arXiv:astro-ph/9710125v1 13 Oct 1997
Environmental Influences on Dark Matter Halos and
Consequences for the Galaxies Within Them
Gerard Lemson1& Guinevere Kauffmann2
1Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel
2Max-Planck Institut f¨ ur Astrophysik, D-85740 Garching, Germany
We use large N-body simulations of dissipationless gravitational clustering in cold
dark matter (CDM) cosmologies to study whether the properties of dark matter halos are
affected by their environment. We look for correlations between the masses, formation
redshifts, concentrations, shapes and spins of halos and the overdensity of their local
environment. We also look for correlations of these quantities with the local tidal field.
Our conclusion is extremely simple. Only the mass distribution varies as a function of
environment. This variation is well described by a simple analytic formula based on
the conditional Press-Schechter theory. We find no significant dependence of any other
halo property on environment. Our results do not depend on our choice of cosmology.
According to current hierarchical models, the structure and evolutionary history of a
galaxy is fully determined by the structure and evolutionary history of the dark halo in
which it is embedded. If these models are correct, clustering variations between galaxies
of differing morphological types, luminosities, colours and surface brightnesses, must all
arise because the halo mass function is skewed towards high mass objects in overdense
regions of the Universe and towards low mass objects in underdense regions.
Keywords: galaxies:haloes; galaxies:formation; galaxies:evolution; cosmology:theory; cosmol-
It is well known that many of the observed properties of galaxies correlate with their envi-
ronment. The most famous of these correlations is the morphology-density relation. Davis
& Geller (1976) showed that elliptical galaxies are more strongly clustered on the sky than
spirals, and Dressler (1980) demonstrated that the elliptical fraction in galaxy clusters is an
increasing function of local density. The dependence of the clustering of galaxies on luminosity
has been much more controversial, but recent analyses of the largest available redshift surveys
confirm that L∗and brighter galaxies have a higher clustering amplitude than low-luminosity
galaxies (Park et al 1994; Loveday et al 1995; Benoist et al 1996; Valotto & Lambas 1997).
Low surface brightness galaxies are also distributed differently from their high surface bright-
ness counterparts. They are less clustered on all scales (Mo, McGaugh & Bothun 1994) and
are particularly isolated on scales smaller than a few Mpc (Bothun et al 1993). Finally, the
star formation histories and the gas fractions of galaxies of nominally the same Hubble type
are influenced by their local environments. Spiral galaxies in dense environments have redder
colours and lower star formation rates that spirals in the field, and often also exhibit truncated
HI disks ( e.g. Kennicutt 1983; Cayette et al 1994).
One popular hypothesis for the origin of clustering differences between galaxies of different
types is that mergers, tidal encounters or interactions with a surrounding gaseous medium
modify galaxy properties. Such interactions are more probable in high-density environments.
It should be noted, however, that differences in the clustering amplitude of galaxies of different
morphologies, luminosities and surface brightnesses persist out to scales where the crossing
time is much larger than the Hubble time (Mo et al 1992; Mo, McGaugh & Bothun 1994),
so it does appear that galaxies bear some imprint of the initial field of density fluctuations in
According to the standard cosmological paradigm, structure in the Universe is built up
through a process of hierarchical clustering. Small-scale fluctuations in the initial density
field are the first to collapse to form bound, virialized objects (or dark matter halos) and these
merge together over time to form more and more massive systems. Galaxies form when gas
cools, settles, and turns into stars at the centres of the halos (White & Rees 1978; White
& Frenk 1991). Galaxy formation models of this type have met with considerable success in
explaining many of the trends seen in the properties of galaxies, both at present day and at
high redshift ( e.g. Lacey et al 1993; Kauffmann, White & Guiderdoni 1993; Cole et al 1994).
Their most serious weakness arises from the fact that star formation and supernova feedback
processes are poorly understood and hence difficult to model in a convincing way.
It should be noted, however, that one fundamental aspect of this picture is that the struc-
ture and evolutionary history of a galaxy is fully determined by the structure and evolutionary
history of its surrounding dark matter halo. The merging history of the halo determines the
rate at which gas will cool and become available for star formation, as well as the frequency
of merging events and the mass distribution of accreted objects. The density profile of the
halo, its shape and its distribution of angular momentum determine the structure, the size
and the rotation curve of the galaxy that forms at its centre (Dalcanton, Spergel & Summers
1997; Mo, Mao & White 1997). If the properties of galaxies are observed to vary as a function
of environment, it follows that the properties of their surrounding halos must also vary with
environment. By understanding exactly which halo properties can be affected by local condi-
tions, one can hope to gain a deeper understanding of the origin of the clustering differences
In this paper, we use numerical simulations of gravitational clustering to study the proper-
ties of dark matter halos as a function of local density. In order to achieve both good statistics
and an accurate treatment of the formation history and internal structure of halos, high reso-
lution simulations are needed, which nevertheless contain a fair sample of the Universe, thus
accounting correctly for the influence of large-scale structure on the formation of the halos.
We look for correlations between a set of present-day halo properties, including their masses,
formation redshifts, concentrations, shapes and spins, with the overdensity of their local en-
vironment. We also look for correlations of these quantities with the surrounding tidal field.
Our conclusion is extremely simple. Only the mass distribution of dark halos varies as a
function of environment. The variation in the mass function is described extremely well by
a simple analytic formula based on the conditional Press-Schechter theory derived by Mo &
White (1996). We find no significant dependence of any other halo property on environment.
Our results do not depend on our choice of cosmology. This leads to the conclusion that the
dependences of galaxy morphology, luminosity, surface brightness and star formation rate on
environment, must all arise because galaxies are preferentially found in higher mass halos in
overdense environments and in lower mass halos in underdense environments.
2 The simulations
The GIF project is a joint effort of astrophysicists from Germany and Israel. Its primary
goal is to study the formation and evolution of galaxies in a cosmological context using semi-
analytical galaxy formation models embedded in large high-resolution N-body simulations.
This is done by constructing merger trees of particle halos from dark-matter only simulations
and placing galaxies into them using a phenomenological modelling (for a detailed description
of this procedure as well as results cf. Kauffmann et al. 1997).
The code used for the GIF simulations is called Hydra. It is a parallel adaptive particle-
particle particle-mesh (AP3M) code (for details on the code cf. Couchman, Thomas, & Pearce
1995; Pearce & Couchman 1997). The current version was developed as part of the VIRGO
supercomputing project and was kindly made available by them for the GIF project. The
simulations were started on the CRAY T3D at the Computer Centre of the Max-Planck
Society in Garching (RZG) on 128 processors. Once the clustering strength required an even
larger amount of total memory, they were transferred to the T3D at the Edinburgh Parallel
Computer Centre (EPCC) and finished on 256 processors.
A set of four simulations with N = 2563and with different cosmological parameters was
run. In this paper we focus on a variant of a cold dark matter model, τCDM, with Ω0= 1,
h = 0.5, σ8= 0.6 and shape parameter Γ = 0.21. A value of Γ = 0.21 is usually preferred
by analyses of galaxy clustering, cf. Peacock & Dodds (1994). This is achieved in the τCDM
model despite Ω0 = 1 and h = 0.5 by assuming that a massive neutrino (usually taken to
be the τ neutrino) had existed during the very early evolution of the Universe and came to
dominate the energy density for a short period. It then decayed into lighter neutrinos which
are still relativistic, thus delaying the epoch when matter again started to dominate over
radiation. The neutrino mass and lifetime are chosen such that Γ = 0.21. For a detailed
description of such a model see White, Gelmini, & Silk (1995). The normalization was chosen
so as to match the abundance of rich clusters (White et al 1993). The simulation box size was
85 h−1Mpc and the particle mass was 2×1010M⊙. In order to be sure that our results are not
sensitive to our choice of cosmological initial conditions, we have also analyzed a low-density
CDM model with Ω0= 0.3, ΩΛ= 0.7, Γ = 0.21, h=0.7 and σ8= 0.9 (ΛCDM). This simulation
has a somewhat larger box size (141 h−1Mpc), but the same particle mass.
Halos were selected from the z = 0 output times in the simulations as follows. First, we
searched for high-density regions using a standard friends-of-friends groupfinder with a linking
length of b = 0.2 times the mean interparticle separation. We then searched for the particle
with the lowest potential energy and adopted its position as the halo centre. The distances
of all the particles to the centre were ordered and the radius of the largest sphere with an
overdensity δ ≥ 200 was defined to be the virial radius of the halo, and the mass contained
within this radius was defined to be the virial mass. Halos were only included in our analysis
if the friends-of-friends mass exceeded 70 particles and the virial mass was between 35% and
95% of the friends-of-friends mass. The upper bound served as a check that the majority
of particles at overdensities greater than ≃ 200 were located by the groupfinder. The lower
bound was chosen to ensure that enough particles were located within the virialized region so
that the various halo quantities could be calculated reliably.
The following quantities were evaluated for each halo:
1. The spin parameter λ = LE1/2/GM5/2, calculated for the particles within the virial
radius, where L and E are the angular momentum and the thermal kinetic energy of
2. The formation redshift zform, defined as the redshift at which the most massive halo
progenitor was half the present-day mass of the halo.
3. Concentration indices c10= r10/rvirand c20= r20/rvir, where r10and r20are the radii
enclosing one tenth and one twentieth of the virial mass respectively. These parameters
provide a coarse measure of the shape of the density profile of the halo.
4. The shape of the halo was determined by diagonalizing the moment of inertia tensor of
halo particles within the virial radius. This gives the principal axes p1≥ p2≥ p3.
The properties of the local environment around each halo were determined as follows. We
evaluated the overdensity δ in spheres of various radii surrounding the halo. We also calculated
overdensities in shells around the halo, thereby excluding the contribution of the halo mass
In order to obtain a measure of the higher order harmonics of the surrounding dark matter
distribution, a density field on a 1283grid (cell size 0.66 h−1Mpc) was obtained from the
simulation using cloud-in-cell (CIC) interpolation. The resulting field was smoothed using a
top-hat smoothing window of 10 h−1Mpc and the first and second-order spatial derivatives
of the density and potential fields were determined using fast Fourier transforms in k-space.
We made use of the following k-space window functions:
• density dipole: iki
• density shear field: −kikj
• potential dipole field (acceleration): −iki/k2
• potential shear field (tidal field): kikj/k2
The resulting fields were interpolated to the positions of the halos using the CIC method.
The shear fields were diagonalized and the eigenvalues ordered.
4.1The mass function of halos versus environment
In an extension of the Press-Schechter theory, Bond et al (1991) and Bower (1991) derive
an expression for the fraction of mass in a region of initial radius R0and linear overdensity
δ0, which at redshift z1is contained in halos of mass M1. If one assumes that the region R0
evolves in size and overdensity according to the spherical top-hat collapse model, it is simple
to derive a formula for the mass function of halos in a present-day region of radius R and
overdensity δ. This was first done by Mo & White (1996). They tested their analytic formula
against results from N-body simulations containing 1283particles.
In figure 1, we show how the shape of the halo mass function changes as a function of local
overdensity δ evaluated within spheres of R= 10 h−1Mpc. The thick solid line shows results
derived from the τCDM simulation at z = 0. The thin solid line shows the analytic prediction
of Mo & White (1996). For comparison, the thick dashed line is the average halo mass
function evaluated from the simulation, multiplied by a factor (1 + δ). The thin dashed line
is the same quantity calculated from the Press-Schechter theory. For clarity, the simulation
and the analytic curves have been offset by 1 decade in the y-direction.
Although it is not apparent as a result of the way we have chosen to present our results,
it should be noted that the halo abundances predicted by the analytic theory exceed those
derived from the simulation by a factor ≃ 1.5 over the range of halo masses shown in figure
1. The magnitude of this offset is in rough agreement with that found by Lacey & Cole
(1994) in their tests of the Press-Schechter formalism. Nevertheless, as can be seen in the
plot, the theory does predict the change in shape of the halo mass function with δ remarkably
well. As can be seen, in low density regions, high-mass halos are underepresented, whereas in
high-density regions the situation is reversed and high-mass halos are overabundant.
4.2 Formation times, Spins, Concentrations and Shapes of Halos
Our principal result is that we find no correlation between the environment of halos and
their formation times, spins, concentrations and shapes in either the τCDM or the ΛCDM
simulations. Out of a large number of possible scatterplots, we have selected the following
few to illustrate this conclusion. Results are shown only for the τCDM model, but plots for
ΛCDM are virtually identical.
Figure 1: The halo mass function in spheres of radius R=10 h−1Mpc and local overdensity
δ. The thick solid line shows results derived from the τCDM simulation at z = 0. The thin
solid line shows the analytic prediction of Mo & White (1996). For comparison, the thick
dashed line is the average halo mass function evaluated from the simulation, multiplied by a
factor (1+δ). The thin dashed line is the same quantity calculated from the Press-Schechter
theory. For clarity, the simulation and the analytic curves have been offset by 1 decade in the
In figure 2 we plot the value of the spin parameter λ against the overdensity in top-hat
spheres of radius 10 h−1Mpc for halos in four different mass bins. This figure shows that the
mean value of λ varies at most very weakly with overdensity or with mass. There does appear
to be a slight tendency for λ to increase with overdensity. These results do not change if we
adopt a different value of the smoothing radius.
In figure 3 we plot formation redshift against the overdensity in a shell between 2 h−1and
5 h−1Mpc surrounding each halo. No dependence of mean formation redshift on overdensity
is seen. Higher mass halos form somewhat later on average, as expected (Lacey & Cole 1993).
Figures 2 and 3 demonstrate that the average spin and formation time of halos are not
correlated with their environments. The next step is to demonstrate that the full probability
distributions are also independent of local overdensity. This is shown in figure 4, where we
plot P(λ) and P(zform), for three different ranges in overdensity δ. As can be seen, the curves
are almost indistinguishable from each other. Again, there is perhaps a slight shift towards
larger spins in overdense regions. The distribution of λ is consistent with that found in earlier
work (e.g. Barnes & Efstathiou 1987).
Finally, figure 5 is a mosaic illustrating a further subset of the possibilities we have explored.
In the two left panels, we show the concentration index c10versus the overdensity evaluated
in spheres of radius 10 h−1Mpc surrounding each halo, and the shape axis ratio p1/p3versus
the tidal field axis ratio (φ,11/φ,33). In the two right panels, we plot the spin parameter and
the formation redshift versus principal axes ratios of the tidal field and the density shear
field respectively. (Note that the apparent gaps in the distribution of λ and p1/p3at small
positive values of φ,11/φ,33result from the ordering of the eigenvalues and the values that the
φ,iiassume in practice.) Once again, no correlations are found, either in the mean values of
the halo quantities, or in their full probability distributions. It is interesting that neither the
spins, nor the distribution of shapes of dark halos, depend on the surrounding tidal field.
5Discussion and Conclusions
In this paper, we have demonstrated that mass is the only halo property that correlates sig-
nificantly with local environment. The dependence of the halo mass function on environment
can be understood in terms of a simple extension of the Press-Schechter theory.
It is interesting to note that the fact that halo formation histories do not depend on
local overdensity, is a prediction of the extended Press-Schechter theory in the excursion set
formulation of Bond et al (1991). In this approach, the standard formulae are obtained by
assuming that the mass of a halo to which any given mass element belongs, can be followed
by studying the behaviour of the initial linear density field as it is smoothed with a succession
of sharp k-space filters. In this model, the history of each mass element of a halo of mass M
(and thus the formation history of the halo itself) is statistically independent of the future
of the element and thus of the halo’s environment. Navarro, Frenk & White (1996) have
demonstrated that the density profile of dark matter halos in CDM-like cosmologies has a
“universal” form, and that the characteristic density of a halo is related simply to its formation
time. It is thus also not surprising that our concentration parameters c10and c20turn out
to be independent of environment. There has also been substantial analytic work concerned
with the angular momentum generated by tidal torques acting on local density maxima in the
Figure 2: The spin parameter λ is plotted against overdensity in spheres of radius R=10 h−1
Mpc for halos in different mass ranges, indicated on each panel as the number of particles
found within the virial radius. The solid line shows the mean value of log(λ) as a function of
δ. Dashed lines indicate the 1σ standard deviation.
Figure 3: The formation redshift is plotted against overdensity in the shell between 2 and 5
h−1Mpc for halos in different mass ranges. For an explanation of the lines, see figure 2.
regions of different overdensity δ. δ is calculated in a 10 h−1Mpc sphere for the left-hand
panel, and in a 2-5 h−1Mpc shell for the right-hand panel. The thick solid line in both panels
represents the full probability distribution. In the left panel, the short dashed, thin solid, and
long-dashed lines are for −1 < δ < −0.1, −0.1 < δ < 0.3, and 0.3 < δ. In the right panel, the
short dashed, thin solid, and long-dashed lines are for −1 < δ < −0.1, −0.1 < δ < 0.6, and
0.6 < δ. Bins in δ have been chosen so that they contain an equal number of haloes.
The probability distribution of the spins and the formation redshifts of halos in
The top left panel shows the concentration index (c10 = R10/Rvir) as a function of δ in a
sphere. The bottom left panel shows the relationship between the shape axis ratio p3/p1and
the the ratio of the first and third eigenvalues of the tidal field. In the top right panel, the spin
parameter is plotted against the ratio of the first and third eigenvalues of the tidal field. In
the bottom right panel, formation redshift is plotted against the ratio of the first and second
eigenvalues of the density shear field. Only halos in the mass range 500-5000 are shown.
A mosaic of four different correlations between halo properties and environment.
linear density field. It has been argued (Blumenthal et al 1984; Hoffman 1986) that the height
of a peak is anticorrelated with its angular momentum, on the basis that high peaks collapse
early, so there is not much time for tidal torques to act. If this was the case, overdense
regions of the Universe, in which high peaks are more probable, should harbor halos with
systematically lower angular momentum. Heavens & Peacock (1988) pointed out that this
effect is counterbalanced by the fact that higher peaks experience greater tidal torques. For
realistic power spectra, they estimate that the two effects should nearly cancel out. Steinmetz
& Bartelmann (1995) obtain a similar result by modelling the formation of a halo as a collapse
of a homogeneous ellipsoid acted upon by tidal shear from the surrounding matter. Both
analyses are consistent with the very weak environment-dependence of spin in our data. The
shapes of dark matter halos have proven more difficult to calculate using analytic methods.
Dubinski (1992) has investigated halo shapes in high-resolution N-body simulations and finds
that there is no relationship between the shape of the initial density peak and the shape of
the final collapsed halo. On the other hand, West, Villumsen & Dekel (1991) and Tormen
(1997) have demonstrated a clear tendency for the major axis of cluster-sized halos to align
with larger-scale structures.
The main astrophysical implication of our conclusion is that clustering differences between
galaxies of different types must arise purely because different galaxies sample different mass
halos. Semi-analytic models of galaxy formation in hierarchical cosmologies demonstrate that
it is possible to explain many of the clustering trends seen in the data in this way (Kauffmann,
Nusser & Steinmetz 1997). In these models, elliptical galaxies form when disk galaxies of
comparable mass merge with each other. These mergers occur preferentially in groups at
redshifts ≃ 1. The groups then coalesce to form clusters and superclusters. As shown by
Kauffmann, White & Guiderdoni (1993) and Baugh, Cole & Frenk(1996), the fraction of
ellipticals increases strongly with halo mass in this picture and ellipticals thus end up more
clustered than spirals. In addition, the luminosity of the central galaxy in a halo scales in
proportion to the halo mass, simply because more gas is able to cool and form stars in more
massive halos. Luminosity-dependent clustering is thus a natural outcome of the models.
Finally, it should also be noted that galaxies are assumed to be stripped of their surrounding
dark matter once they have been accreted by a larger system, such as a group or cluster. They
thus lose their supply of new cold gas, their star formation rates decline, and their stellar
populations redden and fade. Red, gas-poor galaxies are thus found predominantly in high
mass halos and are again predicted to be more clustered. The semi-analytic models at present
provide no explanation for the clustering differences between galaxies of different surface
brightnesses. One might speculate that low surface-brightness disks are more fragile and are
more easily destroyed in high-density environments. More detailed modelling is necessary,
however, before a conclusion can be drawn as to whether this will work in practice.
We thank Simon White, Adi Nusser and Joe Silk for helpful discussions.
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