The resummation model in FLAMINGO: precisely predicting matter power suppression from observed halo baryon fractions

TL;DR

Baryonic feedback alters the matter power spectrum, complicating cosmological parameter inference from Stage-IV surveys. The authors develop an improved resummation model that, with zero free parameters, converts observed halo baryon fractions into a precise suppression signal by leveraging a universal relation between retained mass and baryon content, calibrated on the FLAMINGO simulations. The approach achieves percent-level accuracy up to $k\approx 10\,h\mathrm{Mpc}^{-1}$ (extendable to $k\approx 25$ with inner-region data) and remains robust across different cosmologies and feedback implementations, including at redshifts $z>0$ with small parameterizations. These results enable direct marginalization over baryon uncertainties and even reconstruction of the DMO halo mass function from observed halos, with a publicly available Python package to facilitate practical application in data analyses.

Abstract

In order to derive unbiased cosmological parameters from Stage-IV surveys, we need models that can predict the matter power spectrum for at least $k\,\lesssim\,10\,h\mathrm{\,Mpc^{-1}}$ with percent-level accuracy. The main challenge in this endeavour is that baryonic feedback significantly redistributes matter on large scales, but to an unknown degree. Here, we present an improved version of the "resummation" model, which maps observed halo baryon fractions of massive haloes ($M_{\mathrm{500c}}\gtrsim 10^{12.5}\, \mathrm{M_\odot}$) to a flexible suppression signal - i.e. the ratio of baryonic to dark-matter-only (DMO) matter power spectra - using zero free parameters. We calibrate this model to the FLAMINGO hydrodynamical simulations, obtaining a typical accuracy of $\lesssim 1\%$ for $k\,\leq\,10\,h\mathrm{\,Mpc^{-1}}$ given mean halo baryon fractions within the spherical overdensity radii $R_{\mathrm{500c}}$ and $R_{\mathrm{200m}}$. When only those within $R_{\mathrm{500c}}$ are available, we still obtain $\lesssim 2\%$ accuracy. We show that given small-scale stellar mass fractions, the model can be extended to yield $\lesssim 3\%$ accurate suppression signals for all scales measured ($k\,\leq\,25\,h\mathrm{\,Mpc^{-1}}$). We also extend the model to redshifts $z>0$. Central to the model is a seemingly mass-independent and feedback-independent relation that allows observed halo masses to be mapped to equivalent DMO halo masses using only observed mean halo baryon fractions, to $\lesssim 1\%$ accuracy. This relation can also be used to retrieve the DMO halo mass function from observed halo masses and baryon fractions with percent-level accuracy, without any assumptions on the strength of feedback. A Python package implementing the resummation model is made publicly available.

Figures (22)

• Figure 1: The halo-matter cross power components for two different overdensity regions for L1_m9_DMO. Here and elsewhere halo mass selection is consistently done based on $M_{\mathrm{500c}}$. Left: The cross power for mass inside overdensity regions with $\Delta=\mathrm{200m}$ with all matter. Also shown are the total halo-matter cross power (dot-dashed orange), the cross power of matter not in (resolved) haloes and all matter (dot-dashed green), and the model predictions for L1_m9 based on these cross-power components and the measured baryon fractions (dashed lines). Right: As on the left-hand side, but for $\Delta=\mathrm{500c}$ and without model predictions. Note that as $\Delta$ increases, the radius decreases, and the total halo contribution becomes smaller while the non-halo contribution becomes larger.
• Figure 2: The retained mass fraction, $f_\mathrm{ret}=M_\mathrm{tot,bar}/M_\mathrm{tot,DMO}$, as a function of the corrected baryon fraction, $f_\mathrm{bc}=(1-\Omega_\mathrm{b}/\Omega_\mathrm{m})/(1-\bar{f}_\mathrm{b})$, for masses measured within $R_{\mathrm{200m}}$ in the FLAMINGO simulations at $z=0$. Every point represents the average in a $0.5\,\mathrm{dex}$ halo mass bin for all halo masses $M_{\mathrm{500c,DMO}}\geq10^{12}\, \mathrm{M_\odot}$. Different colours represent simulations with different cosmologies and/or feedback. Dotted lines show $f_\mathrm{ret}=f_\mathrm{bc}$, which is the expected retained mass fraction inside a fixed radius if only baryonic mass is removed. The solid lines show the best fit of equation \ref{['eq:fretained']} to all simulations simultaneously. All halo average retained mass fractions for all simulations lie within $1$ percentage point of this line (indicated by dashed lines). Left: Haloes are matched between the hydro simulation and DMO; both $f_\mathrm{bc}$ and $f_\mathrm{ret}$ are measured at a DMO halo mass $M_i$. Right: The measured $f_\mathrm{bc}$ for hydro halo mass $M_i$ translates to an $f_\mathrm{ret}$ for DMO halo mass $M_i$. This version of the relation is most readily applicable to observations.
• Figure 3: As Fig. \ref{['fig:fret_fbc_200m_fixedhaloes_fixedmass']}, but for $\Delta=\mathrm{500c}$. While the scatter increases as we look deeper into the haloes, equation \ref{['eq:fretained']} still describes the retained mass fractions of all relations very well. Note that L2p8_m9 (square symbols), which has by far the best statistics at equal resolution because it has $(2.8)^3\times$ the volume of the other simulations shown, always falls well within $1$ percentage point of the fit for all halo masses. The most massive haloes have retained mass fractions significantly above unity, meaning that cooling and contraction have increased the mass inside these overdensity regions relative to DMO.
• Figure 4: Results of fitting the universal function -- equation \ref{['eq:fretained']} combined with equation \ref{['eq:universal_params']} -- to L2p8_m9 for the corrected baryon fractions and retained mass fractions in seven different overdensity regions, for eighteen $0.2\,\mathrm{dex}$ halo mass bins in the range $\log_{10}(M_{\mathrm{500c}}/[\mathrm{M}_\odot])=[12.0,15.6]$, at $z=0$. Like in the right-hand panels of Fig. \ref{['fig:fret_fbc_200m_fixedhaloes_fixedmass']} and Fig. \ref{['fig:fret_fbc_500c_fixedhaloes_fixedmass']}, baryon fractions measured at a hydro halo mass $M$ are mapped to retained fractions at the same DMO halo mass $M$ in the fit. Even though we again see that the scatter increases as we look deeper into haloes, the function reproduces the retained mass fractions on all scales remarkably well for all halo masses probed here.
• Figure 5: Model predictions compared to the true suppression signal for FLAMINGO simulation L1_m9, which is shown as a solid red line. Red dotted lines show the $\pm1$ percentage-point band around the true suppression signal. The vanDaalen2020 relation, based on the mean baryon fraction in haloes of mass $M_{\mathrm{500c}}\approx 10^{14}\,\mathrm{M_\odot}/h$ and fit to simulation results up to $k\,{=}\,{1}\,h\mathrm{\,Mpc^{-1}}$, is shown as a dashed cyan line for comparison. The results from the resummation model are shown as dot-dashed lines. Using only the mean halo baryon fractions within $R_{\mathrm{200m}}$ as input reproduces the true suppression signal within $1$ percentage point up to $k\,{\approx}\,{3}\,h\mathrm{\,Mpc^{-1}}$, while using only those within $R_{\mathrm{500c}}$ follows the true signal down to smaller scale but underpredicts the amount of suppression on large scales. However, combining baryon fractions from both regions, as described in §\ref{['subsec:modelannulus']}, allows us to reproduce the true signal to within $1$ percentage point up to $k\,{\approx}\,{10}\,h\mathrm{\,Mpc^{-1}}$.