Smooth sailing or ragged climb? -- Increasing the robustness of power spectrum de-wiggling and ShapeFit parameter compression

TL;DR

The paper investigates how to robustly extract ShapeFit’s broadband slope parameter $m$ from galaxy power spectra in the presence of BAO wiggles. It systematically compares 13 de-wiggling methods, showing ~2% de-wiggling systematics but up to ~50% variation in $m$ across methods; to improve robustness, it explores non-local derivative schemes and post-processing (notably Savitzky–Golay smoothing) and derives a conservative systematic uncertainty $\sigma_{m,\mathrm{syst}} = 0.023 |m| + 0.001$. The authors propose a practical, consistent approach (TANH fixed or SG-smoothed with a local derivative) to estimate $m$ with minimized bias across common cosmologies, while highlighting caveats for models that alter early-time physics (e.g., Early Dark Energy). The resulting framework supports reliable cosmological inferences from current and upcoming surveys, with a clear recipe for incorporating $m$-level systematics into the inference pipeline.

Abstract

The baryonic features in the galaxy power spectrum offer tight, time-resolved constraints on the expansion history of the Universe but complicate the measurement of the broadband shape of the power spectrum, which also contains precious cosmological information. In the context of ShapeFit, the broadband information is compressed into a single parameter, the slope of the power spectrum at the pivot scale, $m$, is sensitive to matter-radiation equality and the baryonic suppression. To calculate this parameter, two steps are necessary: 1) smoothing the power spectrum to remove the baryonic oscillations and 2) calculating the derivative of the power spectrum ratio at the pivot scale. In this work we compare thirteen methods designed to separate the broadband and oscillating components and examine their performance. The systematic uncertainty between different de-wiggling procedures is at most $2$%, depending on the scale. For the obtained slope, we show that the de-wiggling procedures impart large 50% differences, but as long as the theory and data pipelines are consistent, this is of no concern for cosmological inference given the precision of existing and ongoing surveys. However, it still motivates the search for more robust ways of extracting the slope. We show that post-processing the power spectrum ratio before taking the derivative makes the slope values far more robust. We further investigate eleven ways of extracting the slope and highlight the two most successful ones. We derive a systematic uncertainty on the slope $m$ of $σ_{m,\mathrm{syst}} = 0.023 |m| + 0.001$ by studying the behavior of the slopes in different cosmologies and the impact in cosmological inference. In cosmologies with a feature in the matter-power spectrum, such as in the early dark energy cosmologies, this systematic uncertainty estimate does not necessarily hold, and further investigation is required.

Figures (32)

• Figure 1: A schematic representation of the relevant information imprinted in the matter power spectrum. The scales relevant to DESI correspond to $0.02h/\mathrm{Mpc}$-$0.2h/\mathrm{Mpc}$DESI:2024hhd, while we use the ShapeFit pivot wavenumber of $0.03h/\mathrm{Mpc}$brieden2021shapefit.
• Figure 2: Left: Power spectra for the fiducial (black) and showcase (green) cosmology (see table \ref{['tab:cosmologies']} for the corresponding cosmological parameters values), and their corresponding de-wiggled power spectra (blue and red dashed lines, respectively). The example de-wiggling algorithm used here is the "Cubic Inflections" algorithm of \ref{['ssec:inflections']}. Right: Ratio of the power spectrum to the no-wiggle power spectrum, for both cosmologies, highlighting the baryonic acoustic oscillations.
• Figure 3: Schematic overview of de-wiggling methods. The red line corresponds to the combined broadband spectrum with additional wiggles (as a power spectrum on the left, and as a correlation function on the right) while blue corresponds to the dewiggled broadband. The green/yellow/orange lines and points show how the corresponding de-wiggled broadband function is obtained. For the "Smoothing" method, the yellow/orange/green lines show sufficiently long smoothing windows at certain locations across the function, whose averages are correspondingly shown as yellow/orange/green points, through which a smooth function is drawn. For the "Fitting" method, the green/orange lines show the mean squared deviation that has to be minimized in any approach that fits a smooth function to the oscillations. For the "Inflections" method, the green point show the inflections of the oscillations (after subtracting a rough broadband trend) through which the blue function is fitted. Finally, for the "Peak removal" method, the correlation function is fitted where there is no BAO peak by a smooth function, resulting in the blue curve directly.
• Figure 4: Ratio of the (shifted) power spectra for the computation of \ref{['eq:mdef']}, using the fiducial and showcase cosmologies of \ref{['tab:cosmologies']}. Left: Absolute ratio, showing the ratio without de-wiggling, and for all the de-wiggling methods the mean, median, and the 10-90% quantile range. Right: Relative ratio, normalized by the median. The dashed vertical grey line represents the pivot wavenumber $k_p$ at which the ShapeFit slope (see \ref{['sec:ShapeFit']}) is evaluated.
• Figure 5: Gradient method for the derivative. Left: Approximation of the logarithm of the ratio used for \ref{['eq:mdef']} (approximation in solid, true function in dashed lines). Right: Derivative of the functional approximation (whose value at the pivot scale $k_p$ is taken as $m$).