We investigate the properties of hybrid gravitational/hydrodynamical simulations, examining both the numerics and the general physical properties of gravitationally driven, hierarchical collapse in a mixed baryonic/dark matter fluid. We demonstrate that, under certain restrictions, such simulations converge with increasing resolution to a consistent solution. The dark matter achieves convergence provided that the relevant scales dominating nonlinear collapse are resolved. If the gas has a minimum temperature (as expected, for example, when intergalactic gas is heated by photoionization due to the ultraviolet background) and the corresponding Jeans mass is resolved, then the baryons also converge. However, if there is no minimum baryonic collapse mass or if this scale is not resolved, then the baryon results err in a systematic fashion. In such a case, as resolution is increased the baryon distribution tends toward a higher density, more tightly bound state. We attribute this to the fact that under hierarchical structure formation on all scales there is always an earlier generation of smaller scale collapses, causing shocks which irreversibly alter the state of the baryon gas. In a simulation with finite resolution we therefore always miss such earlier generation collapses, unless a physical scale is introduced below which such structure formation is suppressed in the baryons. We also find that the baryon/dark matter ratio follows a characteristic pattern, such that collapsed structures possess a baryon enriched core (enriched by factors ∼2similar-toabsent2\sim 2∼ 2 or more over the universal average) which is embedded within a dark matter halo, even without accounting for radiative cooling of the gas. The dark matter is unaffected by changing the baryon distribution (at least in the dark matter dominated case investigated here), allowing hydrodynamics to alter the distribution of visible material in the universe from that of the underlying mass.

Cosmology: theory - Hydrodynamics - Large scale structure of the universe - Methods: numerical

Hydrodynamics is thought to play a key role in the formation of the visible structures in the universe, such as bright galaxies and hot intracluster gas. For this reason there is a great deal of interest in incorporating hydrodynamical effects into cosmological structure formation simulations in order to make direct, quantitative comparisons of such simulations to observed data. In addition to gravitation, a cosmological hydrodynamical simulation must minimally account for pressure support, shock physics, and radiative cooling, as these are the fundamental physical processes thought to play a dominant role in the formation of large, bright galaxies (White & Rees 1978). There is already a bewildering array of such studies published, including Cen & Ostriker (1992a,b), Katz, Hernquist, & Weinberg (1992), Evrard, Summers, & Davis (1994), Navarro & White (1994), and Steinmetz & Müller (1994), to page_seo_title merely a few. In order to appreciate the implications of such ambitious studies, it is important that we fully understand both the physical effects of hydrodynamics under a cosmological framework and the numerical aspects of the tools used for such investigations. Basic questions such as how the baryon to dark matter ratio varies in differing structures (galaxies, clusters, and filaments) and exactly how this is affected by physical processes such as shock heating, pressure support, or radiative cooling remain unclear. It is also difficult to separate real physical effects from numerical artifacts, particularly given the current limitations on the resolution which can be achieved. For example, in a recent study of X-ray clusters Anninos & Norman (1996) find the observable characteristics of a simulated cluster to be quite resolution dependent, with the integrated X-ray luminosity varying as Lx∝(Δx)-1.17proportional-tosubscriptݐ¿ݑ¥superscriptΔݑ¥1.17L_x\propto(\Delta x)^-1.17italic_L start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT ∝ ( roman_Δ italic_x ) start_POSTSUPERSCRIPT - 1.17 end_POSTSUPERSCRIPT, core radius rc∝(Δx)0.6proportional-tosubscriptݑŸݑsuperscriptΔݑ¥0.6r_c\propto(\Delta x)^0.6italic_r start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ∝ ( roman_Δ italic_x ) start_POSTSUPERSCRIPT 0.6 end_POSTSUPERSCRIPT, and emission weighted temperature TX∝(Δx)0.35proportional-tosubscriptݑ‡ݑ‹superscriptΔݑ¥0.35T_X\propto(\Delta x)^0.35italic_T start_POSTSUBSCRIPT italic_X end_POSTSUBSCRIPT ∝ ( roman_Δ italic_x ) start_POSTSUPERSCRIPT 0.35 end_POSTSUPERSCRIPT (where ΔxΔݑ¥\Delta xroman_Δ italic_x is the gridcell size of the simulation). In a study of the effects of photoionization on galaxy formation, Weinberg, Hernquist, & Katz (1996) find that the complex interaction of numerical effects (such as resolution) with microphysical effects (such as radiative cooling and photoionization heating) strongly influences their resulting model galaxy population. In this paper we focus on separating physical from numerical effects in a series of idealized cosmological hydrodynamical simulations. This study is intended to be an exploratory survey of hydrodynamical cosmology, similar in spirit to the purely gravitational studies of Melott & Shandarin (1990), Beacom et al. (1991), and Little, Weinberg, & Park (1991).

We will examine the effects of pressure support and shock heating in a mixed baryonic/dark matter fluid undergoing gravitationally driven hierarchical collapse. This problem is approached with two broad questions in mind: how stable and reliable is the numerical representation of the system, and what can we learn about the physics of such collapses? These questions have been investigated for purely gravitational systems in studies such as those mentioned above. In those studies numerically it is found that the distribution of collisionless matter converges to consistent states so long as the nonlinear collapse scale is resolved. Such convergence has not been demonstrated for collisional systems, however. It is not clear that hydrodynamical simulations will demonstrate such convergence in general, nor if they do that the nonlinear scale is the crucial scale which must be resolved. Hydrodynamical processes are dominated by localized interactions on small scales, allowing the smallest scales to substantially affect the state of the baryonic gas. As an example, consider a collisional fluid undergoing collapse. Presumably such a system will undergo shocking near the point of maximal collapse, allowing a large fraction of the kinetic energy of the gas to be converted to thermal energy. In a simple case such as a single plane-wave perturbation (the Zel’dovich pancake collapse), the obvious scale which must be resolved is the scale of the shock which forms around the caustic. However, in a hierarchical structure formation scenario there is a hierarchy of collapse scales, and for any given resolution limit there is always a smaller scale which will undergo nonlinear collapse. The subsequent evolution of the gas could well depend upon how well such small scale interactions are resolved, and changes in the density and temperature of gas on small scales could in turn influence how it behaves on larger scales (especially if cooling is important).

In this paper we examine a series of idealized experiments, evolving a mixed fluid of baryons and collisionless dark matter (dark matter dominated by mass), coupled gravitationally in a flat, Einstein-de Sitter cosmology. The mass is seeded with Gaussian distributed initial density perturbations with a power-law initial power spectrum. We perform a number of simulations, varying the resolution, the initial cutoff in the density perturbation spectrum, and the minimum allowed temperature for the baryons. Enforcing a minimum temperature for the baryons implies there will be a minimal level of pressure support, and therefore a minimum collapse scale (the Jeans mass), below which the baryons are pressure supported against collapse. From the numerical point of view, performing a number of simulations with identical initial physical conditions but varying resolution allows us to unambiguously identify resolution effects. By enforcing a Jeans mass for the baryons we introduce an intrinsic mass scale to the problem, which may or may not be resolved in any individual experiment. The hope is that even if the gas dynamical results do not converge with increasing resolution in the most general case, the system will converge if the fundamental Jeans mass is resolved.

The effects of the presence (or absence) of a baryonic Jeans mass also raises interesting physical questions. Though we simply impose arbitrary minima for the baryon temperatures here, processes such as photoionization enforce minimum temperatures in the real universe by injecting thermal energy into intergalactic gas. The Gunn-Peterson test indicates that the intergalactic medium is highly ionized (and therefore at temperatures T≳104greater-than-or-equivalent-toݑ‡superscript104T\gtrsim 10^4italic_T ≳ 10 start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPTK) out to at least z≲5less-than-or-similar-toݑ§5z\lesssim 5italic_z ≲ 5. Shapiro, Giroux, & Babul (1994) discuss these issues for the intergalactic medium. The dark matter, however, is not directly influenced by this minimal pressure support in the baryons, and therefore is capable of collapsing on arbitrarily small scales. Pressure support provides a mechanism to separate the two species, and since the dark matter dominates the mass density it can create substantial gravitational perturbations on scales below the Jeans mass. While there are many studies of specific cosmological models with detailed microphysical assumptions, the general problem of the evolution of pressure supported baryons in the presence of nonlinear dark matter starting from Gaussian initial conditions has not been investigated in a systematic fashion.

This paper is organized as follows. In §2 we discuss the particulars of how the simulations are constructed and performed. In §3 we characterize the numerical effects we find in these simulations, and in §4 we discuss our findings about the physics of this problem. Finally, §5 summarizes the major results of this investigation.

2 The Simulations

A survey such as this optimally requires a variety of simulations in order to adequately explore the range of possible resolutions and input physics. Unfortunately, hydrodynamical cosmological simulations are generally quite computationally expensive, and therefore in order to run a sufficiently broad number of experiments we restrict this study to 2-D simulations. There are two primary advantages to working in 2-D rather than 3-D. First, parameter space can be more thoroughly explored, since the computational cost per simulation is greatly reduced and a larger number of simulations can be performed. Second, working in 2-D enables us to perform much higher resolution simulations than are feasible in 3-D. While the real universe is 3-D and we must therefore be cautious about making specific quantitative predictions based on this work, 2-D experiments can be used to yield valuable qualitative insights into the behaviour of these systems. For similar reasons Melott & Shandarin (1990) and Beacom et al. (1991) also utilize 2-D simulations in their studies of purely gravitational dynamics.

The 2-D simulations presented here can be interpreted as a slice through an infinite 3-D simulation (periodic in (x,y)ݑ¥ݑ¦(x,y)( italic_x , italic_y ) and infinite in zݑ§zitalic_z). The particles interact as parallel rods of infinite length, obeying a gravitational force law of the form Fgrav∝1/rproportional-tosubscriptݐ¹grav1ݑŸ\mbox$F_\mbox\scriptsize grav$\propto 1/ritalic_F start_POSTSUBSCRIPT grav end_POSTSUBSCRIPT ∝ 1 / italic_r. The numerical technique used for all simulations is SPH (Smoothed Particle Hydrodynamics) for the hydrodynamics and PM (Particle-Mesh) for the gravitation. The code and technique are described and tested in Owen et al. (1996), so we will not go into much detail here. We do note, however, that while our code implements ASPH (Adaptive Smoothed Particle Hydrodynamics) as described in our initial methods paper, we are not using the tensor smoothing kernel of ASPH for this investigation, but rather simple SPH. The results should be insensitive to such subtle technique choices since the goal is to compare simulation to simulation, so we employ simple SPH in order to separate our findings from questions of technique.

All simulations are performed under a flat, Einstein-de Sitter cosmology, with 10% baryons by mass (Ωbary=0.1,Ωdm=0.9,Λ=0formulae-sequencesubscriptΩbary0.1formulae-sequencesubscriptΩdm0.9Λ0\mbox$\Omega_\mbox\scriptsize bary$=0.1,\mbox$\Omega_\mbox\scriptsize dm% $=0.9,\Lambda=0roman_Ω start_POSTSUBSCRIPT bary end_POSTSUBSCRIPT = 0.1 , roman_Ω start_POSTSUBSCRIPT dm end_POSTSUBSCRIPT = 0.9 , roman_Λ = 0). Thus the mass density is dominated by the collisionless dark matter, which is linked gravitationally to the collisional baryons. The baryon and dark matter particles are initialized on the same perturbed grid, with equal numbers of both species. Therefore, initially all baryons exactly overlie the dark matter particles, and only hydrodynamical effects can separate the two species. The baryon/dark matter mass ratio is set by varying the particle mass associated with each species. The initial density perturbation spectrum is taken to be a power-law P(k)=⟨|δρ(k)/ρ¯|2⟩∝knݑƒݑ˜delimited-⟨⟩superscriptݛ¿ݜŒݑ˜¯ݜŒ2proportional-tosuperscriptݑ˜ݑ›P(k)=\langle|\delta\rho(k)/\bar\rho|^2\rangle\propto k^nitalic_P ( italic_k ) = ⟨ | italic_δ italic_ρ ( italic_k ) / over¯ start_ARG italic_ρ end_ARG | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⟩ ∝ italic_k start_POSTSUPERSCRIPT italic_n end_POSTSUPERSCRIPT up to a cutoff frequency kcsubscriptݑ˜ݑk_citalic_k start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT. Note that since these are 2-D simulations, for integrals over the power spectrum this is equivalent in the 3-D to a power spectrum of index n-1ݑ›1n-1italic_n - 1. In this paper we adopt a “flat” (n=0ݑ›0n=0italic_n = 0) 2-D spectrum

P2-D(k)=Anorm{k0 : k≤kc ⇒ P3-D(k)=k-10 : k>kc,subscriptݑƒ2-Dݑ˜subscriptݐ´normcasessuperscriptݑ˜0:formulae-sequenceݑ˜subscriptݑ˜ݑ⇒subscriptݑƒ3-Dݑ˜superscriptݑ˜10:ݑ˜subscriptݑ˜ݑ\mbox{$P_{\mbox{\scriptsize 2-D}}$}(k)=\mbox{$A_{\mbox{\scriptsize norm}}$}% \left\{\begin{array}[]{l@{\quad: \quad}l}k^{0}&k\leq k_{c}\quad\Rightarrow% \quad\mbox{$P_{\mbox{\scriptsize 3-D}}$}(k)=k^{-1}\\ 0&k>k_{c},\end{array}\right.italic_P start_POSTSUBSCRIPT 2-D end_POSTSUBSCRIPT ( italic_k ) = italic_A start_POSTSUBSCRIPT norm end_POSTSUBSCRIPT { start_ARRAY start_ROW start_CELL italic_k start_POSTSUPERSCRIPT 0 end_POSTSUPERSCRIPT : end_CELL start_CELL italic_k ≤ italic_k start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ⇒ italic_P start_POSTSUBSCRIPT 3-D end_POSTSUBSCRIPT ( italic_k ) = italic_k start_POSTSUPERSCRIPT - 1 end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL 0 : end_CELL start_CELL italic_k >italic_k start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT , end_CELL end_ROW end_ARRAY (1)
where Anormsubscriptݐ´normA_{\mbox{\scriptsize norm}}italic_A start_POSTSUBSCRIPT norm end_POSTSUBSCRIPT normalizes the power-spectrum. Note that using a flat cosmology and power-law initial conditions implies these simulations are scale-free, and should evolve self-similarly in time. We can choose to assign specific scales to the simulations in order to convert the scale-free quantities to physical units. All simulations are halted after 60 expansion factors, at which point the nonlinear scale (the scale on which δρ/ρ∼1similar-toݛ¿ݜŒݜŒ1\delta\rho/\rho\sim 1italic_δ italic_ρ / italic_ρ ∼ 1) is roughly 1/8 of the box size.

The Jeans length is the scale at which pressure support makes the gas stable against the growth of linear fluctuations due to self-gravitation - the Jeans mass is the amount of mass contained within a sphere of diameter the Jeans length. The Jeans length λJsubscriptݜ†ݐ½\lambda_{J}italic_λ start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT and mass MJsubscriptݑ€ݐ½M_{J}italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT are defined by the well known formula (Binney & Tremaine 1987)

λJ=(πcs2Gρ)1/2,subscriptݜ†ݐ½superscriptݜ‹superscriptsubscriptݑݑ 2ݐºݜŒ12\lambda_{J}=\left(\frac{\pi c_{s}^{2}}{G\rho}\right)^{1/2},italic_λ start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT = ( divide start_ARG italic_π italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_G italic_ρ end_ARG ) start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT , (2)

MJ=4π3ρ(12λJ)3=πρ6(πcs2Gρ)3/2,subscriptݑ€ݐ½4ݜ‹3ݜŒsuperscript12subscriptݜ†ݐ½3ݜ‹ݜŒ6superscriptݜ‹superscriptsubscriptݑݑ 2ݐºݜŒ32M_{J}=\frac{4\pi}{3}\rho\left(\frac{1}{2}\lambda_{J}\right)^{3}=\frac{\pi\rho}% {6}\left(\frac{\pi c_{s}^{2}}{G\rho}\right)^{3/2},italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT = divide start_ARG 4 italic_π end_ARG start_ARG 3 end_ARG italic_ρ ( divide start_ARG 1 end_ARG start_ARG 2 end_ARG italic_λ start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT = divide start_ARG italic_π italic_ρ end_ARG start_ARG 6 end_ARG ( divide start_ARG italic_π italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG italic_G italic_ρ end_ARG ) start_POSTSUPERSCRIPT 3 / 2 end_POSTSUPERSCRIPT , (3)
where ρݜŒ\rhoitalic_ρ is the mass density and cssubscriptݑݑ c_{s}italic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT the sound speed. The baryons are treated as an ideal gas obeying an equation of state of the form P=(γ-1)uρݑƒݛ¾1ݑ¢ݜŒP=(\gamma-1)u\rhoitalic_P = ( italic_γ - 1 ) italic_u italic_ρ, where PݑƒPitalic_P is the pressure and uݑ¢uitalic_u is the specific thermal energy. Enforcing a minimum specific thermal energy (and therefore temperature) in the gas forces a minimum in the sound speed cs2=γP/ρ=γ(γ-1)usuperscriptsubscriptݑݑ 2ݛ¾ݑƒݜŒݛ¾ݛ¾1ݑ¢c_{s}^{2}=\gamma P/\rho=\gamma(\gamma-1)uitalic_c start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = italic_γ italic_P / italic_ρ = italic_γ ( italic_γ - 1 ) italic_u, which therefore implies we have a minimum Jeans mass through equation (3). Note that ρݜŒ\rhoitalic_ρ is the total mass density (baryons and dark matter), since it is the total gravitating mass which counts, and therefore MJsubscriptݑ€ݐ½M_{J}italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT as expressed in equation (3) represents the total mass contained within r≤λJ/2ݑŸsubscriptݜ†ݐ½2r\leq\lambda_{J}/2italic_r ≤ italic_λ start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT / 2. If we want the total baryon mass contained within this radius, we must multiply MJsubscriptݑ€ݐ½M_{J}italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT by Ωbary/ΩsubscriptΩbaryΩ\mbox{$\Omega_{\mbox{\scriptsize bary}}$}/\Omegaroman_Ω start_POSTSUBSCRIPT bary end_POSTSUBSCRIPT / roman_Ω.

It is also important to understand how the mass resolution is set for the baryons by the SPH technique. This is not simply given by the baryon particle mass, since SPH interpolation is a smoothing process typically extending over spatial scales of several interparticle spacings. In general the mass resolution for the hydrodynamic calculations can be estimated as the amount of mass enclosed by a typical SPH interpolation volume. If the SPH smoothing scale is given by hℎhitalic_h and the SPH sampling extends for ηݜ‚\etaitalic_η smoothing scales, then the mass resolution MRsubscriptݑ€ݑ…M_{R}italic_M start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT is given by

MR=43π(ηh)3ρ.subscriptݑ€ݑ…43ݜ‹superscriptݜ‚ℎ3ݜŒM_{R}=\frac{4}{3}\pi(\eta h)^{3}\rho.italic_M start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT = divide start_ARG 4 end_ARG start_ARG 3 end_ARG italic_π ( italic_η italic_h ) start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_ρ . (4)
This is probably something of an overestimate, since the weight for each radial shell in this interpolation volume (given by the SPH sampling kernel WݑŠWitalic_W) falls off smoothly towards r=ηhݑŸݜ‚ℎr=\eta hitalic_r = italic_η italic_h, but given the other uncertainties in this quantity equation (4) seems a reasonable estimate. Note that the resolution limit for the SPH formalism is best expressed in terms of a mass limit, appropriate for SPH’s Lagrangian nature. For this reason we choose to express the Jeans limit in terms of the Jeans mass (eq. [3]) throughout this work, as the Jeans limit can be equally expressed in terms of a spatial or a mass scale. In N-body work it is common to express the mass resolution of an experiment in units of numbers of particles. In our simulations we use a bi-cubic spline kernel which extends to η=2ݜ‚2\eta=2italic_η = 2 smoothing lengths, and initialize the smoothing scales such that the smoothing scale hℎhitalic_h extends for two particle spacings. We therefore have a mass resolution in 2-D of roughly 50 particles, or equivalently in 3-D roughly 260 particles.

We perform simulations both with and without a minimum temperature (giving Jeans masses MJ=0subscriptݑ€ݐ½0M_{J}=0italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT = 0, MJ>0subscriptݑ€ݐ½0M_{J}>0italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT >0), at three different resolutions (N=Nbary=Ndm=642ݑsubscriptݑbarysubscriptݑdmsuperscript642N=\mbox{$N_{\mbox{\scriptsize bary}}$}=\mbox{$N_{\mbox{\scriptsize dm}}$}=64^{2}italic_N = italic_N start_POSTSUBSCRIPT bary end_POSTSUBSCRIPT = italic_N start_POSTSUBSCRIPT dm end_POSTSUBSCRIPT = 64 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, 1282superscript1282128^{2}128 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, and 2562superscript2562256^{2}256 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT), and for three different cutoffs in the initial perturbation spectrum (kc=32subscriptݑ˜ݑ32k_{c}=32italic_k start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = 32, 64, and 128). The initial density perturbations are initialized as Gaussian distributed with random phases and amplitudes, but in such a manner that all simulations have identical phases and amplitudes up to the imposed cutoff frequency kcsubscriptݑ˜ݑk_{c}italic_k start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT. The cutoff frequencies are the subset of kc∈(32,64,128)subscriptݑ˜ݑ3264128k_{c}\in(32,64,128)italic_k start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ∈ ( 32 , 64 , 128 ) up to the Nyquist frequency for each resolution kNyq=N1/2/2subscriptݑ˜Nyqsuperscriptݑ122\mbox{$k_{\mbox{\scriptsize Nyq}}$}=N^{1/2}/2italic_k start_POSTSUBSCRIPT Nyq end_POSTSUBSCRIPT = italic_N start_POSTSUPERSCRIPT 1 / 2 end_POSTSUPERSCRIPT / 2, so for each resolution we have kc(N=642)=32subscriptݑ˜ݑݑsuperscript64232k_{c}(N=64^{2})=32italic_k start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_N = 64 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) = 32, kc(N=1282)∈(32,64)subscriptݑ˜ݑݑsuperscript12823264k_{c}(N=128^{2})\in(32,64)italic_k start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_N = 128 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ∈ ( 32 , 64 ), and kc(N=2562)∈(32,64,128)subscriptݑ˜ݑݑsuperscript25623264128k_{c}(N=256^{2})\in(32,64,128)italic_k start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ( italic_N = 256 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ∈ ( 32 , 64 , 128 ). For each value of the minimum temperature we therefore have a grid of simulations which either have the same input physics at differing resolutions (i.e., kc=32subscriptݑ˜ݑ32k_{c}=32italic_k start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT = 32 for N∈[642,1282,2562]ݑsuperscript642superscript1282superscript2562N\in[64^{2},128^{2},256^{2}]italic_N ∈ [ 64 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , 128 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , 256 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ]), or varying input physics at fixed resolution (i.e., N=2562ݑsuperscript2562N=256^{2}italic_N = 256 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT for kc∈[32,64,128]subscriptݑ˜ݑ3264128k_{c}\in[32,64,128]italic_k start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ∈ [ 32 , 64 , 128 ]). This allows us to isolate and study both numerical and physical effects during the evolution of these simulations. In total we discuss twelve simulations.

For the simulations with a minimum temperature, there is an ambiguity in assigning a global Jeans mass with that temperature. The density in equation (3) is formally the local mass density, and therefore the Jeans mass is in fact position dependent through ρ(r→)ݜŒ→ݑŸ\rho(\vec{r})italic_ρ ( over→ start_ARG italic_r end_ARG ). Throughout this work we will refer to the Jeans mass at any given expansion factor as the Jeans mass defined using a fixed minimum temperature and the average background density, making this mass scale a function of time only. This is equivalent to taking the zeroth order estimate of MJsubscriptݑ€ݐ½M_{J}italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT, giving us a well defined characteristic mass scale. In terms of this background density, Figure 6 shows the baryon Jeans mass (in units of the resolved mass via equation [4]) as a function of expansion. Note that for a given simulation MRsubscriptݑ€ݑ…M_{R}italic_M start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT remains fixed, and it is the Jeans mass which grows as MJ∝ρ-1/2∝a3/2proportional-tosubscriptݑ€ݐ½superscriptݜŒ12proportional-tosuperscriptݑŽ32M_{J}\propto\rho^{-1/2}\propto a^{3/2}italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT ∝ italic_ρ start_POSTSUPERSCRIPT - 1 / 2 end_POSTSUPERSCRIPT ∝ italic_a start_POSTSUPERSCRIPT 3 / 2 end_POSTSUPERSCRIPT. It is apparent that the N=2562ݑsuperscript2562N=256^{2}italic_N = 256 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT simulations resolve the Jeans mass throughout most of the evolution, the N=1282ݑsuperscript1282N=128^{2}italic_N = 128 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT simulations resolve MJsubscriptݑ€ݐ½M_{J}italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT by a/ai∼15similar-toݑŽsubscriptݑŽݑ–15a/a_{i}\sim 15italic_a / italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ 15, and the N=642ݑsuperscript642N=64^{2}italic_N = 64 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT simulation does not approach MJ/MR∼1similar-tosubscriptݑ€ݐ½subscriptݑ€ݑ…1M_{J}/M_{R}\sim 1italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT / italic_M start_POSTSUBSCRIPT italic_R end_POSTSUBSCRIPT ∼ 1 until the end of our simulations at a/ai∼60similar-toݑŽsubscriptݑŽݑ–60a/a_{i}\sim 60italic_a / italic_a start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ∼ 60. The specific value of Tminsubscriptݑ‡ݑšݑ–ݑ›T_{min}italic_T start_POSTSUBSCRIPT italic_m italic_i italic_n end_POSTSUBSCRIPT used in this investigation is chosen to yield this behaviour. We discuss physically motivated values for this minimum temperature in §5.

3 Numerical Resolution and the Jeans Mass

3.1 Dark Matter

We will begin by examining the dark matter distribution, as this is a problem which has been examined previously. Figures 6a and b show images of the dark matter overdensity ρdm/ρ¯dmsubscriptݜŒdmsubscript¯ݜŒdm\mbox{$\rho_{\mbox{\scriptsize dm}}$}/\mbox{$\bar{\rho}_{\mbox{\scriptsize dm}% }$}italic_ρ start_POSTSUBSCRIPT dm end_POSTSUBSCRIPT / over¯ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT dm end_POSTSUBSCRIPT for the MJ=0subscriptݑ€ݐ½0M_{J}=0italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT = 0 simulations. In order to fairly compare with equivalent images of the SPH baryon densities, the dark matter information is generated by assigning a pseudo-SPH smoothing scale to each dark matter particle, such that it samples roughly the same number of neighboring dark matter particles as the SPH smoothing scale samples in the baryons. We then use the normal SPH summation method to assign dark matter densities, which are used to generate these images. The panels in the figure are arranged with increasing simulation resolution NݑNitalic_N along rows, and increasing cutoff frequency kcsubscriptݑ˜ݑk_{c}italic_k start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT down columns. The diagonal panels represent each resolution initialized at its Nyquist frequency for P(k)ݑƒݑ˜P(k)italic_P ( italic_k ). Note that the physics of the problem is constant along rows, and numerics is constant along columns. If resolution were unimportant, the results along rows should be identical. Likewise, since the numerics is held constant along columns, only physical effects can alter the results in this direction.

Comparing the dark matter densities along the rows of Figure 6a, it is clear that the structure becomes progressively more clearly defined as the resolution increases. This is to be expected, since the higher resolution simulations can resolve progressively more collapsed/higher density structures. The question is whether or not the underlying particle distribution is systematically changing with resolution. In other words, do the simulations converge to the same particle distribution on the scales which are resolved? Figure 6b shows this same set of dark matter overdensities for the MJ=0subscriptݑ€ݐ½0M_{J}=0italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT = 0 simulations, only this time each simulation is degraded to an equivalent N=642ݑsuperscript642N=64^{2}italic_N = 64 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT resolution and resampled. This is accomplished by selecting every nݑ›nitalic_nth node from the higher resolution simulations, throwing away the rest and suitably modifying the masses and smoothing scales of the selected particles. Note that now the dark matter distributions look indistinguishable for the different resolution experiments, at least qualitatively. This similarity implies that the high frequency small scale structure has minimal effect on the larger scales resolved in this figure. Looking down the columns of Figure 6a it is clear that increasing kcsubscriptݑ˜ݑk_{c}italic_k start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT does in fact alter the dark matter particle distribution, such that the large scale, smooth filaments are progressively broken up into smaller clumps aligned with the overall filamentary structure. These differences are lost in the low-res results of Figure 6b, implying that these subtle changes do not significantly affect the large scale distribution of the dark matter.

In Figures 6a and b we show the mass distribution functions for the dark matter overdensity f(ρdm/ρ¯dm)ݑ“subscriptݜŒdmsubscript¯ݜŒdmf(\mbox{$\rho_{\mbox{\scriptsize dm}}$}/\mbox{$\bar{\rho}_{\mbox{\scriptsize dm% }}$})italic_f ( italic_ρ start_POSTSUBSCRIPT dm end_POSTSUBSCRIPT / over¯ start_ARG italic_ρ end_ARG start_POSTSUBSCRIPT dm end_POSTSUBSCRIPT ). Figure 6a includes all particles from each simulation (as in Figure 6a), while Figure 6b is calculated for each simulation degraded to equivalent N=642ݑsuperscript642N=64^{2}italic_N = 64 start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT resolutions (comparable to Figure 6b). The panels are arranged as in Figure 6, with MJ=0subscriptݑ€ݐ½0M_{J}=0italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT = 0 and MJ>0subscriptݑ€ݐ½0M_{J}>0italic_M start_POSTSUBSCRIPT italic_J end_POSTSUBSCRIPT >0 overplotted as different line types. It is clear that the varying Jeans mass in the baryons has negligible effect on the dark matter, a point we will return to in §4. The full resolution results of Figure 6a show a clear trend for a larger fraction of the mass to lie at higher densities with increasing resolution. There is also a similar though weaker trend with increasing kcsubscriptݑ˜ݑk_{c}italic_k start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT. However, examining the resampled results of Figure 6b it appears that the results of all simulations converge, b

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