Kinetic field theory: Higher-order perturbation theory

We give a detailed exposition of the formalism of kinetic field theory (KFT) with emphasis on the perturbative determination of observables. KFT is a statistical nonequilibrium classical field theory based on the path integral formulation of classical mechanics, employing the powerful techniques developed in the context of quantum field theory to describe classical systems. Unlike previous work on KFT, we perform the integration over the probability distribution of initial conditions in the very last step. This significantly improves the clarity of the perturbative treatment and allows for physical interpretation of intermediate results. We give an introduction to the general framework, but focus on the application to interacting $N$-body systems. Specializing the results to cosmic structure formation, we reproduce the linear growth of the cosmic density fluctuation power spectrum on all scales from microscopic, Newtonian particle dynamics alone.

Figure 1

The density evolution for zeroth and first order perturbation theory in blue and red curves, respectively. The initial spatial distribution at $t=0$ has mean value $\mu =2$ and standard deviation $\sigma =2$ (leftmost Gaussian). The momentum distribution has mean value $\lambda =2$ and standard deviation $\xi =1$, such that in every unit of time the density distribution moves to the right by two units and broadens. This is precisely the behavior in zeroth order perturbation theory (blue curves). The first order correction counteracts the diffusion due to the encoded attractive interaction (red curves) [19].

Figure 2

Evolution of the toy model system over time (left to right). The upper row shows the distribution in phase space (density indicated by color), and the lower row the corresponding spatial density function. The singular density function at $t_c$ (middle column) is a result of the spatial projection rather than an inherent singularity of the system [19].

Figure 3

Evolution of the amplitude of the power spectrum against scale factor $a$ for the cosmological standard model. The lowest curve is the zeroth order approximation, and the curves above add incrementally the higher-order corrections. Shown are the curves up to 15th order in perturbation theory. The highest-order curves are visibly indistinguishable from the expected linear growth given by the dashed black line [19].

Figure 4

Left panel: Relative difference between the expected linear growth of the power spectrum and the perturbative approximations for up to 15th order. The $x$ axis is scale factor $a$. Hence this panel effectively shows the residuals of Fig. 3. Right panel: As the left panel, but the relative difference is plotted against the perturbation order at final scale factor $a=1$. Percent-level accuracy (indicated by the gray dashed curve) is achieved by going to at least 12th perturbation order [19].