theory (hereafter LDT) can be used reliably to access the
one-point statistics of the cosmic density field on mildly nonlinear
scales111Beyond the mildly nonlinear regime, phenomenological approaches based
on the halo model (Thiele et al., 2020), hierarchical or lognormal models
(Valageas et al., 2004; Barber et al., 2004; Repp & Szapudi, 2017; Bernardeau,
2022; Uhlemann et al., 2023) are needed.(Bernardeau & Reimberg, 2016). Through
projection, it can be connected to observables such as cosmic shear fields. A
natural approach is to rely directly on the CGF rather than the PDF, because
the theoretical prediction for the latter must be obtained from the CGF
through a (not always tractable) complicated inverse Laplace transform.
Moreover, cosmological analyses of real data, relying for example on Monte-
Carlo Markov chains, require a huge number of evaluations of a theoretical
model for different cosmologies, reinforcing the importance of efficiency.

In this work, we focus on exploring the cosmological information content of
the CGF of the weak lensing convergence, but the results should be generally
applicable to other cosmological fields. For weak lensing and photometric
clustering observables, the CGF is particularly advantageous. The widely
applied Limber approximation assumes that the correlations of the underlying
matter density field along the line of sight are negligible compared to those
in the transverse directions, which means that individual redshift slices can
be treated as statistically independent (Bernardeau & Valageas, 2000;
Barthelemy et al., 2020). This makes the CGF a much more practical statistical
tool than the PDF, because the CGFs in the individual redshift slices can
simply be summed up. This is to be contrasted with the PDF, as Barthelemy et
al. (2022) showed that the computational cost of calculating joint PDFs
between redshift bins of source galaxies from large deviation theory rapidly
increases with the number of redshift bins, to the point of becoming
intractable for the exploitation of upcoming surveys like Euclid (Laureijs et
al., 2011).

The main goal of this paper is to demonstrate how the CGF can be exploited to
capture the same cosmological information as the corresponding PDF (in the
same way that the two-point correlation function and the power spectrum should
lead to identical cosmological constraints), and to perform a simple
cosmological Fisher forecast for the weak lensing convergence field. The goal
is to provide a preliminary proof of concept for eventual joint CGF analyses
of lensing and clustering fields. In this example, we use the large deviations
formalism to predict the theoretical convergence CGF and the corresponding
PDF. In recent work, Breysse et al. (2022) exploited an estimator based on the
characteristic function to demonstrate its potential for extracting cross
correlations between fields. The characteristic function is the Fourier
transform of the PDF, that is, the MGF with a purely imaginary parameter, so
the two approaches are not dissimilar.

Although, similarly to the CGF and with similar references, the history of the
PDF in cosmology is quite old, recent motivation for the application of PDFs
can be found in, for example, Boyle et al. (2021), in which the authors made
use of a Fisher analysis approach to perform a forecast for the LDT-based
theoretical model of the convergence PDF. It was found that the convergence
PDF yields significantly tighter constraints on cosmological parameters,
including the equation of state of dark energy and the neutrino mass, than the
two-point correlation function. Patton et al. (2017) used numerical
simulations to demonstrate that the weak lensing convergence PDF provides
information complementary to that of the cosmic shear two-point correlation
function, also in the presence of weak lensing systematics. This
complementarity between two-point and non-Gaussian statistics is even more
relevant in the presence of shape noise. Liu & Madhavacheril (2019) also used
numerical simulations to demonstrate the power of the tomographic weak lensing
PDF in helping to constrain the total neutrino mass.

From the perspective of theoretical modelling, there exist numerous works
based on numerical simulations or, more recently, the halo model as in Thiele
et al. (2020). Such approaches have the advantage of being appropriate for
modelling smaller scales but require phenomenological input. We will focus
instead here on "from first principles" approaches that are applicable to
somewhat larger scales. As early as Bernardeau & Valageas (2000) and its
follow-up papers, hierarchical models for the statistics of the matter field
were used to construct the PDF of the aperture mass. These works were later
re-interpreted in terms of a large deviation principle, paving the way to
extensions to more realistic observables (e.g. Reimberg & Bernardeau, 2018).
The general formalism involving large deviation theory was first used
explicitly for the clustering of the matter field (see for example Bernardeau
& Reimberg, 2016; Uhlemann et al., 2016). At roughly the same time as Reimberg
& Bernardeau (2018), the works of Bernardeau & Valageas (2000) also inspired
Friedrich et al. (2018) in their modelling of the PDF of tangential shear
profiles. The most recent works on the PDF of the convergence and the aperture
mass within this large deviation framework are Barthelemy et al. (2020) and
Barthelemy et al. (2021), which take into account the full geometry of the
light-cone and probe the validity regime of the formalism with the help of
numerical simulations.

We organise this paper as follows. In Section 2, we define the CGF and review
our theoretical model of weak lensing convergence one-point statistics from
LDT. In Section 3, we determine an appropriate sampling for the CGF to
minimise the impact of rare events to provide robust cosmological analysis. In
Section 4, we examine the covariance matrix of the sampled CGF and perform
Fisher forecasts to demonstrate that the PDF and CGF contain equal
cosmological information. We provide some discussion in Section 5 and conclude
in Section 6. In Appendix A, we validate our estimator for the CGF, and in
Appendix B, we discuss the conditioning of the CGF covariance matrix.

##  2 Theoretical modelling of the convergence field

###  2.1 Useful statistical definitions

Throughout this work, we make use of different statistical quantities that we
briefly introduce here for clarity. From the PDF, ${\mathcal{P}}_{X}$, of some
continuous random variable $X$, one can define the moment generating function
(MGF) as the Laplace transform of the PDF

$M_{X}(\lambda)=\mathbb{E}\left(e^{\lambda
X}\right)=\int_{-\infty}^{+\infty}e^{\lambda x}{\mathcal{P}}_{X}(x){\rm d}x,$
(1)

or equivalently as the expectation value222Note that we make use throughout
this work of the ergodicity hypothesis, in which one assumes that ensemble
averages are equivalent to spatial averages
($\mathbb{E}(.)\rightarrow\left\langle.\right\rangle$) over one realisation of
a random field at one fixed time. This requires that spatial correlations
decay sufficiently rapidly with separation so that one has access to many
statistically independent volumes in one realisation. of the random variable
$e^{\lambda X}$. Note however that the existence of an MGF is not guaranteed
for all possible random fields. For example, the MGF of a strictly lognormal
field is undefined for real positive $\lambda$. For the MGF to exist (in cases
where the PDF also exists), the PDF needs to decay faster than the exponential
for the integral in equation (1) to exist. In the case of the cosmic density
PDF, as computed within the large deviation framework in this paper, there
actually exists a critical positive real value $\lambda_{c}$ – hereafter
dubbed the critical point – beyond which the MGF is not defined. In practice
and for a field sampled within a finite volume, the MGF along the real axis
will always exist and will simply tend towards $e^{\lambda X_{\rm max}}$ for
$\lambda\geq\lambda_{c}$, where $X_{\rm max}$ is the maximum value of $X$ in
the finite field.

The moment generating function, as its name implies, can be used to find the
moments of the distribution as can be seen from the series expansion of the
expectation of $e^{\lambda X}$,

$\displaystyle M_{X}(\lambda)$ $\displaystyle=\\!\mathbb{E}\left(e^{\lambda
X}\right)\\!$
$\displaystyle=\sum_{n=0}^{+\infty}\frac{\lambda^{n}\mathbb{E}\left(X^{n}\right)}{n!},$
(2)

so that the $n$th derivative of the moment generating function in $\lambda=0$
is equal to the $n$th-order moment, $\mathbb{E}\left(X^{n}\right)$. The
logarithm of the MGF is the CGF

$\phi(\lambda)=\log(M(\lambda))=\log(\mathbb{E}(e^{\lambda
X}))=\sum_{n=1}^{+\infty}k_{n}\frac{\lambda^{n}}{n!}\,,$ (3)

where $k_{n}$ are the cumulants (i.e. the connected moments) of the
distribution and where we have dropped the subscript $X$ for clarity. Note
that, in this work, we directly measure the CGF of a given field using this
equation. For each value of $\lambda$, we calculate $\exp{(\lambda X)}$ for
the entire field and then take the logarithm of the mean.

The reduced cumulants are defined as

$S_{n}=\frac{k_{n}}{k_{2}^{n-1}},\ n\geq 1$ (4)

where $k_{2}$ is the variance. These are of importance in the context of
cosmological structure formation because the $S_{n}$ of the cosmic matter
density field have been shown to be independent of the variance (and therefore
redshift) down to mildly non-linear scales (Peebles, 1980; Baugh et al.,
1995). We thus also define the scaled cumulant generating function (SCGF
hereafter) as

$\varphi(\lambda)=\lim_{k_{2}\rightarrow
0}\sum_{n=1}^{+\infty}S_{n}\,\frac{\lambda^{n}}{n!}=\lim_{k_{2}\rightarrow
0}k_{2}\,\phi\left(\frac{\lambda}{k_{2}}\right),$ (5)

which we also extrapolate to non-zero values of the variance (i.e. evaluate at
finite $k_{2}$ on our chosen smoothing scale). One can reconstruct the PDF
from the CGF using an inverse Laplace transform (inverting equation (1))

${\mathcal{P}}(x)=\int_{-i\infty}^{+i\infty}\frac{{\rm d}\lambda}{2\pi
i}\,\text{exp}\left(-\lambda x+\phi(\lambda)\right)\,.$ (6)

###  2.2 Large deviation theory of the matter density field

The large deviation theory (LDT) framework in large-scale structure has mainly
been used to access the one-point statistics of the smoothed density field
(see, for example, Bernardeau & Reimberg, 2016; Uhlemann et al., 2016;
Uhlemann et al., 2018b) and by projection the convergence and aperture-mass
fields (Barthelemy et al., 2020, 2021; Boyle et al., 2021). It has also been
extended to the joint distribution between densities measured at a some
distance (Codis et al., 2016b) and projected quantities between different
source bins (Barthelemy et al., 2022). The results are most simply applied for
highly symmetrical window functions such as two- or three-dimensional top
hats, but can be generalised to other smoothing schemes (Bernardeau &
Reimberg, 2016; Reimberg & Bernardeau, 2018; Barthelemy et al., 2021). We
begin here by recalling some of the results of LDT for the one-point
statistics of the matter density contrast smoothed in two-dimensional disks
(which replicates the dynamics within long cylinders), which in turn will
allow us to compute the one-point statistics of projected quantities like the
convergence field.

A set of random variables $\\{\rho^{\epsilon}\\}_{\epsilon}$ with PDF
${\mathcal{P}}_{\epsilon}(\rho^{\epsilon})$ is said to satisfy a large
deviation principle if the limit

$\Psi_{\rho^{\epsilon}}(\rho^{\epsilon})=-\lim_{\epsilon\rightarrow
0}\epsilon\log\left[{\mathcal{P}}_{\epsilon}(\rho^{\epsilon})\right]$ (7)

exists, where $\epsilon$ is the driving parameter. $\Psi$ is known as the rate
function of $\rho^{\epsilon}$ and describes the exponential decay of its PDF.
The driving parameter $\epsilon$ indexes the random variable with respect to
some evolution, for example an evolution in time. In the case of the matter
density field smoothed on a single scale $R$, this driving parameter is the
variance, which acts as a clock from initial to late times
($\epsilon\equiv\sigma^{2}_{R}$). We now omit the $\epsilon$ sub/superscripts
in our notation for simplicity.

The existence of a large deviation principle for the random variable $\rho$
implies that its SCGF $\varphi_{\rho}$ is given through Varadhan’s theorem as
the Legendre-Fenchel transform of the rate function $\Psi_{\rho}$ (Ellis,
1985; Touchette, 2011)

$\varphi_{\rho}(\lambda)=\sup_{\rho}\,\left[\lambda\rho-\Psi_{\rho}(\rho)\right],$
(8)

where the Legendre-Fenchel transform reduces to a simple Legendre transform
when $\Psi_{\rho}$ is convex. In that case,

$\varphi_{\rho}(\lambda)=\lambda\rho-\Psi_{\rho}(\rho),$ (9)

where $\rho$ is a function of $\lambda$ through the stationary condition
333The $\rho_{c}$ value at which $\Psi_{\rho}$ ceases to be convex leads to a
$\lambda_{c}$ value that corresponds to the critical point mentioned in the
discussion of equation (1).

$\lambda=\frac{\partial\Psi_{\rho}(\rho)}{\partial\rho}.$ (10)

Another consequence of the large-deviation principle is the so-called
contraction principle. This principle states that for a random variable $\tau$
satisfying a large deviation principle and related to $\rho$ through the
continuous mapping $f$, the rate function of $\rho$ can be computed as

$\Psi_{\rho}(\rho)=\inf_{\tau:f(\tau)=\rho}\Psi_{\tau}(\tau).$ (11)

This is called the contraction principle because $f$ can be many-to-one, in
which case we are contracting information about the rate function of one
random variable down to the other. In physical terms, this states that an
improbable fluctuation of $\rho$ is brought about by the most probable of all
improbable fluctuations of $\tau$.

For the case of the matter density field, the rate function of the late-time
density field at different scales can be computed from the initial conditions
if the most likely mapping between the two is known – that is, if one is able
to identify the leading field configuration that will contribute to the
infimum of equation (11). In cylindrically symmetric configurations, as for a
disk of radius $R$ at redshift $z$ (in 2D space) or alternatively a very long
3D cylinder centered on this disk, the most likely mapping between final and
initial conditions should preserve the symmetry444This is only true for a
certain range of density contrasts around zero, very much sufficient for our
purposes. However, one could note that there are counter-examples in which the
spherical or cylindrical symmetry does not lead to spherical/cylindrical
collapse being the most likely dynamics, for example in 1D for very high
values of the density. (Valageas, 2002), which leads to initial conditions
also being cylindrically symmetric and the dynamics between the two being
those of cylindrical collapse.

Thus, starting from Gaussian initial conditions,555Primordial non-
Gaussianities can also straightforwardly be accounted for in this formalism as
shown by Uhlemann et al. (2018a). the rate function of the linear field is
simply given by a quadratic term. The rate function of the late-time density
field in a disk of radius $R$ is then given by

$\Psi_{\rm cyl}(\rho)=\sigma^{2}_{R}\frac{\bar{\tau}^{2}}{2\sigma^{2}_{r}},$
(12)

where $\sigma^{2}_{R}$ — our driving parameter — is the variance of the non-
linear density field in the disk, $\sigma^{2}_{r}$ is the variance of the
linear density field inside the initial disk (before collapse) of radius
$r=R\,\rho^{1/2}$ (from mass conservation), and $\bar{\tau}$ is the linear
density contrast obtained through the most probable mapping between the linear
and late-time density fields. This mapping is given by 2D spherical
(cylindrical) collapse, for which an accurate parametrisation is given by
(Bernardeau, 1995)

$\zeta(\bar{\tau})=\rho=\left(1-\frac{\bar{\tau}}{\nu}\right)^{-\nu}.$ (13)

In the spirit of previous works involving the density filtered in spherical
cells, the value of $\nu$ in this parametrisation of $\zeta$ is chosen to be
$\nu=1.4$, so as to reproduce the value of the tree-order skewness in
cylinders as computed from perturbation theory (Uhlemann et al., 2018b).

Then, as a straightforward consequence of the contraction principle, the rate
function given by equation (12) is also the rate function of any monotonic
transformation of $\rho$, so that for the density contrast $\delta=\rho-1$, we
have $\Psi_{\delta}(\delta)=\Psi_{\rho}(\rho(\delta))$. Thus, plugging
equation (12) into equation (9) gives us the SCGF of the matter density
contrast in a disk at redshift $z$.

Finally, one of the key aspects of the large deviation formalism in the
cosmological context is that we apply the result for the SCGF beyond the
$\sigma^{2}_{R}\rightarrow 0$ limit. This allows us to extrapolate to the CGF
of the real density field by rescaling the SCGF by the driving parameter (the
non-linear variance) at the scale and redshift being considered, such that

$\phi_{\rm cyl}(\lambda)=\frac{1}{\sigma^{2}_{R}}\varphi_{\rm
cyl}(\lambda\sigma^{2}_{R}).$ (14)

This is physically meaningful because the reduced cumulants $S_{n}$ from the
cylindrical/spherical collapse dynamics have been shown to be very robust over
a large range of scales and redshifts down to mildly non-linear scales
($\gtrsim 5$ Mpc/$h$ at $z\gtrsim 0$ (see for example Figure 2 in Uhlemann et
al. (2016) and Figure A1 in Uhlemann et al. (2018b)), so that re-scaling by
the non-linear variance allows access to the full one-point statistics of the
non-linear density field. In LDT terms, the SCGF given by the large deviation
principle is the well-defined asymptotic form taken by the cumulants of the
field in the regime where the variance goes to zero, and we simply keep this
form for our predictions of the non-linear CGF.

Finally, note that although equations (9) and (12) have been known and used
for three decades in the context of counts-in-cells statistics (that is, the
density field filtered in 3D top-hat windows — see for example Bernardeau,
1994; Valageas, 1998), their re-derivation through large deviation statistics
is more general and allows one to set up a framework for the computation of
different probabilities in the cosmological context.

###  2.3 From matter density to convergence

Let us recall that for a flat cosmology, the convergence $\kappa$ can be
interpreted as a line-of-sight projection of the matter density distribution
between the observer and the source and can be written as (Mellier, 1999)

$\kappa({\bm{\vartheta}})=\int_{0}^{z_{s}}{\rm d}z\frac{{\rm d}\chi}{{\rm
d}z}\,\omega(\chi,\chi_{s})\,\delta(\chi,\chi{\bm{\vartheta}}),$ (15)

where $\chi$ is the comoving radial distance, $z_{s}$ is the source redshift,
and $\chi_{s}$ is the cosmology-dependent distance associated with the source
redshift. The lensing kernel $\omega$ is then defined as

$\omega(\chi,\chi_{s})=\frac{3\,\Omega_{m}\,H_{0}^{2}}{2\,c^{2}}\,\frac{\chi(\chi_{s}-\chi)}{\chi_{s}}\,(1+z(\chi)).$
(16)

Under the small-angle/Limber approximation, it can be shown that correlators
of the (smoothed) convergence field can be seen as a juxtaposition of the 2D
correlators of the underlying density field, as if each 2D slice along the
line of sight is statistically independent of the others (Bernardeau &
Valageas, 2000; Barthelemy et al., 2020). In terms of the one-point statistics
of the smoothed $\kappa$ field within a top-hat window function of angular
radius $\theta$, this translates to saying that the CGF of $\kappa$ is a sum
along the line of sight of the CGF of independent 2D slices of the matter
density contrast:666Rigorously, this result applies for a juxtaposition of
very long cylinders centered on the slices and of length $L\rightarrow\infty$.
Since the symmetry and thus the most likely dynamics of these long cylinders
are the same as for a 2D slice in a 2D space, and since the results are
independent of $L$, we refer to ‘2D slices’ for clarity. This emphasises that
correlations along the line of sight are negligible compared to those in the
transverse directions.

$\phi_{\kappa,\theta}(\lambda)=\int_{0}^{\chi_{s}}{\rm d}\chi\,\phi_{{\rm
cyl}}^{<\chi\theta}(\omega(\chi,\chi_{s})\lambda,\chi(z)),$ (17)

where $\phi_{{\rm cyl}}^{<\chi\theta}$ is the CGF of the density contrast
filtered within a disk of radius $\chi(z)\theta$ so as to reproduce the
geometry of the light cone.

Equation (17) thus reduces the complexity of the problem down to computing the
one-point statistics of the 2D matter density in each two-dimensional slice
(or equivalently within long 3D cylinders at the same redshift) along the line
of sight, which we have already done in Section 2.2. Using these results, we
can then build the non-linear CGF of the convergence field. Note that equation
(17) highlights the nice property of the projected CGF being expressible
simply as a sum of independent redshift slices. This is an important property
when considering more complicated joint distributions, in which the only
modification in this integral would be the replacement of the
$\omega(\chi,\chi_{s})\lambda$ term by a term depending on more than one
$\lambda$ variable. This form is much simpler than that of the corresponding
multi-variate PDF (Barthelemy et al., 2022).

Finally, it is important when working with the CGF to know the approximate
location of the (theoretical) critical point of the convergence field,
$\lambda_{c}$. First, the critical points $\delta_{c}$ in each redshift slice
are calculated by finding where the second derivative of the rate function
becomes equal to zero. The corresponding critical $\lambda$ values can be
obtained by applying the stationary condition (Eq. (10)). The minimum
$\lambda_{c}$ along the line of sight is taken as the critical point of the
convergence CGF.

##  3 The Cumulant Generating Function as an Alternative Observable

Figure 1: The impact of small tail cuts in $\kappa$ on the corresponding CGF
($z_{s}=2$, $\theta=10$ arcmin). The red line shows the CGF calculated from
the large deviation formalism, with the imposed error bars measured from the
Takahashi simulation convergence maps. The blue line shows the CGF obtained
from a Laplace transform of the PDF cut to exclude $\kappa$ values with
probability densities lower than $10^{-3}$ ($\kappa\in[-0.045;0.07]$). The
green line shows the residual between the CGF obtained from the cut CGF and
the original. This roughly shows what happens on a simulated small convergence
map where rare events are not present, and demonstrates that using either the
CGF or PDF as an observable should yield similar information content, because
the largest deviations are seen far in the negative $\lambda$ tail, which we
will see is not needed to extract cosmological information. The vertical
dotted lines indicate the section of the CGF we eventually use for our Fisher
analysis.

Although the large deviations formalism presented in Section 2 initially
provides predictions for the full CGF, all previous work has focused on the
corresponding PDF (as a complicated inverse Laplace transform (6)) or on
individual cumulants (as Taylor coefficients of the CGF (3)). The main reason
for this seems to be the intuitiveness of these quantities — the PDF is easily
interpreted as a simple histogram and the cumulants are visualisable as
properties of the shape of the PDF. However, the CGF is clearly also an
observable (the log of the expectation value in equation (3)).

Beyond simplicity, another motivation for concerning ourselves with the CGF
instead of the PDF is as follows. In recent work, Barthelemy et al. (2022)
explored the computational complexity involved in calculating the joint PDF
for a tomographic weak lensing survey. Despite some simplification strategies,
an exact calculation of the joint PDF remains computationally expensive, with
the computational time required increasing rapidly with the number of source
redshift bins considered. However, this complexity mostly comes from the fact
that the correlation structure between bins is expressed in a very complicated
manner for the PDF, while it is much more transparent for the CGF. As a
consequence, working with the CGF directly should significantly improve the
efficiency of our calculations. This is particularly important for
cosmological forecasts and analyses, where it is desirable to be able to
rapidly produce a large number of predictions for different sets of
cosmological parameters. Additionally, note that although Barthelemy et al.
(2022) focused explicitly on joint-lensing statistics between redshift bins,
their result could be summarised by stating that the Limber approximation
allows drastic simplifications of the correlation structure between CGFs, and
generalised to the case of the (fully projected or tomographic) density field
at different redshifts. This could be applied to joint analyses of lensing and
clustering, making the inclusion of their full joint distributions more
feasible.

For the CGF to be a robust observable for cosmological forecasting, it has to
be demonstrated that it changes minimally when very rare events are excluded,
in order to allow for measurement limitations and small survey sizes. Figure 1
compares a CGF calculated directly from the LDT formalism (red) with one
Laplace-transformed from a truncated PDF (blue), with the rare-event tails cut
out to exclude values of $\kappa$ with probability densities below $10^{-3}$.
The result is very satisfactory, with very little difference seen between the
two. The residual is shown in green, with error bars derived from 108
convergence maps from the Takahashi simulations (see Section 3.2). The maximal
difference of a few percent occurs at very low $\lambda$ values that we will
see later are not needed to extract cosmological information. Note that this
difference at low $\lambda$ is expected, because the asymptotic behaviour of
the negative tail of the CGF is $\propto\lambda\kappa_{\rm min}$, where
$\kappa_{\rm min}$ is either the minimum theoretical value the convergence can
take (when the density contrast equals -1 all along the line of sight), or the
minimum value of $\kappa$ after the cut or in a small simulated field.

###  3.1 Relating $\kappa$ and $\lambda$ with the saddle-point approximation

One primary goal of this work is to demonstrate that for practical purposes
the PDF and CGF contain equal information, by determining how to accurately
translate information content between the two. While the PDF is generally
considered in $\kappa$ bins of (usually equal) finite width, the CGF is
sampled at individual $\lambda$ values. We can apply a saddle-point
approximation to the inverse Laplace transform in equation (6) such that

Figure 2:  The relationship between values of $\kappa$ and $\lambda$ inferred
from the saddle-point approximation (18). The blue section shows the region of
$\lambda$ values that we use in practice in our Fisher forecasts, covering the
range from $\kappa_{\rm min}$ to zero. As explained later, we find that using
only the negative part of the CGF is sufficient to reproduce the constraints
from the bulk of the PDF. The denser dotted line at $\lambda=125$ represents
the theoretical critical point. The theoretical PDF is shown in the background
in grey to provide intuition for the corresponding abundance of $\kappa$
values.

$\mathcal{P}(\kappa)\approx\frac{1}{\sqrt{2\pi}}\sqrt{\frac{\partial^{2}\psi(\kappa)}{\partial\kappa^{2}}}\exp(-\psi(\kappa)),$
(18)

where $\phi_{\kappa}(\lambda)=-\psi(\kappa)+\lambda\kappa$ with the stationary
condition $\lambda=\partial\psi(\kappa)/\partial\kappa$. This gives an
approximate direct relationship between values of $\kappa$ and $\lambda$,
which we use as a guideline for predicting appropriate values of $\lambda$ to
sample to obtain approximately the same information from the CGF as from a
given PDF. This relationship is illustrated in Figure 2. The blue section
highlights the range of $\lambda$ values we ultimately use in our Fisher
forecasts (corresponding to $\kappa$ values between the $\kappa_{\rm min}$ we
adopt for the PDF and zero). The choice to use zero as the upper limit in
$\lambda$ instead of a $\lambda$ value derived from $\kappa_{\rm max}$ is
explained in Section 4.2, while its impact in terms of cosmological
information is discussed in Section 4.3.

###  3.2 Measuring the CGF in Simulations

To validate our model for the CGF and to generate a covariance matrix, we use
a set of weak lensing convergence maps generated from ray-traced N-body
simulations as part of the Takahashi simulation suite (Takahashi et al.,
2017). We have 108 maps for a fixed source redshift of $z_{s}\approx 2$. We
smooth the maps with a top-hat filter of radius 10 arcmin in harmonic space.
We extract the PDF in 50 linearly-spaced bins in the range $-0.1\leq\kappa\leq
0.1$ and the CGF for real values of $\lambda$ in the range
$-1000\leq\lambda\leq 150$ with a spacing of 5, having calculated that the
theoretical critical point in the CGF lies at approximately $\lambda=125$. The
CGF is calculated directly as we define it in equation (3) – for each value of
$\lambda$, we take the logarithm of the expectation value of
$\exp(\lambda\kappa)$, it is neither reconstructed from the PDF nor from
cumulants. Figure 3 shows the theoretical and measured PDFs (upper panel)
alongside the residual between them in green; the corresponding result for the
CGF is shown in the lower panel. We also demonstrate in Appendix A that our
estimator for the CGF is robust and discuss its validity and potential biases.

Figure 3: Upper panel: A comparison of the theoretically modelled PDF and that
from the Takahashi simulations (for fixed source redshift $z_{s}\approx 2$ and
a smoothing scale of 10 arcmin). The theoretical and simulated PDFs are shown
in blue (see the left y-axis). The residual is shown in green (see the right
y-axis) and shows percent-level accuracy within the region considered. The
vertical dotted lines indicate the portion of the PDF used for the Fisher
matrix forecast (which is decided based on a $\chi^{2}$ test to ensure the
Gaussianity of the data vector). The horizontal grey bands represent a 2.5%
and 5% deviation. Lower panel: A comparison of the corresponding CGFs. The
theoretical and simulated CGFs are shown in blue (see the left y-axis). The
theoretical CGF is shown up to $\lambda=125$, which approximately corresponds
to the critical point beyond which the CGF develops linear asymptotic
behaviour (evolving $\propto\lambda\kappa_{\rm max}$) with rapidly increasing
error bars. This behaviour is shown as measured from the maps by the blue
points. The error bars on the residual become larger around $\lambda=100$,
indicating that the critical value in the measured CGF is lower than that
predicted by the theory, as expected. The residual between the theoretical and
measured CGFs is shown in green (see the right y-axis).

##  4 Cosmological Analysis

###  4.1 Covariance matrix

Figure 4 shows the covariance matrix of the CGF as measured from 108 Takahashi
simulation convergence maps. The lower $\lambda$ limit is determined from the
saddle-point approximation (18) and our corresponding lower $\kappa$ limit for
the PDF (see Figure 3 for precise values), and the upper limit goes a little
beyond the critical point. Note that the colour bar range includes only
positive values (on a logarithmic scale) because the points in the CGF are
always positively correlated with each other. This can be understood
intuitively as a consequence of the positive correlations between the
cumulants (due to their dependence on the variance) and the polynomial form of
the CGF seen in equation (3). We see a divergence in the covariance for high
$\lambda$ values beyond the critical point, as we expect, and which
corresponds to the large error bars beyond the critical point seen in Figure
3.

In Figure 5 we show a correlation matrix for a data vector that combines CGFs
for two smoothing scales ($\theta=7.5^{\prime}$ and
$\theta=10^{\prime}$).777Note that in previous work (Boyle et al., 2021), it
was shown that combining PDFs with different smoothing scales provided a
significant enhancement in cosmological information. While the covariance
matrix is used in Fisher matrix forecasts, the correlation matrix has the
advantage of being more visually intuitive for quantities of different
magnitudes. The upper left panel corresponds to the covariance matrix seen in
Figure 5. We see that the two CGFs are also very strongly correlated with each
other, as is the case for the corresponding PDFs. We also note that beyond the
critical point, the CGF values are only weakly correlated with those before
the critical point but very strongly correlated with each other. This is as we
expect, because the behaviour of the CGF here is determined only by the
maximum $\kappa$ value in a particular map. We note that the critical point in
the maps occurs at lower $\lambda$ (around 100, though a precise estimation
does not exactly make sense in a finite volume) than in the theory (where we
predict $\lambda_{\rm crit}=125$).

Figure 4: The covariance matrix of the CGF as measured for a sample of
$\lambda$ values. The lower $\lambda$ value of -200 is extracted from the
lowest $\kappa$ value used for the PDF using the saddle-point approximation
(18). The critical point is at approximately 125 for the theoretical CGF but
is lower in the actual maps. We can observe a number of distinctive features.
First, the values of $\phi(\lambda)$ are all positively correlated with each
other (note that the colour bar only covers correlation coefficients between
zero and one and has a logarithmic scale). Second, the large covariance values
for higher $\lambda$ values correspond to the expected behaviour of errors
beyond the critical point. Figure 5: The correlation matrix for CGFs on two
smoothing scales: 10 arcmin on the top left and 7.5 arcmin on the bottom
right, with the cross-correlation between the two scales shown in the two
other panels. We note that the top-left panel corresponds to the case shown in
the covariance matrix in Figure 4. We emphasise the very strong correlation
between the negative-$\lambda$ CGF points and between the two CGF smoothing
scales. We also note the very strong correlation between points beyond the
critical point and the very weak correlation between these points and the rest
of the CGF — both behaviours are expected because these points are sensitive
only to the maximum values of $\kappa$ in the individual maps.

###  4.2 Ensuring a Gaussian-distributed data vector

Fisher matrix forecasts rely on the assumption that the observable data vector
is Gaussian distributed. We now need a robust test to determine whether this
is the case for the CGF, and indeed whether it is the case for all values of
$\lambda$. The $\chi^{2}$ test is a very useful tool for this purpose, as was
demonstrated, for example, in Friedrich et al. (2021b). If the measurements of
the elements of a data vector $\bm{s}$, with $\hat{{s}}_{i}$ one particular
measurement, are Gaussian distributed, then the distribution of values of
$\chi^{2}_{i}$ defined as

$\chi^{2}_{i}=(\hat{s}_{i}-\mu)C^{-1}(\hat{s}_{i}-\mu),$ (19)

where $\mu$ is the mean measured data vector, should follow a $\chi^{2}$
distribution for the appropriate number of degrees of freedom (the number of
elements in the data vector). Next, one can draw random Gaussian data vectors
from the covariance matrix and use them in equation (19) to see if these
follow a similar distribution. If so, we are assured that the data vector
considered is reasonably Gaussian-distributed.

The above approach limits us to 108 samples for the inferred distribution,
which can be a little limiting when trying to determine how well the result
matches a $\chi^{2}$. As an alternative, one can combine pairs of measurements
to increase the number of samples to $108\cdot 107/2$ while preserving the
covariance, as

$\chi^{2}_{ij}=\frac{1}{2}(\hat{s}_{i}-\hat{s}_{j})C^{-1}(\hat{s}_{i}-\hat{s}_{j}).$
(20)

This approach is adopted here and leads to a visually better result for a
limited number of measurements.

Figure 6 shows a sample $\chi^{2}$ test for the PDF, including 13 $\kappa$
bins between $\kappa_{\rm min}=-0.02$ and $\kappa_{\rm max}=0.026$. This is a
relatively small number of bins, but is chosen to keep the $\chi^{2}$ test
reliable as our covariance matrix is based on a rather small number of
convergence maps (108 from the Takahashi simulations). Uhlemann et al. (2023)
showed that a robust covariance matrix can be extracted for twice the number
of bins in the same $\kappa$ range with 500 maps. We see that our result
coincides reasonably well with a $\chi^{2}$ distribution for 13 degrees of
freedom.

Figure 7 shows a number of $\chi^{2}$ tests for the CGF. The lower value of
$\lambda$ is fixed at $-205$ (as inferred from the saddle point approximation
and the value of $\kappa_{\rm min}$ used for the PDF) and various values are
chosen for the upper limit $\lambda_{\rm max}$, with $\lambda_{\rm max}$
increasing on moving from left to right. Note that $\lambda=-5$ is the last
negative $\lambda$ point we can sample because we sample $\lambda$ with a
spacing of 5, and we never include $\lambda=0$ because its value is fixed at
$\phi(0)=0$ by definition. We see that including points close to the critical
point significantly degrades the performance of the $\chi^{2}$ test. This is
probably because the behaviour of the CGF at high $\lambda$ is determined by
large $\kappa$ values, and therefore by rare events that are less likely to be
Gaussian-distributed. In Figure 8, we show the actual distributions of the
maximum $\lambda$ values for each case in Figure 7. We see that these are
reasonably Gaussian up until about $\lambda=65$, which is consistent with the
result seen for the $\chi^{2}$ tests. Overall, we see that the $\chi^{2}$ test
works best when only the negative-$\lambda$ part of the CGF is sampled
($\lambda_{\rm max}=-5$), and as we will see shortly, this part of the CGF
alone is sufficient to provide equally strong cosmological constraints to the
PDF.

There are additional reasons to restrict ourselves to the negative-$\lambda$
region of the CGF. The value of the critical point varies with the angular
scale and source redshift considered, and since our theoretical model tends to
overpredict its value — typically because the cylindrical collapse dynamics
underpredicts the convergence skewness, and thus underpredicts the high-
density tail of the PDF which leads to a larger $\lambda_{c}$ — it is safer to
take a conservative approach and adopt $\lambda_{\rm max}=0$. On a more
theoretical level, the approximation of cylindrical collapse as a proxy for
the non-linear dynamics should break down for $\lambda>0$ (and even more when
approaching $\lambda_{c}$) in the CGF as argued for instance by Valageas
(2002), although we find it still performs well when comparing to simulations
(see Fig. 3). In the next section, we will explain why we do not expect
excluding the positive tail of the CGF to lead to a significant loss of
cosmological information.

In Figure 9, therefore, we show $\chi^{2}$ tests for only the negative part of
the CGF ($-205\leq\lambda\leq-5$) while varying the number of $\lambda$ values
sampled. The $\chi^{2}$ works well for a range of samplings, but works less
well for 11 points. Moreover, we show in Section 4.5 that a Fisher forecast
with 11 or more points is unstable. In Appendix B, we show that this is due to
the covariance matrix being ill-conditioned because of the high level of
correlation between points in the CGF. A ridge-regression scheme can be
applied to resolve this, as demonstrated in the appendix.

In practice, as we will see in Section 4.5, it is nevertheless possible to
obtain cosmological constraints competitive with the PDF with quite a small
number of data points from the CGF (e.g. three points can provide equivalent
constraints to the PDF for two parameters).

Finally, in Figure 10, we show a $\chi^{2}$ test for a data vector consisting
of two CGFs with different smoothing scales. The same values of $\lambda$ are
chosen for both CGFs, and the total number of points indicated in the legend
refers to the total number of $\lambda$ points sampled over the two CGFs — for
example, $N_{\rm points,total}$ of six means three $\lambda$ points are
sampled from each CGF. In this case, we see that the $\chi^{2}$ test performs
well up to and including a total number of sampled points of 14 (seven per
CGF).

Figure 6: $\chi^{2}$ test for the PDF. There are 13 $\kappa$ bins between
$\kappa_{\rm min}=-0.02$ and $\kappa_{\rm max}=0.026$. The $\chi^{2}$ test
performs well, indicating that the data vector is Gaussian-distributed. Figure
7: $\chi^{2}$ tests for the CGF. The different instances are shifted
rightwards by an arbitrary amount for clearer viewing. The lowest value of
$\lambda$ sampled (-205) is determined from the minimum value of $\kappa$ used
in the PDF analysis using the saddle-point approximation. The success of the
$\chi^{2}$ test seems relatively insensitive to the choice of $\lambda_{\rm
min}$. The maximum value of $\lambda$ is varied with the number of points
sampled being fixed. The $\chi^{2}$ test is less successful as $\lambda_{\rm
max}$ is pushed to higher values and closer to the critical point. Figure 8:
Distributions of $\phi(\lambda)$ for the $\lambda$ values corresponding to the
maximum values in Figure 7. The distributions become more and more non-
Gaussian on approaching the critical point, with a very clear non-Gaussianity
for $\lambda=95$, which explains the result seen in Figure 7.  Figure 9: As
Figure 7, but in this case the maximum value of $\lambda$ is fixed and the
total number of points in the data vector is varied. The $\chi^{2}$ test seems
to become less successful as the number of points is increased. This is a
result of the covariance matrix of the CGF being ill-conditioned — a problem
that becomes worse as points are more closely sampled (see main text). This is
discussed further in Appendix B. Figure 10: A $\chi^{2}$ test for a data
vector consisting of two CGFs for different smoothing scales
($\theta=7.5^{\prime}$ and $\theta=10^{\prime}$). Here $N_{\rm points,total}$
refers to the total number of points sampled from the two CGFs, so $N_{\rm
points,total}=6$ refers to three points being sampled from each CGF. We see
that the $\chi^{2}$ test generally performs quite well but somewhat worsens as
the number of points is increased.

###  4.3 The Negative-$\lambda$ CGF

In the previous section, we provided a number of reasons for using only the
negative tail of the CGF as a cosmological forecasting observable. We can now
explain why using only the negative part of the CGF to derive cosmological
information should not lead to significant information loss.

Figure 11: Simple Fisher forecasts for three cosmological parameters (which
require combining PDFs from two smoothing scales to provide meaningful
constraints) using only half of the PDF (the negative-$\kappa$ part, black)
and the full PDF (grey). We see that the constraints from using only half of
the PDF are almost equal to those from the full PDF.

The first thing to note is that formally, the entire statistical information
is located in the vicinity of $\lambda=0$ (one way to understand this is that
the cumulants can be defined as the derivatives of the CGF around the zero
point), and we can probe this region using the negative tail of the CGF alone.
Secondly, according to the $\kappa-\lambda$ relationship shown in Figure 2, a
cosmological analysis using only the negative-$\lambda$ part of the CGF should
be similar (not identical because Figure 2 is only an approximation) to an
analysis only using the negative-$\kappa$ part of the PDF. We show in Figure
11 that cosmological constraints using only the negative-$\kappa$ half of the
PDF provide almost equivalent constraints to the full bulk of the PDF. We find
that the 1-$\sigma$ constraints on the parameters are only about 10% weaker
when using only the negative half of the PDF. We assume that this is because
for a positively-skewed PDF, the PDF maximum is located within the
negative-$\kappa$ region, so the variance and skewness (the moments from which
almost all of the information comes) can be quite well constrained from this
portion. We can therefore expect that the constraints from the
negative-$\lambda$ CGF capture most of the information one could extract from
the $\kappa$-PDF as computed in the LDT framework. Note that this remains a
statement about $\kappa$ values in the bulk of the PDF and thus not applicable
to cosmological information that could be extracted from extremely non-linear
and rare events. Moreover, we see in Figure 3 that LDT seems to underpredict
the positive part of the CGF and that the prediction for the negative portion
is much more accurate. The same behaviour can be observed at the level of the
PDF. This might contribute to the lower weight of the positive part of the
CGF.

###  4.4 Derivatives with respect to cosmology

Figure 12: The derivatives of the CGF (upper panel) and PDF (lower panel) with
respect to $\Omega_{cdm}$ (red), $\sigma_{8}$ (blue) and $w_{0}$ (green) as
predicted from the LDT formalism. In both cases, we see that the shapes of the
derivatives for $\Omega_{cdm}$ and $\sigma_{8}$ are very similar, which is
consistent with the well-known degeneracy between the two parameters. Note
that these derivatives give no indication of the measurability of the relative
effects of the cosmological parameters on the observables, and are intended
only to show that the relative trends for the PDF and CGF derivatives are
consistent.

Figure 12 shows the derivatives of both the theoretical CGF (lower panel) and
PDF (upper panel) with respect to $\Omega_{\rm cdm}$ (red), $\sigma_{8}$
(blue) and $w_{0}$ (green). In both cases, we see that the shapes of the
derivatives are quite similar for $\Omega_{\rm cdm}$ and $\sigma_{8}$ — this
reflects the well-known weak lensing degeneracy between these two parameters.
We also see in both cases that the primary change is a change in the variance,
from the parabola-like derivative for the PDF (an increase in the variance
leads to a lower peak and higher tails), and with the derivatives for the CGF
looking something like quadratic functions of $\lambda$ in the lower panel.

###  4.5 Fisher Forecasts

In this section, we perform a number of Fisher forecasts to demonstrate
conclusively that the weak lensing convergence CGF can be used to extract an
identical amount of cosmological information to the PDF. A Fisher forecast for
a set of cosmological parameters, $\bm{\theta}$, given an observed data
vector, $\bm{s}$, is defined as

$F_{ij}=\sum_{\alpha,\beta}\frac{\partial
s_{\alpha}}{\partial\theta_{i}}C^{-1}_{\alpha\beta}\frac{\partial
s_{\beta}}{\partial\theta_{j}}~{},$ (21)

where $s_{\alpha}$ is element $\alpha$ of $\bm{s}$, the cosmological
parameters are indexed as $i,j$ and $C_{\alpha,\beta}$ is the covariance
matrix of $\bm{s}$, which is assumed to be cosmology-independent.

The Fisher matrix allows us to determine the error contours on a set of
cosmological parameters under the assumption that the likelihood is Gaussian,
which should be assured by the relevant $\chi^{2}$ test. The inverse of the
Fisher matrix gives the parameter covariance. The error on the parameter
$\theta_{i}$, marginalised over all other parameters, is given by

$\delta\theta_{i}\geq\sqrt{\left(F^{-1}\right)_{ii}}\ .$ (22)

As we estimate the covariance matrix in this work from a limited number of
covariance maps (108), it is critical to correct for the effect of noise on
the matrix inversion process. This is done by multiplying by the
Kaufman–Hartlap factor (Kaufman, 1967; Hartlap et al., 2006),

$h=(N_{\rm map}-2-N_{\rm points})/(N_{\rm map}-1).$ (23)

Here $N_{\rm points}$ can be the number of points sampled for the CGF or the
number of bins considered for the PDF, i.e. it is the length of the given data
vector. Based on the results of the previous sections, we take a conservative
approach and use only the negative part of the CGF to produce Fisher matrix
forecasts. In Figure 13, we compare a number of two-parameter Fisher forecasts
for the CGF with the corresponding PDF forecast, varying the number of points
sampled in the CGF. We see clearly that including only negative values of
$\lambda$ up to reasonably close to the zero point produces a result that very
closely agrees with the constraint from the PDF (the grey dashed line). Moving
beyond the zero point (not shown) tightens these constraints, but we know that
this information may not be appropriately distributed from the $\chi^{2}$
tests in the last section. The minimum value of $\lambda$ is determined from
the minimum value of $\kappa$ considered in the PDF forecast using the saddle-
point relation between $\kappa$ and $\lambda$, but the results are not very
sensitive to this precise choice.

We also see in Figure 13 that the CGF forecast performs very well even for a
data vector with only three points (at least in this simple two parameter case
— of course, with an increasing number of cosmological parameters, more points
will be needed to allow meaningful constraints). The results are consistent up
until a data vector of length nine, but beyond this point the agreement breaks
down as the $\chi^{2}$ test fails (see Figure 9).

In Figure 14, we extend to a cosmological forecast for three parameters
(adding $w_{0}$). This requires combining PDFs / CGFs from two different
smoothing scales (7.5 and 10 arcmin) to provide meaningful constraints,
because the cosmological information from both statistics is provided
overwhelmingly by the first two cumulants (effectively the variance and
skewness) and two measurements are insufficient to break degeneracies between
three parameters. Once again, we see good agreement between the results from
the CGF for different samplings and the PDF.

In general, as already found in Boyle et al. (2021), the cosmological
information from convergence one-point statistics is dominated by the
information contained in the variance and skewness. However, note that in the
case of joint analyses (combining lensing and clustering or different redshift
bins), the number of terms required to include all the cumulants of a given
order (e.g. order three for the generalisation of the skewness) increases
dramatically with the number of random variables. Additionally, cumulants of
order higher than three in these scenarios also contain powers of the random
variables smaller than three and may therefore contain additional cosmological
information. Thus, as advocated throughout this work, it makes more sense to
directly use the full CGF in such joint-analyses settings.

Figure 13: Fisher matrix forecasts for the CGF sampled at several equidistant
points between $\lambda_{\rm min}=-205$ (determined from $\kappa_{\rm min}$ in
the PDF forecast using the saddle-point approximation) and the final point we
measure before zero, $\lambda_{\rm max}=-5$. A comparison forecast for the PDF
is shown as a grey dashed line. Sampling three points of the CGF (red) is
sufficient to accurately replicate the PDF contour in this simple two-
parameter case. The result is consistent as more data points are added (up to
nine). Beyond this point, the ill-conditioning of the covariance matrix causes
the results to become unreliable — see Appendix B. Figure 14: The Fisher
matrix forecast expanded to capture three cosmological parameters. In this
case, two smoothing scales are combined to provide sufficient information —
7.5 and 10 arcmin. We see that the results from the CGF and PDF are highly
consistent.

##  5 Discussion

###  5.1 Applicability of the CGF

In the previous section, we showed that the CGF can be used as an alternative
observable to the PDF that provides roughly equivalent cosmological
constraints. In this section, we will discuss in which scenarios the CGF is
preferable to the PDF — in analogy to the complementarity between power
spectra and correlation functions in two-point analyses. We recommend
considering turning to the CGF in cases where the joint one-point statistics
of random variables are considered, either because it can be practically time-
consuming and numerically intensive to perform a multi-dimensional inverse
Laplace transform to extract the PDF, or physically favourable for the
correlation structure between the random variables considered. One example of
the former case is the calculation of the joint statistics between different
smoothing scales for either matter or lensing fields in the context of LDT.
One example of the latter case is considering joint statistics between lensing
fields in different tomographic bins or joint statistics between lensing and
clustering. In these contexts, the Limber approximation is generally applied,
meaning that the slices of the density field along the line of sight are
considered statistically independent. This means that random variables are
only correlated if they consist of functions of overlapping density slices,
which makes their correlation structure extremely straightforward for the CGF
(see appendix A of Barthelemy et al. (2022)).

As an example of this complexity, the BNT transform (or nulling strategy)
(Bernardeau et al., 2014) is an invertible linear application that maximally
decorrelates lensing fields measured in different tomographic bins. We can
consider sets of three random variables correlated two by two, with the first
and third variables only being correlated through the second one. This
simplified correlation structure is then particularly easily described by
relying on CGFs instead of PDFs. This simplified correlation structure also
inevitably leads to a simplification in the computation of the full joint
statistics, independently of how they are calculated, making it almost
mandatory for the efficient estimation of cosmological parameters.

###  5.2 Observational and systematic effects

The observational and systematical issues facing measurements of the CGF are
generally the same as for the PDF. Rare events can be excluded from the CGF by
choosing a limited range of $\lambda$ values, as for the PDF, in which we
generally make cuts in the high and low $\kappa$ tails. The cuts in $\lambda$
chosen can be roughly related to $\kappa$ cuts in the corresponding PDF using
the saddle-point approximation, as explained in Section 3.1, just as a rough
link can be made between an angular scale cut in a correlation function and a
cut in harmonic space $\ell$ in the corresponding power spectrum. As for the
PDF, we have seen that the $\chi^{2}$ test helps one sample the CGF in such a
way as to ensure a Gaussian likelihood, and cutting the PDF and CGF based on
the $\chi^{2}$ test leads to consistent cosmological constraints. The addition
of statistically-independent statistics like shape noise to the CGF is
straightforward — for example, shape noise can be directly added to the
cumulants. Masks in the field of view present only a limited complication to
counts-in-cells statistics of direct observables such as the shear field, as
shown by Friedrich et al. (2018), because masks do not meaningfully affect
local quantity measurements. Masks do, however, complicate the reconstruction
of the convergence field, but it has been at least empirically shown on
numerical simulations that the effect of this on the scaled cumulant
generating function (equation (5)) is minimal (Gatti et al., 2020; Gatti et
al., 2021), such that mask effects can be captured through their impact on the
variance. This suggests a possible advantage to making use of the CGF as the
basis for analysis of the convergence field.

Finally, in the hypothetical absence of a theoretical model, one possible
disadvantage of the CGF compared to the PDF would be that the CGF likelihood
tends to be less Gaussian than that of the PDF in small area maps (e.g. $\sim$
25 square degrees). This means a large set of large-area simulated maps would
be required to calculate the covariance. In addition, the fact that the
elements of the CGF covariance matrix are very correlated can make it more
prone to numerical issues in its estimation and inversion, as discussed in
Appendix B. However, note that we also demonstrate in Appendix B that in
practice, fast lognormal models can also be used to estimate the covariance
matrix of the CGF.

##  6 Conclusions

The value of one-point statistics as a complement to two-point statistics has
been clearly documented (e.g. Friedrich et al., 2018; Uhlemann et al., 2020b;
Boyle et al., 2021). Besides carrying additional cosmological information,
one-point statistics have a number of valuable properties: they are easily
measured and can be theoretically modelled from first principles, it is
straightforward to remove the influence of very non-linear or rare events, and
it is easy to only consider the highest signal-to-noise part of the signal. Up
until now, works aiming to exploit the full hierarchy of one-point cumulants
of cosmological fields have focused on the probability distribution function
(PDF).

In this work, we have used the weak lensing convergence as an example to
demonstrate for the first time how the CGF can be used as an alternative
cosmological observable as a probe of one-point statistics. We have shown that
the likelihood of the CGF restricted to the negative axis is Gaussian and that
a sampling of this section of the CGF can be used to extract the same amount
of cosmological information as is contained in the bulk of the corresponding
PDF. We have performed simple Fisher matrix forecasts to demonstrate this
agreement, leveraging theoretical predictions for how the CGF depends on
cosmology.

The key figures that summarise our results are as follows. In Figure 1, we
showed that the range of the CGF we sample for our Fisher matrix forecasts is
very insensitive to rare events. In Figure 9, we showed that the data vector
we consider is Gaussian distributed and therefore suitable for Fisher matrix
forecasts. In Figures 13 and 14, we showed that equivalent results can be
obtained from Fisher forecasts using either the PDF or CGF, including for
multiple parameters and combinations of smoothing scales.

The aim of this work has been to provide proof of concept for the exploitation
of the CGF as a cosmological probe. The most logical next step would be to
approach more complex scenarios in which we expect the CGF to be particularly
useful, including tomographic analyses and studies of cross-correlated lensing
and galaxy clustering fields. Extending our formalism to the galaxy density
field would require a treatment of galaxy bias and shot-noise. At the level of
the 1-point PDF these have been incorporated into the LDT framework by
Friedrich et al. (2021b) and their results can be translated to the CGF level.
Another natural extension would be the application of our theoretical model of
the CGF to real data — the efficiency with which the CGF can be modelled using
LDT makes it particularly appealing for MCMC analyses. In this case, one would
need to ensure the survey-specific accuracy of the theoretical model to avoid
a survey-dependent bias in the estimation of cosmological parameters and their
confidence intervals. We emphasise that the work presented here is not
intended to represent a realistic CGF-based analysis that can be applied to
survey data. A primary purpose for validating the feasibility of CGF-based
analyses is to allow for more efficient evaluations of a ‘nulled’ 1-point
statistic, the result of applying the BNT transform (Bernardeau et al., 2014)
to a combination of tomographic measurements with appropriate weights to
cancel out small-scale contributions. Nulling has been shown to dramatically
improve the accuracy of the theoretical PDF from LDT Barthelemy et al. (2020)
and is the foundation for using CGFs as a way to reduce the complexity of the
numerical computation of the theory data vector in a tomographic setting
(Barthelemy et al., 2022). In practical applications, we envision the use of
the CGF as a probe of cosmological parameters in conjunction with 2-point
statistics and 3$\times$2 point analyses.

## Acknowledgements

ABo and SC’s work is supported by the SPHERES grant ANR-18-CE31-0009 of the
French Agence Nationale de la Recherche, ABa’s by the ORIGINS excellence
cluster and SC’s by Fondation MERAC. CU is supported by the STFC Astronomy
Theory Consolidated Grant ST/W001020/1 from UK Research & Innovation. This
work has made use of the Infinity Cluster hosted by Institut d’Astrophysique
de Paris. We thank Stéphane Rouberol for running smoothly this cluster for us.
We thank Francis Bernardeau for helpful discussions and comments. ABo thanks
L. Boyle and O. Boyle for fruitful discussions.

## Data Availability

There is no new data associated with this article.

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##  Appendix A Validity and bias of the CGF estimator

The literature from fields making use of the cumulant generating function (see
for example Knight et al., 1997; Ghosh & Beran, 2006) does not to our
knowledge make use of a more refined estimator for the cumulant generating
function than the one we use throughout this paper.888There exist, however,
unbiased estimators for the individual cumulants called
https://mathworld.wolfram.com/k-Statistic.htmlk-statistics that may be used to
correct the CGF to leading order. We believe the following shows this is not
necessary for current and next generation galaxy surveys. However, though the
estimator for the moment generating function $\langle
e^{\lambda\kappa}\rangle$ is unbiased, taking the natural logarithm of this
estimator does not lead to an unbiased estimator of the log of the moment
generating function (the CGF). The goal of this section is thus to more
quantitatively estimate the (leading order) bias of the CGF estimator and show
that it is negligible for our purposes. We denote $\hat{\phi}$ and $\hat{M}$
our estimators for respectively the CGF and MGF. To leading order, the CGF
bias can be directly related to the variance of $\phi(\lambda)$ as:

$\displaystyle\hat{\phi}$ $\displaystyle=\ln(\hat{M})$ (24)
$\displaystyle\approx\ln(M+\hat{\delta M})$ (25) $\displaystyle\approx\ln
M+\ln\left(1+\frac{\hat{\delta M}}{M}\right)$ (26)
$\displaystyle\approx\phi+\frac{\hat{\delta
M}}{M}-\frac{1}{2}\left(\frac{\hat{\delta M}}{M}\right)^{2}.$ (27)

And since the estimator for the MGF is unbiased, $\langle\hat{\delta
M}\rangle=0$, and

$\displaystyle\Rightarrow\langle\hat{\phi}-\phi\rangle$
$\displaystyle=-\frac{1}{2}\left\langle\left(\frac{\hat{\delta
M}}{M}\right)^{2}\right\rangle$ (28)
$\displaystyle=-\frac{1}{2}\left\langle\left(\hat{\delta\phi}\right)^{2}\right\rangle$
(29) $\displaystyle=-\frac{1}{2}\mathrm{Var}(\hat{\phi}).$ (30)

The bias thus highly depends on the survey size999Note especially that the
bias relative to the standard deviation
$\frac{\langle\hat{\phi}-\phi\rangle}{\mathrm{Std}(\hat{\phi})}\approx-\frac{1}{2}\mathrm{Std}(\hat{\phi})\propto\sqrt{\frac{1}{A_{\mathrm{survey}}}}$.
but is easily given by the diagonal of the covariance matrix.

To confirm our derivation, we also measure from the Takahashi simulation the
jackknife (leading order) bias of the CGF estimator. We thus measure the
unbiased estimator of the MGF from the $n=$108 Takahashi convergence maps used
throughout this work, smoothed on a scale of $10$ arcmin. Our estimator of the
CGF is then given by

$\hat{\phi}(\lambda)=\log\left(\frac{1}{n}\sum_{i=1}^{i=n}\hat{M}_{i}(\lambda)\right).$
(31)

We also define $n$ different estimators for the CGF, each given by

$\hat{\phi}_{j}(\lambda)=\log\left(\frac{1}{(n-1)}\sum_{i=1;i\neq
j}^{i=n}\hat{M}_{i}(\lambda)\right),$ (32)

and the jackknife estimator, which is given by

$\hat{\phi}_{\rm
jack}(\lambda)=\frac{1}{n}\sum_{j=1}^{j=n}\hat{\phi}_{j}(\lambda).$ (33)

From the jackknife estimator, we can obtain the exact leading order bias
($\mathcal{O}\frac{1}{n}$), which is given by

${\rm bias}_{\phi}=\langle\hat{\phi}-\phi\rangle=n\left((n-1)(\hat{\phi}_{\rm
jack}-\hat{\phi})\right),$ (34)

where the overall $n$ factor comes from the fact that the bias is estimated
for a survey size of $n$ independent full-skies and we want the bias for only
one.

Thus using either equation (30) or (34), we can correct our estimator of the
CGF and determine the magnitude of the effect on the overall measurement. In
Figure 15, we first compare (green line and points) the two expressions we
obtained for the bias and see that they are virtually identical. We also plot
the two resulting estimators of the CGF — biased and unbiased. We see that the
bias for a full-sky study is totally negligible, and could easily be corrected
for smaller area surveys.

Figure 15: The unbiased (red) and biased (blue) CGF estimators (see the left
y-axis). The bias itself is shown in green (see the right y-axis). We see that
the bias is totally negligible in the range of $\lambda$ values we consider in
our analysis. The green points show half the variance of $\phi(\lambda)$, i.e.
the diagonal in Figure 4 divided by a factor of two. As demonstrated in the
text, this matches the bias very well.

##  Appendix B COVARIANCE MATRIX CONDITIONING

Figure 16: A comparison between the mean CGF from FLASK (from 100 shifted
lognormal convergence maps) and the theoretical CGF for the same cosmology
from LDT.

Figure 17: (Upper panel) The correlation matrices of the CGF from the
Takahashi simulations (top right) and the FLASK code (bottom left). Both are
extracted from 108 maps. (Lower panel) The diagonals of the corresponding
covariance matrices. There are clear differences between the two covariance
matrices for $\lambda>0$ (including anti-correlations between points in the
FLASK case), but the agreement for $\lambda<0$ is reasonably good.

In previous work (Boyle et al., 2021), we made use of the FLASK code (Xavier
et al., 2016) to rapidly generate large numbers of lognormal convergence maps
to rapidly produce covariance matrices for the PDF. The CGF is undefined for
positive $\lambda$ in a strictly lognormal field, but the smoothing we impose
on the FLASK maps, along with the original finite resolution and sky area,
cause a deviation from a perfect lognormal distribution. In any case, we have
conservatively used only the negative-$\lambda$ CGF throughout most of this
paper.

In Figure 16, we compare the mean CGF from 100 FLASK realisations to the
theoretical prediction from LDT. We see reasonably good agreement, especially
considering that the FLASK realisations underpredict the kurtosis by
construction. The lognormal fields generated by FLASK are defined by the input
angular power spectrum (which we generate using the Boltzmann code CLASS, Blas
et al., 2011) and a shift parameter that is a function of the desired skewness
(measured directly from the smoothed Takahashi simulation maps). The shift
parameter is given by

$\displaystyle\lambda$
$\displaystyle=\frac{\sigma}{\tilde{\mu}_{3}}\left(1+y(\tilde{\mu}_{3})^{-1}+y(\tilde{\mu}_{3})\right)-\langle\kappa\rangle,$
(35) $\displaystyle y(\tilde{\mu}_{3})$
$\displaystyle=\sqrt[3]{\frac{2+\tilde{\mu}_{3}^{2}+\tilde{\mu}_{3}\sqrt{4+\tilde{\mu}_{3}^{2}}}{2}}\,,$

where $\langle\kappa\rangle$ is the mean (zero for the $\kappa$ field),
$\sigma$ is the variance and $\tilde{\mu}_{3}$ is the skewness. We calculate
the shift parameter for the Takahashi cosmology at $z_{s}=2$ to be 0.085.

In Figure 17, we show correlation matrices generated from the sets of
Takahashi and FLASK convergence maps (108 maps each). We see that the
agreement in the negative-$\lambda$ region seems reasonably strong by eye, but
there are clear differences for positive $\lambda$. We also see anti-
correlations between negative and positive $\lambda$ values in the FLASK case,
while all of the points in the Takahashi case are positively correlated. The
lower panel of Figure 17 shows the diagonals of the corresponding covariance
matrices. We see reasonable agreement for negative $\lambda$ (taking into
account that there are some differences in the mean CGFs as seen in Figure
16).

In total, we generated 500 FLASK convergence maps with our specifications. One
might expect that a covariance matrix generated from this significantly larger
number of maps would allow for accurate Fisher matrix forecasts with a finer
sampling of the CGF than was possible with the original 108 Takahashi maps.
However, we find that the covariance matrices for the CGF are generally ill-
conditioned, with very large condition numbers once the sampling of the CGF
reaches a certain density. This is not an issue for the PDF but is somewhat
predictable for the CGF, because the points in the CGF are very strongly
correlated with each other. The condition number of the CGF covariance
matrices is found to decrease only minimally even when significantly
increasing the number of maps used. Naturally, the magnitude of this problem
increases considerably as one considers more and more points from the CGF, as
points that are closer together in $\lambda$ are even more strongly
correlated.

There are a number of approaches one can take to improve the conditioning of a
covariance matrix, such as restructuring the data vector into a form that
decreases the correlation between its individual components. However, the
simplest approach seems to be to define a small positive constant $\alpha$
multiplied by the identity matrix that we add to the estimated covariance
matrix. This has the effect of ensuring the eigenvalues are sufficiently
different from zero for a stable numerical inversion. This process, called
‘ridge regression’, has been exploited in a few other works for other
cosmological observables, e.g. Matthews & Newman (2012). The constant $\alpha$
should be chosen from the numerical error we expect on the small eigenvalues
of our covariance matrix so that the shift we impose just ensures they are no
longer compatible with zero. To estimate it, we compute the standard deviation
$\hat{\sigma}_{\rm jack}$ of the jacknife estimator $\hat{J}_{\rm ack}$ of the
eigenvalues of the covariance matrix and determine the minimal $\alpha$ value
for which $\hat{J}_{\rm ack}-\hat{\sigma}_{\rm jack}$ is always positive.
Using 108 jackknife replicates based on the 108 Takahashi realisations, we
find that a good value is $\alpha\sim 5\cdot 10^{-16}$. Finally, we note that
since this method increases the smallest eigenvalues of the covariance matrix,
it can only decrease the final constraints on the cosmological parameters even
if the value of $\alpha$ is not ideal. This makes this a conservative approach
for cosmological analysis.

Figure 18: Fisher forecasts for the PDF (grey) and CGF (green) using the
covariance matrix constructed from 108 Takahashi convergence maps (solid) and
500 FLASK lognormal convergence maps (dotted). Eleven points are sampled from
the CGF between $\lambda_{\rm min}$ of -205 and zero. We take steps to reduce
the conditioning number of the CGF covariance, which allows it to be stably
inverted (see main text), even for a fine sampling of points of the CGF. We
see that the Fisher forecasts are all mutually consistent — the PDF and CGF
provide approximately the same constraints, as do the Takahashi and FLASK
covariances.

In Fig. 18, we show Fisher forecasts for the PDF (grey) and CGF (green) using
both the Takahashi (solid, 108 maps) and FLASK (dotted, 500 maps) covariance
matrices. The approach described above is used to improve the conditioning of
the CGF covariance matrices, allowing us to produce consistent Fisher
forecasts even when sampling a large number of points (11) from the CGF. We
see excellent agreement between all four cases. This also confirms that the
FLASK code can be used to rapidly generate covariance matrices for the weak
lensing convergence CGF, which is helpful because such covariances can be
combined with theoretical derivatives (e.g. from LDT) to produce cosmological
forecasts without the need for any input from expensive N-body simulations.