The effect of baryons on the variance and the skewness of the mass distribution in the universe at small scales
aa r X i v : . [ a s t r o - ph . C O ] M a r Mon. Not. R. Astron. Soc. , 000–000 (0000) Printed 31 October 2018 (MN L A TEX style file v2.2)
The effect of baryons on the variance and the skewness ofthe mass distribution in the universe at small scales
T. Guillet ⋆ , R. Teyssier , and S. Colombi . Service d’astrophysique, CEA/Saclay, F-91191 Gif-sur-Yvette, France Universit¨at Z¨urich, Institute f¨ur Theoretische Physik, Winterthurerstrasse 190, CH-8057 Z¨urich Institut d’Astrophysique de Paris, CNRS UMR 7095 & UPMC, 98bis, bd Arago, F-75014 Paris
31 October 2018
ABSTRACT
We study the dissipative effects of baryon physics on cosmic statistics at smallscales using a cosmological simulation of a (50 Mpc /h ) volume of the universe. TheMareNostrum simulation was performed using the AMR code ramses , and includesmost of the physical ingredients which are part of the current theory of galaxy for-mation, such as metal-dependent cooling and UV heating, subgrid modelling of theISM, star formation and supernova feedback. We re-ran the same initial conditionsfor a dark matter only universe, as a reference point for baryon-free cosmic statistics.In this paper, we present the measured small-scale amplification of σ and S dueto baryonic physics and their interpretation in the framework of the halo model. Asshown in recent studies, the effect of baryons on the matter power spectrum can beaccounted for at scales k . h. Mpc − by modifying the halo concentration param-eter. We propose to extend this result by using a composite halo profile, which is alinear combination of a NFW profile for the dark matter component, and an expo-nential disk profile mimicking the baryonic component at the heart of the halo. Thishalo profile form is physically motivated, and depends on two parameters, the massfraction f d of baryons in the disk, and the ratio λ d of the disk’s characteristic scaleto the halo’s virial radius. We find this composite profile to reproduce both the small-scale variance and skewness boosts measured in the simulation up to k ∼ h. Mpc − for physically meaningful values of the parameters f d and λ d . Although simulationslike the one presented here usually suffer from various problems when compared toobservations, our modified halo model could be used as a fitting model to improve thedetermination of cosmological parameters from weak lensing convergence spectra andskewness measurements. Key words:
Gravitational lensing – cosmological parameters – Galaxy: disk –Hydrodynamics – large-scale structure of Universe
One of the great challenges in modern cosmology is to under-stand the nature of dark energy, which is believed to domi-nate the energy budget in the universe ( ∼ Λ directly im-pacts the rate of structure formation at recent epochs, themass distribution and its time evolution bear the signatureof the dark energy content of the universe. Cosmic shearmeasurements provide an independent method of probingthe total mass distribution at large scales. Combined withphotometric redshifts, it is even possible to extract the 3D ⋆ E-mail: [email protected] matter distribution and reconstruct the matter power spec-trum at different epochs. Comparing these measurementsto theoretical predictions will set strong constraints on thecosmological parameters (e.g., Hu & Tegmark 1999; Huterer2002; Amara & Refregier 2006; Albrecht & Bernstein 2007),and among them both the equation of state w of dark energyand its possible evolution w ′ with redshift.The cosmic shear convergence power spectrum P κ de-pends on the total matter power spectrum P , which con-tains the information about w and w ′ through the growthrate of structures. Extracting the dark energy equation ofstate from weak lensing signals therefore requires a goodtheoretical model for P , with a typical accuracy of a fewpercent up to angular scales of about 10’ (see for exam-ple Refregier 2003; Bartelmann & Schneider 2001, for a re- c (cid:13) T. Guillet, R. Teyssier and S. Colombi view). Substantial work has been done to measure andpredict the dark matter power spectrum from collision-less N -body simulations (see, e.g., Efstathiou & Eastwood1981; Jenkins et al. 1998). Semi-analytic models for the darkmatter power spectrum have also been proposed, reachingthe percent level accuracy (Hamilton et al. 1991; Jain et al.1995; Peacock & Dodds 1996; Smith et al. 2003). While thedistribution of total matter is likely to closely follow darkmatter at large scales, dissipative physics is expected tomodify the power spectrum at small scales, and thereforepossibly interfere with weak lensing measurements.The interest for the effects of baryons on the con-vergence power spectrum has led to the development ofsemi-analytic halo models to estimate effect of cold (White2004) and hot (Zhan & Knox 2004) gas on the total matterpower spectrum. More recently, numerical simulations havebeen carried out to complement those semi-analytical results(Jing et al. 2006; Rudd et al. 2008). While the exact effect ofthe baryons differ quantitatively between different models,the models and simulations agree qualitatively on a boostof the total matter power spectrum due to cold baryons atsmall scales. At k ∼ h. Mpc − , this amplification has beenfound to be of around 10% at z = 0. Our theoretical under-standing of galaxy formation is however far from being com-plete. Current numerical simulations that include variouscomplex baryons physical processes suffer from the so-called overcooling problem (Blanchard et al. 1992; Cole 1991), withtoo many baryons condensing into gaseous and stellar diskswhen compared to observational constraints. Statistical ef-fects measured in Galaxy formation simulations, includingthe one used in the present work, are therefore likely over-estimated. If we can account for the effect of baryons at therequired accuracy in this extreme case, real datasets will beprobably even easier to deal with.We still need a flexible and accurate tool to accountfor the effect of baryons on the statistical properties ofthe matter density field in a parametrised model. The halomodel has been developed in the last decade to meet thesegoals. The halo model is based on the idea that the mat-ter distribution in the universe can be described as a col-lection of individual halos, in which baryonic structuressuch as galaxies form (Neyman & Scott 1952; White & Rees1978). Scherrer & Bertschinger (1991) proposed a formalismto compute correlation functions of the continuous densityfield from a model of discrete virialized halos. Since then,there has been notable contributions and refinements tothe halo model approach, such as Ma & Fry (2000); Seljak(2000) and subsequent developments (see Cooray & Sheth2002 for a review in the context of large scale structure).As the halo model has proved to be a successful frame-work for describing statistical properties of the dark mat-ter density field in the non-linear regime, there has beenalso interest in extending it to baryons in the context of theSunyaev-Zeldovich effect (Refregier & Teyssier 2002) and ofthe galaxy distribution (Seljak 2000). In previous studies,White (2004) and Zhan & Knox (2004) have used the halomodel with a baryonic component to describe the effect ofcold and hot gas respectively from a semi-analytical stand-point. More recently, Rudd et al. (2008) have shown that thehalo model can be used in a self-consistent way to describethe amplification of the power spectrum caused by baryonsas measured in cosmological simulations. They proposed to modify the concentration parameter mass dependence of thedark matter halos to account for the collapse of baryons atsmall scale, leading to more concentrated halos.In this paper, we would like to extend the previous mod-els for cosmic statistics to smaller scales, where baryons arelikely to dominate the total mass distribution. For that pur-pose, we use the results of a recent, high-resolution, cos-mological simulation, featuring state-of-the-art galaxy for-mation physics, thanks to the Horizon collaboration . Thesimulation was performed on the MareNostrum supercom-puter at the Barcelona Supercomputer Centre using the ramses code (Teyssier 2002), including a detailed treatmentof metal–dependent gas cooling, UV heating, star formation,supernovae feedback and metal enrichment.The simulation data are compared to the analytical pre-diction of a modified halo model , taking into account smallscale baryons physics in an ad-hoc way by adding to thehalo mass profile a small baryonic component, modelled asan exponential disk with mass fraction and scale length asthe only 2 additional free parameters. This approach, whichmodifies the shape of the halo profile, is in essence similarto the one of White (2004), which we use as a starting pointfor our theoretical model to compare against our numericalsimulation. In contrast to previous studies, we also computethe effect of baryons on the skewness of the mass distribu-tion. It has been shown that measuring the third momentof the cosmic shear is of paramount importance, since it canbreak the degeneracy in the cosmological parameters esti-mation based on the power spectrum alone, and reduce thecorresponding error bars by a factor of 2 (Bernardeau et al.1997; Jain & Van Waerbeke 2000; Takada et al. 2000). Us-ing only the two additional parameters of the model, wecan fit the simulation data with great accuracy, for boththe power spectrum and the skewness. This has importantconsequences for future weak lensing surveys, since the diskparameters of the model could be fitted together with thecosmological parameters, promoting baryons physics froma mere systematic effect to an additional probe of the un-derlying cosmological model. Within the modified concen-tration model of Rudd et al. (2008), statistical bias effectshave been studied by Zentner et al. (2008), and further byHearin & Zentner (2009) in the context of the test of generalrelativity by weak lensing surveys. We have performed a cosmological simulation of unprece-dented scale, using 2048 processors of the MareNostrumcomputer installed at the Barcelona Supercomputing Centrein Spain. We have used intensively the AMR code ramses (Teyssier 2002) for 4 weeks dispatched over one full year.This effort is part of a consortium between the Horizonproject in France and the MareNostrum galaxy formationproject in Spain . The originality of this project relies onusing a lot of (if not all) physical ingredients that are partof the current theory of galaxy formation, and at the same c (cid:13)000
One of the great challenges in modern cosmology is to under-stand the nature of dark energy, which is believed to domi-nate the energy budget in the universe ( ∼ Λ directly im-pacts the rate of structure formation at recent epochs, themass distribution and its time evolution bear the signatureof the dark energy content of the universe. Cosmic shearmeasurements provide an independent method of probingthe total mass distribution at large scales. Combined withphotometric redshifts, it is even possible to extract the 3D ⋆ E-mail: [email protected] matter distribution and reconstruct the matter power spec-trum at different epochs. Comparing these measurementsto theoretical predictions will set strong constraints on thecosmological parameters (e.g., Hu & Tegmark 1999; Huterer2002; Amara & Refregier 2006; Albrecht & Bernstein 2007),and among them both the equation of state w of dark energyand its possible evolution w ′ with redshift.The cosmic shear convergence power spectrum P κ de-pends on the total matter power spectrum P , which con-tains the information about w and w ′ through the growthrate of structures. Extracting the dark energy equation ofstate from weak lensing signals therefore requires a goodtheoretical model for P , with a typical accuracy of a fewpercent up to angular scales of about 10’ (see for exam-ple Refregier 2003; Bartelmann & Schneider 2001, for a re- c (cid:13) T. Guillet, R. Teyssier and S. Colombi view). Substantial work has been done to measure andpredict the dark matter power spectrum from collision-less N -body simulations (see, e.g., Efstathiou & Eastwood1981; Jenkins et al. 1998). Semi-analytic models for the darkmatter power spectrum have also been proposed, reachingthe percent level accuracy (Hamilton et al. 1991; Jain et al.1995; Peacock & Dodds 1996; Smith et al. 2003). While thedistribution of total matter is likely to closely follow darkmatter at large scales, dissipative physics is expected tomodify the power spectrum at small scales, and thereforepossibly interfere with weak lensing measurements.The interest for the effects of baryons on the con-vergence power spectrum has led to the development ofsemi-analytic halo models to estimate effect of cold (White2004) and hot (Zhan & Knox 2004) gas on the total matterpower spectrum. More recently, numerical simulations havebeen carried out to complement those semi-analytical results(Jing et al. 2006; Rudd et al. 2008). While the exact effect ofthe baryons differ quantitatively between different models,the models and simulations agree qualitatively on a boostof the total matter power spectrum due to cold baryons atsmall scales. At k ∼ h. Mpc − , this amplification has beenfound to be of around 10% at z = 0. Our theoretical under-standing of galaxy formation is however far from being com-plete. Current numerical simulations that include variouscomplex baryons physical processes suffer from the so-called overcooling problem (Blanchard et al. 1992; Cole 1991), withtoo many baryons condensing into gaseous and stellar diskswhen compared to observational constraints. Statistical ef-fects measured in Galaxy formation simulations, includingthe one used in the present work, are therefore likely over-estimated. If we can account for the effect of baryons at therequired accuracy in this extreme case, real datasets will beprobably even easier to deal with.We still need a flexible and accurate tool to accountfor the effect of baryons on the statistical properties ofthe matter density field in a parametrised model. The halomodel has been developed in the last decade to meet thesegoals. The halo model is based on the idea that the mat-ter distribution in the universe can be described as a col-lection of individual halos, in which baryonic structuressuch as galaxies form (Neyman & Scott 1952; White & Rees1978). Scherrer & Bertschinger (1991) proposed a formalismto compute correlation functions of the continuous densityfield from a model of discrete virialized halos. Since then,there has been notable contributions and refinements tothe halo model approach, such as Ma & Fry (2000); Seljak(2000) and subsequent developments (see Cooray & Sheth2002 for a review in the context of large scale structure).As the halo model has proved to be a successful frame-work for describing statistical properties of the dark mat-ter density field in the non-linear regime, there has beenalso interest in extending it to baryons in the context of theSunyaev-Zeldovich effect (Refregier & Teyssier 2002) and ofthe galaxy distribution (Seljak 2000). In previous studies,White (2004) and Zhan & Knox (2004) have used the halomodel with a baryonic component to describe the effect ofcold and hot gas respectively from a semi-analytical stand-point. More recently, Rudd et al. (2008) have shown that thehalo model can be used in a self-consistent way to describethe amplification of the power spectrum caused by baryonsas measured in cosmological simulations. They proposed to modify the concentration parameter mass dependence of thedark matter halos to account for the collapse of baryons atsmall scale, leading to more concentrated halos.In this paper, we would like to extend the previous mod-els for cosmic statistics to smaller scales, where baryons arelikely to dominate the total mass distribution. For that pur-pose, we use the results of a recent, high-resolution, cos-mological simulation, featuring state-of-the-art galaxy for-mation physics, thanks to the Horizon collaboration . Thesimulation was performed on the MareNostrum supercom-puter at the Barcelona Supercomputer Centre using the ramses code (Teyssier 2002), including a detailed treatmentof metal–dependent gas cooling, UV heating, star formation,supernovae feedback and metal enrichment.The simulation data are compared to the analytical pre-diction of a modified halo model , taking into account smallscale baryons physics in an ad-hoc way by adding to thehalo mass profile a small baryonic component, modelled asan exponential disk with mass fraction and scale length asthe only 2 additional free parameters. This approach, whichmodifies the shape of the halo profile, is in essence similarto the one of White (2004), which we use as a starting pointfor our theoretical model to compare against our numericalsimulation. In contrast to previous studies, we also computethe effect of baryons on the skewness of the mass distribu-tion. It has been shown that measuring the third momentof the cosmic shear is of paramount importance, since it canbreak the degeneracy in the cosmological parameters esti-mation based on the power spectrum alone, and reduce thecorresponding error bars by a factor of 2 (Bernardeau et al.1997; Jain & Van Waerbeke 2000; Takada et al. 2000). Us-ing only the two additional parameters of the model, wecan fit the simulation data with great accuracy, for boththe power spectrum and the skewness. This has importantconsequences for future weak lensing surveys, since the diskparameters of the model could be fitted together with thecosmological parameters, promoting baryons physics froma mere systematic effect to an additional probe of the un-derlying cosmological model. Within the modified concen-tration model of Rudd et al. (2008), statistical bias effectshave been studied by Zentner et al. (2008), and further byHearin & Zentner (2009) in the context of the test of generalrelativity by weak lensing surveys. We have performed a cosmological simulation of unprece-dented scale, using 2048 processors of the MareNostrumcomputer installed at the Barcelona Supercomputing Centrein Spain. We have used intensively the AMR code ramses (Teyssier 2002) for 4 weeks dispatched over one full year.This effort is part of a consortium between the Horizonproject in France and the MareNostrum galaxy formationproject in Spain . The originality of this project relies onusing a lot of (if not all) physical ingredients that are partof the current theory of galaxy formation, and at the same c (cid:13)000 , 000–000 aryons and statistics of the mass distribution time cover a large enough volume to provide a fair sampleof the universe, especially at redshifts above 1.The physical processes we have included in our simula-tion are described now in more detail. We have consideredmetal-dependent cooling and UV heating using the Hardtand Madau background model (see Ocvirk et al. 2008).We have incorporated a simple model of supernova feed-back and metal enrichment using the implementation de-scribed in Dubois & Teyssier (2008). For high-density re-gions, we have considered a polytropic equation of stateto model the complex, multi-phase and turbulent struc-ture of the ISM (Yepes et al. 1997; Springel & Hernquist2003) in a simplified form (see Schaye & Dalla Vecchia 2008;Dubois & Teyssier 2008): the ISM is defined as a gas witha density above n ≈ . / cc. Star formation has alsobeen included, for ISM gas only ( n H > n ), by spawn-ing star particles at a rate consistent with the Kennicuttlaw derived from local observations of star-forming galaxies.In more mathematical terms, we have ˙ ρ ⋆ = ρ gas /t ⋆ where t ⋆ = ( n H /n ) − / t and t = 8 Gyr. Recast in units of thelocal free-fall time, this corresponds to a star formation ef-ficiency of 5%. The simulation was started with a base gridof 1024 cells and the same number of dark matter parti-cles, and the grid was progressively refined, on a cell-by-cellbasis, when the local number of particles exceeded 10. Asimilar criterion was used for the gas, implementing whatis called a quasi-Lagrangian refinement strategy. Five ad-ditional levels of refinement were considered, providing aresolution between 1 and 2 kpc physical at all times. In thisway, our spatial resolution is consistent with the angular res-olution used to derive the Kennicutt law from observations.On the other hand, we are not in a position to resolve thescale height of thin cold disks so that the detailed galacticdynamics might be affected by resolution effects.The simulation was run for a ΛCDM universe withΩ m = 0 .
3, Ω Λ = 0 .
7, Ω b = 0 . H = 70 km/s/Mpc, σ = 0 . /h . Our dark matterparticle mass ( m p ≈ × M ⊙ /h ), our spatial resolution (1kpc physical ) and our box size make this simulation ideallysuited to study the formation of galaxies within dark mat-ter halos, from dwarf– to Milky-Way–sized objects at highredshift. For large galaxies, we can nicely resolve the radialextent of the disk (not its vertical extent), while for smallgalaxies, we can resolve the gravitational contraction of thecooling gas, but barely the final disk. The simulation wasstopped at redshift z ≈ . × , and the total number of AMRcells was above 5 × .To quantify the effects of baryons on statistical proper-ties of the mass distribution, the MareNostrum run, whichincludes dissipative physics, was re-run from the same ini-tial conditions with baryon mass converted to dark matter.This latter dark matter only (henceforth DMO) simulationserves as a reference for statistical quantities without thepresence of dissipative physics. Both runs were carried outup to redshift z = 2, which we will consider in the rest ofthis paper. It is already late enough to witness interestingstructures such as well-formed galaxy disks. http://astro.ft.uam.es/˜marenostrum Meaningful statistics of the density field can be extractedfrom different statistical quantities, such as the n -point cor-relation functions, the density PDF, or the one-point cumu-lants. By far, the easiest quantities to measure are the one-point moments S p ( R ), i.e. the p -th order cumulant of thesmoothed density field as a function of the smoothing scale R . The S p parameters have also been studied extensively,whether from a theoretical standpoint (Balian & Schaeffer1989; Szapudi & Szalay 1993), in the perturbation theoryframework (Bernardeau 1994), or in simulations and obser-vations (see, e.g., Colombi et al. 2000; Marinoni et al. 2008).For the 50 Mpc /h box of MareNostrum, we have computedthe statistics for scales ranging from 15 kpc /h to 2 Mpc /h .With weak lensing applications in mind, we are primar-ily interested in the total mass statistics. In the case of thedissipative simulation, this requires a consistent treatmentof both dark matter particles and gas cells.The total local density in the dissipative simulation canbe written ρ tot = ρ g + ρ DM + ρ s , (1)where ρ g , ρ DM and ρ s are the local gas, dark matter and stardensities respectively. However, because of the different na-ture of the gas (which is a continuous cell-based quantity),and stars and CDM (which are modelled as collisionless par-ticles), the three matter components should be treated sepa-rately. One could simply evaluate ρ DM and ρ s by binning theparticles into cells to obtain a local estimate of the densities,and then simply calculate ρ tot as in Eq. 1 and computing itsmoments. However, as we discuss below, the discrete natureof particles mandates a special treatment, and we chose in-stead to compute the moments and cross-correlations of thedifferent species separately, and then reconstruct the mo-ments of the total density field as we now describe.Obtaining the moments of the gas distribution involvesno theoretical difficulty. The gas density of the whole sim-ulation box is mapped onto a 2048 grid from the AMRcells using a donnercell scheme, where the resulting value ineach cartesian cell is directly copied from the finest AMRcell covering it. To determine the moments of the smoothedgas density field for a given comoving smoothing radius R ,we compute the average of the density over cubic regionsof volume πR . We restrict ourselves to values of R corre-sponding to smoothing boxes which span an integer numberof base grid cells. The average densities in such cubic re-gions are computed using a fast convolution algorithm (seee.g. Blaizot et al. 2006), and the moments over all such re-gions are then evaluated. Since the simulation directly pro-vides the continuous gas density ρ g , this prescription yieldsunbiased estimates of the moments of the gas distribution.Particles require a somewhat more careful treat-ment. The statistics of particle distributions are readilystudied using a counts-in-cells analysis (see for exampleBalian & Schaeffer 1989; Bouchet & Hernquist 1992; Sheth1996). The idea is to count particles within the same cu-bic cells of scale R used for the smoothing of the gas den-sity. The particle counting is implemented by first bin-ning the particles into the base grid using a nearest gridpoint (NGP) scheme (Hockney & Eastwood 1981), and thencounting particles in the cubic regions, again by using fast c (cid:13) , 000–000 T. Guillet, R. Teyssier and S. Colombi convolution. This is indeed equivalent to computing a localparticle density by NGP, and then performing the R -scalesmoothing. In this case, however, simply computing the mo-ments of the resulting data will introduce shot noise effects(Szapudi & Szalay 1993; Bernardeau et al. 2002).It is possible to correct for these effects using factorialmoments . Let us consider a continuous field ρ sampled by afinite collection of particles. Given a cell of size R and volume v = R , we call ˜ ρ = v R v ρ ( x )d x the average value of ρ over the cell. Equivalently, ˜ ρ is the value of ρ smoothed atscale R at the centre of the cell. The local Poisson samplinghypothesis (see e.g. Bernardeau et al. 2002) states that thedistribution of the number N of particles in the cell followsa Poisson probability mass function of mean ˜ ρv/m p , where m p is the mass of a single particle. Letting( N ) k ≡ N ( N − · · · ( N − k + 1) , (2)the factorial moments are defined as F k ≡ (cid:10) ( N ) k (cid:11) P = h N ( N − · · · ( N − k + 1) i P , (3)where N is the cell particle count, and h . . . i P denotes theaverage over the Poisson distribution of the particle sam-pling. It has been shown (Szapudi & Szalay 1993) that the F k yield unbiased estimators of the moments of the under-lying smoothed density field ˜ ρ at the scale of the cell size,in the sense that˜ ρ k = (cid:16) m p v (cid:17) k F k = (cid:16) m p v (cid:17) k (cid:10) ( N ) k (cid:11) P . (4)Let us now consider the density fields smoothed at scale R for the gas, dark matter and stars, ρ g , ρ DM and ρ s re-spectively. For the sake of readability, we shall drop thetilde notation, and the density fields are to be understood assmoothed at the scale R in the rest of this section. Using Eq.1, we can express the moments of the smoothed total den-sity field ρ tot as a function of the moments and correlationsof the individual species using the multinomial theorem: (cid:10) ρ k tot (cid:11) = X k + k + k ≤ k (cid:18) kk , k , k (cid:19) (cid:10) ρ k g ρ k DM ρ k s (cid:11) . (5)In this equation, h· · ·i denotes the ensemble average overall realisations of the underlying density fields, not to beconfused with the average h· · ·i P over particle samplings fora fixed realisation of ρ introduced in equation 3.Provided we can compute all correlations of the form (cid:10) ρ k g ρ k DM ρ k s (cid:11) , we are now in position to reconstruct the mo-ments of the smoothed total density field. Under the localPoisson sampling hypothesis, one can deduce from Eq. 3 theidentity: (cid:10) ρ k g ρ k DM ρ k s (cid:11) = (cid:16) m DM v (cid:17) k (cid:16) m s v (cid:17) k (cid:10) ρ k g ( N DM ) k ( N s ) k (cid:11) , (6)which involves the definition (2), and where m DM and m s are the dark matter and star particle masses. Since the Pois-son processes of the counts-in-cells for the different particlespecies are independent of each other, Eq. 6 involves no ap-proximation, even though the underlying fields ρ DM and ρ s are correlated.From the moments (5), we can straightforwardly com-pute the moments of the total matter overdensity (cid:10) δ k tot (cid:11) = (cid:10) ( ρ tot / ¯ ρ tot − k (cid:11) from the binomial theorem. We can finally compute the S k parameters, which aredefined as S k ( R ) ≡ (cid:10) δ k ( R ) (cid:11) c h δ ( R ) i k − c , (7)where the c subscripts denote the connected moments ofthe smoothed density field, whose generating function is thelogarithm of the generating function of the (cid:10) δ k (cid:11) . Because of the particular significance of the 3D total matterpower spectrum P ( k ) in the convergence power spectrum,we have also measured P ( k ) in the dissipative and DMOsimulations, in addition to the one-point statistics. The vari-ance of the total matter density field smoothed at scale R can be expressed as: σ ( R ) = 12 π Z d kk k P ( k ) | W ( kR ) | , (8)where W is the Fourier transform of a spherical top-hatwindow function with volume unity: W ( x ) = 3 x (sin x − x cos x ) . (9)Various sophisticated techniques for estimating the powerspectrum have been proposed, especially for correcting massassignment and sampling effects (Jing 2005; Cui et al. 2008;Colombi et al. 2009). Since the 2-point correlation function(or, equivalently, the power spectrum) is not our primaryinterest in this paper, we have settled for a simple methodwhich we expect to yield reasonable results, even if not asaccurate as our one-point moments measurements.The gas and particles densities were mapped onto a2048 base grid and added up, using donnercell for thegas and NGP binning for the dark matter particles. Thespectrum is computed using FFT folding (see Jenkins et al.1998; Colombi et al. 2009) and corrected for the NGP con-volution and shot noise bias effects (Hockney & Eastwood1981). Because of cooling, the baryons will condense to form densestructures such as disks at the centre of dark matter ha-los. Figure 1 shows a density map of one of the biggestMareNostrum halos, together with contours of the densityratio ρ bar /ρ CDM . The effect of cooling can be seen as densebaryon-dominated regions at the cores of the halos and halosubstructures.The small-scale baryonic features directly impact thedensity statistics at small scales: as the smoothing scale de-creases, the disks become more and more apparent in thedensity PDF as peaks in the high-density regions. We canexpect this process to broaden the distribution, thereby in-creasing the variance, and as only the higher-density regionsare affected, the skewness should also increase.The computed variance σ and skewness S from theMareNostrum dissipative and DMO simulations is presentedon Fig. 2. Comparing the DMO simulation (solid black) withthe total matter in the dissipative run (blue dashes), we in-deed note that the presence of baryonic physics dramatically c (cid:13) , 000–000 aryons and statistics of the mass distribution − − R [Mpc/h]10 σ ( R ) − − R [Mpc/h]10 S ( R ) Figure 2.
Variance (left) and skewness (right) of the smoothed density field of different species at z = 2, as a function of the smoothingscale in the MareNostrum dissipative and DMO simulations. The solid line shows σ and S for dark matter in the DMO simulation,while the dashed and dotted lines correspond to the dissipative simulation: short dashes for dark matter, long dashes for total matter,and dots for baryons. The error bars on the DMO data are one-sigma wide and determined by the subvolumes method as described inthe text. amplifies both σ and S at small R . At k ∼ h. Mpc − ,the power spectrum boost reaches about 35% (see Fig. 4),most of which is caused by cold baryons (stars). Becauseour study is carried out at z = 2, precise comparisons withprevious results of Jing et al. (2006); Rudd et al. (2008) aredifficult. Note however that we observe the same qualita-tive effects. The variance plot on Fig. 2 also demonstratesthe presence of CDM adiabatic contraction (Gnedin et al.2004). As the gas cools down, its contraction drags the darkmatter into local potential wells created by dense baryonclumps. This effect results in a net condensation of the darkmatter, whose effect on variance can be seen by comparingthe DMO run (solid black curve) with CDM of the dissi-pative run (short-dashed curve). Both observed boosts anddark matter contraction effects are well in accordance withthe results presented in Weinberg et al. (2008).Because of the relatively small size of the MareNostrumsimulation box, the results presented on Fig. 2 are contam-inated to some degree by cosmic variance and finite volumeeffects. We have estimated those effects in the MareNostrumDMO simulation. Note that the rigorous determination oferror bars is beyond the scope of this article, and we do notexpect baryons to modify those uncertainties significantly.The cosmic variance and finite volume effects on thestatistical quantities were sampled by three different inde-pendent methods. We have run a set of 10 smaller 256 cosmological simulations up to z = 2 with the same box sizeand power spectrum as the MareNostrum box, only with dif-fering random phases. The statistical quantities were thencomputed on each box, and the variance of those quanti-ties over the 10 boxes were used as a first estimate of theMareNostrum cosmic variance effects. While such ensemble simulations are easy to carry out, this method is known tounderestimate the actual cosmic variance, as all the reali-sations of the initial density field are constrained: the totalbox matter density is fixed to the background matter den-sity of the universe. In addition, this method cannot be usedto determine the variance at small scales because of the lowresolution of the ensemble simulations. Relative cosmic er-ror derived from this set of simulations is presented on Fig.3 (dashed curve). The FORCE code (FORtran for CosmicErrors Colombi & Szapudi 2001), implements the results ofSzapudi et al. (1999) and provides cosmic variance estima-tions given the values of the density cumulants. The corre-sponding cosmic error, based on the MareNostrum DMO cu-mulants, is shown as the solid curves on Fig. 3. This estima-tion relies on a perturbative expansion which breaks downwhen relative errors approach unity. As the MareNostrumerrors range from about 5% to 30%, the FORCE computa-tion still holds, but the quality of the estimation is impacted,especially at small scales where the errors on high-order cu-mulants increase. To confirm the FORCE results at smallscales, we have studied the variance of the statistical quan-tities over a random sample of cubic subvolumes of size ℓ . Let ǫ X ( ℓ, R ) = p var(X(R)) / ¯ X ( R ) be the relative cosmic errorof a statistical quantity X at scale R defined on subvolumesof size ℓ . To obtain the cosmic variance of the whole simula-tion box (i.e., ǫ X ( L, R ) for all R ), we computed ǫ X ( ℓ, R ) for ℓ = L/ , L/ , L/
32 and extrapolated in ℓ to ℓ = L assum-ing the power-law form ǫ X ( ℓ, R ) = ǫ X ( L, R )( ℓ/L ) η . This lastestimation of the error is represented on Figure 3 in dottedlines. None of these methods ensures accurate determinationof the errors over the whole range of scales, however, theypaint a clear picture of cosmic variance in the MareNostrum c (cid:13) , 000–000 T. Guillet, R. Teyssier and S. Colombi
Figure 1.
Map of the projected dark matter density of oneof the largest halos in the MareNostrum simulation ( R vir =0 .
59 Mpc /h comoving shown as the dashed circle) at z = 2.The contours represent isovalues of the baryon to dark matterdensity ratio ρ bar /ρ CDM . The outer black contours correspond to ρ bar = 0 . ρ CDM , while the inner red contours delimit equal den-sities regions. The total matter density is baryon-dominated atsmall scales well within the halo core. The bright central galaxyclearly stands out of the halo substructures, whose distributionwithin the halo remains mainly unaffected by the presence ofbaryons (see Weinberg et al. 2008). simulations. As can be seen on Fig. 2, the observed boosts in σ and S are well above cosmic variance effects. Note thatscales comparable to the MareNostrum box size correspondto a patch of z = 0 . σ /σ are mostly devoid of finite volume contam-ination. For the rest of this paper, we will therefore onlyconsider such amplification ratios (or “boosts”) for the vari-ance and skewness of the total matter in the dissipative runwith respect to the dark matter of the DMO run. The halo model provides a well-tested and flexible frame-work for the study of the properties of matter distribu-tion in non-linear stages of gravitational collapse. Whilefirst studied in the context of the galaxy distribution(Neyman & Scott 1952), it has become a full-fledged and now mature tool for cosmological statistics through signifi-cant contributions and improvements to its various ingredi-ents.Attempting to reproduce the effect of baryons onthe boost factors of variance and skewness requires us tomodel both the DMO and dissipative matter distributions.Rudd et al. (2008) have shown that modifying the halo con-centration relation can account for the variance amplifica-tion at scales k . h. Mpc − . In this paper, we will use astandard halo model to describe the dark matter of the DMOrun. We base our halo profile for the total mass on the DMOhalo model, but instead of modifying only c ( M ), we proposeto also modify the halo profile itself. As discussed previously,the quantity of interest is the boost of the statistics (i.e. theamplification of the variance and skewness witnessed on thetotal matter halo model with respect to the reference halomodel). We now briefly describe the different key ingredientswhich take part in the computation of σ ( R ) and S ( R ) inthe halo model.Statistical description of the density field as a set ofvirialized halos requires the specification of the mass distri-bution of the halos (the mass function), their density profilesand associated mass parametrisation, and a model for halo-halo correlations.A simple model for the halo mass function was givenby Press & Schechter (1974) based on the spherical col-lapse model. Since then, there has been more convincingderivations of the Press-Schechter result, as well as attemptsto take into account non-symmetric collapses and tidal ef-fects (Bond et al. 1991; Audit et al. 1997; Sheth et al. 2001;Sheth & Tormen 2002). These studies resulted in otherparametrizations, such as the Sheth-Tormen mass function(Sheth & Tormen 1999).In this study, we use the Sheth-Tormen prescription forthe halo mass function, as it turns out to be a better fit toour simulations than the Press-Schechter form. In normal-ized units, the Sheth-Tormen mass function reads: m ¯ ρ n ( m ) d m = A ( p ) r qπ (cid:16) (cid:0) qν (cid:1) − p (cid:17) × ν exp (cid:18) − qν (cid:19) d νν , (10)where ν ≡ δ c /σ . δ c ≈ .
68 is the collapse density threshold inthe spherical collapse model, and p ≈ . , A ( p ) ≈ . , q ≈ . . M ⊙ . While the cutoff has little effecton the variance as computed by the halo model, the skew-ness drops significantly at large scales, resulting in a bet-ter fit against the measured S . This is not surprising sincehigh-order moments at large scale are sensitive to rare events(such as massive halos, e.g. Colombi et al. 1994), which arenot well sampled by the MareNostrum box.For the DMO model halo profile, we use the standardNFW form (Navarro et al. 1997): u NFW ( r | M ) ∝ x − (1 + x ) − , x ≡ c ( M ) rR vir (11)and u ( r | M ) is normalized such that R u ( r | M ) d r = 1. Our c (cid:13) , 000–000 aryons and statistics of the mass distribution − − R [Mpc /h ]10 − − σ ( R )r e l a t i v ec o s m i ce rr o r − − R [Mpc /h ]10 − − S ( R )r e l a t i v ec o s m i ce rr o r Figure 3.
Estimates for the relative cosmic errors ∆ σ /σ and ∆ S /S for each method described in the text. The dashed curvescorrespond to the 10 ensemble simulations, the solid curves to results of the FORCE code, and the dotted curves to the subvolumesestimation. halo virial radius R vir is defined such that, for a halo of mass M , we have M = π ¯ ρR ∆, with ∆ = 200. Note that themass M of a halo is related to its comoving Lagrangian size R by M = π ¯ ρR . The NFW model has proved to fit nu-merical dark matter profiles over a large range of masses andradii with some dispersion in the concentration parameter c ( M ) (Kravtsov et al. 1998; Jing 2000). The central loga-rithmic slope of dark matter profiles, which is − c ( M ) is parametrized inour model according to the result of Bullock et al. (1999): c ( M, z ) = c z (cid:16) MM ∗ (cid:17) b , c ≈ , b ≈ − . . (12)This power-law model has been found to be a good fitto numerical simulations also for dark energy cosmologies(Dolag et al. 2004).Following Scherrer & Bertschinger (1991), we can ex-press the density 2-point correlation function ξ ( r ) as: ξ ( r ) = ξ h ( r ) + ξ hh ( r ) , (13)where ξ h represents the contribution to the correlationfunction from mass within the same halo, and ξ hh containsthe contribution from mass located in different halos. ξ h is essentially the autocorrelation of the halo profile,and its contribution to the power spectrum is: P h ( k ) = Z n ( m ) (cid:18) m ¯ ρ (cid:19) | u ( k | m ) | d m, (14)where u ( k | m ) is the Fourier transform of the halo profile for a halo of mass m . We compute the halo-halo contributionfollowing Mo & White (1996) and its subsequent extensions(Mo et al. 1997; Sheth & Lemson 1999; Sheth & Tormen1999). Assuming deterministic biasing on large scales, wecan write the ξ hh contribution from two halos of masses M and M as: ξ hh ( r | M , M ) = b ( M ) b ( M ) ξ ( r ) ≈ b ( M ) b ( M ) ξ lin ( r ) , (15)where ξ lin is the matter correlation function from linear the-ory. This prescription is accurate at large scales, and consis-tent with the choice of mass function provided the bias b ( M )is computed from f ( ν ) as prescribed in Mo et al. (1997).Now in possession of a halo model for ξ ( r ) (and there-fore its Fourier transform P ( k )), we can evaluate σ ( R ) us-ing Eq. 8. While in principle the halo model ingredients presented sofar fully determine the statistics of the density field, addi-tional work is needed to extract S ( R ).At large enough scales, the one-point statistics S k may be computed using perturbation theory (Fry 1984;Juszkiewicz et al. 1993; Bernardeau 1994; Bernardeau et al.2002), which yields S PT3 = 347 + γ, (16)where γ = d ln σ ( R ) / d ln R . However, in the MareNostrumsimulation at z = 2, PT is only expected to be valid at scalesgreater than ∼ /h . A first interesting refinement tak-ing discrete halos into account is the Poisson cluster model,where halo-halo correlations are neglected and profiles are c (cid:13) , 000–000 T. Guillet, R. Teyssier and S. Colombi assumed to be point-like (Sheth 1996). Halo profiles, how-ever, are responsible for most of the behaviour of small-scales statistics, and thus neither perturbation theory andthe point-cluster model are appropriate for our study.Fortunately, the full computation of the higher-order cumulants S k in the halo model was developed inScoccimarro et al. (2001). Following the authors, we define: u m ( R, ν ) ≡ Z k d k π [ u ( k | ν )] m | W ( kR ) | (17) A i,j ( R ) ≡ Z d νf ( ν ) b i ( ν ) u ( R, ν ) [ u ( R, ν )] j (cid:18) M ¯ ρ (cid:19) j +1 , (18)where R is such that δ c /σ ( R ) = ν . In these notations, thethird cumulant of the density field in the halo model writes (cid:10) δ (cid:11) c = S PT3 σ + 3 σ A , + A , . (19) We have tested some families of halo profiles to attempt toreproduce the observed effect of baryons on the statisticsof the density field. The reference halo model for the DMOsimulation is based on a NFW profile with the commonlyused c ( M ) relationship of Bullock et al. (1999) as written inEq. 12.As suggested by previous numerical studies (Rudd et al.2008), an increase in c and a steeper concentration slope b are expected to reproduce – at least partially – the in-crease in power at small scales due to baryonic physics andradiative processes in particular. We have accordingly triedto adjust the concentration parameters with a NFW profileto obtain a good match for both the variance and skewnessat small scales. The power spectrum, variance and skewnessboosts for a NFW-based model with parameters comparableto Rudd et al. (2008) ( c = 20 , b = − .
15) are presented asthe dotted curves on Fig. 4 and 5. This model reproducesthe MareNostrum variance and power spectrum boosts downto a scale of about 0 . /h . At smaller scales however,the halo model underestimates the variance amplification. Alarge part of this discrepancy is likely due to the differencein the simulation codes and physical modelling between thetwo studies. Note however that the skewness S of this halomodel lacks much of the measured small-scale amplification,as can be seen on Fig. 5. The distinctive bend is also not re-produced at all, which suggests the profile form distributesmatter too evenly across scales.With the partial success of this profile, one might ex-pect NFW profiles with higher concentrations to yield betterfits. It turns out however that reasonably fitting the vari-ance boost at small scale requires very high values of c ,exceeding 30. Such high values of the concentration param-eter are too high to be accepted as physically meaningful.Yet more importantly, while increasing c will indeed boostthe variance, it fails to reproduce at all the correspondingsmall-scale skewness amplification. This can be seen on Fig.5, and the S boost of a pure NFW halo model remainsessentially flat for varying values of c , with a very weakdependence.This leads us to believe that, while the NFW pro-file with adjusted concentration parameters has merits in − k [ h Mpc − ]0246810 P t o t( k ) / P D M O ( k ) − − . . . . . . Figure 4.
Relative power spectrum amplification due to baryonsat z = 2. The solid curve is the measured power spectrum, thedotted curve is a NFW profile with c = 20, b = − .
15, and thedashed curve is the halo model with our composite halo profile. ℓ − . − . . . . A h a l o ( ℓ ) / A d a t a ( ℓ ) − Figure 6.
Error on the amplification of the 3D total mass powerspectrum at z = 2 for the halo models represented as a functionof the angular mode ℓ in the flat sky approximation. The dashedcurve is the reference DMO model (i.e. without any boost), thedotted curve is the pure NFW model with modified c ( M ), and thesolid curve is the composite halo model amplification. The lightand dark shaded areas are estimates of the expected experimentalerrors on C ℓ for the CFHT Wide-field and LSST experiments,respectively. c (cid:13) , 000–000 aryons and statistics of the mass distribution − − R [Mpc /h ]10 − σ t o t( R ) / σ D M O ( R ) − − R [Mpc /h ]10 − S , t o t( R ) / S , D M O ( R ) Figure 5.
Effect of baryons on the variance and the skewness S boost factors, as measured on the MareNostrum simulation (solidcurve) and modelled by a NFW profile with c = 20, b = − .
15 (dotted curve), and the composite profile (dashed curve). modelling the variance amplification caused by dissipativephysics, it can only paint a limited picture of the statisticalproperties of the density field in the presence of baryons.As increasing c essentially amounts to concentrating morematter within the central region of the halos, we naturallyturn to other centrally-concentrated halo profiles.One way to concentrate more matter within the centreregion is by using families of profiles with steeper centralcusps than NFW of the form: u α ( x ) = x − α (1 + x ) α − , (20)where α = 1 yields an NFW profile. We have tested thisfamily of profiles on a wide range of values 1 ≤ α ≤ .
5. Foreach value of α , we attempted to find an best-fitting valueof ( c , b ), again by exploring the parameter space. It is inter-esting to note that high values of α , in the range [2 . , . σ small-scale steepen-ing and a strong S amplification. Isothermal ( α = 2) pro-files are known to be a good description of the total densityin haloes hosting elliptical galaxies (see e.g. Gavazzi et al.2007; van de Ven et al. 2009). In the case of the MareNos-trum simulation however, this property seems coincidental,as the simulated physics form no truly elliptical galaxy com-parable to observations. Moreover, the residuals of the best σ and S fits for such profiles cast doubt on the legitimacyof the analytical form u α for the statistical analysis of thesimulation. A good candidate profile which is both centrally-concentrated and physically-motivated is a composite haloprofile (see White 2004; Zhan & Knox 2004), parametrized by the dimensionless parameters f d and λ d : u f d ,λ d ( r | M ) = (1 − f d ) u NFW ( r | M )+ f d u exp ,λ d ( r | M ) , (21)where u exp ,λ d is a spherically averaged exponential disk pro-file with length scale r d proportional to the halo’s virial ra-dius: u exp ,λ d ( r | M ) ∝ exp( − r/r d ) r/r d , r d ≡ λ d R vir . (22)The dimensionless parameter λ d is essentially the spin pa-rameter of the halo, and defines the disk scale r d . The profile u f d ,λ d features a central r − cusp and behaves like the NFWprofile for radii bigger than the disk length scale r d . How-ever, because of the profile normalization, it concentratesmore mass within the central exponential than a pure NFW. u f d ,λ d can be seen as a halo profile concentrating a fraction f d of the mass within a central exponential disk profile, andthe remaining 1 − f d in a standard NFW component.This form of composite profile is physically motivated.The total mass distribution in group-sized halos is known tobe well described by a halo component and a concentratedcomponent corresponding to the bright central galaxy (see,e.g., Dubinski 1998). The presence of baryons does not fun-damentally change the diffuse halo component: the distri-bution of satellite galaxies within halos is very similar tothe halo occupation distribution of dark matter substruc-tures in pure N -body simulations (see Weinberg et al. 2008).This suggests keeping a NFW profile to account for the darkmatter, diffuse gas and halo substructures, while introduc-ing a spiked central component mimicking the bright centralgalaxy’s disk. We may expect this NFW profile to be moreconcentrated than in the dark matter only case, becauseof the adiabatic contraction of the CDM due to the pres-ence of baryons. This will therefore lead to an increase of c (cid:13) , 000–000 T. Guillet, R. Teyssier and S. Colombi c in the c ( M ) relationship of equation (12). For the com-posite profile, f d is to be understood as the fraction of thetotal halo mass which resides in the galactic disk in theform of baryons. As most formed galaxies found at z = 2 inMareNostrum simulation are spirals, we restrict ourselves inthis paper to an exponential disk profile for the central com-ponent. We believe this form captures the essential featuresof the dense central baryonic regions which are importantfor the halo model. It also places interesting constraints onthe profile parameters f d and λ d , as mass fractions and an-gular momenta of disks are well-studied, both theoreticallyand observationally. We further assume both λ d and f d to beindependent of halo mass. The assumption that the disk sizeis a fixed fraction of R vir corresponds to the singular isother-mal sphere model of disk formation (see Mo et al. 1998). Wepostpone refinements of this model to future work.Here again, we explored the ( f d , λ d ) parameter space tofind a good fit to the MareNostrum data. Our best modelhas parameters: c = 13 . b = − . f d = 0 . λ d = 0 .
025 (23)The corresponding power spectrum, variance and skewnessboosts are represented on Fig. 4 and 5 as dashed curves.This halo profile reproduces accurately both the measured σ and S amplifications down to the smallest scales. With a base grid resolution and particle count of 1024 and abox size of 50 Mpc /h , the MareNostrum simulation resolvesthe length and mass scales of galactic disks while also pro-viding a volume large enough for cosmological studies. Thismakes it particularly suitable for the study of the effect ofbaryonic physics on cosmic statistics. Such an intermediatebox size, however, will be affected at both small and largescales by resolution and finite volume effects.At very small scales, counts-in-cells measurements areexpected to suffer from shot noise, as the density field is sam-pled by a finite number of particles. This translates into bothstatistical variance and bias at small scales, if using naivestatistical estimators for the moments of the density field.Assuming particles trace the density field as a local Poissonprocess, it can be shown, however, that factorial momentsdefined in Eq. 3 are unbiased estimators (Szapudi & Szalay1993; Bernardeau et al. 2002). We thus do not expect ourmeasurements to be affected by Poisson noise at small scales.On large scales, the results will be contaminated bycosmic variance and finite volume effects. In cosmologicalsimulations, statistical quantities are usually computed bytaking the spatial average – instead of ensemble average –of local quantities over the single simulated volume. Thisprescription is only appropriate for scales corresponding towavenumbers k for which the simulation provides sufficientindependent samples. For a box of a given size L , the sam-pling of large scales k with πk approaching L suffers from thedecreasing number of independent modes. The low numberof modes at low wavenumbers introduces variance on large scale quantities, which is purely statistical in nature. As pre-sented in earlier, we have measured the cosmic variance ofthe whole MareNostrum box, and while conservative esti-mates for the errors range from 5%–30% depending on thestatistic and estimation method, it is our understanding thatcosmic variance does not fundamentally affect our result. Aspreviously mentioned, we have minimized the effect of cos-mic errors on our conclusions by only considering ratios ofstatistical quantities from simulations run with the same setof random phases.We have shown that, although different halo profiles candescribe variance amplification due to dissipative physics atsmall scale by merely modifying the concentration param-eter c ( M ), the third moment S introduces additional con-straints on the inner profiles which cannot be reproducedby changing c ( M ) alone. The distinctive slope of S ( R ) atsmall scales seems characteristic of a higher mass concen-tration towards the core than NFW. Unsurprisingly, profileswith a core or relatively weak central density peaks do notdescribe well the effective total matter distribution in thepresence of baryons.Instead, we have found that using a superposition ofa NFW profile and an exponential profile yields realisticvariance amplification and S gain for reasonable values ofthe concentration parameters c and b , disk mass fraction f d and disk scale λ d . One should note that the values ofthe best-fitting λ d and f d parameters are in good agreementwith the expected physical properties of the galaxies of thesimulated MareNostrum universe. The f d = 0 .
09 value isquite compatible with observed and predicted baryon diskmass fractions (see, e.g., Somerville et al. 2008).In this model, we chose not to introduce any mass orredshift dependence in f d and λ d . For the latter, this as-sumption is supported in part by the weak dependence onmass of r d /R vir (Somerville et al. 2008). On the other hand,for f d , a proper model should account for the variationof M/L as a function of halo mass in real galaxies (see,e.g., Yang et al. 2003). We postpone this more elaborate ap-proach to a future paper.The modified c and b parameters of Eq. 23 for theNFW profile correspond to a more concentrated CDM com-ponent than in the pure DMO model. This is in accordance,both qualitatively and quantitatively, with the results ofRudd et al. (2008). It is interesting to note that the vari-ance boost caused by the NFW component is of the sameorder of magnitude as the adiabatic contraction effect visibleon Fig. 2, albeit slightly weaker. This supports the idea thatthe composite halo profile concentrates a significant fractionof the halo’s baryonic mass within the central disk, while theremaining halo gas essentially follows the NFW componentwhich accounts for the cold dark matter. This last CDMcomponent “feels” the presence of the hot gas through theprocess of adiabatic contraction.This suggests that both variance and skewness of thedensity field can be estimated at small scale within theframework of the halo model by using a composite halo pro-file. While the halo model is a valuable tool for the studyof theoretical power spectra, the accuracy requirements ofprecision cosmology are arguably too stringent to considerdirectly fitting cosmological parameters to observed cosmicstatistics. The halo model has merit however, as it allows c (cid:13) , 000–000 aryons and statistics of the mass distribution one to study the dependence of cosmic shear with baryonicfeatures such as galaxy disk masses and sizes. ACKNOWLEDGEMENTS
The authors thankfully acknowledge the computer re-sources, technical expertise and assistance provided by theBarcelona Supercomputing Centre – Centro Nacional deSupercomputaci´on ( ). The dark mat-ter only re-run was carried out at the CEA CCRT com-puting centre ( ). We are grate-ful to J. Fry and D. Weinberg for useful discussions on thehalo model. We would also like to thank A. Refregier andA. Amara for their helpful insights and comments on theweak lensing aspects of this study.
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