Perturbative interaction approach to cosmological structure formation

A new approach to cosmological perturbation theory has been recently introduced by Bartelmann et al., relying on nonequilibrium statistical theory of classical particles, and treating the gravitational interaction perturbatively. They compute analytic expressions for the nonlinear matter power spectrum, to first order in the interaction, and at one-loop order in the linear power spectrum. The resulting power spectrum is well behaved even at large wave numbers and seems in good agreement with results from numerical simulations. In this paper, we rederive their results concisely with a different approach, starting from the implicit integral solution to particle trajectories. We derive the matter power spectrum to first order in the interaction, but to arbitrary order in the linear power spectrum, from which the one-loop result follows. We also show that standard linear perturbation theory can only be recovered at infinite order in the gravitational interaction. At finite order in the interaction, we find that the linear power spectrum is systematically and significantly underestimated. A comprehensive study of the convergence of the theory with the order of the interaction for nonlinear scales will be the subject of future work.

Figure 1

Variance of the displacement field per axis, per logarithmic $k$ interval, at redshift zero: ${\mathrm{\Delta }}_{\psi }^2\equiv \frac{1}{3}k{P}_{\mathrm{L}}(k)/(2{\pi }^2)$. The linear power spectrum was obtained with Camb [13] for cosmological parameters consistent with Planck results [14].

Figure 2

Linear density field up to order $n$ in the interaction: for $n=0,1,2,3$ and $\infty$ from bottom to top. The normalization is such that the linear growth factor (corresponding to $n\le \infty$) is unity at redshift zero. The starting conformal time ${\eta }_{*}$ corresponds to redshift $z_{*}=99$. Cosmological parameters consistent with Planck results are used.