Kinetic Field Theory: Second-order perturbation theory

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Astoundingly successful as it is, the cosmological standard model leaves us with two difficult and fundamentally important puzzles, viz. the dark matter and the dark energy, whose origin, composition, and place in the framework of physics are enigmatic. Large cosmological surveys are being and are about to be undertaken with the major goals of mapping and quantifying the dark-matter distribution and finding out about a possible evolution of the dark-energy density with time. Precise cosmological inference from these surveys hinges on the depth and precision of our understanding of the late-time, non-linear evolution of cosmic structures. While numerical simulations have returned a wealth of impressive and important results, calculating higher-order statistical properties of large-scale cosmic structures at late times is still a forbiddingly time-consuming task.Recently, we have developed a novel analytic approach to non-linear cosmic structure formation, based on a non-equilibrium kinetic field theory (KFT). This theory has several important advantages compared to more conventional approaches. In particular, it operates on the microscopic degrees of freedom of classical particles in phase space. Since the Hamiltonian flow in phase space is bijective and diffeomorphic, the notorious shell-crossing problem is absent from our approach. In its present form, the theory is free of adjustable parameters. We have shown in earlier work that even first-order perturbation theory applied to KFT reproduces the non-linear power spectrum of cosmic density fluctuations very well to wave numbers of k ~ 10 h/Mpc at the present cosmic epoch. We could further show how the central mathematical object of KFT, its generating functional, can be fully factorized into terms of a standard form.Based on this factorization, a diagrammatic approach to perturbation theory developed therefrom, and a first version of a symbolic computer code for constructing these diagrams, we propose here to calculate the second-order perturbation terms for the non-linear power spectrum of cosmic density fluctuations. Main challenges will be the reliable and fast numerical integration of the generic expressions appearing in the factorization of the generating functional, and the calculation of the one- and two-loop integrals appearing in the perturbation series. The essential goals of the proposed research are to finalize and test our symbolic computer code for perturbation diagrams, to supply it with a code for numerical integration of perturbation terms, and to evaluate, analyze and compare the contributions of these terms to the non-linearly evolved cosmic power spectrum. Based on the quality of the first-order perturbation theory in KFT, we are confident that we can analytically calculate accurate non-linear power spectra reaching wave numbers of k ~ 20-50 h/Mpc at the present epoch.