Cosmological Shock Waves in the Large Scale Structure of the
Universe: Non-gravitational Effects
Hyesung Kang1, Dongsu Ryu2, Renyue Cen3, and J. P. Ostriker3
ABSTRACT
Cosmological shock waves result from supersonic flow motions induced by hi-
erarchical clustering of nonlinear structures in the universe. These shocks govern
the nature of cosmic plasma through thermalization of gas and acceleration of
nonthermal, cosmic-ray (CR) particles. We study the statistics and energetics of
shocks formed in cosmological simulations of a concordance ΛCDM universe, with
a special emphasis on the effects of non-gravitational processes such as radiative
cooling, photoionization/heating, and galactic superwind feedbacks. Adopting
an improved model for gas thermalization and CR acceleration efficiencies based
on nonlinear diffusive shock acceleration calculations, we then estimate the gas
thermal energy and the CR energy dissipated at shocks through the history of
the universe. Since shocks can serve as sites for generation of vorticity, we also
examine the vorticity that should have been generated mostly at curved shocks in
cosmological simulations. We find that the dynamics and energetics of shocks are
governed primarily by the gravity of matter, so other non-gravitational processes
do not affect significantly the global energy dissipation and vorticity generation
at cosmological shocks. Our results reinforce scenarios in which the intraclus-
ter medium and warm-hot intergalactic medium contain energetically significant
populations of nonthermal particles and turbulent flow motions.
Subject headings: cosmic rays – large-scale structure of universe – methods: nu-
merical – shock waves – turbulence
1Department of Earth Sciences, Pusan National University, Pusan 609-735, Korea:
kang@uju.es.pusan.ac.kr
2Department of Astronomy & Space Science, Chungnam National University, Daejeon 305-764, Korea:
ryu@canopus.cnu.ac.kr
3Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544-1001, USA:
cen@astro.pricenton.edu, jpo@astro.pricenton.edu
http://arxiv.org/abs/0704.1521v1
– 2 –
1. Introduction
Astrophysical plasmas consist of both thermal particles and nonthermal, cosmic-ray
(CR) particles that are closely coupled with permeating magnetic fields and underlying
turbulent flows. In the interstellar medium (ISM) of our Galaxy, for example, an approx-
imate energy equipartition among different components seems to have been established,
i.e., εtherm ∼ εCR ∼ εB ∼ εturb ∼ 1 eV cm
−3 (Longair 1994). Understanding the complex
network of physical interactions among these components constitutes one of fundamental
problems in astrophysics.
There is substantial observational evidence for the presence of nonthermal particles and
magnetic fields in the large scale structure of the universe. A fair fraction of X-ray clusters
have been observed in diffuse radio synchrotron emission, indicating the presence of GeV
CR electrons and µG fields in the intracluster medium (ICM) (Giovannini & Feretti 2000).
Observations in EUV and hard X-ray have shown that some clusters possess excess radia-
tion compared to what is expected from the hot, thermal X-ray emitting ICM, most likely
produced by the inverse-Compton scattering of cosmic background radiation (CBR) pho-
tons by CR electrons (Fusco-Femiano et al. 1999; Bowyer et al. 1999; Berghöfer et al. 2000).
Assuming energy equipartition between CR electrons and magnetic fields, εCRe ∼ εB ∼
0.01−0.1eV cm−3 ∼ 10−3−10−2εtherm can be inferred in typical radio halos (Govoni & Feretti
2004). If some of those CR electrons have been energized at shocks and/or by turbulence,
the same process should have produced a greater CR proton population. Considering the
ratio of proton to electron numbers, K ∼ 100, for Galactic CRs (Beck & Kraus 2005), one
can expect εCRp ∼ 0.01 − 0.1εtherm in radio halos. However, CR protons in the ICM have
yet to be confirmed by the observation of γ-ray photons produced by inelastic collisions
between CR protons and thermal protons (Reimer et al. 2003). Magnetic fields have been
also directly observed with Faraday rotation measure (RM). In clusters of galaxies strong
fields of a few µG strength extending from core to 500 kpc or further were inferred from
RM observations (Clarke et al. 2001; Clarke 2004). An upper limit of . µG was imposed
on the magnetic field strength in filaments and sheets, based the observed limit of the RMs
of quasars outside clusters (Kronberg 1994; Ryu et al. 1998).
Studies on turbulence and turbulent magnetic fields in the large scale structure of the
universe have been recently launched too. XMM-Newton X-ray observations of the Coma
cluster, which seems to be in a post-merger stage, were analyzed in details to extract clues
on turbulence in the ICM (Schuecker et al. 2004). By analyzing pressure fluctuations, it was
shown that the turbulence is likely subsonic and consistent with Kolmogoroff turbulence.
RM maps of clusters have been analyzed to find the power spectrum of turbulent magnetic
fields in a few clusters (Murgia et al. 2004; Vogt & Enßlin 2005). While Murgia et al. (2004)
– 3 –
reported a spectrum shallower than the Kolmogoroff spectrum, Vogt & Enßlin (2005) argued
that the spectrum could be consistent with the Kolmogoroff spectrum if it is bended at a
few kpc scale. These studies suggest that as in the ISM, turbulence does exist in the ICM
and may constitute an energetically non-negligible component.
In galaxy cluster environments there are several possible sources of CRs, magnetic
fields, and turbulence: jets from active galaxies (Kronberg et al. 2004; Li et al. 2006), ter-
mination shocks of galactic winds driven by supernova explosions (Vök & Atoyan 1999),
merger shocks (Sarazin 1999; Gabici & Blasi 2003; Fujita et al. 2003), structure formation
shocks (Loeb & Waxmann 2000; Miniati et al. 2001a,b), and motions of subcluster clumps
and galaxies (Subramanian et al. 2006). All of them have a potential to inject a similar
amount of energies, i.e., E ∼ 1061 − 1062 ergs into the ICM. Here we focus on shock scenar-
Astrophysical shocks are collisionless shocks that form in tenuous cosmic plasmas via col-
lective electromagnetic interactions between gas particles and magnetic fields. They play key
roles in governing the nature of cosmic plasmas: i.e., 1) shocks convert a part of the kinetic
energy of bulk flow motions into thermal energy, 2) shocks accelerate CRs by diffusive shock
acceleration (DSA) (Blandford & Ostriker 1978; Blandford & Eichler 1987; Malkov & Drury
2001), and amplify magnetic fields by streaming CRs (Bell 1978; Lucek & Bell 2000), 3)
shocks generate magnetic fields via the Biermann battery mechanism (Biermann 1950;
Kulsrud et al. 1997) and the Weibel instability (Weibel 1959; Medvedev et al. 2006), and 4)
curved shocks generate vorticity and ensuing turbulent flows (Binney 1974; Davies & Widrow
2000).
In Ryu et al. (2003) (Paper I), the properties of cosmological shock waves in the inter-
galactic medium (IGM) and the energy dissipations into thermal and nonthermal components
at those shocks were studied in a high-resolution, adiabatic (non-radiative), hydrodynamic
simulation of a ΛCDM universe. They found that internal shocks with low Mach numbers
of M . 4, which formed in the hot, previously shocked gas inside nonlinear structures,
are responsible for most of the shock energy dissipation. Adopting a nonlinear DSA model
for CR protons, it was shown that about 1/2 of the gas thermal energy dissipated at cos-
mological shocks through the history of the universe could be stored as CRs. In a recent
study, Pfrommer et al. (2006) identified shocks and analyzed the statistics in smoothed par-
ticle hydrodynamic (SPH) simulations of a ΛCDM universe, and found that their results
are in good agreement with those of Paper I. While internal shocks with lower Mach num-
bers are energetically dominant, external accretions shocks with higher Mach numbers can
serve as possible acceleration sites for high energy cosmic rays (Kang et al. 1996, 1997;
Ostrowski & Siemieniec-Ozieblo 2002). It was shown that CR ions could be accelerated up
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to ∼ Z × 1019eV at cosmological shocks, where Z is the charge of ions (Inoue et al. 2007).
Ryu et al. (2007) (Paper II) analyzed the distribution of vorticity, which should have
been generated mostly at cosmological shock waves, in the same simulation of a ΛCDM
universe as in Paper I, and studied its implication on turbulence and turbulence dynamo.
Inside nonlinear structures, vorticity was found to be large enough that the turn-over time,
which is defined as the inverse of vorticity, is shorter than the age of the universe. Based on
it Ryu et al. (2007) argued that turbulence should have been developed in those structures
and estimated the strength of the magnetic field grown by the turbulence.
In this paper, we study cosmological shock waves in a new set of hydrodynamic sim-
ulations of large structure formation in a concordance ΛCDM universe: an adiabatic (non-
radiative) simulation which is similar to that considered in Paper I, and two additional
simulations which include various non-gravitational processes (see the next section for de-
tails). As in Papers I and II, the properties of cosmological shock waves are analyzed, the
energy dissipations to gas thermal energy and CR energy are evaluated, and the vorticity
distribution is analyzed. We then compare the results for the three simulations to highlight
the effects of non-gravitational processes on the properties of shocks and their roles on the
cosmic plasmas in the large scale structure of the universe.
Simulations are described in §2. The main results of shock identification and properties,
energy dissipations, and vorticity distribution are described in §3, §4, and §5, respectively.
Summary and discussion are followed in §6.
2. Simulations
The results reported here are based on the simulations previously presented in Cen & Ostriker
(2006). The simulations included radiative processes of heating/cooling, and the two sim-
ulations with and without galactic superwind (GSW) feedbacks were compared in that pa-
per. Here an additional adiabatic (non-radiative) simulation with otherwise the same setup
was performed. Hereafter these three simulations are referred as “Adiabatic”, “NO GSW”,
and “GSW” simulations, respectively. Specifically, the WMAP1-normalized ΛCDM cosmol-
ogy was employed with the following parameters: Ωb = 0.048, Ωm = 0.31, ΩΛ = 0.69,
h ≡ H0/(100 km/s/Mpc) = 0.69, σ8 = 0.89, and n = 0.97. A cubic box of comoving size
85 h−1Mpc was simulated using 10243 grid zones for gas and gravity and 5123 particles for
dark matter. It allows a uniform spatial resolution of ∆l = 83 h−1kpc. In Papers I and II,
an adiabatic simulation in a cubic box of comoving size 100 h−1Mpc with 10243 grid zones
and 5123 particles, employing slightly different cosmological parameters, was used. The
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simulations were performed using a PM/Eulerian hydrodynamic cosmology code (Ryu et al.
1993).
Detailed descriptions for input physical ingredients such as non-equilibrium ioniza-
tion/cooling, photoionization/heating, star formation, and feedback processes can be found
in earlier papers (Cen et al. 2003; Cen & Ostriker 2006). Feedbacks from star formation
were treated in three forms: ionizing UV photons, GSWs, and metal enrichment. GSWs
were meant to represent cumulative supernova explosions, and modeled as outflows of sev-
eral hundred km s−1. The input of GSW energy for a given amount of star formation was
determined by matching the outflow velocities computed for star-burst galaxies in the sim-
ulation with those observed in the real world (Pettini et al. 2002)(see also Cen & Ostriker
2006, for details).
Figure 1 shows the gas mass distribution in the gas density-temperature plane, fm(ρgas, T ),
and the gas mass fraction as a function of gas temperature, fm(T ), at z = 0 for the three
simulations. The distributions are quite different, depending primarily on the inclusion of
radiative cooling and photoionization/heating. GSW feedbacks increase the fraction of the
WHIM with 105 < T < 107K, and at the same time affect the distribution of the warm/diffuse
gas with T < 105.
3. Properties of Cosmological Shock Waves
We start to describe cosmological shocks by briefing the procedure by which the shocks
were identified in simulation data. The details can be found in Paper I. A zone was tagged as
a shock zone currently experiencing shock dissipation, whenever the following three criteria
are met: 1) the gradients of gas temperature and entropy have the same sign, 2) the local
flow is converging with ~∇ · ~v < 0, and 3) |∆ log T | ≥ 0.11 corresponding to the temperature
jump of a shock with M ≥ 1.3. Typically a shock is represented by a jump spread over
2 − 3 tagged zones. Hence, a shock center was identified within the tagged zones, where
~∇ ·~v is minimum, and this center was labeled as part of a shock surface. The Mach number
of the shock center, M , was calculated from the temperature jump across the entire shock
zones. Finally to avoid confusion from complex flow patterns and shock surface topologies
associated with very weak shocks, only those portions of shock surfaces with M ≥ 1.5 were
kept and used for the analysis of shocks properties.
Figure 2 shows the locations of identified shocks in a two-dimensional slice at z = 0 in the
GSW simulation. The locations are color-coded according to shock speed. As shown before
in Paper I, external accretion shocks encompass nonlinear structures and reveal, in addition
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to cluster complexes, rich topology of filamentary and sheet-like structures in the large scale
structure. Inside the nonlinear structures, there exist complex networks of internal shocks
that form by infall of previously shocked gas to filaments and knots and during subclump
mergers, as well as by chaotic flow motions. The shock heated gas around clusters extends
out to ∼ 5 h−1Mpc, much further out than the observed X-ray emitting volume.
In the GSW simulation, with several hundred km s−1 for outflows, the GSW feedbacks
affected most greatly the gas around groups of galaxies, while the impact on clusters with
kT & 1 keV was minimal. In Figure 3 we compare shock locations in a region around
two groups with kT ∼ 0.2 − 0.3 keV in the three simulations. It demonstrates that GSW
feedbacks pushed the hot gas out of groups with typical velocities of ∼ 100 km s−1 (green
points). In fact the prominent green balloons of shock surfaces around groups in Figure 2
are due to GSW feedbacks (see also Figure 4 of Cen & Ostriker 2006).
In the left panels of Figure 4 we compare the surface area of identified shocks, normalized
by the volume of the simulation box, per logarithmic Mach number interval, dS(M)/d logM
(top), and per logarithmic shock speed interval, dS(Vs)/d log Vs (bottom), at z = 0 in the
three simulations. Here S and Vs are given in units of (h
−1Mpc)−1 and km s−1. The
quantity S provides a measure of shock frequency or the inverse of the mean comoving
distance between shock surfaces. The distributions of dS(M)/d logM for the NO GSW and
GSW simulations are similar, while that for the Adiabatic simulation is different from the
other two. This is mainly because the gas temperature outside nonlinear structures is lower
without photoionization/heating in the Adiabatic simulation. As a result, external accretion
shocks tend to have higher Mach number due to colder preshock gas. The distribution of
dS(Vs)/d log Vs, on the other hand, is similar for all three simulations for Vs > 15 km s
−1. For
Vs < 15 km s
−1, however, there are more shocks in the Adiabatic simulation (black points in
Figure 3). Again this is because in the Adiabatic simulation the gas temperature is colder in
void regions, and so even shocks with low speeds of Vs < 15 km s
−1 were identified in these
regions. The GSW simulation shows slightly more shocks than the NO GSW simulation
around Vs ∼ 100 km s
−1, because GSW feedbacks created balloon-shaped surfaces of shocks
with typically those speeds (green points in Figure 3).
For identified shocks, we calculated the incident shock kinetic energy flux, Fφ = (1/2)ρ1V
where ρ1 is the preshock gas density. We then calculated the kinetic energy flux through
shock surfaces, normalized by the volume of the simulation box, per logarithmic Mach num-
ber interval, dFφ(M)/d logM , and per logarithmic shock speed interval, dFφ(Vs)/d logVs. In
the right panels of Figure 4, we compare the flux at z = 0 in the three simulations. Once
again, there are noticeable differences in dFφ(M)/d logM between the Adiabatic simulation
and the other two simulations, which can be interpreted as the result of ignoring photoion-
– 7 –
ization/heating in the gas outside nonlinear structures in the Adiabatic simulation. GSW
feedbacks enhance only slightly the shock kinetic energy flux for Vs ∼ 100−300 km s
−1, as can
be seen in the plot of dFφ(Vs)/d logVs. Yet, the total amount of the energy flux is expected
to be quite similar for all three simulations. This implies that the overall energy dissipation
at cosmological shocks is governed mainly by the gravity of matter, and that the inclusion of
various non-gravitational processes such as radiative cooling, photoionization/heating, and
GSW feedbacks have rather minor, local effects.
We note that a temperature floor of Tfloor = TCBR was used for the three simulations in
this work, while Tfloor = 10
4 K was set in paper I. It was because in Paper I only an adiabatic
simulation was considered and the 104 K temperature floor was enforced to mimic the effect
of photoionization/heating on the IGM. However we found that when the same temperature
floor is enforced, the statistics of the current Adiabatic simulation agree excellently with
those of Paper I. Specifically, the shock frequency and kinetic energy flux, dS(M)/d logM
and dFφ(M)/d logM , for weak shocks with 1.5 ≤ M . 3 are a bit higher in the current
Adiabatic simulation, because of higher spatial resolution. But the total kinetic energy flux
through shock surfaces, Fφ(M > 1.5), agrees within a few percent. On the other hand, In
Paper I we were able to reasonably distinguish external and internal shocks according to the
preshock temperature, i.e., external shocks if T1 ≤ Tfloor and internal shocks if T1 > Tfloor. We
no longer made such distinction in this work, since the preshock temperature alone cannot
tell us whether the preshock gas is inside nonlinear structures or not in the simulations with
radiative cooling.
4. Energy dissipation by Cosmological Shock Waves
The CR injection and acceleration rates at shocks depend in general upon the shock
Mach number, field obliquity angle, and the strength of the Alfvén turbulence responsible
for scattering. At quasi-parallel shocks, in which the mean magnetic field is parallel to the
shock normal direction, small anisotropy in the particle velocity distribution in the local fluid
frame causes some particles in the high energy tail of the Maxwellian distribution to stream
upstream (Giacalone et al. 1992). The streaming motions of the high energy particles against
the background fluid generate strong MHD Alfvén waves upstream of the shock, which in turn
scatter particles and amplify magnetic fields (Bell 1978; Lucek & Bell 2000). The scattered
particles can then be accelerated further to higher energies via Fermi first order process
(Malkov & Drury 2001). These processes, i.e., leakage of suprathermal particles into CRs,
self-excitation of Alfvén waves, amplification of magnetic fields, and further acceleration
of CRs, are all integral parts of collisionless shock formation in astrophysical plasmas. It
– 8 –
was shown that at strong quasi-parallel shocks, 10−4− 10−3 of the incoming particles can be
injected into the CR population, up to 60% of the shock kinetic energy can be transferred into
CR ions, and at the same time substantial nonlinear feedbacks are exerted to the underlying
flow (Berezhko et al. 1995; Kang & Jones 2005).
At perpendicular shocks with weakly perturbed magnetic fields, on the other hand, par-
ticles gain energy mainly by drifting along the shock surface in the ~v× ~B electric field. Such
drift acceleration can be much more efficient than the acceleration at parallel shocks (Jokipii
1987; Kang et al. 1997; Ostrowski & Siemieniec-Ozieblo 2002). But the particle injection
into the acceleration process is expected to be inefficient at perpendicular shocks, since the
transport of particles normal to the average field direction is suppressed (Ellison et al. 1995).
However, Giacalone (2005) showed that the injection problem at perpendicular shocks can be
alleviated substantially in the presence of fully turbulent fields owing to field line meandering.
As in Paper I, the gas thermalization and CR acceleration efficiencies are defined as
δ(M) ≡ Fth/Fφ and η(M) ≡ FCR/Fφ, respectively, where Fth is the thermal energy flux
generated and FCR is the CR energy flux accelerated at shocks. We note that for gasdy-
namical shocks without CRs, the gas thermalization efficiency can be calculated from the
Rankine-Hugoniot jump condition, as follows:
δ0(M) =
eth,2 − eth,1
, (1)
where the subscripts 1 and 2 stand for preshock and postshock regions, respectively. The
second term inside the brackets subtracts the effect of adiabatic compression occurred at a
shock too, not just the thermal energy flux entering the shock, namely, eth,1v1.
At CR modified shocks, however, the gas thermalization efficiency can be much smaller
than δ0(M) for strong shocks with large M , since a significant fraction of the shock kinetic
energy can be transferred to CRs. The gas thermalization and CR acceleration efficiencies
were estimated using the results of DSA simulations of quasi-parallel shocks with Bohm
diffusion coefficient, self-consistent treatments of thermal leakage injection, and Alfvén wave
propagation (Kang & Jones 2007). The simulations were started with purely gasdynamical
shocks in one-dimensional, plane-parallel geometry, and CR acceleration was followed by
solving the diffusion-convection equation explicitly with very high resolution. Shocks with
Vs = 150 − 4500 km s
−1 propagating into media of T1 = 10
4 − 106 K were considered.
After a quick initial adjustment, the postshock states reach time asymptotic values and the
CR modified shocks evolve in an approximately self-similar way with the shock structure
broadening linearly with time (refer Kang & Jones 2007, for details). Given this self-similar
nature of CR modified shocks, we calculated time asymptotic values of δ(M) and η(M) as
the ratios of increases in the gas thermal and CR energies at shocks to the kinetic energy
– 9 –
passed through the shocks at the termination time of the DSA simulations. As in Eq. (1),
the increase of energies due to adiabatic compression was subtracted.
Figure 5 shows δ(M) and η(M) estimated from DSA simulations and their fittings for
the cases with and without a preexisting CR component. The fitting formulae are given
in Appendix A. Without a preexisting CR component, gas thermalization is more efficient
than CR acceleration at shocks with M . 5. However, it is likely that weak internal shocks
propagate through the IGM that contains CRs accelerated previously at earlier shocks.
In that case, shocks with preexisting CRs need to be considered. Since the presence of
preexisting CRs is equivalent to a higher injection rate, CR acceleration is more efficient in
that case, especially at shocks with M . 5 (Kang & Jones 2003). In the bottom panel the
efficiencies for shocks with PCR/Pg ∼ 0.3 in the preshock region are shown. For comparison,
δ0(M) for shocks without CRs is also drawn. Both δ(M) and η(M) increase with Mach
number, but η(M) asymptotes to ∼ 0.55 while δ(M) to ∼ 0.30 for strong shocks with
M & 30. So about twice more energy goes into CRs, compared to for gas heating, at strong
shocks.
The efficiencies for the case without a preexisting CR component in the upper panel
of Figure 5 can be directly compared with the same quantities presented in Figure 6 of
Paper I. In Paper I, however, the gas thermalization efficiency was not calculated explicitly
from DSA simulations, and hence δ0(M) for gasdynamic shocks was used. It represents
gas thermalization reasonably well for weak shocks with M . 2.5, but overestimates gas
thermalization for stronger CR modified shocks. Our new estimate for η(M) is close to that
in Paper I, but a bit smaller, especially for shocks with M . 30. This is because inclusion of
Alfvén wave drift and dissipation in the shock precursor reduces the effective velocity change
experienced by CRs in the new DSA simulations of Kang & Jones (2007).
A note of caution for η(M) should be in order. As outlined above, CR injection is
less efficient and so the CR acceleration efficiency would be lower at perpendicular shocks,
compared to at quasi-parallel shocks. CR injection and acceleration at oblique shocks are not
well understood quantitatively. And the magnetic field directions at cosmological shocks are
not known. Considering these and other uncertainties involved in the adopted DSA model,
we did not attempt to make further improvements in estimating δ(M) and η(M) at general
oblique shocks. But we expect that an estimate at realistic shocks with chaotic magnetic
fields and random shock obliquity angles would give reduced values, rather than increased
values, for η(M). So η(M) given in Figure 5 may be regarded as upper limits.
By adopting the efficiencies in Figures 5, we calculated the thermal and CR energy fluxes
dissipated at cosmological shocks, dFth(M)/d logM , dFth(Vs)/d logVs, dFCR(M)/d logM
and dFCR(Vs)/d log Vs, using Fth = Fφδ(M) and FCR = Fφη(M), in the same way we
– 10 –
calculated dFφ(M)/d logM and dFφ(Vs)/d logVs in the previous section. We then integrated
from z = 5 to z = 0 the shock kinetic energy passed and the thermal and CR energies
dissipated through shock surfaces as follows:
dYi(X)
d logX
Eth,0
∫ z=0
dFi[X, z(t)]
d logX
dt, (2)
where the subscript i ≡ φ, th, or CR stands for the kinetic, thermal, or CR energies fluxes,
the variable X is either M or Vs, and Eth,0 is the total gas thermal energy at z = 0 inside
the simulation box normalized by its volume.
Figure 6 shows the resulting dYi(M)/d logM and dYi(Vs)/d logVs and their cumulative
distributions, Yi(> M) and Yi(> Vs), for the GSW simulation. Weak shocks with M . 4 or
fast shocks with Vs & 500 km s
−1 are responsible most for shock dissipations, as already noted
in Paper I. While the thermal energy generation peaks at shocks in the range 1.5 . M . 3,
the CR energy peaks in the range 2, 5 . M . 4 if no preexisting CRs are included or in
the range 1.5 . M . 3 if preexisting CRs of PCR/Pg ∼ 0.3 in the preshock region are
included. With our adopted efficiencies, the total CR energy accelerated and the total gas
thermal energy dissipated at cosmological shocks throughout the history of the universe
are compared as YCR(M ≥ 1.5) ∼ 0.5Yth(M ≥ 1.5), when no preexisting CRs are present.
With preexisting CRs in the preshock region, the CR acceleration becomes more efficient,
so YCR(M ≥ 1.5) ∼ 1.7Yth(M ≥ 1.5), i.e., the total CR energy accelerated at cosmological
shocks is estimated to be 1.7 times the total gas thermal energy dissipated. We note here
again that these are not meant to be very accurate estimates of the CR energy in the IGM,
considering the difficulty of modeling shocks as well as the uncertainties in the DSA model
itself. However, they imply that the IGM and the WHIM, which are bounded by strong
external shocks with high M and filled with weak internal shocks with low M , could contain
a dynamically significant CR population.
5. Vorticity Generation at Cosmological Shock Waves
Cosmological shocks formed in the large scale structure of the universe are by nature
curved shocks, accompanying complex, often chaotic flow patterns. It is well known that vor-
ticity, ~ω = ∇×~v, is generated at such curved oblique shocks (Binney 1974; Davies & Widrow
2000). In Paper II, the generation of vorticity behind cosmological shocks and turbulence
dynamo of magnetic fields in the IGM were studied in an adiabatic ΛCDM simulation. In
this study we analyzed the distribution of vorticity in the three simulations to assess quanti-
tatively the effects of non-gravitational processes. Here we present the magnitude of vorticity
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with the vorticity parameter
τ(~r, z) ≡ tage(z)ω(~r, z) =
tage(z)
teddy(~r, z)
, (3)
where tage(z) is the age of the universe at redshift z. With teddy = 1/ω interpreted as
local eddy turnover time, τ represents the number of local eddy turnovers in the age of
the universe. So if τ ≫ 1, we expect that turbulence has been fully developed after many
turnovers.
Figure 7 shows fluid quantities and shock locations in a two-dimensional slice of (21.25 h−1Mpc)2,
delineated by a solid box in Figure 2, at z = 0 in the GSW simulations. The region contains
two clusters with kT ∼ 1− 2 keV in the process of merging. Bottom right panel shows that
vorticity increases sharply at shocks. The postshock gas has a larger amount of vorticity
than the preshock gas, indicating that most, if not all, of the vorticity in the simulation was
produced at shocks.
Figure 8 shows the gas mass distribution in the gas density-vorticity parameter plane,
fm(ρgas, τ), (upper panel) and the gas mass fraction per logarithmic τ interval, dfm(τ)/d log τ ,
(bottom panel) for the three simulations. The most noticeable point in the upper panel is
that vorticity is higher at the highest density regions with ρ̃ ≡ ρgas/〈ρgas〉 & 10
3 in the NO
GSW and GSW simulations than in the Adiabatic simulation. This is due to the additional
flow motions induced by cooling. Inclusion of GSW feedbacks, on the other hand, does not
alter significantly the overall distribution in the gas density-vorticity parameter plane. The
bottom panel indicates that cooling increased the mass fraction with large vorticity τ & 10,
while reduced the mass fraction with 1 . τ . 10. GSW feedbacks increased slightly the
mass fraction with 1 . τ . 10, which corresponds to the gas in the regions outskirts of
groups that expand further out due to GSWs (i.e., balloons around groups). But overall we
conclude that the non-gravitational processes considered in this paper have limited effects
on vorticity in the large scale structure of the universe.
We note that the highest density regions in the NO GSW and GSW simulations have
τ ∼ 30 on average. As described in details in Paper II, such values of τ imply that local
eddies have turned over many times in the age of the universe, so that the ICM gas there
has had enough time to develop magnetohydrodynamic (MHD) turbulence. So in those
regions, magnetic fields should have grown to have the energy approaching to the turbulent
energy. On the other hand, the gas with 1 . ρ̃ . 103, mostly in filamentary and sheet-like
structures, has 0.1 . τ . 10. MHD turbulence should not have been fully developed there
and turbulence growth of magnetic fields would be small. Finally in the low density void
regions with ρ̃ . 1, vorticity is negligible with τ . 0.1 on average, as expected.
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6. Summary
We identified cosmological shock waves and studied their roles on cosmic plasmas in
three cosmological N-body/hydrodynamic simulations for a concordance ΛCDM universe in
a cubic box of comoving size 85 h−1Mpc: 1) adiabatic simulation (Adiabatic), 2) simulation
with radiative cooling and photoionization/heating (NO GSW), and 3) same as the second
simulation but also with galactic superwind feedbacks (GSW). The statistics and energetics
of shocks in the adiabatic simulation are in an excellent agreement with those of Paper I
where an adiabatic simulation with slightly different cosmological parameters in a cubic box
of comoving size 100 h−1Mpc was analyzed.
Photoionization/heating raised the gas temperature outside nonlinear structures in the
NO GSW and GSW simulations. As a result, the number of identified shocks and their
Mach numbers in the NO GSW and GSW simulations were different from those in the Adi-
abatic simulation. GSW feedbacks pushed out gas most noticeably around groups, creating
balloon-shaped surfaces of shocks with speed Vs ∼ 100 km s
−1 in the GSW simulation. How-
ever, those have minor effects on shock energetics. The total kinetic energy passed through
shock surfaces throughout the history of the universe is very similar for all three simula-
tions. So we conclude that the energetics of cosmological shocks was governed mostly by
the gravity of matter, and the effects non-gravitational processes, such as radiative cooling,
photoionization/heating, and GSW feedbacks, were rather minor and local.
We estimated both the improved gas thermalization efficiency, δ(M), and CR acceler-
ation efficiency, η(M), as a function shock Mach number, from nonlinear diffusive shock
simulations for quasi-parallel shocks that assumed Bohm diffusion for CR protons and in-
corporated self-consistent treatments of thermal leakage injection and Alfvén wave propa-
gation (Kang & Jones 2007). The cases without and with a preexisting CR component of
PCR/Pg ∼ 0.3 in the preshock region were considered. At strong shocks, both the injection
and acceleration of CRs are very efficient, and so the presence of a preexisting CR component
is not important. At shocks with with M & 30, about 55 % of the shock kinetic energy goes
into CRs, while about 30 % becomes the thermal energy. At weak shocks, on the other hand,
without a preexisting CR component, the gas thermalization is more efficient than the CR
acceleration. But the presence of a preexisting CR component is critical at weak shocks,
since it is equivalent to a higher injection rate and the CR acceleration becomes more effi-
cient with it. As a result, η(M) is higher than δ(M) even at shocks with M . 5. However,
at perpendicular shocks, the CR injection is suppressed, and so the CR acceleration could
be less efficient than at parallel shocks. Thus our CR shock acceleration efficiency should be
regarded as an upper limit.
With the adopted efficiencies, the total CR energy accelerated at cosmological shocks
– 13 –
throughout the history of the universe is estimated to be YCR(M ≥ 1.5) ∼ 0.5 Yth(M ≥ 1.5),
i.e., 1/2 of the total gas thermal energy dissipated, when no preexisting CRs are present.
With a preexisting CR component of PCR/Pg ∼ 0.3 in the preshock region, YCR(M ≥ 1.5) ∼
1.7 Yth(M ≥ 1.5), i.e., the total CR energy accelerated is estimate to be 1.7 times the total
gas thermal energy dissipated. Although these are not meant to be very accurate estimates of
the CR energy in the ICM, they imply that the ICM could contain a dynamically significant
CR population.
We also examined the distribution of vorticity inside the simulation box, which should
have been generated mostly at curved cosmological shocks. In the ICM, the eddy turn-over
time, teddy = 1/ω, is about 1/30 of the age of the universe, i.e., τ ≡ tage/teddy ∼ 30. In
filamentary and sheet-like structures, τ ∼ 0.1− 10, while τ . 0.1 in void regions. Radiative
cooling increased the fraction of gas mass with large vorticity τ & 10, while reduced the
mass fraction with 1 . τ . 10. GSW feedbacks increased slightly the mass fraction with
1 . τ . 10. Although the effects of these non-gravitation effects are not negligible, the
overall distribution of vorticity are similar for the three simulations. So we conclude that the
non-gravitational processes considered in this paper do not affect significantly the vorticity
in the large scale structure of the universe.
HK was supported in part by KOSEF through Astrophysical Research Center for the
Structure and Evolution of Cosmos (ARCSEC). DR was supported in part by a Korea Re-
search Foundation grant (KRF-2004-015-C00213). RC was supported in part by NASA grant
NNG05GK10G and NSF grant AST-0507521. The work of HK and DR was also supported in
part by Korea Foundation for International Cooperation of Science & Technology (KICOS)
through the Cavendish-KAIST Research Cooperation Center.
A. Fitting Formulae for δ(M) and η(M)
The gas thermalization efficiency, δ(M), and the CR acceleration efficiency, η(M), for
the case without a preexisting CR component (in upper panel of Figure 5) are fitted as
follows:
for M ≤ 2
δ(M) = 0.92 δ0 (A1)
η(M) = 1.96× 10−3(M2 − 1) (A2)
for M > 2
δ(M) =
(M − 1)n
– 14 –
a0 = −4.25, a1 = 6.42, a2 = −1.34, a3 = 1.26, a4 = 0.275 (A4)
η(M) =
(M − 1)n
b0 = 5.46, b1 = −9.78, b2 = 4.17, b3 = −0.334, b4 = 0.570 (A6)
The efficiencies for the case with a preexisting CR component (in bottom panel of Figure
5) are fitted as follows:
for M ≤ 1.5
δ(M) = 0.90 δ0 (A7)
η(M) = 1.025 δ0 (A8)
for M > 1.5
δ(M) =
(M − 1)n
a0 = −0.287, a1 = 0.837, a2 = −0.0467, a3 = 0.713, a4 = 0.289 (A10)
η(M) =
(M − 1)n
(A11)
b0 = 0.240, b1 = −1.56, b2 = 2.80, b3 = 0.512, b4 = 0.557 (A12)
Here δ0(M) is the gas thermalization efficiency at shocks without CRs, which was cal-
culated from the Rankine-Hugoniot jump condition, (black solid line in Figure 5):
δ0(M) =
γ(γ − 1)M2R
2γM2 − (γ − 1)
(γ + 1)
(A13)
γ + 1
γ − 1 + 2/M2
(A14)
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– 18 –
Fig. 1.— Top panels: Gas mass distribution in the gas density-temperature plane at z = 0
for the Adiabatic, NO GSW, and GSW simulations. Bottom panel: Gas mass fraction as a
function of gas temperature at z = 0 for the three simulations.
– 19 –
Fig. 2.— Two-dimensional slice of (85 h−1Mpc)2 showing shock locations at z = 0 in
the GSW simulation, which are color-coded according to shock speed as follows: black for
Vs < 15 km s
−1, blue for 15 ≤ Vs < 65 km s
−1, green for 65 ≤ Vs < 250 km s
−1, red for
250 ≤ Vs < 1000 km s
−1, and magenta for Vs ≥ 1000 km s
−1. A blown-up image of the box
(dashed line) in the upper right corner is shown in Figure 3, while a blown-up image of the
box (solid line) around two merging clusters is shown in Figure 7.
– 20 –
Fig. 3.— Two-dimensional slice of (21.25 h−1Mpc)2 showing shock locations at z = 0 in
the Adiabatic, NO GSW and GSW simulations. The locations are color-coded according to
shock speed. Two groups in the GSW simulation have kT ∼ 0.2− 0.3 keV.
– 21 –
Fig. 4.— Left panels: Inverse of the mean comoving distance between shock surfaces as a
function of Mach number M (top) and shock speed Vs (bottom) at z = 0 for the Adiabatic
(solid line), NO GSW (dashed line), and GSW (dotted line) simulations. Right panels:
Kinetic energy flux per comoving volume passing through shock surfaces in units of 1040 ergs
s−1 (h−1Mpc)−3 as a function of M (top) and Vs (bottom). Note that the bottom two panels
have different ranges of abscissa.
– 22 –
Fig. 5.— Gas thermalization efficiency, δ(M), and CR acceleration efficiency, η(M), as
a function of Mach number. Red and blue dots are the values estimated from numerical
simulations based on a DSA model and red and blue lines are the fits. The top panel shows
the case without preexisting CRs, while the bottom panel shows the case with preexisting
CRs at a level of PCR/Pg ∼ 0.3 in the preshock region. Black solid line is for the gas
thermalization efficiency for shocks without CRs.
– 23 –
0 1 2 3 4
0 1 2 3 4
1.5 2 2.5 3 3.5
1.5 2 2.5 3 3.5
Fig. 6.— Left panels: Shock kinetic energy passed, dYφ (dotted line), thermal energy dis-
sipated, dYth (dashed line), and CR energy dissipated, dYCR (solid line), through surfaces
of cosmological shocks with Mach number between logM and logM + d(logM) (top) and
through surfaces of cosmological shocks with shock speed between log Vs and log Vs+d(log Vs)
(bottom), integrated from z = 5 to z = 0. Red and magenta lines are the CR energy for the
cases without and with preexisting CRs, respectively. Blue and green lines are the thermal
energy for the cases without and with preexisting CRs, respectively. The thermal energy
expected to be dissipated at cosmological shocks without CRs (long dashed cyan line) is also
plotted for comparison. Right panels: Cumulative energy distributions, Yi(> M) (top) and
Yi(> Vs) (bottom), for Mach number greater than M and for shock speed greater than Vs.
The energies are normalized by the gas thermal energy at z = 0 inside the simulation box
– 24 –
Fig. 7.— Two-dimensional slice of (21.25 h−1Mpc)2 around two merging clusters with kT ∼
1−2 keV at z = 0 in the GSW simulation. Distributions of gas density (top left), temperature
(top right), shock locations (bottom left), and vorticity (bottom right) are shown. In the
gas density, temperature, and vorticity distributions, back, blue and red contours represent
regions of low, middle, and high values, respectively.
– 25 –
Fig. 8.— Top panels: Gas mass distribution in the gas mass density-vorticity parameter
plane at z = 0 for the Adiabatic, NO GSW, and GSW simulations. The vorticity parameter
is defined as τ = ωtage(z), where ω = |~∇×~v| and tage(z) is the age of the universe at redshift
z. Bottom panel: Gas mass fraction as a function of vorticity parameter at z = 0 for the
three simulations.
	Introduction
	Simulations
	Properties of Cosmological Shock Waves
	Energy dissipation by Cosmological Shock Waves
	Vorticity Generation at Cosmological Shock Waves
	Summary
	Fitting Formulae for (M) and (M)

  Cosmological shock waves result from supersonic flow motions induced by
hierarchical clustering of nonlinear structures in the universe. These shocks
govern the nature of cosmic plasma through thermalization of gas and
acceleration of nonthermal, cosmic-ray (CR) particles. We study the statistics
and energetics of shocks formed in cosmological simulations of a concordance
$\Lambda$CDM universe, with a special emphasis on the effects of
non-gravitational processes such as radiative cooling, photoionization/heating,
and galactic superwind feedbacks. Adopting an improved model for gas
thermalization and CR acceleration efficiencies based on nonlinear diffusive
shock acceleration calculations, we then estimate the gas thermal energy and
the CR energy dissipated at shocks through the history of the universe. Since
shocks can serve as sites for generation of vorticity, we also examine the
vorticity that should have been generated mostly at curved shocks in
cosmological simulations. We find that the dynamics and energetics of shocks
are governed primarily by the gravity of matter, so other non-gravitational
processes do not affect significantly the global energy dissipation and
vorticity generation at cosmological shocks. Our results reinforce scenarios in
which the intracluster medium and warm-hot intergalactic medium contain
energetically significant populations of nonthermal particles and turbulent
flow motions.

Introduction
Astrophysical plasmas consist of both thermal particles and nonthermal, cosmic-ray
(CR) particles that are closely coupled with permeating magnetic fields and underlying
turbulent flows. In the interstellar medium (ISM) of our Galaxy, for example, an approx-
imate energy equipartition among different components seems to have been established,
i.e., εtherm ∼ εCR ∼ εB ∼ εturb ∼ 1 eV cm
−3 (Longair 1994). Understanding the complex
network of physical interactions among these components constitutes one of fundamental
problems in astrophysics.
There is substantial observational evidence for the presence of nonthermal particles and
magnetic fields in the large scale structure of the universe. A fair fraction of X-ray clusters
have been observed in diffuse radio synchrotron emission, indicating the presence of GeV
CR electrons and µG fields in the intracluster medium (ICM) (Giovannini & Feretti 2000).
Observations in EUV and hard X-ray have shown that some clusters possess excess radia-
tion compared to what is expected from the hot, thermal X-ray emitting ICM, most likely
produced by the inverse-Compton scattering of cosmic background radiation (CBR) pho-
tons by CR electrons (Fusco-Femiano et al. 1999; Bowyer et al. 1999; Berghöfer et al. 2000).
Assuming energy equipartition between CR electrons and magnetic fields, εCRe ∼ εB ∼
0.01−0.1eV cm−3 ∼ 10−3−10−2εtherm can be inferred in typical radio halos (Govoni & Feretti
2004). If some of those CR electrons have been energized at shocks and/or by turbulence,
the same process should have produced a greater CR proton population. Considering the
ratio of proton to electron numbers, K ∼ 100, for Galactic CRs (Beck & Kraus 2005), one
can expect εCRp ∼ 0.01 − 0.1εtherm in radio halos. However, CR protons in the ICM have
yet to be confirmed by the observation of γ-ray photons produced by inelastic collisions
between CR protons and thermal protons (Reimer et al. 2003). Magnetic fields have been
also directly observed with Faraday rotation measure (RM). In clusters of galaxies strong
fields of a few µG strength extending from core to 500 kpc or further were inferred from
RM observations (Clarke et al. 2001; Clarke 2004). An upper limit of . µG was imposed
on the magnetic field strength in filaments and sheets, based the observed limit of the RMs
of quasars outside clusters (Kronberg 1994; Ryu et al. 1998).
Studies on turbulence and turbulent magnetic fields in the large scale structure of the
universe have been recently launched too. XMM-Newton X-ray observations of the Coma
cluster, which seems to be in a post-merger stage, were analyzed in details to extract clues
on turbulence in the ICM (Schuecker et al. 2004). By analyzing pressure fluctuations, it was
shown that the turbulence is likely subsonic and consistent with Kolmogoroff turbulence.
RM maps of clusters have been analyzed to find the power spectrum of turbulent magnetic
fields in a few clusters (Murgia et al. 2004; Vogt & Enßlin 2005). While Murgia et al. (2004)
– 3 –
reported a spectrum shallower than the Kolmogoroff spectrum, Vogt & Enßlin (2005) argued
that the spectrum could be consistent with the Kolmogoroff spectrum if it is bended at a
few kpc scale. These studies suggest that as in the ISM, turbulence does exist in the ICM
and may constitute an energetically non-negligible component.
In galaxy cluster environments there are several possible sources of CRs, magnetic
fields, and turbulence: jets from active galaxies (Kronberg et al. 2004; Li et al. 2006), ter-
mination shocks of galactic winds driven by supernova explosions (Vök & Atoyan 1999),
merger shocks (Sarazin 1999; Gabici & Blasi 2003; Fujita et al. 2003), structure formation
shocks (Loeb & Waxmann 2000; Miniati et al. 2001a,b), and motions of subcluster clumps
and galaxies (Subramanian et al. 2006). All of them have a potential to inject a similar
amount of energies, i.e., E ∼ 1061 − 1062 ergs into the ICM. Here we focus on shock scenar-
Astrophysical shocks are collisionless shocks that form in tenuous cosmic plasmas via col-
lective electromagnetic interactions between gas particles and magnetic fields. They play key
roles in governing the nature of cosmic plasmas: i.e., 1) shocks convert a part of the kinetic
energy of bulk flow motions into thermal energy, 2) shocks accelerate CRs by diffusive shock
acceleration (DSA) (Blandford & Ostriker 1978; Blandford & Eichler 1987; Malkov & Drury
2001), and amplify magnetic fields by streaming CRs (Bell 1978; Lucek & Bell 2000), 3)
shocks generate magnetic fields via the Biermann battery mechanism (Biermann 1950;
Kulsrud et al. 1997) and the Weibel instability (Weibel 1959; Medvedev et al. 2006), and 4)
curved shocks generate vorticity and ensuing turbulent flows (Binney 1974; Davies & Widrow
2000).
In Ryu et al. (2003) (Paper I), the properties of cosmological shock waves in the inter-
galactic medium (IGM) and the energy dissipations into thermal and nonthermal components
at those shocks were studied in a high-resolution, adiabatic (non-radiative), hydrodynamic
simulation of a ΛCDM universe. They found that internal shocks with low Mach numbers
of M . 4, which formed in the hot, previously shocked gas inside nonlinear structures,
are responsible for most of the shock energy dissipation. Adopting a nonlinear DSA model
for CR protons, it was shown that about 1/2 of the gas thermal energy dissipated at cos-
mological shocks through the history of the universe could be stored as CRs. In a recent
study, Pfrommer et al. (2006) identified shocks and analyzed the statistics in smoothed par-
ticle hydrodynamic (SPH) simulations of a ΛCDM universe, and found that their results
are in good agreement with those of Paper I. While internal shocks with lower Mach num-
bers are energetically dominant, external accretions shocks with higher Mach numbers can
serve as possible acceleration sites for high energy cosmic rays (Kang et al. 1996, 1997;
Ostrowski & Siemieniec-Ozieblo 2002). It was shown that CR ions could be accelerated up
– 4 –
to ∼ Z × 1019eV at cosmological shocks, where Z is the charge of ions (Inoue et al. 2007).
Ryu et al. (2007) (Paper II) analyzed the distribution of vorticity, which should have
been generated mostly at cosmological shock waves, in the same simulation of a ΛCDM
universe as in Paper I, and studied its implication on turbulence and turbulence dynamo.
Inside nonlinear structures, vorticity was found to be large enough that the turn-over time,
which is defined as the inverse of vorticity, is shorter than the age of the universe. Based on
it Ryu et al. (2007) argued that turbulence should have been developed in those structures
and estimated the strength of the magnetic field grown by the turbulence.
In this paper, we study cosmological shock waves in a new set of hydrodynamic sim-
ulations of large structure formation in a concordance ΛCDM universe: an adiabatic (non-
radiative) simulation which is similar to that considered in Paper I, and two additional
simulations which include various non-gravitational processes (see the next section for de-
tails). As in Papers I and II, the properties of cosmological shock waves are analyzed, the
energy dissipations to gas thermal energy and CR energy are evaluated, and the vorticity
distribution is analyzed. We then compare the results for the three simulations to highlight
the effects of non-gravitational processes on the properties of shocks and their roles on the
cosmic plasmas in the large scale structure of the universe.
Simulations are described in §2. The main results of shock identification and properties,
energy dissipations, and vorticity distribution are described in §3, §4, and §5, respectively.
Summary and discussion are followed in §6.
2. Simulations
The results reported here are based on the simulations previously presented in Cen & Ostriker
(2006). The simulations included radiative processes of heating/cooling, and the two sim-
ulations with and without galactic superwind (GSW) feedbacks were compared in that pa-
per. Here an additional adiabatic (non-radiative) simulation with otherwise the same setup
was performed. Hereafter these three simulations are referred as “Adiabatic”, “NO GSW”,
and “GSW” simulations, respectively. Specifically, the WMAP1-normalized ΛCDM cosmol-
ogy was employed with the following parameters: Ωb = 0.048, Ωm = 0.31, ΩΛ = 0.69,
h ≡ H0/(100 km/s/Mpc) = 0.69, σ8 = 0.89, and n = 0.97. A cubic box of comoving size
85 h−1Mpc was simulated using 10243 grid zones for gas and gravity and 5123 particles for
dark matter. It allows a uniform spatial resolution of ∆l = 83 h−1kpc. In Papers I and II,
an adiabatic simulation in a cubic box of comoving size 100 h−1Mpc with 10243 grid zones
and 5123 particles, employing slightly different cosmological parameters, was used. The
– 5 –
simulations were performed using a PM/Eulerian hydrodynamic cosmology code (Ryu et al.
1993).
Detailed descriptions for input physical ingredients such as non-equilibrium ioniza-
tion/cooling, photoionization/heating, star formation, and feedback processes can be found
in earlier papers (Cen et al. 2003; Cen & Ostriker 2006). Feedbacks from star formation
were treated in three forms: ionizing UV photons, GSWs, and metal enrichment. GSWs
were meant to represent cumulative supernova explosions, and modeled as outflows of sev-
eral hundred km s−1. The input of GSW energy for a given amount of star formation was
determined by matching the outflow velocities computed for star-burst galaxies in the sim-
ulation with those observed in the real world (Pettini et al. 2002)(see also Cen & Ostriker
2006, for details).
Figure 1 shows the gas mass distribution in the gas density-temperature plane, fm(ρgas, T ),
and the gas mass fraction as a function of gas temperature, fm(T ), at z = 0 for the three
simulations. The distributions are quite different, depending primarily on the inclusion of
radiative cooling and photoionization/heating. GSW feedbacks increase the fraction of the
WHIM with 105 < T < 107K, and at the same time affect the distribution of the warm/diffuse
gas with T < 105.
3. Properties of Cosmological Shock Waves
We start to describe cosmological shocks by briefing the procedure by which the shocks
were identified in simulation data. The details can be found in Paper I. A zone was tagged as
a shock zone currently experiencing shock dissipation, whenever the following three criteria
are met: 1) the gradients of gas temperature and entropy have the same sign, 2) the local
flow is converging with ~∇ · ~v < 0, and 3) |∆ log T | ≥ 0.11 corresponding to the temperature
jump of a shock with M ≥ 1.3. Typically a shock is represented by a jump spread over
2 − 3 tagged zones. Hence, a shock center was identified within the tagged zones, where
~∇ ·~v is minimum, and this center was labeled as part of a shock surface. The Mach number
of the shock center, M , was calculated from the temperature jump across the entire shock
zones. Finally to avoid confusion from complex flow patterns and shock surface topologies
associated with very weak shocks, only those portions of shock surfaces with M ≥ 1.5 were
kept and used for the analysis of shocks properties.
Figure 2 shows the locations of identified shocks in a two-dimensional slice at z = 0 in the
GSW simulation. The locations are color-coded according to shock speed. As shown before
in Paper I, external accretion shocks encompass nonlinear structures and reveal, in addition
– 6 –
to cluster complexes, rich topology of filamentary and sheet-like structures in the large scale
structure. Inside the nonlinear structures, there exist complex networks of internal shocks
that form by infall of previously shocked gas to filaments and knots and during subclump
mergers, as well as by chaotic flow motions. The shock heated gas around clusters extends
out to ∼ 5 h−1Mpc, much further out than the observed X-ray emitting volume.
In the GSW simulation, with several hundred km s−1 for outflows, the GSW feedbacks
affected most greatly the gas around groups of galaxies, while the impact on clusters with
kT & 1 keV was minimal. In Figure 3 we compare shock locations in a region around
two groups with kT ∼ 0.2 − 0.3 keV in the three simulations. It demonstrates that GSW
feedbacks pushed the hot gas out of groups with typical velocities of ∼ 100 km s−1 (green
points). In fact the prominent green balloons of shock surfaces around groups in Figure 2
are due to GSW feedbacks (see also Figure 4 of Cen & Ostriker 2006).
In the left panels of Figure 4 we compare the surface area of identified shocks, normalized
by the volume of the simulation box, per logarithmic Mach number interval, dS(M)/d logM
(top), and per logarithmic shock speed interval, dS(Vs)/d log Vs (bottom), at z = 0 in the
three simulations. Here S and Vs are given in units of (h
−1Mpc)−1 and km s−1. The
quantity S provides a measure of shock frequency or the inverse of the mean comoving
distance between shock surfaces. The distributions of dS(M)/d logM for the NO GSW and
GSW simulations are similar, while that for the Adiabatic simulation is different from the
other two. This is mainly because the gas temperature outside nonlinear structures is lower
without photoionization/heating in the Adiabatic simulation. As a result, external accretion
shocks tend to have higher Mach number due to colder preshock gas. The distribution of
dS(Vs)/d log Vs, on the other hand, is similar for all three simulations for Vs > 15 km s
−1. For
Vs < 15 km s
−1, however, there are more shocks in the Adiabatic simulation (black points in
Figure 3). Again this is because in the Adiabatic simulation the gas temperature is colder in
void regions, and so even shocks with low speeds of Vs < 15 km s
−1 were identified in these
regions. The GSW simulation shows slightly more shocks than the NO GSW simulation
around Vs ∼ 100 km s
−1, because GSW feedbacks created balloon-shaped surfaces of shocks
with typically those speeds (green points in Figure 3).
For identified shocks, we calculated the incident shock kinetic energy flux, Fφ = (1/2)ρ1V
where ρ1 is the preshock gas density. We then calculated the kinetic energy flux through
shock surfaces, normalized by the volume of the simulation box, per logarithmic Mach num-
ber interval, dFφ(M)/d logM , and per logarithmic shock speed interval, dFφ(Vs)/d logVs. In
the right panels of Figure 4, we compare the flux at z = 0 in the three simulations. Once
again, there are noticeable differences in dFφ(M)/d logM between the Adiabatic simulation
and the other two simulations, which can be interpreted as the result of ignoring photoion-
– 7 –
ization/heating in the gas outside nonlinear structures in the Adiabatic simulation. GSW
feedbacks enhance only slightly the shock kinetic energy flux for Vs ∼ 100−300 km s
−1, as can
be seen in the plot of dFφ(Vs)/d logVs. Yet, the total amount of the energy flux is expected
to be quite similar for all three simulations. This implies that the overall energy dissipation
at cosmological shocks is governed mainly by the gravity of matter, and that the inclusion of
various non-gravitational processes such as radiative cooling, photoionization/heating, and
GSW feedbacks have rather minor, local effects.
We note that a temperature floor of Tfloor = TCBR was used for the three simulations in
this work, while Tfloor = 10
4 K was set in paper I. It was because in Paper I only an adiabatic
simulation was considered and the 104 K temperature floor was enforced to mimic the effect
of photoionization/heating on the IGM. However we found that when the same temperature
floor is enforced, the statistics of the current Adiabatic simulation agree excellently with
those of Paper I. Specifically, the shock frequency and kinetic energy flux, dS(M)/d logM
and dFφ(M)/d logM , for weak shocks with 1.5 ≤ M . 3 are a bit higher in the current
Adiabatic simulation, because of higher spatial resolution. But the total kinetic energy flux
through shock surfaces, Fφ(M > 1.5), agrees within a few percent. On the other hand, In
Paper I we were able to reasonably distinguish external and internal shocks according to the
preshock temperature, i.e., external shocks if T1 ≤ Tfloor and internal shocks if T1 > Tfloor. We
no longer made such distinction in this work, since the preshock temperature alone cannot
tell us whether the preshock gas is inside nonlinear structures or not in the simulations with
radiative cooling.
4. Energy dissipation by Cosmological Shock Waves
The CR injection and acceleration rates at shocks depend in general upon the shock
Mach number, field obliquity angle, and the strength of the Alfvén turbulence responsible
for scattering. At quasi-parallel shocks, in which the mean magnetic field is parallel to the
shock normal direction, small anisotropy in the particle velocity distribution in the local fluid
frame causes some particles in the high energy tail of the Maxwellian distribution to stream
upstream (Giacalone et al. 1992). The streaming motions of the high energy particles against
the background fluid generate strong MHD Alfvén waves upstream of the shock, which in turn
scatter particles and amplify magnetic fields (Bell 1978; Lucek & Bell 2000). The scattered
particles can then be accelerated further to higher energies via Fermi first order process
(Malkov & Drury 2001). These processes, i.e., leakage of suprathermal particles into CRs,
self-excitation of Alfvén waves, amplification of magnetic fields, and further acceleration
of CRs, are all integral parts of collisionless shock formation in astrophysical plasmas. It
– 8 –
was shown that at strong quasi-parallel shocks, 10−4− 10−3 of the incoming particles can be
injected into the CR population, up to 60% of the shock kinetic energy can be transferred into
CR ions, and at the same time substantial nonlinear feedbacks are exerted to the underlying
flow (Berezhko et al. 1995; Kang & Jones 2005).
At perpendicular shocks with weakly perturbed magnetic fields, on the other hand, par-
ticles gain energy mainly by drifting along the shock surface in the ~v× ~B electric field. Such
drift acceleration can be much more efficient than the acceleration at parallel shocks (Jokipii
1987; Kang et al. 1997; Ostrowski & Siemieniec-Ozieblo 2002). But the particle injection
into the acceleration process is expected to be inefficient at perpendicular shocks, since the
transport of particles normal to the average field direction is suppressed (Ellison et al. 1995).
However, Giacalone (2005) showed that the injection problem at perpendicular shocks can be
alleviated substantially in the presence of fully turbulent fields owing to field line meandering.
As in Paper I, the gas thermalization and CR acceleration efficiencies are defined as
δ(M) ≡ Fth/Fφ and η(M) ≡ FCR/Fφ, respectively, where Fth is the thermal energy flux
generated and FCR is the CR energy flux accelerated at shocks. We note that for gasdy-
namical shocks without CRs, the gas thermalization efficiency can be calculated from the
Rankine-Hugoniot jump condition, as follows:
δ0(M) =
eth,2 − eth,1
, (1)
where the subscripts 1 and 2 stand for preshock and postshock regions, respectively. The
second term inside the brackets subtracts the effect of adiabatic compression occurred at a
shock too, not just the thermal energy flux entering the shock, namely, eth,1v1.
At CR modified shocks, however, the gas thermalization efficiency can be much smaller
than δ0(M) for strong shocks with large M , since a significant fraction of the shock kinetic
energy can be transferred to CRs. The gas thermalization and CR acceleration efficiencies
were estimated using the results of DSA simulations of quasi-parallel shocks with Bohm
diffusion coefficient, self-consistent treatments of thermal leakage injection, and Alfvén wave
propagation (Kang & Jones 2007). The simulations were started with purely gasdynamical
shocks in one-dimensional, plane-parallel geometry, and CR acceleration was followed by
solving the diffusion-convection equation explicitly with very high resolution. Shocks with
Vs = 150 − 4500 km s
−1 propagating into media of T1 = 10
4 − 106 K were considered.
After a quick initial adjustment, the postshock states reach time asymptotic values and the
CR modified shocks evolve in an approximately self-similar way with the shock structure
broadening linearly with time (refer Kang & Jones 2007, for details). Given this self-similar
nature of CR modified shocks, we calculated time asymptotic values of δ(M) and η(M) as
the ratios of increases in the gas thermal and CR energies at shocks to the kinetic energy
– 9 –
passed through the shocks at the termination time of the DSA simulations. As in Eq. (1),
the increase of energies due to adiabatic compression was subtracted.
Figure 5 shows δ(M) and η(M) estimated from DSA simulations and their fittings for
the cases with and without a preexisting CR component. The fitting formulae are given
in Appendix A. Without a preexisting CR component, gas thermalization is more efficient
than CR acceleration at shocks with M . 5. However, it is likely that weak internal shocks
propagate through the IGM that contains CRs accelerated previously at earlier shocks.
In that case, shocks with preexisting CRs need to be considered. Since the presence of
preexisting CRs is equivalent to a higher injection rate, CR acceleration is more efficient in
that case, especially at shocks with M . 5 (Kang & Jones 2003). In the bottom panel the
efficiencies for shocks with PCR/Pg ∼ 0.3 in the preshock region are shown. For comparison,
δ0(M) for shocks without CRs is also drawn. Both δ(M) and η(M) increase with Mach
number, but η(M) asymptotes to ∼ 0.55 while δ(M) to ∼ 0.30 for strong shocks with
M & 30. So about twice more energy goes into CRs, compared to for gas heating, at strong
shocks.
The efficiencies for the case without a preexisting CR component in the upper panel
of Figure 5 can be directly compared with the same quantities presented in Figure 6 of
Paper I. In Paper I, however, the gas thermalization efficiency was not calculated explicitly
from DSA simulations, and hence δ0(M) for gasdynamic shocks was used. It represents
gas thermalization reasonably well for weak shocks with M . 2.5, but overestimates gas
thermalization for stronger CR modified shocks. Our new estimate for η(M) is close to that
in Paper I, but a bit smaller, especially for shocks with M . 30. This is because inclusion of
Alfvén wave drift and dissipation in the shock precursor reduces the effective velocity change
experienced by CRs in the new DSA simulations of Kang & Jones (2007).
A note of caution for η(M) should be in order. As outlined above, CR injection is
less efficient and so the CR acceleration efficiency would be lower at perpendicular shocks,
compared to at quasi-parallel shocks. CR injection and acceleration at oblique shocks are not
well understood quantitatively. And the magnetic field directions at cosmological shocks are
not known. Considering these and other uncertainties involved in the adopted DSA model,
we did not attempt to make further improvements in estimating δ(M) and η(M) at general
oblique shocks. But we expect that an estimate at realistic shocks with chaotic magnetic
fields and random shock obliquity angles would give reduced values, rather than increased
values, for η(M). So η(M) given in Figure 5 may be regarded as upper limits.
By adopting the efficiencies in Figures 5, we calculated the thermal and CR energy fluxes
dissipated at cosmological shocks, dFth(M)/d logM , dFth(Vs)/d logVs, dFCR(M)/d logM
and dFCR(Vs)/d log Vs, using Fth = Fφδ(M) and FCR = Fφη(M), in the same way we
– 10 –
calculated dFφ(M)/d logM and dFφ(Vs)/d logVs in the previous section. We then integrated
from z = 5 to z = 0 the shock kinetic energy passed and the thermal and CR energies
dissipated through shock surfaces as follows:
dYi(X)
d logX
Eth,0
∫ z=0
dFi[X, z(t)]
d logX
dt, (2)
where the subscript i ≡ φ, th, or CR stands for the kinetic, thermal, or CR energies fluxes,
the variable X is either M or Vs, and Eth,0 is the total gas thermal energy at z = 0 inside
the simulation box normalized by its volume.
Figure 6 shows the resulting dYi(M)/d logM and dYi(Vs)/d logVs and their cumulative
distributions, Yi(> M) and Yi(> Vs), for the GSW simulation. Weak shocks with M . 4 or
fast shocks with Vs & 500 km s
−1 are responsible most for shock dissipations, as already noted
in Paper I. While the thermal energy generation peaks at shocks in the range 1.5 . M . 3,
the CR energy peaks in the range 2, 5 . M . 4 if no preexisting CRs are included or in
the range 1.5 . M . 3 if preexisting CRs of PCR/Pg ∼ 0.3 in the preshock region are
included. With our adopted efficiencies, the total CR energy accelerated and the total gas
thermal energy dissipated at cosmological shocks throughout the history of the universe
are compared as YCR(M ≥ 1.5) ∼ 0.5Yth(M ≥ 1.5), when no preexisting CRs are present.
With preexisting CRs in the preshock region, the CR acceleration becomes more efficient,
so YCR(M ≥ 1.5) ∼ 1.7Yth(M ≥ 1.5), i.e., the total CR energy accelerated at cosmological
shocks is estimated to be 1.7 times the total gas thermal energy dissipated. We note here
again that these are not meant to be very accurate estimates of the CR energy in the IGM,
considering the difficulty of modeling shocks as well as the uncertainties in the DSA model
itself. However, they imply that the IGM and the WHIM, which are bounded by strong
external shocks with high M and filled with weak internal shocks with low M , could contain
a dynamically significant CR population.
5. Vorticity Generation at Cosmological Shock Waves
Cosmological shocks formed in the large scale structure of the universe are by nature
curved shocks, accompanying complex, often chaotic flow patterns. It is well known that vor-
ticity, ~ω = ∇×~v, is generated at such curved oblique shocks (Binney 1974; Davies & Widrow
2000). In Paper II, the generation of vorticity behind cosmological shocks and turbulence
dynamo of magnetic fields in the IGM were studied in an adiabatic ΛCDM simulation. In
this study we analyzed the distribution of vorticity in the three simulations to assess quanti-
tatively the effects of non-gravitational processes. Here we present the magnitude of vorticity
– 11 –
with the vorticity parameter
τ(~r, z) ≡ tage(z)ω(~r, z) =
tage(z)
teddy(~r, z)
, (3)
where tage(z) is the age of the universe at redshift z. With teddy = 1/ω interpreted as
local eddy turnover time, τ represents the number of local eddy turnovers in the age of
the universe. So if τ ≫ 1, we expect that turbulence has been fully developed after many
turnovers.
Figure 7 shows fluid quantities and shock locations in a two-dimensional slice of (21.25 h−1Mpc)2,
delineated by a solid box in Figure 2, at z = 0 in the GSW simulations. The region contains
two clusters with kT ∼ 1− 2 keV in the process of merging. Bottom right panel shows that
vorticity increases sharply at shocks. The postshock gas has a larger amount of vorticity
than the preshock gas, indicating that most, if not all, of the vorticity in the simulation was
produced at shocks.
Figure 8 shows the gas mass distribution in the gas density-vorticity parameter plane,
fm(ρgas, τ), (upper panel) and the gas mass fraction per logarithmic τ interval, dfm(τ)/d log τ ,
(bottom panel) for the three simulations. The most noticeable point in the upper panel is
that vorticity is higher at the highest density regions with ρ̃ ≡ ρgas/〈ρgas〉 & 10
3 in the NO
GSW and GSW simulations than in the Adiabatic simulation. This is due to the additional
flow motions induced by cooling. Inclusion of GSW feedbacks, on the other hand, does not
alter significantly the overall distribution in the gas density-vorticity parameter plane. The
bottom panel indicates that cooling increased the mass fraction with large vorticity τ & 10,
while reduced the mass fraction with 1 . τ . 10. GSW feedbacks increased slightly the
mass fraction with 1 . τ . 10, which corresponds to the gas in the regions outskirts of
groups that expand further out due to GSWs (i.e., balloons around groups). But overall we
conclude that the non-gravitational processes considered in this paper have limited effects
on vorticity in the large scale structure of the universe.
We note that the highest density regions in the NO GSW and GSW simulations have
τ ∼ 30 on average. As described in details in Paper II, such values of τ imply that local
eddies have turned over many times in the age of the universe, so that the ICM gas there
has had enough time to develop magnetohydrodynamic (MHD) turbulence. So in those
regions, magnetic fields should have grown to have the energy approaching to the turbulent
energy. On the other hand, the gas with 1 . ρ̃ . 103, mostly in filamentary and sheet-like
structures, has 0.1 . τ . 10. MHD turbulence should not have been fully developed there
and turbulence growth of magnetic fields would be small. Finally in the low density void
regions with ρ̃ . 1, vorticity is negligible with τ . 0.1 on average, as expected.
– 12 –
6. Summary
We identified cosmological shock waves and studied their roles on cosmic plasmas in
three cosmological N-body/hydrodynamic simulations for a concordance ΛCDM universe in
a cubic box of comoving size 85 h−1Mpc: 1) adiabatic simulation (Adiabatic), 2) simulation
with radiative cooling and photoionization/heating (NO GSW), and 3) same as the second
simulation but also with galactic superwind feedbacks (GSW). The statistics and energetics
of shocks in the adiabatic simulation are in an excellent agreement with those of Paper I
where an adiabatic simulation with slightly different cosmological parameters in a cubic box
of comoving size 100 h−1Mpc was analyzed.
Photoionization/heating raised the gas temperature outside nonlinear structures in the
NO GSW and GSW simulations. As a result, the number of identified shocks and their
Mach numbers in the NO GSW and GSW simulations were different from those in the Adi-
abatic simulation. GSW feedbacks pushed out gas most noticeably around groups, creating
balloon-shaped surfaces of shocks with speed Vs ∼ 100 km s
−1 in the GSW simulation. How-
ever, those have minor effects on shock energetics. The total kinetic energy passed through
shock surfaces throughout the history of the universe is very similar for all three simula-
tions. So we conclude that the energetics of cosmological shocks was governed mostly by
the gravity of matter, and the effects non-gravitational processes, such as radiative cooling,
photoionization/heating, and GSW feedbacks, were rather minor and local.
We estimated both the improved gas thermalization efficiency, δ(M), and CR acceler-
ation efficiency, η(M), as a function shock Mach number, from nonlinear diffusive shock
simulations for quasi-parallel shocks that assumed Bohm diffusion for CR protons and in-
corporated self-consistent treatments of thermal leakage injection and Alfvén wave propa-
gation (Kang & Jones 2007). The cases without and with a preexisting CR component of
PCR/Pg ∼ 0.3 in the preshock region were considered. At strong shocks, both the injection
and acceleration of CRs are very efficient, and so the presence of a preexisting CR component
is not important. At shocks with with M & 30, about 55 % of the shock kinetic energy goes
into CRs, while about 30 % becomes the thermal energy. At weak shocks, on the other hand,
without a preexisting CR component, the gas thermalization is more efficient than the CR
acceleration. But the presence of a preexisting CR component is critical at weak shocks,
since it is equivalent to a higher injection rate and the CR acceleration becomes more effi-
cient with it. As a result, η(M) is higher than δ(M) even at shocks with M . 5. However,
at perpendicular shocks, the CR injection is suppressed, and so the CR acceleration could
be less efficient than at parallel shocks. Thus our CR shock acceleration efficiency should be
regarded as an upper limit.
With the adopted efficiencies, the total CR energy accelerated at cosmological shocks
– 13 –
throughout the history of the universe is estimated to be YCR(M ≥ 1.5) ∼ 0.5 Yth(M ≥ 1.5),
i.e., 1/2 of the total gas thermal energy dissipated, when no preexisting CRs are present.
With a preexisting CR component of PCR/Pg ∼ 0.3 in the preshock region, YCR(M ≥ 1.5) ∼
1.7 Yth(M ≥ 1.5), i.e., the total CR energy accelerated is estimate to be 1.7 times the total
gas thermal energy dissipated. Although these are not meant to be very accurate estimates of
the CR energy in the ICM, they imply that the ICM could contain a dynamically significant
CR population.
We also examined the distribution of vorticity inside the simulation box, which should
have been generated mostly at curved cosmological shocks. In the ICM, the eddy turn-over
time, teddy = 1/ω, is about 1/30 of the age of the universe, i.e., τ ≡ tage/teddy ∼ 30. In
filamentary and sheet-like structures, τ ∼ 0.1− 10, while τ . 0.1 in void regions. Radiative
cooling increased the fraction of gas mass with large vorticity τ & 10, while reduced the
mass fraction with 1 . τ . 10. GSW feedbacks increased slightly the mass fraction with
1 . τ . 10. Although the effects of these non-gravitation effects are not negligible, the
overall distribution of vorticity are similar for the three simulations. So we conclude that the
non-gravitational processes considered in this paper do not affect significantly the vorticity
in the large scale structure of the universe.
HK was supported in part by KOSEF through Astrophysical Research Center for the
Structure and Evolution of Cosmos (ARCSEC). DR was supported in part by a Korea Re-
search Foundation grant (KRF-2004-015-C00213). RC was supported in part by NASA grant
NNG05GK10G and NSF grant AST-0507521. The work of HK and DR was also supported in
part by Korea Foundation for International Cooperation of Science & Technology (KICOS)
through the Cavendish-KAIST Research Cooperation Center.
A. Fitting Formulae for δ(M) and η(M)
The gas thermalization efficiency, δ(M), and the CR acceleration efficiency, η(M), for
the case without a preexisting CR component (in upper panel of Figure 5) are fitted as
follows:
for M ≤ 2
δ(M) = 0.92 δ0 (A1)
η(M) = 1.96× 10−3(M2 − 1) (A2)
for M > 2
δ(M) =
(M − 1)n
– 14 –
a0 = −4.25, a1 = 6.42, a2 = −1.34, a3 = 1.26, a4 = 0.275 (A4)
η(M) =
(M − 1)n
b0 = 5.46, b1 = −9.78, b2 = 4.17, b3 = −0.334, b4 = 0.570 (A6)
The efficiencies for the case with a preexisting CR component (in bottom panel of Figure
5) are fitted as follows:
for M ≤ 1.5
δ(M) = 0.90 δ0 (A7)
η(M) = 1.025 δ0 (A8)
for M > 1.5
δ(M) =
(M − 1)n
a0 = −0.287, a1 = 0.837, a2 = −0.0467, a3 = 0.713, a4 = 0.289 (A10)
η(M) =
(M − 1)n
(A11)
b0 = 0.240, b1 = −1.56, b2 = 2.80, b3 = 0.512, b4 = 0.557 (A12)
Here δ0(M) is the gas thermalization efficiency at shocks without CRs, which was cal-
culated from the Rankine-Hugoniot jump condition, (black solid line in Figure 5):
δ0(M) =
γ(γ − 1)M2R
2γM2 − (γ − 1)
(γ + 1)
(A13)
γ + 1
γ − 1 + 2/M2
(A14)
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– 18 –
Fig. 1.— Top panels: Gas mass distribution in the gas density-temperature plane at z = 0
for the Adiabatic, NO GSW, and GSW simulations. Bottom panel: Gas mass fraction as a
function of gas temperature at z = 0 for the three simulations.
– 19 –
Fig. 2.— Two-dimensional slice of (85 h−1Mpc)2 showing shock locations at z = 0 in
the GSW simulation, which are color-coded according to shock speed as follows: black for
Vs < 15 km s
−1, blue for 15 ≤ Vs < 65 km s
−1, green for 65 ≤ Vs < 250 km s
−1, red for
250 ≤ Vs < 1000 km s
−1, and magenta for Vs ≥ 1000 km s
−1. A blown-up image of the box
(dashed line) in the upper right corner is shown in Figure 3, while a blown-up image of the
box (solid line) around two merging clusters is shown in Figure 7.
– 20 –
Fig. 3.— Two-dimensional slice of (21.25 h−1Mpc)2 showing shock locations at z = 0 in
the Adiabatic, NO GSW and GSW simulations. The locations are color-coded according to
shock speed. Two groups in the GSW simulation have kT ∼ 0.2− 0.3 keV.
– 21 –
Fig. 4.— Left panels: Inverse of the mean comoving distance between shock surfaces as a
function of Mach number M (top) and shock speed Vs (bottom) at z = 0 for the Adiabatic
(solid line), NO GSW (dashed line), and GSW (dotted line) simulations. Right panels:
Kinetic energy flux per comoving volume passing through shock surfaces in units of 1040 ergs
s−1 (h−1Mpc)−3 as a function of M (top) and Vs (bottom). Note that the bottom two panels
have different ranges of abscissa.
– 22 –
Fig. 5.— Gas thermalization efficiency, δ(M), and CR acceleration efficiency, η(M), as
a function of Mach number. Red and blue dots are the values estimated from numerical
simulations based on a DSA model and red and blue lines are the fits. The top panel shows
the case without preexisting CRs, while the bottom panel shows the case with preexisting
CRs at a level of PCR/Pg ∼ 0.3 in the preshock region. Black solid line is for the gas
thermalization efficiency for shocks without CRs.
– 23 –
0 1 2 3 4
0 1 2 3 4
1.5 2 2.5 3 3.5
1.5 2 2.5 3 3.5
Fig. 6.— Left panels: Shock kinetic energy passed, dYφ (dotted line), thermal energy dis-
sipated, dYth (dashed line), and CR energy dissipated, dYCR (solid line), through surfaces
of cosmological shocks with Mach number between logM and logM + d(logM) (top) and
through surfaces of cosmological shocks with shock speed between log Vs and log Vs+d(log Vs)
(bottom), integrated from z = 5 to z = 0. Red and magenta lines are the CR energy for the
cases without and with preexisting CRs, respectively. Blue and green lines are the thermal
energy for the cases without and with preexisting CRs, respectively. The thermal energy
expected to be dissipated at cosmological shocks without CRs (long dashed cyan line) is also
plotted for comparison. Right panels: Cumulative energy distributions, Yi(> M) (top) and
Yi(> Vs) (bottom), for Mach number greater than M and for shock speed greater than Vs.
The energies are normalized by the gas thermal energy at z = 0 inside the simulation box
– 24 –
Fig. 7.— Two-dimensional slice of (21.25 h−1Mpc)2 around two merging clusters with kT ∼
1−2 keV at z = 0 in the GSW simulation. Distributions of gas density (top left), temperature
(top right), shock locations (bottom left), and vorticity (bottom right) are shown. In the
gas density, temperature, and vorticity distributions, back, blue and red contours represent
regions of low, middle, and high values, respectively.
– 25 –
Fig. 8.— Top panels: Gas mass distribution in the gas mass density-vorticity parameter
plane at z = 0 for the Adiabatic, NO GSW, and GSW simulations. The vorticity parameter
is defined as τ = ωtage(z), where ω = |~∇×~v| and tage(z) is the age of the universe at redshift
z. Bottom panel: Gas mass fraction as a function of vorticity parameter at z = 0 for the
three simulations.
	Introduction
	Simulations
	Properties of Cosmological Shock Waves
	Energy dissipation by Cosmological Shock Waves
	Vorticity Generation at Cosmological Shock Waves
	Summary
	Fitting Formulae for (M) and (M)
