# Yourfeed: Towards open science and interoperable systems for social media

Ziv Epstein1*, Hause Lin2,3*
1MIT Media Lab, 2MIT Sloan, 3University of Regina
{zive<EMAIL_ADDRESS>
*These authors contributed equally

###### Abstract

> Existing social media platforms (SMPs) make it incredibly difficult for
> researchers to conduct studies on social media, which in turn has created a
> knowledge gap between academia and industry about the effects of platform
> design on user behavior. To close the gap, we introduce Yourfeed, a research
> tool for conducting ecologically valid social media research. We introduce
> the platform architecture, as well key opportunities such as assessing the
> effects of exposure of content on downstream beliefs and attitudes,
> measuring attentional exposure via dwell time, and evaluating heterogeneous
> newsfeed algorithms. We discuss the underlying philosophy of
> interoperability for social media and future developments for the platform.

## Introduction

Today, more than half of the population of earth are on social media. As a
tool for individuals to gain information about the world and connect with
others, and as a platform for researchers to build a generalizable
understanding about human nature (Lazer et al., 2020), the possibilities for
social media to provide meaningful information and facilitate collective
action are nearly endless. Yet the corporatized modern platforms that exist
today have in many ways fallen short of this lofty vision.

For one, the profit incentive of social media platforms (SMPs) has prioritized
designs and interfaces that put the priorities of the company above the user
(Lorenz-Spreen et al., 2020). Newsfeed algorithms optimized for engagement
(e.g. Facebook (Oremus et al., 2021)) or dwell (e.g. TikTok (Smith, 2021;
Team, 2021)) keep users on-platform and in turn may drive the spread of
misinformation (Hao, 2021), while many users are unaware of the fact that
algorithms are dictating what they see (Eslami et al., 2015). Dark patterns
(UX/UI elements that subtly trick you (Gray et al., 2018), for examples see
Harris (2016)) abound on user interfaces. Platforms control user data and
content policies, with little transparency or recourse for how these policies
are created and enforced (Sadowski et al., 2021).

In addition, the culture and structure of social media platforms have
drastically limited researchers’ ability to use them to generate public
knowledge about their workings Greene et al. (2022); Bail et al. (2022).
Internally, SMPs have sophisticated experimental frameworks for understanding
the effect of algorithms and interfaces on user behavior, which they use to
further refine their platforms for profit. Yet these insights are held behind
walled gardens of NDAs and industry secrets. Sometimes, the results of these
studies are publicly shared, such as a 61-million person experiment to
increase voter turnout (Bond et al., 2012), or the infamous experiment of
emotional contagion (Kramer et al., 2014). In both cases, there was major
public backlash about the impact of these systems, and the questionable ethics
of such experiments (Jouhki et al., 2016; Zittrain, 2014; Zuboff, 2015).

For academics attempting to ask similar questions, they must rely on crude
alternatives. One is partnerships with the companies, where the SMP grants the
researchers access to user data to write public facing papers. However to
ensure privacy, these datasets are often aggregated to a point that they are
of limited use, and these partnerships might stall for political reasons
(Doctorow, 2019). Furthermore, the SMP providing the data can set the agenda,
and differentially favor academics whose research interests align with their
strategy (Hegelich, 2020). Another approach is social media field experiments,
where academics attempt to leverage platform APIs to conduct experiments with
real users (Mosleh et al., 2022). These field experiments can however violate
platforms Terms and Services, “poison the well” by undermining trust in
digital ecosystems (Pennycook et al., 2021), and are often challenging to
replicate as platforms change their policies to adapt. A final approach is
controlled laboratory experiments with survey tools like Qualtrics. This
paradigm allows researchers to run RCTs to understand the effect of design
interventions on user behavior (see e.g. Pennycook and Rand (2021)). However,
the context of Qualtrics leaves a lot to be desired for drawing generalizable
inferences about the impact of interfaces and algorithms on user behavior (Lin
et al., 2021). In particular, the standard Qualtrics approach presents social
media posts one at a time, rather than in a newsfeed layout. This feed layout,
however, is a critical design feature of modern social media platforms, and
therefore Qualtrics omits key attentional features of online social media use
and consumption.

Figure 1: Overview of the platform architecture

To close the knowledge gap between the platforms and the public, we introduce
Yourfeed, a modular tool for ecologically valid social media research.
Yourfeed mirrors the attentional properties of social media by providing a
feed layout for content. In addition, it tracks and surfaces the dwell time of
each post, a key metric from industry that is hard to measure on traditional
survey platforms like Qualtrics. Yourfeed is designed to be modular and
interoperable, allowing researchers to upload their own components, such as
stimulus sets, ranking algorithms, or design interventions. In this paper, we
introduce the Yourfeed system, with several key features and studies
researchers could run. We conclude with the implications of Yourfeed on
building public knowledge about platforms impact user behavior, and how this
knowledge might help nudge platforms to open up their policies and surfaces
for a more democratic approach.

Figure 2: Top left: Dwell and share data for a user as they scroll through
Yourfeed. A red dot corresponds to posts the user decided to share, while a
hollow red dot corresponds to posts the user shared and then unshared. Top
right: Dwell and like data for a user as they scroll through Yourfeed. Bottom
left: Dwell and bookmark data for a user as they scroll through Yourfeed.
Bottom right: Average dwell time of 100 users over the course of a browsing
session (Entity order = 0 is the top of the newsfeed, entity order=100 is the
bottom).

## System Overview

Yourfeed is an experimental tool that allows researchers to conduct randomized
studies in a feed environment (see Figure 1). To do so, researchers specify a
series of conditions (e.g. 1 to n) on the create-studies page and a series of
attributes for each condition. Once a set of experimental parameters has been
submitted, Yourfeed generates a unique URL for that specific experiment. When
a user first visits Yourfeed at that URL, the Experiment Parameter Randomizer
assigns that user to a condition (the yellow box in Figure 1), and then
renders the feed interface using the attributes for that condition. In
particular, there are several types of attributes used to generate the feed
interface for a given user. The first are the entity sets, which are sets of
newsfeed items to display in the feed. Experimenters can select these sets for
each condition, and also specify how many items from a given entity set are
displayed. For each user, the items from the specified entity sets constitute
that user’s inventory. This inventory is then ordered by a feed algorithm,
which can be a user-specified blackbox algorithm (the blue box in Figure 1).
The default algorithm simply orders the feed items in a random order, but
researchers can upload their own bespoke algorithms in a standardized format.
The sorted inventory is then populated with engagement scores (e.g. the number
of likes or posts, the pink box in Figure 1). These engagement scores could be
omitted if specified by the experiment parameters, sampled randomly for each
post for each user (e.g. as down in Avram et al. (2020)), or actually show the
number of likes or shares from other participants in the study. Finally, the
sorted inventory populated with engagement is then displayed in the feed
interface, which is additionally styled according to experimental parameters
(e.g. with advertisements, a modal intervention, or a Instagram/Facebook
skin). Following the feed interaction, participants proceed to a survey
environment where post-task demographics and beliefs/attitudes can be
assessed. Researchers can also upload their own entity sets at the upload
page, and download the data for their experiments at the download-data page.
The platform is available online at https://www.yourfeed.social.

The affordances of Yourfeed allow researchers to ask research questions that
would be difficult otherwise. Below, we enumerate several key features of
Yourfeed and example research questions and studies that can be conducted with
those features.

### More ecologically valid studies on social media behavior and attitudes

There are many studies researchers can conduct on survey platforms like
Qualtrics that can be directly transported into Yourfeed with added ecological
validity. One class of studies involves measuring the causal effect of
exposure of content on downstream beliefs and attitudes. Since researchers can
specify different entity sets for different conditions, they can randomly
assign participants to exposure of different topics, then follow up with the
same set of survey questions as dependent variables. For example, a researcher
might randomly vary the amount of content related to climate change, and look
at how that influences how important participants perceive climate change to
be. A second class of studies involves measuring the impact of a design
intervention on sharing behavior. For example, a researcher might introduce an
interstitial accuracy prompt at a random point in the feed, and look how this
interstitial impacts sharing discernment (both the magnitude and duration of
the treatment effect).

An important implication of Yourfeed’s design is that share rates on the
platform are lower. In a recent Yourfeed experiment with 876 participants
recruited from Prolific, the overall average share rate was 7.7%, overall
average like rate was 12.1%, compared to a share rate of 32.5% and a like rate
of 31.% on Qualtrics (using data from Epstein et al. (2021)). This suggests
that share rates from Qualtrics, while correlated with actual sharing (Mosleh
et al., 2020), may be systematically larger than they would be on platform.

### Quantifying attention dynamics via dwell time

A key limitation of survey experiments on engagement with social media is that
they often show social media posts one by one, which forces attentional
capture on each one. Such setups are contrived and differ dramatically from
the design of feed-based platforms, where some posts grab attention while
others are scrolled past. Building on the metrics and practices of the online
advertising industry (Hwang, 2020), Yourfeed measures the amount of time a
user spends looking at a particular post by recording the amount of time that
post is viewable (e.g. dwell time). A researcher could treat this value itself
as an outcome variable, and look at which user- and item-level covariates are
predictive of different attention dynamics.

To underscore the kinds of data Yourfeed collects and generates, we present
pilot data from a study conducted on Yourfeed (run in June 2022 with N=876
participants recruited from Prolific). As shown in Figure 2, Yourfeed provides
granular data about how long each participants dwelled on a given post
(entity), and how they engaged with each. We also find that dwell time
decreases over a browsing session, whereby participants spend the most time on
the top item in their feed, and the least on the bottom.

### Serving and evaluating heterogeneous newsfeed algorithms

Recently, Gizmodo made public a series of Facebook papers which elucidate the
inner workings of how Facebook operates. Among these are experiments that
compare algorithmic ranking to chronological newsfeeds (Cameron et al., 2022).
There has been growing concern about the effects of algorithmic ranking on
user behavior (Eckles, 2022; Sassine et al., 2022), and political information
flow (Reuning et al., 2022) but a lack of methods for studying these effects
directly has resulted in leaked internal SMP documents being a key method for
public knowledge on the topic.

Just as entity sets or design interventions can be randomly assigned to
participants, so can feed algorithms. This feature allows researchers to
causally assess the effects of different algorithms on user behavior. For
example, a growing body of work has recommended newsfeed algorithms that are
more aligned with social priorities (Cen and Shah, 2021; Bhadani et al.,
2022), and Yourfeed provides a platform for evaluating the effects of these
algorithms on user behavior.

In addition to randomly assigning participants to different algorithms,
Yourfeed could randomly assign participants to different worlds, each of which
evolve independently (Salganik et al., 2006; Epstein et al., 2021). This
feature allows researchers to quantify the variation introduced by such
algorithms via counterfactuals, and to measure ecosystem-level outcomes, like
the diversity of content (Epstein et al., 2021).

As an example, a researcher could create a study with three conditions: a
control condition where items are randomly ordered, a second condition with a
newsfeed that optimizes and sorts by engagement (e.g. shares, like Facebook),
and third condition with a newsfeed that optimizes by dwell time (like
TikTok). For the two algorithmic conditions, participants could be assigned to
one of 10 worlds which evolve independently. In particular, for a given user
the order of the newsfeed is determined solely by the patterns of engagement
(in the second condition) or dwell (in the third condition) by other
participants assigned to that user’s world. The researcher could compute the
diversity of content across each world, and therefore assess the causal effect
of the ranking algorithms on diversity across worlds.

## Related Work

Many recent papers have developed custom platforms to explore behavioral
aspects of social media. Avram et al. (2020) developed Fakey, a simulated
newsfeed game that over 8,500 users played. They randomized the engagement
metrics associated with each post, and find that exposure to social engagement
metrics increases vulnerability to misinformation. Roozenbeek and Van der
Linden (2019) developed Bad News, a browser based that over 15,000 users
engaged with. They found that users’ ability to detect misinformation
increases after gameplay, across demographic subgroups. Epstein et al. (2021)
developed Meet the Ganimals, an online platform for generating and curating
AI-generated hybrid animals (Epstein et al., 2020), that over 10,000 players
engaged with. OpenHuman (Greshake Tzovaras et al., 2019) is a community-based
platform that gives users higher levels of personal data access and control.

Other experimental platforms have explored participatory algorithm design and
algorithmic auditing. Bhargava et al. (2019) developed Gobo, an experimental
system for exploring user control of invisible algorithms in social media.
Gobo allowed users to control the ranking of posts in their newsfeed with
manual controls, such as the seriousness or rudeness of posts, or the gender
of the poster. Turingbox (Epstein et al., 2018) was an experimental two-sided
market where contributors could upload algorithms for evaluations, and social
scientists could evaluate them, towards understanding the consequences of
their behavior (Epstein et al., 2018).

Perhaps most closely related to Yourfeed are tools developed for researchers
to conduct social media research. The Mock Social Media Website Tool (Jagayat
et al., 2021) is an open-source software for studying social media.
Participants in the MSMW can like posts, share posts on a timeline, add their
own posts and reply to other posts. The (Mis)information Game (Butler et al.,
2022) is an online testing platform that allows researchers to customize
posts, source information and engagement information. Unlike a feed
enviornment, (Mis)information Game presents posts one at time for users to
interact with and comment on. Community Connect (Mahajan et al., 2021) is an
open-source and customizable platform for experimentation, designed for
controlled information flow across groups through bridge nodes.

## Discussion

There are several components that constitute modern social media platforms.
The first is the underlying social data on which the platform is based, such
as the social graph and posts or comments. The second is the recommender
algorithm which for a given user’s inventory serves a subset of that content
to their feed. The third and final is a “skin” or interface by which this feed
content is displayed and users interact with the medium. The current model of
social media offers a monolithic experience with each of these components
molded together, with no opportunities for users to select alternative
components, like newsfeed algorithms or skins. This walled-garden approach
means that SMPs can carefully design these components with their incentives in
mind (e.g. maximizing profit via engagement optimization) and serve them to
users who have no choice but use them if they want to access the platforms.
Yourfeed offers an alternative, interoperable vision of social media. Rather
than a cloister, there could exist a rich ecology of third-party newsfeed
algorithms and skins, each designed with a different perspective and goal in
mind. Users could access the same underlying data, but specify the particular
components that satisfy their goals. On one hand, this would augment user
autonomy by providing an expressive grammar by which to interact with social
media. And on the other hand it would spur innovation as developers compete to
make newsfeed algorithms and skins that users actually want to adopt. The
economics of this paradigm involve unseating the dominance that current SMPs
have on the market, which is exciting future work beyond the scope of this
white paper.

Along the lines of this call for interoperable systems are questions of who
should own the underlying data. Currently, the data is centrally owned and
stored by the SMPs themselves. However, there is a growing body of work that
suggests that the data can be owned and stored by the communities that create
that data, in data cooperatives Greshake Tzovaras et al. (2019); Pentland and
Hardjono (2020) or a pluralistic social network (Chow, 2022; Zignani et al.,
2018; Zulli et al., 2020).

There are several limitations to the existing instantiation of Yourfeed.
First, like its survey antecedents, there is no explicitly “social” components
of the browsing experience: users have no profile picture or other identifying
characteristics, posts have no author, and perhaps most critically, clicking
share does not in fact share that post with anyone. These decisions are
intentional, as we focused primarily on capturing the attentional affordances
of browsing a feed, not the social affordances. However, we hope the next
version of Yourfeed accommodates more of these features. In addition, Yourfeed
caters to a research audience who recruits participants from online labor
markets like Amazon’s Mechanical Turk or Lucid. As such, interacting with the
platform is part of a transactional task for money, rather than the in situ
way most people interact with social media. In the future, we hope Yourfeed
and tools like it that 1) offer radical alternatives to the social media
status quo and 2) prioritize experimentation and public knowledge over profit
will become more preferable for social media users, and therefore generate
organic participation.

Building on a legacy of speculative design (Dunne and Raby, 2013), Yourfeed
offers a low-friction sandbox for designers, scientists, and citizens to
prototype their own ideas for how social media should look and operate. We
hope the development of such affirmative visions will bootstrap all three
prongs of reform, and in turn help social media become a tool for helping
society solve complex challenges.

## References

  * (1)
  * Avram et al. (2020) Mihai Avram, Nicholas Micallef, Sameer Patil, and Filippo Menczer. 2020.  Exposure to social engagement metrics increases vulnerability to misinformation.  _arXiv preprint arXiv:2005.04682_ (2020).
  * Bail et al. (2022) Chris Bail et al. 2022\.  Social-media reform is flying blind.  _Nature_ 603, 7903 (2022), 766–766.
  * Bhadani et al. (2022) Saumya Bhadani, Shun Yamaya, Alessandro Flammini, Filippo Menczer, Giovanni Luca Ciampaglia, and Brendan Nyhan. 2022.  Political audience diversity and news reliability in algorithmic ranking.  _Nature Human Behaviour_ 6, 4 (2022), 495–505.
  * Bhargava et al. (2019) Rahul Bhargava, Anna Chung, Neil S Gaikwad, Alexis Hope, Dennis Jen, Jasmin Rubinovitz, Belén Saldías-Fuentes, and Ethan Zuckerman. 2019.  Gobo: A system for exploring user control of invisible algorithms in social media. In _Conference Companion Publication of the 2019 on Computer Supported Cooperative Work and Social Computing_. 151–155.
  * Bond et al. (2012) Robert M Bond, Christopher J Fariss, Jason J Jones, Adam DI Kramer, Cameron Marlow, Jaime E Settle, and James H Fowler. 2012\.  A 61-million-person experiment in social influence and political mobilization.  _Nature_ 489, 7415 (2012), 295–298.
  * Butler et al. (2022) Lucy Butler, Padraig Lamont, Dean Law Yim Wan, Toby Prike, Mehwish Nasim, Bradley Walker, Nicolas Fay, and Ullrich Ecker. 2022\.  The (Mis)Information Game: A Social Media Simulator.  PsyArxiv.  https://psyarxiv.com/628wc/
  * Cameron et al. (2022) Dell Cameron, Shoshana Wodinsky, and Mark DeGeurin. 2022\.  We’re Publishing the Facebook Papers. Here’s How Facebook Killed News Feed Fixes Over Fear of Conservative Backlash.  Web.
  * Cen and Shah (2021) Sarah Cen and Devavrat Shah. 2021.  Regulating algorithmic filtering on social media.  _Advances in Neural Information Processing Systems_ 34 (2021), 6997–7011.
  * Chow (2022) Wes. Chow. 2022\.  A Sociotechnical Framework for a Pluralistic Social Network.  Github.  https://github.com/wesc/sf-psn
  * Doctorow (2019) Cory Doctorow. 2019\.  Facebook promised to provide academics data to study disinformation, but their foot-dragging has endangered the whole project.  Boingbong.  https://boingboing.net/2019/12/12/gdpr-all-purpose-excuse.html
  * Dunne and Raby (2013) Anthony Dunne and Fiona Raby. 2013.  _Speculative everything: design, fiction, and social dreaming_.  MIT press.
  * Eckles (2022) Dean Eckles. 2022\.  Algorithmic transparency and assessing effects of algorithmic ranking.  (2022).
  * Epstein et al. (2020) Ziv Epstein, Océane Boulais, Skylar Gordon, and Matt Groh. 2020.  Interpolating GANs to scaffold autotelic creativity.  _arXiv preprint arXiv:2007.11119_ (2020).
  * Epstein et al. (2021) Ziv Epstein, Matthew Groh, Abhimanyu Dubey, and Alex Pentland. 2021.  Social influence leads to the formation of diverse local trends.  _Proceedings of the ACM on Human-Computer Interaction_ 5, CSCW2 (2021), 1–18.
  * Epstein et al. (2018) Ziv Epstein, Blakeley H Payne, Judy Hanwen Shen, Casey Jisoo Hong, Bjarke Felbo, Abhimanyu Dubey, Matthew Groh, Nick Obradovich, Manuel Cebrian, and Iyad Rahwan. 2018\.  TuringBox: An experimental platform for the evaluation of AI systems. In _IJCAI 2018_. International Joint Conferences on Artificial Intelligence, 5826–5828.
  * Eslami et al. (2015) Motahhare Eslami, Aimee Rickman, Kristen Vaccaro, Amirhossein Aleyasen, Andy Vuong, Karrie Karahalios, Kevin Hamilton, and Christian Sandvig. 2015\.  ” I always assumed that I wasn’t really that close to [her]” Reasoning about Invisible Algorithms in News Feeds. In _Proceedings of the 33rd annual ACM conference on human factors in computing systems_. 153–162.
  * Gray et al. (2018) Colin M Gray, Yubo Kou, Bryan Battles, Joseph Hoggatt, and Austin L Toombs. 2018.  The dark (patterns) side of UX design. In _Proceedings of the 2018 CHI conference on human factors in computing systems_. 1–14.
  * Greene et al. (2022) Travis Greene, David Martens, and Galit Shmueli. 2022\.  Barriers to academic data science research in the new realm of algorithmic behaviour modification by digital platforms.  _Nature Machine Intelligence_ 4, 4 (2022), 323–330.
  * Greshake Tzovaras et al. (2019) Bastian Greshake Tzovaras, Misha Angrist, Kevin Arvai, Mairi Dulaney, Vero Estrada-Galiñanes, Beau Gunderson, Tim Head, Dana Lewis, Oded Nov, Orit Shaer, et al. 2019\.  Open Humans: A platform for participant-centered research and personal data exploration.  _GigaScience_ 8, 6 (2019), giz076.
  * Hao (2021) Karen Hao. 2021\.  How Facebook got addicted to spreading misinformation.  _MIT Technology Review_ (2021).
  * Harris (2016) Tristan Harris. 2016\.  How Technology is Hijacking Your Mind — from a Magician and Google Design Ethicist.  Medium.
  * Hegelich (2020) Simon Hegelich. 2020\.  Facebook needs to share more with researchers.  _Nature_ 579, 7800 (2020), 473–474.
  * Hwang (2020) Tim Hwang. 2020\.  _Subprime attention crisis: Advertising and the time bomb at the heart of the Internet_.  FSG originals.
  * Jagayat et al. (2021) Arvin Jagayat, Gurkaran Boparai, Carson Pun, and Becky Choma. 2021\.  Mock Social Media Website Tool.  https://docs.studysocial.media
  * Jouhki et al. (2016) Jukka Jouhki, Epp Lauk, Maija Penttinen, Niina Sormanen, and Turo Uskali. 2016.  Facebook’s emotional contagion experiment as a challenge to research ethics.  _Media and Communication_ 4 (2016).
  * Kramer et al. (2014) Adam DI Kramer, Jamie E Guillory, and Jeffrey T Hancock. 2014\.  Experimental evidence of massive-scale emotional contagion through social networks.  _Proceedings of the National Academy of Sciences_ 111, 24 (2014), 8788–8790.
  * Lazer et al. (2020) David MJ Lazer, Alex Pentland, Duncan J Watts, Sinan Aral, Susan Athey, Noshir Contractor, Deen Freelon, Sandra Gonzalez-Bailon, Gary King, Helen Margetts, et al. 2020\.  Computational social science: Obstacles and opportunities.  _Science_ 369, 6507 (2020), 1060–1062.
  * Lin et al. (2021) Hause Lin, Kaitlyn M Werner, and Michael Inzlicht. 2021\.  Promises and perils of experimentation: The mutual-internal-validity problem.  _Perspectives on Psychological Science_ 16, 4 (2021), 854–863.
  * Lorenz-Spreen et al. (2020) Philipp Lorenz-Spreen, Stephan Lewandowsky, Cass R Sunstein, and Ralph Hertwig. 2020.  How behavioural sciences can promote truth, autonomy and democratic discourse online.  _Nature human behaviour_ 4, 11 (2020), 1102–1109.
  * Mahajan et al. (2021) Khyati Mahajan, Sourav Roy Choudhury, Sara Levens, Tiffany Gallicano, and Samira Shaikh. 2021\.  Community connect: A mock social media platform to study online behavior. In _Proceedings of the 14th ACM International Conference on Web Search and Data Mining_. 1073–1076.
  * Mosleh et al. (2020) Mohsen Mosleh, Gordon Pennycook, and David G Rand. 2020\.  Self-reported willingness to share political news articles in online surveys correlates with actual sharing on Twitter.  _Plos one_ 15, 2 (2020), e0228882.
  * Mosleh et al. (2022) Mohsen Mosleh, Gordon Pennycook, and David G Rand. 2022\.  Field experiments on social media.  _Current Directions in Psychological Science_ 31, 1 (2022), 69–75.
  * Oremus et al. (2021) Will Oremus, Chris Alcantara, Jeremy Merrill, and Artur Galocha. 2021.  How Facebook shapes your feed: The evolution of what posts get top billing on users’ news feeds, and what gets obscured.  Web.  https://www.washingtonpost.com/technology/interactive/2021/how-facebook-algorithm-works/
  * Pennycook et al. (2021) Gordon Pennycook, Ziv Epstein, Mohsen Mosleh, Antonio A Arechar, Dean Eckles, and David G Rand. 2021.  Shifting attention to accuracy can reduce misinformation online.  _Nature_ 592, 7855 (2021), 590–595.
  * Pennycook and Rand (2021) Gordon Pennycook and David G Rand. 2021.  The psychology of fake news.  _Trends in cognitive sciences_ 25, 5 (2021), 388–402.
  * Pentland and Hardjono (2020) Alex Pentland and Thomas Hardjono. 2020.  2\. Data Cooperatives.  In _Building the New Economy_. PubPub.
  * Reuning et al. (2022) Kevin Reuning, Anne Whitesell, and A Lee Hannah. 2022\.  Facebook algorithm changes may have amplified local republican parties.  _Research & Politics_ 9, 2 (2022), 20531680221103809\.
  * Roozenbeek and Van der Linden (2019) Jon Roozenbeek and Sander Van der Linden. 2019.  Fake news game confers psychological resistance against online misinformation.  _Palgrave Communications_ 5, 1 (2019), 1–10.
  * Sadowski et al. (2021) Jathan Sadowski, Salomé Viljoen, and Meredith Whittaker. 2021.  Everyone should decide how their digital data are used—Not just tech companies.
  * Salganik et al. (2006) Matthew J Salganik, Peter Sheridan Dodds, and Duncan J Watts. 2006.  Experimental study of inequality and unpredictability in an artificial cultural market.  _science_ 311, 5762 (2006), 854–856.
  * Sassine et al. (2022) Jad Sassine, M Amin Rahimian, and Dean Eckles. 2022\.  Influence of Repetition through Limited Recall. In _Proceedings of the International AAAI Conference on Web and Social Media_ , Vol. 16. 863–872.
  * Smith (2021) Ben Smith. 2021\.  How TikTok Reads Your Mind.  Web.  https://www.nytimes.com/2021/12/05/business/media/tiktok-algorithm.html
  * Team (2021) WSJ Team. 2021.  Inside TikTok’s Algorithm: A WSJ Video Investigation.  Web.  https://www.wsj.com/articles/tiktok-algorithm-video-investigation-11626877477
  * Zignani et al. (2018) Matteo Zignani, Sabrina Gaito, and Gian Paolo Rossi. 2018\.  Follow the “mastodon”: Structure and evolution of a decentralized online social network. In _Proceedings of the International AAAI Conference on Web and Social Media_ , Vol. 12. 541–550.
  * Zittrain (2014) Jonathan Zittrain. 2014\.  Facebook could decide an election without anyone ever finding out.  _New Republic_ 1 (2014).
  * Zuboff (2015) Shoshana Zuboff. 2015\.  Big other: surveillance capitalism and the prospects of an information civilization.  _Journal of information technology_ 30, 1 (2015), 75–89.
  * Zulli et al. (2020) Diana Zulli, Miao Liu, and Robert Gehl. 2020.  Rethinking the “social” in “social media”: Insights into topology, abstraction, and scale on the Mastodon social network.  _New Media & Society_ 22, 7 (2020), 1188–1205.


11institutetext: Université Paris-Saclay, Université Paris Cité, CEA, CNRS,
AIM, 91191, Gif-sur-Yvette, France
11email<EMAIL_ADDRESS>22institutetext: Universitäts-
Sternwarte, Fakultät für Physik, Ludwig-Maximilians-Universität München,
Scheinerstraße 1, 81679 München, Germany 33institutetext: Institutes of
Computer Science and Astrophysics, Foundation for Research and Technology
Hellas (FORTH), Greece

# Theoretical wavelet $\ell_{1}$-norm from one-point PDF prediction

Vilasini Tinnaneri Sreekanth 11 Sandrine Codis 11 Alexandre Barthelemy 22
Jean-Luc Starck 1133

(Received ?; accepted ?)

###### Abstract

Context. Weak gravitational lensing, resulting from the bending of light due
to the presence of matter along the line of sight, is a potent tool for
exploring large-scale structures, particularly in quantifying non-
Gaussianities. It stands as a pivotal objective for upcoming surveys. In the
realm of current and forthcoming full-sky weak-lensing surveys, the
convergence maps, representing a line-of-sight integration of the matter
density field up to the source redshift, facilitate field-level inference,
providing an advantageous avenue for cosmological exploration. Traditional
two-point statistics fall short of capturing non-Gaussianities, necessitating
the use of higher-order statistics to extract this crucial information. Among
the various higher-order statistics available, the wavelet $\ell_{1}$-norm has
proven its efficiency in inferring cosmology (Ajani et al. (2021)). However,
the lack of a robust theoretical framework mandates reliance on simulations,
demanding substantial resources and time.

Aims. Our novel approach introduces a theoretical prediction of the wavelet
$\ell_{1}$-norm for weak lensing convergence maps, grounded in the principles
of Large-Deviation theory. This method builds upon recent work by Barthelemy
et al. (2021), offering a theoretical prescription for aperture mass one-point
probability density function.

Methods. We present, for the first time, a theoretical prediction of the
wavelet $\ell_{1}$-norm for convergence maps, derived from the theoretical
prediction of their one-point probability distribution. Additionally, we
explore the cosmological dependence of this prediction and validate the
results on simulations.

Results. A comparison of our predicted wavelet $\ell_{1}$-norm with
simulations demonstrates a high level of accuracy in the weakly non-linear
regime. Moreover, we show its ability to capture cosmological dependence,
paving the way for a more robust and efficient parameter inference process.

###### Key Words.:

higher order statistics – $\kappa$-PDF – $\ell_{1}$-norm– large deviation
theory– non-gaussianities

##  1 Introduction

Our present comprehension of the genesis of large-scale structure (Peebles
1980) posits that it emerged through gravitational instability driven by
primordial fluctuations in matter density. Realistic models detailing
structure formation prescribe an initial spectrum of perturbations reflecting
the primordial spectrum, characterised by small fluctuations at large scales.
In an LCDM Universe, the variance of the density fluctuations, denoted by
$\sigma(R)^{2}$, is inversely proportional to (some power of) scales so that
at small scales, the variance is large and non-linear effects become
important.

Hence, there are two limiting regimes: the linear regime which is
characterized by $\sigma^{2}(R)<<1$, and the non-linear regime which is given
by $\sigma^{2}(R)>>1$. In particular, if 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, manifesting as higher-order connected correlations in
the non-linear regime (Bernardeau et al. 2002).

One of the powerful methods, to probe these non-linearities present in the
large-scale structure (hereafter LSS) is to look at the distortions of the
distant images of galaxies. Distortions happen because of gravitational
lensing, which is the phenomenon in which the paths of the photons are bent
due to the presence of 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, requiring statistical analysis
over a vast number of galaxies to detect a signal. Analyzing 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, readers are referred to (Mandelbaum (2018);Kilbinger (2018)),
and Kilbinger (2015).

In this era of precision cosmology, with past, present, and future surveys
such as the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS) (Heymans
et al. 2012), Hyper Suprime-Cam (HSC) (Mandelbaum & Collaboration 2017),
Euclid (Laureijs et al. 2011), LSST ((Ivezić et al. 2019)), to name a few, we
now have access to the non-Gaussian part of cosmological signals, which are
induced by the non-linear evolution of the structures on small scales. All of
these experiments, consider weak gravitational lensing to be a key probe –
jointly with galaxy clustering – to explore the current unanswered questions
in Cosmology like 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 like the mean
matter density, the amplitude of matter fluctuations (Troxel & Ishak 2015).

The conventional approach to infer the cosmology from the data involves the
computation of the two-point statistics, which has been employed with
remarkable success in the past (Munshi et al. 2008; Kilbinger 2015; Bartelmann
& Maturi 2017; Hildebrandt et al. 2017) ((Euclid Collaboration et al. 2024;
Loureiro et al. 2023; Ingoglia et al. 2022)). However, it is not sufficient if
we want to probe the non-Gaussianities present (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, there arises a necessity to complement this approach with an
alternative higher-order statistic capable of effectively capturing the non-
Gaussian features embedded in the structure. Higher-order 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), Minkowski functional (Kratochvil et al. 2012;
Parroni et al. 2020), three-point statistics (Takada & Jain 2004; Semboloni et
al. 2011; Rizzato et al. 2019) and wavelet and scattering transform (Ajani et
al. 2021; Cheng & Ménard 2021) amongst many others, can better probe the non-
Gaussian structure of the Universe and provide additional constraints on the
cosmological parameters. Another example was given in Ajani et al. (2021)
employing the starlet $\ell_{1}$-norm that 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
$\ell_{1}$-norm of a starlet, which is a type of wavelet that uses
$B3$-spline, offers an efficient multiscale computation of all map pixels
including the under-densities and over-densities distribution, partially
probing similar information to peak and void counts.

The Probability Distribution Function (hereafter PDF) of the weak lensing
convergence map ($\kappa$) serves as another valuable repository of
cosmological information that has drawn a lot of interest in recent years,
with many theoretical and numerical works including (Bernardeau & Valageas
2001; Liu & Madhavacheril 2019; Barthelemy et al. 2021). The convergence field
$\kappa$ represents the weighted projection of matter density fluctuations
along the line of sight and its higher-order correlations offer a promising
avenue for addressing challenges inherent in standard weak lensing analyses,
particularly those related to degeneracies in the two-point correlation
function (2PCF).

The one-point $\kappa$-PDF statistic, obtained by quantifying smoothed
$\kappa$ field values within predefined apertures or cells, presents a
practical advantage as its measure is rather simple. This simplicity contrasts
with other non-Gaussian probes used for studying the weak lensing 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 $\kappa$ field as well as the aperture mass (Barthelemy
et al. 2021). Notably, this one-point PDF could be directly linked to the
wavelet coefficients at different scales (Ajani et al. 2021).

Let us emphasize that similarly to the convergence PDF, all of the above-
mentioned non-Gaussian statistics, which probe deviations from Gaussian
behaviour, are also commonly computed using the convergence maps. However,
these convergence maps are not observed directly and are reconstructed from
the reduced shear maps. The shear maps since its first detection two decades
ago (Bacon et al. 2000; Kaiser et al. 2000; Waerbeke et al. 2000) have been a
powerful cosmological probe. However, due to its spin-2 nature, it is rather
difficult to obtain higher-order summary statistics from shear. While the
convergence maps in principle contain the same information as the shear maps
(Schneider et al. 2002; Shi et al. 2011), the compression of the lensing
signal is greater in convergence maps compared to the shear field, resulting
in 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 ill-defined 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, let us note that due to the non-Gaussian nature of the weak-
lensing signal at small scales, employing mass-inversion methods with
smoothing or de-noising for regularization may not be optimal (Starck et al.
2021; Jeffrey et al. 2020).

The objective of this paper is to present, for the first time, the prediction
of the wavelet $\ell_{1}$-norm derived from theoretical predictions of the PDF
of convergence maps. The paper is organized as follows: In Sect. 2.1, we begin
by revisiting weak lensing convergence, followed by an introduction to the
wavelets and wavelet $\ell_{1}$-norm in Sect. 2. This is followed by the
introduction of the LDT formalism for the $\kappa$-PDF and extending this to
the wavelet coefficients, in Sect. 3. Section 3.3 delves into the derivation
of this wavelet $\ell_{1}$-norm for the wavelet coefficients from theory.
Subsequently, we present and discuss the results in Sect. 4, concluding with
the summarizing of our findings in Sect. 5.

##  2 Wavelet $\ell_{1}$-norm: definition and measurements

We introduce in this section first the expression for the convergence maps and
move to a more general definition of the wavelets and the wavelet
$\ell_{1}$-norm and see how it is related to the PDF.

###  2.1 Convergence maps

Let us start with the expression for convergence, which is given by a
projection of density along the comoving coordinates, weighted by a lensing
kernel involving the comoving distances. It is given by (Mellier 1999)

$\kappa(\bm{\vartheta})=\int_{0}^{\chi_{s}}{\rm
d}\chi\omega(\chi,\chi_{s})\delta(\chi,\mathcal{D}\bm{\vartheta}),$ (1)

where $\chi$ is the comoving radial distance – $\chi_{s}$ the radial distance
of the source – that depends on the cosmological model, and $\mathcal{D}$ is
the comoving angular distance given by

$\mathcal{D}(\chi)\equiv\left\\{\begin{aligned}
{\frac{\sin(\sqrt{K}\chi)}{\sqrt{K}}}{\text{ for }K>0}\\\ {\chi\qquad}{\text{
for }K=0}\\\ {\frac{\sinh(\sqrt{-K}\chi)}{\sqrt{-K}}}{\text{ for
}K<0}\end{aligned}\right.,$ (2)

with $K$ the constant space curvature. The lensing kernel $\omega$ is defined
as

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

where, $c$ is the speed of light, $\Omega_{m}$ the matter density parameter
and $H_{0}$ is the values of bubble constant at redshift $z=0$. The shear
field $\gamma(\theta)$ which is interpretable directly from the observation is
related to the weak lensing convergence $\mathcal{\kappa}$ through mass
inversion (Starck et al. 2006; Martinet et al. 2015). The convergence mass
maps are in principle constructed only up to a mass sheet degeneracy. However,
constructing a mass map by convolving the $\kappa$ 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 $\kappa$ into different
maps called wavelet coefficients $w_{\theta_{j}}$, i.e. ${\cal
W}\kappa=\left\\{w_{\theta_{1}},\dots,w_{\theta_{j}},\dots
w_{\theta_{J}}\right\\}$ where $\theta_{j}$ is the angular scale, and
$\theta_{J}$ 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 $\Upsilon$, and noting
$\Upsilon_{\theta_{j},x,y}(m,n)=\Upsilon\left(\frac{m-x}{\theta_{j}},\frac{n-y}{\theta_{j}}\right)$
the dilated wavelet function at scale $\theta_{j}$ and at spatial location
$(x,y)$, the wavelet coefficients $w_{\theta_{j}}$ are computed as the inner
products

$\displaystyle w_{\theta_{j}}(x,y)$ $\displaystyle=$
$\displaystyle<\kappa,\Upsilon_{\theta_{j},x,y}>$ $\displaystyle=$
$\displaystyle\sum_{m}\sum_{n}\kappa(m,n)\Upsilon\left(\frac{m-x}{\theta_{j}},\frac{n-y}{\theta_{j}}\right),$

The wavelet function $\Upsilon$ is often chosen to be derived from the
difference of two resolutions, for example as
$\Upsilon(x,y)=4\xi(2x,2y)-\xi(x,y)$ as utilized in the starlet wavelet
transform (Starck et al. 2015), where the function $\xi$ is called the scaling
function, typically a low-pass 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
$\xi_{\theta_{j},x,y}(m,n)=\xi\left(\frac{m-x}{\theta_{j}},\frac{n-y}{\theta_{j}}\right)$.
Employing dyadic scales, i.e. $\theta_{j}=2^{j-1}\theta_{1}$, 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 localized in both the spatial and
the Fourier domains, thus suitable for analysing the lensing signal’s
structures at various scales. An important advantage of wavelet analysis,
compared to a standard multi-resolution 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 peak count analysis was almost diagonal, and
neglecting the off-diagonal terms has little impact on the cosmological
parameters estimation. This would not be the case with a multi-scale Gaussian
analysis. Other advantages are that very fast algorithms exist allowing us to
compute all scales with a low complexity, and also that wavelet functions are
exactly equivalent to traditional weak-lensing aperture mass functions which
have been used for decades (Leonard et al. 2012) (aperture mass is just the
convergence/shear within a compensated filters, 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 $\ell_{1}$-norm of the wavelet scales is very efficient in
constraining cosmological parameters. In Ajani et al. (2023), a toy model was
utilized, incorporating a mock source catalogue for weak lensing and a mock
lens catalogue for galaxy clustering sourced from the cosmo-SLICS ((Harnois-
Déraps et al. 2019)) simulations to mimic the KiDS-1000 data survey properties
described in Giblin et al. (2018) and Hildebrandt et al. (2021). The study
examined forecasts concerning the matter density parameter $\Omega_{m}$, the
matter fluctuation amplitude $\sigma_{8}$, the dark energy equation of state
$w_{0}$, and the reduced Hubble constant $h$. It was observed that the wavelet
$\ell_{1}$-norm demonstrated superior performance compared to peaks, multi-
scale peaks, or a combination thereof (Ajani et al. 2023). Therefore, we
propose to build the theory for wavelet $\ell_{1}$-norm, which could be used
for constraining the cosmological parameters with upcoming surveys.

###  2.3 Wavelet $\ell_{1}$-norm

To measure the wavelet $\ell_{1}$-norm from a $\kappa$ map, we first obtain
the wavelet scale by convolving the map with the wavelet function, as given in
Eq. 2.2.

We then compute 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 $\\{B_{i}\\}_{1\leq i\leq N}$.

One can then obtain the normalised111In the literature, the common practice
has been to use an un-normalised version of the $\ell_{1}$-norm, but for an
easier comparison with prediction and simulation we choose here to normalize
this quantity. For the sake of simplicity, we will call it $\ell_{1}$-norm in
this paper.  wavelet $\ell_{1}$-norm by extracting the $\ell_{1}$-norm of the
histogram such that,

$\ell_{1}^{\theta_{j},i}=\sum_{k=1}^{\\#coef(S_{\theta_{j},i})}|S_{\theta_{j},i}[k]|\leavevmode\nobreak\
/N/\Delta B,$ (6)

where the set of coefficients at scale ${\theta_{j}}$ and amplitude bin $i$,
$S_{\theta_{j},i}=\left\\{w_{\theta_{j}}\leavevmode\nobreak\
/\leavevmode\nobreak\ \leavevmode\nobreak\
w_{\theta_{j}}(x,y)\in[B_{i},B_{i+1}]\right\\}$, depicts the wavelet
coefficients $w_{\theta_{j}}$ having an amplitude within the bin
$[B_{i},B_{i+1}]$, and the pixel indices are given by $(x,y)$ and $N$ is the
total number of pixels and $\Delta B$ is the uniform bin width. In other
words, for each bin, the number of pixels $k$, that fall in the bin is
collected and the absolute values of these pixels are summed to obtain the
$\ell_{1}$-norm at this bin $i$.

As we already highlighted before, this definition enables 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. It
has the advantage of being a multiscale approach. In addition, the robustness
of $\ell_{1}$-norm statistics is also well established for decades in the
statistical literature (Huber 1987; Giné et al. 2003).

It turns out that the $\ell_{1}$-norm of the wavelet coefficients can be
related directly to the PDF of the wavelet coefficients via the following
relation

$\ell_{1}^{\theta_{j},i}=P(w_{j})^{i}\times|B^{i}|,$ (7)

which states that the $\ell_{1}$-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 $i$.

###  2.4 Gaussian $\ell_{1}$-norm

Let us now look at the anticipated wavelet $\ell_{1}$-norm expected from a
Gaussian distribution. The generic shape of a wavelet $\ell_{1}$-norm obtained
from a Gaussian PDF

$P(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^{2}\right)$
(8)

with mean $\mu=0$ and $\sigma=0.1$ using the relation shown in Eq. (7) is
demonstrated in Fig. 1.

Figure 1: This figure illustrates the PDF and the $\ell_{1}$-norm for a
Gaussian distribution in blue and the non-Gaussian distribution in orange. On
the left, we present the Probability Distribution Function (PDF) for the two
distributions, and on the right, we display the derived $\ell_{1}$-norm of
these PDFs. Let us note that the heights of the peaks are the same for a
Gaussian distribution but this does not hold for a non-Gaussian PDF.

Here, we see that the height of the two peaks obtained for the wavelet
$\ell_{1}$-norm for the Gaussian (blue solid line) distribution are equal as
expected. But, as the Universe evolves, non-linearities develop, which means
that the convergence field $\kappa$ that we are interested in can no longer be
modelled using a Gaussian model. These non-linearities would lead to asymmetry
which would also be evident from the peaks of the wavelet $\ell_{1}$-norm,
which will no longer be corresponding. This is demonstrated in Fig. 1 for a
test case scenario, where we have applied a non-linear exponential
transformation, to the Gaussian field to obtain a non-Gaussian field (orange
solid line) and obtained the PDF and the $\ell_{1}$-norm. Here we would like
to emphasize that the usual way to extract this non-linear information is
either by relying on heavy $\mathcal{N}$-body simulations or by using an
approximation, as is done in (Tessore et al. 2023). But, thanks to LDT, we now
have a theoretically motivated way to obtain the PDF in the mildly non-linear
regime, using which we will show in this paper that we can also obtain the
wavelet $\ell_{1}$-norm in the mildly non-linear regime.

##  3 Large Deviation Theory formalism

Let us now, look at the application of 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 undergoes variation. To see how to derive the wavelet
$\ell_{1}$-norm based on the LDT formalism, we first need to recap how to
obtain the PDF from the LDT formalism.

###  3.1 LDT for the matter field

The application of LDT to the field 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. 2014).
Barthelemy et al. (2020) then provided an LDT-based prediction for the top-
hat-filtered weak-lensing 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 for the interested readers. In this section, we simply recall the
main equation that will then be needed for us to derive the wavelet
$\ell_{1}$-norm prediction.

Within the context of the application of LDT in Cosmology, there are three
main steps involved. The first is the derivation of the rate function, which
is obtained directly from the first principles of Cosmology. Using the
contraction principle in the LDT formalism, we can connect the statistics of
the late-time non-linear densities to the earlier time density field as soon
as 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, …).

In particular, the result for convergence reads

$P(\kappa)=\int_{-i\infty}^{+i\infty}\frac{d\lambda}{2\pi
i}\exp(-\lambda\kappa+\phi_{\kappa,\theta}(\lambda))$ (9)

where, $P(\kappa)$ is the PDF of the convergence field $\kappa$, and
$\phi_{\kappa,\theta}$ is the cumulant generating function (CGF) of the field
$\kappa$ at angular scale $\theta$. The derivation of the CGF in the above
equation and PDF from there is explained in more detail in Appendix A.

###  3.2 Extending to wavelet coefficients

From the PDF, we can now compute the $\ell_{1}$-norm. As was given in the
Sect. 2, a wavelet can be written as a function of scaling functions. Though
various wavelet filters exists in literature, in this study, we employ a
function of concentric disks to construct the wavelet filter
$\Upsilon_{\theta_{1}}$.

The scaling function $\xi$ is then given by a spherical top hat filter and is
given by

$\hat{\xi}_{\theta_{i}}=2J_{1}(\theta_{i}l)/(\theta_{i}l)$ (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 (black solid line) that is used in this work, which is
given as a function of two scaling functions (orange and blue solid lines) at
different scales.

Figure 2: The compensated filter (depicted by the black solid line) is derived
through a difference of two top-hat filters at different scales, as described
in Eq. (5) and Eq. (10). The blue and orange solid lines represent the
individual spherical filters obtained at radii $\theta_{1}$ and
$\theta_{2}=2\theta_{1}$, respectively. For visualization purposes, the
compensated filter is multiplied by $-1$.

We now apply the LDT formalism described previously to the wavelet
coefficients described in Eq. (5) and obtain the PDF of the wavelet
coefficients ($w_{\theta_{1}}$) by using the scales
$\theta_{1},\theta_{2}=2\theta_{1}$ as

$P(w_{\theta_{1}})=\int_{-i\infty}^{+i\infty}\frac{d\lambda}{2\pi
i}\exp\left(-\lambda w_{\theta_{1}}+\phi_{w_{\theta_{1}}}(\lambda)\right).$
(11)

We refer the readers to Appendix A, for more details on the derivation of the
CGF in the above equation and the resulting PDF. We emphasize that
$w_{\theta_{1}}$ is essentially the standard aperture mass aperture mass
defined on the convergence field using a difference of two top-hat filters, as
was discussed previously.

###  3.3 Wavelet $\ell_{1}$-norm predicted via Large Deviation Theory

In the previous sections, we saw how the LDT formalism can be applied to
derive the PDF of the wavelet coefficients. We will now see how to derive the
wavelet $\ell_{1}$-norm from the PDF of the wavelet coefficients
$P(w_{\theta_{1}})$. This wavelet $\ell_{1}$-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 it
corresponds to, as is given in Equation (7). Extending this to the PDF of the
wavelet coefficients from the LDT formalism, we get the final equation for bin
$i$

$\ell_{1}^{i}=P^{i}(w_{\theta_{1}})\times|w_{\theta_{1}}^{i}|.$ (12)

As a first approximation, one can assume that the PDF is Gaussian, which is
the case when we consider large scale or early time in the standard model of
Cosmology. In the subsequent section, we look at obtaining the $\ell_{1}$-norm
for a Gaussian case first and then highlight the motivation to go beyond
Gaussian models.

##  4 Confronting $\ell_{1}$-norm prediction to simulations

###  4.1 Measurements from simulation

To demonstrate the accuracy of the theory for realistic surveys, we compare
our prediction results with full-sky simulations of Takahashi et al. (2017) to
avoid additional errors from small-scale patch sizes. The simulations of
Takahashi et al. (2017) provides full sky lensing maps at a fixed cosmology
222http://cosmo.phys.hirosaki-u.ac.jp/takahasi/allsky_raytracing/ each for two
grid resolutions: $4096^{2}$ and $8192^{2}$. The data sets include the full
sky maps at intervals of $150h^{-1}$ comoving radial distance from redshifts
$z=0.05$ to $5.3$. We use the simulated convergence maps that are directly
provided as products of these simulations. The smoothing of the convergence
maps is implemented as a direct convolution of the convergence map with the
wavelet filter at a given angular scale. Once the filtered field is obtained,
we measure the sum of the absolute values of the pixels of the smoothed
$\kappa$ mass map in linearly spaced bins of the range of $\kappa$ to obtain
the wavelet $\ell_{1}$-norm.

###  4.2 Obtaining the prediction

To obtain the prediction of the wavelet $\ell_{1}$-norm for each of the
cosmologies used here, we first use CAMB (Lewis & Challinor 2011) to get the
linear and non-linear(Halofit takahashi (Takahashi et al. 2012)) matter power
spectra. We also use CAMB to obtain the comoving radial distances $\chi(z)$.
The density CGF (Eq. (24)) is calculated for redshift slices between $z=0$ and
the source redshift $z_{s}$ using the full Halofit power spectrum as input.
These projected CGFs are then rescaled by the measured variance
$\sigma^{2}_{M_{ap},sim}$ (see Appendix A for more details), before going
through an inverse Laplace transform to get the final PDF. This predicted PDF
is used to derive the predicted wavelet $\ell_{1}$-norm as given in Eq. (12).

###  4.3 Validating the prediction with simulation

In Figures 3 and 4, we show the predicted wavelet $\ell_{1}$-norm and the one
measured from the simulated convergence map for different source redshifts
$z_{s}\approx 1.2,1.4,2.0$ at
$\theta=$20\text{\,}\mathrm{\SIUnitSymbolArcminute}$$ and for different scales
$\theta=$15\text{\,}\mathrm{\SIUnitSymbolArcminute}$,$18\text{\,}\mathrm{\SIUnitSymbolArcminute}$,$20\text{\,}\mathrm{\SIUnitSymbolArcminute}$$
at source redshift $z_{s}=1.423$ respectively. On the bottom panels, we show
the residuals. For comparison, we display the PDF and the residuals of a
Gaussian distribution (dash-dot line) with the same mean and variance as the
simulation. The Gaussian PDF is obtained using Eq. (8), and the
$\ell_{1}$-norm is derived as explained previously in Eq. (12). In both plots,
the $x$-axis for the residuals is scaled by the standard deviation. The shaded
regions in the colors correspond to the $3\sigma$ region obtained from the
error bars.

The error bars are obtained by taking the standard deviation of the values of
the wavelet $\ell_{1}$-norm for 10 different patches of the full sky Takahashi
map in each of the bins.

Let us first concentrate on the Gaussian curves. We see a clear mismatch with
the Gaussian prediction as expected since the measurements show a clear
asymmetry unless the Gaussian model which by definition is symmetric.

Let us now focus on the LDT prediction that goes beyond the Gaussian regime.
In this case, we observe a better agreement between the simulation and the
predicted wavelet $\ell_{1}$-norm results to percent levels (which corresponds
to the simulation error bars in shaded area) within the $2-\sigma$ region
around zero on the $x$-axis. This demonstrates the predictive accuracy of the
results derived from the LDT formalism. It is important to note that this
$2-\sigma$ specification refers to the range on the $x$-axis, rather than
representing error bars around the mean signal value.

In these plots, we indeed clearly see that LDT is capable of capturing the
asymmetry of the two peaks hence valuable information about the non-
gaussianities. 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 ($\sigma^{2}$) and reduced skewness
($S_{3}$) which, given that the mean convergence is zero, are obtained as

$\displaystyle\sigma^{2}$ $\displaystyle=E\left[\kappa^{2}\right]$ (13)
$\displaystyle S_{3}$ $\displaystyle=\frac{E[\kappa^{3}]}{\sigma^{4}}$ (14)

where,

$E[\kappa^{n}]=\int P(\kappa)\kappa^{n}d\kappa.$ (15)

Table 1: The cumulants (standard deviation $\sigma$, skewness $S_{3}$)
obtained from the PDF from the LDT prediction and the values obtained from the
Takahashi simulation at source redshift $z_{s}\approx 2.05$ for scales 15, 18,
and 20 arcminutes. The corresponding PDFs are shown in Figure 4
$\begin{array}[]{|p{0.25\linewidth}|l|c|c|c|r|}\hline\cr Data&\sigma^{2}\times
10^{-5}&S_{3}\\\ \hline\cr Prediction 15'&2.92&-26.3\\\ \hline\cr Simulation
15'&2.92&-42.8\\\ \hline\cr Prediction 18'&2.61&-25.94\\\ \hline\cr Simulation
18'&2.61&-38.2\\\ \hline\cr Prediction 20'&2.44&-25.7\\\ \hline\cr Simulation
20'&2.44&-35.4\\\ \hline\cr\end{array}$

Figure 3: Top panel: Predicted (solid) $\ell_{1}$-norm as compared to
measurements in simulation (dot markers) for an inner radius
$\theta_{1}=$20\text{\,}\mathrm{\SIUnitSymbolArcminute}$$ and different source
redshifts $z_{s}=1.21,1.43,2.05$, displayed with blue, orange, and green lines
respectively. Red dash-dotted lines show the Gaussian prediction for
reference. The vertical dotted and dot-dash lines correspond to the $1\sigma$
and $2\sigma$ 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 dash-dotted plots illustrate the
residual of the $\ell_{1}$-norm derived from the Gaussian PDF with the same
mean and variance as the simulation PDF. The shaded region indicates the
$3\sigma$ region around the error bars for each redshift. The prediction is in
good agreement with the measurements up to approximately 2 sigma and remains
within percent levels.

Figure 4: Top panel: Predicted (solid) $\ell_{1}$-norm as compared to
measurements in simulation (dot markers)for different inner radii
$\theta_{1}=$15\text{\,}\mathrm{\SIUnitSymbolArcminute}$,$18\text{\,}\mathrm{\SIUnitSymbolArcminute}$,$20\text{\,}\mathrm{\SIUnitSymbolArcminute}$$
and single source redshift $z_{s}=1.43$, displayed with blue, orange, and
green lines respectively. Red dash-dotted lines show the Gaussian prediction
for reference. The vertical dotted and dot-dash lines correspond to the
$1\sigma$ and $2\sigma$ 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 dash-dotted plots
illustrate the residual of the $\ell_{1}$-norm derived from the Gaussian PDF
with the same mean and variance as the simulation PDF. The shaded region
indicates the $3\sigma$ region around the error bars for each inner radius.
The prediction is in good agreement with the measurements up to approximately
2 sigma and remains within percent levels.

From Table 1, we see that the variance from the prediction and simulation for
different scales match perfectly, which we expect given the re-scaling by the
power spectrum that we perform (see appendix A). As expected from the
formalism, we see that the reduced skewness $(S_{3})$ increases with
decreasing smoothing scale. We also notice that the $\ell_{1}$-norm for the
positive bins is not at all equal to the negative bins, suggesting how
prominent the non-Gaussian features of the mass map are in this regime. It was
shown in Barthelemy et al. (2020), that the theoretical precision achievable
is constrained by the mixing of scales when integrating the density field
along the line of sight since highly non-linear scales are not precisely
captured by the spherical collapse formalism. Consequently, weak lensing
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). That is also evident when we look at Fig. 3 and Fig. 4,
where we do see that the level of accuracy increases as we move to higher
redshifts and higher smoothing scales. To bypass this issue, a rising approach
is to use the so-called Bernardeau-Nishimichi-Taruya (BNT) transform
(Bernardeau et al. 2014), which enables more accurate theoretical predictions
by narrowing the range of physical scales contributing to the signal.

###  4.4 Cosmology dependence of PDF and wavelet $\ell_{1}$-norm

In this section, we explore the cosmological dependence of the predicted pdf
of the wavelet coefficients and $\ell_{1}$-norm. We would like to emphasize
that since the wavelet decomposition is analogous to using an aperture mass
map filtering, we henceforth can also call the PDF of the wavelet coefficients
as the aperture mass PDF From the equation of the rate function given in Eq.
(22), the cosmology dependence of the aperture mass PDF in LDT enters through
the scale-dependence of, the non-linear power-spectrum, 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 would also be
cosmology-dependent. In the context of applying LDT to predict the wavelet
$\ell_{1}$-norm and PDFs, based on the equations introduced above, we could
assume that it should be able to efficiently capture the cosmology dependence.
Indeed, the cosmology dependence of the one- scale one-point convergence PDF
has been studied in detail in Boyle et al. (2021) which showed that LDT
captures very accurately the cosmology dependence. In this work, we extend
this study for the case of the PDF of the wavelet coefficients and the wavelet
$\ell_{1}$-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,
since the wavelet $\ell_{1}$-norm is directly dependent on the PDF, we could
naively expect it to depend on the cosmology too, and encapsulate the
dependence.

Since we are interested in obtaining the derivatives with respect to
cosmological parameters, we need a simulation suite that is available for
different cosmologies. For this, we use the Cosmological Massive Neutrino
simulations (MassiveNus) (Liu et al. 2018) suite to see how well the theory
can capture the dependence on cosmology. These simulations are released by the
Columbia Lensing Group 333http://columbialensing.org/. The MassiveNuS
simulations encompass 101 different cosmological models at source redshifts
$z_{s}={0.5,1.0,1.5,2.0,2.5}$, by varying three parameters: the neutrino mass
sum $M_{\nu}$, the total matter density $\Omega_{m}$, and the primordial power
spectrum amplitude $A_{s}$. For each redshift, there are 10,000 distinct map
realizations generated through random rotation and translation of the initial
n-body box then stitched together to reconstruct pseudo-independent light
cones that are unlikely to cross the same structures. Each $\kappa$ map
contains $512^{2}$ pixels, covering a total angular area of $12.25$ square
degrees, spanning a range of $\ell\in[100,37,000]$ with a pixel resolution of
0.4 arcminutes. The measurement of the wavelet $\ell_{1}$-norm is obtained
using the same process as was described previously in Sec. 4.1. Additionally,
the measurement of the PDF is derived from the histogram of the binned and
smoothed map. To obtain the derivatives, we use the model with parameters
$\Omega_{m}=0.30$, $A_{s}=2.1\times 10^{-9}$, and $M_{\nu}=0.0$ eV as the
fiducial model (model 1b in Liu et al. (2018)) and obtain 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 to 6. For the derivative
with respect to $M_{\nu}$, we employ a model with the same $\Omega_{m}$ and
$A_{s}$ values but with $M_{\nu}=0.0,\text{eV}$ and then compute 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 $\ell$, which does influence us much because of the
scales we are looking at, and at low $\ell$ which invites caution when
choosing our largest scales. We typically try to maintain a factor of 20
between the physical scale of the box and the largest physical scale probed by
our filters.

It is evident from Figures 5 and 6, that the prediction effectively captures
the changes in cosmology, demonstrating good agreement with the results. The
errorbars are obtained by taking the error of mean of 3000 simulations, which
are also used to obtain the mean of the measurements.

Table 2: The cosmological parameters used to obtain the derivatives of the
PDF.
$\begin{array}[]{|p{0.3\linewidth}|l|c|r|}\hline\cr\text{Parameters}&\text{lower
bound}&\text{fiducial}&\text{upper bound}\\\
\hline\cr$\Omega_{m}$&0.28&0.3&0.31\\\ $A_{s}\times 10^{-9}$&2.0&2.1&2.17\\\
$M_{\nu}$&0.09&0.0&0.11\\\ \hline\cr\end{array}$

Figure 5: Derivative of the PDF with respect to $M_{\nu}$ (blue), $\Omega_{m}$
(orange) and $A_{s}$ (green). The solid lines show the derivatives obtained
from prediction and the dotted lines show the derivatives from simulation. It
is obtained at source redshift $z_{s}=2$ and inner radius
$\theta_{1}=$22.5\text{\,}\mathrm{\SIUnitSymbolArcminute}$$. The results for
simulation are obtained from the MassiveNus simulation suite, by averaging the
results over 10,000 simulations.

Figure 6: Derivative of the wavelet $\ell_{1}$-norm with respect to $M_{\nu}$
(blue), $\Omega_{m}$ (orange) and $A_{s}$ (green). The solid lines show the
derivatives obtained from prediction, and the dotted lines show the
derivatives from simulation. The predicted wavelet $\ell_{1}$-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
$\theta_{1}=$22.5\text{\,}\mathrm{\SIUnitSymbolArcminute}$$. The results for
simulation are obtained from the MassiveNus simulation suite, by averaging the
results over 10,000 simulations.

As demonstrated in Boyle et al. (2021) for the case of a 1-cell PDF, our
results here demonstrate that the theory captures well the effects of adding
massive neutrinos to cosmology, extending the results also for the PDF of the
wavelet coefficients.

###  4.5 Reproducible research

In the spirit of open research, the code related to reproducing the plots in
this paper is available at github. A part of this Python code that computes
the aperture mass/wavelet coefficient PDF was inspired by the public
Mathematica code L2DT used in Barthelemy et al. (2021).

##  5 Discussion and Conclusions

In this work, we have used the LDT to build upon the previous work by
Barthelemy et al. (2021) and introduced a formalism to predict the
$\ell_{1}$-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 non-linear regime. The approach of Barthelemy et al. (2021)
incorporates the geometric and time-evolution 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 enables us to treat redshift slices as statistically
independent. It was found that the most likely non-linear dynamics of the
matter density field filtered in concentric disks can be well approximated for
small variances by the cylindrical collapse model, thus allowing for LDT to be
applied in this context. From there, because the wavelet $\ell_{1}$-norm can
be directly related to the multi-scale one-point statistics of the
convergence, we have shown in this paper that a theory-based prediction for
the wavelet $\ell_{1}$-norm is tractable.

The robust validation of our predictions against the simulations, in Figs. 4
and 3, highlighted the reliability and applicability of our theoretical
framework. Specifically, our theory aligns with the simulations within percent
level accuracy across the examined range of source redshifts $z_{s}{\approx
1-2}$ and angular scales $\theta_{1}{\approx 15-20}$. This quantitative
assessment affirms the predictive precision of our model. Obviously going
towards smaller scales or lower redshifts would lead to a larger departure
between simulations and theory.

Figure 6, further emphasizes the cosmological dependencies and the very good
performance of the theoretical prediction, when compared with that of the
simulation. Since the simulations have validated the theoretical predictions,
it suggests that our model could be applied to investigate a broad range of
cosmological parameters, while bypassing the need for high computing resources
and providing a theoretical tool for cosmology inference that is robust and
fast, without the 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 estimations when using the simulation data. The
use of a theory-based approach has the added benefit of reducing the
artificial biases in 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 non-Gaussian information, this also pushes the model
deeper into the nonlinear regime. This effect can be mitigated by adjusting
the smoothing scale accordingly. We would also like to emphasize the
usefulness of the wavelet coefficients of the convergence maps, instead of the
convergence maps or the shear maps directly. Convergence maps already reduce
computation expense, because of the presence of 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 multi-scale analysis method, which is
well suited to constrain 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 (non-linear) scales that is
inherent to such quantities projected along the past light cone and makes
standard perturbative approaches inaccurate even on relatively large angular
scales. (Bernardeau et al. 2014) To overcome this issue, an emerging method is
to utilize the BNT transform which allows for more precise theoretical
predictions by reducing the range of physical scales that contribute to the
signal. This could be applied in our context and enables even more accurate
predictions for analysing tomographic surveys. Investigating the performance
of this approach is left for future works, together with the impact of
systematics that would require 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 $\ell_{1}$-norm summary statistic proposed here presents
several advantages in the realm of cosmological parameter inference: i) it
provides a fast ($\sim 1$ minute) calculation; ii) it does not rely on a heavy
numerical simulation; iii) it captures well the cosmology dependence; and iv)
wavelet $\ell_{1}$-norm could lead to competitive or even tighter constraints
as demonstrated by previous works. We hence have at hand a promising method
that could enable a comprehensive multi-scale analysis of cosmic shear
datasets in the mildly non-linear regime.

###### 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 (ANR-22-CE31-0014-01 TOSCA and
ANR-18-CE31-0009 SPHERES). AB’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 AST-1210877, NSF AST-140041, and NASA ATP-80NSSC18K1093. 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.

## References

  * Ajani et al. (2023) Ajani, V., Harnois-Déraps, J., Pettorino, V., & Starck, J.-L. 2023, A&A, 672, L10
  * Ajani et al. (2020) Ajani, V., Peel, A., Pettorino, V., et al. 2020, Phys. Rev. D, 102, 103531
  * Ajani et al. (2021) Ajani, V., Starck, J.-L., & Pettorino, V. 2021, A&A, 645, L11
  * Bacon et al. (2000) Bacon, D. J., Refregier, A. R., & Ellis, R. S. 2000, MNRAS, 318, 625
  * Bartelmann & Maturi (2017) Bartelmann, M. & Maturi, M. 2017, Scholarpedia, 12, 32440
  * Barthelemy et al. (2021) Barthelemy, A., Codis, S., & Bernardeau, F. 2021, MNRAS, 503, 5204
  * Barthelemy et al. (2020) Barthelemy, A., Codis, S., Uhlemann, C., Bernardeau, F., & Gavazzi, R. 2020, MNRAS, 492, 3420
  * Bernardeau et al. (2002) Bernardeau, F., Colombi, S., Gaztañaga, E., & Scoccimarro, R. 2002, Physics Reports, 367, 1–248
  * Bernardeau et al. (2014) Bernardeau, F., Nishimichi, T., & Taruya, A. 2014, MNRAS, 445, 1526
  * Bernardeau et al. (2014) Bernardeau, F., Pichon, C., & Codis, S. 2014, Phys. Rev. D, 90, 103519
  * Bernardeau & Reimberg (2016) Bernardeau, F. & Reimberg, P. 2016, Phys. Rev. D, 94, 063520
  * Bernardeau & Valageas (2000) Bernardeau, F. & Valageas, P. 2000, A&A, 364, 1
  * Bernardeau & Valageas (2001) Bernardeau, F. & Valageas, P. 2001, Construction of the one-point PDF of the local aperture mass in weak lensing maps
  * Boyle et al. (2021) Boyle, A., Uhlemann, C., Friedrich, O., et al. 2021, MNRAS, 505, 2886
  * Cheng & Ménard (2021) Cheng, S. & Ménard, B. 2021, MNRAS, 507, 1012
  * Euclid Collaboration et al. (2024) Euclid Collaboration, Deshpande, A. C., Kitching, T., et al. 2024, A&A, 684, A138
  * Fageot et al. (2014) Fageot, J., Bostan, E., & Unser, M. 2014, in 2014 IEEE International Conference on Image Processing (ICIP), 6096–6100
  * Gatti et al. (2020) Gatti, M., Chang, C., Friedrich, O., et al. 2020, MNRAS, 498, 4060
  * Giblin et al. (2018) Giblin, B., Heymans, C., Harnois-Déraps, J., et al. 2018, MNRAS, 480, 5529
  * Giné et al. (2003) Giné, E., Mason, D. M., & Zaitsev, A. Y. 2003, The Annals of Probability, 31, 719
  * Harnois-Déraps et al. (2019) Harnois-Déraps, J., Giblin, B., & Joachimi, B. 2019, A&A, 631, A160
  * Heymans et al. (2012) Heymans, C., Van Waerbeke, L., Miller, L., et al. 2012, MNRAS, 427, 146
  * Hildebrandt et al. (2021) Hildebrandt, H., van den Busch, J. L., Wright, A. H., et al. 2021, A&A, 647, A124
  * Hildebrandt et al. (2017) Hildebrandt, H., Viola, M., Heymans, C., et al. 2017, MNRAS, 465, 1454
  * Huber (1987) Huber, P. J. 1987, Computational statistics & data Analysis, 5, 255
  * Huterer (2010) Huterer, D. 2010, General Relativity and Gravitation, 42, 2177–2195
  * Ingoglia et al. (2022) Ingoglia, L., Covone, G., Sereno, M., et al. 2022, MNRAS, 511, 1484
  * Ivezić et al. (2019) Ivezić, Ž., Kahn, S. M., Tyson, J. A., et al. 2019, ApJ, 873, 111
  * Jeffrey et al. (2020) Jeffrey, N., Lanusse, F., Lahav, O., & Starck, J.-L. 2020, MNRAS, 492, 5023
  * Kaiser et al. (2000) Kaiser, N., Wilson, G., & Luppino, G. A. 2000, Large-Scale Cosmic Shear Measurements
  * Kilbinger (2015) Kilbinger, M. 2015, Reports on Progress in Physics, 78, 086901
  * Kilbinger (2018) Kilbinger, M. 2018, Cosmological parameters from weak cosmological lensing
  * Kratochvil et al. (2012) Kratochvil, J. M., Lim, E. A., Wang, S., et al. 2012, Phys. Rev. D, 85, 103513
  * Kruse & Schneider (1999) Kruse, G. & Schneider, P. 1999, MNRAS, 302, 821
  * Laureijs et al. (2011) Laureijs, R., Amiaux, J., Arduini, S., et al. 2011, Euclid Definition Study Report
  * Leonard et al. (2012) Leonard, A., Pires, S., & Starck, J.-L. 2012, MNRAS, 423, 3405–3412
  * Lesgourgues & Pastor (2012) Lesgourgues, J. & Pastor, S. 2012, Advances in High Energy Physics, 2012, 1–34
  * Lewis & Challinor (2011) Lewis, A. & Challinor, A. 2011, CAMB: Code for Anisotropies in the Microwave Background, Astrophysics Source Code Library, record ascl:1102.026
  * Li et al. (2019) Li, Z., Liu, J., Zorrilla Matilla, J. M., & Coulton, W. R. 2019, Phys. Rev. D, 99, 063527
  * Lin & Kilbinger (2015) Lin, C.-A. & Kilbinger, M. 2015, A&A, 583, A70
  * Liu et al. (2018) Liu, J., Bird, S., Matilla, J. M. Z., et al. 2018, Journal of Cosmology and Astroparticle Physics, 2018, 049
  * Liu & Madhavacheril (2019) Liu, J. & Madhavacheril, M. S. 2019, Phys. Rev. D, 99, 083508
  * Liu et al. (2015a) Liu, J., Petri, A., Haiman, Z., et al. 2015a, Phys. Rev. D, 91, 063507
  * Liu et al. (2015b) Liu, X., Pan, C., Li, R., et al. 2015b, MNRAS, 450, 2888
  * Loureiro et al. (2023) Loureiro, A., Whiteaway, L., Sellentin, E., et al. 2023, The Open Journal of Astrophysics, 6
  * Mandelbaum (2018) Mandelbaum, R. 2018, Annual Review of Astronomy and Astrophysics, 56, 393–433
  * Mandelbaum & Collaboration (2017) Mandelbaum, R. & Collaboration, H. S.-C. H. 2017, in American Astronomical Society Meeting Abstracts# 229, Vol. 229, 226–02
  * Martinet et al. (2015) Martinet, N., Bartlett, J. G., Kiessling, A., & Sartoris, B. 2015, A&A, 581, A101
  * Mead et al. (2021) Mead, A. J., Brieden, S., Tröster, T., & Heymans, C. 2021, MNRAS, 502, 1401
  * Mellier (1999) Mellier, Y. 1999, ARA&A, 37, 127
  * Mukhanov (2005) Mukhanov, V. 2005, Physical Foundations of Cosmology (Cambridge: Cambridge Univ. Press)
  * Munshi et al. (2008) Munshi, D., Valageas, P., van Waerbeke, L., & Heavens, A. 2008, Phys. Rep, 462, 67
  * Parroni et al. (2020) Parroni, C., Cardone, V. F., Maoli, R., & Scaramella, R. 2020, A&A, 633, A71
  * Peebles (1980) Peebles, P. J. E. 1980, The large-scale structure of the universe
  * Peel et al. (2017) Peel, A., Lin, C.-A., Lanusse, F., et al. 2017, A&A, 599, A79
  * Peel et al. (2018) Peel, A., Pettorino, V., Giocoli, C., Starck, J.-L., & Baldi, M. 2018, A&A, 619, A38
  * Petri (2016) Petri, A. 2016, Astronomy and Computing, 17, 73
  * Reimberg & Bernardeau (2018) Reimberg, P. & Bernardeau, F. 2018, Phys. Rev. D, 97
  * Rizzato et al. (2019) Rizzato, M., Benabed, K., Bernardeau, F., & Lacasa, F. 2019, MNRAS, 490, 4688
  * Schneider et al. (2002) Schneider, P., van Waerbeke, L., Kilbinger, M., & Mellier, Y. 2002, A&A, 396, 1
  * Semboloni et al. (2011) Semboloni, E., Schrabback, T., van Waerbeke, L., et al. 2011, MNRAS, 410, 143
  * Shi et al. (2011) Shi, X., Schneider, P., & Joachimi, B. 2011, A&A, 533, A48
  * Starck et al. (2015) Starck, J.-L., Murtagh, F., & Fadili, J. 2015, Sparse image and signal processing: Wavelets and related geometric multiscale analysis, second edition, 1–423
  * Starck et al. (2006) Starck, J. L., Pires, S., & Réfrégier, A. 2006, A&A, 451, 1139
  * Starck et al. (2021) Starck, J. L., Themelis, K. E., Jeffrey, N., Peel, A., & Lanusse, F. 2021, A&A, 649, A99
  * Takada & Jain (2004) Takada, M. & Jain, B. 2004, MNRAS, 348, 897
  * Takahashi et al. (2017) Takahashi, R., Hamana, T., Shirasaki, M., et al. 2017, The Astrophysical Journal, 850, 24
  * Takahashi et al. (2012) Takahashi, R., Sato, M., Nishimichi, T., Taruya, A., & Oguri, M. 2012, ApJ, 761, 152
  * Takahashi et al. (2012) Takahashi, R., Sato, M., Nishimichi, T., Taruya, A., & Oguri, M. 2012, The Astrophysical Journal, 761, 152
  * Tessore et al. (2023) Tessore, N., Loureiro, A., Joachimi, B., von Wietersheim-Kramsta, M., & Jeffrey, N. 2023, The Open Journal of Astrophysics, 6
  * Touchette (2009) Touchette, H. 2009, Physics Reports, 478, 1
  * Troxel & Ishak (2015) Troxel, M. & Ishak, M. 2015, Physics Reports, 558, 1, the intrinsic alignment of galaxies and its impact on weak gravitational lensing in an era of precision cosmology
  * Uhlemann et al. (2018) Uhlemann, C., Pichon, C., Codis, S., et al. 2018, MNRAS, 477, 2772
  * Valageas (2002) Valageas, P. 2002, A&A, 382, 412
  * Varadhan (1984) Varadhan, S. R. S. 1984, Large Deviations and Applications (Society for Industrial and Applied Mathematics)
  * Waerbeke et al. (2000) Waerbeke, L. V., Mellier, Y., Erben, T., et al. 2000, Detection of correlated galaxy ellipticities on CFHT data: first evidence for gravitational lensing by large-scale structures
  * Weinberg et al. (2013) Weinberg, D. H., Mortonson, M. J., Eisenstein, D. J., et al. 2013, Physics reports, 530, 87

##  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 non-equilibrium systems.
For a detailed overview, readers are referred to Touchette (2009).

The utilization of 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. (2014)). Barthelemy et al. (2020) provided an LDT-based
prediction for the top-hat-filtered weak-lensing 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 $i$)
$\\{\rho^{\epsilon}_{i}\\}$, $1\leq i\leq N$ with joint PDF
$\mathcal{P}_{\epsilon}(\\{\rho^{\epsilon}_{i}\\})$ is satisfied if the
following limit exists

$\psi_{\\{\rho^{\epsilon}_{i}\\}}(\\{\rho^{\epsilon}_{i}\\})=-\lim_{\epsilon\to
0}\epsilon\log[\mathcal{P}_{\epsilon}(\\{\rho_{i}^{\epsilon}\\})].$ (16)

If such limit exists, $\psi_{\\{\rho^{\epsilon}_{i}\\}}$ is called the rate
function.

Here, $\epsilon$ 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 ${\rho_{i}}$, then Varadhan’s theorem
allows us to connect its Scaled Cumulant Generating Function (SCGF)
$\varphi_{{\rho_{i}}}$ to the rate function $\psi_{{\rho_{i}}}$ through a
Legendre-Fenchel transform

$\varphi_{\\{\rho_{i}\\}}(\\{\lambda_{i}\\})=\sup_{\\{\rho_{i}\\}}\Bigl{[}\sum_{i}\lambda_{i}\rho_{i}-\psi_{\\{\rho_{i}\\}}(\\{\rho_{i}\\})\Bigr{]}.$
(17)

This Legendre-Fenchel transform reduces to Legendre transform if the rate
function is convex

$\varphi_{\\{\rho_{i}\\}}(\\{\lambda_{i}\\})=\sum_{i}\lambda_{i}\rho_{i}-\psi_{\\{\rho_{i}\\}}(\\{\rho_{i}\\}),$
(18)

where $\lambda_{i}$ and $\rho_{i}$ are one by one related via the stationary
condition

$\lambda_{k}=\frac{\partial\psi_{\\{\rho_{i}\\}}(\\{\rho_{i}\\})}{\partial\rho_{k}},\forall
k\in\\{1,\dots,N\\}.$ (19)

Another consequence of the large-deviation principle is the so-called
contraction principle. This principle suggests that when dealing with a set of
random variables ${\tau_{i}}$ following a LDP and connected to ${\rho_{i}}$
through the continuous mapping $f$, the rate function of ${\rho_{i}}$ can be
determined as follows

$\psi_{\\{\rho_{i}\\}}(\\{\rho_{i}\\})=\inf_{\\{\tau_{i}\\}:f(\\{\tau_{i}\\})=\\{\rho_{i}\\}}\psi_{\\{\tau_{i}\\}}(\\{\tau_{i}\\}).$
(20)

In more tangible terms, this statement suggests that an uncommon change in the
behavior of ${\rho_{i}}$ is predominantly influenced by the most probable
variation among all unlikely changes in ${\tau_{i}}$.

In the context of cosmology, if we take $\rho_{k}$ to denote the late-time
densities, then $\bar{\tau_{k}}$, the most likely initial field configuration,
is obtained through the most probable mapping between the linear and late-time
fields. In 2D, this most likely dynamics is given by cylindrical collapse,
which is known to be well-fitted by

$\zeta(\bar{\tau_{k}})=\rho_{k}=\Bigl{(}1-\frac{\bar{\tau_{k}}}{\nu}\Bigr{)}^{-\nu}.$
(21)

The choice of $\nu$ as 1.4 is made to match the calculated value of the third-
order 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
late-time field configuration, we can compute the rate function for the late-
time density field. It is expressed as follows

$\psi({\rho_{i}})=\frac{\sigma^{2}_{R1}}{2}\sum_{k,j}\Xi_{kj}({\tau_{i}})\bar{\tau_{k}}\bar{\tau_{j}}.$
(22)

Here, $\sigma^{2}_{R_{1}}$ is the driving parameter and is determined by the
variance within the largest disk $R_{1}$. $\Xi_{kj}$ 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 top-hat
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. 7. This means that the cumulants can be written as

Figure 7: 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.

$\bigl{<}M_{ap}^{p}\bigr{>}_{c}=\int_{0}^{\chi_{s}}d\chi\omega(\chi,\chi_{s})\bigl{<}(\delta_{<\mathcal{D}(\chi)\theta_{2}}-\delta_{<\mathcal{D}(\chi)\theta_{1}})^{p}\bigr{>}_{c},$
(23)

where
$\delta_{<\mathcal{D}(\chi)\theta_{2}}-\delta_{<\mathcal{D}(\chi)\theta_{1}}$
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 $\chi$. This simplifies the problem greatly, now
having to calculate just the one-point statistics related to the density slope
within each two-dimensional slice 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 non-linear
prescription coming from the Halofit to model the non-linear covariance. In
this paper, we use the Halofit-Takahashi version Takahashi et al. (2012) to
compute the covariances.

Now using Eq. (18), and Eq. (22), and since we need the density slope between
two concentric disks, we get the SCGF as

$\varphi_{\delta_{2}-\delta_{1}}(\lambda)=\varphi_{\delta_{1},\delta_{2}}(-\lambda,\lambda)$
(24)

To ensure that this choice does not lead to any discrepancies with the
numerical simulations, we also rescale the projected CGF given in Eq. (18)
with the measured variance $\sigma_{M_{ap},sim}^{2}$ instead of the one
computed from Halofit $\sigma_{M_{ap},hfit}^{2}$ as is given below

$\phi_{M_{ap}}(\lambda)=\frac{\sigma_{M_{ap},hfit}^{2}}{\sigma_{M_{ap},sim}^{2}}\phi_{M_{ap}}\left(\lambda\frac{\sigma_{M_{ap},sim}^{2}}{\sigma_{M_{ap},hfit}^{2}}\right)$
(25)

Once the CGF of individual slices is obtained, we can use Eq. (23) to get the
CGF (the details of this derivation is given in Barthelemy et al. (2021)) of
the lensing aperture mass as

$\phi_{\kappa,\theta_{1},\theta_{2}}(\lambda)=\int_{0}^{\chi_{s}}d\chi
w^{p}(\chi,\chi_{s})\phi_{\delta
1,\delta_{2},M_{ap}}(w(\chi,\chi_{s})\lambda,\mathcal{D}(\chi)\theta
1,\mathcal{D}(\chi)\theta_{2}).$ (26)

After calculating the CGF, the inverse Laplace transform can be used to obtain
the convergence PDF

$P(\kappa)=\int_{-i\infty}^{+i\infty}\frac{d\lambda}{2\pi
i}\exp(-\lambda\kappa+\phi_{\kappa,\theta}(\lambda)).$ (27)

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 $\ell_{1}$-norm
using the Eq. (12) as demonstrated in the Sect. 3.3.