Launched in 2023, the Euclid satellite is surveying the sky in optical and infrared wavelengths to create an unprecedented map of the Universe's large-scale structure. A cornerstone of its mission is the measurement of weak gravitational lensing—subtle distortions in the shapes of distant galaxies. This phenomenon is a powerful cosmological probe, capable of tracing the evolution of dark matter and helping to distinguish between dark energy and modified gravity theories.
Traditionally, cosmologists have analyzed weak lensing data using second-order statistics (like the power spectrum) paired with a Gaussian likelihood model. This established approach, however, faces significant challenges:
- Loss of Information: Second-order statistics fully capture information only if the underlying matter distribution is Gaussian. In reality, the cosmic web is highly structured, with clusters, filaments, and voids, making this approach inherently lossy.
- Complex Covariance: The method requires estimating a covariance matrix, which is both cosmology-dependent and non-Gaussian. This necessitates running thousands of computationally intensive N-body simulations for each model, a massive and often impractical undertaking.
- Systematic Errors: Incorporating real-world complications—such as survey masks, intrinsic galaxy alignments, and baryonic feedback—into this framework is notoriously difficult.

In response to these limitations, a new paradigm has emerged: likelihood-free inference via forward modelling. This technique bypasses the need for a covariance matrix by directly comparing real data to synthetic observables generated from a forward model. Its advantages are profound: it eliminates the storage and computational burden of massive simulation sets, naturally incorporates high-order statistical information, and can seamlessly integrate systematic effects. However, this new method has its own hurdles: it demands immense GPU resources to process Euclid-sized surveys, and its conclusions are only as reliable as the simulations it uses, potentially leading to circular debates if simulations and observations disagree.

A recent breakthrough (Tinnaneni Sreekanth, 2024) offers a compelling path forward. This work provides the first theoretical framework to directly predict key wavelet statistics of weak lensing convergence maps—exactly the kind Euclid will produce—for any given set of cosmological parameters. It has been shown in Ajani et al (2021) that the wavelet coefficient L1-norm is extremely powerful to constraint the cosmological parameters. This innovation promises to harness the power of advanced, non-Gaussian statistics without the traditional computational overhead, potentially unlocking a new era of precision cosmology. We have demonstrated that this theoretical prediction can be used to build a highly efficient emulator (Tinnaneri Sreekanth et al, 2025), dramatically accelerating the computation of these non-Gaussian statistics. However, it is crucial to note that this emulator, in its current stage, provides only the mean statistic and does not include cosmic variance. As such, it cannot yet be used for full statistical inference on its own. 

This PhD thesis aims to revolutionize the analysis of weak lensing data by constructing a complete, end-to-end framework for likelihood-free cosmological inference. The project begins by addressing the core challenge of stochasticity: we will first calculate the theoretical covariance of wavelet statistics, providing a rigorous mathematical description of their uncertainty. This model will then be embedded into a stochastic map generator, creating realistic mock data that captures the inherent variability of the Universe.
To ensure our results are robust, we will integrate a comprehensive suite of systematic effects—such as noise, masks, intrinsic alignments, and baryonic physics—into the forward model. The complete pipeline will be integrated and validated within a simulation-based inference framework, rigorously testing its power to recover unbiased cosmological parameters. The culmination of this work will be the application of our validated tool to the Euclid weak lensing data, where we will leverage non-Gaussian information to place competitive constraints on dark energy and modified gravity.

References
V. Ajani, J.-L. Starck and V. Pettorino, 'Starlet l1-norm for weak lensing cosmology', Astronomy and Astrophysics,  645, L11, 2021.
V. Tinnaneri Sreekanth, S. Codis, A. Barthelemy, and J.-L. Starck, 'Theoretical wavelet l1-norm from one-point PDF prediction', Astronomy and Astrophysics,  691, id.A80, 2024.
V. Tinnaneri Sreekanth, J.-L. Starck and S. Codis, 'Generative modeling of convergence maps based in LDT theoretical prediction', Astronomy and Astrophysics,  701, id.A170, 2025.