Simulations of Baryon Acoustic Oscillations I: Growth of Large-Scale Density Fluctuations

TL;DR

This work investigates how accurately large-scale density perturbations are evolved in cosmological $N$-body simulations, with a focus on baryon acoustic oscillations (BAO). It shows that finite box size causes nonlinear mode coupling among a few largest-scale modes, leading to systematic deviations from linear growth and artificial oscillations in single realizations. The authors develop a second-order perturbation theory model with the kernel $F_2$ that quantitatively reproduces these deviations, providing a practical correction to recover linear BAO features in individual realizations. They find that the dispersion from finite-mode coupling scales as $\propto L^{-3/2}\Delta k^{-1/2}$ and remains non-negligible even for volumes of a few hundred Mpc, implying that percent-level BAO precision requires very large simulations ($L>2\,h^{-1}$Gpc) or higher-order corrections.

Abstract

We critically examine how well the evolution of large-scale density perturbations is followed in cosmological $N$-body simulations. We first run a large volume simulation and perform a mode-by-mode analysis in three-dimensional Fourier space. We show that the growth of large-scale fluctuations significantly deviates from linear theory predictions. The deviations are caused by {\it nonlinear} coupling with a small number of modes at largest scales owing to finiteness of the simulation volume. We then develop an analytic model based on second-order perturbation theory to quantify the effect. Our model accurately reproduces the simulation results. For a single realization, the second-order effect appears typically as ``zig-zag'' patterns around the linear-theory prediction, which imprints artificial ``oscillations'' that lie on the real baryon-acoustic oscillations. Although an ensemble average of a number of realizations approaches the linear theory prediction, the dispersions of the realizations remain large even for a large simulation volume of several hundred megaparsecs on a side. For the standard $Λ$CDM model, the deviations from linear growth rate are as large as 10 percent for a simulation volume with $L = 500h^{-1}$Mpc and for a bin width in wavenumber of $Δk = 0.005h$Mpc$^{-1}$, which are comparable to the intrinsic variance of Gaussian random realizations. We find that the dispersions scales as $\propto L^{-3/2} Δk^{-1/2}$ and that the mean dispersion amplitude can be made smaller than a percent only if we use a very large volume of $L > 2h^{-1}$Gpc. The finite box size effect needs to be appropriately taken into account when interpreting results from large-scale structure simulations for future dark energy surveys using baryon acoustic oscillations.

Figures (8)

• Figure 1: We plot the evolution of the power spectrum from the initial epoch (black line) to $z=3$ (green), $z=1$ (blue), and $z=0$ (purple). The measured power spectrum is divided by the no-wiggle model of Eisenstein & Hu (1998).We subtract the intrinsic deviations from the input power spectrum at the initial epoch. The numbers indicate integer sums of $n_1^2+n_2^2+n_3^2$ of wavenumber vectors. The dashed lines are the one-loop power spectra at each redshift (see text).
• Figure 2: Evolution of the deviation of the power amplitude with respect to the linear theory prediction. The dots are the measurements from our simulation, and red solid lines are the model prediction using the second-order perturbation theory. The integers denote $n^2=n_1^2+n_2^2+n_3^3$ of wavenumbers, and the figures show different range of $n^2$, $n^2=1-8$ (upper left panel), $n^2=9-16$ (upper right panel), $n^2=17-24$ (lower left panel), and $n^2=25-32$ (lower right panel).
• Figure 3: Same as Fig.\ref{['mode_evolv']}, but for phase evolution in units of radians. We plot the results only for modes with $n_1 \geq n_2 \geq n_3$.
• Figure 4: The amplitude dispersions of the $100$ realizations at $z=0$ for $L=500 h^{-1}$ Mpc (top), $1 h^{-1}$ Gpc (middle), and $2 h^{-1}$ Gpc (bottom). The grey dots with error bars are for the un-binned data, while the black big symbols are for the binned data of $\Delta k=0.005 h$/Mpc. The value of $k$ for the binned data is the weighted mean of $k$ with the number of wavenumbers in the bin. The dashed lines are the theoretical prediction.
• Figure 5: The amplitude dispersions calculated from our simulation outputs (filled circle $\bullet$) and the theoretical predictions (solid lines). We also show the dispersions due to the initial Gaussian distribution (dashed lines). The vertical dotted line is the position of the BAO first peak.