Perturbation theory with dispersion and higher cumulants: Framework and linear theory

The standard perturbation theory (SPT) approach to gravitational clustering is based on a fluid approximation of the underlying Vlasov-Poisson dynamics, taking only the zeroth and first cumulant of the phase-space distribution function into account (density and velocity fields). This assumption breaks down when dark matter particle orbits cross and leads to well-known problems, e.g., an anomalously large backreaction of small-scale modes onto larger scales that compromises predictivity. We extend SPT by incorporating second and higher cumulants generated by orbit crossing. For collisionless matter, their equations of motion are completely fixed by the Vlasov-Poisson system, and thus we refer to this approach as Vlasov Perturbation Theory (VPT). Even cumulants develop a background value, and they enter the hierarchy of coupled equations for the fluctuations. The background values are in turn sourced by power spectra of the fluctuations. The latter can be brought into a form that is formally analogous to SPT, but with an extended set of variables and linear as well as nonlinear terms, that we derive explicitly. In this paper, we focus on linear solutions, which are far richer than in SPT, showing that modes that cross the dispersion scale set by the second cumulant are highly suppressed. We derive stability conditions on the background values of even cumulants from the requirement that exponential instabilities be absent. We also compute the expected magnitude of averaged higher cumulants for various halo models and show that they satisfy the stability conditions. Finally, we derive self-consistent solutions of perturbations and background values for a scaling universe and study the convergence of the cumulant expansion. The VPT framework provides a conceptually straightforward and deterministic extension of SPT that accounts for the decoupling of small-scale modes.

Figure 1

Normalized cumulants ${\overline{\mathcal{E}}}_4,{\overline{\mathcal{E}}}_6,{\overline{\mathcal{E}}}_8$ (bottom to top) for Evans infinite halos as a function of halo-shape parameter $q$. Dashed lines denote negative values. When halos are spherical ($q=1$), the distribution function becomes Gaussian and the density profile isothermal.

Figure 2

Normalized cumulants ${\overline{\mathcal{E}}}_4,{\overline{\mathcal{E}}}_6,{\overline{\mathcal{E}}}_8$ (bottom to top) for NFW halos as a function of halo concentration parameter $c$. Dashed lines denote negative values. For each case we show results for two values of the anisotropy parameter, $\beta =0$ (lower value) and $\beta =1/2$ (higher value). The left panel corresponds to integrating out to the virial radius, whereas the right panel corresponds to when we associate with the halo a region of 1.5 times the virial radius.

Figure 3

Linear kernels $F_{1,a}(k,\eta )$ that describe the suppression relative to SPT for $a=\delta$, $\theta$ when taking the second cumulant into account. In addition, the rescaled scalar perturbation mode ${\overline{g}}_k=g_k/\epsilon$ of the dispersion tensor possesses a growing mode, shown for $a=\overline{g}$, with $F_{1,\overline{g}}\to 2/(2+\alpha )$ for $k\to 0$. Here $\alpha =2$ and $k_{\sigma }=1/\sqrt{\epsilon (\eta )}$ such that the time-dependence is scaled out.

Figure 4

Linear VPT kernels $F_{1,a}(k,\eta )$ when taking the second, third and fourth cumulant into account, respectively. The upper panel shows the case $\overline{\omega }=1$, and the lower $\overline{\omega }=-1$, for $\delta$ and $\theta$. Furthermore, $\alpha =2$ and $k_{\sigma }=1/\sqrt{\epsilon (\eta )}$.

Figure 5

Linear VPT kernel $F_{1,\overline{g}}(k,\eta )$ when taking the second, third and fourth cumulant into account, respectively, for $\overline{\omega }=1$ and $\alpha =2$.

Figure 6

Linear VPT kernel $F_{1,\delta }(k,\eta )={\mathcal{C}}_{0,0}$ when taking perturbation modes ${\mathcal{C}}_{\ell ,2n}$ of cumulants up to $\ell +2n\le c_{\max }$ into account. For this figure we set $\alpha =2$ and ${\mathcal{E}}_{2n}=0$ with $2n\ge 4$. As before, the time-dependence of the linear kernel is scaled out by normalizing the wavenumber to $k_{\sigma }=1/\sqrt{\epsilon }$. Note how the inclusion of higher cumulants only enhances the suppression of UV modes.

Figure 7

Constraints on the non-Gaussianity of the distribution function from requiring stability of the linear solutions. The top panel shows the parameter region allowed by stability within the plane spanned by the dimensionless 4th and 6th cumulant expectation values $({\overline{\mathcal{E}}}_4,{\overline{\mathcal{E}}}_6)$, while the bottom panel shows constraints on the 4th and 8th cumulant expectation values $({\overline{\mathcal{E}}}_4,{\overline{\mathcal{E}}}_8)$. In each case, these are obtained from the evolution equations for perturbations modes of cumulants up to $c_{\max }$ as given in the legend. Note that ${\overline{\mathcal{E}}}_6$ is relevant for $c_{\max }\ge 5$, and ${\overline{\mathcal{E}}}_8$ for $c_{\max }\ge 7$. The gray dashed and black lines show the expectation from the Evans and NFW halo models, see Sec. 3. The fact that the backreaction on linear modes from dispersion and higher cumulants expected from halos is broadly stable is reassuring, making linearized VPT a good starting point for a perturbative expansion.

Figure 8

Velocity dispersion scale $k_{\sigma }={\epsilon }^{-1/2}$ relative to the nonlinear scale $k_{nl}$ for power-law initial spectrum $P_0\propto k^{n_s}$, obtained when using the linear approximation for the source term for $\epsilon$, and neglecting third and higher cumulants.

Figure 9

Velocity dispersion scale $k_{\sigma }={\epsilon }^{-1/2}$ relative to the nonlinear scale $k_{nl}$ for power-law initial spectrum $P_0\propto k^{n_s}$, obtained from a self-consistent solution when including cumulant perturbations up to order $c_{\max }$, and for $n_s=-1$, 0, 1, 2, respectively. The corresponding self-consistent solutions for the fourth, sixth, and eighth cumulant expectation values are shown in Table 2.

Figure 10

The distribution function monopole for NFW halos with $\beta =1/2$ (solid) compared to a Gaussian distribution with the same second cumulant (dashed), at the scale radius of the halo $x\equiv cr/r_{\mathrm{vir}}=1$. The left panel shows $f^{(0)}(p)$ as a function of momentum $p$ in units of $p_0\equiv \sqrt{Gmf(c)c/r_{\mathrm{vir}}}$ while the right panel shows $4\pi p^2{f}^{(0)}(p)$.