Large-N expansions applied to gravitational clustering

TL;DR

The paper develops a path-integral formulation for gravitational clustering in the hydrodynamical (single-stream) limit and applies two one-loop large-$N$ expansions to compute two-point statistics of density and velocity fields for Gaussian initial conditions. The direct steepest-descent expansion yields a non-linear propagator $R$ with persistent oscillations and an envelope near the linear prediction, while the 2PI effective-action expansion produces damped oscillations and a more coupled evolution for the two-point function $G$, often improving agreement in the quasi-linear regime. Both schemes reproduce standard perturbation theory at one-loop and provide partial resummations that extend applicability into the weakly non-linear regime without relying on $N$-body simulations, offering a framework to gauge the range of validity of different expansions. The formalism is readily applied to a $\Lambda$CDM background and holds potential for extensions to the Vlasov equation or other effective descriptions, which could aid interpretation of weak-lensing and BAO measurements that probe weakly non-linear scales.

Abstract

We develop a path-integral formalism to study the formation of large-scale structures in the universe. Starting from the equations of motion of hydrodynamics (single-stream approximation) we derive the action which describes the statistical properties of the density and velocity fields for Gaussian initial conditions. Then, we present large-N expansions (associated with a generalization to N fields or with a semi-classical expansion) of the path-integral defined by this action. This provides a systematic expansion for two-point functions such as the response function and the usual two-point correlation. We present the results of two such expansions (and related variants) at one-loop order for a SCDM and a LCDM cosmology. We find that the response function exhibits fast oscillations in the non-linear regime with an amplitude which either follows the linear prediction (for the direct steepest-descent scheme) or decays (for the 2PI effective action scheme). On the other hand, the correlation function agrees with the standard one-loop result in the quasi-linear regime and remains well-behaved in the highly non-linear regime. This suggests that these large-N expansions could provide a good framework to study the dynamics of gravitational clustering in the non-linear regime. Moreover, the use of various expansion schemes allows one to estimate their range of validity without the need of N-body simulations and could provide a better accuracy in the weakly non-linear regime.

Figures (39)

• Figure 1: The self-energy terms $\Sigma_{0;ij}^+(k)$ of eq.(\ref{['S0pS0m']}) as a function of wavenumber $k$. We display the four components $\Sigma_{0;11}^+$ (solid line), $\Sigma_{0;21}^+$ (dot-dashed line), $\Sigma_{0;12}^+$ (dotted line) and $\Sigma_{0;22}^+$ (dashed line). They are all negative except for $\Sigma_{0;11}^+$ which is positive at $k < 0.2 h$ Mpc$^{-1}$ and negative at higher $k$. All terms are of the same magnitude and grow roughly as $k^2$ in agreement with eq.(\ref{['S01122']}).
• Figure 2: The self-energy $\Sigma_0(k;\eta_1,\eta_2)$ as a function of forward time $\eta_1$, for $\eta_2=-1.4$ (i.e. $z_2=3$) and wavenumbers $k=0.1,1$ and $10 \times h$ Mpc$^{-1}$. The line styles are as in Fig. \ref{['figS0pk']}. The diagonal components vanish for $\eta_1<\eta_2$ whereas the off-diagonal components vanish for $\eta_1\leq\eta_2$. All terms are negative except for $\Sigma_{0;11}$ which becomes positive at $\eta_1>-0.8$ for $k=0.1 h$ Mpc$^{-1}$.
• Figure 3: The self-energy $\Sigma_0(k;\eta_1,\eta_2)$ as a function of backward time $\eta_2$, for $\eta_1=0$ (i.e. $z_1=0$) and wavenumbers $k=0.1,1$ and $10\times h$ Mpc$^{-1}$. The line styles are as in Fig. \ref{['figS0pk']}. All terms are negative except for $\Sigma_{0;11}$ which becomes positive at $\eta_2<-0.6$ for $k=0.1 h$ Mpc$^{-1}$.
• Figure 4: The self-energy $\Pi_0(k)$ as a function of wavenumber $k$. At high $k$ all components are very close and positive. At $k\leq 0.06 h$ Mpc$^{-1}$ the off-diagonal components become negative.
• Figure 5: The non-linear response $R(k;\eta_1,\eta_2)$ as a function of forward time $\eta_1$, for $\eta_2=-1.4$ (i.e. $z_2=3$) and wavenumbers $k=1$ (left panel) and $10\times h$ Mpc$^{-1}$ (right panel). We also plot the linear response $R_L$ which shows a simple exponential growth.