The effect of dark matter resolution on the collapse of baryons in high redshift numerical simulations

1 MB·English

by  John A. Regan

Mon.Not.R.Astron.Soc.000,000–000 (0000) Printed24March2015 (MNLATEXstylefilev2.2)
The effect of dark matter resolution on the collapse of
baryons in high redshift numerical simulations
5
John A. Regan1⋆, Peter H. Johansson1 & John H. Wise2
1
0
2 1Department of Physics, University of Helsinki, Gustaf H¨allstro¨min katu 2a, FI-00014 Helsinki, Finland
2Centerfor Relativistic Astrophysics, Georgia Institute of Technology, 837 State Street, Atlanta, GA 30332, USA
r
a
M

3 24March2015
2

] ABSTRACT
A
We examine the impactofdarkmatter particle resolutiononthe formationofa bary-
G onic core in high resolution adaptive mesh refinement simulations. We test the effect
. thatbothparticlesmoothingandparticlesplittinghaveonthehydrodynamicproper-
h
tiesofacollapsinghaloathighredshift(z >20).Furthermore,wevarythebackground
p
field intensity, with energy below the Lyman limit (< 13.6 eV), as may be relevant
-
o forthe caseofmetal-free starformationandsuper-massiveblackhole seedformation.
r We find that using particle splitting methods greatly increases our particle resolution
t
s without introducing any numerical noise and allows us to achieve converged results
a over a wide range of external background fields. Additionally, we find that for lower
[
valuesofthebackgroundfieldalowerdarkmatterparticlemassisrequired.Wedefine

2 theradiusofthecoreasthepointatwhichtheenclosedbaryonicmassdominatesover
v the enclosed dark matter mass. For our simulations this results in Rcore ∼5 pc. We
0 find that in order to produce convergedresults which are not affected by dark matter
5 particles requires that the relationship Mcore/MDM >100.0 be satisfied, where Mcore
6 is the enclosed baryon mass within the core and MDM is the minimum dark matter
5
particle mass. This ratio should provide a very useful starting point for conducting
0
convergence tests before any production run simulations. We find that dark matter
.
1 particle smoothing is a useful adjunct to already highly resolved simulations.
0
5 Key words: Cosmology:theory–large-scalestructure–firststars,methods:numer-
1 ical
:
v
i
X
r 1 INTRODUCTION Dark matter is modelled as a collisionless fluid and in
a
principle can be solved using the collisionless Boltzmann
Gravitational collapse of dark matter has been well
equation (e.g. Shlosman et al. 1979; Rasio et al. 1989;
studied over the past four decades with increas-
Hozumi 1997). However, the computational demands of
ingly high resolution N-body numerical simulations
solving this six dimensional equation means that instead
(Daviset al. 1985; Frenk et al. 1988; Warren et al. 1992;
dark matter is typically treated in a Monte-Carlo fashion
Cen et al. 1994; Gelb & Bertschinger 1994; Navarro et al.
(however see recent work by Hahn & Angulo 2015) and
1996, 1997; Hernquist et al. 1996; Jenkins et al.
represented by a set of collisionless particles which interact
2001; Wambsganss et al. 2004; Springel et al. 2008;
with each other gravitationally. Solving the gravitational
Diemand et al. 2008; Bosch et al. 2014) leading to the con-
interactions of a many-body system is then dealt with in
firmation that the large-scale structure of the Universe is
one of two ways, direct summation (e.g. von Hoerner 1960;
built up through hierarchical collapse of small dark matter
Aarseth 1963; von Hoerner 1963; Steinmetz 1996) or more
haloes. Itiswithinthegravitationalpotentialwellsofthese
commonly, in cosmological simulations, using an approxi-
dark matter structures that galaxies and their associated
mate treatment involving a Tree algorithm (Barnes & Hut
activegalacticnucleiandquasarsform(White& Rees1978;
1986, 1989) or a particle mesh algorithm (Efstathiou et al.
White& Frenk 1991; Springel et al. 2005; Croton et al.
1985; Hockney & Eastwood 1988) - see also Bertschinger
2006;Johansson et al. 2012; Vogelsberger et al. 2014).
(1998) for a comprehensive review.
The subsequent addition of gas dynamics to the
simulationsincreasesthecomplexityofthesimulationssub-
⋆ E-mail:john.regan@helsinki.fi
(cid:13)c 0000RAS
2 J.A. Regan et al.
stantially,requiringnotjust adetailedtreatmentofthegas galaxy. This is particularly crucial as correctly accounting
(fluid)interaction withthedarkmatter,butalso ofarange for the cooling and heating effect becomes extremely
of physical processes involving the gas including radiative important in determining the characteristics of the final
cooling, star formation and the tracking of individual object. In thispaper we study thisexact effect.
chemical species. Incorporating the gas dynamics into the It is also worth noting that apart from the negative
simulation usually also takes the form of one of two mech- impactthatpoorlyresolveddarkmatterparticlesmayhave
anisms - smoothed particle hydrodynamics (SPH) (Lucy on the baryonic physics an additional numerical artefact
1977; Gingold & Monaghan 1977) or grid based methods may occur if a dark matter particle flies into the high
(Ryuet al. 1990; Cen et al. 1990; Stone& Norman 1992; density region from a lower density region. This “flyby”
Bryan et al. 1995) wherethehydrodynamicalequationsare andtheperturbingeffectsofitslargemassmaytheninduce
solvedonthegridusingfinitedifferencetechniquespossibly incorrect physical conditions. While we do not specifically
involvingdeforming oradaptivemeshes(e.g. Springel2010; deal with this issue in this study it can be alleviated using
Bryan et al. 2014). particle smoothing (see §2.4) which will necessarily prevent
Here we focus on the effect the mass resolution of the adverse effects from a large dark matter mass within the
darkmatterparticlescanhaveonthegasdynamicswithina baryon dominated core.
collapsing halo where thegas densities can exceed the dark We vary the mass resolution of the dark matter
matter densities by several orders of magnitudeas happens particles that surround the collapsing core in an effort
in the centres of newly forming galaxies, mini-haloes and to identify any spurious heating effects that the dark
clusters of galaxies. In order to conduct this case study we matter has on the collapsing core and identify resolution
use theEnzo code (see §2 for details on Enzo) to follow the guidelines. Furthermore, we vary the background radiation
collapseofgaswithincollapsinghaloesearlyintheUniverse fieldresultingintheformation(andcollapse)ofmini-haloes
atatimewhenthecoolingofthegasisdrivenbyprimordial and haloes with virial temperatures T > 104 K and again
processes and in which the gas has not yet been enriched test for dark matter mass resolution convergence.
with metals from previous episodes of star formation. This The paper is laid out as follows: in §2 we describe
caseisrelevantforsimulationsinvolvingthefirstgeneration the numerical approach used and the various techniques
ofstars(PopIIIstars)(Abelet al.2002;Bromm et al.2002; employed that can affect the dark matter resolution;
Yoshidaet al. 2003; Bromm & Larson 2004; Yoshida et al. in §3 we describe the characteristics of the simulations
2006; O’Shea & Norman 2008; Yoshidaet al. 2008; used in this study; in §4 we describe the results of our
Wise & Abel 2008; O’Shea & Norman 2008; Turk et al. numerical simulations; in §5 we discuss the parameters
2009; Bromm et al. 2009; Clark et al. 2011a; Stacy et al. for choosing the correct dark matter particle mass and in
2010, 2012; Susaet al. 2014; Stacy & Bromm 2014; §6 we present our conclusions. Throughout this paper we
Hirano et al. 2014) or for the case of direct collapse assume a standard ΛCDM cosmology with the following
models of supermassive black holes (Shlosman et al. 1989; parameters (Planck Collaboration et al. 2014, based on the
Haiman & Loeb 2001; Oh & Haiman 2002;Bromm & Loeb latest Planck data), ΩΛ,0 = 0.6817, Ωm,0 = 0.3183, Ωb,0 =
2003;Haiman 2006;Begelman et al.2006;Wise et al. 2008; 0.0463, σ8 = 0.8347 and h = 0.6704. We further assume
Regan & Haehnelt 2009a,b; Regan et al. 2014a,b) which a spectral index for the primordial density fluctuations of
arealsothoughttoformoutofpristinegasathighredshift. n=0.9616.
Previous studies of dark matter resolution have indi-
cated the negative numerical effects that dark matter can
have on the collapse of baryons in the context of galaxy
formation. Steinmetz& White (1997) showed that using 2 NUMERICAL SETUP
dark matter particles with masses above a critical mass
limit (their work was focused in the context of galaxy We have used the publicly available adaptive mesh re-
formation) introduced spurious two body heating effects finement (AMR) code Enzo2. AMR codes are a popular
toolforstudyingthegravitationalandhydrodynamicalcol-
which they found completely suppressed the effects of
radiative cooling. Although their numerical tests were lapse of astrophysical objects (although see for example
carried out using the SPH technique they argued that Yoshidaet al. 2006, Clark et al. 2011b & Mayeret al. 2014
amongothersforcollapsesimulationsconductedusingSPH
the results are valid in the case of grid based approaches
techniques)duetotheirinherentabilitytoextendovermany
also given the modelling of the dark matter component is
similar in both instances. And while other studies have magnitudes in dynamic range. Therefore using Enzo as the
codewith whichtoperform thisstudyisan obviouschoice.
noted the effect that dark matter mass resolution plays in
Furthermore,weuseversion3.03 whichisthebleedingedge
galaxy formation simulations (e.g. Machacek et al. 2001;
Kaufmann et al. 2006) no specific study has been made, version of the code incorporating a range of new features
(e.g. Goldbaum et al. in prep).
to the best of our knowledge, on the effect of dark matter
resolution on the collapsing core1 within an embryonic Enzo wasoriginallydevelopedbyGregBryanandMike
Norman at the University of Illinois (Bryan & Norman
1995, 1997; Norman & Bryan 1999; O’Sheaet al. 2004;
1 Thecoreradiusisheredefinedastheradiusatwhichthebary- Bryan et al. 2014). The gravity solver in Enzo uses an N-
onicenclosedmassexceedsthedarkmattermassforthefirsttime. Body adaptive particle-mesh technique (Efstathiou et al.
Typically,foroursimulationsthisoccursbetween1and10parsecs
fromthepointofhighestdensity.Thisdefinitioniscloselyrelated
to the Jeans length, λJ, of the gas for a gas with an isothermal 2 http://enzo-project.org/
temperatureof8000Kandnumberdensity, n∼1×106 cm−3. 3 Changeset: 6037:1c232317e50d
(cid:13)c 0000RAS,MNRAS000,000–000
3
104 0.30
3)− 122.7 M 122.7 M
m 9.5 M ⊙ 0.25 9.5 M ⊙
c s
( ⊙ e ⊙
nsity 103 0.7 M⊙ erenc 0.20 0.7 M⊙
ber De102 ss Diff 0.15
m a
M 0.10
u
N d
atter 101 close 0.05
M n
E
k 0.00
r
a
D
100 −0.05
101 102 101 102
Radius (pc) Radius (pc)
Figure1.ThelefthandpanelshowsthesphericallyaverageddarkmatterdensityprofileforadarkmatteronlysimulationofHaloA.
Theoutputistaken ataredshiftofz=23andthedensityisscaledbythemassofthehydrogenatom toenableadirectcomparison
with the gas density. The lines arefor a minimum dark matter mass of 122.7 M⊙, 9.5M⊙ and 0.7 M⊙. The righthand panel shows
thedifferencebetween theenclosedmassprofileforthesamehaloatthesameoutputtime.Thedifferences arecalculated againstthe
highestresolutionsimulation(i.e.thedarkmatter simulationwithM=0.7M⊙).
1985; Hockney& Eastwood 1988; Couchman 1991) while concentrate our study on radiation fields with 80 J21 and
thehydrodynamicsareevolvedusingthepiecewiseparabolic 500 J21 and then in §4.3 we explore the effect of varying
method combined with a non-linear Riemann solver for the external field. The shape of the spectrum is fully de-
shock capturing. The AMR methodology allows for addi- scribedbythetemperatureoftheblackbodyradiation, T
eff
tional finer meshes to be laid down as the simulation runs and the amplitude of the fluctuation, κ J21, where κ is any
to enhancethe resolution in a given, userdefined,region. scalar quantity. We normalise the external radiation field
For our simulations we set the maximum refinement at the hydrogen ionisation edge. In our simulations we al-
level to 18. Refinement is triggered in Enzo when the re- low the external field to affect the rates of three species -
finementcriteria areexceeded.Therefinementcriteria used H−,H2,H+2, the rates for the three species are given below
in this work were based on three physical measurements: (Abelet al. 1997):
(1) The dark matter particle over-density, (2) The baryon
4π 13.6 B(E,T ) σ(E)
oinvterro-dduencesitayddanitdio(n3a)ltmheesJheeasnwshleenngtthh.eTohveerfi-drestnstiwtyo(cρrim∆teeρarnia) kH− = h Z0.76 J21B(13.6,Tefefff) E dE [s−1]
of a grid cell with respect to the mean density exceeds
8m.0umfoMrabsasrFyoornRseafinnde/moernDtEMxp.oFnuerntthepramraomree,tweretsoet−t0h.e1Mmianki-- kH2 =1.38×109J21BB((E13L.W6,,TTeff)) [s−1] (1)
eff
ingthesimulation super-Lagrangian andthereforereducing
thethresholdforrefinementashigherdensitiesarereached.
Fleonrgtthhetofinbael1c6riitnertihaewseersuents.the number of cells per Jeans kH2+ = 4hπ Z2.1635.6J21BB(1(E3.6,T,Tefefff))σ(EE)dE [s−1]
where kX is the photoionisation rate for a given
species, J21 is the external field with implied units of
2.1 Chemical Modelling 10−21 erg cm−2 s−1 Hz−1 sr−1,σ(E) is thecross section at
agivenenergy(frequency)andB(E,T )istheblackbody
We adopt a nine species model for the chemical network eff
used including H,H+,He,He+,He++,e−,H2,H+2 and H−. tshpeecetnruemrgyatinantheenLerygmy,anE-,Wteermnpererbaatunrde,(1T2e.f8feVan)d. TEhLeWinis-
The gas is allowed to cool radiatively during the collapse.
tegration limits are in eV and do not exceed the hydrogen
Additionally,weincorporateanexternalradiationfieldwith
ionisation threshold of 13.6 eV.
anenergybelowthehydrogenionisationedgeintothesimu-
lationsconductedhere.Specificallyweincorporateradiation
energies from 0.76 eV up to 13.6 eV which affects thethree
2.2 Realisations
species H−,H2,H+2 as described in more detail below. The
intensity of thebackground radiation is varied overa range Using the initial conditions generator MUSIC (Hahn & Abel
ofvaluesbetween0and500×10−21erg cm−2 s−1 Hz−1 sr−1 2011) we generated over 5000 random dark matter reali-
withtheeffectivetemperatureoftheblackbodytemperature sations with Enzo using a 2563 grid. Each simulation was
set to T = 50000 K in all cases. We use the variable J21 as run until a redshift of z = 30 with an inline “Friends of
shorthand for 10−21 erg cm−2 s−1 Hz−1 sr−1. We initially Friends” halo finder. Several realisations were identified as
(cid:13)c 0000RAS,MNRAS000,000–000
4 J.A. Regan et al.
0.5
MM==112222..77MM⊙⊙ MM==99..55MM⊙⊙ MM==00..77MM⊙⊙ P
r
o
je
0.2 cte
d
D
e
n
0.1 sit
y
(
g
c
m
0.05 −
250pc 2
)
Figure 2. Each panel shows a dark matter density projection of the 0.5 kpc (physical) surroundingthe collapsing halo at a redshift
ofz=23.Thelefthandpanelhasaminimumdarkmatter massofMDM∼122.7M⊙,themiddlepanelhasaminimumdarkmatter
mass of MDM ∼9.5M⊙, and the righthand panel has a minimumdark matter mass of MDM ∼0.7 M⊙. Each projection is centred
onthepointofmaximumdensityinthesimulationwithadarkmatterparticlemassofMDM∼122.7M⊙.
having a high sigma peak (σ > 4.0 at z > 30, see for ex- Particles are either split once or twice. Each splitting
ampleRegan et al. 2014a for detailson thedefinitionof σ). reduces the particle mass by a factor of 13 producing 13
Three realisations were found from the initial dark matter new child particles in the process and so also increasing
runs and a single halo realisation was used to conduct this the computational demands of the simulation. We follow
study-labeledHaloA.Itshouldbenotedthatwehavealso Kitsionas & Whitworth(2002)inchoosingtospliteachpar-
conducted resolution tests on the two other haloes found - entparticle into13 childparticles, thechoice of 13is some-
Halo B andHalo C -howevertheresultswerebroadlysim- what arbitrary but attempts to takeinto account two com-
ilar.WediscussthecharacteristicsofHaloB andHaloC in peting arguments - 1) there should not be too large a dif-
§4.4 but focus the bulkof this studyaround Halo A. ference between the child mass and theparent mass and 2)
Each realisation wasthen resimulated with baryonsin- the collective density distribution of the family of children
cluded and with different levels of nested grids and hence should approximate the spherically symmetric density dis-
dark matter particle resolution. We ran simulations utilis- tribution of theparent.
ing one, two and three levels of grid nesting. Each nested We test for convergence using both one and two itera-
grid reduces the dark matter particle mass resolution by a tions of splitting. We split the particles in a subregion sur-
factor of eight. Memory limitations restrict the number of roundingthepointofmaximumdensity.Thelocationofthe
nestedgridstoamaximumofthree.Forthisstudyweusea point of maximum density is found from previous simula-
boxsizeof2.0h−1 Mpcwhichallowsustofindlargehaloes tions where particle splitting was not used. This allows for
(highsigmapeaks)whilealsorestrictingthedynamicrange the increased mass resolution to be targeted in the region
required by the simulation. This means that the maximum surrounding the collapse. We select a region of 43.75 h−1
darkmatterparticleresolutionthatEnzo canachieveinthe kpc(comoving)withinwhichtodothefirstiterationofpar-
simulation is MDM ∼ 103 M⊙. Enzo does not dynamically ticle splitting. The second splitting, if performed, is done
refinetheparticlemassduringthesimulationwhenrefining within a region half this size - 21.88 h−1 kpc (comoving).
itsgrids.Inordertorefinethedarkmatterparticlesfurther This allows for maximum dark matter resolution alongside
we employ a Particle Splittingprocedure. maximum grid resolution.
Inordertoinvestigateanysystematiceffectonthedark
matter particle properties we initially conducted dark mat-
teronlysimulationswherewesplittheparticlesataredshift
2.3 Particle Splitting
ofz=40andsubsequentlylookedatthehighestdensityre-
Inordertoincreasetheresolution ofthedarkmatterparti- gion approximately 80 Myrs later at a redshift of z = 23.
cles still furtherwe employ theparticle splitting techniques Intherunswithbaryonsthecollapseoccursataredshiftof
of Kitsionas & Whitworth (2002), several other authors approximatelyz=23orearlierandsoexaminingtheform-
havepreviouslyadoptedthesametechnique(Yoshida et al. ing dark matter halo at a redshift of z=23 is appropriate.
2006;Greif et al.2011;Kim et al.2011;Hirano et al.2014). InFigure1weshowthedarkmatterdensityprofilesand
Wesplitparticlesataredshiftofz =40soastominimisethe enclosedmassprofilesforHaloAataredshiftofz=23.The
effectsofanynoiseintroducedbythesplittingtechnique.In profiles are all centred at the point of maximum density as
particlesplitting, thedistribution of matterwithinaregion foundinthesimulationwhereparticlesplittingwasnotcon-
assumesthattheregionhasauniformmassdistributionand ducted(i.e.thesimulationwithadarkmatterparticlemass
sononewperturbationmodesareadded.Splittingthepar- of MDM = 122.7 M⊙). In the left hand panel the density
ticles at a redshift of 40 ensures that we approximate this profile shows clearly that the differences between the run
case and furthermore we test for convergence as described with noparticle splitting (blueline) and therunswith par-
below.
(cid:13)c 0000RAS,MNRAS000,000–000
5
Table 1.HaloDetails
Halo Namea MTotb MDMc MBaryond Re200 Tvfir ngH,max
HaloA 2.3×107 2.0×107 3.0×106 0.35 9718 1.05×109
HaloB 2.3×107 2.0×107 3.0×106 0.34 9098 1.01×109
HaloC 1.4×107 1.2×107 1.6×106 0.30 7402 1.18×109
Notes: The above table contains the simulation namea, the total massb (gas & dark matter) at the virial radius4 [M⊙],
the enclosed massindarkmatterc at thevirialradius[M⊙],the enclosedmass inbaryonsd at thevirialradius[M⊙],the
virialradiuse [kpc],the virialtemperaturef [K],andthemaximum gas numberdensityg inthe halo[cm−3].Allunits are
physical,unlessexplicitlystatedotherwise.
ticlesplitting(cyanandmagentalines)arenegligiblewhich 2.5 External Radiation Fields
givesusconfidencethatthedarkmattersplittingintroduces
As noted above we concentrate our study on ex-
little or noadditional perturbingeffects tothe dark matter
component.Thisshouldnotbesurprisingsincethesplitting ternal radiation fields with values of 80 J21 and
occursearlyinthehaloformationprocessatatimewhenthe 500 J21 both of which may be relevant for study-
ing the direct collapse of super-massive black holes
non-lineareffectsofthegravitationalcollapsearestillnegli-
at high redshift (Shlosman et al. 1989; Begelman et al.
gible.Tofurtheremphasisethepointweshowthedifference
2006; Dijkstra et al. 2008; Regan & Haehnelt 2009b,a;
betweentheenclosedmassprofilesforthesameoutputs.We
Shanget al. 2010; Johnson et al. 2013; Latif et al. 2013,
normalise against thethehighest resolution simulation (i.e.
2014a,b;Agarwal et al.2014;Regan et al.2014a,b).Avalue
the simulation with a dark matter particle mass of MDM
= 0.7 M⊙). The differences are negligible at large radii (R of 80 J21 sets the field to where H2 formation is inhibited
butnotcompletelydestroyed,thecollapseofthehalothere-
& 100 pc) and increase to only a factor of . 0.3 at small
foretakesplaceinanenvironmentwherethecollapseisnot
scales.
Figure 2shows a density projection of thedark matter isothermal. A value of 500 J21 on the other hand results
in an isothermal collapse. By examining both scenarios the
densityalongthex-axisforthishalo(HaloA)foreachcase.
effectsofdifferentdarkmatterresolutionsonthedirectcol-
The left hand panel has a minimum dark matter particle
lapse model can be fully assessed.
mass of MDM = 122.7 M⊙, the middle panel has a mini-
Wefurtherassess theimpact ofsmaller radiation fields
mum dark matter mass of MDM = 9.5 M⊙ and the right
handpanelhasaminimumdarkmattermassofMDM=0.7 on the collapse. We use field strengths of both 1 J21 and
M⊙. The halo becomes smoother as the dark matter reso- 10 J21 and we also test the case where no background ex-
lutionisincreased butotherwise remainsvisuallythesame. ists (i.e. 0 J21). All of these fields may be relevant for the
case where metal-free stars form (e.g. Bromm et al. 1999;
The spatial scale in each case is given in the left panel and
Abelet al. 2002; Bromm et al. 2002). A comparison of the
corresponds to 250 pc physical.
affect of different field strengths is discussed in §4.3.
Wedonot examinetheaffect of changing theH2 three
body formation rates in this work and instead use the
2.4 Particle Smoothing
rates of Martin et al. (1996) throughout. This topic has
In regions of high density (resolution) the cell size can be- beenexaminedcomprehensivelybybothTurk et al.(2011a)
come very small and the particles can begin to experience andBovino et al.(2014).TheMartin et al.(1996)ratesare
spurious two body effects. In order to avoid such spurious found to provide the most consistent fit from the work
interactions the dark matter particles can be “smoothed” of Turk et al. (2011a), are density dependent and follow
onto the grid once the cell size drops below a given thresh- closely the rates of Forrey (2013) which are advocated by
old. In Enzo thisis controlled viatheMaximumParticleRe- Bovino et al. (2014).
finementLevel parameter. Once the cell size drops below a
userdefinedthreshold thedarkmattermass is interpolated
onto the grid with some smoothing length h (which will al-
ways be quoted in comoving units). While computationally 3 SIMULATION DETAILS
inefficient to do, it is done in only a very small region and
The primary goal of this study is to evaluate the impact, if
soitseffect ontheoverallcomputational timeissmall. The
any,of the dark matter resolution on the results of the col-
darkmatternoweffectivelybehavesasacontinuous,rather
lapseofthegas. Thegeneral halocharacteristics ofHaloA,
thandiscrete,distributionintheseveryhighdensityregions
Halo B and Halo C are given in Table 1. The study focuses
and the effects of (large) dark matter particles coming into
onHaloAforthemostpartwiththecharacteristicsofHalo
closecontactisalleviated.Inthisstudywesetthethreshold
B and Halo C only discussed in §4.4.
value of h to 2.8 comoving parsecs (equivalent to a refine-
Each(re)simulationofHaloAdiffersineitherthemin-
ment level of 12). We run simulations where thesmoothing
imum dark matter particle size or the minimum smoothing
is initiated and also where the smoothing is not activated.
In the runs without the smoothing the smoothing length is
effectively the minimum cell size (i.e. the cell size at maxi-
mum refinement level) which in the case of our simulations 4 Thevirialmassisdefinedas200timesthemeandensityofthe
is 0.04 comoving parsecs at level 18. Universeinthiscase.
(cid:13)c 0000RAS,MNRAS000,000–000
6 J.A. Regan et al.
-2 -1 0 1 2 -2 -1 0 1 2
10 10 10 10 10 10 10 10 10 10
2
10
4
) 10 o
K
1 i
10 t
( a
R
e
r 100 s
u s
a
t M
a 3
10 -1
r 6646 M , z = 23.71 10 d
e
⊙ e
p 830 M , z = 24.13 s
m 103 M⊙, z = 23.43 Halo A 10-2 lo
e 8.0 M ⊙, z = 23.50 h = 0.04 pc nc
T ⊙ 80 J E
0.6 M , z = 23.46 21 10-3
⊙
4
) 10 o
K
1 i
10 t
( a
R
e
r 100 s
u s
a
t M
a 3
10 -1
r 6646 M , z = 23.79 10 d
e
⊙ e
p 830 M , z = 24.25 s
m 103 M⊙, z = 23.42 Halo A 10-2 lo
e 8.0 M ⊙, z = 23.50 h = 2.8 pc nc
T ⊙ 80 J E
0.6 M , z = 23.42 21 10-3
2 ⊙
10
-2 -1 0 1 2 -2 -1 0 1 2
10 10 10 10 10 10 10 10 10 10
Radius (pc) Radius (pc)
Figure 3. Halo A: In the top two panels no smoothing of the dark matter particles is activated and so the smoothing length is the
same as the minimum cell size -0.04 pc (comoving). In the bottom two panels the smoothing is activated at a refinement level of 12
(2.8pccomoving).Thebottomrightpanelshowstheencloseddarkmattermassdividedbytheenclosedbaryonmass-MDM/MBaryon.
Seetextforfurtherdetailsontheplotcharacteristics.
length of the dark matter particle. Furthermore, the exter- 4 RESULTS
nal radiation field that a simulation is exposed to is varied
4.1 Halo A - Varying the Mass Resolution and
between 0 J21 and 500 J21. The simulations are automati-
Smoothing Lengths
cally halted once the maximum refinement level is reached.
Thereforeeachsimulatedhalo,whichisexposedtothesame Webegin by examining the temperature profile for the col-
external radiation field, also has a very similar maximum lapsed haloes for the different dark matter mass resolu-
density and comparisons aremade at thismaximum refine- tionsandsmoothinglengths.Inthefollowingcasesthefield
mentleveloutput.Eachsimulationisthereforecomparedat strength is set to either 80 J21 or 500 J21, we will examine
a very similar stage in its evolution. thecase of lower background radiation intensities in §4.3.
Thesmoothinglengthsforeachrunareseteitheratthe In Figure 3 we show the results for Halo A when the
maximum refinement level which corresponds to 0.04 (re- external field is fixed at 80 J21. The top row in the panel
finement level 18) comoving pc or they are smoothed at a shows simulations where no smoothing of the dark mat-
refinementlevelof12whichcorrespondsto2.80comovingpc ter particles occurs. The bottom panel shows simulations
(see §2.4 for more details on setting the particle smoothing wherethesmoothingisactivatedatarefinementlevelof12
lengths). Each parameter used in each simulation is clearly (2.8comovingparsecs).Eachsimulationwasrunwithdiffer-
markedin theleft handplot window ofeach figure.Topro- ent minimum dark matter particle resolution ranging from
mote clarity we refer to a simulation with a dark matter MDM = 6646 M⊙ to MDM = 0.6 M⊙. Looking at the left
particle resolution of 0.6 M⊙ as S06, a simulation with a hand column where we have plotted the temperature pro-
dark matter particle resolution of 8.0 M⊙ as S8, etc. files, centred on the densest point in the simulation, we see
that down to approximately 1 parsec from the centre the
(cid:13)c 0000RAS,MNRAS000,000–000
7
-2 -1 0 1 2 -2 -1 0 1 2
10 10 10 10 10 10 10 10 10 10
2
10
4
) 10 o
K
1 i
10 t
( a
R
e
r 100 s
u s
a
t 6646 M , z = 23.32 M
a 103 ⊙ -1
r 830 M , z = 23.21 10 d
e
⊙ e
p 103 M , z = 23.00 s
m ⊙ Halo A 10-2 lo
8.0 M , z = 22.92 c
e h = 0.04 pc n
⊙
T 0.6 M , z = 22.94 500 J E
21 -3
⊙ 10
4
) 10 o
K
1 i
10 t
( a
R
e
r 100 s
u s
a
t 6646 M , z = 23.33 M
a 103 ⊙ -1
r 830 M , z = 23.14 10 d
e
⊙ e
p 103 M , z = 22.98 s
m ⊙ Halo A 10-2 lo
8.0 M , z = 22.91 c
e h = 2.8 pc n
⊙
T 0.6 M , z = 22.92 500 J E
21 -3
⊙ 10
2
10
-2 -1 0 1 2 -2 -1 0 1 2
10 10 10 10 10 10 10 10 10 10
Radius (pc) Radius (pc)
Figure 4.Halo A:SameasFigure3except theexternal radiationfieldissetto500J21.
smoothed and non-smoothed simulations agree very well. approximately 1 parsec from the maximum gas density but
Within the core of the halo differences begin to appear. In its mass dominates the region). The results from S830 and
the non-smoothed simulations (top left panel) the two low- S6646arethereforelikelytobespuriousasthedarkmatter
est resolution simulations show a hotter core compared to resolution in each case is insufficient and the dark matter
the higher resolution simulations (i.e. those with a particle massendsupdominatingthecentralpotential.Theheating
resolution MDM ≤103 M⊙). In the smoothed runsthe low- of the core, seen in theleft hand panel, should therefore be
est resolution simulation has a temperature in the core of treatedwithcautionduetothepresenceofperturbingforces
Tcore ∼6000Kwhilealltheothersimulationsconvergetoa from these large dark matter particles.
core temperature Tcore ∼2000 K - note also that S8 shows Forthecaseofthesmoothed simulations weagain plot
some significant heating within the core at a radius of ap- the dark matter to gas enclosed mass ratios in the bottom
proximately 0.05 pc. It should also be noted here that we rightpanel.InthiscaseweagainseethatS6646isdarkmat-
do not include the effects of radiation self-shielding in our ter dominated and that S830 is only marginally gas domi-
simulationswhichmaybeimportantinthecoreofthehalo. nated.Thehigherresolutionsimulations(S103,S8andS06)
The right hand column yields a partial explanation for on the other hand are all strongly gas dominated. Again in
this. Considering first the non-smoothed runs in the top this case S6646 and S830 should be treated with caution
right panel - in both S830 and S6646 the core is dark mat- (even though S830 shows converged results).
ter dominated. The right hand panel plots the ratio of the Regarding the temperature dichotomy within the core
enclosed dark matter mass to the enclosed gas mass. Val- seen in the left hand column, the result is not as bad as it
uesgreaterthan1.0 indicateregions wherethedarkmatter firstappearsandreflectsthetimeatwhichwechoosetoex-
dominates compared to the gas. In simulations S830 and aminethecore. ExcludingS830 andS6646 weseethatS06,
S6646theregioninsideapproximately1parsecisdarkmat- S8andS103allagreequitewell(withthepossibleexception
ter dominated (in S6646 a single dark matter particle sits of the smoothed S8 simulation which shows a bump at ap-
(cid:13)c 0000RAS,MNRAS000,000–000
8 J.A. Regan et al.
proximately 0.05 pc). However, allowing the simulations to Equation 4 can then be used to determine the radius at
runatthemaximumrefinementlevelforashorttimelonger which,foragivendarkmatterparticlemass, thedarkmat-
resultsinthecoreheatingupineachcase.Thisresultsfrom ter(under-)resolutioncanbegintoinfluencethebaryontem-
thecollisional dissociation ofH2 whichpushesthecoreonto perature. Wewill return to this point in §4.3.
theatomiccoolingtrackonceagain-thisisshownintheleft If we compare the non-smoothed simulations to the
panelofFigure9anddiscussedinmoredetailinAppendixA smoothed ones when theradiation field isset to80 J21 (i.e.
(as it lies somewhat outside thescope of thecurrent work). upperandlowerrows ofFigure3)weseethatthetempera-
Wefurthercautionthereaderthatrunningatthemaximum ture profiles in the left hand panels both show a significant
refinementlevelmeansweareartificallypreventingcollapse scatter.Themassratiosarequalitativelysimilar(righthand
which may induce unphysical results and that further re- panels) in both the smoothed and non-smoothed runs with
search needs to be done to examine this scenario in more thehigherresolution(S103,S8andS06)wellconvergedbut
detail. withS6646andS830showingsignificantscatterbetweenthe
Strongconvergencebetween S06, S8and S103is there- smoothed and non-smoothed runs. For example, the S830
foreachieved,whentheexternalfieldissetto80J21,butthe runinthesmoothedsimulationismarginallygasdominated
studyindicatesthatthelowerresolutionsimulationsdissoci- whileinthenon-smoothedcaseS830isstronglydarkmatter
atetheH2 faster compared to thehigher resolution simula- dominated. The overall effect of smoothing when the radi-
tions, driven by spurious dark matter effects, which impact ation field is set to 80 J21 is therefore marginal, the higher
thehydrodynamics. resolution simulations are already well converged and so
In Figure 4 we show the results when the background smoothingonlyimprovestheconvergenceslightly.Thelower
radiationissetto500J21 withallotherparametersremain- resolution runs are dark matter dominated and smooth-
ing unchanged. In this case the external radiation field is ingcannot preventthisalthough smoothingmay somewhat
strongenoughtoreducetheH2 fractionbyseveralordersof lessen theadverse effects.
magnitudewithin thehalo. The left handcolumn of Figure In comparing the cases where the external radiation
4 shows the temperature profiles which show highly con- field is stronger with J set to 500 J21 the effects are only
verged results for both the non-smoothed (top panel) and slightly more encouraging. In the 500 J21 case the gas is
smoothed (bottom panel) for even the lowest dark matter warmerandhencethesoundspeediscorrespondinglyhigher
resolutions. The right hand column shows the dark mat- meaningthatthediscretenesseffects,quantifiedbyV are
disc
ter to gas enclosed mass ratio. The S6646 simulations are somewhat lessened.This hastheeffectthat forthetemper-
dark matter dominated in both cases (and yet the temper- ature profiles the results appear completely converged as
ature profiles remain converged) but theS830 simulation is the effects of the radiation overwhelm any spurious dark
moregasdominatedinthesmoothed simulationscompared matter effects. Nonetheless in both the 80 J21 case and the
to the non-smoothed case. The higher resolution runs also 500 J21 case the S6646 simulation is strongly dark matter
show stronger convergencewhen smoothed. dominated and smoothing thedark matter ontothegrid at
this stage only alleviates slightly this particular ill effect.
However, in the case of S830 we observe that while in the
4.2 Halo A - Comparing Smoothed and
non-smoothed case the core is marginally gas dominated it
non-Smoothed Simulations
is significantly gas dominated in the smoothed case - al-
The effect of smoothing the particles onto the grid is most though the ratio remains above 0.1. Finally similar to the
noticeable in the low resolution simulations (i.e. S6646 & 80J21 casethesimulationswithaparticlemassMDM .103
S803) while the effects become less pronounced once the M⊙ are relatively well converged in boththesmoothed and
darkmatterparticlemassdropsorequivalentlyoncethedis- non-smoothed cases although thesmoothed case does show
cretisationvelocity,V ,becomeslessthanthelocalsound less deviation at thesmallest scales.
disc
speed (equal to a few km/s at these temperatures). We de- Therefore,theresultisthatsmoothingtendstoimprove
fineV roughlyasthevelocitythatadarkmatterparticle convergence only in isolated cases or when the dark matter
disc
can impart ontoits surroundings resolution is already high. However, smoothing poorly re-
solved dark matter particles onto the grid will likely have
V = GMDM (2) no positive effect and it is therefore more important to ad-
disc
r Rs equatelyresolve agivenregion with asufficiently high dark
whereGistheusualgravitationalconstant,MDMisthemass matter mass resolution and then to smooth the particles
of a single dark matter particle and Rs is the smoothing onto the grid to remove any lingering discrete effects from
length. This relationship can also be recast in terms of a thedark matter particles.
discretisation temperature resulting in theequation
Tdisc= µmHGMDM (3) 4.3 AHmaloplAitu-dCehanging the External Field
r Rs kb
where µ is the mean molecular weight, set to be 1.22 in Sofar we haveselected only external fields of 80 J21 or 500
thisstudy,andkb istheBoltzmannconstant.Thisequation J21. These fields are relevant for investigating the case of
can be furtherexpanded by replacing RS with max(R, RS) a collapsing halo exposed to a nearby strong UV source as
where R is any radius becoming mayberequiredinthedirectcollapseformationmodel(e.g.
Dijkstra et al. 2008; Visbal et al. 2014). We now turn our
T (R)= µmHGMDM (4) attention to investigating the effects of other external field
disc
smax(R,RS) kb amplitudes which may be expected at this redshift. In par-
(cid:13)c 0000RAS,MNRAS000,000–000
9
-2 -1 0 1 2 3 -2 -1 0 1 2 3
10 10 10 10 10 10 10 10 10 10 10 10
)
K 101
(
e 103 0
r 10 o
u
i
t t
a -1 a
er 0 J21, z = 33.04 10 R
p 102 1 J , z = 30.90
21
m Halo A 10 J , z = 25.43 10-2
21
Te h = 0.04 pc 80 J21, z = 23.43
MDM=103M 500 J21, z = 23.00 10-3
104 ⊙
)
K 101
(
e 103 0
r 10 o
u
i
t t
a -1 a
er 0 J21, z = 32.31 10 R
p 102 1 J , z = 30.41
21
m Halo A 10 J , z = 25.35 10-2
21
Te h = 0.04 pc 80 J21, z = 23.50
MDM=8M 500 J21, z = 22.92 10-3
104 ⊙
-2 -1 0 1 2 3
10 10 10 10 10 10
)
K 101
(
e 103 0
r 10 o
u
i
t t
a -1 a
er 0 J21, z = 31.79 10 R
p 102 1 J , z = 28.96
21
m Halo A 10 J , z = 25.13 10-2
21
Te h = 0.04 pc 80 J21, z = 23.46
MDM=0.6M 500 J21, z = 22.94 10-3
⊙
-2 -1 0 1 2 3 -2 -1 0 1 2 3
10 10 10 10 10 10 10 10 10 10 10 10
Radius (pc) Radius (pc)
Figure 5. Halo A: Each row shows the temperature profile and the ratio of the dark matter enclosed mass to the baryon enclosed
mass. The y-axis label is simply identified as “Ratio” for convenience. The top row shows the case where the particle mass is set to
MDM = 103 M⊙ and no smoothing is employed. The dark matter mass resolution then varies between each row with every other
parameter remainingunchanged. In each individual panel the external field varies between 0J21 and 500 J21. The dotted lineinthe
lefthandpanelsisthe“discretisationtemperature”asdefinedinequation4.Atagivenscaleitshowstheeffectadarkmatterparticle
ofagivenmasscanhaveonthebaryontemperature.
ticular,wechoosethreefurtherfieldamplitudesof0J21(i.e. that the simulations with lower field amplitudes (0 J21, 1
nobackground radiation), 1 J21 and 10 J21.To test for any J21 and 10 J21) all show a strong temperature increase to-
discreteness effects from under-resolved dark matter parti- wards the core of the halo. As the mass resolution is in-
cles we do not smooth the particles onto the grid so the creased (middle and bottom rows) this phenomenon disap-
smoothing length is set to the cell size of the finest grid - pearsandthetemperatureprofilesconvergetotemperatures
0.04 comoving parsecs in each case. close to T ∼ 1000 K as would be expected for the forma-
In Figure 5 we have plotted the radial profiles of the tion of metal-free stars at this redshift (Abelet al. 2002;
temperatureandthedarkmattermasstogasratiowiththe Yoshidaet al. 2006, 2008; Bromm et al. 2009; Smith et al.
mass resolution increasing between each panel as indicated 2009;Turk et al.2009;Stacy et al.2010;Clark et al.2011a;
in theright hand column of thefigure. Stacy & Bromm 2014).
Examining first the top panel where the dark mat- In the lowest mass resolution simulations this spurious
ter particle resolution is set to MDM = 103 M⊙ we see behaviour can be explained by examining the dark matter
(cid:13)c 0000RAS,MNRAS000,000–000
10 J.A. Regan et al.
-2 -1 0 1 2 -2 -1 0 1 2
10 10 10 10 10 10 10 10 10 10
4 4
10 10
) )
K K
( (
e e
r 103 103 r
u u
t t
a a
r r
e e
p 102 0 J21, z = 33.47 0 J21, z = 32.61 102 p
m 1 J , z = 30.52 1 J , z = 30.09 m
21 21
e 10 J21, z = 25.63 Halo B 10 J21, z = 24.86 e
T 80 J21, z = 23.39 h = 0.04 pc 80 J21, z = 23.40 T
500 J21, z = 23.06 MDM=103M MDM=0.6M 500 J21, z = 23.09
⊙ ⊙
4 4
10 10
) )
K K
( (
e e
r 103 103 r
u u
t t
a a
r r
e e
p 102 0 J21, z = 32.11 0 J21, z = 32.04 102 p
m 1 J , z = 29.86 1 J , z = 29.85 m
21 21
e 10 J21, z = 26.98 Halo C 10 J21, z = 26.76 e
T 80 J21, z = 24.24 h = 0.04 pc 80 J21, z = 24.23 T
1 500 J21, z = 23.20 MDM=103M MDM=0.6M 500 J21, z = 23.15 1
10 ⊙ ⊙ 10
-2 -1 0 1 2 -2 -1 0 1 2
10 10 10 10 10 10 10 10 10 10
Radius (pc) Radius (pc)
Figure 6.Halo B & Halo C:Thetemperature profileforhaloes B & Careshown fortwo differentdarkmatter particleresolutions.
HaloBis showninthe toprow whilehaloC isdisplayedinthe bottom row.Theparticleresolutionissetat MDM = 103M⊙ inthe
lefthandcolumnwhileitissettoMDM =0.6M⊙ intherighthandcolumn. Thedotted lineineachpanel isouranalytical estimate
ofthediscretisationtemperature, Tdisc(R).
to gas ratio in the right hand column. In the lowest mass similar effect isseen in themiddleleft panelwherethepar-
resolution simulations (MDM = 103 M⊙, top right panel) ticlemassissetto(MDM =8M⊙).Inthebottomleftpanel
thesimulations withexternalradiation amplitudesof0J21, the Tdisc(R) function does not intersect any of the temper-
1 J21 and 10 J21 are all dark matter dominated (or only ature profile lines. This indicates that at this particle res-
mildly gas dominated) within the core of the halo resulting olution we would not expect any impact from dark matter
in spurious heating of the gas due to the presence of large heating effects at the spatial scales probed in our simula-
darkmatterparticles(wesawthisbeforeinFigure3forex- tions and this is exactly what we see.
ample). As we increase the dark matter particle resolution Takingthefieldstrengthof1J21 asthefiducialcasewe
we see this behaviour disappear as the dark matter to gas findthatthecore(<1pc)ofthesimulationwasresolvedby
ratio drops significantly and the temperature profiles con- only 36 dark matter particles (with a V ∼18 km/s and
disc
verge to T ∼1000 K as expected. Tdisc∼47000 K)whenthemassresolution wassettoMDM
In the temperature panels (left hand panels) we have = 103 M⊙. When the resolution was increased to MDM =
also overplotted our analytical estimation of the discreti- 8 M⊙ the number of dark matter particles in the core in-
sation temperature, T (R), (see equation 4). This gives creased to 473 particles (V ∼ 5 km/s and T ∼ 3600
disc disc disc
the scale at which we expect thedark matter particle mass K)andincreasingtheresolutiontoMDM =0.6M⊙ resulted
to start to impact (negatively) on the temperature. In the inthecorebeingresolvedby3072particles(V ∼1km/s
disc
top left panel we see that the large dark matter particle and T ∼300 K).
disc
mass used here (MDM = 103 M⊙) means that the Tdisc(R) Thesmallerhaloes,thosewithagascorelessthan∼104
function intersects the 0 J21 profile very close to where the M⊙,whichcollapsewhentheexternalfieldamplitudeisrel-
baryon temperature starts to show a spurious increase. A atively small require a much higher dark matter resolution.
(cid:13)c 0000RAS,MNRAS000,000–000