Free Cosmic Density Bispectrum on Small Scales

Abstract

We study the asymptotic behaviour of the free, cold-dark matter density fluctuation bispectrum in the limit of small scales. From an initially Gaussian random field, we draw phase-space positions of test particles which then propagate along Zel'dovich trajectories. Only initial momentum-momentum correlation are considered, making the formulas identical to the typical Zel'dovich approximation. A suitable expansion of the initial momentum auto-correlations of these particles leads to an asymptotic series whose lower-order power-law exponents we calculate. The dominant contribution has an exponent of $-11/2$. For triangle configurations with zero surface area, this exponent is even enhanced to $-9/2$. These power laws can only be revealed by a non-perturbative calculation with respect to the initial power spectrum. They are valid for a general class of initial power spectra with a cut-off function, required to enforce convergence of its moments. We then confirm our analytic results numerically. Finally, we use this asymptotic behaviour to investigate the shape dependence of the bispectrum in the small-scale limit, and to show how different shapes grow over cosmic time. These confirm the usual model of gravitational collapse within the Zel'dovich picture.

Figures (5)

• Figure 1: The left panel shows the correlation functions $a_1$ and $a_2$ (cf. Eqs. (\ref{['eq:bispec:a1']}), (\ref{['eq:bispec:a2']})). The vertical dashed line indicates the heuristic value up to which point the asymptotic description (cf. Eqs. (\ref{['eq:a1:definition']})-(\ref{['eq:a2:definition']})) is used for the functions instead of their full integral expressions. In the right panel, the important combination $a_1+\sigma_1^2/3$ is shown as a naive addition and using the asymptotic expansion to provide numerical stability for small $q$. As later computations require this stability for very small $q$, an implementation of these functions using asymptotic descriptions is important. The initial power spectrum is taken as the Bardeen power spectrum bard86 using an exponential cut-off with $k_s=10^2 \, h\operatorname{Mpc}^{-1}$.
• Figure 2: The absolute value of the integrand of the bispectrum (cf. Eq. (\ref{['eq:bispec:general']})) is shown in the three panels. In the top left one, the integrand is plotted along the $q_{12x}$ axis which is non-oscillating. The bottom panel shows the integrand along the $q_{13z}$ axis, on which the behaviour is expected to go with $\sim\exp(-x^4)$. The right panel shows the integrand along a random direction in $\textbf{q}$-space. The Bardeen power spectrum bard86 and an equilateral $\textbf{k}$ configuration are used. The integrand has been transformed based on a transformation of the Hessian of the exponent of the integrand of the bispectrum. The dashed lines show the integrand's asymptotic approximation by the splitting lemma. For large triangle sidelengths $k$, the agreement of the approximation and the original integrand can be seen in all directions.
• Figure 3: The left panel shows the numerical integration result of the free bispectrum, with asymptotic approximations for $k\rightarrow 0$ and $k\rightarrow\infty$. The right panel shows the reduced bispectrum, where the bispectrum is normalized with the square of the linearly evolved power spectrum. The bottom figures show the relative deviation with respect to the numerical result. The $\textbf{k}$ vector configuration used is an equilateral triangle with side length $k$. The initial power spectrum is taken as the Bardeen power spectrum bard86 using an exponential cut-off with $k_s=10^2 \, h\operatorname{Mpc}^{-1}$.
• Figure 4: Both panels show the dependence of the reduced bispectrum $Q$ on the $\textbf{k}$ configuration. The left panel is an isosceles configuration, with changing angle $\pi-\alpha$ between $\textbf{k}_2$ and $\textbf{k}_3$ while keeping the total circumference $U=\lvert\textbf{k}_1\rvert+\lvert\textbf{k}_2\rvert+\lvert\textbf{k}_3\rvert=4000 h\text{Mpc}^{-1}$ constant. The right panel keeps $\lvert\textbf{k}_2\rvert$ and $\lvert\textbf{k}_3\rvert$ constant, with $\lvert\textbf{k}_2\rvert=2 \lvert\textbf{k}_3\rvert = 1000h\text{Mpc}^{-1}$. The reduced bispectrum in the large-scale approximation $Q_{k\rightarrow 0}$ is rather insensitive to cosmology, specifically there is no dependence on time or overall $k$-scale in this lowest order approximation bern02. The values of the large-scale approximation had to be rescaled such that both results fit in the same plot.
• Figure 5: The figure gives the value of the reduced bispectrum for two configurations as a function of time $\tau_2^2=t^2\sigma_2^2$. The first configuration is the equilateral one, corresponding to cosmic filaments. The second one is a parallel one, corresponding to pancakes lewi11. The large-scale approximation, to lowest order, is completely insensitive to the used cosmological model and time, in these $\textbf{k}$ configurations. The small-scale asymptotics gives an interesting growth of different structures within the Zel'dovich picture. The dots in the plot indicate what would correspond to the current cosmic time, given different values of $\sigma_2^2$ due to different cut-offs $k_s$. The squares indicate the maximum amplitude, which are shown to be independent of the higher moments $\sigma_{n>2}^2$.