A universal model for halo concentrations
Benedikt Diemer
1,2,3
and Andrey V. Kravtsov
2,3,4
ApJ 799, 108 • arXiv 1407.4730 • python code at benediktdiemer.com/code
A universal model
Our model differs from all other models in the literature. For example, our
model predicts an upturn at high masses (contrary to models based on
halo samples where unrelaxed halos are excluded), and very low
concentrations where the power spectrum is steep.
10
4
4
4
3
1
2
⌫200c
3
4
cvir
3
1
2
3
c200c
0.1
1
2
0.0
3
4
0.1
1010 1011 1012 1013 1014 1015
M200c (h 1 M )
c200m
4
1
⌫vir
2
3
4
⌫200m
What parameters influence concentration?
Given the remaining non-universality in the c-ν relation, we are looking for a
second parameter besides ν that affects concentration. When comparing the
c-ν relation in two self-similar cosmologies with Ωm=1 and power-law spectra
with slopes n=-2 and n=-2.5 (maroon and pink lines in the top right figure).
Clearly, n has a large impact on c at fixed ν. The ΛCDM relations lie in between
the n=-2 and n=-2.5 curves, roughly the range of n in a ΛCDM power
spectrum. Thus, we postulate that n is a second parameter influencing
concentration, and that concentration can be expressed as a universal
function of only peak height and the power spectrum slope.
Our model makes some noteworthy predictions. First, c increases at large
peak heights (see also Prada et al. 2012). This upturn is due to unrelaxed halos
(Ludlow et al. 2012), meaning
This work, z = 30
models based on halo samples
This work, n = 3.5
where unrelaxed halos were
Bullock + 2001, z = 30
Zhao + 2009, z = 30
excluded do not show this upturn
Prada + 2012, z = 30
6
(e.g., Correa et al. 2015).
Simulation, z = 26 (Diemand + 2005)
Second, out model predicts that
concentration decreases for very
steep power spectra, without a
particular
floor
value.
This
prediction can be tested in Earthmass halos at z~30 (blue lines in
the figure to the right). Such halos
have been shown to have
extremely low concentration, in
agreement with our model.
1) ITC Fellow, Harvard-Smithsonian Center for
Astrophysics; [email protected]
2) Department of Astronomy and Astrophysics,
The University of Chicago
3) Kavli Institute for Cosmological Physics
4) Enrico Fermi Institute, The University of Chicago
Simulation, z = 31 (Anderhalden + 2013)
Simulation, z = 32 (Ishiyama 2014)
c200c
5
c200c
0.1
Model predictions at high masses and redshifts
10
5
3
3
0.5
4
5
3
0.1
3
= 0.010
= 0.5
= 1.0
= 1.5
= 2.0
= 4.0
= 6.0
4
0.0
2
⌫200c
4
15
5
1
⌫200c
c
c
10
= 0.015
= 0.5
= 1.0
= 1.510
= 2.0
= 4.0
= 6.0
c
z
z
z
z
z
z
z
15
c/cfit
Concentration exhibits a non-trivial dependence on mass, redshift, and
cosmological parameters. However, it has been shown that the relation
between c and peak height, ν, is almost universal with redshift (e.g., Prada et
al. 2012). The figure below shows the c-ν relation for three definitions of the
virial radius. The c200c-ν relation is most universal with redshift, while the other
definitions, cvir and c200m, experience a spurious evolution at low redshift,
namely when dark energy becomes important and ρm or ρc diverge.
1
Are concentrations universal?
3
z
z
z
z
z
z
z
5
5
c200c
We show that these deviations can be explained by a residual dependence
of concentration on the local slope of the matter power spectrum, n. We
construct a universal function of only two variables, c200c(ν,n), that
accurately describes concentrations across 22 orders of magnitude in mass
and over a wide range of redshifts and cosmologies.
= 0.0
= 1.0
= 2.0
= 4.0
= 6.0
4
1
Analyzing a large suite of cosmological simulations, we find that, at fixed ν,
the inner density profiles are most universal across redshift in units of R200c.
This implies that the relation between c and ν is most universal when c is
defined as c200c=R200c/rs. However, sizeable deviations from universality in
the c200c-ν relation remain.
⇤CDM, z
⇤CDM, z
⇤CDM, z
⇤CDM, z
⇤CDM, z
n = 2.0
n = 2.5
c/cfit
Numerous theoretical arguments and simulation results indicate that the
density profiles of halos (and hence their concentrations) are closely related
to their mass assembly history which is expected to be a universal function
of peak height, ν=δc/σ(M). Thus, we expect that concentration, c, should be
close to universal, i.e. independent of redshift and cosmology, at fixed ν.
10
We fit our simulation results with
a function c200c(ν,n), namely a
double
power
law
whose
normalization and location of the
minimum vary with n. With only
seven free parameters, this model
fits ΛCDM concentrations to
about 5% accuracy from z=0 to
z=6 (figure below; the solid lines
show simulation results, the
dashed lines our model). We
provide the python code colossus
to evaluate our model for
arbitrary cosmology, halo mass,
redshift, and mass definition.
c200c
Abstract
4
2
0
1
2
3
⌫
References
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§
§
Correa et al. 2015, MNRAS 452, 1217
Diemer & Kravtsov 2015, ApJ 799, 108
Ludlow et al. 2012, MNRAS 427, 1322
Prada et al. 2012, MNRAS 423, 3018
Zhao et al. 2003, MNRAS 339, 12