Stable clustering, the halo model and nonlinear cosmological power spectra

TL;DR

The paper tests longstanding predictions for nonlinear gravitational clustering by running high-resolution, scale-free N-body simulations that reveal stable clustering is not generally realized on small scales. It critiques the HKLM scaling and demonstrates how halo mergers modify haloes, leading to a shallower nonlinear power than classic stable-clustering expectations. By combining halo-model concepts with HKLM-like scaling, the authors derive a new fitting function that accurately matches both scale-free and CDM simulations, outperforming the PD96 and JMW95 prescriptions. The work provides practical tools and data to predict nonlinear power spectra for general curved spectra, with implications for interpreting weak lensing and galaxy clustering measurements. It also opens avenues for higher-order statistics using the halo-model framework.

Abstract

We present the results of a large library of cosmological N-body simulations, using power-law initial spectra. The nonlinear evolution of the matter power spectra is compared with the predictions of existing analytic scaling formulae based on the work of Hamilton et al. The scaling approach has assumed that highly nonlinear structures obey `stable clustering' and are frozen in proper coordinates. Our results show that, when transformed under the self-similarity scaling, the scale-free spectra define a nonlinear locus that is clearly shallower than would be required under stable clustering. Furthermore, the small-scale nonlinear power increases as both the power-spectrum index n and the density parameter Omega decrease, and this evolution is not well accounted for by the previous scaling formulae. This breakdown of stable clustering can be understood as resulting from the modification of dark-matter haloes by continuing mergers. These effects are naturally included in the analytic `halo model' for nonlinear structure; using this approach we are able to fit both our scale-free results and also our previous CDM data. This approach is more accurate than the commonly-used Peacock--Dodds formula and should be applicable to more general power spectra. Code to evaluate nonlinear power spectra using this method is available from http://as1.chem.nottingham.ac.uk/~res/software.html Following publication, we will make the power-law simulation data available through the Virgo website http://www.mpa-garching.mpg.de/Virgo

Figures (16)

• Figure 1: Slices showing the growth of structure in the glass $n=-2$ simulation (left column) and 'grid-start' $n=-1.5$ simulation (right column). All of the slices are of thickness $L/10$. From the $n=-2$ simulation we show expansion factors $a=0.2, 0.45$ and $0.55$, and from the $n=-1.5$ simulation we show epochs $a=0.25,0.63$ and $1.0$. The normalization of the final states in the $n=-2$ and $n=-1.5$ runs were $\Delta^2(2\pi/L,a=1.0)= 0.133$ and $0.046$, respectively.
• Figure 2: Same as Fig. \ref{['slice1']}, but this time showing the comoving projection of particles in the glass $n=-1$ simulation (left column) and glass $n=0$ simulation (right column). From the $n=-1.0$ simulation we show epochs $a=0.25, 0.63$ and $0.83$, and from the $n=0$ simulation we show expansion factors $a=0.1, 0.3$ and $0.5$. The normalization of the final states in the $n=-1$ and $n=0$ runs were $\Delta^2(2\pi/L,a=1.0)= 0.017$ and $0.003$, respectively.
• Figure 3: (Top) The glass-discreteness corrected (squares) and uncorrected power spectrum (stars) of the glass $n=-2$ simulation at an epoch $a=0.55$. We show the linear fluctuation spectrum (dashed line), which demonstrates that the box scale mode is still linear, the nonlinear spectrum according to the scaling formula of PD96 (thin solid line), a shot noise spectrum (dot-dashed line) and our two power-law discreteness model outlined in the text (thick solid line). (Bottom) Three epochs from the early stages of the same $n=-2$ simulation. From bottom to top epochs are $a=0.025$ (squares), 0.1 (circles) and 0.3 (stars). This demonstrates that the discreteness spectrum does not evolve and also that the linear spectrum has been correctly established early on. Again, the lines are as in the top panel, with the thick solid line representing our fit to the discreteness spectrum.
• Figure 4: (Top) Comparison of discreteness uncorrected power spectra measured from the quiet start (squares) and glass start (stars) $n=-2$ simulations at epochs $a=0.4$ and 0.55. Line styles are as in Fig. \ref{['glass']}. (Bottom) Comparison of discreteness corrected power spectra for the same outputs from the two simulations.
• Figure 5: Nonlinear power plotted against linear power (points) for the four scale-free simulations. For clarity, the data have been separated from each other by one order of magnitude in the $y$-direction, with the $n=0$ data untranslated. To determine the linear power given a nonlinear data point, the appropriate linear scale is required. In the HKLM method, this is found using the transformation $k_{{{\rm L}}}=[1+\Delta^2_{{{\rm NL}}}(k_{{{\rm NL}}})]^{-1/3}k_{{{\rm NL}}}$. The solid line represents the fitting formula for the Einstein--de Sitter models presented in Appendix \ref{['appHKLM']}; the dashed line represents the PD96 fitting formula; the dotted lines are the fits using the formula of JMW95.