Nonlocal Gravity Study Defines Effective Delta Sources and Newtonian Limit with Exponential Form Factors

The nature of gravity at its most fundamental level remains a key question in physics, and scientists continually explore modifications to Einstein’s theory to address persistent mysteries about the universe. Thomas M. Sangy from Universidade Federal de Juiz de Fora, Nicolò Burzillà from INFN Laboratori Nazionali di Frascati, and Breno L. Giacchini from Charles University, along with their colleagues, investigate the behaviour of gravity when considering ‘nonlocal’ effects, where gravity’s influence isn’t strictly limited by immediate proximity. Their work focuses on a specific class of nonlocal gravity models and reveals how these models behave in weak gravitational fields, similar to those found around everyday objects and within our solar system. By comparing different representations of gravitational sources and potentials, the team demonstrates that even with complex modifications to gravity, solutions remain stable and physically realistic, offering valuable insights into potential extensions of Einstein’s theory and paving the way for exploring subtle corrections to our understanding of the universe.

Modified Gravity, Regular Black Holes, and Dark Matter

This research program comprehensively investigates modified gravity theories, regular black holes, and their potential connection to dark matter. Researchers explore theories beyond General Relativity, aiming to resolve singularities, such as those found at black hole centers, and potentially explain dark matter’s nature. They examine higher-derivative and nonlocal gravity, allowing for the construction of regular black hole solutions, often involving modifications to the Schwarzschild metric. The team also investigates whether spacetime exhibits noncommutative properties at extremely small scales, potentially influencing black hole solutions and quantum gravity effects.

A significant portion of the research focuses on the mathematical tools required to solve complex equations and analyze resulting solutions. Researchers utilize special functions, including hypergeometric and Fox H-functions, to represent solutions in closed form. Integral transforms and asymptotic expansions evaluate integrals and approximate solutions, while numerical simulations model the behavior of these modified gravity theories and compare them with observational data. This combination of analytical and computational techniques allows for a thorough investigation of theoretical predictions.

The research connects to cosmology and the search for dark matter candidates. Researchers explore whether regular black holes or other compact objects formed in modified gravity theories could contribute to the universe’s dark matter content, specifically considering stringballs and Planckballs. They investigate the cosmological implications of these modified gravity theories, such as their effect on the expansion of the universe and the formation of large-scale structures. Researchers employed an effective source formalism to analyze effective sources, mass functions, and Newtonian potentials, providing a comprehensive understanding of how these models behave in weak-field scenarios. They developed representations of these quantities using series, integrals, and special functions, alongside approximations to explore the dependence on model parameters. To determine the Newtonian potential, the team utilized the Fox H-function, enabling solutions under specific conditions.

They derived a closed-form solution for a particular case, expressed as an arctangent function, and calculated the power series representation of the potential using the Fourier transform method. Furthermore, the heat kernel method was implemented, recognizing the close relationship between the effective source and the heat kernel of a specific operator, allowing the potential to be expressed in terms of a generalized error function. The study also explored the behavior of the models as a parameter approaches infinity. Researchers found that the form factor converges to a rectangle function, though not uniformly, requiring careful consideration when using it in Fourier integral representations. By utilizing power series representations and properties of the Gamma function, they derived a closed-form solution for the effective source, expressed in terms of sine and cosine functions. This allowed for a detailed analysis of the mass function, revealing that it does not converge to a constant value at large distances, unlike models with finite parameters, and demonstrating the absence of a Newtonian singularity at the origin.

Nonlocal Gravity Resolves Singularities, Ensures Renormalizability

Scientists investigated nonlocal gravitational models, focusing on their ability to resolve singularities present in general relativity and ensure renormalizability, a crucial property for a consistent quantum theory of gravity. The research centers on modifications to the standard Einstein-Hilbert action, incorporating exponential form factors to alter the propagation of gravity and potentially resolve problematic high-energy behavior. The team explored actions that include terms quadratic in curvature, modified by infinite-derivative operators, aiming to formulate theories that are ghost-free and renormalizable. The study demonstrates that introducing nonlocal terms, specifically exponential form factors, significantly modifies the tree-level propagator, improving its ultraviolet behavior without introducing unwanted ghost degrees of freedom.

Measurements confirm that these models exhibit regular Newtonian-limit solutions, meaning the solutions remain finite and avoid curvature singularities. Specifically, the team found that for form factors with a parameter greater than one, the Newtonian potential oscillates, yet the effective masses remain positive, ensuring physical viability. Further analysis revealed that the linearized solutions are indeed regular, with all curvature invariants remaining finite, a consequence of the form factor growing faster than any polynomial for large values of a parameter. The research builds upon previous work demonstrating the ability of these models to admit nonsingular bouncing cosmological solutions, potentially offering a resolution to the Big Bang singularity. By employing an effective source formalism, the team compared key quantities, such as effective sources, mass functions, and Newtonian potentials. The study demonstrates that oscillations in the Newtonian potential arise only under specific conditions, yet remarkably, the effective masses remain positive despite these oscillations, ensuring the solutions remain physically plausible. Furthermore, the team calculated the leading logarithmic correction to the Newtonian potential within these models, providing a more refined understanding of gravitational interactions. The work establishes a generalized Poisson equation applicable to these nonlocal models, achieved through the introduction of spin-s potentials which separate the contributions of different gravitational sectors. Future research may focus on extending these findings to more complex scenarios.