PHYSICAL REVIEW D 90, 043524 (2014)

Scattering, damping, and acoustic oscillations: Simulating the structure of

dark matter halos with relativistic force carriers
Matthew R. Buckley,1 Jesús Zavala,2 Francis-Yan Cyr-Racine,3,4 Kris Sigurdson,5 and Mark Vogelsberger6
1
Department of Physics and Astronomy, Rutgers University, Piscataway, New Jersey 08854, USA
2
Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30,
2100 Copenhagen, Denmark
3
NASA Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, USA
4
California Institute of Technology, Pasadena, California 91125, USA
5
Department of Physics and Astronomy, University of British Columbia, Vancouver,
British Columbia V6T 1Z1, Canada
6
Department of Physics, Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
(Received 28 May 2014; published 20 August 2014)
We demonstrate that self-interacting dark matter models with interactions mediated by light particles can
have significant deviations in the matter power spectrum and detailed structure of galactic halos when
compared to a standard cold dark matter scenario. While these deviations can take the form of suppression
of small-scale structure that are in some ways similar to that of warm dark matter, the self-interacting
models have a much wider range of possible phenomenology. A long-range force in the dark matter can
introduce multiple scales to the initial power spectrum, in the form of dark acoustic oscillations and an
exponential cutoff in the power spectrum. Using simulations we show that the impact of these scales can
remain observationally relevant up to the present day. Furthermore, the self-interaction can continue to
modify the small-scale structure of the dark matter halos, reducing their central densities and creating a dark
matter core. The resulting phenomenology is unique to these type of models.

DOI: 10.1103/PhysRevD.90.043524 PACS numbers: 98.80.-k, 95.35.+d

I. INTRODUCTION self-interactions keep the dark matter in kinetic equilibrium

with itself until comparatively late in the Universe’s
Dark matter is one of the clearest signals of the existence
evolution. If these interactions are mediated by a light
of new physics beyond the Standard Model of particle
(relativistic) force carrier whose energy density is large
physics. The agreement between observations and the
predictions of cold dark matter with a cosmological compared to that of dark matter, then the sound speed in the
constant (ΛCDM) [1] gives solid evidence of the gravita- dark matter fluid remains high and sound waves can
tional interaction between dark matter and the Standard propagate long distances, potentially leaving signatures
Model baryons. Astrophysical observations provide only on cosmological and astrophysical scales. This can also
one further piece of information about the nature of dark occur if the self-interaction is mediated by a force carrier
matter: namely that it is “cold,” or at most “warm,” in that it that also couples dark matter to a Standard Model relativ-
cannot have been highly relativistic during structure for- istic species, such as the neutrinos [24–33]. Though not all
mation in the early universe (see e.g. [2]). However, self-interacting dark matter (SIDM) models possesses such
discrepancies between simulations and observations of a light particle, it is not uncommon [34–74]. As two of the
dark matter structure at small scales [3–15] have led to authors have previously noted [75], the large sound speed
an interest in models with large dark matter self-scattering of the dark matter sector in these models allows for “dark”
cross sections. Though it is possible that baryonic physics acoustic oscillations (DAOs, in analogy to the more
can resolve the issue [16–23], such models can provide a familiar baryon acoustic oscillations) that can leave detect-
rich range of phenomenology beyond those exhibited in able imprints in the anisotropies of the cosmic microwave
cold dark matter (CDM) and so are a subject of much background (CMB) and in the distribution of matter and
theoretical and observational interest. galaxies throughout the Universe.
Given the large cross sections that would be required to In addition, collisional damping between the light
significantly alter the halo properties of dark matter through mediator and the dark matter can erase primordial structure
self-interactions (comparable to the nuclear cross sections below some critical scale set by the diffusion length of the
of the visible sector), it is necessary to reconsider the early force carrier at the temperature of kinetic decoupling. To
universe cosmology realized in such models. While the first order, one might naively expect such a model to have
cold dark matter would still decouple from the standard a phenomenology qualitatively similar to warm dark matter
model plasma as in a typical dark matter scenario, the (WDM) [76–78], resulting primarily in a suppressed

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MATTHEW R. BUCKLEY et al. PHYSICAL REVIEW D 90, 043524 (2014)
number of dark matter halos below some critical mass. This it is an experimental fact [81] that the dark matter-baryon
has been pointed out in the context of model building spin-independent scattering cross sections at galactic
[24,26,32,48], but the full range of phenomenology has not velocities (∼200 km=s) are constrained to be less than
been pursued and very few cosmological simulations have 10−43 –10−45 cm2 for dark matter with masses between
been performed including these effects [79,80]. As we ∼10–104 GeV, the limits we can place on the self-
demonstrate in this paper, models with long-range forces interactions of dark matter are far weaker. From the
have a number of unique properties beyond the exponential separation between the baryonic gas and dark matter in
suppression of small-scale structure. These differences can the Bullet Cluster (a system of colliding galaxy clusters),
lead to significant deviations from both collisionless CDM the self-scattering cross section per unit mass must be less
and WDM predictions. Indeed, a SIDM model with light than ∼1 cm2 =g for relative velocities of order 1000 km=s
mediators can have novel effects on the small-scale matter [82,83]. For a 100 GeV dark matter particle (a represen-
power spectrum, the dark matter halo mass function, and tative mass for a thermal neutralino), this is an upper
the internal structure of surviving halos. We will demon- bound of only ∼2 × 10−22 cm2. This is orders of magnitude
strate the range of possible behavior of SIDM models with
above the cross sections with nucleons probed by direct
a light force carrier using cosmological N-body simulations
detection, and much larger than the annihilation cross
where the initial conditions are allowed to deviate from that
section in the early universe implied by a cold thermal
of CDM, and include the effects of momentum exchange
relic (∼10−36 cm2 ).
through self-interactions on the internal structure of the
The “crisis in small-scale structure” [3–5,7–10] has
dark matter halos.
In the next section, we discuss the physics of SIDM with focused attention on SIDM models with cross sections
a relativistic force carrier in the formation and evolution of per mass of approximately 1 cm2 =g for dark matter with
galactic structure from the early universe till the present. velocities of 30–60 km=s, sufficient to alter the structure of
We will focus on three effects: dark matter halos on the scale of dwarf galaxies (that is,
(i) the exponential suppression of primordial structure with masses of 109–10 M⊙ ) [84]. Relatively speaking, this
from collisional damping, cross section is enormous, comparable to a nuclear scatter-
(ii) the imprint of dark acoustic oscillations in the initial ing cross section in the Standard Model. While it is not
matter power spectrum, and required, such a large cross section makes it reasonable to
(iii) the modification of dark matter halo properties due consider the existence of light force carriers in the dark
to the self-interaction of dark matter. sector. This could occur, for example, in dark atom models
This section will also introduce the details of the “dark interacting through dark photons [34,48,50–57], in mirror
atom” model we adopt as a specific example of a SIDM dark matter models [35–42,58,59,85], or dark matter
model with a long-range mediator. In Sec. III, we outline coupled to Standard Model photons [60–67], neutrinos
the cosmological simulation methods used as well as the [24–33], or to some other new light particle [68–73,86–88].
parameters of the dark atom, cold dark matter, and warm A SIDM model where the interactions are mediated by a
dark matter benchmark models. The results of convergence force carrier that is relativistic during the thermal freeze-out
tests for our simulations are included in the Appendix. In in the early universe can have significant deviations in the
Sec. IV we show the results of our simulations, demon- present-day galactic structure compared to both noninter-
strating the present-day effects of both the exponential acting CDM as well as SIDM without a relativistic
cutoff and the DAOs in the matter power spectrum. This mediator. There are three main effects that we will explore
provides a sampling of the range of possible effects from in this paper: an exponential suppression of the initial
DAOs, as the dark matter microphysics is varied. Finally in power spectrum of dark matter halos, DAOs in the dark
Sec. V, we demonstrate the modifications of the dark matter matter thermal bath leaving imprints on the initial matter
halo structure due to the both the self-interactions and the power spectrum as well as the galaxy correlation function,
changes in the initial power spectrum. Though this paper and modifications of the structure of dark matter halos
does not cover the full set of available phenomenology that from energy transport made possible by dark matter self-
can arise from SIDM models with long-range forces, the scattering. The importance of each of these effects varies
range of simulations provides a representative sample of depending on the parameters of the SIDM model. The first
what is possible. and second effects are determined by the mass and
couplings of the relativistic force carrier’s interaction with
II. SELF-INTERACTIONS AND the dark matter in the early universe, while the third effect is
STRUCTURE FORMATION due to the self-scattering cross section as the primordial
Though the theory space of dark matter candidates is dark matter halos evolve forward in time, combined with
vast, most of the proposed dark matter particles have delayed formation for small halos. We first introduce our
extremely small interactions in the present Universe, both benchmark model for SIDM with a relativistic force carrier,
internally and between themselves and the baryons. While the dark atom model, and then discuss each of the possible

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effects in turn, and give an estimate of the range of T̂ gvis ðTÞgdark ðT f Þ 1=3
ξðTÞ ≡ ¼ : ð1Þ
parameters over which they are relevant. T gvis ðT f Þgdark ðTÞ

A. Dark atoms As g is a strictly increasing function of T, ξ < 1 unless there

While there are many SIDM scenarios that mediate the are a large number of degrees of freedom available in the
self-scattering interaction through a light force carrier, in dark sector that decouple after the freeze-out of dark matter.
this paper we will use the dark atom model as a benchmark The ratio ξ in ADM is therefore set by model-dependent
[34–42,48,50–59]. In this model, dark matter is composed assumptions in the UV completion of the theory which are
of two different particles, oppositely charged under a new beyond the scope of this work.
long-range Abelian gauge group Uð1ÞD , mediated by a As we will be interested in the kinetic decoupling of dark
dark photon ϕ. The two particles, analogous to the proton radiation from the massive dark matter in the early
and electron, form a bound state at low energies. universe, as well as the self-scattering of the dark atoms
Depending on the details of the dark atom parameter in the present day, we require several cross sections for this
space, this results in most of the dark matter being model. As the Universe cools, the dark radiation will
neutral under Uð1ÞD, and so avoid possible constraints remain in kinetic equilibrium with the dark matter by a
on Uð1ÞD -charged dark matter from dark magnetic fields in combination of Compton scattering off of the charged P
galaxies [72]. For in depth investigations of dark atom and E, and by Rayleigh scattering off of the Uð1ÞD -neutral
phenomenology, we refer the reader to Refs. [52–54,89]. dark atoms. The two relevant cross sections are [53]
The full range of behaviors available to these models is

much broader than the subject of this paper, so for our m2E
σ Compton ¼ 1 þ 2 σ Thomson ; ð2Þ
benchmark scenarios we only present results relevant to our mP
present work.
4
The fundamental parameter space of atomic dark matter 4 T̂
σ Raleigh ≈ 32π σ Thomson ðT̂ ≪ BD Þ; ð3Þ
(ADM) are the masses mP and mE of the oppositely BD
charged “proton” P and “electron” E analogues (by
definition, mE < mP ), the coupling parameter gD (alter- 8πα2D
natively, the dark fine structure constant αD ), and the dark σ Thomson ¼ : ð4Þ
3m2E
photon mass mϕ (massless for an unbroken gauge group).
We will assume that both of these fundamental particles are As the Universe cools, the fraction xD of ionized E and P
fermionic. Though it is possible for ADM models to decreases [53], and so the relative importance of the two
accommodate both asymmetric and thermal (symmetric) scattering modes changes. Since Compton scattering scales
production mechanisms, for simplicity we assume that the as xD it is most important before dark atom recombination,
dark matter was produced in some asymmetric process while Raleigh scattering, scaling as 1 − xD , can only
[90–110]. This implies that the number densities of P̄ and Ē become relevant after the onset of dark recombination.
are much lower than that of P and E. To study the effects of particle collisions inside dark
As the temperature of the dark sector decreases, the P matter halos, we also need to determine the elastic self-
and E particles form bound states H analogous to a scattering cross section of the dark atoms in the present
hydrogen atom. As a result, there are additional derived Universe. In the literature, self-interacting dark matter
parameters of the theory: mass of the bound state mD , the models are usually characterized by their momentum-
binding energy BD, and the temperature of the dark sector transfer cross section:
T̂. For this paper, we will make the assumption that the dark
Z
photon mass mϕ is negligibly small. This allows the dark dσ
atoms to retain the familiar atomic structure of standard σ tr ≡ ð1 − cos θÞ DM dΩ; ð5Þ
dΩ
baryonic hydrogen, albeit with modified energy levels, and
facilitate the computation of their thermal and ionization where dσ DM =dΩ is the differential scattering cross section
history [53]. for dark matter. The angular kernel suppresses irrelevant
Recall that the temperature of the dark sector T̂ will in forward scattering events while giving maximum weight
general be lower than the temperature T of the visible to backward scatterings. For identical particles colliding in
sector, as the visible sector will be reheated as particles the center-of-mass frame (such as two dark atoms), these
freeze-out of thermal equilibrium. It is T̂, rather than T, two processes should be indistinguishable and the above
which will set the number density and velocity of the dark expression overestimates the actual cross section. However,
matter and dark radiation. If the temperature-dependent since a dark matter “particle” in a typical N-body simu-
number of degrees of freedom in the two sectors are gvis and lation is composed of an extremely large number of actual
gdark , and the two sectors were last in chemical equilibrium dark matter particles, there is a fair amount of uncertainty
at the freeze-out temperature T f , then about which microphysical cross section best describes the

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MATTHEW R. BUCKLEY et al. PHYSICAL REVIEW D 90, 043524 (2014)
collisions of these dark matter macroparticles. To facilitate significantly change the momentum of a dark matter
comparison with previous works on SIDM, we choose particle. In dark atom models, the relevant cross sections
to parametrize our benchmark models in terms of the are those of Eqs. (2) and (3).
commonly accepted σ tr . As the Universe expands and cools, primordial pertur-
Due to the complex internal structure of atoms, their bations enter the horizon, each characterized by the wave
elastic scattering cross section generally has a highly non- number k. As long as the dark radiation has not decoupled
trivial energy and angular dependence, with many resonan- and the perturbation’s wavelength is comparable to the
ces appearing as the collision energy is varied [54]. Since our diffusion length of the dark radiation, that mode will be
goal in this work is to explore the possible interplay between damped, and as a result, no structure of that scale or smaller
dark matter self-interaction and the modified matter power can be seeded. This will appear as a suppression in the
spectrum caused by the relativistic force carrier, we consider power spectrum of density perturbations of scale k. This
simplified prescriptions for the dark matter scattering cross suppression shares some features with that found in models
section. First, we assume that all of the dark matter is made of WDM. In the latter case, the suppression is due to the
of neutral dark atoms (that is, we ignore any residual high-velocity WDM free-streaming out of the initial over-
ionization left over after dark recombination) and we also densities (collisionless damping).
neglect inelastic scattering processes that could excite or However, unlike the WDM scenario, the matter power
even ionize the dark atoms. Such inelastic processes can lead spectrum for SIDM with a relativistic force carrier has
to a very rich phenomenology on subgalactic scales (see e.g. significant nontrivial structures. These are the result of
Refs. [111–113]), but it is beyond the scope of this paper to acoustic oscillations in the dark matter-dark radiation
analyze their consequences in detail. Second, we model system. On length scales larger than the typical dark
neutral dark atom collisions with a hard-sphere scattering radiation mean free path, the dark matter and dark radiation
process with a constant cross section. While this choice is can be considered as a single nearly perfect fluid. As long
not necessarily self-consistent within our dark atom model, it as the momentum-transfer rate between the dark radiation
allows us to cleanly explore any possible connection and the dark matter is large compared to the Hubble rate,
between the oscillations and damping in the matter power the relativistic pressure of the dark fluid leads to a restoring
spectrum and the self-scattering of dark matter inside halos force that effectively opposes the gravitational growth of
and is acceptable as long as the range of velocities being dark matter overdensities and allow the propagation of
considered is not too large. Our specific choices for the cross longitudinal sound waves in the dark fluid. These sounds
sections are discussed in Sec. II C. waves propagate through the cosmos until the epoch of
dark matter kinematic decoupling [estimated by Eq. (6)] at
B. Structure formation and evolution up to decoupling which point the pressure support falters and dark radiation
Returning for a moment to a general SIDM model, we begins to free-stream out of dark matter fluctuations. Much
consider the effect of an additional relativistic particle in the like the case of their baryonic counterparts, the matter
early universe. We will assume that the dark matter is in distribution retains a memory of these DAOs which appears
thermal equilibrium, and therefore the present-day abun- as oscillations in the matter power spectrum or as a distinct
dance of dark matter is set either by thermal freeze-out or sound horizon in the matter correlation function.
some asymmetric production mechanism [101,105,106]. The specific shape of the SIDM matter power spectrum
Until the light force carrier ϕ kinetically decouples or the and correlation function is mostly governed by the relative
temperature drops below the carrier mass mϕ , the sound size of the dark matter sound horizon rDAO and of the
speed in the dark matter bath remains relativistic, and the diffusion (Silk) damping scale rSD at the epoch of kin-
primordial perturbations in the dark sector undergo colli- ematic decoupling. The comoving length scale is related to
sional (Silk) damping on scales below the dark radiation’s the comoving wave number k ¼ π=r. These scales are to a
diffusion length. The power spectra we will consider are good approximation given by [75]
calculated using the full Boltzmann formalism [114,115],
pffiffiffiffiffiffi
but some intuition can be obtained by considering a 4ξ2 Ωγ
simplified version [116]. rDAO ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3H0 ΩDM Ωm
Dark matter D is in kinetic equilibrium with the dark pffiffiffiffiffiffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
radiation ϕ until the rate Γ of significant energy exchange γ DM Ωr þ Ωm aD þ Ωm þ γ DM aD
× ln pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ; ð7Þ
between the two particle baths is less than the Hubble time γ DM Ωr þ Ωm
H−1 :
where we have defined
ΓðTÞ ¼ nϕ ðTÞhσviDϕ v2D ≲ HðTÞ−1 : ð6Þ
3ΩDM
The extra factors of dark matter velocity vD are present to γ DM ≡ ;
account for the multiple scatters that are necessary to 4ξ4 Ωγ

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sDAO wDAO
1000
100

1
3

3
mD 1 GeV 0.01 mD 1 TeV
Mpc h

Mpc h
0.001 D 0.008 D 0.009
BD 1 keV BD 1 keV
0.5 6 0.5
0 10 0
6
Pk

Pk
10
Self Interacting DM Self Interacting DM
Warm DM Analogue 10 Warm DM Analogue
9 Silk Damping Envelope 10
10 Silk Damping Envelope
Cold DM Cold DM
12 14
10 10
10 4 0.001 0.01 0.1 1 10 100 10 4 0.001 0.01 0.1 1 10 100
k h Mpc k h Mpc

FIG. 1 (color online). Comparison between the linear matter power spectra as a function of wave number k for SIDM with a light
mediator (here, dark atoms) and that of WDM with a free-streaming length comparable to the sound horizon of the former. We also
display the standard matter power spectrum for cold collisionless dark matter as well as a fit to the Silk damping envelope of SIDM. The
left panel displays the benchmark model for which rDAO ≫ rSD (strong DAO), while the right panel shows the scenario for which
rDAO ∼ rSD (weak DAO). Here, ξ0 ≡ ξðT CMB;0 Þ.

and (ii) rSD ∼ rDAO : For these models, the dark radiation

1=2 diffusion scale is comparable to the dark matter
4a3 m sound horizon at decoupling, leading to a sub-
rSD ≈ π pffiffiffiffiffi D D : ð8Þ
81H0 Ωr ΩDM ρcrit σ Compton stantial damping of the acoustic oscillations in the
dark plasma. This diffusion damping significantly
Here, aD stands for the scale factor at the epoch of dark broadens and dilutes the dark matter sound hori-
matter kinematic decoupling, and H0 is the present-day zon, leaving only the small-scale suppression of
Hubble constant. ΩDM , Ωγ , Ωr , and Ωm stand for the energy structure as the key signature of these models.
density in dark matter, photons, radiation (including neu- In this case, we expect the matter power spectrum
trinos and dark radiation), and nonrelativistic matter, to display only a handful of strongly damped
respectively, all in units of the critical density of the oscillations.
Universe, ρcrit . We note that both Eqs. (7) and (8) were Since models falling into the first category (which we will
derived in the tightly coupled regime of the dark plasma call “strong” DAO) are characterized by two distinct scales,
which is valid until close to the epoch of kinematic they contrast significantly with WDM theories whose
decoupling. We immediately see that the dark matter sound cosmological behavior is uniquely determined by their
horizon and its Silk damping scale depend on different free-streaming length. On the other hand, the cosmological
combinations of the cosmological and SIDM parameters, observables of models falling into the second category
implying that these two scales could be somewhat inde- (“weak” DAO) are mostly determined by one scale, the Silk
pendently varied. We can thus identify two kinds of damping length, implying that these scenarios might be
models:1 harder to distinguish from a WDM model once nonlinear
(i) rSD ≪ rDAO : In this case, diffusion damping is evolution is taken into account. That said, the dark matter
ineffective on scales close to the dark matter sound self-interactions will generically lead to a different internal
horizon. This results in a sharp and localized sound structure for dark matter halos (as we will discuss next),
horizon that is imprinted in the dark matter density giving us another handle to distinguish the SIDM scenarios
field after kinematic decoupling. In terms of the from WDM models. The N-body simulations presented in
matter power spectrum, these models display a the next few sections aim at determining whether such
significant number of nearly undamped acoustic distinction is possible and also whether the sound horizon
oscillations before Silk damping becomes effective and Silk damping scale remain separately imprinted in the
and dramatically reduces power on small length nonlinear matter field.
scales. Generically, this category encompasses mod- In Fig. 1, we show the linear power spectrum of CDM,
els with large values of σ Compton =mD . compared to that of a dark atom model, with two
benchmark parameter sets that exemplify strong (left
1 panel) and weak (right panel) DAOs. The power spec-
We note that the case rSD > rDAO is ill defined since this
requires the dark radiation to be effectively decoupled from dark trum is calculated using the full Boltzmann equations for
matter, in which case no sound wave can propagate in the dark dark matter coupled to dark radiation [53]. The two
medium. parameters sets are

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MATTHEW R. BUCKLEY et al. PHYSICAL REVIEW D 90, 043524 (2014)
10 sDAO 100 wDAO

1
10
Distance Mpc h

Distance Mpc h
0.1 rDAO
1

0.01
0.1 rDAO
0.001 mD 1 GeV mD 1 TeV
D 0.008 0.01 D 0.009
10 4 BD 1 keV BD 1 keV
rSD 0 0.5 0.001 rSD 0.5
0
aD aD
5
10
8 7 6 5 4 8 7 6 5 4
10 10 10 10 10 10 10 10 10 10
a a

FIG. 2 (color online). Evolution of the size of the dark matter sound horizon (rDAO ) and of the Silk damping scale (rSD ) as a function of
the cosmological scale factor a. The vertical dashed line denotes the epoch of kinematic decoupling. The grayed regions denote where
our calculation of these scales breaks down. The left panel displays the strong DAO model for which rDAO ≫ rSD always, while the right
panel shows the weak DAO model with rDAO ∼ rSD at a ¼ aD . Here, ξ0 ≡ ξðT CMB;0 Þ.

Strong DAO : mD ¼ 1 GeV; αD ¼ 8 × 10−3 ; corresponding size. The mass of dark matter enclosed
today by wave number k is approximately
BD ¼ 1 keV; ξðT CMB;0 Þ ¼ 0.5 ð9Þ

−3
k
Weak DAO : mD ¼ 1 TeV; αD ¼ 9 × 10 ; −3 MðkÞ ≈ ð1012 M⊙ Þ : ð11Þ
Mpc−1
BD ¼ 1 keV; ξðT CMB;0 Þ ¼ 0.5; ð10Þ
For comparison, in supersymmetric models with a
where T CMB;0 is the temperature of the CMB today. In “standard” neutralino dark matter candidate, the mass
this paper, we will denote the two models as ADMsDAO cutoff in the power spectrum is set by the temperature at
and ADMwDAO . We note that both models considered in which the dark matter kinetically decouples from the
this work are in agreement with the cosmological con- relativistic Standard Model neutrinos. Under reasonable
straints presented in Ref. [75]. In the ADMsDAO case, we assumptions for the neutralino physics, this occurs around
observe that the power spectrum displays a number of T ∼ 30 MeV [116–119]. The physical Jeans wave number,
nearly undamped oscillations before the Silk damping setting the scale at which perturbations will begin gravi-
cutoff (dot-dashed damping envelope) becomes important tational collapse (assuming sound speed vs ) is
on smaller scales. In contrast, for the ADMwDAO case
even the first oscillation is strongly Silk damped as

4πρðTÞ 1=2
compared to the CDM amplitude. In both cases, we kJ ¼ : ð12Þ
observe that the overall shape of the linear matter power m2Pl v2s
spectrum of SIDM models with long-range forces sig-
nificantly departs from that of WDM and CDM (also Here ρðTÞ is the total energy density of the Universe at
shown in Fig. 1) on small length scales, but is otherwise temperature T. Assuming that the Universe is radiation
identical to CDM on larger cosmological scales. The dominated at this point in its history, the Jeans wave
evolution of the two key scales, rSD and rDAO , as a number for such models is kJ ∼ 106 Mpc−1 , and so dark
function of the scale factor a is shown in Fig. 2. The matter halo masses extend down to ≲10−6 M⊙ . In the dark
scale factors of kinetic decoupling aD , used in Eqs. (7) atom SIDM model with self-scattering cross sections large
and (8), are also shown as vertical dashed lines. As enough to affect the evolution of galactic structures, the
expected, ðrDAO =rSD Þja¼aD ≫ 1 in the strong DAO case, decoupling temperature (or mediator mass) is expected to
while ðrDAO =rSD Þja¼aD ∼ 1 in the weak DAO case. be much lower, in the range of keV or tens of eV, resulting
In this work, we are interested in the impact of the dark in a suppression of structure on the scale of dwarf galaxies
matter microphysics (through its effect on the matter power or larger. As can be seen in Fig. 1, the characteristic scale at
spectrum and the self-scattering cross section) on the which the power spectrum deviates from that of CDM is
number density and distribution of small-scale structure 1–10h=Mpc−1 for our two benchmark scenarios. From the
in the Universe. It is therefore useful to convert the length previous arguments, for the dark atom models under
scales rDAO and rSD (or, their equivalent wave numbers) consideration here, we expect suppression of dark matter
into the mass of a collapsed dark matter halo of the structure to begin at scales between 109 and 1012 M⊙ ,

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SCATTERING, DAMPING, AND ACOUSTIC … PHYSICAL REVIEW D 90, 043524 (2014)
though we point out that the lower end of this mass range resonances appearing as the collision energy is varied. As
will be below the resolution limit of our simulations. our interest in this paper is merely demonstrating the
The scales over which these ADM models deviate from general properties of SIDM models coupled to a light
the predictions of CDM are constrained primarily by mediator, our simulation suite does not have the sufficient
measurements of the Lyman-α forest [120–122], though resolution that would require a more detailed treatment of
uncertainties exist in the conversion between the primordial the v dependence of σ tr .
power spectrum and the observations [123]. As pointed out Very roughly, observations exclude values of σ tr =mD
in Ref. [53], the addition of ADM would require hydro- large enough to cause much larger than one scattering per
dynamical simulations of the ADM cosmology to accu- particle per dynamical time of the relevant system. From
rately apply the Lyman-α constraints. With this caveat in observations of the Bullet Cluster, σ tr =mD < 1.25 cm2 =g
mind, the Lyman-α data sees no deviation from ΛCDM for velocities on the order of 1000 km=s [82,83]. The shape
on scales larger than k < 2h Mpc−1 [124] or 5h Mpc−1 of galaxy clusters and massive elliptical galaxies also
[122,125,126]. This corresponds to a minimum halo mass indicate that σ tr =mD ≲ 1 cm2 =g for similar characteristic
of 1011–13 M⊙ . The lower edge of this mass range would be velocities [128]. To create observable cored profiles for
in tension with our ADM power spectra, though again, care dwarf galaxies (v ∼ 30–60 km=s), σ tr =mD must be larger
must be taken in directly extrapolating the bounds to than 0.1 cm2 =g [132], while a minimum of 0.6 cm2 =g is
scenarios with new dark matter physics. needed to reduce the central masses these dwarfs [84]
sufficiently to solve the “too big to fail” problem [9].
C. Effects of self-interactions on post-decoupling For our weak DAO benchmark model, analytical esti-
halo evolution mates of the dark matter transfer cross section over mass
(see e.g. Ref. [54]) show that it is negligibly small for the
As covered in the previous two subsections, the presence
typical velocities of interest (v ∼ 100–1000 km=s) that we
of a light particle coupled to the dark matter in the early
are able to resolve with our simulations. For instance,
universe impacts the matter power spectrum, imprinting it
ðσ tr =mD Þweak ∼ 0.1 cm2 =g at v ¼ 220 km=s while it drops
with both DAOs and a damping scale. Once the dark matter
to ðσ tr =mD Þweak ∼ 5 × 10−3 cm2 =g at v ¼ 500 km=s,
decouples from the dark radiation, the interaction mediated
where v is the relative velocity of colliding particles.
by the light particle causes the dark matter to evolve
Such a small cross section over mass is mainly caused
forward as a self-interacting cold dark matter particle
by the large dark atom mass in this model (mD ¼ 1 TeV)
(albeit one with modified initial conditions). If the self-
which suppresses the number density of dark matter
interaction is sufficiently large, it will effect energy and
particles. We therefore neglect2 collisions between dark
momentum transfer in dark matter halos, transforming
matter particles for the weak DAO case, but we emphasize
central density cusps into cores [5], changing velocity
that they would become important in higher resolution
distributions and smoothing velocity profiles [127–129],
simulations. For the strong DAO benchmark case, we have
and reducing the triaxiality of galaxies [130]. Some have
the opposite problem that the transfer cross section over
claimed that SIDM models can bring simulation and
mass is much larger than the bounds mentioned above at
observation into closer alignment [5,84,127,128], though
the velocities of interest. To avoid simulating a grossly
this is a subject of active debate. We are interested in the
unrealistic case, we instead assign to this model a constant
interplay of the SIDM scattering effects with the alteration
value for the transfer cross section over mass that is in the
of the initial matter power spectrum.
interesting range to potentially address the small-scale
The figure of merit for SIDM models is the momentum-
astrophysical problems:
transfer cross section per dark matter mass, σ tr =mD , where
σ tr is given in Eq. (5). As a general rule, any “realistic” ðσ tr =mD Þstrong ¼ 1 cm2 =g: ð13Þ
model of SIDM will have σ tr be a nontrivial function of the
relative velocity v. For example, scattering of particles We note that despite our apparent ad hoc choices above,
charged under an unbroken Uð1Þ gauge field is propor- there are a large number of dark atom models that naturally
tional to v−4 . Models with a massive mediator have a more have σ tr =mD ∼ 0.1–1 cm2 =g at velocities relevant to astro-
complicated velocity dependence. If the spectrum admits physical objects such as dwarf galaxies and clusters.
bound states, resonances can develop, leading to σ tr varying However, these models typically have Silk damping mass
greatly as a function of v [30,54,131–133]. This is scales M SD < 109 M⊙ (see Eq. (11), which would be much
equivalent to the Sommerfeld enhancement which is more computationally expensive. Nevertheless, we expect
possible in dark matter annihilation [134]. the general conclusions drawn from our current N-body
As these examples indicate, it is not uncommon for simulations to be broadly applicable to fully realistic dark
a model of SIDM to have transfer cross sections with
a complicated dependence on v. As discussed in 2
We have explicitly verified that, on the mass scales probed by
Sec. II A, our benchmark dark atom model certainly has our simulations, the collisions have negligible effects for the
such a nontrivial functional form, with many molecular weak DAO case.

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TABLE I. Simulation suite discussed in this work. Only one simulation has self-scattering (ADMsDAO ), the rest
are collisionless. The second column characterizes the relevance of dark acoustic oscillations to the Silk damping
scale. The third column gives an estimate of the scale at which the non-CDM models’ power spectra is exponentially
suppressed as compared to the CDM benchmark. This is the collisionless damping scale for the WDM model, and
the rSD scale for the ADM and ADM-derived models.

Name σ tr =mD ½cm2 =g
ðrDAO =rSD Þja¼aD Suppression scale [Mpc/h]
CDM   
WDM   2.1a
ADMwDAO (nc) 0.1b 1.07 1.5
ADMsDAOenv (nc)   0.12
ADMsDAO 1.0c 17.9 0.12
ADMsDAO (nc)  17.9 0.12
a
Chosen such that it matches the DAO scale of ADMsDAO .
b
Actual value of σ tr =mD evaluated v ¼ 220 km=s, but not used in our simulations since it is too small to be
relevant.
c
Independent of velocity.

atom simulations as well as to other SIDM models coupled subscript “nc.” In the weak DAO case, only noncollisional
to a light force carrier, and future work at higher resolution runs are performed since the relevant σ tr =mD is too small to
could probe this interesting range of dark atom parameters. have significant effects at the resolved scales of the
simulation, as mentioned in the previous section. Thus,
III. SIMULATION OF DARK ATOMS for ADMwDAO, only the effects of the modifications to the
primordial power spectrum are probed in this work, though
Having introduced the three handles (rDAO , rSD , and future simulations at higher resolution may resolve the
σ tr =mD ) that together can distinguish a SIDM model with a effects of a nonzero transfer cross section.
long-range force from CDM, WDM, or “regular” SIDM For each of these models, we choose a flat cosmology
models, we wish to investigate the observable differences that is consistent3 with the results from the Planck mission
between these sets of models. To do so, we use fully [135] with a corresponding linear power spectrum (at
cosmological N-body simulations of dark matter to com- z ¼ 0) as shown in Fig. 1. These cases have the following
pare the properties of dark matter halos formed in each cosmological parameters (of relevance for the simulations):
scenario, which emphasize different phenomenological Ωm ¼ 0.305, ΩΛ ¼ 0.695, H0 ¼ 100h km s−1 Mpc−1 with
aspects of long-range SIDM. h ¼ 0.696, ns ¼ 0.97 and σ 8 ¼ 0.86, where ns is the
The two dark atom models we consider are given by the spectral index of the primordial power spectrum, and σ 8
strong and weak DAO parameter sets given by Eqs. (9) and is the rms amplitude of linear mass fluctuations in
(10). As can be seen in Fig. 2, the value of rDAO for these 8h−1 Mpc spheres at redshift zero.
two models is similar. To demonstrate the importance of the The simulations follow the growth of dark matter
two scales rDAO and rSD inherent in the ADM models, we structure from z ¼ 127 to z ¼ 0 in a cubic box of
compare to an additional two sets of models which have size L ¼ 64h−1 Mpc with 5123 simulation particles starting
power spectra suppressed at a single scale when compared from the same initial conditions (save for the varying power
to the CDM scenario. First, a WDM model is used, chosen spectra across models) with a fixed comoving softening
to have a free-streaming damping scale equivalent to the length (Plummer equivalent), ϵ ∼ 2.8h−1 kpc. The dark
rDAO of ADMsDAO (which is also close to the sound matter particle mass is mp ∼ 1.65 × 108 h−1 M ⊙ and the
horizon of ADMwDAO ). Next, a modified ADM model is Nyquist frequency is ∼25h Mpc−1 . For the ADM simu-
constructed, the power spectrum of which is fit to the lations, an algorithm for elastic isotropic self-scattering is
damping envelope of the ADMsDAO model. This model is implemented on top of the N-body code GADGET-3 for
somewhat ad hoc, as—lacking acoustic oscillations—it is gravitational interactions (last described in Ref. [136]) as
not a model of atomic dark matter that can be realized with described in detail in Ref. [127]. The algorithm is based on
any choice of parameters. However, it isolates the effect of a Monte Carlo approach to represent the microphysical
the single rSD scale, and so is useful for comparison scattering process in the macroscopic context of the
purposes. simulation. The main properties of our simulation suite
In addition, for the strong DAO model, we run simu- are given in Table I. The last column of this table gives
lations both with and without the self-interactions between an exponential suppression scale, which is set by the
dark matter particles. This allows us to isolate the effect of
collisions on the structure and number of dark matter halos.
3
The noncollisional simulations will be denoted by the Taking into account the extra dark radiation in our model.

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FIG. 3 (color online). Evolution of the dimensionless matter power spectrum Δ2 ðkÞ ≡ k3 PðkÞ for the WDM (upper left, dotted red
line), ADMwDAO (upper right, dashed blue line), and ADMsDAO (lower panel, solid blue line) simulations. In each case, the CDM power
spectrum is shown in black for comparison. We display the fully nonlinear Δ2 ðkÞ evaluated at 6 different redshifts from z ¼ 10 to z ¼ 0.
The dotted vertical lines at large and small k values denote the Nyquist frequency of our simulation box and the largest scale
(fundamental mode) probed by our simulations, respectively.

collisionless damping scale for WDM, rSD for ADMsDAO IV. EFFECTS OF ACOUSTIC OSCILLATIONS AND
and ADMwDAO , and the fitted envelope (effectively rSD ) for THE DAMPING SCALE
ADMsDAOenv . Using convergence tests, we show in the
Using our simulations, we can now investigate the
Appendix that the effective halo mass resolution for our
simulations is better than 1011 h−1 M ⊙ and that inner nonlinear evolution of the matter power spectrum in
densities of dark matter halos can be trusted at radius of SIDM models with a light force carrier. Given the
∼3ϵ, or ∼8.4h−1 kpc. resolved scales of these simulations, and the relatively

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FIG. 4 (color online). Projected dark matter density at z ¼ 0 in a slice of thickness 20h−1 Mpc through the full box (64h−1 Mpc on a
side) of four of our simulations, which have 5123 particles. Ordered from top left to bottom right, according to the abundance of
low-mass halos: CDM, ADMsDAO , ADMwDAO and WDM (see Table I).

low, velocity-independent self-scattering cross section We display dimensionless nonlinear power spectra
employed, we do not expect to see significant deviations [Δ2 ðkÞ ≡ k3 PðkÞ] evaluated at six different redshifts rang-
in the number density of halos between the collisional and ing from z ¼ 10 to z ¼ 0. For both the strong and weak
noncollisional simulations. As a result, this section is most DAO cases, we observe that the nonlinear evolution
devoted to the observable results of the initial matter power progressively erases the acoustic oscillations and regener-
spectrum being imprinted with the scales rSD and rDAO . At ates power on scales initially affected by DAOs. At redshift
the resolution of these simulations, nonzero self-interaction zero, the nonlinear matter power spectrum of our ADM
cross sections can alter the inner structure of low-mass models closely resemble that of CDM, except for a modest
halos, which we will explore in Sec. V. suppression on scales k ≳ 5h Mpc−1 (in the regime
Figure 3 shows the redshift evolution of the nonlinear dominated by correlations among dark matter particles
matter power spectrum for three of our non-CDM simu- within individual halos, the 1-halo term). A suppression is
lations, in each case comparing it to that of CDM. also observed for our WDM benchmark at z ¼ 0 for large

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wave numbers, but its magnitude is slightly larger than in
the ADMsDAO case, in line with our expectations given the
absence of acoustic oscillations in this model. At first sight,
it thus seems that nonlinearities erase the distinction
between rDAO and rSD in Δ2 ðkÞ at low redshifts in
SIDM models with relativistic force carriers, effectively
replacing these two quantities by a single effective damping
scale. However, as we discuss below, the actual situation is
more subtle and interesting.
At larger redshifts, the nonlinear matter power spectra
shown in Fig. 3 significantly depart from that of CDM.
Indeed, while the power spectra of our three benchmark
models were largely indistinguishable at z ¼ 0 for
k ≲ 5h Mpc−1 , we note that they all display very different
shapes at z ¼ 5 on the scales probed by our simulations.
This indicates that the structure formation history in each
model is in general quite different, leading to distinct
predictions about the structure of the high-redshift universe
as will soon become apparent in our study of the halo mass
function. Essentially, low-mass halos (≲1013 h−1 M⊙ ) in
FIG. 5 (color online). Differential halo mass function (number
our two ADM simulations form later than in the CDM case density of dark matter halos per unit logarithmic mass) as a
but earlier than in the WDM analogue model, implying that function of halo mass at z ¼ 0 for the simulation suite listed in
the densities of ADM halos will be somewhat in between Table I. The statistical error bars are Poissonian. The clear upturn
these two limiting cases (as, for the moment, we are not at ∼1011 h−1 M ⊙ for WDM is due to spurious halos formed due to
discussing the impact of self-scattering). Among other discreteness effects associated to the sharp truncation of the
effects, this has implications for the reionization history power spectrum at small scales. The vertical dotted red line marks
of the Universe in SIDM models with light mediators and this spurious transition for the WDM case using the formula from
could potentially be probed with high-redshift tracers of the Ref. [145]. This mass also serves as a conservative limit for
convergence of the mass function for all simulations. The dotted
density field such as the 21-cm line [137–141].
black vertical line marks the mass where halos have 100 particles.
In Fig. 4, we give a visual impression of the simulations
at z ¼ 0 by showing the projected dark matter distribution
within a slice that is 20h−1 Mpc thick. The color scale is construct spherical density profiles. We then define the
arranged in such a way that regions of higher density appear virial radius (r200 ) and mass (M 200 ) of the halo as the radius
as bright magenta. The simulations are ordered from top where the mean overdensity is 200 times the critical
left to bottom right according to their abundance of low- density, and the mass internal to this radius. We caution
mass halos. We only show four of our simulations since the that these choices imply that, at low masses for the non-
cases of ADMsDAOenv ðncÞ and ADMsDAO ðncÞ are very CDM models, some of the objects that we define as “halos”
similar to CDM and ADMsDAO , respectively. It is already are in reality structures that are in early stage of collapse or
clear at a visual level that the ADM simulations preserve protohalos that are not yet fully virialized [144]. We thus
the large-scale structure of CDM but with a deficit of low- expect the mass functions shown in this section to be
mass halos. The case of the WDM simulation is of course conservative upper limits on the actual mass function of
more dramatic given the large scale at which the power virialized dark matter halos.
spectrum has been truncated. Numerical artifacts due to the discreteness of the density
We show in Fig. 5 the number density of dark matter field in simulations are prevalent well above the interpar-
halos as a function of halo mass at z ¼ 0 (differential halo ticle separation (dp ) whenever there is a sharp cutoff in the
mass function) for dark matter models drawn from the power spectrum. This situation is well known in WDM N-
simulation suite described in Table I. We emphasize again body simulations [6,145,146] where spurious halos domi-
that the initial conditions in each simulation (CDM, WDM, nate below a limiting mass of M dis ¼ 10.1ΩDM ρcrit dp k−2 peak ,
ADMwDAO , etc.) are the same except for the input linear where kpeak is the wave number where Δ2 ðkÞ reaches its
spectra shown in Fig. 1. Halos are identified using the maximum. We clearly confirm this mass scaling for the
friends-of-friends (FOF) algorithm [142] with a linking WDM case, M dis ∼ 1011 h−1 M⊙ (red vertical dotted line in
length of b ¼ 0.2. Afterwards, each FOF halo is searched Fig. 5). For our ADM simulations, discreteness effects are
for self-bound substructures using the SUBFIND algorithm an issue at masses at least a factor of a few lower, which is
[143]. With this algorithm we can identify the center of the already close to our limit to resolve dark matter halos
gravitational potential for each halo, which we use to reliably, as indicated by the vertical black dotted line

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FIG. 6 (color online). Ratio of the differential mass functions of the WDM (left), ADMwDAO (center), and ADMsDAO (right)
simulations to that of CDM as a function of mass at different redshifts. Statistical error bars are shown for z ¼ 3 for reference purposes.
The vertical dotted line in the left panel marks the mass where spurious halos due to numerical artifacts dominate the mass function
(same as in Fig. 5). This mass, ∼1011 h−1 M ⊙ , is also an effective convergence mass for all our simulations.

denoted “100mp ” in Fig. 5. In any case, at all redshifts, we number as our WDM benchmark. This is reflected in the
can put a conservative limit of convergence of the mass z ¼ 0 mass spectrum as both simulations have suppressed
function at M 200 ∼ 1011 h−1 M ⊙ for all simulations (see the halo abundance below the same mass scale. However, due
Appendix). to the acoustic oscillations, the power is not exponentially
The CDM simulation serves as the default scenario suppressed at larger wave numbers, and so the initial
against which the WDM and ADM models can be suppression of low-mass halos is not as severe as with
compared. We show in Fig. 6 the ratio of the halo mass WDM, although their number density never fully reaches
functions of the WDM (left panel), ADMwDAO (middle that of the CDM benchmark. The formation of small
panel), and ADMsDAO (right panel) simulations to that of structures is delayed relative to CDM, although not as
CDM over a range of redshifts. Several things are immedi- dramatically as in the WDM case. This implies that
ately apparent from Figs. 5 and 6. The WDM simulation ADMsDAO halos with masses corresponding to scales
has the largest deviation from the CDM baseline, starting at between rDAO and rSD are typically denser than their
high halo masses (∼4 × 1012 h−1 M⊙ at z ¼ 0). The sup- WDM counterparts and thus less likely to be tidally
pression of the halo masses in the strong DAO scenario disrupted in the later universe (although self-scattering
appears at similarly high halo masses, but critically, more changes this picture somewhat as we show in the next
halos in this mass range are formed in this simulation than section).
in either the WDM or weak DAO cases. The weak DAO Moreover, we observe in Fig. 6 that the overall shape of
simulation has large suppression of halos formed at masses the ADMsDAO halo mass function departs significantly
smaller than ∼1012 h−1 M ⊙ . from that of either WDM or ADMwDAO whose deviation
This pattern is as expected given the initial power spectra from CDM is monotonically increasing towards low halo
shown in Fig. 1. WDM has essentially no power on small masses. We instead see that the initial suppression of the
scales in the initial matter power spectrum, which results in mass function around the scale corresponding to
large suppression of low-mass halos in the early universe. rDAO ðM 200 ∼ 4 × 1012 h−1 M⊙ Þ is followed by an increase
This is borne out in the simulations: looking at the at masses below M 200 ≲ 3 × 1011 h−1 M ⊙, before becoming
evolution of the WDM mass function over redshift (left suppressed again on scales affected by Silk damping. This
panel, Fig. 6), we see that low-mass halos are not initially behavior of the ADMsDAO mass function, which is apparent
present in large numbers, but develop later. The low-mass over the broad range of redshifts displayed in Fig. 6, seems
halos form later than would occur in CDM; as the average to indicate that the two distinct scales characterizing SIDM
dark matter density in the Universe is lower at the time of models with a light force carrier can remain imprinted in
formation, the resulting low-mass halos have lower central the mass distribution of objects populating the Universe.
densities than one would expect in CDM. A similar effect While further work is required, it is interesting that such
occurs in the ADM models, The impact on halo density nontrivial structures set by the early cosmology of the
profiles will be explored in more detail in Sec. V. SIDM models could survive to the present day.
Our model of ADM with strong DAO has a power The strong DAO envelope simulation (ADMsDAOenv in
spectrum which deviates from CDM at the same wave Fig. 5) shows no reduction of the mass function until near

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11 −1
10 h M⊙ , which is in line with our expectation as its spectra. This is expected, as the self-scattering cross section
initial matter power spectrum is not exponentially sup- of this model is too small to significantly reduce the number
pressed until k ∼ 10h Mpc−1 . This scale is also where we of massive halos. While nonzero σ tr can evaporate small
expect the acoustic oscillations of the strong DAO model to halos, those are far below the resolution limit of our
be severely damped. We indeed observe that the mass simulations.
functions of our ADMsDAOenv and ADMsDAO simulations The weak DAO simulation shows deviations from the CDM
become similar for masses below ∼2 × 1010 h−1 M⊙, in line predictions which become significant on slightly smaller
with our expectations. Unfortunately, halos at this mass mass scales compared to ADMsDAO ð∼1012 h−1 M⊙ Þ.
scale are poorly resolved in our numerical experiments and Again, this is as expected from the initial matter power
further work is needed to unambiguously determine that the spectrum, which begins to diverge from CDM at wave
suppression at these scales is indeed caused by Silk numbers of a few h=Mpc. This deviation starts small and
damping of the initial dark matter density field. We also increases significantly at lower (∼3 × 1011 h−1 M ⊙ ) halo
note that the collisional and noncollisional simulations of masses, although not as quickly as in the WDM case.
ADMsDAO have identical late-time nonlinear matter power Since our weak DAO model is characterized by

FIG. 7 (color online). Radial density profiles for four example halos at z ¼ 0 with masses of M ¼ 1.4 × 1014 , 1.7 × 1013 , 5.5 × 1012 ,
and 1.6 × 1012 M ⊙ (from top left to bottom right). The vertical dashed line marks the resolution limit 3ϵ of our simulation, where
ϵ ∼ 2.8h−1 kpc is the Plummer-equivalent softening length. Notice how only the simulation with self-interactions develops cored dark
matter profiles, clearly resolved at the highest masses. The remaining simulations have less dense low-mass (close to the filtering mass
scale) halos than CDM due to the modification of their initial power spectrum. Although less dense than CDM, these halos still have a
steep NFW-like inner density profile. The value of the virial mass in each legend is that of the CDM simulation.

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rSD ∼ rDAO , we would naturally expect the acoustic oscil- this self-interaction modification of the profile appears to
lations to play a subdominant role (compared to the strong extend down to the smallest mass halos, the radius at which
DAO case) in preventing a strong suppression of the mass the deviation from NFW occurs drops below the resolution
function. Looking at Fig. 6 and comparing the WDM and of our simulations.
ADMwDAO cases, we observe a milder suppression of the mass Looking now at the lower mass halos shown in Fig. 7, we
function in the weak DAO case, indicating that the severely see that those formed by the simulations with a large
damped acoustic oscillations in this model indeed have an exponential suppression in the primordial power spectrum
important effect. (WDM, ADMwDAO , and ADMsDAO in both collisional and
In summary, we see that the rich phenomenology of noncollisional runs) have suppressed densities when com-
SIDM coupled to a relativistic force carrier in the early pared to CDM. This effect is absent for the 1.4 × 1014 M ⊙
universe leads to observables that are both qualitatively and halos. Note that this highest mass range corresponds to
quantitatively different than both WDM and CDM. The wave numbers of k < 1 h=Mpc, where all the simulations
nonlinear evolution of the matter power spectrum in each have power spectra that are identical to CDM. These halos
model leads to a unique structure formation history and are therefore assembled at the same time as in the CDM
overall behavior of the halo mass function. For the first benchmark, which is reflected in their density profiles. The
time, we have characterized the shape of the mass function WDM simulation, with the greatest suppression of power in
of SIDM models coupled to a light mediator and shown it to the initial conditions, is seen to have the greatest reduction
have features that cannot be easily mimicked by a warm or in dark matter density in the present-day low-mass halos.
cold dark matter scenario. Both the collisional and noncollisional ADMsDAO runs
have a similar reduction of density at this mass scale,
V. INNER HALO DENSITIES: EFFECT OF although only the collisional run has the deviation from
SELF-INTERACTIONS NFW profiles in the inner slope of halos, as discussed
We now turn from the abundance of dark matter halos to previously.
their internal structure. In this paper, we examine only the This reduction in halo density in simulations with
dark matter density profiles, leaving additional properties suppressed small-scale power (relative to CDM) can be
such as velocity distributions for future work. As men- seen statistically in Fig. 8. Here, for each halo within a
tioned previously, SIDM models with long-range forces
can modify the dark matter profiles in two possible ways.
The first is the (by now well known) effect of nonzero
σ tr =mD , which allows for energy transport within the dark
matter halo itself. Energy transfer from collisions in the
high density central region can transform a cuspy Navarro-
Frenk-White (NFW) profile [147] into a cored profile. As
only the ADMsDAO simulation was performed with colli-
sional dark matter, this is the only set of halos in which this
effect could be seen.
In addition, the initial suppression of small scales in the
WDM and ADM models results in low-mass halos forming
later in the Universe’s evolution, as mentioned in the
previous section. As the dark matter density is lower at
later times, this delay in halo formation at small scales
results in less dense halos. However, this does not change
the shape of the density profiles, resulting in the low-mass
halos being less dense, but still NFW-like (e.g. for
WDM, [78]).
The combination of the two ways of modifying the dark
matter density profiles can be seen in Fig. 7, where we plot
the dark matter density as a function of radius for halos of FIG. 8 (color online). Plot of the maximum circular velocity
(V max ) versus the radius at this maximum (Rmax ) for all main
mass 1.4 × 1014 , 1.7 × 1013 , 5.5 × 1012 and 1.6 × 1012 M⊙ .
(central) halos for the different simulations. For the CDM case,
As can be seen, the collisional simulation ADMsDAO has a we show the median and 1σ regions of the distribution (shaded
profile that deviates significantly from a cuspy NFW, area). All CDM halos with masses around 1011 M ⊙ , 1012 M⊙ and
especially at the largest halo masses. As the noncollisional 1013 M ⊙ are shown with small dots. The other lines are the
ADMsDAO simulation shows no such modification for the medians of the distributions in each simulations (the spread is
highest mass halos, this effect can be unambiguously similar to CDM in all cases). The upturn at lower velocities,
associated with the dark matter self-interaction. Though V max ∼ 50 km=s, is created by resolution effects.

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simulation, we plot the maximum velocity V max (a measure the ADMsDAO simulation significantly reduces the central
of the total enclosed mass) versus the radius Rmax at which densities relative to the noncollisional run.
this maximum velocity is found. The black solid line and We note that given the scale of the power spectrum cutoff
shaded area show the median and 1σ regions of the CDM of the WDM case and assuming a thermal relic scenario,
distributions. For the other simulations, only the median is the impact of the primordial thermal velocities of the dark
drawn. This figure demonstrates more clearly that the matter particles might be relevant to structure formation
concentration of halos from simulations with suppressed and evolution. We have not however attempted to include
primordial power spectra is much lower than that predicted such velocities since the proper implementation scheme is
by CDM. An effect caused by the delayed formation of still unclear. This is not expected to be important for the
smaller halos. Again, we see the largest deviation for the halo mass function since the typical collapse velocities are
WDM scenario, where the power spectrum is suppressed much larger than the thermal velocity dispersion in the case
the most at small scales. we have considered. Our WDM model is actually close to
More importantly, notice that in Fig. 8 the effect of self- the one presented in Ref. [144], where this statement is
interactions is not visible (compare the collisional and discussed quantitatively. The inner WDM halo densities
noncollisional ADMsDAO lines). This is because the bulk might still be affected by thermal velocities (developing a
of the collisions occur well within Rmax . The effects of self- dark matter core, see e.g. Refs. [78,148]). However, the
interactions are apparent only at inner radii as shown in region where this is important is expected to be below the
Fig. 9, where we plot the density at a fixed radius versus resolution of our simulations, i.e., for the WDM case we
V max . The central radius chosen is R ¼ 3ϵ ∼ 8h−1 kpc, have studied here, the expected core is much smaller than
where the density profiles for all cases studied here are the one developed in the ADMsDAO case (see Ref. [148]).
sufficiently converged (see the Appendix). Although all non- Moreover, as we mentioned before, it is still controversial
CDM simulations are less dense than CDM at this radius, the how to properly include thermal velocities in simulations.
effect of collisions is the most dramatic for most massive
halos. Again, the models with the greatest initial suppression VI. CONCLUSIONS
of power on small scales have the greatest reduction in
density. Furthermore, the addition of self-interactions in Though not necessary for a model of SIDM, the large
transfer cross sections needed to significantly alter the
profile of dark matter halos can easily be realized via a new
light-mediator force coupling to the dark matter. If this
force carrier is light enough (≲MeV), then it acts as a long-
range force during the early universe, keeping the dark
matter and dark radiation in kinetic equilibrium with a
relativistic sound speed. This introduces a collisional
damping scale into the dark sector (rSD ), which would
suppress the formation of halos below a critical size, set by
the dark sector parameters. Furthermore, the dark radiation
bath allows for dark acoustic oscillations which introduces
an additional scale, the sound horizon of dark matter
(rDAO ), into the initial matter power spectrum. As the
DAO scale depends on a different combination of dark
sector parameters, the relationship between the two key
scales (rDAO and rSD ) can vary greatly across models. In
addition, the nonzero momentum-transfer cross section
between dark matter particles in the present day can
significantly alter the density profiles of dark matter
halos and, if large enough, reduce the number density of
low-mass halos.
Though previous works [24,26,32,48] have commented
FIG. 9 (color online). Dark matter density at a halo-centric on these effects individually, no full cosmological simu-
radius of 3ϵ, where ϵ is the Plummer-equivalent softening length
lations have been performed until now. In this paper, we
of the simulation, as a function of the maximum rotational
velocity V max of the halo. This radius is roughly the radius down
performed an initial exploration of the phenomenology
to which we can trust our simulations. The thick lines are the available to SIDM models with a long-range force, using
median of the distributions for each case. For the CDM and the dark atom model as a benchmark. Though we restricted
ADMsDAO cases we also show the 1σ region of the distributions; ourselves to relatively large dark matter halo masses, we
the spread for the other cases is similar. A given value of V max have demonstrated that the multiple scales inherent in this
roughly correspond to a given virial mass (see Fig. 8). class of models can lead to observable modifications of the

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MATTHEW R. BUCKLEY et al. PHYSICAL REVIEW D 90, 043524 (2014)
halo properties, which are distinct from either CDM or under a contract with the National Aeronautics and Space
WDM scenarios. In particular, we have demonstrated that Administration. The research of K. S. is supported in part
the imprint of the Silk (collisional) damping scale and the by a National Science and Engineering Research
DAO scale can survive in the differential halo mass Council (NSERC) of Canada Discovery Grant. The Dark
spectrum to z ¼ 0. Furthermore, even without the effects Cosmology Centre is funded by the DNRF. J. Z. is
of nonzero scattering cross section, the density profiles of supported by the EU under a Marie Curie International
low-mass halos are altered, as the delayed collapse results Incoming Fellowship, Contract No. PIIF-GA-2013-
in a suppression of the inner halo densities. With suffi- 627723. The simulations in this paper were carried out
ciently large self-interactions, the inner slopes of the on the Gardar supercomputer supported by the Nordic High
density profiles can also be modified; turning NFW-like Performance Computing (NHPC).
halos into cored-like systems.
The suppression of halos on smaller mass scales could be APPENDIX: CONVERGENCE TESTS
probed by future strong lensing studies [149–159]. As can
be seen, SIDM with long-range force carriers can modify To show that the results we have presented are not
the mass spectrum away from CDM in a quantitatively strongly affected by the resolution of our simulations, we
different manner than WDM. As seen in Fig. 6, echoes of performed convergence tests for the halo mass function and
the initial DAO structure are visible in the mass functions of for the halo density profiles. For each of the simulations we
a strong DAO ADM model even at low redshifts. With ran a complementary set of identical simulations with a
future lensing measurements probing the low-mass halo factor of 8 fewer particles than used in our primary results,
regime, it may be possible to not only discriminate WDM i.e., 2563 particles rather than 5123 . With such a test we can
models from CDM, but also to find evidence for long-range establish conservative levels of convergence for our higher
dark mediators through their effects on the multi-epoch resolution simulations.
halo mass function. Figure 10 shows the differential halo mass function at
In this first exploratory paper, we have restricted our- different redshifts for the main simulations in our work: all
selves to studying SIDM scenarios which alter relatively simulations which appear in Table I with the exceptions
large (∼1011–12 M ⊙ ) halos. As such, we do not address in of ADMsDAO ðncÞ, which has a mass function almost
detail the potential of such models to resolve the various identical to ADMsDAO , and ADMsDAOenv (nc), which is
outstanding crises in small-scale structure faced by CDM very similar to CDM, except at the smallest masses
[3–15]. Straightforward extrapolation of our results to (M 200 ≲ 1011 h−1 M⊙ ). The high (low) resolution level is
smaller values of rSD and rDAO would indicate that the shown with a thick (thin) line. Except for the WDM case,
addition of long-range forces to models of self-interaction even the low resolution simulations are essentially con-
would, in addition to the well-known formation of dark verged at M200 ∼ 1011 h−1 M ⊙ . This is quantified with the
matter cores in low-mass halos [84,127,128], also reduce vertical dotted lines, which show the value of the virial
the abundance of low-mass halos, while leaving larger mass where the convergence in the mass function is ≲20%
objects relatively unchanged. However, further work is at all redshifts. At larger masses, errors in the mass function
needed to confirm this intuition, likely requiring higher are essentially dominated by counting statistics. The
resolution simulations, as well as the important addition of vertical lines are therefore quite conservative limits of
velocity-dependent cross sections and baryonic physics. the virial mass (M200 ) where the mass function is not
Given the novel effects of these long-range force models, affected by numerical resolution for our highest resolution
such models and simulations are likely to provide unique runs. In the case of the WDM simulations, convergence is
and interesting cosmologies that can be directly compared harder to achieve due to the strong impact of discreteness
and constrained by observations in the near future. effects. In these cases the minimum mass we can trust is
reliably given by the formula given in Ref. [145], which
implies a mass resolution improvement that scales as N 1=3 .
ACKNOWLEDGMENTS
For our high resolution simulation, this limiting mass is
M. R. B. would like to thank Alyson Brooks for useful ≲1011 h−1 , while for the low resolution version is a factor of
discussion and comments. M. R. B. and J. Z. thank the 2 larger. These expectations appear with dotted lines in the
Aspen Center for Physics, where the initial conversations upper left of the WDM panel in Fig. 10. Our simulations
that led to their interest in this project were held. The clearly confirm that this is the appropriate limit of con-
Aspen Center for Physics is supported by the National vergence for the mass function.
Science Foundation under Grant No. 1066293. The work of We therefore conclude that the results presented in this
F. Y. C. R. was performed in part at the California Institute paper regarding the mass function are numerically con-
of Technology for the Keck Institute for Space Studies, verged at M 200 ∼ 1011 h−1 M⊙ at all redshifts. Except for
which is funded by the W. M. Keck Foundation. Part of the the WDM simulation, this is a conservative limit of
research described in this paper was carried out at the Jet convergence, that is nevertheless sufficient to support
Propulsion Laboratory, California Institute of Technology, our main conclusions.

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SCATTERING, DAMPING, AND ACOUSTIC … PHYSICAL REVIEW D 90, 043524 (2014)

FIG. 10 (color online). Differential halo mass function (number density of dark matter halos per unit logarithmic mass) as a function of
halo mass at different redshifts for four of our simulations according to the legend (see Table I). Our highest resolution levels (5123
particles) are shown with thick lines while lower resolution versions (2563 particles) of the same simulations are shown with thinner
lines. The Poissonian statistical error bars are shown for both resolution levels. For the simulations: CDM (top left), ADMsDAO (top
right), and ADMwDAO (bottom left), we mark the value of M 200 where the convergence of the mass function, at the lowest resolution
level, is better than 20% at all redshifts. In the case of the WDM simulation (bottom right), the vertical dotted lines mark the masses
where discreteness effects are important (at z ¼ 0) according to the formula given by Ref. [145]. Our simulations are clearly consistent
with the upturn expected at these masses due to spurious halos. Overall, all our high resolution simulations have a mass function that is
converged at M 200 ∼ 1011 h−1 M ⊙ , which is sufficient for the main conclusions of our paper.

Figure 11 shows a statistical convergence test for the resolution simulations. Four dark matter models are shown:
inner density of dark matter halos at a fixed radius of three CDM and ADMsDAO (shifted vertically up by a factor of 3)
times the Plummer-equivalent softening length of the low in the left and WDM and ADMwDAO (shifted vertically up
resolution simulation: 3ϵLOW ¼ 16.7h−1 kpc. This radius by a factor of 2.5) in the right. We can see that for CDM, the
was chosen so that we can assess if the densities extracted densities are essentially converged at this radius across all
from the high resolution simulations are to be trusted at the the mass (V max ) range, although at the low end, the spread
corresponding radius of 3ϵHI ¼ 8.4h−1 kpc, which is the in the distribution is clearly larger in the low resolution
radius we use in Fig. 9. The plot shows the distribution case. This is because the values of V max and Rmax start to be
(median and 1σ regions) of the densities at a radius 3ϵLOW affected importantly by resolution at V max ∼ 200 km=s for
as a function of the maximum circular velocity V max of the the low resolution simulations. The WDM and ADMwDAO
halos in the high (solid lines) and low (dashed lines) show a similar level of convergence as CDM in this plot,

043524-17
MATTHEW R. BUCKLEY et al. PHYSICAL REVIEW D 90, 043524 (2014)

FIG. 11 (color online). Dark matter density at a fixed radius (3ϵLOW ) from the center of each halo as a function of its maximum
rotational velocity V max. The Plummer-equivalent softening length for our low resolution simulations (2563 particles), ϵLOW , is
5.6h−1 kpc. We show the results for four of our simulations, two in each panel: CDM and ADMsDAO in the left and WDM and
ADMwDAO in the right. For clarity, the cases of ADMsDAO and ADMwDAO have been shifted vertically by a factor of 3, 2.5, respectively.
The high (low) resolution case is shown with solid (dashed) lines. We show the median and 1σ region of the distributions for each case
(shown also as shaded areas for the high resolution case). The convergence in the median of the distributions is within the 1σ statistical
spread for all cases, except at lowest V max (mass) values for WDM.

except at the low end where, for a fixed value of V max , the impact on the collision frequency that is used in the
central densities are systematically underestimated in the algorithm for self-scattering. SIDM simulations (with a
low resolution case. In the case of ADMsDAO , there is a CDM power spectrum), have also shown that inner halo
small underestimation of the central densities across all densities are only minimally affected at a radius of 3ϵ for
scales in the low resolution case, which signals that the simulations where the spatial resolution varies by a factor of
inner densities have not fully converged at 3ϵLOW . However ∼6 (see Fig. 9 of [127]).
this underestimation is minimal, clearly less than the 1σ We are therefore confident that for all our simulations,
statistical spread of the distribution. The lack of full the dark matter densities at a radius of 3ϵ are reliable,
convergence in the ADMsDAO case is caused by the fact particularly for high mass halos, which is the regime where
that a poorly resolved initial density cusp of a halo has an we draw our main conclusions.

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