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

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 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. The ShapeFit parameter compression aims to compute the slope $m$ of the power spectrum at large scales, sensitive to matter-radiation equality and the baryonic suppression. We show that the de-wiggling procedures impart large (50%) differences on the obtained slope values, but as long as the theory and data pipelines are set up consistently, this is of no concern for cosmological inference given the precision of existing and on-going 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 $\sigma_{m,\mathrm{syst}} = 0.023|m| + 0.001$ by studying the behavior of the slopes in different cosmologies within and beyond $\Lambda$CDM 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.