Large-scale structure perturbation theory without losing stream crossing

We suggest an approach to perturbative calculations of large-scale clustering in the Universe that includes from the start the stream crossing (multiple velocities for mass elements at a single position) that is lost in traditional calculations. Starting from a functional integral over displacement, the perturbative series expansion is in deviations from (truncated) Zel’dovich evolution, with terms that can be computed exactly even for stream-crossed displacements. We evaluate the one-loop formulas for displacement and density power spectra numerically in 1D, finding dramatic improvement in agreement with N-body simulations compared to the Zel’dovich power spectrum (which is exact in 1D up to stream crossing). Beyond 1D, our approach could represent an improvement over previous expansions even aside from the inclusion of stream crossing, but we have not investigated this numerically. In the process we show how to achieve effective-theory-like regulation of small-scale fluctuations without free parameters.

Figure 1

One loop suppression of displacement power, in 1D, for $n=2$. Dotted lines show the suppression of the linear power by $W(k)\equiv \exp (-k^2{c}^2/2)$, for the value of $c$ that sets the one loop correction to zero in the $k\to 0$ limit (black) or a smaller value for contrast (yellow), i.e., these are effectively truncated Zel’dovich, while the denominator is bare Zel’dovich. Solid adds the 1-loop correction, i.e., dotted is the leading order result and the difference between solid and dotted is the correction. For $n=2$ the correction in the $c\to 0$ limit is rapidly divergent. Note that the preferred $c$ is fixed entirely by the principle that the correction goes to zero in the low-$k$ limit—it is not a free parameter. In fact, we do not have simulations of the 1D displacement power, so the black curve is literally a (lowest order) prediction.

Figure 2

One loop suppression of displacement power, in 1D, for $n=1$, similar to Fig. 1.

Figure 3

One loop suppression of displacement power, in 1D, for $n=0.5$, similar to Fig. 1.

Figure 4

One loop suppression of displacement power, in 1D, for $n=0$.

Figure 5

One loop density power, in 1D, for $n=2$, relative to the initial power law. Solid black shows the final one loop prediction, using the value of $c$ fixed internally by the displacement power calculation as shown in Fig. 1. Black dotted shows the Zel’dovich power for the same $c$, i.e., the difference is the happily small one loop correction. Red dashed are the N-body results of [55], who say that their $n=2$ simulations had not fully converged. E.g., the points at low $k$ are almost surely a couple percent lower where they would agree well with our prediction (see the smaller-$n$ figures that follow where they did not have this issue). For comparison, we show the same quantities for $c=0.1$, where the one loop calculation makes a valiant effort to correct the massive failure of Zel’dovich, but it is clear that $c=0.6$ is a much better starting point for a perturbative expansion.

Figure 6

One loop density power, in 1D, for $n=1$, relative to the initial power law, similar to Fig. 5. Note that the values of $c$ used in all figures were computed literally “blind” to the simulation results—it was clear that $W(k)$ was needed to keep blue spectra from diverging (and from common-sense understanding of the physics), and the exact numerical values used here were settled by the displacement power calculation before the density power was ever computed.

Figure 7

One loop density power, in 1D, for $n=0.5$, relative to the initial power law, similar to Fig. 5.

Figure 8

One loop density power, in 1D, for $n=0$, relative to the initial power law, similar to Fig. 5.