Deconstructing Baryon Acoustic Oscillations: A Comparison of Methods

TL;DR

This paper quantitatively compares three BAO analysis strategies—Full power spectrum $P(k)$, wiggles-only, and spherical-harmonic $C(\ell)$—using Fisher-matrix forecasts for two DETF-style surveys to constrain a 7-parameter cosmology with evolving dark energy. It shows that the dark energy Figure of Merit can vary by up to a factor of $\sim 35$ across methods and depends strongly on the chosen wavenumber range, with full broadband information offering the strongest constraints but higher susceptibility to systematics. The results reveal a hierarchy in constraining power ($P(k)$ highest, wiggles-only lowest, $C(\ell)$ intermediate) and demonstrate that Planck priors significantly reduce method-to-method differences, underscoring the value of joint BAO+CMB analyses. Overall, the work highlights the need to specify the BAO analysis method in forecasts and provides guidance on how non-linear cutoffs and survey design influence cosmological constraints.

Abstract

The Baryon Acoustic Oscillations (BAOs) or baryon wiggles which are present in the galaxy power spectrum at scales 100-150Mpc/h are powerful features with which to constrain cosmology. The potential of these probes is such that these are now included as primary science goals in the planning of several future galaxy surveys. However, there is not a uniquely defined BAO Method in the literature but a range of implementations. We study the assumptions and cosmological performances of three different BAO methods: the full Fourier space power spectrum [P(k)], the `wiggles only' in Fourier space and the spherical harmonics power spectrum [C(l)]. We contrast the power of each method to constrain cosmology for two fiducial surveys taken from the Dark Energy Task Force (DETF) report and equivalent to future ground and space based spectroscopic surveys. We find that, depending on the assumptions used, the dark energy Figure of Merit (FoM) can change by up to a factor of 35 for a given fiducial model and survey. We compare our results with the DETF implementation and, discuss the robustness of each probe, by quantifying the dependence of the FoM with the wavenumber range. The more information used by a method, the higher its statistical performance, but the higher its sensitivity to systematics and implementations details.

Figures (5)

• Figure 1: Illustration of the main building blocks of the linear galaxy power spectrum. Panel A: The observed linear galaxy power spectrum. This includes the linear galaxy bias, the smooth power spectrum (broad band power), baryonic wiggles and linear redshift distortions. In linear theory the redshift evolution of the galaxy power spectrum depends solely on the linear growth factor (see equation \ref{['clustering:eq:real']}); we illustrate this by comparing the linear galaxy power spectrum at redshift $z=0$ (solid blue line) and redshift $z=0.5$ (dot-dashed red line). Panels B, C and D illustrate the different building blocks of the galaxy power spectrum. Panel B: The smooth part of the galaxy power spectrum contains information in the shape and normalization of the power spectrum; it can be calculated using the analytic formula given by Eisenstein & Hu (1999). Panel C: The ratio (blue solid line) of the full power spectrum and the smooth part of the power spectrum reveals the residual baryonic wiggles. The dashed line corresponds to no baryonic wiggles. Panel D: The ratio (blue solid line) of the radial galaxy power spectrum ($\mu = 1$ in equation \ref{['clustering:eq:biasdef']}) to the tangential spectrum ($\mu = 0$) illustrates the scale-independent effect of linear redshift distortions. Linear redshift distortions add power in the radial direction. The dashed line corresponds to no redshift distortions.
• Figure 2: The value of $k_{\rm max}$ as a function of redshift z. The solid line (diamonds) corresponds $\sigma(R)<0.20$ and $k_{\rm max}<0.25 h{\rm Mpc}^{-1}$. The dashed and dot-dashed lines correspond respectively to deviations around the central line, specifically $k_{\rm max}(z)\cdot 0.5$ (dot-dashed) and $k_{\rm max}(z)\cdot 1.2$ (dashed) respectively and are used in section \ref{['forecasts:kmax']}.
• Figure 3: Marginalised 1$\sigma$ constraints (without Planck priors) on dark energy equation of state parameters $w_0$ and $w_a$ for the three different BAO methods: Red (dotted): Full P(k), Blue (solid): C(l), Green (hashed: BAO 'wiggles only'. TOP: ground based survey (BAO-IIIS-o). BOTTOM: space-based survey (BAO-IVS-o).
• Figure 4: Figure of Merit (FoM) as a function of normalized $k_{\rm max}$ for the full power spectrum $P(k)$ [dotted, diamonds], the spherical harmonic $C(\ell)$ [solid, squares] and the 'wiggles only' [dot-dashed, triangles] methods - for the space based survey BAO-IVS-o. The normalized values of $k_{\rm max}$ are taken from Figure \ref{['implementation:fig:kmax']}. The left-hand panel corresponds to constraints from the BAO methods without any priors, the right-hand panel includes CMB constraints from Planck.
• Figure 5: Factor of improvement of the Figure of Merit (FoM) as a function of normalized $k_{\rm max}$ for the full power spectrum $P(k)$ [dotted, diamonds], the spherical harmonic $C(\ell)$ [solid, squares] and the 'wiggles only' [dot-dashed, triangles] methods - for the space based survey BAO-IVS-o. The improvement is quantified relatively to the value of the FoM at $k^{\rm Norm}_{\rm max} = 0.5\cdot k_{\rm max}^{\rm fid}$. The normalized values of $k_{\rm max}$ are taken from Figure \ref{['implementation:fig:kmax']}. The left-hand panel corresponds to constraints from the BAO methods without any priors, the right-hand panel includes CMB constraints from Planck. In the left panel (without Planck priors), it is clear that the FoM for the $P(k)$ methods evolves more rapidly with $k_{\rm max}$ than the $C(\ell)$ method, which in turn evolves more rapidly than the 'wiggles only' method. When Planck priors are included, the FoM from the $C(\ell)$ method evolves more rapidly than the other two methods. This may be do to different parameter degeneracies. In this case, the FoM from the $P(k)$ method evolves more rapidly than the that for the 'wiggles only' method at high $k_{\rm max}$.