Calibrating the Baryon Oscillation Ruler for Matter and Halos

TL;DR

This study quantifies how nonlinear gravitational evolution and halo bias shift and broaden the baryon acoustic oscillation (BAO) feature, using perturbation theory plus large-volume N-body simulations. The key finding is that a second-order term, $P_{22}$, drives most of the shift by acting like a scaled derivative of the linear power spectrum $P_L$, and that halos imprint bias-dependent shifts that can be captured with Eulerian/Lagrangian bias frameworks and a calibrated $b_1$–$b_2$ relation. The authors propose a corrected halo/galaxy BAO template that incorporates damping and a $P_{22}$-derived correction, enabling unbiased recovery of the acoustic scale $\alpha$ and reducing systematics below planned survey thresholds. They also extend the results to redshift space and to $\text{ΛCDM}$ cosmologies, outlining practical implications for Stage III/IV BAO experiments and highlighting avenues for further refinement of bias modeling and additional physics.

Abstract

We characterize the nonlinear evolution of the baryon acoustic feature as traced by the dark matter and halos, using a combination of perturbation theory and N-body simulations. We confirm that the acoustic peak traced by the dark matter is both broadened and shifted as structure forms, and that this shift is well described by second-order perturbation theory. These shifts persist for dark matter halos, and are a simple function of halo bias, with the shift (mostly) increasing with increasing bias. Extending our perturbation theory results to halos with simple two parameter bias models (both in Lagrangian and Eulerian space) quantitatively explains the observed shifts. In particular, we demonstrate that there are additional terms that contribute to the shift that are absent for the matter. At z=0 for currently favored cosmologies, the matter shows shifts of ~0.5%, b=1 halos shift the acoustic scale by ~0.2%, while b=2 halos shift it by ~0.5%; these shifts decrease by the square of the growth factor at higher redshifts. These results are easily generalized to galaxies within the halo model, where we show that simple galaxy models show marginally larger shifts than the correspondingly biased halos, due to the contribution of satellites in high mass halos. While our focus here is on real space, our results make specific predictions for redshift space. For currently favored cosmological models, we find that the shifts for halos at z=0 increase by ~0.3%; at high z, they increase by ~0.5% D^2. Our results demonstrate that these theoretical systematics are smaller than the statistical precision of upcoming surveys, even if one ignored the corrections discussed here. Simple modeling, along the lines discussed here, has the potential to reduce these systematics to below the levels of cosmic variance limited surveys.

Figures (13)

• Figure 1: The mass functions at $z=0$ for our $c$CDM (triangles) and $\Lambda$CDM (squares) cosmologies, along with two commonly used fitting functions due to Press & Schechter PS (dotted, blue) and Sheth & Tormen STbias (dashed, red). Since we are using the sum of the particles in a FoF group for our definition of mass, we do not expect perfect agreement with either fitting function. The simulations for the $\Lambda$CDM cosmology are discussed in §\ref{['sec:implications']}.
• Figure 2: The nonlinear matter power spectrum at $z=0$ (red, upper) and $z=1$ (blue, middle), compared to the linear theory (black, lower) for our $c$CDM and $\Lambda$CDM models. The $z=1$ power spectra have been scaled by $1/D^2$ to match the other power spectra on large scales and the $\Lambda$CDM spectra have been offset (vertically) for clarity. Note the strong damping of the oscillations and the large excess power on small scales in the evolved fields.
• Figure 3: The shift in the acoustic scale, $\alpha-1$ vs. redshift for the mass (black triangles) and $\nu=1.9$ halos (red squares) using Eq. (\ref{['eqn:ESW']}) as the template. Also shown are the best fit power-laws (dashed) and the expectations of perturbation theory [$\alpha-1\propto D^2(z)$] (dotted). The points and curves for the halos are shifted by $\delta z = 0.05$ for clarity.
• Figure 4: The out-of-phase contribution predicted by perturbation theory well approximates the derivative of the acoustic signal. The points plot $P_{22}$, while the line is a scaled version of $dP_L/d\ln k$ with the scaling given in the inset. The smooth components of both curves have been subtracted by fitting a cubic spline to the data. All curves are for $z=0$.
• Figure 5: The $\Delta\chi^2$ for fits of our $z=0$ mass power spectrum to the functional form of Eq. \ref{['eqn:ESW']} (dashed red), to the form including $P_{22}$ (solid black) and marginalizing over the amplitude of the $P_{22}$ term (dashed blue). There are $60$ degrees of freedom in the fit, and in each case the best fit is a reasonable fit. Note that Eq. \ref{['eqn:ESW']} gives a biased acoustic scale, including the $P_{22}$ term eliminates the bias, and allowing the amplitude of the $P_{22}$ term to float results in very weak constraints.