Perturbation theory with dispersion and higher cumulants: Nonlinear regime

We present nonlinear solutions of Vlasov perturbation theory (VPT), describing gravitational clustering of collisionless dark matter with dispersion and higher cumulants induced by orbit crossing. We show that VPT can be cast into a form that is formally analogous to standard perturbation theory (SPT), but including additional perturbation variables, nonlinear interactions, and a more complex propagation. VPT nonlinear kernels have a crucial decoupling property: for fixed total momentum, the kernels become strongly suppressed when any of the individual momenta cross the dispersion scale into the nonlinear regime. This screening of UV modes allows us to compute nonlinear corrections to power spectra even for cosmologies with very blue power-law input spectra, for which SPT diverges. We compare predictions for the density and velocity divergence power spectra as well as the bispectrum at one-loop order to $N$-body results in a scaling universe with spectral indices $-1\le n_s\le +2$. We find a good agreement up to the nonlinear scale for all cases, with a reach that increases with the spectral index $n_s$. We discuss the generation of vorticity as well as vector and tensor modes of the velocity dispersion, showing that neglecting vorticity when including dispersion would lead to a violation of momentum conservation. We verify momentum conservation when including vorticity, and compute the vorticity power spectrum at two-loop order, necessary to recover the correct large-scale limit with slope $n_w=2$. Comparing to our $N$-body measurements confirms the cross-over from $k^4$ to $k^2$ scaling at large scales. Our results provide a proof-of-principle that perturbative techniques for dark matter clustering can be systematically improved based on the known underlying collisionless dynamics.

Figure 1

Integrand of the one-loop contribution to the density power spectrum in VPT and SPT, for a scaling universe with power-law input spectrum $P_0\propto k^{n_s}$ with spectral indices $n_s=2,1,0,-1$ shown in the four panels. In SPT the integrand grows as $F_3(\mathit{k},\mathit{q},-\mathit{q})P_0(q)d^{3}q\sim q^{n_s+1}d\ln q$ for large loop wave number $q$, i.e., the well-known suppression $F_3\propto k^2/q^2$ due to momentum conservation is not sufficient to render the one-loop integral UV finite for all $n_s\ge -1$. In contrast, the physically expected screening of UV modes within the full collisionless Vlasov-Poisson dynamics is captured by VPT, compensating the increase for large $q$ for blue spectra with high $n_s$ and leading to a finite integral with asymptotic decay of the integrand as $F_{1,\delta }(k,\eta )F_{3,\delta }(\mathit{k},\mathit{q},-\mathit{q},\eta )P_0(q)d^{3}q\sim q^{-2}d\ln q$ independently of the spectral index $n_s$. The sum of linear and one-loop contributions within VPT matches well with $N$-body results (see Fig. 10, including details on the cumulant hierarchy truncation used for this figure). Note that the $1/q^2$ scaling within VPT is universal and occurs for all truncations considered in this work, see Fig. 2 and Eq. (62) for details.

Figure 2

Nonlinear kernels $F_{2,a}(\mathit{k}-\mathit{q},\mathit{q},\eta )$ (left column) and $F_{3,a}(\mathit{k},\mathit{q},-\mathit{q},\eta )$ (right column) versus $q=|\mathit{q}|$ in VPT (blue lines) compared to SPT (black dashed) for the density contrast $a=\delta$ (top row) and velocity divergence $a=\theta$ (bottom row). For the background dispersion we chose $\epsilon (\eta )={\epsilon }_0{e}^{\alpha \eta }$ as well as $\omega (\eta )=\overline{\omega }\times \epsilon (\eta {)}^2$, with parameters as indicated in the figures, and $\eta =0$. We show the dependence on $q$ relative to the dispersion scale $k_{\sigma }$, such that the result applies to any value of ${\epsilon }_0=1/k_{\sigma }^2$. The blue lines show second and third cumulant approximations (see Table 2), which are both suppressed compared to the EdS-SPT kernels for $q\gtrsim k_{\sigma }$ (shown in black dashed lines). The thin black solid line shows the analytical result of Eq. (57) at first order in $\epsilon$.

Figure 3

Nonlinear kernels $F_{3,a}(\mathit{k},\mathit{q},-\mathit{q},\eta )$ versus $q=|\mathit{q}|$ for $a=\delta$, $\theta$ as in Fig. 2, but comparing the impact of various approximations related to vorticity, as well as vector and tensor modes of the velocity dispersion tensor (see Table 2). Including vorticity backreaction has a significant impact on $F_{3,\delta }$ within the most relevant regime $k_{\sigma }\lesssim q\lesssim 2k_{\sigma }$ where the suppression of VPT relative to SPT sets in [($\mathbf{sw}$) vs ($\mathbf{s}$) in upper panel].

Figure 4

Nonlinear kernels $F_{3,a}(\mathit{k},\mathit{q},-\mathit{q},\eta )$ versus $k=|\mathit{k}|$ for $a=\delta$, $\theta$ (top) and the scalar perturbations $a=g,\delta \epsilon$ of the stress tensor (bottom). $F_{3,\delta }\propto k^2$ for $k\to 0$ is ensured by mass and momentum conservation, while for $F_{3,\theta }$ it is a consequence of the symmetry $\mathit{q}\to -\mathit{q}$, see Sec. 3b. In contrast, the kernels of the velocity dispersion modes approach a constant at low $k$.

Figure 5

Nonlinear kernels $F_{3,a}(\mathit{k}-\mathit{p}-\mathit{q},\mathit{p},\mathit{q},\eta )$ for $a=\delta$ (upper panel) and $a=\theta$ (lower panel) versus $k=|\mathit{k}|$, for fixed $\mathit{p}$ and $\mathit{q}$ at $\eta =0$, and assuming $\epsilon ={\epsilon }_0{e}^{\alpha \eta }$ with $\alpha =4/3$. The blue lines show the ($\mathbf{s}$), ($\mathbf{sw}$), and ($\mathbf{sv}$) approximation schemes, respectively, for ($\mathbf{cum}\mathbf{3}+$) with $\overline{\omega }=1$. The black dashed lines show the EdS-SPT kernels $F_3$ and $G_3$ for comparison. For $F_{3,\delta }$, a spurious linear scaling with $k$ occurs in the limit $k\to 0$ when vorticity is neglected ($\mathbf{s}$), while the $k^2$ scaling ensured by mass and momentum conservation is realized for all other approximation schemes that include vorticity. For $F_{3,\theta }$, the linear scaling with $k$ is a physical effect, and occurs independently of the approximation scheme as long as $\mathit{p}+\mathit{q}\ne 0$. For the figure we chose cosines $c_{kp}=0.3$, $c_{kq}=0.5$, $c_{pq}=0.875$, $p=0.5k_{\sigma }$, $q=0.5k_{\sigma }$ where $k_{\sigma }=1/\sqrt{{\epsilon }_0}$.

Figure 6

Nonlinear VPT kernels describing the generation of vorticity $F_{n,w_i}$ (left column) and the vector mode of the stress tensor $k^2{F}_{n,{\nu }_i}$ (right column), from nonlinear coupling of two ($n=2$, upper row) or three ($n=3$, lower row) perturbation modes of the initial linear density field. The kernels are shown for various approximation schemes as indicated in the legend (see Table 2). For $n=2$ we also show the analytical results at first order in $\epsilon$ (black solid line), as well as a line $\propto k^2$ (black dashed) for comparison. Note that $F_{2,w_i}$ points in the direction perpendicular to the plane spanned by $\mathit{k}$ and $\mathit{q}$, and we show its projection on $\mathit{k}\times \mathit{q}/|\mathit{k}\times \mathit{q}|$. For $n=3$ we show two components $F_{3,w_i}$, $i=1$, 2, perpendicular to $\mathit{k}$ (see Appendix pp4 for details), finding $F_{3,w_i}\propto k$ for $k\to 0$, similarly to $\theta$ (see Sec. 3b). The impact of the vector mode on vorticity [($\mathbf{sw}$) vs ($\mathbf{svt}$) in the left panels] is noticeable at $k\gtrsim k_{\sigma }$, while the impact of third cumulant perturbations is relatively mild for vorticity as well as vector mode kernels (($\mathbf{cum}\mathbf{2}$) vs ($\mathbf{cum}\mathbf{3}+$) in the upper row). The analytical results Eqs. (90), (92) at first order in $\epsilon$ are shown as thin black lines, and agree well for $k\ll k_{\sigma }$. In SPT all these kernels are zero. For the figure we chose $\epsilon ={\epsilon }_0{e}^{\alpha \eta }$ with $\alpha =4/3$, and $c_{kp}=0.3$, $c_{kq}=0.5$, $c_{pq}=0.875$, $p=k_{\sigma }$, $q=0.5k_{\sigma }$, where $k_{\sigma }=1/\sqrt{{\epsilon }_0}$.

Figure 7

Second order kernel of the tensor perturbation $t_{ij}$ of the velocity dispersion that describes the generation of tensor modes from coupling two scalar modes, captured by VPT. We show the dimensionless kernel $k^2{F}_{2,t_{\mathrm{eff}}}(\mathit{k}-\mathit{q},\mathit{q})$ versus $k$ in two approximations (see legend), as well as the analytical result at first order in $\epsilon$ (black solid line). The kernel scales as $k^2$ for $k\to 0$ (black dashed line).

Figure 8

Mass function $\nu f(\nu )$ fits to simulation measurements (see Appendix pp7) as a function of $\widehat{m}=m/m_{*}$ for $n_s=-1$, 0, 1, 2 (top to bottom from the low-mass end).

Figure 9

Density power spectrum $P_{\delta \delta }(k,\eta )$ in one-loop (solid) and linear (dashed) approximation for various values of the background dispersion ${\epsilon }_0=1/k_{\sigma }^2$ for a power law input spectrum $P_0\propto k^{n_s}$ with $n_s=0$ and $\epsilon (\eta )={\epsilon }_0{e}^{4\eta /(n_s+3)}$. The linear and one-loop result contains the numerical kernels $F_{1,\delta }(k,\eta )$, $F_{2,\delta }(\mathit{k}-\mathit{q},\mathit{q},\eta )$ and $F_{3,\delta }(\mathit{k},\mathit{q},-\mathit{q},\eta )$ computed taking up to third cumulant scalar perturbations as well as vorticity, vector and tensor modes of the stress tensor into account ($\mathbf{cum}\mathbf{3}+$, $\mathbf{svt}$). We set $\overline{\omega }=0.668$ (see text and [6]) for the fourth cumulant expectation value $\omega (\eta )=\overline{\omega }\times \epsilon (\eta {)}^2$.

Figure 10

Comparison of $P_{\delta \delta }$ in linear (dashed) and one-loop (solid) approximation in VPT within the ($\mathbf{cum}\mathbf{3}+$, $\mathbf{svt}$) scheme to $N$-body results (red circles), for $n_s=2,1,0,-1$, as noted. In absence of a precise knowledge of the background dispersion, we adjusted $k_{\sigma }/k_{\mathrm{nl}}$ by a one-parameter fit up to $k\le k_{\max }=0.6k_{\mathrm{nl}}$, with best-fit values given in the legend for each case. Notably, these values of $k_{\sigma }/k_{\mathrm{nl}}$ are consistent with our theoretical expectations given in Sec. 5. We use the same fixed values of $k_{\sigma }/k_{\mathrm{nl}}$ as given here when comparing VPT and $N$-body results for velocity spectra and bispectra below (for each approximation and $n_s$, respectively) such that for all these statistics there is only a single free parameter that is adjusted once to a common value.

Figure 11

Left panel: dependence of the density power spectrum $P_{\delta \delta }(k,\eta )$ in one-loop (solid) and linear (dashed) order on the approximation schemes ($\mathbf{s}$), ($\mathbf{sw}$), ($\mathbf{sv}$), ($\mathbf{svt}$) quantifying the impact of successively including vorticity ($\mathbf{sw}$), vector ($\mathbf{sv}$) and tensor ($\mathbf{svt}$) modes. Note that ($\mathbf{sv}$) and ($\mathbf{svt}$) are essentially indistinguishable. Right panel: second versus third cumulant approximations ($\mathbf{cum}\mathbf{2}$) vs ($\mathbf{cum}\mathbf{3}+$), with three choices of $\overline{\omega }$ for the latter and a common value $k_{\sigma }=1.31k_{\mathrm{nl}}$ (see Fig. 10) in all cases.

Figure 12

One-loop density power spectrum $P_{\delta \delta }(k,\eta )$ compared to $N$-body results for $n_s=2,1,0,-1$ and three approximation schemes. Taking only scalar perturbation modes into account ($\mathbf{cum}\mathbf{3}+$, $\mathbf{s}$) leads to significant deviations from our fiducial scheme ($\mathbf{cum}\mathbf{3}+$, $\mathbf{svt}$) with vorticity, vector and tensor modes. On the contrary, the second cumulant approximation ($\mathbf{cum}\mathbf{2}$, $\mathbf{svt}$) gives results close to ($\mathbf{cum}\mathbf{3}+$, $\mathbf{svt}$). This means the impact of the third cumulant is mild whereas neglecting the backreaction of vorticity+vector(+tensor) modes would require a notably different $k_{\sigma }/k_{\mathrm{nl}}$ which generally makes the agreement worse for $k\lesssim k_{\mathrm{nl}}$ (gray lines). It is therefore important to take this backreaction into account, i.e. use the ($\mathbf{cum}\mathbf{2}/\mathbf{3}+$, $\mathbf{svt}$) (or ($\mathbf{cum}\mathbf{2}/\mathbf{3}+$, $\mathbf{sv}$), see Fig. 11) schemes.

Figure 13

Velocity divergence and cross power spectra $P_{\delta \theta }$ (first row) and $P_{\theta \theta }$ (second row) for $n_s=2$ (left column) and $n_s=0$ (right column). Red circles with error bars show the corresponding $N$-body results. The linear (one-loop) approximation within the ($\mathbf{cum}\mathbf{3}+$, $\mathbf{svt}$) scheme of VPT is shown with dashed (solid) lines, with $k_{\sigma }/k_{\mathrm{nl}}$ fixed to the same value as for $P_{\delta \delta }$ (see Fig. 10) for each case, respectively. Therefore, these can be regarded as unique predictions of VPT as there is no free parameter being fit to the measurements.

Figure 14

Equilateral bispectrum at tree-level (dashed) and one-loop (solid) VPT compared to $N$-body results for $n_s=2,1,0,-1$, respectively. The bispectra are normalized to the SPT tree-level bispectrum. In each case $k_{\sigma }/k_{\mathrm{nl}}$ is fixed to the same value as for $P_{\delta \delta }$ (see Fig. 10). Therefore, these can be regarded as unique predictions of VPT as there is no free parameter being fit to the measurements.

Figure 15

Vorticity power spectrum $P_{w_i{w}_i}$ within VPT in one-loop (dashed) and two-loop (solid) approximation, for $n_s=2,1,0,-1$. The scaling $P_{w_i{w}_i}\propto k^2$ occurs only starting at two-loop order, while the one-loop result scales as $k^4$ for small $k$. The figure shows the dependence of the one- and two-loop pieces on $k_{\sigma }/k_{\mathrm{nl}}$, see Eq. (127). The error bars correspond to the Monte Carlo integration error (visible only at low $k$).

Figure 16

Vorticity power spectrum $P_{w_i{w}_i}$ within VPT in one-loop (dashed) and two-loop (solid) approximation, for $n_s=0$. The generic scaling $P_{w_i{w}_i}\propto k^2$ for low $k$ is obtained starting at two-loop only, since the one-loop piece is suppressed as $k^4$. The figure shows the difference between the ($\mathbf{cum}\mathbf{2}$, $\mathbf{sw}$) and ($\mathbf{cum}\mathbf{2}$, $\mathbf{sv}$) scheme (see Table 2), i.e., the backreaction of the vector modes of the velocity dispersion on vorticity.

Figure 17

Comparison of the vorticity power spectrum $P_{w_i{w}_i}$ in VPT at two-loop order (solid line) to $N$-body results (red circles) for $n_s=0$ (top) and $n_s=-1$ (bottom). The cross-over from $k^4$ to $k^2$ scaling for low $k$ is clearly observed in both the $N$-body and two-loop VPT results. Note that the absolute normalization of the $N$-body vorticity power spectrum is sensitive to mass resolution (see Appendix pp8). For $n_s=-1$ we rescaled the $N$-body result by a factor 4 in order to ease comparison of the shape. The dashed and dot-dashed lines show the one- and two-loop contributions, respectively. The value of $k_{\sigma }/k_{\mathrm{nl}}$ is fixed to the one obtained from $P_{\delta \delta }$ in the same VPT approximation, two-loop ($\mathbf{cum}\mathbf{2}$, $\mathbf{sv}$), see Eq. (124).

Figure 18

Tadpole diagram contributing to the one-loop power spectrum (left), and a corresponding diagram with an insertion of the source term $Q(\eta ){\delta }^{(3)}(\mathit{k})$ contributing to the equation of motion for the perturbations (right). Both diagrams cancel each other, to ensure that the perturbation around the background maintains a vanishing mean value. It is equivalent to skip the right diagram, and use the prescription to set the vertices to zero when the ingoing lines exactly add up to zero, thereby eliminating also the left diagram.

Figure 19

Generation of vorticity/vector perturbations at one-loop (left diagram) and backreaction on scalar power spectra (right diagram). Here $v_i\in \{w_i,{\nu }_i\}$ corresponds to dashed lines, and solid lines represent any scalar perturbations $s\in \{\delta ,\theta ,g,\delta \epsilon ,A,\pi ,\chi \}$ (possibly of different type for each line). The open square denotes the initial power spectrum, and the filled circles are nonlinear vertices.

Figure 20

Self-similarity of the density power spectrum for different spectral indices ($n_s=-1$ to $n_s=2$ from top to bottom at high-$k$). For each spectral index, different colors denote different outputs.

Figure 21

Self-similarity of the halo mass function for different spectral indices ($n_s=-1$ to $n_s=2$ from top to bottom at low $\widehat{m}=m/m_{*}$). For each spectral index, different colors denote different outputs.

Figure 22

Self-similarity of the velocity divergence power spectrum for different spectral indices ($n_s=-1$ to $n_s=2$ from top to bottom at high-$k$). For each spectral index, different colors denote different outputs.

Figure 23

Convergence study of the vorticity power spectrum measurements for $n_s=-1$ (left) and $n_s=0$ (right). Each line shows a single output (averaged over two pair-fixed realizations), starting from early outputs (highest amplitude at low-$k$) down to the last output. The dashed line denotes $P_{ww}\propto k^2$ expected as $k\to 0$. Convergence is achieved for the last 3–4 outputs for $n=-1$, which agree with each other at the 10% level. For $n_s=0$ the situation is less clear, as expected from the higher values of $\lambda k_{\mathrm{peak}}$ for this spectral index, but the last two outputs are fairly consistent with each other.