Monge Ampère gravity: from the large deviation principle to cosmological simulations through optimal transport

Abstract: We study Monge-Amp`ere gravity (MAG) as an effective theory of cosmological structure formation through optimal transport theory. MAG is based on the Monge-Amp`ere equation, a nonlinear version of the Poisson equation, that relates the Hessian determinant of the potential to the density field. We explain how MAG emerges from a conditioned system of independent and indistinguishable Brownian particles, through the large deviation principle, in the continuum limit. To numerically explore this highly non-linear theory, we develop a novel N-body simulation method based on semi-discrete optimal transport. Our results obtained from the very first N-body simulation of Monge-Amp`ere gravity with over 100 millions particles show that on large scales, Monge-Amp`ere gravity is similar to the Newtonian gravity but favours the formation of anisotropic structures such as filaments. At small scales, MAG has a weaker clustering and is screened in high-density regions. Although here we study the Monge-Amp`ere gravity as an effective rather than a fundamental theory, our novel highly-performant optimal transport algorithm can be used to run high-resolution simulations of a large class of modified theories of gravity, such as Galileons, in which the equations of motion are second-order and of Monge-Amp`ere type.

• The paper establishes a theoretical foundation for MAG by deriving it from the large deviation principle applied to mass-preserving maps.
• It introduces a novel simulation method using semi-discrete optimal transport in large-scale N-body simulations with over 100 million particles.
• The results indicate that MAG yields cosmic structures similar to Newtonian predictions on large scales but shows a natural screening effect in dense regions.

Monge Ampère Gravity: Bridging Large Deviations and Cosmological Simulations through Optimal Transport

The research paper titled "Monge Ampère gravity: from the large deviation principle to cosmological simulations through optimal transport" presents an exploration into Monge-Ampère Gravity (MAG) as an effective theory for cosmological structure formation. This paper introduces innovative computational techniques to model MAG using semi-discrete optimal transport, while drawing fundamental links between advanced mathematical frameworks and astrophysical phenomena.

MAG extends the classical Newtonian framework of gravity by implementing the Monge-Ampère equation, which is a nonlinear partial differential equation relating the Hessian determinant of a potential to the density field. This contrasts with the Poisson equation that sums eigenvalues (trace) instead of their product. These additional terms lead to intriguing features such as the development of anisotropic structures and screening effects in high-density regions.

Key Contributions and Findings

• Theoretical Foundation: MAG is derived from a conditioned system of Brownian particles through the large deviation principle, which is analogical to an emergent theory in statistical mechanics. This approach links cosmological structure formation to optimal mass transport problems by considering mass-preserving maps.
• Novel Simulation Approach: The paper introduces an efficient numerical method for simulating MAG using N-body simulations of over 100 million particles. This method relies on semi-discrete optimal transport, allowing for computational tractability even with highly non-linear dynamics.
• Large-scale Cosmic Structure Formation: Numerical simulations demonstrated that MAG produces large-scale cosmic structures similar to those predicted by traditional Newtonian gravity. However, on smaller scales, MAG predicts less clustering due to a non-diverging potential, resulting in a natural screening in dense regions.
• Symmetry and Anisotropy: The symmetry properties of the Monge-Ampère equation, notably its invariance under affine transformations, contribute to the stability and abundance of anisotropic structures like filaments and ellipsoidal haloes.

Technical Insights

• Numerical Implementation: The paper's simulation methodology effectively tackles the nonlinear Monge-Ampère equation by employing the semi-discrete Kantorovich dual formulation. This provides improved convergence guarantees via efficient numerical solvers such as the KMT-Newton algorithm within a cosmological context.
• Complexity and Performance: While solving the Monge-Ampère equation is inherently more complex than simulating Newtonian gravity, the use of optimal transport reduces operational complexity, especially in the semi-discrete setting. The simulations leverage multi-core processors to handle high computational demands effectively.

Implications and Future Research

• Modified Theories of Gravity: MAG as portrayed in this paper contributes substantially to the study of alternative gravity theories beyond the standard Newtonian paradigm. It emerges as a tool for probing and challenging the boundaries of existing gravitational theories such as Galileons.
• Astrophysical Applications: The algorithms devised in this study may find utility in reconstructing the initial conditions of the universe and refining models of cosmological evolution, providing enhanced insights into the physics of structure formation at both large and small scales.
• Theoretical Exploration: The results warrant further investigation into the implications of such emergent gravitational models on cosmological phenomena. This includes cross-validation with observational data to assess the empirical accuracy of MAG within different astrophysical contexts.

In conclusion, this study offers a comprehensive exploration of Monge-Ampère gravity, proposing new pathways to simulate cosmic structures with rigorous mathematical underpinnings. It opens up significant opportunities for extending the frontiers of cosmological research, blending theoretical insights with cutting-edge computational methodologies.