Gravitational Turbulence: the Small-Scale Limit of the Cold-Dark-Matter Power Spectrum

TL;DR

The paper establishes a universal small-scale tail for the cold dark matter power spectrum, showing $P(k)\propto k^{-d}$ for $k\gg k_{\rm nl}$ in $d$ dimensions. It derives this result via two non-perturbative frameworks: (i) a phase-space turbulence picture based on the Vlasov--Poisson system, where Jeans collapse drives a cascade of the quadratic Casimir invariant $C_2$ under a critical balance between linear phase-mixing and nonlinear tidal effects, and (ii) a saddle-point analysis of the kinetic-field-theory representation of $P(k)$, which yields the same scaling from stationary points with relative coordinates and momenta $q,p\sim k^{-1}$. The analyses hinge on the coldness of the distribution, ensuring a phase-space description remains valid through stream-crossing, and they predict a scale-dependent flux of $C_2$ distinct from Kolmogorov-type turbulence. 1D Vlasov--Poisson simulations corroborate the $k^{-d}$ scaling and illustrate the phase-space cascade, with the predictions truncated by the thermal scale $v_{\rm th}$. The findings have potential implications for dark-matter halo modeling and effective field theories of large-scale structure, offering a physical underpinning for the small-scale power-law tail.

Abstract

The matter power spectrum, $P(k)$, is one of the fundamental quantities in the study of large-scale structure in cosmology. Here, we study its small-scale asymptotic limit, and show that for cold dark matter in $d$ spatial dimensions, $P(k)$ has a universal $k^{-d}$ asymptotic scaling with the wave-number $k$, for $k \gg k_{\rm nl}$, where $k_{\rm nl}^{-1}$ denotes the length scale at which non-linearities in gravitational interactions become important. We propose a theoretical explanation for this scaling, based on a non-perturbative analysis of the system's phase-space structure. Gravitational collapse is shown to drive a turbulent phase-space flow of the quadratic Casimir invariant, where the linear and non-linear time scales are balanced, and this balance dictates the $k$ dependence of the power spectrum. A parallel is drawn to Batchelor turbulence in hydrodynamics, where large scales mix smaller ones via tidal interactions. The $k^{-d}$ scaling is also derived by expressing $P(k)$ as a phase-space integral in the framework of kinetic field theory, which is analysed by the saddle-point method; the dominant critical points of this integral are precisely those where the time scales are balanced. The coldness of the dark-matter distribution function - its non-vanishing only on a $d$-dimensional sub-manifold of phase-space - underpins both approaches. The theory is accompanied by $1\mathrm{D}$ Vlasov-Poisson simulations, which confirm it.

Figures (6)

• Figure 1: Colour plots of the distribution function showing the time evolution of three cold streams, with $v_{\rm th}/v_0 = 5\times 10^{-3}$. The colour-bar in each panel extends from $0$ to $\max f$ at the corresponding time in the simulation. See text and appendix \ref{['appendix: simulation set up']} for details. A video is available in this https://ybginat.com/index.php/gravitational-turbulence/.
• Figure 2: Plots of the contributions to the non-linear term in equation \ref{['eqn: f^2 transport equation']}, for the simulation shown in figure \ref{['fig:results time evolution']}. Defining $\mathcal{N}(\mathbf{k},\mathbf{k}',\mathbf{s}) \equiv \left\langle\Im \left[ \hat{\mathbf{g}}(\mathbf{k}')\hat{f}(\mathbf{k}-\mathbf{k}',\mathbf{s})\hat{f}^*(\mathbf{k},\mathbf{s})\right]\right\rangle_{\rm T}$, the left panel displays $\left\vert\mathcal{N}\right\vert/\max_{k'}\left\{|\mathcal{N}|\right\}$, at $sv_0 = 0.52$, as a function of both $k$ and $k'$ (the average is calculated as in § \ref{['subsec:phase space cascade']}). The Batchelor peak is very prominent at low $k'$, and another peak is visible on the diagonal $k' = k$, corresponding to the limit (ii) of $\mathcal{N}$. In this paper we only require the limit (ii) when integrated over $s$ (in § \ref{['subsec: source CDM']} below); to gauge the validity of the approximation in that case, we show in the right panel shows the integral of $s\mathcal{N}$ over $0\leq s \leq k\tau_0/2$ (in logarithmic scale), normalised in the same way, exhibiting the same two prominent peaks.
• Figure 3: A contour plot of the time-averaged power spectrum $\langle |\hat{f}|^2(t,k,s)/C_2(t)\rangle_{\rm T}$, for the same simulation as in figure \ref{['fig:results time evolution']} (note that $C_2$ decays because of collisions and finite-grid effects, so it is sensible to normalise the power spectrum by $C_2$ at every time). The line $s=k\tau_0$ is also plotted in blue.
• Figure 4: The time evolution of the power spectra $\langle |\hat{f}|^2/C_2(t) \rangle_{10}$ for the simulation illustrated in figure \ref{['fig:results time evolution']} (early times: blue; late times: red). The time averaging was over $10$ simulation outputs (corresponding to time windows of duration $1.36\tau_0$). Left panel: the $k$-spectrum at $s=0.52/v_0$. Right panel: the $s$-spectrum at $k=k_0$. Both spectra are compensated by the expected asymptotics (multiplied by $k^{1}$ and $s^{1}$, respectively). A $k^{-1}$ power law establishes itself quickly, extending to $k\sim 200k_0$, which is of the order of $k_{\rm c}(v_{\rm th}^{-1})$, as expected; similarly, the $s^{-1}$ power law extends to $sv_0\sim 200$. The steeper spectra at smaller scales are the scalings that prevail below the thermal scale (see appendix \ref{['subsec:simulation results']}).
• Figure 5: The power spectra for the same simulation as in figure \ref{['fig:results time evolution']}, but now averaged over $t\in [0,30]\tau_0$, as a proxy for the ensemble average. Unlike in figure \ref{['fig:spectra time evolution']}, they are uncompensated, and display the theoretical $k^{-d}$ and $s^{-d}$ scalings explicitly ($d=1$).