On the Robustness of the Acoustic Scale in the Low-Redshift Clustering of Matter

TL;DR

This paper analyzes how non-linear structure formation alters the Baryon Acoustic Oscillation signature at low redshift, arguing that the dominant non-linear effect arises from differential motions of pairs separated by the acoustic scale. It develops a Lagrangian-displacement formalism, calibrated by a modest set of simulations, to model the degradation of the acoustic signal as a convolution with a pair-displacement distribution, and shows that the signature remains robust because the beat frequency is very small and the main peak occurs at large separations. Analytical (Zel'dovich) estimates combined with N-body results demonstrate that the displacement distributions are nearly Gaussian and that redshift-space distortions can be incorporated as anisotropic broadening. For biased tracers, local-bias theory predicts only small first-order shifts, with potential sub-percent biases that can be further constrained with simulations; overall, the acoustic scale remains a reliable standard ruler, though precise calibration will require large-volume surveys and tailored simulations. The proposed framework enables accurate forecasting without requiring prohibitively large simulations, by focusing on the displacement kernel that governs non-linear smearing of the acoustic peak.

Abstract

We discuss the effects of non-linear structure formation on the signature of acoustic oscillations in the late-time galaxy distribution. We argue that the dominant non-linear effect is the differential motion of pairs of tracers separated by 150 Mpc. These motions are driven by bulk flows and cluster formation and are much smaller than the acoustic scale itself. We present a model for the non-linear evolution based on the distribution of pairwise Lagrangian displacements that provides a quantitative model for the degradation of the acoustic signature, even for biased tracers in redshift space. The Lagrangian displacement distribution can be calibrated with a significantly smaller set of simulations than would be needed to construct a precise power spectrum. By connecting the acoustic signature in the Fourier basis with that in the configuration basis, we show that the acoustic signature is more robust than the usual Fourier-space intuition would suggest because the beat frequency between the peaks and troughs of the acoustic oscillations is a very small wavenumber that is well inside the linear regime. We argue that any possible shift of the acoustic scale is related to infall on 150 Mpc scale, which is O(0.5%) fractionally at first-order even at z=0. For the matter, there is a first-order cancellation such that the mean shift is O(10^{-4}). However, galaxy bias can circumvent this cancellation and produce a sub-percent systematic bias.

Figures (6)

• Figure 1: Snapshots of evolution of the radial mass profile versus comoving radius of an initially point-like overdensity located at the origin. All perturbations are fractional for that species; moreover, the relativistic species have had their energy density perturbation divided by 4/3 to put them on the same scale. The black, blue, red, and green lines are the dark matter, baryons, photons, and neutrinos, respectively. The redshift and time after the Big Bang are given in each panel. The units of the mass profile are arbitrary but are correctly scaled between the panels for the synchronous gauge. a) Near the initial time, the photons and baryons travel outwards as a pulse. b) Approaching recombination, one can see the wake in the cold dark matter raised by the outward going pulse of baryons and relativistic species. c) At recombination, the photons leak away from the baryonic perturbation. d) With recombination complete, we are left with a CDM perturbation towards the center and a baryonic perturbation in a shell. e) Gravitational instability now takes over, and new baryons and dark matter are attracted to the overdensities. f) At late times, the baryonic fraction of the perturbation is near the cosmic value, because all of the new material was at the cosmic mean. These figures were made by suitable transforms of the transfer functions created by CMBfast Sel96zZal00.
• Figure 2: The distribution of pairwise Lagrangian displacements for particles initially separated by $100h^{-1}{\rm\,Mpc}$. The left panel is in real space; the right panel is in redshift space. Both are at redshift $z=0.3$. The plot is shown as the log of probability versus the square of the separation so that a Gaussian distribution would be a straight line. The distribution in the radial direction is slightly skew; we fold the distribution at zero and show the infalling and outflowing distribution as separate lines. In real space, the distribution is nearly Gaussian; in redshift space, it is slightly cuspier. The displacement in the radial direction has slightly more variance than that in the tangential direction. For the redshift-space plot, the displacement is always in the direction along the line of sight. The displacements in the direction across the line of sight are of course identical to those in real space.
• Figure 3: The rms Lagrangian displacement in the radial direction as a function of redshift for initial separations of $100h^{-1}{\rm\,Mpc}$. The upper points are for redshift space when the initial separation is along the line of sight. The lower points are for real space. The lines are a model in which the real-space result varies as the growth function $D$, while the redshift-space result varies as $D(1+f)$, where $f = d\ln D/d\ln a$. The residuals to the fit are $\sim\!1\%$, although our simulations may not be accurate to this level due to their limited mass resolution.
• Figure 4: The power in the simulations compared to the model. The left column shows real space; the right column shows redshift space. The top row is $z=0.3$, the middle $z=1$, and the bottom row is $z=3$. In all panels, the simulation power has been fit over the range $0<k<0.4h{\rm\,Mpc}^{-1}$ to a model of $b^2 P_{\rm smear} + a_0 + a_2 k^2 + a_4 k^4 +a_6 k^6$, where $P_{\rm smear}$ is the smeared linear power spectrum $P_{\rm linear}$, with the small scales restored by the no-wiggle form. What is plotted is the residual $P_{\rm res} = (P_{\rm measured} - a_0-a_2 k^2-a_4 k^4 -a_6 k^6)/b^2$ divided by the linear power spectrum. In detail, we divide by the initial power spectrum of the simulation, so that the variations in the initial amplitudes of particular modes cancel out; this doesn't change the sense of the plot but tightens the residuals slightly. The smeared model ( red) has been similarly divided by the linear power spectrum. If the linear spectrum were fully intact, it would be flat in this plot. Erasure of the acoustic peaks produces oscillations in this plot. One sees that the model provides an excellent match to the observed degradation.
• Figure 5: The correlation function in the simulations compared to the model with the small-scale linear power restored. The left panel shows $z=0.3$; the right, $z=1$. In both panels, the simulation data is the solid black line, the linear correlation function is the short-dashed thin line, and the model correlation function is the long-dashed red line. We have not removed any broad-band nuisance spectra in making this figure.