Issue 
A&A
Volume 691, November 2024



Article Number  A80  
Number of page(s)  11  
Section  Cosmology (including clusters of galaxies)  
DOI  https://doi.org/10.1051/00046361/202450061  
Published online  31 October 2024 
Theoretical wavelet ℓ_{1}norm from onepoint probability density function prediction
^{1}
Université ParisSaclay, Université Paris Cité, CEA, CNRS, AIM,
91191
GifsurYvette,
France
^{2}
UniversitätsSternwarte, Fakultät für Physik, LudwigMaximiliansUniversität München,
Scheinerstraße 1,
81679
München,
Germany
^{3}
Institutes of Computer Science and Astrophysics, Foundation for Research and Technology Hellas (FORTH),
Heraklion,
Crete,
Greece
^{★} Corresponding author; vilasini.tinnanerisreekanth@cea.fr
Received:
21
March
2024
Accepted:
19
August
2024
Context. Weak gravitational lensing, which results from the bending of light by matter along the line of sight, is a potent tool for exploring largescale structures, particularly in quantifying nonGaussianities. It is a pivotal objective for upcoming surveys. In the realm of current and forthcoming fullsky weaklensing surveys, convergence maps, which represent a lineofsight integration of the matter density field up to the source redshift, facilitate fieldlevel inference. This provides an advantageous avenue for cosmological exploration. Traditional twopoint statistics fall short of capturing nonGaussianities, necessitating the use of higherorder statistics to extract this crucial information. Among the various available higherorder statistics, the wavelet ℓ_{1} norm has proven its efficiency in inferring cosmology. However, the lack of a robust theoretical framework mandates reliance on simulations, which demand substantial resources and time.
Aims. Our novel approach introduces a theoretical prediction of the wavelet ℓ_{1}norm for weaklensing convergence maps that is grounded in the principles of largedeviation theory. This method builds upon recent work and offers a theoretical prescription for an aperture mass onepoint probability density function.
Methods. We present for the first time a theoretical prediction of the wavelet ℓ_{1}norm for convergence maps that is derived from the theoretical prediction of their onepoint probability distribution. Additionally, we explored the cosmological dependence of this prediction and validated the results on simulations.
Results. A comparison of our predicted wavelet ℓ_{1} norm with simulations demonstrates a high level of accuracy in the weakly nonlinear regime. Moreover, we show its ability to capture cosmological dependence. This paves the way for a more robust and efficient parameterinference process.
Key words: gravitational lensing: weak / cosmology: miscellaneous / cosmology: theory / dark matter / largescale structure of Universe
© The Authors 2024
Open Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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1 Introduction
Our current comprehension of the genesis of the largescale structure (Peebles 1980) posits that it emerged through gravitational instability driven by primordial fluctuations in matter density. Realistic models detailing the structure formation prescribe an initial spectrum of perturbations that reflected the primordial spectrum, characterised by small fluctuations at large scales. In a LambdaCold Dark Matter (LCDM) Universe, the variance of the density fluctuations, denoted by σ(R)^{2}, is inversely proportional to (some power of) the scales so that at small scales, the variance is large, and nonlinear effects become important.
Hence, there are two limiting regimes: the linear regime, which is characterised by σ^{2}(R) ≪ 1, and the nonlinear regime, which is given by σ^{2}(R) ≫ 1. In particular, when the primordial density fluctuations are Gaussian, then they remain Gaussian in the subsequent linear regime, and Fourier modes evolve independently. However, the coupling between different Fourier modes becomes significant and plays a pivotal role in modifying the statistical properties, which manifest as higherorder connected correlations in the nonlinear regime (Bernardeau et al. 2002).
One of the powerful methods for probing these nonlinearities in the largescale structure (hereafter LSS) is to study the distortions in distant images of galaxies. Distortions are caused by gravitational lensing, which is the phenomenon that bends the photon paths through massive objects, such as galaxies or galaxy clusters. Weak gravitational lensing denotes that except for rare cases of strong lensing, these distortions are often subtle and require a statistical analysis over a vast number of galaxies for a signal to be detected. The analysis of these distortions provides a unique opportunity to investigate the distribution of matter in the Universe and infer cosmological parameters. For a more comprehensive review of weak lensing, we refer to Mandelbaum (2018); Kilbinger (2018), and Kilbinger (2015).
In this era of precision cosmology, with past, present, and future surveys such as the CanadaFranceHawaii Telescope Lensing Survey (CFHTLenS) (Heymans et al. 2012), the Hyper SuprimeCam (HSC) (Mandelbaum & Collaboration 2017), Euclid (Laureijs et al. 2011), Legacy Survey of Space and Time (LSST) (Ivezić et al. 2019), we now have access to the nonGaussian part of cosmological signals, which are induced by the nonlinear evolution of the structures on small scales. All of these experiments consider weak gravitational lensing to be a key probe, jointly with galaxy clustering, for exploring the currently unanswered questions in cosmology, such as the neutrino mass sum (Lesgourgues & Pastor 2012; Li et al. 2019), the nature of dark energy, and dark matter (Huterer 2010), and it offers substantial constraints on standard cosmological parameters such as the mean matter density and the amplitude of matter fluctuations (Troxel & Ishak 2015).
The conventional approach to inferring the cosmology from the data involves the computation of the twopoint statistics, which has been employed with remarkable success (Munshi et al. 2008; Kilbinger 2015; Bartelmann & Maturi 2017; Hildebrandt et al. 2017; Euclid Collaboration 2024; Loureiro et al. 2023; Ingoglia et al. 2022). However, it is not sufficient when we wish to probe nonGaussianities (Weinberg et al. 2013). This limitation arises from the construction of the power spectrum, which considers information solely from the norm of wave vectors while neglecting phase information, thereby discarding a significant aspect of structural details. Consequently, this approach must be complemented with an alternative higherorder statistic than can effectively capture the nonGaussian features embedded in the structure. Higherorder statistics such as peak counts (Kruse & Schneider 1999; Liu et al. 2015a,b; Lin & Kilbinger 2015; Peel et al. 2017; Ajani et al. 2020), higher moments (Petri 2016; Peel et al. 2018; Gatti et al. 2020), the Minkowski functional (Kratochvil et al. 2012; Parroni et al. 2020), threepoint statistics (Takada & Jain 2004; Semboloni et al. 2011; Rizzato et al. 2019), and wavelet and scattering transform (Ajani et al. 2021; Cheng & Ménard 2021) can better probe the nonGaussian structure of the Universe and provide additional constraints on the cosmological parameters. Another example was given in Ajani et al. (2021), who employed the starlet ℓ_{1}norm, which has the advantage of being easy to measure and was claimed to encompass even more cosmological information than the power spectrum or peak and void counts in the setting considered. The ℓ_{1}norm of a starlet, which is a type of wavelet that uses B3spline, offers an efficient multiscale computation of all map pixels including the under and overdensity distribution, which partially probe similar information to peak and void counts.
The probability distribution function (hereafter PDF) of the weaklensing convergence map (κ) serves as another valuable repository of cosmological information that has drawn much interest in recent years, with many theoretical and numerical works, including (Bernardeau & Valageas 2000; Liu & Madhavacheril 2019; Barthelemy et al. 2021). The convergence field κ represents the weighted projection of matter density fluctuations along the line of sight, and its higherorder correlations offer a promising avenue for addressing challenges inherent in standard weaklensing analyses, particularly those related to degeneracies in the twopoint correlation function (2PCF).
The onepoint κPDF statistic, obtained by quantifying smoothed κ field values within predefined apertures or cells, presents a practical advantage as its measure is rather simple. This simplicity contrasts with other nonGaussian probes used for studying the weaklensing convergence field, such as the bispectrum (which involves counting triangular configurations) or Minkowski functionals (a topological measurement). Previous research has successfully devised a precise theoretical model, grounded in large deviation theory (LDT), for both the cumulant generating function and the PDF of the lensing κ field as well as the aperture mass (Barthelemy et al. 2021). Notably, this onepoint PDF can be directly linked to the wavelet coefficients at different scales (Ajani et al. 2021).
We emphasise that similar to the convergence PDF, all of the abovementioned nonGaussian statistics that probe deviations from Gaussian behaviour are also commonly computed using convergence maps. However, these convergence maps are not observed directly and are reconstructed from reduced shear maps. Since their first detection two decades ago (Bacon et al. 2000; Kaiser et al. 2000; Waerbeke et al. 2000), shear maps have been a powerful cosmological probe. However, due to its spin2 nature, it is rather difficult to obtain higherorder summary statistics from shear. While convergence maps in principle contain the same information as shear maps (Schneider et al. 2002; Shi et al. 2011), the compression of the lensing signal is greater in convergence maps than in the shear field, resulting in an easier extraction and reduced computational costs, with the caveat that the reconstruction of the convergence maps is not perfectly solved and is a very complex illdefined inverse problem. Convergence maps emerge as a novel tool, potentially offering additional constraints that complement those derived from the shear field. However, accessing this information is not straightforward and implies the use of a reconstruction method (or mass inversion). In particular, we note that due to the nonGaussian nature of the weaklensing signal at small scales, it may not be optimal to employ massinversion methods with smoothing or denoising for regularisation (Starck et al. 2021; Jeffrey et al. 2020).
The objective of this paper is to present for the first time the prediction^{1} of the wavelet ℓ_{1}norm derived from theoretical predictions of the PDF of convergence maps. The paper is organised as follows: in Sect. 2.1, we begin by revisiting weaklensing convergence, followed by an introduction to the wavelets and wavelet ℓ_{1}norm in Sect. 2. This is followed by the introduction of the LDT formalism for the κPDF and extending this to the wavelet coefficients in Sect. 3. Section 3.3 derives this wavelet ℓ_{1}norm for the wavelet coefficients from theory. Subsequently, we present and discuss the results in Sect. 4, and we conclude by summarising our findings in Sect. 5.
2 Wavelet ℓ_{1}norm: Definition and measurements
We first introduce in this section the expression for the convergence maps and then define the wavelets and the wavelet ℓ_{1}norm more generally to relate it to the PDF.
2.1 Convergence maps
We start with the expression for convergence, which is given by the projection of density along the comoving coordinates, weighted by a lensing kernel involving the comoving distances. It is given by (Mellier 1999) $$\kappa (\mathit{\vartheta})={{{\displaystyle \int}}^{\text{}}}_{0}^{{\chi}_{s}}\text{d}\chi \omega \left(\chi ,{\chi}_{s}\right)\delta (\chi ,\mathcal{D}\mathit{\vartheta}),$$(1)
where χ is the comoving radial distance (χ_{s} the radial distance of the source) that depends on the cosmological model, and Ɗ is the comoving angular distance, given by $$\mathcal{D}(\chi )\equiv \{\begin{array}{c}\frac{\mathrm{sin}(\sqrt{K}\chi )}{\sqrt{K}}\text{for}K0\\ \chi \text{\hspace{1em}for}K=0,\\ \frac{\mathrm{sinh}(\sqrt{K}\chi )}{\sqrt{K}}\text{for}K0\end{array}$$(2)
with K the constant space curvature. The lensing kernel ω is defined as $$\omega \left(\chi ,{\chi}_{s}\right)=\frac{3{\text{\Omega}}_{m}{H}_{0}^{2}}{2{c}^{2}}\frac{\mathcal{D}(\chi )\mathcal{D}\left({\chi}_{s}\chi \right)}{\mathcal{D}\left({\chi}_{s}\right)}(1+z(\chi )),$$(3)
where c is the speed of light, Ω_{m} is the matter density parameter, and H_{0} is the value of the hubble constant at redshift z = 0. The shear field γ(θ), which can be directly interpreted from the observation, is related to the weaklensing convergence κ through mass inversion (Starck et al. 2006; Martinet et al. 2015). The convergence mass maps are in principle only constructed up to a mass sheet degeneracy. However, constructing a mass map by convolving the κ map with a radically symmetrical compensated filter makes the statistics insensitive to mass sheet degeneracy.
2.2 Wavelets
A wavelet transform enables us to decompose an image κ into different maps called wavelet coefficients ${w}_{{\theta}_{j}}$, that is, ${\mathcal{W}}_{K}=\left\{{w}_{{\theta}_{1}},\dots ,{w}_{{\theta}_{j}},\dots {w}_{{\theta}_{J}}\right\}$ where θ_{j} is the angular scale, and θ_{J} is the largest scale used in the analysis. The wavelet coefficients are the values of the wavelet scales at the pixel coordinates (x, y). Considering a wavelet function Υ, and noting $\text{\hspace{0.05em}}{\text{\Upsilon}}_{{\theta}_{j},x,y}(m,n)=\text{\Upsilon}\left(\frac{mx}{{\theta}_{j}},\frac{ny}{{\theta}_{j}}\right)$ the dilated wavelet function at scale θ_{j} and at spatial location (x, y), the wavelet coefficients ${w}_{{\theta}_{j}}$ are computed as the inner products $$\begin{array}{l}{w}_{{\theta}_{j}}(x,y)=<\kappa ,{\Upsilon}_{{\theta}_{j},x,y}>\\ \text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}={\displaystyle \sum _{m}{\displaystyle \sum _{n}\kappa}}(m,n)\Upsilon \left(\frac{mx}{{\theta}_{j}},\frac{ny}{{\theta}_{j}}\right).\end{array}$$(4)
The wavelet function Υ is often chosen to be derived from the difference of two resolutions, for example as Υ(x, y) = 4ξ(2x, 2y) − ξ(x, y) as used in the starlet wavelet transform (Starck et al. 2015), where the function ξ is called the scaling function, typically a lowpass filter. In our case, we write the wavelet coefficients as $${w}_{{\theta}_{j}}(x,y)=<\kappa ,{\xi}_{{\theta}_{j+1},x,y}><\kappa ,{\xi}_{{\theta}_{j},x,y}>,$$(5)
with ${\xi}_{{\theta}_{j},x,y}(m,n)=\xi \left(\frac{mx}{{\theta}_{j}},\frac{ny}{{\theta}_{j}}\right)$. Employing dyadic scales, i.e. θ_{j} = 2^{j−1}θ_{1}, enables us to have very fast algorithms through the use of a filter bank (see Starck et al. 2015 for more details).
A wavelet acts as a mathematical function localised in both the spatial and the Fourier domains, and it is thus suitable for analysing the lensing signal structures at various scales. An important advantage of the wavelet analysis compared to a standard multiresolution analysis through the use of a set of Gaussian functions of different sizes is that it decorrelates the information. As an example, Ajani et al. (2023) have shown that the covariance matrix derived from a wavelet peakcount analysis was almost diagonal, and neglecting the offdiagonal terms has little impact on the cosmological parameter estimation. This would not be the case with a multiscale Gaussian analysis. Other advantages are that very fast algorithms exist that allow us to compute all scales with a low complexity, and also that wavelet functions are exactly equivalent to traditional weaklensing aperture mass functions, which have been used for decades (Leonard et al. 2012) (the aperture mass is just the convergence or shear within a compensated filter, which is one property of wavelets).
Several statistics have been derived from wavelet coefficients in the past, such as cumulants (up to order 6) (Fageot et al. 2014) or peak counts (Ajani et al. 2020). It has recently been shown (Ajani et al. 2021) that the ℓ_{1}norm of the wavelet scales is very efficient in constraining cosmological parameters. Ajani et al. (2023) used a toy model that incorporated a mock source catalogue for weak lensing and a mock lens catalogue for galaxy clustering sourced from the cosmoSLICS (HarnoisDéraps et al. 2019) simulations to mimic the KiDS1000 data survey properties described in Giblin et al. (2018) and Hildebrandt et al. (2021). The study examined forecasts of the matter density parameter Ω_{m}, the matter fluctuation amplitude σ_{8}, the dark energy equation of state w_{0}, and the reduced Hubble constant h. It was observed that the wavelet ℓ_{1}norm performed better than peaks, multiscale peaks, or a combination thereof (Ajani et al. 2023). Therefore, we propose to build a theory for the wavelet ℓ_{1}norm that can be used to constrain the cosmological parameters of upcoming surveys.
2.3 Wavelet ℓ_{1}norm
To measure the wavelet ℓ_{1}norm from a κ map, we first obtained the wavelet scale by convolving the map with the wavelet function, as given in Eq. (5).
We then computed a histogram of the values of the wavelet scale at each pixel coordinate, ${w}_{{\theta}_{j}}(x,y)$, using a specific binning with bin edges denoted {B_{i}}_{1≤i≤N}.
We then obtained the normalised^{2} wavelet ℓ_{1}norm by extracting the ℓ_{1}norm of the histogram such that $${\ell}_{1}^{{\theta}_{j},i}={\displaystyle \sum _{k=1}^{\#coef\left({S}_{{\theta}_{j},i}\right)}\left{S}_{{\theta}_{j},i}[k]\right}/N/\text{\Delta}B,$$(6)
where the set of coefficients at scale θ_{j} and amplitude bin i, ${S}_{{\theta}_{j},i}=\left\{{w}_{{\theta}_{j}}/{w}_{{\theta}_{j}}(x,y)\in \left[{B}_{i},{B}_{i+1}\right]\right\}$, depicts the wavelet coefficients ${w}_{{\theta}_{j}}$ with an amplitude within the bin [B_{i}, B_{i+1}], the pixel indices are given by (x, y), N is the total number of pixels, and ∆B is the uniform bin width. In other words, for each bin, the number of pixels k that fall in the bin was collected, and the absolute values of these pixels were summed to obtain the ℓ_{1}norm at this bin i.
As we already highlighted before, this definition enabled us to gather the data represented by the absolute values of every pixel of the map, rather than solely describing it through the identification of local minima or maxima. This has the advantage of being a multiscale approach. In addition, the robustness of ℓ_{1}norm statistics has also been well established for decades in the statistical literature (Huber 1987; Giné et al. 2003).
The ℓ_{1}norm of the wavelet coefficients can be directly related to the PDF of the wavelet coefficients via the following relation: $${\ell}_{1}^{{\theta}_{j},i}=P{\left({w}_{j}\right)}^{i}\times \left{B}^{i}\right,$$(7)
which states that the ℓ_{1}norm for a given scale j and at a given bin i (B_{i}) can be related to its PDF by multiplying the PDF value of the bin i by the absolute value of the bin i.
Fig. 1 Probability density function and ℓ_{1}norm for a Gaussian distribution (blue) and the nonGaussian distribution (orange). On the left, we present the PDF for the two distributions, and on the right, we display the derived ℓ_{1}norm of these PDFs. The peak heights are the same for a Gaussian distribution, but this does not hold for a nonGaussian PDF. 
2.4 Gaussian ℓ_{1}norm
We studied the anticipated wavelet ℓ_{1}norm expected from a Gaussian distribution. The generic shape of a wavelet ℓ_{1}norm obtained from a Gaussian PDF, $$P(x)=\frac{1}{\sigma \sqrt{2\pi}}\mathrm{exp}\left(\frac{1}{2}{\left(\frac{x\mu}{\sigma}\right)}^{2}\right),$$(8)
with a mean µ = 0 and σ = 0.1 using the relation shown in Eq. (7) is demonstrated in Fig. 1.
The heights of the two peaks obtained for the wavelet ℓ_{1}norm for the Gaussian (solid blue line) distribution are equal, as expected. Because the Universe evolves, however, nonlinearities develop, which means that the convergence field κ that we are interested in can no longer be modelled using a Gaussian model. These nonlinearities would lead to an asymmetry that would also be evident from the peaks of the wavelet ℓ_{1}norm, which will no longer correspond. This is demonstrated in Fig. 1 for a testcase scenario, where we applied a nonlinear exponential transformation to the Gaussian field to obtain a nonGaussian field (solid orange line) and obtained the PDF and the ℓ_{1}norm. We emphasise that the usual way to extract this nonlinear information is either by relying on heavy 𝒩body simulations or by using an approximation, as was done in Tessore et al. (2023). With LDT, we now have a theoretically motivated way to obtain the PDF in the mildly nonlinear regime, with which we show that we can also obtain the wavelet ℓ_{1}norm in the mildly nonlinear regime.
3 Formalism of the large deviation theory
We applied the LDT to the convergence field. LDT, as explored in earlier works (Varadhan 1984), examines the rate at which the probabilities of specific events exponentially decrease as a key parameter of the problem varies. To derive the wavelet ℓ_{1}norm based on the LDT formalism, we first need to recapitulate how the PDF is obtained from the LDT formalism.
3.1 Large deviation theory for the matter field
The application of LDT to the field of Large Scale Structure cosmology has systematically been developed in recent years and is applied in the specific context of cosmic shear observations in this work. (Bernardeau & Reimberg 2016) clarified the application of the theory to the cosmological density field, establishing its link to earlier studies focused on cumulant calculations and modelling the matter PDF through perturbation theory and spherical collapse dynamics (Valageas 2002; Bernardeau et al. 2014b). Barthelemy et al. (2020) then provided an LDTbased prediction for the tophatfiltered weaklensing convergence PDF on mildly nonlinear scales, building upon earlier work by (Bernardeau & Valageas 2000).
A more detailed derivation of the specific equations used is presented in the Appendix A. In this section, we recall the main equation with which we derive the wavelet ℓ_{1}norm prediction.
Application of LDT in cosmology involves three main steps. The first step is the derivation of the rate function, which is directly obtained from the first principles of cosmology. Using the contraction principle in the LDT formalism, we can connect the statistics of the latetime nonlinear densities to the earliertime density field when the most likely mapping between the two is known. Previous works have shown that assuming spherical collapse for this mapping provides us with a very accurate prediction for cosmic field PDF (density, velocity, convergence, etc.).
In particular, the result for convergence reads $$P(\kappa )={{{\displaystyle \int}}^{\text{}}}_{i\infty}^{+i\infty}\frac{\text{d}\lambda}{2\pi i}\mathrm{exp}\left(\lambda \kappa +{\varphi}_{\kappa ,\theta}(\lambda )\right),$$(9)
where P(κ) is the PDF of the convergence field κ, and ϕ_{κ,θ} is the cumulant generating function (CGF) of the field κ at angular scale θ. The derivation of the CGF in the above equation and the PDF from there is explained in more detail in the Appendix A.
3.2 Extension to wavelet coefficients
From the PDF, we can now compute the ℓ_{1}norm. As described in the Sect. 2, a wavelet can be written as a function of scaling functions. Although various wavelet filters exists in literature, we employed a function of concentric disks to construct the wavelet filter ${\text{\Upsilon}}_{{\theta}_{1}}$.
The scaling function ξ is then given by a spherical tophat filter and is given by $${\widehat{\xi}}_{{\theta}_{i}}=2{J}_{1}\left({\theta}_{i}l\right)/\left({\theta}_{i}l\right),$$(10)
where J_{1} is the first Bessel function of the first kind. This is applied, as shown in Eq. (5), to obtain our compensated filter. In Fig. 2 we show the compensated filter (solid black line) we used, which is given as a function of two scaling functions (solid orange and blue lines) at different scales.
We applied the LDT formalism described previously to the wavelet coefficients described in Eq. (5) and obtained the PDF of the wavelet coefficients $\left({w}_{{\theta}_{1}}\right)$ by using the scales θ_{1},θ_{2} = 2θ_{1} as $$P\left({w}_{{\theta}_{1}}\right)={{{\displaystyle \int}}^{\text{}}}_{i\infty}^{+i\infty}\frac{\text{d}\lambda}{2\pi i}\mathrm{exp}\left(\lambda {w}_{{\theta}_{1}}+{\varphi}_{{w}_{{\theta}_{1}}}(\lambda )\right).$$(11)
We refer to the Appendix A for more details about the derivation of the CGF in the above equation and the resulting PDF. We emphasise that ${w}_{{\theta}_{1}}$ is essentially the standard aperture mass defined on the convergence field using the difference of two tophat filters, as was discussed previously.
Fig. 2 Compensated filter (solid black line), derived through the difference of two tophat filters at different scales, as described in Eqs. (5) and (10). The solid blue and orange lines represent the individual spherical filters obtained at radii θ_{1} and θ_{2} = 2θ_{1}, respectively. For visualisation purposes, the compensated filter is multiplied by −1. 
3.3 Wavelet ℓ_{1}norm predicted via large deviation theory
In the previous sections, we showed that the LDT formalism can be applied to derive the PDF of the wavelet coefficients. We derive the wavelet ℓ_{1}norm from the PDF of the wavelet coefficients $P\left({w}_{{\theta}_{1}}\right)$ here. This wavelet ℓ_{1}norm given in Eq. (6), which is the sum of the absolute values of the pixels in each bin, is equivalent to multiplying the counts/PDF value by the absolute value of the bin to which it corresponds, as given in Eq. (7). Extending this to the PDF of the wavelet coefficients from the LDT formalism, we obtain the final equation for bin i, $${\ell}_{1}^{i}={P}^{i}\left({w}_{{\theta}_{1}}\right)\times \left{w}_{{\theta}_{1}}^{i}\right.$$(12)
As a first approximation, we assume that the PDF is Gaussian, which is the case when we consider large scales or early times in the standard model of cosmology. In the subsequent section, we first obtain the ℓ_{1}norm for a Gaussian case and then highlight the motivation to go beyond Gaussian models.
4 Comparing ℓ_{1}norm prediction to simulations
4.1 Measurements from simulation
To demonstrate the accuracy of the theory for realistic surveys, we compared our prediction results with fthe ullsky simulations of Takahashi et al. (2017) to avoid additional errors from smallscale patch sizes. The simulations of Takahashi et al. (2017) provide fullsky lensing maps at a fixed cosmology^{3} each for two grid resolutions: 4096^{2} and 8 1 92^{2}. The data sets include the fullsky maps at intervals of 150 h^{−1} comoving radial distance from redshifts z = 0.05–5.3. We used the simulated convergence maps that are directly provided as products of these simulations. The convergence maps were smoothed as a direct convolution of the convergence map with the wavelet filter at a given angular scale. After we obtained the filtered field, we measured the sum of the absolute values of the pixels of the smoothed κ mass map in linearly spaced bins of the range of κ to obtain the wavelet ℓ_{1}norm.
4.2 Obtaining the prediction
To obtain the prediction of the wavelet ℓ_{1}norm for each of the cosmologies used here, we first used CAMB (Lewis & Challinor 2011) to obtain the linear and nonlinear (Halofit takahashi (Takahashi et al. 2012)) matter power spectra. We also used CAMB to obtain the comoving radial distances χ(z). The density CGF (Eq. (A.9)) was calculated for redshift slices between z = 0 and the source redshift z_{s} using the full Halofit power spectrum as input. These projected CGFs were then rescaled by the measured variance ${\sigma}_{{M}_{ap},sim}^{2}$ (see the Appendix A for more details), before going through an inverse Laplace transform to obtain the final PDF. This predicted PDF was used to derive the predicted wavelet ℓ_{1}norm as given in Eq. (12).
4.3 Validating the prediction with simulation
In Figures 3 and 4, we show the predicted wavelet ℓ_{1}norm and the one measured from the simulated convergence map for different source redshifts z_{s} ≈ 1.2, 1.4, and 2.0 at θ = 20′ and for different scales θ = 15′, 18′, and 20′ at source redshift z_{s} = 1.423, respectively. In the bottom panels, we show the residuals. For comparison, we display the PDF and the residuals of a Gaussian distribution (dashdotted line) with the same mean and variance as the simulation. The Gaussian PDF was obtained using Eq. (8), and the ℓ_{1}norm was derived as explained previously in Eq. (12). In both plots, the xaxis for the residuals is scaled by the standard deviation. The shaded regions in the colours correspond to the 3σ region obtained from the error bars.
The error bars were obtained by taking the standard deviation of the values of the wavelet ℓ_{1}norm for ten different patches of the fullsky Takahashi map in each of the bins.
We first concentrated on the Gaussian curves. The Gaussian prediction is a clear mismatch, as expected, because the measurements show a clear asymmetry for the Gaussian model, which by definition is symmetric.
We then focused on the LDT prediction, which goes beyond the Gaussian regime. In this case, the agreement between the simulation and the predicted wavelet ℓ_{1}norm results is better at the percent level (which corresponds to the simulation error bars in the shaded area) within the 2σ region around zero on the xaxis. This demonstrates the predictive accuracy of the results derived from the LDT formalism. It is important to note that this 2σ specification refers to the range on the xaxis and does not represent error bars around the mean signal value.
These plots clearly show that LDT captures the asymmetry of the two peaks well and therefore providees valuable information about the nonGaussianities. To better assess this aspect, we also show in Table 1 the cumulants obtained for the three scales at source redshift z_{s} = 1.423. The cumulants shown here are the variance (σ^{2}) and reduced skewness (S_{3}), which, given that the mean convergence is zero, were obtained as $${\sigma}^{2}=E\left[{\kappa}^{2}\right]$$(13) $${S}_{3}=\frac{E\left[{\kappa}^{3}\right]}{{\sigma}^{4}},$$(14)
where $$E\left[{\kappa}^{n}\right]={\displaystyle \int P}(\kappa ){\kappa}^{n}\text{d}\kappa .$$(15)
Table 1 shows that the variance from the prediction and simulation for different scales matches perfectly, which we expect given the rescaling by the power spectrum that we performed (see Appendix A). As expected from the formalism, the reduced skewness (S_{3}) increases with decreasing smoothing scale. We also note that the ℓ_{1} norm for the positive bins is not at all equal to the negative bins, suggesting how prominent the nonGaussian features of the mass map are in this regime. Barthelemy et al. (2020) reported that the theoretical precision that can be achieved is constrained by the mixing of scales when integrating the density field along the line of sight since highly nonlinear scales are not precisely captured by the spherical collapse formalism. Consequently, weaklensing statistical measures are often approached through more phenomenological methods, such as halo models. These models are capable of incorporating baryonic physics, which becomes crucial at smaller scales, as highlighted by Mead et al. (2021). This is also evident in Figs. 3 and 4, where the level of accuracy increases at higher redshifts and higher smoothing scales. To bypass this issue, a rising approach is to use the socalled BernardeauNishimichiTaruya (BNT) transform (Bernardeau et al. 2014a), which enables more accurate theoretical predictions by narrowing the range of physical scales that contribute to the signal.
Fig. 3 Comparison of the predicted wavelet ℓ_{1} norm to the simulations at different redshifts. Top panel: predicted (solid) ℓ_{1}norm as compared to measurements in simulation (dots) for an inner radius θ_{1} = 20′ and different source redshifts z_{s} = 1.21,1.43 and 2.05, displayed with blue, orange, and green lines, respectively. The dashdotted red lines show the Gaussian prediction for reference. The vertical dotted and dotdashed lines correspond to the 1σ and 2σ regions around the mean of the ${w}_{{\theta}_{1}}$ for each of the cases considered. Bottom panel: residual of the prediction relative to the simulation (dotted lines). For reference, the dashdotted plots illustrate the residual of the ℓ_{1} norm derived from the Gaussian PDF with the same mean and variance as the simulation PDF. The shaded region indicates the 3σ region around the error bars for each redshift. The prediction agrees well with the measurements up to approximately 2σ and remains within percent levels. 
Fig. 4 Comparison of the predicted wavelet ℓ_{1} norm to the simulations at different scales. Top panel: predicted (solid) ℓ_{1} norm as compared to measurements in simulation (dots) for different inner radii θ_{1} = 15′, 18′, and 20′ and a single source redshift z_{s} = 1.43, displayed with blue, orange, and green lines, respectively. The dashdotted red lines show the Gaussian prediction for reference. The vertical dotted and dotdashed lines correspond to the 1σ and 2σ regions around the mean of the ${w}_{{\theta}_{1}}$ for each of the case considered. Bottom panel: residual of the prediction relative to the simulation (dotted lines). For reference, the dashdotted plots illustrate the residual of the ℓ_{1} norm derived from the Gaussian PDF with the same mean and variance as the simulation PDF. The shaded region indicates the 3σ region around the error bars for each inner radius. The prediction agrees well with the measurements up to approximately 2σ and remains within percent levels. 
Cumulants (standard deviation σ, skewness S_{3}) obtained from the PDF from the LDT prediction and the values obtained from the Takahashi simulation at source redshift z_{s} ≈ 2.05 for scales of 15, 18, and 20 arcminutes.
4.4 Cosmology dependence of the probability distribution function and the wavelet ℓ_{1} norm
In this section, we explore the cosmological dependence of the predicted pdf of the wavelet coefficients and ℓ_{1}norm. We emphasise that because the wavelet decomposition is analogous to using an aperture mass map filtering, we henceforth also call the PDF of the wavelet coefficients the aperture mass PDF. From the equation of the rate function given in Eq. (A.7), the cosmology dependence of the aperture mass PDF in LDT enters through the scale dependence of the nonlinear powerspectrum, the dynamics of the spherical collapse, and the lensing kernel. Moreover, the presence of massive neutrinos, if any, also affects the aperture mass as it enters the lensing kernel, as given in Eq. (3) by contributing to the total matter density budget. All of this inevitably means that the summary statistics are also cosmologydependent. In the context of applying LDT to predict the wavelet ℓ_{1} norm and PDFs, based on the equations introduced above, we assume that it should be able to efficiently capture the cosmology dependence. The cosmology dependence of the onescale onepoint convergence PDF was studied in detail by Boyle et al. (2021), who showed that LDT captures the cosmology dependence very accurately. We extend this study for the case of the PDF of the wavelet coefficients and the wavelet ℓ_{1} norm of aperture mass maps. As for the one scale case, the rate function for the multi scale also depends on the variance and the dynamics of the spherical collapse, which means that the PDF of the wavelet coefficients would also be expected to depend (similarly) on the cosmology. Additionally, because the wavelet ℓ_{1} norm is directly dependent on the PDF, we could naively expect it to depend on the cosmology as well, and encapsulate the dependence.
We are interested in obtaining the derivatives with respect to cosmological parameters. We therefore need a simulation suite that is available for different cosmologies. For this, we used the cosmological massive neutrino simulations (MassiveNus) (Liu et al. 2018) suite to determine how well the theory captures the dependence on cosmology. These simulations were released by the Columbia Lensing Group^{4}. The MassiveNuS simulations encompass 101 different cosmological models at source redshifts z_{s} = 0.5, 1.0, 1.5, 2.0, and 2.5 by varying three parameters: the neutrino mass sum M_{v}, the total matter density Ω_{m}, and the primordial power spectrum amplitude A_{s}. For each redshift, 10000 distinct map realisations are generated through random rotation and translation of the initial nbody box and are then stitched together to reconstruct pseudoindependent light cones that are unlikely to cross the same structures. Each k map contains 512^{2} pixels, covering a total angular area of 12.25 square degrees, spanning a range of ℓ ∈ [100,37 000] with a pixel resolution of 0.4 arcminutes. The wavelet ℓ_{1} norm is measured using the same process as described previously in Sect. 4.1. Additionally, the measurement of the PDF is derived from the histogram of the binned and smoothed map. To obtain the derivatives, we used a model with the parameters Ω_{m} = 0.30, A_{s} = 2.1 × 10^{−9}, and M_{v} = 0.0 eV as the fiducial model (model 1b in Liu et al. (2018)) and obtained the derivatives of the PDF for each of the parameters. The different parameters are shown in Table 2. The outcomes are illustrated in Figures 5–6. For the derivative with respect to M_{v}, we employed a model with the same Ω_{m} and A_{s} values, but with M_{v} = 0.0, eV and then computed the derivative. However, it is important to note that due to resolution and finite volume effects, the simulation power spectrum of the convergence maps exhibits a deficit in power at high ℓ, which strongly affects our results because of the scales we considered, and at low ℓ, which invites caution when choosing our largest scales. We typically tried to maintain a factor of 20 between the physical scale of the box and the largest physical scale probed by our filters.
Figures 5 and 6 clearly show that the prediction effectively captures the changes in cosmology and agrees well with the results. The error bars were obtained by taking the error of the mean of 3000 simulations, which were also used to obtain the mean of the measurements.
As demonstrated in Boyle et al. (2021) for the case of a onecell PDF, our results here demonstrate that the theory captures the effects of adding massive neutrinos to cosmology well and extend the results to the PDF of the wavelet coefficients as well.
Cosmological parameters used to obtain the derivatives of the PDF.
Fig. 5 Derivative of the PDF with respect to M_{v} (blue), Ω_{m} (orange), and A_{s} (green). The solid lines show the derivatives obtained from the prediction, and the dotted lines show the derivatives from the simulation. It is obtained at source redshift z_{s} = 2 and inner radius θ_{1} = 22.5′. The results for the simulation are obtained from the MassiveNus simulation suite by averaging the results over 10 000 simulations. 
4.5 Reproducible research
In the spirit of open research, the code for reproducing the plots in this paper is available at github. The part of this Python code that computed the aperture mass/wavelet coefficient PDF was inspired by the public Mathematica code L2DT used in Barthelemy et al. (2021).
Fig. 6 Derivative of the wavelet ℓ_{1} norm with respect to M_{v} (blue), Ω_{m} (orange), and A_{s} (green). The solid lines show the derivatives obtained from the prediction, and the dotted lines show the derivatives from the simulation. The predicted wavelet ℓ_{1} norm is derived from the predicted PDF, as was explained in previous sections. It is obtained at source redshift z_{s} = 2 and inner radius θ_{1} = 22.5′. The results for the simulation are obtained from the MassiveNus simulation suite by averaging the results over 10 000 simulations. 
5 Discussion and conclusions
We have used the LDT to build upon previous work by Barthelemy et al. (2021) and introduced a formalism to predict the ℓ_{1} norm of the wavelet coefficients of the lensing field and their cosmological dependence for any given redshift and scales as long as they are in the mildly nonlinear regime. The approach of Barthelemy et al. (2021) incorporated geometric and timeevolution aspects within the light cone by considering that the correlations of the underlying matter density field along the line of sight are negligible compared to transverse directions. This Limber approximation enabled us to treat redshift slices as statistically independent. We found that the most likely nonlinear dynamics of the matter density field filtered in concentric disks can be well approximated for small variances by the cylindrical collapse model, thus allowing us to apply LDT in this context. From there, because the wavelet ℓ_{1} norm can be directly related to the multiscale onepoint statistics of the convergence, we showedr that a theorybased prediction for the wavelet ℓ_{1} norm is tractable.
The robust validation of our predictions against the simulations is shown in Figs. 4 and 3. This highlighted the reliability and applicability of our theoretical framework. Specifically, our theory aligns with the simulations within a percentlevel accuracy in the examined range of source redshifts z_{s} ≈ 1–2 and angular scales θ_{1} ≈ 15–20. This quantitative assessment affirms the predictive precision of our model. Smaller scales or lower redshifts would clearly lead to a larger departure between simulations and theory.
Figure 6 further emphasises the cosmological dependencies and the very good performance of the theoretical prediction when compared with that of the simulation. Since the simulations validated the theoretical predictions, this suggests that our model can be applied to investigate a broad range of cosmological parameters while bypassing the need for highcost computing resources and providing a theoretical tool for cosmology inference that is robust and fast, without loss of quality. This paves the way for a more comprehensive and faster forecast analysis based on theory without having to rely on expensive numerical simulations. In addition, we noted in Figures 5 and 6 that there is numerical noise in the simulation results (fitted lines). This can lead to biases in parameter estimates when using the simulation data. A theorybased approach has the added benefit of reducing the artificial biases in the parameter inference, which was also noted in Boyle et al. (2021).
Another point to be noted is that although lower source redshifts are expected to generate more nonGaussian information, this also pushes the model deeper into the nonlinear regime. This effect can be mitigated by adjusting the smoothing scale accordingly. We also emphasise the usefulness of the wavelet coefficients of the convergence maps, instead of directly using the convergence maps or the shear maps. Convergence maps already reduce computational expense through a more compressed lensing signal when compared to the shear map. However, the use of the wavelet scales and/or coefficients provides us with a multiscale analysis method that is well suited to constraining the cosmology.
Moreover, wavelet coefficients are invariant under the mass sheet degeneracy, which renders the link to the measured shear data more straightforward.
More generally, one of the issues with the theoretical modelling of weak lensing we currently face is the mixing of (nonlinear) scales that is inherent in such quantities projected along the past light cone and makes standard perturbative approaches inaccurate even at relatively large angular scales. (Bernardeau et al. 2014a) To overcome this issue, an emerging method is the BNT transform, which allows more precise theoretical predictions by reducing the range of physical scales that contribute to the signal. This can be applied in our context and enables even more accurate predictions for analysing tomographic surveys. An investigation of the performance of this approach is left for future works, together with the impact of systematics, which would require us to devise specific simulated data, as well as additional theoretical development.
We conclude that the novel approach for obtaining the theoretical prediction of the wavelet ℓ_{1}norm summary statistic proposed here presents several advantages in the realm of cosmological parameter inference: (i) it provides a fast (about one minute) calculation, (ii) it does not rely on a heavy numerical simulation, (iii) it captures the cosmology dependence well, and (iv) the wavelet ℓ_{1} norm might lead to competitive or even tighter constraints, as demonstrated by previous works. This promising method might enable a comprehensive multiscale analysis of cosmic shear datasets in the mildly nonlinear regime.
Data availability
The code for reproducing the plots in this paper is available at https://github.com/vilasinits/LDT_2cell_l1_norm
Acknowledgements
This work was funded by the TITAN ERA Chair project (contract no. 101086741) within the Horizon Europe Framework Program of the European Commission, and the Agence Nationale de la Recherche (ANR22CE31001401 TOSCA and ANR18CE310009 SPHERES). A.B.’s work is supported by the ORIGINS excellence cluster. We thank Jia Liu and the Columbia Lensing group http://columbialensing.org) for making the MassiveNus (Liu et al. 2018) simulations available. The creation of these simulations is supported through grants NSF AST1210877, NSF AST140041, and NASA ATP80NSSC18K1093. We thank New Mexico State University (USA) and Instituto de Astrofisica de Andalucia CSIC (Spain) for hosting the Skies & Universes site for cosmological simulation products. We also thank C. Uhlemann, O. Friedrich, Lina Castiblanco, and Martin Kilbinger for insightful discussions.
Appendix A Large Deviation Theory
A.1 LDT for the matter field
The computations detailed in this section draw heavily from the formulations provided in Boyle et al. (2021) and Barthelemy et al. (2021). Here, we will succinctly restate the key equations and direct the reader to the comprehensive treatments in Boyle et al. (2021), Barthelemy et al. (2021), and Reimberg & Bernardeau (2018).
The LDT, as explored in earlier works Varadhan (1984), examines the rate at which the probabilities of specific events diminish as a key parameter of the problem undergoes variation. Widely applied in various mathematical and theoretical physics domains, this theory is particularly prominent in statistical physics, covering both equilibrium and nonequilibrium systems. For a detailed overview, readers are referred to Touchette (2009).
The Large Deviation Principle (LDP) in the domain of Large Scale Structure cosmology has been systematically developed in recent years and will be applied in the specific context of cosmic shear observations in this work. Bernardeau & Reimberg (2016) clarified the application of the theory to the cosmological density field, establishing its link to earlier studies focused on cumulant calculations and modelling the matter PDF through perturbation theory and spherical collapse dynamics (Valageas (2002); Bernardeau et al. (2014b)). Barthelemy et al. (2020) provided an LDTbased prediction for the tophatfiltered weaklensing convergence PDF on mildly nonlinear scales, building upon earlier work by Bernardeau & Valageas (2000).
The LDP applied to the matter density field relies on three key aspects:
Defining a rate function for variables in the initial field configuration. In this context, we opt for Gaussian initial conditions and establish their covariance matrix.
Describing the relationship between the initial field configuration (representing the mass profile) and the resulting mass profile, based on 2D cylindrical collapse or a suitable approximation.
Using these foundations to express observable quantities, like a map created with a specific filter, as functional expressions that depend on both the final and initial mass profile.
A LDP for matter densities at multiple scales (indexed by i) $\left\{{\rho}_{i}^{\u03f5}\right\}$, 1 ≤ i ≤ N with joint PDF $\mathcal{P}{\u03f5}_{}\left(\left\{{\rho}_{i}^{\u03f5}\right\}\right)$ is satisfied if the following limit exists $${\psi}_{\left\{{\rho}_{i}^{\u03f5}\right\}}\left(\left\{{\rho}_{i}^{\u03f5}\right\}\right)=\underset{\u03f5\to 0}{\mathrm{lim}}\u03f5\mathrm{log}\left[{\mathcal{P}}_{\u03f5}\left(\left\{{\rho}_{i}^{\u03f5}\right\}\right)\right].$$(A.1)
If such limit exists, ${\psi}_{\left\{{\rho}_{i}^{\u03f5}\right\}}$ is called the rate function.
Here, ϵ is a driving parameter, indicating the set of random variables linked to a specific evolutionary process. For the matter density field at a single scale, this parameter reflects its variance, marking time from initial stages to later times. In joint statistics involving concentric disks of matter, the common driving parameter can be selected as the variance at any radius or scale. This consistency arises because all variances behave the same as they approach zero. In this limit, they directly scale with the square of the growth rate of structure in the linear regime.
If LDP holds for the random variable ρ_{i}, then Varadhan’s theorem allows us to connect its Scaled Cumulant Generating Function (SCGF) ${\phi}_{{\rho}_{i}}$ to the rate function ${\psi}_{{\rho}_{i}}$ through a LegendreFenchel transform $${\phi}_{\left\{{\rho}_{i}\right\}}\left(\left\{{\lambda}_{i}\right\}\right)=\underset{\left\{{\rho}_{i}\right\}}{\mathrm{sup}}\left[{\displaystyle \sum _{i}{\lambda}_{i}}{\rho}_{i}{\psi}_{\left\{{\rho}_{i}\right\}}\left(\left\{{\rho}_{i}\right\}\right)\right].$$(A.2)
This LegendreFenchel transform reduces to Legendre transform if the rate function is convex $${\phi}_{\left\{{\rho}_{i}\right\}}\left(\left\{{\lambda}_{i}\right\}\right)={\displaystyle \sum _{i}{\lambda}_{i}}{\rho}_{i}{\psi}_{\left\{{\rho}_{i}\right\}}\left(\left\{{\rho}_{i}\right\}\right),$$(A.3)
where λ_{i} and ρ_{i} are one by one related via the stationary condition $${\lambda}_{k}=\frac{\partial {\psi}_{\left\{{\rho}_{i}\right\}}\left(\left\{{\rho}_{i}\right\}\right)}{\partial {\rho}_{k}},\forall k\in \left\{1,\dots ,N\right\}.$$(A.4)
Another consequence of the largedeviation principle is the socalled contraction principle. This principle suggests that when dealing with a set of random variables τ_{i} following a large deviation principle LDP and connected to ρ_{i} through the continuous mapping f, the rate function of ρ_{i} can be determined as follows $${\psi}_{\left\{{\rho}_{i}\right\}}\left(\left\{{\rho}_{i}\right\}\right)=\underset{\left\{{\tau}_{i}\right\}:f\left(\left\{{\tau}_{i}\right\}\right)=\left\{{\rho}_{i}\right\}}{\mathrm{inf}}{\psi}_{\left\{{\tau}_{i}\right\}}\left(\left\{{\tau}_{i}\right\}\right).$$(A.5)
In more tangible terms, this statement suggests that an uncommon change in the behavior of ρ_{i} is predominantly influenced by the most probable variation among all unlikely changes in τ_{i}.
In the context of cosmology, if we take ρ_{k} to denote the latetime densities, then ${\overline{\tau}}_{k}$, the most likely initial field configuration, is obtained through the most probable mapping between the linear and latetime fields. In 2D, this most likely dynamics is given by cylindrical collapse, which is known to be wellfitted by $$\zeta \left({\overline{\tau}}_{{}_{k}}\right)={\rho}_{k}={\left(1\frac{{\overline{\tau}}_{{}_{k}}}{v}\right)}^{v}.$$(A.6)
The choice of v as 1.4 is made to match the calculated value of the thirdorder skewness of the matter density contrast in cylinders from perturbation theory, as presented in Uhlemann et al. (2018). The standard result for spherical collapse dynamics in 1 to 3D can be found in Mukhanov (2005).
Now that we know the most likely connection between initial conditions and the latetime field configuration, we can compute the rate function for the latetime density field. It is expressed as follows $$\psi \left({\rho}_{i}\right)=\frac{{\sigma}_{R1}^{2}}{2}{\displaystyle \sum _{k,j}{\Xi}_{kj}}\left({\tau}_{i}\right){\overline{\tau}}_{k}{\overline{\tau}}_{j}.$$(A.7)
Here, ${\sigma}_{{R}_{1}}^{2}$ is the driving parameter and is determined by the variance within the largest disk R_{1}. Ξ_{kj} is the inverse covariance matrix between the linear density field inside the initial disks of radii ${R}_{k}{\rho}_{k}^{1/2}$.
A.2 Applying LDT to aperture mass PDF
Following Barthelemy et al. (2021), an accurate modelling of the tophat smoothed weak lensing convergence can be obtained (Limber) approximating that the convergence field is an assembly of statistically independent infinitely long cylinders of the underlying matter density contrast. Those cylinders are centred on slices along the line of sight, and they reduce to those 2D slices as illustrated in Fig. A.1. This means that the cumulants can be written as $${\langle}_{{M}_{ap}^{p}}={{{\displaystyle \int}}^{\text{}}}_{0}^{{\chi}_{s}}d\chi \omega (\chi ,\chi s){\left({\delta}_{<\mathcal{D}(\chi ){\theta}_{2}}{\delta}_{<\mathcal{D}(\chi ){\theta}_{1}}\right)}^{p}{\rangle}_{c},$$(A.8)
where ${\delta}_{<\mathcal{D}\left(\chi \right){\theta}_{2}}{\delta}_{<\mathcal{D}(\chi ){\theta}_{1}}$ is a random variable that defines the slope between two concentric disks of radii $\mathcal{D}(\chi ){\theta}_{2}$ and $\mathcal{D}(\chi ){\theta}_{1}$ at comoving radial distance χ. This simplifies the problem greatly, now having to calculate just the onepoint statistics related to the density slope within each twodimensional slice along the line of sight.
Fig. A.1 Schematic view of the procedure to obtain the aperture mass map following (Barthelemy et al. 2021). The projected quantities can be inferred as a superposition of the underlying 3D density field along the line of sight. 
As is explained in Appendix A of Barthelemy et al. (2021), the choice of driving parameter is not predicted by theory and is left as a free parameter. However, when computing the joint statistics of the density fields at different scales, this choice prevents us from imposing correct quadratic contributions in the CGF. This leads us to use the full nonlinear prescription coming from the Halofit to model the nonlinear covariance. In this paper, we use the HalofitTakahashi version Takahashi et al. (2012) to compute the covariances.
Now using Eq. (A.3), and Eq. (A.7), and since we need the density slope between two concentric disks, we get the SCGF as $${\phi}_{{\delta}_{2}{\delta}_{1}}(\lambda )={\phi}_{{\delta}_{1},{\delta}_{2}}(\lambda ,\lambda )$$(A.9)
To ensure that this choice does not lead to any discrepancies with the numerical simulations, we also rescale the projected CGF given in Eq. (A.3) with the measured variance ${\sigma}_{{M}_{ap},sim}^{2}$ instead of the one computed from Halofit ${\sigma}_{{M}_{ap},hf\text{\hspace{0.17em}}it}^{2}$ as is given below $${\varphi}_{{M}_{ap}}(\lambda )=\frac{{\sigma}_{{M}_{ap},hfit}^{2}}{{\sigma}_{{M}_{ap},sim}^{2}}{\varphi}_{{M}_{ap}}\left(\lambda \frac{{\sigma}_{{M}_{ap},sim}^{2}}{{\sigma}_{{M}_{ap},hfit}^{2}}\right)$$(A.10)
Once the CGF of individual slices is obtained, we can use Eq. (A.8) to get the CGF (the details of this derivation is given in Barthelemy et al. (2021)) of the lensing aperture mass as $${\varphi}_{\kappa ,{\theta}_{1},{\theta}_{2}}(\lambda )={{{\displaystyle \int}}^{\text{}}}_{0}^{{\chi}_{s}}d\chi {w}^{p}\left(\chi ,{\chi}_{s}\right){\varphi}_{\delta 1,{\delta}_{2},{M}_{ap}}\left(w\left(\chi ,{\chi}_{s}\right)\lambda ,\mathcal{D}(\chi )\theta 1,\mathcal{D}(\chi ){\theta}_{2}\right).$$(A.11)
After calculating the CGF, the inverse Laplace transform can be used to obtain the convergence PDF $$P(\kappa )={{{\displaystyle \int}}^{\text{}}}_{i\infty}^{+i\infty}\frac{d\lambda}{2\pi i}\mathrm{exp}\left(\lambda \kappa +{\varphi}_{\kappa ,\theta}(\lambda )\right).$$(A.12)
As described in more detail in Barthelemy et al. (2021), the inverse Laplace transform we need to perform assumes that the CGF is defined in the complex plane along the path of integration. Unfortunately, the use of numerical results for the matter power spectra prevents us to perform this continuation from the real axis. As a result, we use an informed fit of the numerical CGF along the real axis with a finite number of coefficients that we then extend to the complex plane. This allows to perform the previous integral.
Now that we have a PDF of the aperture mass map P(M_{ap})/alternatively the wavelet coefficients, we could use that to obtain the wavelet ℓ_{1}norm using the Eq. (12) as demonstrated in the Sect. 3.3.
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Code publicly available at GitHub: https://github.com/vilasinits/LDT_2cell_l1_norm
All Tables
Cumulants (standard deviation σ, skewness S_{3}) obtained from the PDF from the LDT prediction and the values obtained from the Takahashi simulation at source redshift z_{s} ≈ 2.05 for scales of 15, 18, and 20 arcminutes.
All Figures
Fig. 1 Probability density function and ℓ_{1}norm for a Gaussian distribution (blue) and the nonGaussian distribution (orange). On the left, we present the PDF for the two distributions, and on the right, we display the derived ℓ_{1}norm of these PDFs. The peak heights are the same for a Gaussian distribution, but this does not hold for a nonGaussian PDF. 

In the text 
Fig. 2 Compensated filter (solid black line), derived through the difference of two tophat filters at different scales, as described in Eqs. (5) and (10). The solid blue and orange lines represent the individual spherical filters obtained at radii θ_{1} and θ_{2} = 2θ_{1}, respectively. For visualisation purposes, the compensated filter is multiplied by −1. 

In the text 
Fig. 3 Comparison of the predicted wavelet ℓ_{1} norm to the simulations at different redshifts. Top panel: predicted (solid) ℓ_{1}norm as compared to measurements in simulation (dots) for an inner radius θ_{1} = 20′ and different source redshifts z_{s} = 1.21,1.43 and 2.05, displayed with blue, orange, and green lines, respectively. The dashdotted red lines show the Gaussian prediction for reference. The vertical dotted and dotdashed lines correspond to the 1σ and 2σ regions around the mean of the ${w}_{{\theta}_{1}}$ for each of the cases considered. Bottom panel: residual of the prediction relative to the simulation (dotted lines). For reference, the dashdotted plots illustrate the residual of the ℓ_{1} norm derived from the Gaussian PDF with the same mean and variance as the simulation PDF. The shaded region indicates the 3σ region around the error bars for each redshift. The prediction agrees well with the measurements up to approximately 2σ and remains within percent levels. 

In the text 
Fig. 4 Comparison of the predicted wavelet ℓ_{1} norm to the simulations at different scales. Top panel: predicted (solid) ℓ_{1} norm as compared to measurements in simulation (dots) for different inner radii θ_{1} = 15′, 18′, and 20′ and a single source redshift z_{s} = 1.43, displayed with blue, orange, and green lines, respectively. The dashdotted red lines show the Gaussian prediction for reference. The vertical dotted and dotdashed lines correspond to the 1σ and 2σ regions around the mean of the ${w}_{{\theta}_{1}}$ for each of the case considered. Bottom panel: residual of the prediction relative to the simulation (dotted lines). For reference, the dashdotted plots illustrate the residual of the ℓ_{1} norm derived from the Gaussian PDF with the same mean and variance as the simulation PDF. The shaded region indicates the 3σ region around the error bars for each inner radius. The prediction agrees well with the measurements up to approximately 2σ and remains within percent levels. 

In the text 
Fig. 5 Derivative of the PDF with respect to M_{v} (blue), Ω_{m} (orange), and A_{s} (green). The solid lines show the derivatives obtained from the prediction, and the dotted lines show the derivatives from the simulation. It is obtained at source redshift z_{s} = 2 and inner radius θ_{1} = 22.5′. The results for the simulation are obtained from the MassiveNus simulation suite by averaging the results over 10 000 simulations. 

In the text 
Fig. 6 Derivative of the wavelet ℓ_{1} norm with respect to M_{v} (blue), Ω_{m} (orange), and A_{s} (green). The solid lines show the derivatives obtained from the prediction, and the dotted lines show the derivatives from the simulation. The predicted wavelet ℓ_{1} norm is derived from the predicted PDF, as was explained in previous sections. It is obtained at source redshift z_{s} = 2 and inner radius θ_{1} = 22.5′. The results for the simulation are obtained from the MassiveNus simulation suite by averaging the results over 10 000 simulations. 

In the text 
Fig. A.1 Schematic view of the procedure to obtain the aperture mass map following (Barthelemy et al. 2021). The projected quantities can be inferred as a superposition of the underlying 3D density field along the line of sight. 

In the text 
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