The contribution of massive haloes to the matter power spectrum in the presence of AGN feedback

The clustering of matter, as measured by the matter power spectrum, informs us about dark matter and cosmology, as well as baryonic effects on the distribution of matter in the universe. Using cosmological hydrodynamical simulations from the cosmo-OWLS and BAHAMAS simulation projects, we investigate the contribution of power in haloes with various masses, defined by particles within some overdensity region, to the full power spectrum, as well as the power ratio between baryonic and dark matter only (DMO) simulations for a matched (between simulations) and an unmatched set of haloes. We find that the presence of AGN feedback suppresses the power on all scales for haloes of all masses examined ($10^{11.25}\leq M_{500,\mathrm{crit}}\leq 10^{14.75}\,\mathrm{M_\odot}/h$), by ejecting matter from within $r_{500,\mathrm{c}}$ to $r_{200,\mathrm{m}}$ and potentially beyond in massive haloes ($M_{500,\mathrm{crit}}\gtrsim 10^{13}\,\mathrm{M_\odot}/h$), and likely impeding the growth of lower-mass haloes as a consequence. A lower AGN feedback temperature drastically changes the behaviour of high-mass haloes ($M_{500,\mathrm{crit}}\geq 10^{13.25}\,\mathrm{M_\odot}/h$), damping the effects of AGN feedback at small scales, $k\,\gtrsim\,4\,h\mathrm{\,Mpc^{-1}}$. For $k\,\lesssim\,3\,h\mathrm{\,Mpc^{-1}}$, group-sized haloes ($10^{14\pm0.25}\, \mathrm{M_\odot}/h$) dominate the power spectrum, while on smaller scales the combined contributions of lower-mass haloes to the full power spectrum rise above that of the group-sized haloes. Finally, we present a model for the power suppression due to feedback, which combines observed mean halo baryon fractions with halo mass fractions and halo-matter cross-spectra extracted from dark matter only simulations to predict the power suppression to percent-level accuracy down to $k\,\approx\,10\,h\mathrm{\,Mpc^{-1}}$ without any free parameters.


INTRODUCTION
Understanding the way matter clusters is an integral piece in furthering our knowledge about the Universe and its inner workings.The clustering of matter governs halo collapse and galaxy formation, along with mergers and other processes.All these processes are heavily dependent on the cosmological parameters, the fundamental parameters describing our universe.These parameters dictate, among other things, the matter density of the universe, its expansion, and, by extension, how much of the matter is formed into haloes and galaxies, along with their formation timescales.
Although the ΛCDM model is the most widely established cosmological model, incorporating cold dark matter and dark energy, its parameters are not set.Observational data from the two largest cosmological probes, the Wilkinson Microwave Anisotropy Probe (WMAP) (see Komatsu et al. 2011;Hinshaw et al. 2013 for the 7-and 9-year results, respectively) and Planck (see Ade ⋆ E-mail: mvanloon98@gmail.comet al. 2014;Planck Collaboration et al. 2016 for the 2013 and 2015 releases, respectively), provide measurements of the cosmic microwave background (CMB) from which the cosmology of our Universe can be derived, but there is growing evidence that there is tension between these early-Universe probes and late-Universe measurements (see Abdalla et al. 2022 for a recent review).The latter type of measurements typically study some aspect of the clustering of matter, for example through weak gravitational lensing (e.g.Heymans et al. 2021;Abbott et al. 2022), and compare this to theoretical models to derive cosmology.One particular measure of clustering is the matter power spectrum, generally given as P (k) (or ∆ 2 (k)): the amplitude squared of matter overdensities as a function of Fourier scale, k, which is given by 2π/λ, where λ refers to the physical scale.
On the largest scales, linear perturbation theory is sufficient to model the power spectrum, originally proposed by Press & Schechter (1974) and extended by Bond et al. (1991).However, on smaller, non-linear scales, governing halo collapse and galaxy formation, linear theory underestimates the power and additional physics needs to be taken into account.A widely accepted model incorporating non-linear scales is the analytical halo model (e.g.Peacock & Smith 2000;Seljak 2000;Cooray & Sheth 2002), which has been added to by many others (e.g.Tinker et al. 2008;Duffy et al. 2008;Semboloni et al. 2013;Mead et al. 2015;Debackere et al. 2020;Mead et al. 2021).As future weak lensing surveys, enabling high precision (< 1%) measurements of (matter) power spectra, draw near, the importance of theoretical understanding and predictions on these non-linear scales increases (e.g.Huterer & Takada 2005;Laureijs 2009;Hearin et al. 2012).
Currently, the precise effects of baryons, e.g. through galaxy formation and evolution, on the matter power spectrum (hereafter referred to as: power spectrum) are still largely undetermined (see Chisari et al. 2019 for a review).Countless complex and coupled processes underlie the formation and evolution of galaxies that often play out on small, not directly observable, scales.Hydrodynamical, cosmological simulations, which can produce a statistically interesting sample of realistic galaxies, offer insights and help to determine which theoretical models best describe observations or which observations are predicted based on theory.The trade-off between resolution and sample size remains a challenge, although increased computational efficiency and so-called 'subgrid models', which allow unresolved physical processes to be taken into account, have come a long way in the last few decades (for a review, see Vogelsberger et al. 2020).
The widely accepted idea that the clustering of all matter is fully determined by that of dark matter was refuted when van Daalen et al. (2011) found that the presence of AGN feedback has a profound impact on the matter power spectrum, using simulations from the OWLS project (Schaye et al. 2010).Not only is there an effect on the clustering of baryons by the material that is ejected out to large radii, the dark matter reacts to these changes by expanding, a phenomenon dubbed 'the back reaction' (e.g.Duffy et al. 2010;van Daalen et al. 2011;Velliscig et al. 2014).This suppression of the matter power spectrum has been confirmed and/or modelled by others since then (e.g.Vogelsberger et al. 2014;Mead et al. 2015;Hellwing et al. 2016;Mummery et al. 2017;Chisari et al. 2018;Schneider et al. 2019;Aricò et al. 2021;Pandey et al. 2023;Salcido et al. 2023).However, the mechanisms causing (AGN) feedback and the consequences on the surrounding material are as of yet inadequately understood in the context of the matter power spectrum to predict the power spectrum to near 1% accuracy over a sufficient range of scales based solely on cosmology (e.g.van Daalen et al. 2020).
In order to make valid predictions of the power spectrum, we need to understand which parts of the matter distribution contribute significantly to the full power spectrum on any given scale.One way to separate the power spectrum into components, is to distinguish between the contributions to the full power spectrum of autopower produced by haloes only, auto-power generated by matter not in haloes and cross-power between halo and non-halo matter.The contribution of these components is influenced by the definition of halo boundary used in a scale-dependent way, as found by van Daalen & Schaye (2015) for the halo auto-power spectrum in dark matter only (DMO) simulations.
Keeping a consistent definition for halo boundaries, e.g. a radius marking a spherical overdensity (SO) region, separate power spectra for haloes of certain masses can be considered, to explore their individual or combined effects to the full power spectrum.Therefore, van Daalen & Schaye (2015) considered the contribution of power in haloes to the full power spectrum in DMO simulations for a varying minimum halo mass.They found that group-sized haloes (with M200,mean ≳10 13.75 M⊙/h) are the main contributors to matter clustering for 2 ≲k ≲ 10 h Mpc −1 , even though they contain only ∼ 13% of the total mass.The deviation between the mass fraction and the contribution to the total power stems from the interplay between the rarity of massive haloes, lowering their contribution to the total power, and the deeper potential wells of massive haloes, which result in increased clustering of the matter, along with their already more clustered environments (i.e.their bias), raising their contribution to the total power (see also e.g.Debackere et al. 2020).Even so, van Daalen & Schaye (2015) found that a significant (> 1%) amount of power is still provided by unresolved haloes (M200,mean ≲10 9.5 M⊙/h) and, potentially, smooth dark matter.
The relevance of the large contribution of group-sized haloes to matter clustering was further confirmed when van Daalen et al. (2020) examined a library of 92 power spectra for cosmological simulations using different galaxy formation models and parameters.They found that the mean baryon fraction of these (M ∼10 14 M⊙/h) haloes can be used to predict the power suppression due to galaxy formation to typically within 1% for k < 1 h Mpc −1 .
Considering the importance of group-sized haloes to clustering measurements and the studying of other physical processes (e.g.McCarthy et al. 2010;Oppenheimer et al. 2021), here we delve deeper into the role of these objects in shaping the matter power spectrum.To this end we utilize the cosmo-OWLS (Le Brun et al. 2014;McCarthy et al. 2014) and BAHAMAS (BAryons and haloes of MAssive Systems, McCarthy et al. 2017) simulations, which were made to study haloes of these masses, and try to quantify the impact of AGN feedback on the contribution of (group-sized) haloes to the matter power spectrum.Furthermore, we explore the influence of AGN feedback on the suppression of power on smaller scales for different components of the power spectrum, creating a model, based on the key element of mass removal from clustered regions, that can predict this suppression to percent-level accuracy down to k ≈ 10 h Mpc −1 in a way that is independent of the galaxy formation recipes used.By furthering our understanding of the contributions haloes of different masses have, and the effects of baryonic feedback processes, we hope this work can provide a stepping stone for future research to increase the precision of theoretical power spectrum predictions on non-linear scales.This paper is structured as follows.In §2 we describe the simulations in more detail, and provide a formal definition of the power spectrum and other methods used to obtain the results.§3 then describes the results, where the contributions of halo auto-power for an unmatched set of r200,m haloes is considered in §3.1.The halo auto-power for a matched set of haloes is similarly considered in §3.2.Using this same matched catalogue, we also explore the clustering contributions of r500,c SO regions instead ( §3.3), consider the results when using a fixed SO radius between simulations ( §3.4), and investigate the effects of varying the AGN feedback temperature ( §3.5).The effects of cosmology on the results is briefly discussed in Appendix B. Other contributions to the power spectrum, i.e. cross terms, are considered in §3.6, based on which we derive a model for power suppression in §4.Finally, §5 provides a summary of our results and a brief discussion of the model's potential.

Name
Abbreviation

Simulations
The results in §3 are based on a set of simulations from the cosmo-OWLS (Le Brun et al. 2014) andBAHAMAS (McCarthy et al. 2017) simulation projects, which were designed to study groupand cluster-sized haloes.Both projects were run on a modified version of GADGET-3, with smoothed particle hydrodynamics (SPH), as described by Springel (2005).The relevant simulation parameters are listed in Table 1 along with the abbreviations of the simulation names, which will be used in the rest of the paper to refer to the simulations.
All simulations used in this study have a volume of (400 h −1 Mpc) 3 and 1024 3 particles (×2 for simulations including baryons, or 2-fluid DMO simulations).These are therefore generally indicated with the suffix L400N1024, but for brevity we will omit this suffix everywhere, except in Appendix A, which studies boxsize effects on the results at a constant resolution.The DMO simulations, as the name indicates, include only dark matter particles.In the REF simulation (cosmo-OWLS), gas and star particles and their relevant physics (e.g.star formation, metal enrichment, supernova feedback) have been added, and the AGN simulations add supermassive black hole particles along with their accretion and feedback to the mix, which are seeded into haloes with > 100 dark matter particles.The particle masses are listed in table 1.The doubling of the number of particles, along with the mass difference between them in a DMO and a baryonic simulation, causes a variation in resolution effects between these simulations on small scales, which also affects the matter clustering.Therefore, in the BAHAMAS project, the 2-fluid DMO simulations split every dark matter particle into two dark matter particles, with the dark matterbaryon mass ratio, and use a separate transfer function for each, to account for this effect and make a comparison on smaller scales between DMO and baryonic simulations more easily interpretable.
For the cosmo-OWLS simulations, the seven-year WMAP cosmological parameters are used (WMAP7, Komatsu et al. 2011), as listed in table 2. The BAHAMAS simulations use the newer, nine-year WMAP results (WMAP9, Hinshaw et al. 2013), also listed in table 2. To investigate whether our results show a dependence on cosmology, these simulations are compared to the simulations run with the 2013 Planck cosmological parameters (Ade et al. 2014 , named 'Planck' in  software package N-GENIC 1 , which was updated to include second-order Lagrangian Perturbation Theory by S. Bird for BA-HAMAS, and using a transfer function as prescribed by Eisenstein & Hu (1999) (cosmo-OWLS) or the Boltzmann code CAMB2 (Lewis et al. 2000) (BAHAMAS), the particles in the simulations are evolved to z = 0 by models, describing physical processes and 'sub-grid' models, describing unresolved, but relevant, physics (for an extended description see Schaye et al. 2010;Le Brun et al. 2014;McCarthy et al. 2017).
Processes that are described by sub-grid models include baryonic feedback, i.e. supernova (SN) and AGN feedback.In their AGN simulations, cosmo-OWLS and BAHAMAS both use thermal AGN feedback, which entails that the feedback is 'accumulated' in the black hole until 1 (cosmo-OWLS) or 20 (BAHAMAS) surrounding particle(s) can be heated by ∆T heat .This heating temperature is a model parameter, shown in table 1, that is varied between simulations, as its 'true' value is unknown.For the B AGN W9 simulation, the parameters governing AGN feedback have been fitted to multiple sets of observational data (see Mc-Carthy et al. 2017 for details).The other BAHAMAS simulations, as well as the cosmo-OWLS AGN simulation, explore different feedback strengths, both weaker and stronger.The cosmo-OWLS 'reference' (REF) simulations include SN feedback but not AGN feedback, causing the well-known 'overcooling' problem (see Mc-Carthy et al. 20102011).

Power spectra
Recall from §1 that the matter power spectrum can be seen as the amplitude squared of the Fourier transform of the matter overdensity field.Most of our results will be shown in terms of the power spectrum, defined as: where k is the Fourier scale, P is the power, V the volume of the simulation (which is (400 h −1 Mpc) 3 everywhere) and δk is the Fourier transform of the density contrast.In Figure 2, an exception is made and P (k)k 1.5 is shown to take out the main k-dependence and make the comparison between lines easier.The shot noise, generated by the 'clustering' of particles with themselves, is subtracted from the power to account for the discreteness of the density field.All auto-power spectra are calculated using a modified version of POWMES (as described in van Daalen & Schaye 2015; see Colombi et al. 2009 for the original version), which takes as input a particle selection, along with a list of parameters and a path to the simulation data, and outputs the power at each scale.
The cross-power spectra, used in §3.6 and §4, are calculated using NBODYKIT (see Hand et al. 2018 for a detailed description).

Selections
For the results in §3, the SUBFIND output of the simulations is first used to divide the haloes based on mass, whereafter the bound and spherical overdensity particles are selected based on their radial distance to the centre of the halo, using PYTHON.
The haloes are defined using cut-off radii where either r200,m or r500,c is chosen, corresponding to the radii wherein the average density is equal to 200 times the mean density (200 × Ωmρcrit) or 500 times the critical density (500 × ρcrit), respectively.However, in order to enable comparison between figures, the halo mass we select on is always chosen as the mass within r500,c (M500,crit= 500 × 4π 3 ρcritr500,c 3 ), regardless of the SO region used in particle selection.Furthermore, a cutoff of 100 particles is enforced to counter resolution effects.This mostly affects the lowest mass bin of figures where haloes are matched between simulations (see §2.2) because not all haloes containing > 100 particles have a > 100 particle match.
The halo mass bins range from M500,crit= 10 11.25 M⊙/h to M500,crit= 10 14.75 M⊙/h with a width of 0.5 dex, to reduce mass evolution effects within the mass bin, while ensuring a large enough sample size.For the figures where haloes are matched between simulations (see §2.2), the halo mass selection is based on the DMO simulation masses, contrary to the unmatched figures where the masses correspond to the simulation shown.

Matching
For some figures in §3 (e.g. Figure 4), the haloes are matched between simulations, which is made possible by the initial conditions of simulations in the same set being identical.For cosmo-OWLS, two sets of matches were made: haloes in C DMO W7 were matched with their equivalents in C AGN W7, and separately with their haloes in C REF W7, for each boxsize (L100, L200, L400), for halo masses 10 11.25 M⊙/h and up.For BAHAMAS, the following sets of matches were made: B DMO W9-B AGN W9, B DMO W9-B AGN7p6 W9, B DMO W9-B AGN8p0 W9 and B DMO PL-B AGN PL, for halo masses ⩾ 10 11.25 M⊙/h.A match between B DMO W9 and B DMO PL for halo masses ⩾ 10 11.25 M⊙/h was also made, to allow for a comparison between cosmologies (see Appendix B).
During the matching, the particle IDs of the 100 most-bound dark matter particles of each halo within the mass range in the 11.5 12.0 12.5 13.0 13.5 14.0 14.5 log 10 (M 500, c [h M −1 ]) DMO simulation, are compared to all the dark matter IDs of the haloes in the same or a neighbouring grid section in the other simulation.The halo where most of these particles are found is placed in a list, where a minimum of 50 particles is required to be considered a match.The particles of the 100 most-bound dark matter particles of these haloes are then found in the DMO simulation in the same way.Furthermore, a halo position check ensures that the matched haloes are < 1 Mpc apart.In the BAHAMAS matching, the computing time is sped up by also imposing a mass limit of a factor 5 within the mass of the halo for haloes in neighbouring grid sections.If a halo pair passes these tests it is considered a match and the index pair is saved.
The fraction of matched haloes as a function of halo mass is given in Figure 1 for the match between B DMO W9 and B AGN W9 (blue line, representative for all BAHAMAS DMObaryon pairs), the match between C DMO W7 and C AGN W7 (green), and the match between B DMO W9 and B DMO PL (red) used in §3.2.For M500,crit⩾10 11.75 M⊙/h, the matched fraction is > 0.95 for all pairs.For the lowest mass bin (M500,crit=10 11.5±0.25 M⊙/h) the matched fraction between different cosmologies is ∼ 0.5, calling for caution in the interpretation.
During the selection of particles for POWMES or NBODYKIT (as described in §2.1.3),the haloes are selected based on their DMO-simulation masses, and only matched haloes are considered.For the results in §3.4, the SO region radii from the corresponding haloes in the DMO simulation are taken as a selection criterion for the AGN simulation haloes.

RESULTS
When considering the effects of galaxy formation on the theoretical matter power spectrum, it is common to consider the ratio of a power spectrum from a hydrodynamical simulation to that of a DMO simulation, so that small (but relevant) deviations from unity become more apparent (see e.g. Figure 2 of van Daalen et al. 2020).In general, the shape of this ratio is a dip downwards from unity for k ≳ 0.1 h Mpc −1 , meaning the power in realistic simulations is suppressed on large and intermediate, non-linear scales, relative to a dark matter only universe, due to matter being removed from clustered regions by feedback processes.This dip flattens out to a suppression of ∼ 10 − 20% depending on the model used (with more effective feedback causing a stronger suppression), followed by a sharp upturn for k ≳ 10 h Mpc −1 into ratios far above unity on small scales.This upturn is caused by the cooling of the baryons into a central galaxy on even smaller scales, which increases the concentration of the inner dark matter halo.
To more precisely understand the physical processes behind the shape of the power spectrum and its suppression due to feedback, we need to dive deeper into the different contributions that make up the power spectrum.Therefore, this section focuses on the contributions of a range of halo masses to the full spectrum and the ratio between baryon and dark matter only simulations to gain more understanding about the role of feedback processes for different halo masses.

Auto-power contributions for non-matched haloes
In this section, the M500,crit halo masses include baryons when present in the simulation.Furthermore, the simulations are not matched, resulting in skewed halo distributions in the mass bins between simulations, as AGN feedback lowers halo masses.§3.2 describes the results that are obtained when the haloes are matched between simulations.
In Figure 2 the power spectra of simulations C AGN W7 (solid), C DMO W7 (transparent) and C REF W7 (transparent and dashed) are shown for a range of halo mass bins between 10 11.25 M⊙/h and 10 14.75 M⊙/h (with a width of 0.5 dex), at z = 0.The power spectra are multiplied by k 1.5 to take out the main k-dependence and to make a comparison between the lines easier.
In the top panel, all particles within r200,m of haloes with masses greater than that indicated in the legend are taken into account.The more haloes are taken into account, the closer the power spectrum is to the full power spectrum (black line).Note that we do not expect to converge to this limit as more haloes are taken into account, since not all matter resides in (resolved, r200,m ) haloes.For all mass bins, the power spectra have a similar shape: the power increases with k, reaches a maximum at the scale at which the (larger) haloes are entered and then decreases (keeping in mind that we multiply P (k) with k 1.5 here).The scale at which the haloes are entered becomes larger with increasing halo mass, effectively moving the peak in the figure towards smaller k.
A comparison of simulations reveals a lower power of C AGN W7 on large scales, compared to both C DMO W7 and C REF W7, due to AGN feedback lowering the halo mass.The haloes that are in a certain C AGN W7 mass bin, are in in a higher mass bin in the C DMO W7 or C REF W7 simulations.As low-mass haloes are more numerous than high-mass haloes, more haloes will be included in the C DMO W7 and C REF W7 mass bins, increasing the power.On smaller scales, C REF W7 power increases more strongly due to over-cooling while C AGN W7 dips under C DMO W7 as the effective feedback heats the gas and pushes baryons to larger scales, the dark matter reacting gravitationally.
In the bottom panel of Figure 2, only particles within r200,m of Halos not matched between simulations M 500, crit = [M 500, crit, low , M 500, crit, low * 10 0.5 ] log10(M500,crit,low h M −1 ) = 11.25 log10(M500,crit,low h M −1 ) = 11.75 log10(M500,crit,low h M −1 ) = 12.25 log10(M500,crit,low h M −1 ) = 12.75 log10(M500,crit,low h M −1 ) = 13.25 log10(M500,crit,low h M −1 ) = 13.75 log10(M500,crit,low h M −1 ) = 14.25 Figure 2. The power spectra of C AGN W7 (solid), C DMO W7 (transparent) and C REF W7 (transparent and dashed) for various halo mass selections, at z = 0.The legend shows which M 500,crit selection is made: M 500,crit ⩾ 10 x−0.25 M ⊙ /h (top panel) or 10 x−0.25 M ⊙ /h ⩽ M 500,crit ⩾ 10 x+0.25 M ⊙ /h (bottom panel), with x ranging from 11.5 to 15 (see top legend for colour scheme).Only particles within the SUB-FIND radius corresponding to a spherical overdensity of 200ρ (r = r 200,m ) are taken into account.The contribution of haloes with M 500,crit = 10 14±0.25 M ⊙ /h to the full spectrum exceeds the contributions of haloes with other masses at k < 3 h Mpc −1 .A comparison between simulations shows that AGN feedback lowers the fraction of mass residing in haloes of all masses, the SO radius, as well as the contribution to the full power spectrum on small scales (k > 8 h Mpc −1 ) of haloes with M 500,crit ≳ 10 11.75 M ⊙ /h.haloes within the indicated mass bin (width of 0.5 dex) are taken into account.The shapes are similar to the top panel, although a comparison between masses reveals that the power on large and intermediate scales increases with halo mass for haloes with M500,crit≲ 10 14.25 M⊙/h.The power of the largest haloes (M500,crit=10 14.5±0.25 M⊙/h, red lines) is lower or equal to the second largest mass bin (M500,crit=10 14±0.25 M⊙/h, orange lines), as was found previously by van Daalen & Schaye (2015).Although cosmic variance plays a minor role in this (as we checked using three B AGN W9 simulations with varying initial conditions, not shown), it is mostly due to cosmic deficiency.In other words, Halos not matched between simulations M 500, crit = [M 500, crit, low , M 500, crit, low * 10 0.5 ] log10(M500,crit,low h M −1 ) = 11.25 log10(M500,crit,low h M −1 ) = 11.75 log10(M500,crit,low h M −1 ) = 12.25 log10(M500,crit,low h M −1 ) = 12.75 log10(M500,crit,low h M −1 ) = 13.25 log10(M500,crit,low h M −1 ) = 13.75 log10(M500,crit,low h M −1 ) = 14.25 Figure 3.The halo clustering contribution as a function of mass, relative to the full matter power spectrum, for C AGN W7 (solid), C DMO W7 (transparent) and C REF W7 (transparent and dashed), at z = 0.The legend and particle selections are the same as in Figure 2. The contribution of haloes with M 500,crit = 10 14±0.25 M ⊙ /h to the full spectrum exceeds the contributions of haloes with other masses for k < 3 h Mpc −1 .AGN feedback lowers the fraction of mass residing in massive haloes (M 500,crit ≳10 13 M ⊙ /h), as well as the contribution to the full power spectrum on small scales k ≳ 8 h Mpc −1 of haloes with M 500,crit ≳ 10 11.75 M ⊙ /h, though resolution effects start to play a significant role at this scale.the 10 14.5±0.25 M⊙/h haloes are so rare that their higher mass and bias no longer wins out, lowering their contribution to the full power spectrum relative to slightly less massive haloes.
For low-mass haloes (M500,crit≲ 10 13 M⊙/h), the peaks of the auto-power spectra from the baryonic simulations lie above those from C DMO W7, whereas those for higher mass haloes lie below.The transition happens around the mass when AGN feedback becomes important, supporting the statement that AGN feedback is the cause.Comparing the bottom to the top panel, we see that C AGN W7 lies below C DMO W7 when all haloes are taken into account, showing again that what happens to clustering in the groups and clusters dominates the effect on the overall power.
To better compare the relative contributions of haloes between the different simulations, it is useful to consider ratios of power spectra instead.In Figure 3, each of the lines portrayed in Figure 2 is divided by the full power spectrum of the accompanying simulation to illustrate the contribution to the full spectrum for each halo mass bin.The masses and line styles (indicating simulations) are as in Figure 2.Here we still take r200,m as the halo boundary.
On large scales, the contribution of haloes to the power flattens off to a fraction of ≈ 0.3 (depending on simulation) when all haloes are taken into account (pink lines).In this regime, the 2halo term dominates: the overdensities are determined by the positions and masses/sizes of the haloes, the clustering in the interior of haloes has a negligible contribution.As we move to higher k (smaller scales), there is a massive rise in contribution around the scales that haloes are entered, which of course differs per mass bin.Along this rise, the 1-halo term starts to dominate over the 2-halo term, as most of the power comes from clustering in the interior of the haloes.Once the halo is entered, a maximum power contribution to the total is reached, after which it declines again for smaller scales.Note that for k > 8 h Mpc −1 resolution effects start to influence the results noticeably, lowering the power.When all haloes are taken into account (pink line, top panel of Figure 3), the contribution to the full power spectrum reaches > 0.9 at k ≈ 3 h Mpc −1 .The remaining percentage of power is provided by a combination of the cross-power between non-halo and halo particles and the autopower in non-halo particles (as seen in §3.6).
The bottom panel of Figure 3 shows the individual contributions of each halo mass bin to the full power spectra of C AGN W7, C REF W7 and C DMO W7.This panel clearly shows that the 10 14±0.25 M⊙/h haloes contribute most to the full power spectrum for k < 3 h Mpc −1 , and lower mass bins take over on smaller scales.Note that resolution effects play some role in this, which generally lower the power on small scales (k ≳ 8 h Mpc −1 ).
On large scales, the fraction of the total mass selected, when selecting the haloes in a certain mass bin, determines the fraction of power to the full power spectrum.As the selection is purely based on halo mass, one might expect all three simulations to lie on top of each other.However, because these are not the same haloes, this is not the case; the discrepancies become larger, as more haloes are taken into account (top panel, k ∼ 0.02 h Mpc −1 ).While C REF W7 and C DMO W7 are in quite good agreement, the total halo contribution on large scales is significantly lower in C AGN W7, as more mass is removed from haloes.Looking at the peak in contribution (when entering the halo) in the top panel of Figure 3 for masses around 10 13.75 M⊙/h (yellow and orange lines), there is a clear horizontal displacement between the power contribution of haloes to the full power spectrum in the C AGN W7 and C DMO W7.This is counter-intuitive, since AGN feedback lowers the halo mass, effectively placing the more extended, denser haloes in a lower mass bin, from which one would expect the peak in C AGN W7 to move to larger scales (smaller k).Looking at the bottom panel, however, we see that the shift does not occur in individual halo mass bins, but is a cumulative effect of feedback reducing the halo mass by different amounts on different mass scales.
We also see that the peak of these haloes in C REF W7 is considerably higher than both other simulations, likely due to extra cross-power between halo particles and the large number of stars that have formed due to overcooling.
When comparing the C AGN W7, C REF W7 and C DMO W7 lines in Figure 3, drawing conclusions is not straightforward because the haloes are not matched between simulations.Baryon cooling strongly peaks the density distribution towards the center of the haloes, changing SO radii and thus the scale at which the halo is entered.Additionally, the removal of mass due to feedback also changes the profiles and therefore the 1-halo contribution, while also lowering the overall power of the haloes and shifting haloes between mass bins.To be able to better distinguish these different effects, in the remainder of §3 we will consider the relative contributions for matched catalogues of haloes.

Auto-power contributions for matched haloes
The left side of Figure 4 shows the power contribution to the full power spectrum for the same mass selections of haloes as in Figure 3, but for a matched set of haloes.The right side shows the ratios between the power spectra of the baryon and DMO simulations (C AGN W7/C DMO W7, solid continuous and C REF W7/C DMO W7, dashed transparent line style) for each mass bin, to illustrate the effect of baryons on the power in haloes for a range of halo masses.
On large scales, small k, the power contribution to the full power spectrum reflects the mass fraction of the selected haloes.Therefore, the top-left panel shows that the masses of the haloes are not the same in the two simulations.When taking only the largest haloes into account (M500,crit⩾10 14.25 M⊙/h, red lines), the masses in the baryon simulations are closest to those in C DMO W7, as seen in the top-right panel at k ≈ 0.02 h Mpc −1 , where the power ratio C AGN W7/C DMO W7 ≈ 0.96.These haloes have such a deep potential well that AGN feedback is damped and does not have a substantial effect.As more lowermass haloes are added to the sample, the power ratio between simulations lowers until it saturates at ≈ 0.88 for M500,crit⩾ 10 12.25 M⊙/h, where the bottom-right panel shows that haloes with masses lower than 10 12.75 M⊙/h either have a similar or a higher C AGN W7/C DMO W7 ratio and the bottom-left panel shows that these mass bins have a much lower contribution to the full power spectrum than the more massive haloes.Consequentially, the C AGN W7/C DMO W7 ratios of the mass bins that take lower mass haloes into account as well as higher mass ones, are determined by the C AGN W7/C DMO W7 ratios of the higher mass haloes up to ∼10 14 M⊙/h and thus lowered by the AGN feedback processes these are subject to.
A possible reason why the simulation ratios of low mass haloes, e.g. 10 12±0.25 M⊙/h (solid purple line, Figure 4), in the bottom-right panel are lower for AGN than REF, could be that these haloes are still indirectly effected by AGN feedback.Lower-mass haloes tend to cluster around high-mass ones, forming large groups.The powerful shock waves caused by AGN in massive haloes can carry out to far beyond these haloes, affecting all the other haloes in the vicinity.This might cause the haloes to lose all their gas due to a shock wave, or in milder cases, the expansion of the central halo weakens its gravitational pull making surrounding objects less strongly bound/expand.A more likely explanation is that the formation and evolution of the haloes were influenced by the AGN at an earlier stage.The gas around galaxies with AGN is much hotter, hindering the accretion and growth of smaller haloes and galaxies.However, this has not been verified in this research.Either way, the effects of AGN feedback are felt by lower-mass galaxies, as can be seen when comparing the solid (C AGN W7/C DMO W7) and transparent dashed (C REF W7/C DMO W7) lines in the bottom right panel of Figure 4.Note that the C REF W7/C DMO W7 ratio is also < 1, due to gas pressure which impedes collapse.For low-mass haloes, M500,crit<10 11.75 M⊙/h, there are few particles making up the halo in these simulations, allowing small discrepancies to have a large influence on the power.We also note that van Daalen & Schaye (2015) found that such haloes are underrepresented at this resolution.
At k of a few, the power spectrum is almost completely determined by the clustering of matter inside haloes, as can be seen by the peak in the top-left panel of Figure 4.Although the figure exhibits the same general shapes as the version without matching (Figure 3), considering matched rather than mass-selected catalogues has significant effects.The different simulations meet at a point, just before the peak, when the halo is entered; this is expected as they are the same haloes, although this would cease to happen if larger amounts of mass were removed.Right after this point, the contribution of haloes to the power in C AGN W7 rises above that of C DMO W7 and C REF W7.This however does not mean that halo matter in the C AGN W7 simulation clusters most strongly on this scale, compared to the other simulations: in fact, the opposite is true.This effect comes from the differences in the full power spectra (through which we are dividing on the left-hand side of the figure), as is verified in the top-right panel of Figure 4.Here we see that the power ratio between C AGN W7 and C DMO W7 (solid lines) is below unity for every mass bin.It is therefore likely the cross-power between matter around haloes and haloes themselves being decreased by the presence of AGN which causes the rise of the peak seen for C AGN W7 in the top-left panel, due to the AGN removing mass from the haloes and depositing it on larger scales, where it clusters less strongly.
Still keeping our focus on the top-right panel, we see that, as expected, the relative auto-power contributions of haloes in different mass ranges compared to C DMO W7 is lower for C AGN W7 than for C REF W7, and that this difference is driven on large scales by the mass difference of ∼10 12.5 M⊙/h-10 14 M⊙/h haloes between these simulations.On small scales (at k ⩾ 3 h Mpc −1 ), where the change in auto-power is a reflection of the change in halo profile, the top-right panel shows a dip for all halo masses in the C AGN W7/C DMO W7 ratio.The C REF W7/C DMO W7 ratio rises above 1 at this scale because the dark matter haloes contract, reacting to the heavily clustered galaxy, and there is no AGN feedback to counter the contraction (the SN feedback being too weak to counter the contraction on the most relevant mass scales).
At 3 < k < 10 h Mpc −1 , the C AGN W7/C DMO W7 power ratio for M500,crit⩾ 10 12.75 M⊙/h haloes drops by ≈ 0.1, because of AGN feedback.At k ⩾ 10 h Mpc −1 , this ratio starts to rise again for all mass bins in the top-right panel, until, at k = 20 h Mpc −1 each reaches more or less the value it had at k ≈ 3 h Mpc −1 .This rise can be attributed to the reaction of dark matter to the behaviour of baryons at small (galaxy) scales (k ∼ 100 h Mpc −1 ), which cluster and increase the power, causing a contraction in the dark matter at larger scales (k ∼ 20 h Mpc −1 ), meaning the AGN effectively change clustering the most at intermediate scales inside haloes, relative to DMO.Despite their low contribution to the full power spectrum (bottom left panel), the lower mass haloes have a part to play in the rise at k ≳ 10 h Mpc −1 , shown by the steeper slope of the C AGN W7/C DMO W7 power ratios at these scales in the top right panel for the lines that take the lower-mass haloes into account.The behaviour of the lower mass galaxies (M500,crit< 10 12.75 M⊙/h) can be explained by their lack of, but vicinity to, galaxies with AGN feedback: they are not affected internally, since they do not host AGN themselves, but their masses are lower than they would have been in a universe without AGN.The solid C AGN W7/C DMO W7 ratios in the bottom right panel follow the same general shapes and have similar slopes We note that resolution effects, coming from the limited number of particles making up a halo, which prohibit measuring the clustering on very small scales and start to play a role at k ≳ 8 h Mpc −1 , lowering the power relative to a higher-resolution simulation.This affects the DMO haloes in cosmo-OWLS most strongly, as these were run with half as many particles as the baryonic simulations, and therefore may affect the ratios shown here.However, the BAHAMAS simulation project uses 2-fluid simulations for DMO, so this effect is compensated (see §2.1.1).As we will see in §3.5, Figure 7 shows that the ratios still exhibit the dip and rise for the BAHAMAS simulations, albeit somewhat differ-ently, confirming that resolution effects are not the leading cause of the small-scale effects presented here.

Varying the SO regions
We now consider how the halo contributions depend on the choice of halo boundary.Figure 5 is similar to Figure 4 for C AGN W7 (solid) and C DMO W7 (transparent), but dashed lines now show the results for a r500,c cut-off radius (dashed) compared to r200,m (continuous).The mass selections are as in Figure 4 and the haloes are matched between the simulations.
The most drastic effect of the change in the SO radius from r200,m to r500,c on the relative halo contribution to the total power spectrum (top-left panel of Figure 5) is a horizontal shift of the  4, but dashed lines now show the results when only particles within r 500,c are included.The peak in halo power contribution is moved to higher k for particles within r 500,c , relative to the results for r 200,m .Although much less mass is included in the smaller r 500,c regions, the power on small scales only drops slightly, as they are highly biased.As the bottom-left panel shows, for both SO regions ∼10 14 M ⊙ /h haloes provide the largest contribution to the total power up to k ∼ a few.From the bottom-right panel, we see that for group-sized haloes the effects of AGN feedback are more extensive within r 500,c , compared to within r 200,m .
peaks towards larger k.The r500,c spherical overdensity region has a smaller radius than r200,m and is often used to explore halo properties through X-ray observations.Direct or indirect effects from baryons are often more apparent when considering this region.Since the peak indicates the scale at which the halo is entered, the shift seen here is expected.Furthermore, on large scales the power contribution to the full power spectrum is lower for r500,c, as less of the mass is taken into account in these calculations.Still, taking C DMO W7 as an example, the particles within r500,c of haloes contribute a maximum fraction of ≈ 0.89 of the full power through their auto-power alone, while the particles within r200,m reach ≈ 0.96 of the full power.In contrast, on large scales, k ∼ 0.02 h Mpc −1 , these SO regions account ≈ 0.06 of the total for r500,c and for ≈ 0.29 of the total for r200,m.Since the difference in power between SO regions on the largest scales is just a reflection of their relative total mass fractions, this indicates that the r500,c regions contain slightly less than half the mass of the r200,m regions.We thus see that almost all power on small scales comes from clustering within the r500,c regions of haloes, even though these regions include much less mass.
The bottom-left panel of Figure 5 shows that for both SO radii, the 10 14±0.25 M⊙/h haloes provide the largest contribution to the full power spectrum for k ≲ 3 h Mpc −1 , although the dominance of these haloes continues up to k ∼ 7 h Mpc −1 for particles within r500,c.
On scales that are inside the halo (k > k peak ), the r500,c halo power spectra exhibit a slight oscillation that is caused by the sharp boundary of r500,c, particularly in the bottom panel.This is roughly equivalent to the oscillations seen in the Fourier transform of a tophat filter, and only more apparent for r500,c than for r200,m due to the higher density at the cut-off radius, and is therefore not a physical effect.
Comparing C AGN W7 and C DMO W7 in the top-left panel of Figure 5, we see that the differences in halo contributions are larger around the peak for r500,c than for r200,m.Specifically, when baryons and AGN are included the peak auto-power contribution of haloes to the matter power spectrum is increased -however, the top-right panel shows that this is again due to the total matter power spectrum decreasing in AGN relative to DMO, rather than the halo power itself.Like in §3.2, we thus conclude that the increase in the relative peak power contributions are due to a decrease in crosspower of haloes with the matter around it.We consider the role of the cross-power further in §3.6 and §4.
Comparing the r200,m and r500,c results in the top-right panel further, we also see that the "bump" seen around k ≈ 3 h Mpc −1 for r200,m for the two most massive halo mass bins (red and orange lines) is not seen for r500,c.This bump is due to the mass moved to the outskirts of the halo by AGN feedback, and the fact that it is not visible for r500,c means that a significant fraction of the mass is being deposited between r500,c and r200,m in the most massive haloes.The bottom-left panel shows that this is mostly due to the contribution of the very most massive haloes probed here (red line).At the same time, from the same panel we see that the ∼10 14.5 M⊙/h haloes show a higher C AGN W7/C DMO W7 ratio for particles within r500,c than particles within r200,m on large scales, around unity for k ≲ 3 h Mpc −1 .This indicates that the mass added to r500,c by the cooling of baryons is balanced by the mass ejected to r >r500,c by AGN feedback for these haloeskeeping in mind that the sizes of these regions may have changed between these simulations as well (see §3.4).We will refer to the region between r500,c and r200,m as the halo annulus, and consider it again in §4.2.
Looking at the other halo mass bins shown in the bottom-right panel of Figure 5, we see that for M500,crit⩾10 12.75 M⊙/h (green, yellow and orange lines), the C AGN W7/C DMO W7 ratios are much lower for r500,c than for r200,m.This is expected, as mass is removed from the central regions by AGN feedback and deposited on larger scales, thereby affecting the mass within r500,c more than that within r200,m.Contrarily, for 10 12.5±0.25M⊙/h haloes (blue), the C AGN W7/C DMO W7 ratio for r500,c lies above that for r200,m, but is still below unity indicating that the entire halo is less massive due to impeded growth or stripping of gas, as mentioned previously, and most of the mass is taken from/not added to the outer region of the halo.For even less massive haloes (pink and purple lines), the C AGN W7/C DMO W7 ratio even exceeds unity for r500,c, indicating that the cooling of baryons dominates over their ejection.
Note that the SO radii change between the simulations as the halo mass changes, limiting the conclusions that can be drawn from this figure concerning where mass is deposited.In §3.4, the radius is kept constant between simulations, to investigate this in more depth.
Finally, we note that the halo auto-power ratios between C AGN W7 and C DMO W7 for all mass bins as shown in the bottom-right panel of Figure 5 are very nearly constant on large scales.For r500,c, this even extends up to k ∼ a few or beyond, depending on halo mass.This indicates that the removal of mass from this region to great approximation determines the change in clustering of these haloes, irrespective of any changes in halo profiles, and that scale-dependent changes in halo clustering (such as seen in the top panels) are simply a consequence of combining different halo mass bins.As we will see in §3.4,this remains true when keeping the SO radius fixed.This suggests that one could accurately predict the total power suppression due to feedback by only knowing the fraction of mass that was removed from certain clustered regions.Based on this, we develop a model to do just that in §4.

Keeping the SO radius fixed between simulations
To get a sense of the how much matter is ejected from which radius by AGN feedback and where it is deposited, not only do the haloes need to be matched, but a constant halo boundary (SO radius) needs to be taken between simulations.This way, the comparison between simulations can be taken one step further, as not only processes within (or outside of) the same haloes, but also within certain regions of space in these haloes can be considered.
In Figure 6, the C AGN W7/C DMO W7 power ratio is displayed for the usual range in halo masses, for a particle selection where either the simulation SO radius (r200,m, continuous lines), or C DMO W7 SO radius (dashed lines) marks the halo boundary.
Although there are slight changes in the ratio between C AGN W7 and C DMO W7 power spectra for the r200,m simulation radii and the C DMO W7 r200,m radii, the C AGN W7/C DMO W7 power ratios of the two regions mostly follow the same trends and have the same shapes (top panel).On large scales, a comparison of the lines tells us that there is less mass in C AGN W7 haloes within the C DMO W7 r200,m radius than in the C AGN W7 SO radius for massive haloes (M500,crit≳10 13.25 M⊙/h), indicating that the C DMO W7 SO radius is smaller.The C AGN W7 r200,m has increased compared to the C DMO W7 r200,m because mass has been ejected from inside this radius to outside it, causing the density profile of the haloes to flatten in C AGN W7.
For lower-mass haloes (M500,crit⩽10 13.25 M⊙/h), the higher C AGN W7/C DMO W7 ratios for the C DMO W7 SO radius at k ∼ 0.02 h Mpc −1 indicate that there is more mass in the C DMO W7 r200,m than in the C AGN W7 r200,m in C AGN W7, implying a larger C DMO W7 r200,m.
The bottom panel of Figure 6 shows the same as the top panel, except that the halo boundary SO radii are the C DMO W7 and simulation r500,c.At this radius, the consequences of changing the radius to the C DMO W7 SO radius have much more impact.Interestingly enough, only haloes with 10 12.25 ≲M500,crit≲ 10 13.75 M⊙/h (blue, green and yellow lines) have an increased C AGN W7/C DMO W7 power ratio for the C DMO W7 r500,c, when compared to the simulation r500,c power ratio, on large scales.The nearest mass bins in both directions (purple and orange lines) suggest no change in C AGN W7/C DMO W7 ratio on large scales and the lowest (10 11.5±0.25 M⊙/h, pink) and highest (10 14.5±0.25 M⊙/h, red) mass haloes have lower C AGN W7/C DMO W7 ratios when choosing the C DMO W7 r500,c instead of the simulation r500,c.This suggests that for intermediate mass haloes, the C DMO W7 r500,c is larger than the C AGN W7 r500,c; for the more massive or lower-mass haloes, the C DMO W7 r500,c is smaller than or equal to the C AGN W7 r500,c.Therefore, the intermediate mass haloes are dominated by mass ejection inside r500,c, while the lower-mass and more massive haloes are dominated by contraction.
The most significant change in the bottom panel is seen for the most massive haloes, where the choice of the simulation or DMO r500,c radius not only shifts the contribution by 10% vertically, but also shows either a large bump or large dip for k ≈ 7 h Mpc −1 , de- M 500, crit = [M 500, crit, low , M 500, crit, low * 10 0.5 ] log10(M500,crit,low h M −1 ) = 11.25 log10(M500,crit,low h M −1 ) = 11.75 log10(M500,crit,low h M −1 ) = 12.25 log10(M500,crit,low h M −1 ) = 12.75 log10(M500,crit,low h M −1 ) = 13.25 log10(M500,crit,low h M −1 ) = 13.75 log10(M500,crit,low h M −1 ) = 14.25 Figure 6.Similar to the bottom-right panel of Figure 5, with results for r 200,m shown in the top panel and those for r 500,c in the bottom panel, but now with two definitions for the SO radii: either the particle selection criterion is as before, that is using the SO radius measured in each simulation (continuous linestyle), or the SO radius as measured in C DMO W7 is used for C AGN W7 as well (dashed linestyle).The change in SO radius between C AGN W7 and C DMO W7 varies between r 200,m and r 500,c and with halo mass, with the r 500,c result being most strongly affected.
pending.This suggests that the DMO r500,c radius is both smaller than the C AGN W7 halo radius for these haloes, and simultaneously such that it captures a lot of the material pushed out by feedback.The r500,c radius in the baryonic simulation, on the other hand, is large enough that the effect of this density bump is effectively smoothed out.To a lesser extent, this effect is also present in the top panel, where the height of the bump at k ≈ 3 h Mpc −1 is increased for the two most massive halo mass bins when the DMO r200,m radius is used.

Varying the feedback temperature
In order to examine the role of the feedback strength on the halo contributions for matched haloes, we can repeat our analysis for the three BAHAMAS simulations where the AGN heating temperature is varied.Figure 7 shows the contribution of haloes to the full power spectrum (left side) and baryon-dark matter power ratios (right side) for matched sets of B DMO W9, B AGN W9, B AGN7p6 W9 and B AGN8p0 W9, for the same range of halo mass bins as the previous figures.The AGN feedback temperature has a relatively large impact on the B AGN W9/B DMO W9 full power spectrum ratio for k ≳ 0.1 h Mpc −1 .However, the changes in the halo mass bin power spectra and the full matter spectra are of similar order, resulting in a relatively small impact of the AGN feedback temperature on the contribution of each halo mass bin to the full power spectrum as can be seen in the top-left and bottomleft panels of Figure 7.The two sets of dashed lines show simulation with a weaker (long-dashed) or a stronger (short-dashed) heating temperature, and show only minor differences with respect to the fiducial feedback strength (solid continuous line).
On large scales, the discrepancies between the contributions to the full power spectrum of B AGN W9, B AGN7p6 W9 and B AGN8p0 W9 are more substantial around M500,crit≈10 12.5−13 M⊙/h (bottom-left panel), suggesting that the masses of these haloes are most sensitive to changes in AGN feedback temperature (as confirmed by the bottom-right panel).
The top-right panel of Figure 7 indicates that the effect of AGN feedback temperature on the power ratios between the AGN simulations and B DMO W9 is quite substantial.Although it is mostly a vertical displacement, where a higher feedback temperature results in a lower AGN/B DMO W9 ratio, there are some changes in AGN/B DMO W9 ratio shape as well.For the most massive haloes (M500,crit⩾10 14.25 M⊙/h, red line), the top and bottom right panels show a clear change in AGN/B DMO W9 ratio slope at k ∼ 4 h Mpc −1 , indicating a change in profile.The B AGN7p6 W9/B DMO W9 power ratio (low AGN feedback temperature) mimics the shapes of the C AGN W7/C DMO W7 power ratio, in the top right panel of Figure 4.This behaviour indicates that these haloes are hardly affected by the AGN feedback: it is damped so aggressively by their deep potential wells, the effects on the haloes are no longer dominating the shape of the B AGN7p6 W9/B DMO W9 power ratio.The B AGN8p0 W9/B DMO W9 ratio (high AGN feedback temperature) for the most massive haloes (M500,crit⩾10 14.25 M⊙/h, red line) follows the slope of the other, less massive haloes.Apparently, at this temperature the potential wells of the haloes are no longer deep enough to produce a significant damping effect.The 10 14±0.25 M⊙/h (orange lines) haloes exhibit a similar, albeit a reduced, effect, as illustrated by the top and bottom right panels.
Finally, looking at the bottom-right panel, we see the same trends as before: increasing/decreasing the feedback temperature mainly lowers/raises the mass and therefore the relative auto-power contribution at each halo mass, where the largest dependence on feedback temperature is seen around ∼10 13 M⊙/h.For the most massive haloes, an effect on the halo profile can also be seen.

Other contributions to the power spectrum
We have thus far only considered auto-power spectra -that is, the clustering of haloes of a given mass relative to their own population.While the auto-power spectrum of all halo particles above a given mass (the relative contribution of which was typically shown in the top panels of the preceding figures) provides the dominant contribution to the total matter power spectrum for k ≳ 0.5 h Mpc −1 , other significant contributions exist, in the form of the auto-power spectrum of non-halo particles (i.e.those that live outside of any particular overdensity region) and the cross-power M 500, crit = [M 500, crit, low , M 500, crit, low * 10 0.5 ] log 10 (M 500, crit, low h M −1 ) = 11.25 log 10 (M 500, crit, low h M −1 ) = 11.75 log 10 (M 500, crit, low h M −1 ) = 12.25 log 10 (M 500, crit, low h M −1 ) = 12.75 log 10 (M 500, crit, low h M −1 ) = 13.25 log 10 (M 500, crit, low h M −1 ) = 13.75 log 10 (M 500, crit, low h M −1 ) = 14.25 4, but showing results for BAHAMAS (B AGN W9, B DMO W9, B AGN7p6 W9 and B AGN8p0 W9) rather than cosmo-OWLS.These simulations shown the change in impact of AGN feedback on haloes of different masses when the strength of the AGN feedback is varied (through the heating temperature).The heating temperature has a large influence on the C AGN W7/C DMO W7 ratios (right side), while the contribution of the halo mass bins to their corresponding full spectra remain similar.spectrum of non-halo and halo particles.If we choose to separate the haloes into different mass bins, the number of auto-and crosspower terms contributing to the matter power spectrum sharply increases, as we need to add cross spectra of particles in haloes of mass A with those in haloes of mass B (etc), and cross spectra with non-halo particles for all mass bins.It is therefore more convenient (and useful) to instead consider the matter power spectrum as a sum of the cross spectra of non-overlapping groups of particlesnamely, haloes in some particular mass bin, and non-halo particles -with all matter.
In Figure 8, we show these cross-power terms for B DMO W9.To better separate halo from non-halo matter, here we again define haloes as r200,m overdensity regions, but still select on their M500,crit mass as before.The halo-matter cross terms are shown as coloured lines for different mass bins, with their to-tal shown as solid gray.Adding the non-halo-matter cross term (where non-halo matter is defined here as all matter not in haloes M500,crit >10 11.25 M⊙/h), shown as dashed grey, yields exactly the total matter power spectrum (black).On large scales, the total halo and non-halo cross terms contribute almost equally to the total matter power spectrum, as they contain a comparable amount of mass.On small scales however, the halo cross term fully dominates, as expected.It should be noted that the division into halo and non-halo matter is a function of resolution, as with increased resolution we expect more diffuse matter to be resolved into lowmass haloes -however, with the current resolution we are already in a regime where the non-halo-matter cross term is subdominant on all scales.
In the following section, we show how we can exploit this Figure 8.The matter power spectrum of B DMO W9 (solid black) and its halo (solid grey) and non-halo (dashed grey) cross-power components, P mh,200m and P mnh,200m respectively (see §4).The mass-weighted halo power in different mass bins, f M,i,200m P mh,i,200m , is shown as solid colored lines.All haloes are selected on their M 500,crit mass.A linear power spectrum (long-dashed purple) is shown for comparison.The halo crosspower dominates on small scales, but on large scales halo and non-halo particles contribute roughly equally to the total power, as the total mass is both components is similar.
division into cross terms to accurately model the power suppression in baryonic simulations.

A "RESUMMATION" MODEL FOR THE BARYONIC SUPPRESSION OF MATTER CLUSTERING
Let us call the cross-power of haloes in mass bin i and all matter, P mh,i,∆ (k).Here we will define haloes as regions with an overdensity ∆ relative to the critical density ρcrit, e.g.∆ = 200Ωm or ∆ = 500.Each halo mass bin will contain some fraction of the total mass in a volume, let us call these fractions fM,i,∆.
We will refer to the matter-matter auto-power spectrum as Pmm(k).The halo cross terms are assumed to be unnormalized with respect to mass -the large-scale (linear) halo bias of haloes in mass bin i, then, is given by bi,∆ = lim In practice, we will calculate the bias by averaging the power ratio on large scales (k ≲ 0.13 h/Mpc), where it is roughly constant.As discussed in §3.6, we can write the matter auto-power spectrum as a sum over all halo-matter cross terms, plus a cross term between all matter and the matter not in haloes.This necessarily includes matter in unresolved haloes.Since haloes are highly biased, this last cross term is expected to be dominated by matter just outside haloes (see also §3.5).We will refer to the cross term of all matter with matter not in (resolved) haloes as P mnh,∆ (k) (note that this term, too, depends on our choice of overdensity region).By definition, then, these cross-power terms must satisfy: where the sum is over all halo mass bins, and thus gives the total contribution of matter in (resolved) haloes.3All quantities considered here can be readily calculated from dark matter only simulations.Figure 8 shows these quantities for B DMO W9.

Accounting for mass loss
Let us now consider what happens to these different terms as matter is redistributed by the processes associated with galaxy formation.While these processes change halo profiles -for example by gas cooling to small scales and forming stars, contracting the inner dark matter halo -the main effect on large scales is caused by removing mass from clustered regions.Therefore, on sufficiently large scales we can approximate the effects of galaxy formation by scaling the mass fractions of haloes by the mass they retained -that is, the ratio of the mass of a feedback-affected halo and its DMO equivalent.
Let us call the mean retained mass fraction for haloes in mass bin i, fret,i,∆.Let us further assume, just for the moment, that the total matter distribution we cross-correlate with is fixed to the DMO distribution.The total contribution to the matter-matter power spectrum of feedback-affected haloes then becomes: where the prime indicates a correction for retained mass.
The mass removed from haloes has to go somewhere, in such a way that the total mass in the volume is conserved.One option is to add this mass to a linear power component -however, the ejected mass is expected to stay around haloes, and therefore still cluster more significantly than linear.We thus add the mass to the non-halo component, which likely was already dominated by mass around haloes.One wrinkle is that we are removing mass from biased regions -we therefore need to take bias into account to ensure that the power at low k after modelling galaxy formation still converges to the original value.The ratio of the corrected non-halo cross-contribution to the original non-halo cross-contribution by definition satisfies: .
If we now consider the low-k limit of this expression, and demand that lim k→0 P ′ mm (k)/Pmm(k) = 1, we find: We thus see that to preserve the power on the largest scale when redistributing mass, we need to not just conserve mass, but the product of mass and bias.Finally, we have to drop our temporary assumption that the total matter distribution we cross-correlate with is held fixed.Luckily, at this point we already know how the contributions from both the halo and non-halo matter distributions transform, and therefore how the total matter contribution transforms.Using double primes to indicate a correction for halo mass loss in both matter components that make up the cross power, we find: The doubly-corrected halo-matter cross power term then becomes P ′′ mh,∆ = q∆P ′ mh,∆ , and similarly P ′′ mnh,∆ = q∆P ′ mnh,∆ .In summary, this model takes dark matter only halo mass fractions, linear biases and (cross-)power spectra, and combines them with the mean fraction of mass retained by haloes that have undergone galaxy formation, relative to their dark matter only equivalent, to predict the change in the total matter power spectrum.This mean fraction of mass retained, fret,i,∆, can be calculated as the total mass ratio of (matched) haloes in a hydrodynamical simulation and its DMO equivalent.Observationally, fret,i,∆ can in principle be derived from the mean observed baryon fractions of haloes of a certain mass, fb,i,∆ , relative to the cosmic baryon fraction.Specifically, under the assumption that all matter removed from the halo was baryonic matter, one would find the retained fraction to be: where fb,i,∆ = M bar,i /Mtot,i is the mean baryon fraction measured in haloes in mass bin i.This can be straightforwardly derived assuming that the "original" baryon fraction of each halo is Ω b /Ωm, and that the retained CDM fraction, equal to fret,i,∆/f bc,i,∆ , is unity.However, in reality the dark matter will respond gravitationally to the loss of mass, and additional mass will be lost as the halo relaxes.The amount of additional mass lost as a consequence of relaxation after a baryonic ejection event will be explored in Wolfs & van Daalen (in prep.).For now, we note that the following linear relation is accurate to ≲ 1% on average for haloes M h,i,500c ≳ 10 12.5 M⊙/h in all cosmo-OWLS and BAHAMAS simulations explored here, regardless of variations in cosmology or feedback strength: with a ≈ 1.768 and b ≈ 0.4206 for ∆ = 200Ωm.For ∆ = 500, the best-fit coefficients are a ≈ 2.311 and b ≈ 0.5251, though we note that for this overdensity radius the variance is significantly larger (rms ≈ 2%).4 Crucially, this relation links the baryon fraction measured at a retained total mass (which is what we observe) to a retained fraction at a DMO total mass -hence no shifting of mass bins is necessary in applying this relationship to equations (3)-( 6).
For lower-mass haloes (M h,i,500c ⩽ 10 12 M⊙/h), the baryon fraction is more challenging to measure observationally.After checking that these haloes play a negligible role in setting the power suppression up to k ≈ 10 h Mpc −1 , we fix the retained fractions for these halo masses to unity.

Combining overdensity regions
In principle, we now have all the ingredients to apply this model in practice for a given choice of overdensity region, using the mean baryon fraction measured inside that region for different halo mass bins to rescale and re-sum DMO cross-power spectra.But, if these measurements are available for multiple overdensity regions -which is now feasible through combinations of X-ray, thermal Sunyaev-Zel'dovich and optical/infrared observations -we can combine this information to model the power spectra more accurately.As we will show, doing so can greatly compensate for the approximation that the halo profiles are fixed.
Say that we measure the total (retained) mass and baryon fraction of a sample of haloes for both ∆ = 500 and ∆ = 200Ωm.We can then calculate the retained fraction for each region with equation ( 9).The net fraction of the "original" DMO mass that has been removed from r500,c, but not r200,m, is then fret,i,200m − fret,i,500c.As shown in §3.3, this fraction is quite significant for massive haloes.We will once more refer to the region r500,c < r < r200,m as the halo annulus, and will indicate it with subscript A. The cross-power contribution of matter in these regions is then given by: ) and the feedback-corrected annulus cross-contribution is then: fret,i,200mfM,i,200m−fret,i,500cfM,i,500c]PmA,i(k).

Model results
A proof of concept is presented in Figure 9. On the lefthand side, we compare the true power suppression measured in B AGN W9 (red line) to the results of our model (orange, green and blue dashed lines).Our "resummation" model takes as input halo baryon fractions measured from B AGN W9 for haloes M500,crit>10 12 M⊙/h and combines these with quantities measured from DMO to predict the power suppression due to galaxy formation.Also shown is the result of applying the van Daalen et al. ( 2020) relation (cyan short-dashed line), which uses only the baryon fraction measured for haloes around M500,crit=10 14 M⊙/h.While our model underpredicts the suppression on large scales when only M500,crit haloes are used (orange), the agreement between the true suppression and our model's prediction when using only M200,mean haloes (green) is remarkable.When information from both overdensity regions is combined as described in §4.2 (blue), the model reproduces the true suppression to < 1% on all scales measured.On the right-hand side of Figure 9, we show the results of applying the same model to each of the cosmo-OWLS and BA-HAMAS simulations explored in the previous sections, combining the M500,crit and M200,mean overdensity regions.We stress that for each simulation the exact same values for parameters a and b of equation ( 9) have been used; the only thing that changes from simulation to simulation are the baryon fractions measured in their haloes (and, when shifting between cosmo-OWLS and BA-HAMAS, the DMO simulation used).The model typically reproduces the true power suppression to within 1% accuracy all the way down to at least k = 8 h Mpc −1 , and almost always within 2%.For the model with the strongest AGN feedback, B AGN8p0 W9, the  2020) (dashed cyan), which was fit to simulations up to k = 1 h Mpc −1 and using only the baryon fraction at a single halo mass.Note that we set the suppression to unity for k < 0.07 h Mpc −1 and smooth the result to counter sampling noise.Right: The same model applied to all simulations explored here.The "resummation" model reproduces the true suppression to well within 1 − 2% on virtually all scales k ⩽ 8 h Mpc −1 , given baryon fractions as a function of halo mass, using only a single set of two fixed parameter values (see equation ( 9)).
model slightly underpredicts the suppression on large scales, which again hints that haloes in this simulation may lose mass from regions larger than r200,m.However, even for this case the model is percent-level accurate.Following the procedure outlined in §4.2, the model is easily extended to larger or smaller overdensity regions, such as M2500,crit, to cover an even larger range in k -as long as baryon fractions can be measured for these regions.
We note that the model is expected to perform slightly worse for cosmo-OWLS, since the DMO simulations of this suite model only a single fluid with a combined dark matter and baryon transfer function, whereas the baryonic simulations separate these into two fluids, and therefore also contain twice as many particles (see Appendix B of van Daalen et al. 2020 for more information).Despite this, the model and simulations show remarkable agreement for this suite as well.
From the results presented in this work, it is clear why the van Daalen et al. ( 2020) relation (shown in cyan in the left-hand panel of Figure 9) was able to accurately fit the power suppression down to k ≈ 1 h Mpc −1 using the baryon fraction of M500,crit∼10 14 M⊙/h haloes: (a) the baryon fraction is strongly correlated with the retained mass fraction of haloes; (b) objects around this mass scale dominate the power contributions on large scales; and (c) the effects of feedback in these haloes are generally highly correlated with those in adjacent mass scales.At the same time, this also makes it clear what the limitations of only using a single baryon fraction are: the relation has to implicitly model the link between 10 14 M⊙/h and other mass scales, capture its scaling with k, and be limited to scales roughly comparable to the size of the halo and above.The former two limitations also made the model less universally applicable, as the power suppression in simulations in which the relation between feedback effects on different mass scales deviated strongly from that of other models were more poorly reproduced (e.g. the original Illustris simulation, see Vogelsberger et al. 2014;van Daalen et al. 2020).In contrast, while the "resummation" model presented here needs additional observationally determined baryon fractions as well as quantities measured from dark matter only simulations as input, it extends to higher k without losing accuracy, could straightforwardly be easily extended to higher redshifts, and has a far less empirical basis.
We have employed several implicit assumptions here, such as that feedback does not significantly change the halo positions or radii.The first is justified (see e.g.van Daalen et al. 2014) while the second is not (see e.g.Velliscig et al. 2014), although it is unclear what the impact of this would be, if any.We also have assumed that the linear halo bias is preserved as haloes lose mass.We have checked this assumption by comparing the bias of matched haloes in our simulations, and found that the relative change in bias is typically ≲ 1% -but future iterations of this model could still take bias changes into account to further improve the performance.Another possible improvement is to reduce the scatter in the fret − f b relation by taking into account a possible halo mass dependence through the mass-dependent concentration of the haloes.As Elbers et al. (in prep.) will show, taking into account the baryon fraction and concentration simultaneously allows one to predict the power suppression more accurately.Still, from the results presented in Figure 9, it would appear that our model is already able to yield highly accurate predictions despite these shortcomings.

SUMMARY AND DISCUSSION
In §3 we presented our findings concerning the contribution of haloes with a range of masses to the full power spectrum and the varying effects of baryons on the power in these haloes, when compared to a dark matter only simulation, as a function of scale using simulations from the cosmo-OWLS and BAHAMAS simulation projects.The contributions of various haloes to the full power spectrum and the baryon/DMO simulation power ratios of these haloes were considered for a matched (and an unmatched) set of haloes, their boundary defined by either their r200,m or r500,c spherical overdensity radius.These were compared to the results for the halo power within a constant SO radius between simulations.Lastly, the influences of variations in AGN feedback temperature on the contributions to the full power spectrum and the baryon/DMO simulation power ratios were also investigated for a matched set of haloes.
Our main findings are as follows: (i) Haloes of around M500,crit∼10 14 M⊙/h provide the dominant contribution to the matter power spectrum up to k ∼ a few.Higher-mass haloes provide a lesser but still very significant contribution on large scales (k ∼ 0.1 h Mpc −1 ), as their rarity becomes more important than their high masses and bias, while lower-mass haloes are less biased on large scales but provide a contribution comparable to ∼10 14 M⊙/h haloes on smaller scales.This is in line with the findings of van Daalen & Schaye (2015).This also means that the effects of galaxy formation on these haloes are a strong indicator for how galaxy formation affects matter clustering as a whole, as found by van Daalen et al. (2020).
(ii) By matching haloes between baryonic and dark matter only simulations, we find that galaxy formation including AGN feedback lowers the masses within r200,m of haloes down to at least M500,crit≈10 11.75 M⊙/h, and therefore also their contributions to the matter power spectrum.On intra-halo scales, the removal of mass from clustered regions by AGN feedback ensures that the contribution of r200,m overdensity regions to the matter power spectrum is smaller than that for a DMO universe up to at least k = 10 h Mpc −1 .Including galaxy formation but not AGN feedback also lowers all contributions, but by less than with AGN feedback, and only up to k ∼ a few, depending on halo mass.
(iii) The halo auto-power for simulations with AGN shows a dip relative to DMO on scales k ≈ 10 h Mpc −1 for massive haloes (M500,crit≳10 12.75 M⊙/h), corresponding to scales internal to the haloes where mass was removed.For lower-mass haloes, this dip is not present, indicating that these haloes do not host (effective) AGN themselves but were affected by them indirectly.This could be due to nearby AGN heating and driving out the gas from these less massive haloes, or by them affecting the mass accretion history of these haloes.
(iv) Even though the auto-power contribution to the matter power spectrum from the same haloes goes down when AGN feedback is included, the peak fractional contribution of haloes ≳10 13.25 M⊙/h to the matter power spectrum is slightly higher than for DMO.This indicates that the contribution of cross-power between matter inside and outside r200,m goes down on scales comparable to this radius.
(v) The choice of halo boundary (r200,m or r500,c ) affects the baryon/DMO power ratios for both massive and lower-mass haloes.Compared to DMO, baryonic haloes with masses ⩽10 12.75 M⊙/h lose a smaller fraction of their mass in their r500,c regions than in their r200,m regions, and the same goes for the most massive haloes (M500,crit>10 14.25 M⊙/h).However, haloes with masses 10 12.75 ⩽M500,crit⩽10 14.25 M⊙/h, in which AGN are most effective, lose more mass from within r500,c than within r200,m.
(vi) By fixing the r500,c radius to the DMO value, we find that the auto-power of haloes with 10 12.25 ≲M500,crit≲ 10 13.75 M⊙/h is increased compared to that within their own r500,c.The inverse is true for much more or much less massive haloes.This suggests that intermediate mass haloes are dominated by mass ejection inside ∼r500,c, while more massive or lower-mass haloes are dominated by contraction.
(vii) The strength of AGN feedback primarily affects the normalization of the halo auto-power contribution as a function of mass, on all scales probed here (k < 10 h Mpc −1 ).Only for the most massive haloes, M500,crit≳10 14 M⊙/h, does the contribution significantly change shape for k ≳ a few, as the AGN feedback affects the halo profiles on these scales.
Furthermore, in §4 we presented a novel model that utilizes the cross-power of halo matter inside spherical overdensity regions with all matter in combination with retained mass fractions to predict the suppression of the total matter power spectrum due to galaxy formation.We found that with a simple, fixed relation between observed halo baryon fractions and retained mass fractions our model can reproduce the power suppression of various simulations to ∼ 1% up to at least k = 8 h Mpc −1 .The best results were obtained if both r200,m and r500,c baryon fractions were available.
The model presented in this work is built on the link between the observed halo baryon fractions and the retained halo mass, rather than the overall suppression of power, and we find strong indications that this link may not (or only weakly) depend on the strength of feedback, allowing measurements of baryon fractions on different scales to set the power suppression in a modelindependent way.There are currently also only two parameters present in our approach, both of which are fixed, and no halo profiles need to be modelled to get accurate predictions up to at least k = 8 h Mpc −1 , which means that cosmological parameters derived from weak lensing observations may have lower uncertainties than in most other approaches, as there are fewer parameters to marginalize over.
The main disadvantage is that our model is built on quantities derived from dark matter simulations, which require significant computational resources -although many of these already exist in the literature for the purposes of weak lensing analysis.However, similar to e.g.baryonification, this shortcoming may be overcome in the future by training a model to produce the halo mass fractions and halo-matter cross-spectra outcomes for such relatively predictable simulations as a function of cosmology, or by using halo model predictions.Exploring the dependence of our approach on cosmology, if any, may also lead to a significant reduction of resources, as fewer dark matter only simulations will be needed if the scaling with cosmology is known.Another disadvantage is that the model requires baryon fractions measured out to r200,m, which are less readily available than those within r500,c.While it is possible to predict one from the other, such a mapping would be model-dependent.However, the availability of SZ measurements of fb,200m is growing, so this may not be an issue for long.Future work will extend the model presented in §4, including to z > 0, and apply it to the Gpc-scale FLAMINGO suite of simulations, recently presented in Schaye et al. (2023).As these simulations contain larger volumes with resolutions similar to BA-HAMAS, as well as additional physics variations, analysing these simulations should lead to a more detailed modelling of the relation between the halo baryon and retained fractions, improving the accuracy of the model.Additionally, folding in other overdensity scales such as r2500,crit or explicitly including galaxies through the measured stellar fractions of halo may allow us to extend the model to higher values of k.

APPENDIX A: BOX-SIZE EFFECTS
When working with simulation data, there are numerical effects on the data that have no physical cause.One source of numerical effects is the limited size of the cosmological volume that is simulated.Apart from the limitation in scales that can be investigated, smaller boxes host fewer haloes disproportionally fewer rare systems, giving larger uncertainties in statistics.
Figure A1 shows the contribution of haloes to the full power spectrum and the ratio of AGN/DMO power for a range of halo masses (the same range as used in the rest of the paper), for cosmo-OWLS simulations with volumes of (400 h −1 Mpc) 3 (continuous lines), (200 h −1 Mpc) 3 (dashed lines) and (100 h −1 Mpc) 3 (dotdashed lines).The resolution has been kept constant between the simulations.The haloes have been matched between the DMO and AGN simulations for each box size.The masses are based on the DMO simulation.
The top-left panel of the figure shows that for most mass bins, the different volumes broadly agree with one another on the M 500, crit = [M 500, crit, low , M 500, crit, low * 10 0.5 ] log 10 (M 500, crit, low h M −1 ) = 11.25 log 10 (M 500, crit, low h M −1 ) = 11.75 log 10 (M 500, crit, low h M −1 ) = 12.25 log 10 (M 500, crit, low h M −1 ) = 12.75 log 10 (M 500, crit, low h M −1 ) = 13.25 log 10 (M 500, crit, low h M −1 ) = 13.75 log 10 (M 500, crit, low h M −1 ) = 14.25   4, except that not only the L400N1024 (continuous), but also the L200N512 (dashed) and the L100N256 (dot-dashed) cosmo-OWLS simulations are shown.The resolution of all simulations is kept fixed as the box size is varied.The largest deviations between the different box sizes are in the most massive haloes, due to cosmic variance being large for these in smaller volumes.contribution of different haloes above a certain mass to the full power spectra.The minor differences can be attributed to cosmic variance as they show no clear trend with increasing box size.The clear exception is the contribution of the most massive haloes (M500,crit⩾10 14.75 M⊙/h) to the full power spectrum in the L100 simulation (red dot-dashed line).This is because these haloes are so rare that they have difficulty forming in the L100 simulation and are therefore underrepresented.The fact that the L200 and L400 simulations converge on the vertical scale shows that these boxes are large enough to give a representational sample.
The bottom-left panel shows convergence for the lower masses (M500,crit≲10 13.75 M⊙/h) between box sizes.The two highest mass bins show large discrepancies.Compensating for the lower contribution to the total of the most massive haloes (M500,crit=10 14.5±0.25 M⊙/h), the second most massive haloes (M500,crit=10 14±0.25 M⊙/h) have a higher contribution to the full power spectrum in the L100 simulation than in L200 or L400.While the L200 and L400 simulations agree on the contribution of 10 14±0.25 M⊙/h haloes to the full power spectrum, the contribution of 10 14.5±0.25 M⊙/h haloes to the total is higher in L200 than in L400.As they both agree on the contribution of haloes with M500,crit⩾10 14.25 M⊙/h (top left panel), it seems a few of the haloes with M500,crit⩾10 14.75 M⊙/h in the L400 simulation are not fully formed in the L200 simulation, and added to the 10 14.5±0.25 M⊙/h mass bin instead, increasing its contribution to the full power spectrum.
The right side of figure A1 shows the AGN/DMO power ratios for each halo mass bin for each volume.The top-right panel shows broad agreement when taking haloes with masses ≳10 13.5 M⊙/h into account.As seen in the left-hand panels however, AGN/DMO ratios of the most massive haloes do not agree across the various box sizes.For haloes with masses ⩾10 13.75 M⊙/h (orange), the M 500, crit = [M 500, crit, low , M 500, crit, low * 10 0.5 ] log 10 (M 500, crit, low h M −1 ) = 11.25 log 10 (M 500, crit, low h M −1 ) = 11.75 log 10 (M 500, crit, low h M −1 ) = 12.25 log 10 (M 500, crit, low h M −1 ) = 12.75 log 10 (M 500, crit, low h M −1 ) = 13.25 log 10 (M 500, crit, low h M −1 ) = 13.75 log 10 (M 500, crit, low h M −1 ) = 14.25   7, except that the B AGN PL and B DMO PL (dashed) have been added to the B AGN W9 and B DMO W9 (continuous) power spectra to allow for a comparison between cosmologies.Both AGN-DMO simulation pairs have been matched and the two DMO simulations have also been matched.The halo mass bins are based on B DMO W9.The influence of cosmology on the contributions of halo power to the full power spectrum and the AGN/DMO ratios of halo power is largest for massive haloes (M 500,crit ≳10 13.25 M ⊙ /h).Furthermore, the Planck cosmology moves power from massive to lower-mass haloes, when compared to the WMAP9 cosmology.
tion of haloes and their AGN/DMO power ratios.The influence of cosmology is much more substantial for the most massive haloes, in their formation and evolution as well as in the influence of AGN feedback on their power suppression in the AGN simulations, when compared to the DMO simulations.As there are many factors that are influenced by cosmology that determine the power in different mass haloes, more extensive research is required to disentangle the effects of the various cosmological parameters and find the one(s) dominating the changes in power.

MFigure 4 .
Figure 4.The left side of the figure is as in Figure 3, except that the haloes are matched between C DMO W7 and the baryonic simulations.Only matched pairs are shown.The right side of the figure shows the power spectra of the C AGN W7 (solid) and C REF W7 (transparent and dashed) simulations divided by the power spectra of C DMO W7, for each halo mass bin.M 500,crit masses are measured in C DMO W7.AGN feedback lowers the masses of all haloes.On small scales, AGN feedback causes a dip in the C AGN W7/C DMO W7 power ratios of massive (M 500,crit ⩾10 12.75 M ⊙ /h) haloes, whereas the lower-mass haloes shown a sharp rise, like all C REF W7/C DMO W7 power ratios.

MFigure 5 .
Figure5.Absolute and relative halo power contributions for C AGN W7 and C DMO W7 as in figure4, but dashed lines now show the results when only particles within r 500,c are included.The peak in halo power contribution is moved to higher k for particles within r 500,c , relative to the results for r 200,m .Although much less mass is included in the smaller r 500,c regions, the power on small scales only drops slightly, as they are highly biased.As the bottom-left panel shows, for both SO regions ∼10 14 M ⊙ /h haloes provide the largest contribution to the total power up to k ∼ a few.From the bottom-right panel, we see that for group-sized haloes the effects of AGN feedback are more extensive within r 500,c , compared to within r 200,m .

Figure 9 .
Figure 9. Left: The true power suppression of B AGN W9 (red) compared to what our "resummation" model predicts.Combining information from both r 200,m and r 500,c gives the best results, although baryon fractions inside r 200,m alone are enough to reproduce the true suppression down to k ≈ 3 h Mpc −1 already.Also shown is the simple relation from van Daalen et al. (2020) (dashed cyan), which was fit to simulations up to k = 1 h Mpc −1 and using only the baryon fraction at a single halo mass.Note that we set the suppression to unity for k < 0.07 h Mpc −1 and smooth the result to counter sampling noise.Right: The same model applied to all simulations explored here.The "resummation" model reproduces the true suppression to well within 1 − 2% on virtually all scales k ⩽ 8 h Mpc −1 , given baryon fractions as a function of halo mass, using only a single set of two fixed parameter values (see equation (9)).

Figure A1 .
Figure A1.Similar to Figure4, except that not only the L400N1024 (continuous), but also the L200N512 (dashed) and the L100N256 (dot-dashed) cosmo-OWLS simulations are shown.The resolution of all simulations is kept fixed as the box size is varied.The largest deviations between the different box sizes are in the most massive haloes, due to cosmic variance being large for these in smaller volumes.

Figure B1 .
Figure B1.Similar to Figure7, except that the B AGN PL and B DMO PL (dashed) have been added to the B AGN W9 and B DMO W9 (continuous) power spectra to allow for a comparison between cosmologies.Both AGN-DMO simulation pairs have been matched and the two DMO simulations have also been matched.The halo mass bins are based on B DMO W9.The influence of cosmology on the contributions of halo power to the full power spectrum and the AGN/DMO ratios of halo power is largest for massive haloes (M 500,crit ≳10 13.25 M ⊙ /h).Furthermore, the Planck cosmology moves power from massive to lower-mass haloes, when compared to the WMAP9 cosmology.

Table 1 .
Relevant simulation parameters for simulations from BAHAMAS (McCarthy et al. 2017) and cosmo-OWLS (Le Brun et al. 2014) that are used in this research.Simulation names (and their abbreviations) include the cosmologies used, along with particle types present (DMO only contains dark matter, REF and AGN are baryonic).All simulations have volumes (400 h −1 Mpc) 3 .The AGN simulations include thermal AGN feedback, with heating temperature ∆T heat , REF only has SN feedback.

Table 2 .
Cosmological table 2) in Figure B1 of Appendix B. Starting from initial conditions generated by V. Springel's parameters from the seven-year WMAP release (Komatsu et al. 2011), the nine-year WMAP release (Hinshaw et al. 2013) and the 2013 Planck release (Ade et al. 2014) used in the simulations from table 1.